Solution Manual for Principles Of Managerial Finance 16th Edition by Chad J. Zutter, Scott Smart

Instructor’s Resource Manual for Principles of Managerial Finance Sixteenth Edition

Chad J. Zutter University of Pittsburgh

Scott B. Smart Indiana University

ISBN-13: 978-0-13-694561-1 ISBN-10: 0-13-694561-9

Chapter 1

The Role and Environment of Managerial Finance

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Table of Contents PART 1 Introduction to Managerial Finance

1

1 The Role of Managerial Finance

3

2 The Financial Market Environment

19

PART 2 Financial Tools

29

3 Financial Statements and Ratio Analysis

31

4 Long- and Short-Term Financial Planning

55

5 Time Value of Money

79

PART 3 Valuation of Securities

119

6 Interest Rates and Bond Valuation

121

7 Stock Valuation

149

PART 4 Risk and the Required Rate of Return

167

8 Risk and Return

169

9 The Cost of Capital

205

PART 5 Long-Term Investment Decisions

231

10 Capital Budgeting Techniques

233

11 Capital Budgeting Cash Flows

261

12 Risk Refinements in Capital Budgeting

293

PART 6 Long-Term Financial Decisions

327

13 Leverage and Capital Structure

329

14 Payout Policy

349

PART 7 Short-Term Financial Decisions

367

15 Working Capital and Current Assets Management

369

16 Current Liabilities Management

383

PART 8 Special Topics in Managerial Finance

399

17 Hybrid and Derivative Securities

401

18 Mergers, LBOs, Divestitures, and Business Failure

421

19 International Managerial Finance

437

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Gitman • Principles of Managerial Finance, Twelfth Edition

Part One Introduction to Managerial Finance Chapters in This Part

Chapter 1

The Role of Managerial Finance

Chapter 2

The Financial Market Environment

Integrative Case 1: Merit Enterprise Corp.

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Chapter 1 The Role of Managerial Finance  Instructor’s Resources Chapter Overview This chapter introduces the field of finance through building-block terms and concepts. The chapter starts by explaining what a firm is and discussing the goals that managers of a firm might pursue. The chapter provides a justification for focusing on shareholders rather than stakeholders broadly, but it also discusses other goals that firms might pursue. The opening section concludes with material on the importance of ethical behavior in business.

The next section discusses the managerial finance function, the key decisions that financial managers make, and the principles that guide their decisions. The discussion draws out distinctions among the overlapping disciplines of finance, economics, and accounting.

The third section describes pros and cons of different legal forms for a business. This section places particular emphasis on differences in taxation of proprietorships, partnerships, and corporations, and it highlights the importance of the marginal tax rate rather than the average tax rate. Next, this section describes the classical principal-agent problem and describes both internal and external corporate governance mechanisms that help manage that problem.

This chapter and the ones to follow stress the important role finance vocabulary, concepts, and tools will play in the professional and personal lives of students—even those choosing other majors, such as accounting, economics information systems, management, marketing, or operations. Whenever possible, personal-finance applications are provided to motivate and illustrate topics. This pedagogical approach should inspire students to master chapter content quickly and easily.

 Suggested Answer to Opener-in-Review Students learned the stock price of Brookdale Senior Living lost 80% of its value from 2015 to 2019, prompting Land and Buildings (a prominent stockholder) to urge the firm sell its real-estate holdings, distribute the anticipated net sales proceeds ($21 cash) to shareholders, and then focus on managing its senior living facilities.

Students were asked whether the proposal would make Brookdale’s shareholders better off if the expected cash proceeds were realized, but stock price dipped to $5 per share.

Before restructuring, an investor with one Brookdale share had $21.35 in total wealth. Afterward, that same investor might have a share worth $5 and $21 in cash—total wealth of $26. The hypothetical shareholder reaped a gain of $4.65 per share or 21.8%. Before the asset sale, with 185.45 million shares outstanding and a share price of $21.35, total shareholder wealth was $3.96 billion. After the sale, with same shares outstanding and wealth per share now $26, shareholder wealth rose to $4.82 billion—a net gain of $0.86 billion.

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Here is a discussion question for the class to motivate future exploration of CEO compensation: Suppose Brookdale’s CEO came up with the asset-sale idea rather than a prominent shareholder, and Brookdale’s board rewarded him with a $1 million dollar bonus—a figure alone that would easily vault the CEO into the top 1% of U.S. income earners. Is the CEO’s compensation excessive?

 Answers to Review Questions 1-1

The goal of a firm, and therefore of all financial managers, is maximizing shareholder wealth. The proper metric for this goal is the price of the firm’s stock. Other things equal, an increasing price per share of common stock relative to the stock market as a whole indicates achievement of this goal.

1-2

Actions that maximize the firm’s current profit may not produce the highest stock price because (1) some firm activities that result in slightly lower profit today generate much larger profits in the future periods (i.e., focusing on current profit overlooks the time value of money); (2) activities that generate higher accounting profits today may not result in higher cash flows to stockholders; and (3) activities that lead to high profits today may involve higher risk, which could result in significant future losses.

1-3

Risk is the chance actual outcomes may differ from expected outcomes. Financial managers must consider risk and return because the two factors tend to have an opposite effect on share price. That is, other things equal, an increase in the risk of cash flows to shareholders will depress firm stock price while higher average cash flows to shareholders will increase stock price.

1-4

Maximizing shareholder wealth does not mean overlooking or minimizing the welfare of other firm stakeholders. Firms with satisfied employees, customers, and suppliers tend to produce higher (or less risky) cash flows for their shareholders compared with companies that neglect non-owner stakeholders.

That said, customers prefer lower prices for firm output, firm employees prefer higher wages, and firm suppliers prefer higher prices for the input goods and services they provide. So actions that produce the highest price of the firm’s stock cannot simultaneously maximize customer, employee, and supplier satisfaction.

1-5

Broadly speaking, the decisions made by financial managers fall under three headings: (i) investment, (ii) capital budgeting, and (iii) working capital. Investment decisions involve the firm’s long-term projects while financing decisions concern the funding of those projects. Working-capital decisions, in contrast are related to the firm’s management of short-term financial resources.

1-6

Financial managers must recognize the tradeoff between risk and return because shareholders prefer higher cash flows but dislike large swings in cash flows. And, as a general rule, actions that boost the firm’s average cash flows also result in greater cash-flow greater volatility. Viewed another way, firm actions to reduce the chance cash flows will be low or negative also tend to reduce average cash flows over time. Understanding this tradeoff is important because shareholders are risk averse. That is, they will only accept larger swings in a firm’s cash flows only if compensated over time with higher average cash flows.

1-7

Finance is often considered applied economics. One reason is firms operate within the larger economy.

More importantly, the bedrock concept in economics—marginal benefit-marginal cost analysis—is also central to managerial finance. Marginal benefit-marginal cost analysis is the notion a firm (or any other economic actor) should take only those actions for which the extra benefits exceed the extra costs.

Nearly, all financial decisions ultimately turn on an assessment of their marginal benefits and marginal costs.

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Chapter 1

The Role of Managerial Finance

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1-8

Accountants and financial managers perform separate but equally important functions for the firm.

Accountants primarily collect and present financial data according to generally accepted financial principles while financial managers make investment, capital-budgeting, and working-capital decisions with financial data. In part because of their different functions, accountants and financial managers log firm revenues and expenses using different conventions. Accountants operate on an accrual basis, recognizing revenues as firm output is sold (whether or not payment is actually received) and firm expenses as incurred. Financial managers, in contrast, focus on actual inflows and outflows of cash, recognizing revenues when physically received and expenses when actually paid.

1-9

Like any economic actor, managers respond to incentives. Managers have a fiduciary duty to maximize shareholder wealth, but as humans, they also have personal goals—such as maximizing their own income, wealth, reputation, and quality of life. If the personal benefits of delivering for shareholders (or the costs of slighting them) are small, a financial manager might opt to further his own interest at the expense of shareholders. For example, CEOs of large firms—those with more sales, assets, employees, etc.—tend to receive more compensation than CEOs of smaller firms. If a CEO has to choose between two operating strategies—one that produces modest growth for his firm but a large jump in current stock price and another that generates rapid growth but a more modest rise in share price—and the firm’s board is not closely monitoring the CEO, she might pursue the high-growth strategy to boost her future compensation. A partial solution to such a problem is a compensation closely linking CEO compensation to firm stock price.

1-10 Sole proprietorships are the most common form of business organization, while corporations tend to be the largest. Large firms tend to organize as corporations to insulate owners from losses (limit liability) and facilitate acquisition of financial capital to fund growth.

1-11 Stockholders are the owners of a corporation. Their ownership (equity) takes the form of common stock or, less frequently, preferred stock. Stockholders elect the board of directors, which has ultimate responsibility for guiding corporate affairs and setting general policy. The board usually comprises key corporate personnel and outside directors. The corporation’s president or chief executive officer (CEO) reports to the board. He or she oversees day-to-day operations subject to the general policies established by the board. The corporation’s owners (shareholders) do not have a direct relationship with management; they provide input by electing board members and voting on major charter issues.

Shareholders receive compensation in two forms: (i) dividends paid on their stock (from corporate earnings) and (ii) capital gains from increases in the price of their shares (which reflect market expectations about future dividends).

1-12 Generally speaking, income from sole proprietorships and partnerships is taxed only once at the individual level; the owner or owners pay personal income tax on their share of firm’s profits. In contrast, corporate income is taxed first at the firm level (via the corporate income tax paid on firm profits) and then again at the personal level (via personal income tax paid on dividends or capital gains enjoyed by shareholders). Under the tax law prevailing in 2020, corporations paid tax at a flat rate of 21%, which means that the average tax rate and the marginal tax rate are the same (21%). Under a progressive tax structure, the tax rates rises with income, so the marginal tax rate generally exceeds the average tax rate.

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1-13 Agency problems arise when managers place personal goals ahead of their duty to shareholders to maximize stock price. The attendant costs are called agency costs. Agency costs can be implicit or explicit; either way they reduce shareholder wealth. An example of an ―implicit‖ agency cost is the dividends or capital gains shareholders miss out on because the firm’s management team pursued a personal interest (like maximizing sales to boost future compensation) rather than maximizing shareholder wealth. Of course, if shareholders sense stock price is not what it should be, they will start monitoring management more closely (as in the chapter opener with Brookdale Senior Living). The expenses associated with greater monitoring are an example of an ―explicit‖ agency cost. Agency problems in a firm can be reduced with a properly constructed and followed corporate-governance structure. Such a structure will feature checks and balances that reduce management’s interest in and ability to deviate from shareholder-wealth maximization. Like all corporate decisions, reducing agency costs is subject to marginal benefit–marginal cost analysis. In other words, the firm should invest in policies to align the incentives of management and shareholders as long as the marginal benefits exceed the marginal costs.

1-14 Firms most commonly try to mitigate agency problems by linking pay to metrics connected with shareholder wealth. Incentive plans tie compensation to share price. For example, the CEO might receive options offering the right to purchase stock at a set price (say current price) any time in the next few years. If the CEO takes actions that subsequently boost share price, she can profit personally by exercising the option—purchasing stock at the set price—and reselling at the higher market price. The higher the firm’s stock price, the more money the CEO can make, so options create a powerful incentive to focus laser-like on shareholder wealth. There is a downside, however. Sometimes general market trends swamp all the good done by management, so even though the CEO obsessed over shareholder wealth, her options proved worthless because a bear market hammered the firm’s stock price. This problem has made performance plans more popular. These plans link compensation with performance measures related to stock price that management can more closely control—such as earnings per share (EPS) and EPS growth. When targets for the performance metrics are attained, managers receive rewards like performance shares and/or cash bonuses.

1-15 If the board of directors fails to keep management focused on shareholder wealth, market forces can apply the necessary pressure. Two such forces are activism by institutional investors (such as Land and Buildings in the chapter opener) and the threat of hostile takeovers. Institutions typically hold large quantities of shares in many corporations. Because of their large stakes, these investors actively monitor management and vote their shares for the benefit of all shareholders. Large institutional investors reduce agency problems by using their voting clout to elect new directors that will make the changes in policies and personnel necessary to get underperforming stock to its highest possible price.

The threat of hostile takeover can also keep management focused on shareholders. Say a firm has a stock price of $15, but that price could be $20 with bold action management is reluctant to take. The lure of a $5 capital gain per share could tempt an outside individual, group of investors or firm not supported by existing management to purchase controlling interest and force the necessary changes.

Incumbent management knows ―necessary changes‖ means unemployment, so the threat of takeover could be enough to align their interests with those of the owners.

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Chapter 1

The Role of Managerial Finance

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 Suggested Answer to Focus on Ethics Box: Do Corporate Executives Have a Social Responsibility?

How would Friedman view a sole proprietor’s use of firm resources to pursue social goals?

In a sole proprietorship, the owner and manager are one in the same. So a manager using firm resources to support social goals would be doing exactly what the owner wanted. Put another way, Friedman would not see a conflict. He did not oppose pursuit of social goals by a firm or individual; he opposed doing so with someone else’s money.

 Suggested Answer to Focus on Practice Box: Must Search Engines Screen Out Fake News?

Is the goal of maximizing shareholder wealth necessarily ethical or unethical?

The ―end‖ of maximizing shareholder wealth is neither ethical nor unethical; it is neutral. But the means employed to pursue the end can be ethical or unethical. For example, taking actions to raise share price in clear violation of U.S. law is unethical—that is to say, wrong even if the violations are not uncovered.

What responsibility, if any, does Google have to help users assess the veracity of online content?

Management’s overriding concern should be shareholder wealth. Knowingly posting content a reasonable person could see is fake harms shareholders by damaging the Google brand, so some due diligence is warranted. How much Google should invest in validating online content depends on the marginal benefits and costs. Specifically, Google should verify as long as the marginal benefit to shareholders exceeds the marginal cost—that is, only as long as the net effect on stock price is positive.

 Suggested Answer to Focus on People/Planet/Profits Box: The Business Roundtable Revisits the Goal of a Corporation What kind of actions could CEOs who are members of the Business Roundtable take that would clearly indicate that their 2019 statement truly represented a break from the shareholder primacy doctrine?

A break from shareholder primacy means not doing things that are good for shareholders or doing things that are not beneficial for shareholders. Doing something that benefits a stakeholder group does not necessarily represent a break from shareholder primacy because sometimes an action that benefits a stakeholder also benefits shareholders. For example, if customers and shareholders place a value on fighting climate change, then a company that makes green investments make may its own shareholders better off while also becoming more green. On the other hand, firms could spend so much on green investments that shareholder value might suffer. That would represent a true break from the shareholder primacy doctrine. Evidence of this might take the form of markets pushing down a firm’s stock price when it announces a major new green investment initiative.

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 Answers to Warm-Up Exercises E1-1

Advantages and disadvantages of partnership versus incorporation (LG 5)

Answer: Each form of business organization has advantages and disadvantages. One advantage of a simple partnership is that each partner’s income is taxed only once as personal income (i.e., subject to the personal income tax). Corporate income, in contrast, is taxed twice—corporate profits will be subject to the corporate income tax, and the dividends and capital gains from each partner’s stock will be taxed as personal income.

Taxation is a key factor in choosing the form of business organization, but two other factors are also important. In a partnership, each partner has unlimited liability and may have to cover debts of other partners, while corporate owners have limited liability that guarantees they cannot lose more than they have invested in the corporation. The third major consideration is ease of transfer of the business. Partnerships are harder to transfer and technically dissolved when a partner dies, while a corporation has an infinite life (absent bankruptcy, merger, or acquisition) with ownership readily transferable through sale of existing shares.

If a third party were asked to decide which legal form of business A&J Tax Preparation should take, it would be useful to have the following information:  Relevant specifics of current personal and corporate income tax codes (such as marginal rates, deductions, etc.)  Expected future changes in tax law  Expected longevity of firm  Age of current owners  Current succession plan  Risk tolerance of owners  Capital needs of firm  Growth prospects of firm  Reasons for each partner’s view on preferred form of ownership

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Chapter 3 Financial Statements and Ratio Analysis

E1-2

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Timing of cash flows (LG 4)

Answer: Based on the information provided, the choice is not obvious. Even though the second project is expected to provide a larger overall increase in earnings, the goal of the firm is maximizing shareholder value (not earnings per se), so the timing and risk of cash flows must be considered to determine which project is superior. For example, even if the second project’s cash flows are higher, they tend to arrive later, so it is not clear whether the second project is preferable to the first.

E1-3

Cash flow vs. profits (LG 4)

Answer: It is not unusual for profitable firms to suffer a cash crunch. This typically happens when expenses must be paid before revenue can be collected. In such cases, the firm must arrange financing to plug the gap between cash inflows and outflows. If cash crunches are regular, management should consider going ahead with the party, particularly if it is important for employee morale (i.e., cancelling might significantly reduce productivity)— provided adequate short-term funding is available. If the crunch is new, larger problems could lie ahead, and funding a party before the cash-flow outlook became clear might expose the firm to financial risk.

E1-4

Sunk costs (LG 5)

Answer: Marginal benefit-marginal cost analysis ignores sunk costs, so the $2.5 million dollars spent over the past 15 years is irrelevant to the current decision. At this point, what matters is whether expected revenues from additional investment exceed expected costs, after adjusting for the risk and timing of cash flows. If so, and funding is available, the investment is sound (irrespective of the specific capital expenditure required). The key to the decision may well lie in the satellite-division manager’s candid assessment that the project has little chance of viability. That assessment suggests additional expenditure is likely to throw good money after bad.

E1-5

Agency costs (LG 6)

Answer: Agency costs arise when one party (principal) designates another party (agent) to act on her behalf and the second party (agent) has latitude to pursue her own interest at the expense of the principal. In a corporation, shareholders are principals and managers agents. If shareholders fail to monitor adequately, managers could focus on personal goals rather than shareholder value. The resulting negative impact on stock price is an example of an agency cost. Another example is the cost of stock options, which focus manager attention on share price but also raise managerial compensation.

In the Donut Shop, Inc. example, the principal is store management, and the agents are employees. As normal humans, employees might prefer talking with other each or taking long breaks to focusing laser-like on customers. Banning tips led to poorer service, which could ultimately drive customers elsewhere and cost store managers their jobs. Tipping, like options, aligns the interests of principals and agents. The prospect of a tip kept employees (agents) focused on customer satisfaction, just as store management (principals) wished.

One potential solution for Donut Shop, Inc., is a profit-sharing plan that includes employees whose behavior reduced customer satisfaction. For the new benefit to be effective, Donut Shop must sell the plan as a replacement for tipping and structure it to provide generous bonuses when profits rise (because profit sharing lacks the immediacy of tips for good service). Perhaps a simpler solution is recognizing the ban on tipping led to customerservice problems in the first place and reversing the policy.

E1-6

Corporate tax liability (LG 5) © 2022 Pearson Education, Inc.

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Answer: The company must pay the corporate tax rate of 21% on both its pretax ordinary income and any capital gain on the sale of the asset. Ross purchased the asset for $125,000 and sold it for $150,000, thereby netting a $25,000 capital gain, so total taxable income is $525,000.

The total tax liability is 21% × $525,000 = $110,250.

 Solutions to Problems P1-1

Liability comparisons (LG 5; Basic) a. Ms. Smith has lost her $47,000 investment and has unlimited personal liability, so she is also liable for the firm’s $108,000 in unpaid debts. b. Ms. Smith has lost her $47,000 investment and shares unlimited liability with her partner, Mr. Brown. Initially, Ms. Smith is liable for $54,000 (50% of total unpaid debts). But if her partner cannot cover half the debt, Ms. Smith is liable for the full amount. c. Ms. Smith has lost her $47,000 investment, but she has limited liability, so she is not personally liable for the firm’s $108,000 in unpaid debts. d. Ms. Smith has limited liability, so she cannot lose more than her $47,000 investment and is not liable for the firm’s $108,000 in unpaid debts.

P1-2

Accrual income versus cash flow for a period (LG 4; Basic) a. Sales Cost of goods sold Net profit

$760,000 300,000 $460,000

b. Cash receipts Cost of goods sold Net cash flow

$690,000 300,000 $390,000

c. The accrual and cash accounting methods show different net profits because the accrual approach counts all $760,000 as revenue, whereas the cash flow approach only counts the $690,000 that the firm collected this year. Both profit figures are informative to the financial manager. The cash flow approach helps the manager make plans for any cash surpluses or shortages that may occur, and the accrual approach gives a good picture for whether the firm is profitable or not as a going concern over time.

P1-3

Personal finance problem: Cash flows (LG 4; Intermediate) a. Total cash inflow: $450  $4,500  $4,950 Total cash outflow: $1,000  $500  $800  $355  $280  $1,200  $222  $4,357 b. Net cash flow: Total cash inflows—Total cash outflows = $4,950  $4,357  $593 c. If Jane is facing a shortage, she could reduce spending on discretionary items such as clothing, dining out, and gas (i.e., travel less). d. Jane should examine anticipated cash flows in other months to verify August is typical.

She may, for instance, discover expenditures not in her August budget—like large quarterly automobile-insurance expenses or large gift purchases in December. To prepare for such outlays, Jane should put the $593 in a bank deposit or money-market account where the funds are readily accessible, and capital losses unlikely. If the $593 will not needed for anticipated bills, Jane should explore longer-term investment options, such as a diversified portfolio of stocks and bonds.

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Chapter 3 Financial Statements and Ratio Analysis

P1-4

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Marginal cost–benefit analysis and the goal of the firm (LG 2 and LG 4; Challenging) a. Marginal benefits of equipment replacement = Benefits of new equipment  Benefits of old equipment $900,000  $300,000  $600,000 b. Marginal cost of equipment replacement = Cost of new equipment – Net salvage value of old equipment $600,000  $250,000  $350,000 c. Net benefits of equipment replacement  Marginal benefits of equipment replacement  Marginal cost of equipment replacement $600,000  $350,000 = $250,000 d. Monsanto should replace the old equipment with the new equipment because the marginal benefits exceed marginal costs.

P1-5

Marginal cost–benefit analysis and the goal of the firm (LG 2 and LG 4; Challenging) a. Marginal benefits of proposed robotics = Benefits of new robotics  Benefits of original robotics $560,000  $400,000  $160,000 b. Marginal cost of proposed robotics = Cost of new robotics – Sales price of current robotics $220,000  $70,000  $150,000 c. Net benefits of new robotics  Marginal benefits of proposed robotics  Marginal cost of proposed robotics $160,000  $150,000 = $10,000 d. Provided cash flows from new and existing robotics are equally risky, Ken Allen should recommend new robotics because the marginal benefits exceed marginal costs. e. Three other important factors are cash-flow risk, cash-flow timing, and interest rates. New technology sometimes presents unique risks—new robotics, for example, could have unanticipated breakdowns that necessitate a recall—so Ken Allen should investigate the riskiness of each cash flow under the marginal-benefit and marginal-cost headings. He should also determine the exact timing of cash inflows/outflows for both options as well as the opportunity cost of funds invested (i.e., the interest rate). Timing and the interest rate are important because the project spans five years, and dollars received/spent today are worth more than dollars received/spent tomorrow.

P1-6

Identifying agency problems, costs, and resolutions (LG 6; Intermediate) a. The agency cost is wages paid to an idle employee whose responsibilities must be covered by someone else. One solution is a time clock everyone must punch when arriving for work, take a lunchbreak, and leave for the day. A punch clock would reduce agency costs by (1) prompting the receptionist to return from lunch on time or (2) reduce wages paid for unproductive time. b. The agency costs are opportunity costs—money budgeted for inflated cost estimates that cannot be used to fund other projects to enhance shareholder wealth. One solution is rewarding managers for accurate cost estimates rather keeping actual costs below their estimates.

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c. The agency cost is lost shareholder wealth; the CEO might agree to sell the firm for less than fair-market value in return for a post-merger position with more income, wealth, power, or visibility. One safeguard is allowing bids from other potential partners once the CEO has publicly disclosed firm interest in merging. Competitive bidding should reveal a merger price fair to shareholders. d. Part-time or temporary workers are less productive than full-time workers for two reasons: (i) new employees must learn their jobs, and (ii) fully trained employees obtain insights about improving efficiency from experience. In the short run, any decline in service caused by part-time or temporary workers would probably not drive branch customers away. And the same revenue with lower costs (from cheaper workers) will, indeed, boost profits. Over the long run, however, consistently less-productive employees will hurt profitability by reducing revenue or raising costs. One solution is rewarding managers with stock for meeting performance targets over a longer horizon (like average branch profit over the past three years).

P1-7

Corporate taxes (LG 5; Basic) a. Firm’s tax liability on $1,863,600 using the corporate tax rate of 21%: Total taxes due  21% × $1,863,600 = $391,356 b. After-tax earnings: $1,863,600 – $391,356 $1,472,244. c.

Average tax rate: $391,356 ÷ $1,863,600  21%.

d. Marginal tax rate is 21%. Notice that the marginal and average tax rates are the same under a flat tax.

P1-8

Marginal tax rates (LG 6; Basic) a. The marginal tax rates at the specified income levels are as follows: Income $15,000 $60,000 $90,000 $150,000 $250,000 $450,000 $1,000,000

Marginal rate 12% 22% 24% 24% 35% 35% 37%

b. The plot shows that as income increases the marginal tax rate increases and peaks at 37% for income in excess of $518,400.

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Chapter 3 Financial Statements and Ratio Analysis

P1-9

Tax liability, marginal tax rate, and average tax rate (LG 6; Basic) a. Tax calculations using Table 1.2: $10,000:

Tax liability: After-tax earnings: Marginal tax rate: Average tax rate:

$80,000:

Tax liability: After-tax earnings: Marginal tax rate: Average tax rate:

$300,000:

Tax liability:

After-tax earnings: Marginal tax rate: Average tax rate: $500,000:

Tax liability:

After-tax earnings: Marginal tax rate: Average tax rate:

(0.10  $9,875)  [0.12  ($10,000 – $9,875)] = $987.50 + $15 = $1,002.50 $10,000 – $1,002.50 = $8,997.50 $9,875 < $10,001 < $40,125 so 12% $1,002.50 ÷ $10,000 10.03% (0.10  $9,875)  [0.12  ($40,125 – $9,875)] + [0.22  ($80,000 – $40,125)] = $987.50 + $3,630 + $8,772.50 = $13,390 $80,000 – $13,390 = $66,610 $40,125 < $80,001 < $85,525 so 22% $13,390 ÷ $80,000 16.74% (0.10  $9,875)  [0.12  ($40,125 – $9,875)] + [0.22  ($85,525 – $40,125)] + [0.24  ($163,525 – $85,525)] + [0.32  ($207,350 – $163,300)] + [0.35  ($300,000 – $207,350)] = $987.50 + $3,630 + $9,988 + $18,666 + $14,096 + $32,427.50 = $79,795 $300,000 – $79,795 = $220,205 $207,350 < $300,001 < $518,400 so 35% $79,795 ÷ $300,000 6.60% (0.10  $9,875)  [0.12  ($40,125 – $9,875)] + [0.22  ($85,525 – $40,125)] + [0.24  ($163,525 – $85,525)] + [0.32  ($207,350 – $163,300)] + [0.35  ($500,000 – $207,350)] = $987.50 + $3,630 + $9,988 + $18,666 + $14,096 + $102,427.50 = $149,795 $500,000 – $149,795 = $350,205 $207,350 < $500,001 < $518,400 so 35% $149,795 ÷ $500,000 .96%

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$1,000,000: Tax liability:

After-tax earnings: Marginal tax rate: Average tax rate: $1,500,000: Tax liability:

After-tax earnings: Marginal tax rate: Average tax rate: $2,000,000: Tax liability:

After-tax earnings: Marginal tax rate: Average tax rate:

(0.10  $9,875)  [0.12  ($40,125 – $9,875)] + [0.22  ($85,525 – $40,125)] + [0.24  ($163,525 – $85,525)] + [0.32  ($207,350 – $163,300)] + [0.35  ($518,400 – $207,350)] + [0.37  ($1,000,000 – 518,400)] = $987.50 + $3,630 + $9,988 + $18,666 + $14,096 + $108,867.50 + $178,192 = $334,427 $1,000,000 – $334,427 = $665,573 $518,400 < $1,000,001 so 37% $334,427 ÷ $1,000,000 .44% (0.10  $9,875)  [0.12  ($40,125 – $9,875)] + [0.22  ($85,525 – $40,125)] + [0.24  ($163,525 – $85,525)] + [0.32  ($207,350 – $163,300)] + [0.35  ($518,400 – $207,350)] + [0.37  ($1,500,000 – 518,400)] = $987.50 + $3,630 + $9,988 + $18,666 + $14,096 + $108,867.50 + $363,192 = $519,427 $1,500,000 – $519,427 = $980,573 $518,400 < $1,500,001 so 37% $519,427 ÷ $1,500,000 .44% (0.10  $9,875)  [0.12  ($40,125 – $9,875)] + [0.22  ($85,525 – $40,125)] + [0.24  ($163,525 – $85,525)] + [0.32  ($207,350 – $163,300)] + [0.35  ($518,400 – $207,350)] + [0.37  ($2,000,000 – 518,400)] = $987.50 + $3,630 + $9,988 + $18,666 + $14,096 + $108,867.50 + $548,192 = $704,427 $2,000,000 – $704,427 = $1,295,573 $518,400 < $2,000,001 so 37% $704,427 ÷ $2,000,000 .22%

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b. To build a plot of the marginal and average tax rates, first highlight the cells containing the partnership earnings before taxes and the cells containing the marginal and average tax rates in part a., and then under the Insert and Charts select Scatter and Scatter with Straight Lines and Markers. Once the graph appears, you can move it, format the axis scales and titles, move the legend, and remove the gridlines.

As pretax partnership income increases, the marginal and average tax rates increase.

However, the average tax rate increases more smoothly and at a slower rate.

P1-10 Double taxation and implied cost of limited liability (LG 5 LG 6; Intermediate) Total tax liability if organized as a sole proprietorship: (0.10  $9,875)  [0.12  ($40,125 – $9,875)] + [0.22  ($85,525 – $40,125)] + [0.24  ($163,525 – $85,525)] + [0.32  ($207,350 – $163,300)] + [0.35  ($518,400 – $207,350)] + [0.37  ($650,000 – 518,400)] = $987.50 + $3,630 + $9,988 + $18,666 + $14,096 + $108,867.50 + $48,692 = $204,927 After-tax income if organized as a sole proprietorship: $650,000 – $204,927 = $445,073 Total tax liability if organized as a corporation: (0.21  $650,000)  [(0.20 + 0.038)  ($650,000 – (0.21  $650,000))] = $136,500 + (0.238  $513,500) = $136,500 + $122,213 = $258,713 After-tax income if organized as a corporation: $650,000 – $258,713 = $391,287

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The implied cost of limited liability is $53,786 because this is how much less after-tax income you have when organized as a corporation.

P1-11 Interest versus dividend income (LG 6; Intermediate) a. The firm faces a 21% flat tax on operating earnings, so 21% × $490,000 = $102,900. b., c.

Before-tax amount Less: Applicable exclusion Taxable amount Tax (21%) After-tax amount

(b) Interest Income $20,000 0 20,000 4,200 15,800

(c) Dividend Income $20,000 10,000 10,000 2,100 17,900

(0.50  $20,000)

d. The after-tax amount of dividends received, $17,900, exceeds the after-tax amount of interest received, $15,800, due to the 50% corporate dividend exclusion. This increases the attractiveness of stock investments by one corporation in another relative to bond investments. e. The firm’s total tax liability is $102,900 from operating earnings, $4,200 from interest earnings, and $2,100 from dividends received for a total of $109,200.

P1-12 Interest versus dividend expense (LG 6; Intermediate) a. EBIT Less interest Pre-tax earnings Less taxes (21%) After-tax earnings (all of this available to common stockholders)

$50,000 12,000 $38,000 7,980 $30,020

b. EBIT Less taxes (21%) After-tax earnings Less preferred div Earnings for common stockholders

$50,000 10,500 $39,500 12,000 $27,500

P1-13 Corporate taxes and the use of debt (LG 5; Intermediate) a. With pre-tax income currently of $200,000, Hemingway’s current annual corporate tax liability is 21% × $200,000 = $42,000. b. Because the corporate tax rate is a flat tax, the average tax rate and the marginal tax rate are the same, 21%. c. If expansion is financed with cash reserves, then taxable income will be $350,000 with a corresponding tax liability of 21% × $350,000 = $73,500. The average tax rate and the marginal tax rate are both 21%.

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d. If expansion is financed with debt financing, taxable income will be $350,000 – $70,000 = $280,000. Taxes owed will equal 21% × ($350,000 – $70,000) = $58,800. Again, the average tax rate and the marginal tax rate are the same (21%) no matter what the income level is under a flat tax. e. Student answers might vary here. Students might say (regardless of the tax law), that income is lower when the company uses debt. That’s true, but again regardless of the tax law, the amount of taxes paid is lower when debt is used. If the value of the company depends on the cash flow that it distributes to ALL investors (not just shareholders), then financing the expansion with debt might be optimal. However, there may be offsetting effects (not mentioned in this chapter) that would negate the tax benefits of debt.

P1-14 ETHICS PROBLEM (Intermediate) Maximizing shareholder wealth subject to ―ethical constraints‖ means pursuing all opportunities to boost stock price consistent with community ethical norms and applicable federal/state laws. ―Community ethical norms‖ refers to prevailing standards about right and wrong. Consistent, knowing violation of such norms can reduce shareholder wealth by prompting stakeholder backlash and punitive government action. For example, in 2017, sexual mistreatment of women in the workplace became an overriding concern for many Americans. Firms with executives guilty of harassing female subordinates were vulnerable to attacks by customers, employees, lawyers, the media, and elected officials. If a firm knew an executive had a history of inappropriate behavior and took no action (believing, perhaps, the executive was irreplaceable), the backlash was even worse when the story inevitably came out. As a result, many high-profile executives were fired to head off customer boycotts, employee defections, hostile-workplace lawsuits, and political retaliation (such as Congressional hearings or targeted legislation) that could hammer the firm’s stock price.

Similarly, abiding by applicable federal and state laws protects shareholders wealth from punitive legal action against the firm and its executives as well as backlash from stakeholders and elected officials.

 Case Case studies are available on www.pearson.com/mylab/finance.

Assessing the Goal of Sports Products, Inc. a.

The primary goal of Sports Products, Inc. should be maximizing shareholder wealth, which means taking all legal and ethical actions to get firm stock price to the highest possible level. Unlike profit maximization, maximizing stock price requires consideration of the level of cash flows (which, unlike profits, can be used to meet firm obligations) as well as the timing and riskiness of those flows.

b.

Yes, there appears to be an agency problem. In this case, the stockholders (owners) of Sports Products are the principals, and company management the agents. Stockholders want the highest possible stock price, but management compensation is directly tied to profits, not share price. So, predictably, company executives have focused on obtaining the highest possible profit, and stock price has languished.

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c.

Sports Products’ approach to pollution control is ethically questionable and harmful to shareholders.

It is unclear whether polluting the stream was intentional or accidental; what is clear from the stateand-local-government lawsuits is the firm violated the law. In the near term, litigation and judgment costs will reduce firm stock price (other things equal). Over the longer term, the related bad publicity could damage Sports Products’ relationships with customers, employees and suppliers— putting further downward pressure on share price. Had the firm been more concerned about shareholder wealth, it would have seen the wisdom in sacrificing some near-term profits to avoid sustained damage to stock price.

d.

The corporate governance system at Sports Products appears weak. A management-compensation system focused on profits, rather than stock price, indicates shareholder welfare is not a firm priority. Another sign of weak governance is management’s willingness to risk an environmental disaster—and the accompanying damage to shareholder wealth—to avoid higher pollution-control costs (and somewhat lower profits).

e.

Recommendations to Sports Products could include the following:  Overhauling management compensation to strengthen incentives to focus on shareholder interests. Specifically, Sports Products should consider distributing stock options to executives or awarding large bonuses based on performance-based metrics related to share price (like earnings per share or growth in earnings per share).  Introducing an explicit system of ―carrots and stocks‖ to reward ongoing management/employee compliance with federal and state laws (particularly those pollution related) and punish transgressions.  Establishing a corporate ethics policy, to be read and signed by all employees, along with a system of ―carrots and stocks‖ to reward ongoing management/employee compliance and punish transgressions.  Recruiting new board members to enact policies to change the corporate culture to focus on shareholder wealth and good corporate citizenship.

 Spreadsheet Exercise Answers to Chapter 1 spreadsheet problem (Space Ace) are available on www.pearson.com/mylab/finance.

 Group Exercise Group exercises are available on www.pearson.com/mylab/finance.

Notes for Adopters Group exercises offer students an opportunity to apply chapter topics in a real-world setting using one fictional and one actual company. Apart from reinforcing learning goals, this approach gives students valuable experience working in teams—as both leader and follower. Assignments can be easily modified to fit an adopter’s course goals. Students should enjoy these exercises; they have less structure than traditional homework and compellingly answer the age-old question: ―Why must I learn this?‖ The first practical issue is assembling groups—should the instructor assign students to groups or let students form their own? This project is semester-long, so group members must work well together for months. If students choose, they are more likely to get along—but at the cost of less intragroup diversity.

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A hybrid strategy is asking students to pair-off and then randomly combining student-selected pairs into larger groups.

The next issue is determining group size and leaders. Exercises generate workloads suitable for three or more students. Larger groups reduce individual workloads but facilitate ―slacking.‖ Apart from missing a learning opportunity, slackers create resentment over unequal contributions to team output. Managing larger groups can also be a challenge for students with little leadership experience. For these reasons, group size should be capped at five. As for selecting CEOs, rotation inside the group gives each student an opportunity to lead.

One final note—exercises were designed to give students the freedom to work largely independent of the instructor. Accordingly, instructions for each assignment are self-explanatory.

Chapter 1 This first chapter asks students to name and describe their fictional firm. They must then justify the decision to go public and discuss different managerial roles within their firm. The group must select a publicly held peer (shadow firm) in a related industry with a wealth of online information (including detailed financials).

The instructor should stress the importance of laboring over initial decisions because later work builds on them. For example, the choice of shadow firm should be weighed carefully because students will apply real-world information about their shadow firm to their fictitious firm. A good first step in narrowing candidates is starting with a familiar industry.

Chapter 2 The Financial Market Environment  Instructor’s Resources Chapter Overview This chapter provides an overview of the institutional framework for channeling funds from net savers to net borrowers. The discussion begins with three basic types of financial institutions— commercial banks, investment banks, and the shadow-banking system. Financial markets more broadly are then introduced along with the distinction between (i) money and capital markets and (ii) primary and secondary markets. This section also discusses the costs of trading in the secondary market. Considerable attention is also focused on the oft-misunderstood topic of ―efficient markets.‖ The third main section of the chapter discusses regulation of financial institutions and financial markets, and that is followed by a fourth section that explains how firms raise equity capital in an IPO and how investment banks assist in that process. The chapter concludes with an exploration of financial markets in crisis, with special emphasis on the role of housing finance in the Financial Crisis and the Great Recession of 2007–09 and on the recent coronavirus pandemic.

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 Suggested Answer to Opener-in-Review Question In the chapter opener, students learned that Softbank paid $1.3 billion for a 7.7% ownership stake in WeWork. These figures imply WeWork was worth $16.88 billion [$1.3 billion  0.077] at the time.

 Answers to Review Questions 2-1

Financial institutions are intermediaries that facilitate the flow of individual, business, and government savings into loans and investments. Broadly speaking, net savers (primarily individuals) prefer low risk and easy access to their money while net borrowers (businesses and government) would like to take risk with the funds and tie them up for a longer term. Financial institutions transform loans and investments into forms savers prefer to hold (such as deposits) or help net borrowers issue debt and equity instruments tailored to saver preferences.

2-2

Overall, the same entities that supply funds—individuals, businesses, and governments—also demand them, so these three groups are all financial-institution customers. That said, the key demanders of funds (net borrowers) are businesses and governments while the key suppliers (net savers) are individuals.

2-3

Commercial banks, investment banks, and the shadow-banking system are all financial institutions.

Broadly speaking, commercial banks transform the deposits of net savers into loans to net borrowers. Investment banks, in contrast, do not ―transform‖ the liquidity and riskiness of financial assets. Instead, they help ―match‖ demanders and issuers of debt and equity instruments.

Specifically, investment banks instruct companies on the best vehicles for raising capital, advise them on mergers/restructuring, and engage in trading and market-making to support their consulting function. Finally, the shadow-banking system performs services for net savers and borrowers similar to commercial banks—but without issuing deposits. By not relying on deposit funding, shadow banks can evade prudential regulation designed to constrain risk-taking by ordinary banks.

2-4

Financial markets facilitate direct interaction of suppliers and demanders of funds. In primary markets, firms sell debt and equity instruments for the first time—a direct exchange between the firm or government issuing securities and the purchasers. An example is Microsoft Corporation selling new shares of common stock to private investors. In secondary markets, investors trade previously issued securities among themselves. An example is an investor buying a share of outstanding Microsoft common stock from another investor through a broker.

Put simply, primary markets feature sales of ―new‖ securities while ―used‖ security transactions take place in secondary markets. Primary and secondary markets have a symbiotic relationship—the easier the resale of a financial asset in a secondary market, the easier the initial sale of that asset in a primary market. Similarly, financial institutions and financial markets are far from independent. Commercial banks, for example, hold large inventories of U.S. Treasury securities to improve the liquidity and risk of their asset portfolio, and strong bank demand makes it easier for the Treasury to sell debt in the first place. Because banks have taken deposits and made loans since the days of goldsmiths in Medieval Europe, they enjoy a comparative advantage in originating and monitoring commercial loans. Aware of this advantage, the capital markets watch bank lending for clues about borrower financial strength.

When a commercial bank announces a new loan to a publicly traded firm, that firm’s stock price typically rises.

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2-5

A private placement is the sale of a new security directly to an investor or a small group of sophisticated investors (such as insurance companies and pension funds). A public offering, in contrast, is the sale of newly issued stock or bonds to the public at large. Firms typically rely on public offerings when they need large sums.

2-6

The money market features trading in short-term, highly marketable debt instruments; ―short term‖ here means an original maturity of one year or less. Money-market instruments typically carry low risk of capital losses. Examples of money-market instruments include U.S. Treasury bills, commercial paper, and negotiable certificates of deposit (issued by large commercial banks). The Eurocurrency market is the international analogue of the U.S. money market. This market features loans of currency held in banks outside the country where it is legal tender.

Participants typically use the Eurocurrency market to evade domestic regulations and tax laws.

The term stems from the European origin of this market; ―Eurocurrency‖ has nothing to do with the euro per se and is no longer specific to Europe.

2-7

The capital market features trading in instruments with original maturities exceeding one year such as bonds and stock (common and preferred). Capital-market instruments are exchanged in broker and dealer markets. In broker markets, a broker coordinates buy and sell orders, executing trades at the midpoint of the bid/ask spread (the highest price a buyer is willing to pay minus the lowest price a seller is willing to accept). The best known broker market is the NYSE. In dealer markets, a market maker executes buy and sell orders using her personal inventory and two distinct trades. For example, an investor might sell the dealer Microsoft stock at the bid price and then, in an independent transaction, another investor would buy Microsoft stock from the dealer at the ask price. ―Ask‖ exceeds ―bid,‖ so the dealer’s reward for maintaining an inventory of Microsoft stock is the opportunity to ―buy low, sell high.‖ The difference, in short, between broker and dealer markets turns on whether traders or dealers provide the liquidity.

2-8

Firms see the capital market as a source of external finance for long-term projects. Put another way, they sell new bonds and stock to raise funds to build factories, launch marketing campaigns, and expand into new markets. Accordingly, they want a liquid market—one ―deep‖ enough to accept newly issued securities at favorable prices. Investors, in contrast, see the capital market as a savings vehicle for long-term needs like retirement. As citizens of the macroeconomy, investors would also like the capital market to steer scarce funds to the most productive uses. To these ends, investors want an efficient capital market—one where securities prices reflect all available information and react swiftly to new information. Capitalmarket efficiency means investors need not waste time trying to identify over or undervalued securities or exploitable patterns in securities prices. Instead, they can maximize long-term returns by putting their savings in diversified mutual funds (i.e., avoiding countless hours studying individual stocks and bonds). Investors will also enjoy higher aggregate growth of output and employment from the spotlight securities prices shine on firms most able to profitably use their savings.

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2-9

The first years of Great Depression featured the worst contraction in American history.

Between August 1929 and March 1933, industrial production fell 52%, the Dow Jones Industrial Average tumbled 89%, unemployment soared to nearly 25%, and roughly 9,000 banks failed (37% of those operating in December 1929). Franklin Roosevelt won the 1932 election with a mandate to restore prosperity and prevent future depressions. Much of the U.S. framework for financial and financial-institution regulation stems from the First New Deal (1933–34). This framework addressed specific factors thought to have caused the slump. To protect depositors from losses in bank failures, the Banking Act of 1933 created federal deposit insurance. To prevent failures in the first place, the Act also barred commercial banks from security underwriting, which was thought to pose dangerous additional risks. To head off fraudulent investment schemes like those preceding the stock-market crash of 1929, the Securities Act of 1933 and Securities Exchange Act of 1934 forced companies wishing to issue public securities to disclose information about their financial condition.

2-10 Both Acts required companies wishing to participate in securities markets to disclose significant information to the public. The Securities Act of 1933 focused on the primary market, compelling sellers of new securities provide reasonably accurate portrayals of their firms to prospective investors. The Securities Exchange Act of 1934, in contrast, regulated trading in secondary markets; forcing publicly traded companies to keep investors informed about firm condition on an ongoing basis. The latter Act also created the Securities Exchange Commission to enforce federal securities laws.

2-11 Angel investors and venture capitalists are both sources of private equity. ―Angels‖ are usually wealthy individuals who fund promising start-ups in return for a slice of firm equity. Venture capitalists, in contrast, are businesses that pool contributions from individuals (often institutional investors like university endowments and pension funds) and invest those funds in promising start-ups. In short, angels pick ―winners‖ themselves whereas venture capitalists pick ―winners‖ for their clients.

2-12 Venture capitalists (VCs) are organized as (i) limited partnerships (most common), (ii) small business investment companies (SBICs), (iii) financial funds, and (iv) corporate funds. The principal difference is how the VC was created. The federal government charters SBICs. Financial institutions (usually commercial banks), in contrast, create financial funds as subsidiaries while nonfinancial firms launch corporate funds, sometimes as subsidiaries. Unlike other VC types, limited partnerships are launched by private individuals. All VCs use a legal agreement to specify deal structure and pricing. Deal structure allocates responsibilities between the start-up and VC and may include constraints on the firm to enhance its chance of success and mitigate VC risk. Pricing depends on the (i) value of the start-up, (ii) perceived risk of its business operations, and (iii) amount of funding needed. In general, VCs provide less funding and require a greater ownership stake when the firm is the early stages of development.

2-13 Firms wishing to go public must (i) secure approval from current shareholders, (ii) obtain certification of the accuracy of their financial documents from company auditors and lawyers, (iii) hire an originating investment bank, (iv) file a registration statement with the Securities and Exchange Commission (SEC), (v) participate in roadshows with the investment bank to spark interest among potential investors and learn about a suitable issuing price, (vi) obtain final SEC approval after the investment bank has finalized issue terms and offer price, and (vii) sell the issue to the investment bank at the guarantee price. The investment bank will then assume the risk of placing the issue with primary-market investors.

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2-14 Broadly speaking, an investment bank facilitates a firm’s issuance of new securities. In a common-stock issue, the bank helps the issuer file a registration statement with the SEC and market the offering to potential investors in a roadshow. The bank also sets the offering price and other terms of the issue. All along the way, the originating investment bank provides advice to help the issuer maximize the volume of funds raised. Finally, the originating bank buys the new securities from the issuer at the guarantee price and then resells the issue to primary-market investors. Sometimes the bank will form a syndicate of other investment banks to share the financial risk of placing the issue.

2-15 Securitization is the process of creating highly liquid marketable securities out of illiquid assets. The first assets securitized on a large scale were residential mortgages—securitizers ―pooled‖ the mortgages and then issued debt claims backed by cash flows from those pools. In other words, the interest and principal on ―mortgage-backed‖ securities (MBSs) paid to investors came from mortgage payments by residential homeowners. Securitization facilitated investment in mortgages by unbundling risk. Lenders might need their funds before the mortgage is repaid or lose money if the homeowner defaults. Securitization allows mortgage originators to earn fees from making the loans but then reduce liquidity and credit risk by selling the mortgage to a securitizer (who, in turn, creates a security with cash flows tailored to the preferences of market investors). Securitizing mortgages promotes efficient risk sharing, which in turn, makes the real-estate sector a more attractive place to invest.

2-16 A mortgage-backed security (MBS) is a debt instrument backed by residential mortgages. ―Backed‖ means principal and interest paid to MBS investors come from payments by residential homeowners with mortgages in the underlying pool. The primary MBS risk is credit risk, the chance homeowners will not make monthly principal and interest payments as stipulated in their mortgage contracts.

2-17 When a home buyer takes out a mortgage, initial equity—the difference between purchase price and mortgage-loan balance—is simply the down payment. Over time, equity will rise as the borrower reduces the mortgage balance with monthly principal and interest payments.

Should housing prices rise, the gap between house value and mortgage balance will widen further—that is to say, home equity rises even faster. If a borrower needs to skip a mortgage payment, the lender will typically allow her to tap equity. Rising prices also imply a vibrant housing market, so a borrower permanently unable to make the monthly payments can easily sell her home to pay off the mortgage.

2-18 A large decline in housing prices could push the value of a borrower’s home below the mortgage balance. With negative equity, the borrower could hold the loss at the original down payment by allowing the lender to foreclose. The only cost would be the negative impact on the borrower’s credit score. But if the decline in housing prices has led many other homeowners to walk away from their mortgages, this borrower may not be too concerned about the blot on her credit report, thinking future lenders will understand the circumstances.

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2-19 The Great Recession of 2007–09 illustrates how a financial-sector crisis can metastasize. In the years running up to the recession, securitizers increasingly pooled mortgage loans to borrowers with less-than-stellar credit. At the time, ―subprime‖ loans seemed relatively low risk because of rapidly rising housing prices. Then, when home prices began to level off (and even dip in some markets), mortgage delinquencies and defaults started climbing. With payments on underlying mortgages falling, the value of mortgage-back securities (MBSs) began to fall as well. Large investment banks (like Lehmann Brothers) and commercial banks (like Citibank) held considerable inventories of now-problematic MBSs. To offset rising MBS losses, commercial banks sharply curbed lending, which produced an economy-wide decline in consumer and investment spending. Investment banks, meanwhile, were large players in the money market—Lehmann, for example, routinely sold a large amount of commercial paper (short-term unsecured corporate debt). When the firm collapsed almost overnight (rendering its commercial paper worthless), the money market froze as investors became wary of all unsecured debt. Now, nonfinancial companies that regularly tapped the money market for short-term funding found themselves in squeeze. They responded by slashing costs and hoarding cash, which put even more downward pressure on economy-wide consumer and investment spending.

 Suggested Answer to Focus on Practice Box: Berkshire Hathaway: Can Buffet Be Replaced?

Thinking about the principal-agent problem from Chapter 1, why might Buffett use different incentive schemes in firms with different growth prospects?

In this Focus on Practice box, the principal is the funding provider—Warren Buffett and Berkshire Hathaway— while the agent is the firm owner/manager receiving the funds. Buffett wants the highest return on his investment over a specific time horizon; the owner/manager may wish to pursue other short-term goals with the money. For a firm in the early stages of development, growth is typically paramount, so Buffett might insist on an incentive scheme rewarding rapid growth of sales rather than profits. [Amazon’s initial business plan, for example, predicted no profit for at least for four to five years.] As the firm matured, Buffett would likely reward earnings growth rather than sales growth.

 Suggested Answer to Focus on Ethics Box: Should Insider Trading Be Legal?

Suppose insider trading were legal. Would it still present an ethical issue for insiders wishing to trade on non-public information?

Yes, even if legal, insider trading could still raise ethical concerns because of potential conflict between an executive’s duty to shareholders and her concern for personal wealth. Suppose, for example, a senior executive with considerable firm stock learned of safety issues with a popular product so serious a massive recall might be necessary. The executive has a fiduciary duty to work with the senior management team on a plan to contain damage to firm stock. Were insider trading legal, she might be tempted to hedge the possibility the plan might fail by dumping her stock quietly before the market became aware of the problem.

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 Answers to Warm-Up Exercises E2-1

Suppliers and demanders of funds (LG 1)

Answer: Individuals as a whole (i.e., the household sector) spend less than they earn and invest the surplus in firms directly (by purchasing their stocks and bonds) or indirectly (through financial institutions —as in making deposits a commercial bank who then lends the funds to firms). If individuals consume more/save less, fewer dollars will be available for investment, thereby driving up the cost of those funds to net borrowers in the form of higher required returns/interest rates. Over time, the rise in returns/rates will reduce investment and economic growth, which means lower growth in incomes and employment.

E2-2

Raising funds (LG 2)

Answer: Gaga can raise the needed $10 million by borrowing from a commercial bank or issuing stocks or bonds in the primary market. To obtain $10 million from a commercial bank, Gaga will likely need an ongoing deposit relationship with that bank. Such a relationship gives the bank low-cost information about Gaga’s cash flows that reduce the cost of lending to the firm. Over time, as Gaga repeatedly borrows and repays the loans, the bank will collect even more information, further reducing the cost of lending. Should Gaga wish to sell bonds or stock to raise the $10 million— that is, tap the financial markets directly for the funding rather than a commercial bank — its first step will be to retain an investment bank for needed expertise, such as advice on what securities to sell and terms to offer. Investment banks offer valuable expertise earned over time through marketmaking/trading activities and advising many firms on securities sales E2-3

Money market vs. capital market (LG 3)

Answer: Short-term, highly liquid, low-risk debt trades in the money market. Reputable firms needing cash for one year or less to fund ongoing operations have traditionally tapped the money market. Suppose a well-known, financially sound firm specializing in recreationalvehicle (RV) sales needs inventory for the summer driving/camping season. The company might sell 90-day commercial paper for money to buy RVs wholesale and then pay off the debt with proceeds from summer sales. Firms sell new bonds and stock in the capital market (where debt and equity with maturities exceeding one year trade) to fund longterm projects like construction of new factories.

E2-4

Biggest benefit of government regulation (LG 4)

Answer: The scale and scope of government involvement in the economy will always be subject to debate, but most economists agree on the need for some financial-sector regulation. Welldesigned regulation promotes confidence in the financial system, and individuals and businesses who trust financial institutions and markets are more likely to save and invest.

More savings and investment, in turn, confers economy-wide benefits through the resulting growth in output, incomes, and employment.

E2-5

Determining net proceeds from stock sale (LG 5)

Answer: Net proceeds = (1,000,000  $20 x 0.95) + (250,000  $20  0.90) = $19,000,000 + $4,500,000 = $23,500,000 E2-6 Mortgage-backed securities (MBSs) (LG 6)

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Answer: Students should start by asking about the following: a. The location of houses securing the underlying mortgages (As the old saying goes, the three most important determinants of real-estate prices are ―location, location, location.‖) b. The percentage of underlying mortgages in foreclosure or ―under water‖ (i.e., with market values below the remaining balance) in the region c. The percentage of underlying mortgages currently delinquent d. Any neighborhood restrictions on renting and about the strength of the regional rental market e. The precedence of MBS investors in bankruptcy (i.e., would other lenders have a senior claim on the houses securing the mortgages?) f. The condition of homes securing the underlying mortgages (e.g., would repairs be needed to sell or rent in the event of foreclosure?) g. The creditworthiness of homeowners still current on their mortgages (i.e., how likely is it borrowers will be unable to make timely payments in the future?) h. The percentage of pool mortgages with adjustable interest rates resetting soon (particularly in a rising rate environment because a reset means borrowers will face higher mortgage payments)

 Solutions to Problems P2-1

Transaction costs (LG3) a. Bid/Ask Spread = Ask Price – Bid Price = $285,909.62 – $285,705.59 = $204.03 b. If TD Ameritrade routes the buy order to the NYSE (a broker market), the market maker could execute the trade at the midpoint of the bid/ask spread. In this transaction, the market maker serves as broker, bringing your buy order together with someone else’s sell order and forgoing the bid/ask spread, so total transactions cost is only the brokerage commission of $5 paid to TD Ameritrade. c. If TD Ameritrade routes the buy order to the NASDAQ (a dealer market), the market maker could execute the order from her own inventory and charge half the bid/ask spread per share. In this case the total transaction costs of $107.01 will include the half-spreads paid to the market maker and the brokerage commission paid to TD Ameritrade.

Transactions costs = (Number of shares × 0.50 × Bid/ask spread) + Brokerage commission Transaction costs = (1 × 0.50 × $204.03) + $5 = $107.01 d. The midpoint of the bid/ask spread is the implied market value of the stock, and the market value of the trade equals the product of the market value of the stock and the number of shares traded.

Midpoint of bid/ask spread = ($285,909.62 + $285,705.59) / 2 = $285,807.605 So, implied market value of the trade = $285,807.605  1 share = $285,807.605

P2-2

Transaction costs (LG 3)

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a. Transactions costs = (Number of shares)  [(0.50) x (Bid/ask spread)] + Brokerage commission $45 = [(1,600)  (0.50)  (Bid/ask spread)] + $25 Bid/ask spread = ($45 – $25) / 800 = $0.025 b. Twitter is listed on the NYSE, a broker market. So, had Charles Schwab routed the order to the NYSE, it could have been executed against a buy order, and total transaction costs would have been only the $25 brokerage commission. But transaction costs included half the bid/ask spread per share traded, so either (i) the order went to the NYSE, no public buy order was available, and the market maker bought the 1,600 shares for her inventory (at a cost of half the bid/ask spread per share) or (ii) Charles Schwab routed the order to a dealer market like NASDAQ, and a market maker added the shares to her inventory (at half the spread per share). c. Transactions costs = (Number of shares)  [(0.50)  (Bid/ask spread)] + Brokerage commission $40.20 = [(1,600)  (0.50)  (Bid/ask spread)] + $25 Bid/ask spread = ($40.20 – $25) / 800 = $0.019 d. Total transactions costs = Transactions costs from sale + Transactions costs from purchase Total transactions costs = $45 + $40.20 = $85.20 Costs could have been reduced by placing both trades online with a request for routing to the NYSE, where the chance of crossing with other public orders is greatest. Had no market maker been necessary, the total round-trip transaction costs would have been $0 because Schwab does not charge a commission for online trades.

P2-3 Initial public offering (LG 5) a. Total proceeds = IPO offer price  IPO shares issued = $36  9,911,434 = $356,811,624 b. Percentage underwriting discount = Underwriting discount / Offer price = $1.80 ÷ $36 = 0.05 or 5% c. Underwriting fee ($) = $1.80  9,911,434 = $17,840,581.

Or Percentage underwriting discount  Total proceeds = 0.05  $356,811,624 = $17,840,581 d. Net proceeds = Total proceeds – Underwriting fee = $356,811,624 – $17,840,581 = $338,971,043 e. IPO underpricing = (Market price – Offer price) / Offer price = ($62 – $36) ÷ $36 = 0.722 or 72.2% f.

Market capitalization = Market price of stock  Number of shares outstanding = $62  (24,070,086 + 232,318,285) = $15,896,079,002

g. Ownership percentage of Class A shares = 24,070,086 ÷ (24,070,086 + 232,318,285) = 0.094 or 9.4% Voting percentage of Class A shares = 24,070,086 ÷ (24,070,086 + 232,318,385  10) = 0.010 or 1.0% P2-4

Initial public offering (LG5)

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a. Total proceeds = IPO offer price  Number of IPO shares issued = $16  6,000,000 = $96,000,000 b. Underwriting fee ($) = 7%  $96,000,000 = $6,720,000 c. Net proceeds = Total proceeds – Underwriting fee = $96,000,000 – $6,720,000 = $89,280,000 d. Market capitalization = Market price of stock  Number of shares outstanding = $20.08  19,189,391 = $385,322,971.28 e. IPO underpricing = (Market price – Offer price) / Offer price = ($20.08 – $16] / $16 = 25.5% f.

P2-5

Positive underpricing indicates that secondary-market investors were willing to pay more for the company’s shares than the IPO offer price that the shares were sold for in the primary market.

ETHICS PROBLEM An ethical issue arises because of access to material nonpublic information and the potential conflict between an insider’s duty to shareholders and concern for personal wealth. For example, suppose an insider knows that their firm plans to acquire another firm and quietly buys shares of the target firm— a move likely to be lucrative because, on average, the stock price of targets jumps on news of an acquisition. In this example, the insider puts personal gain ahead of shareholder welfare. Other market participants might observe the insider’s behavior and buy shares of the target firm as well—thereby boosting the target’s share price and raising the cost of the acquisition.

 Case Case studies are available on www.pearson.com/mylab/finance.

Pros and Cons of Being Publicly Listed a.

Going public will enable Robo-Tech to raise more external capital without additional bankruptcy risk. [Unlike creditors, shareholders cannot take the firm to bankruptcy court if expected dividends are not paid.] Going public will also allow the company to continue operating after Mr. Bradley (the owner/CEO) retires or dies and, before then, insulate him from personal liability for Robo-Tech debts. Finally, going public will give Mr. Bradley a chance to sell personal shares to cash in on his work building the firm or diversify his wealth. [Currently, his human capital and financial wealth are both largely tied up Robo-Tech. After the IPO, Mr. Bradley could sell some Robo-Tech shares and invest in the stocks and bonds of companies in other industries.]

b.

The disadvantages of going public include (i) more burdensome SEC reporting requirements, (ii) potential dilution of Mr. Bradley’s managerial control (e.g., if the IPO left him with fewer than 50% of Robo-Tech shares, a takeover artist could purchase controlling interest and force his removal as CEO), and finally (iii) double taxation of Mr. Bradley’s income (i.e., Robo-Tech will pay corporate income tax on firm profits, and Mr. Bradley will pay personal income tax on dividends/capital gains from company stock).

c.

Robo-Tech is probably too small to meet NYSE and NASDAQ listing requirements. The firm will probably trade over-the-counter or on regional exchanges.

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d.

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If the capital market is efficient, the price of Robo-Tech stock will provide an unbiased estimate of firm value. Efficiency also implies movements in stock price following news about the company will also be unbiased. Robo-Tech will, therefore, have an external real-time ―report card‖ on management actions. For example, suppose extensive research led management to believe moving several U.S. plants to Latin America would create value for shareholders. If firm stock dipped on the announcement, other things equal, management would know the market did not share their enthusiasm.

 Spreadsheet Exercise Answers to Chapter 2’s Velocity Financial spreadsheet problem are available on www.pearson.com/mylab/finance.

 Group Exercise There is no group exercise for Chapter 2.

 Integrative Case 1: Merit Enterprise Corp. a. Option 1 is borrowing $4 billion from JPMorgan Chase (or a syndicate of banks). The pros are the benefits of not going public. Going public means costly SEC disclosure requirements and potentially less scope for current owners to run the company over the long run. [Indeed, if current owners found themselves with fewer than 50% of Merit shares after the IPO, an outsider could purchase controlling interest and remove them from management.] Finally, going public means subjecting current owners to higher taxes—they would face corporate-income tax on Merit profits as well as personal-income tax on dividends/capital gains from Merit stock. The cons of option 1 include the short-run loss of some control to JPMorgan Chase (or the syndicate). For example, bank lenders of such a large sum would insist on restrictive covenants to limit Merit’s discretion in using the funds. And, if a loan payment were missed, bank lenders (unlike shareholders) could take the firm to bankruptcy court. b. Option 2 is going public to raise the needed $4 billion. The pros are the benefits of going public: (i) access to more external capital over time, (ii) insulation of Merit owners from personal liability for firm debts, (iii) opportunities for Merit owners to sell some of their ownership stake to increase consumption or diversify wealth, (iv) extension of Merit’s operating life beyond that of the current owners, and (v) greater flexibility in compensation (e.g., an IPO means Merit could use stock options) to attract more talented executives. The pros are the benefits of not ceding greater short-term control over firm decisions to bank lenders (see previous answer). c. Sara should choose the option that maximizes the wealth of the current owners. Student answers here will vary because the problem statement really does not make it clear whether one option dominates the others.

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Part Two Financial Tools Chapters in This Part Chapter 3

Financial Statements and Ratio Analysis

Chapter 4

Long- and Short-Term Financial Planning

Chapter 5

Time Value of Money

Integrative Case 2: Track Software Inc.

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Interest Rates and Bond Valuation

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Chapter 3 Financial Statements and Ratio Analysis  Instructor’s Resources Chapter Overview This chapter examines the four key components of the stockholders’ report: the income statement, the balance sheet, the statement of retained earnings, and the statement of cash flows. All major items on the income statement and balance sheet are reviewed along with rules for consolidating foreign and domestic financial statements (FASB No. 52). Next, the discussion turns to use of incomestatement/balance-sheet figures to assess a firm’s financial condition. Three types of comparative analysis are noted—cross-sectional, time-series, and combined—and specific ratios for such analysis are presented for five perspectives on firm condition—liquidity, activity, debt, profitability, and market. Each ratio is illustrated using the recent public financial statements of Target Corporation.

The meaning of deviations of performance ratios from industry benchmarks (as well as differences across industries) is also explored. The chapter ends with a complete (cross-sectional and time-series) ratio analysis of Target. The DuPont system is integrated into the example to show how profit margin, sales volume, and leverage interact to determine return on equity.

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 Suggested Answer to Opener-in-Review Students were told Gap Inc. reported $4.516 billion in current assets and $3.209 billion in current liabilities for 2020 and that the year before current assets and current liabilities were $4.251 billion and $2.174 billion, respectively. Students were asked to calculate the current ratios for the two years and comment on the change the company experienced in 2020.

The Gap saw a significant decline in liquidity in 2020, which combined with falling profitability and rising leverage is worrisome.

 Answers to Review Questions 3-1

Generally accepted accounting principles (GAAP), the Financial Accounting Standards Board (FASB), and the Public Company Accounting Oversight Board (PCAOB) all play significant roles in the financial reporting of publicly traded firms. GAAP refers to the basic guidelines firms should use in preparing and maintaining financial records and reports; these guidelines are authorized by the FASB— the accounting profession’s rule-setting body. The PCAOB is a notfor-profit corporation that oversees auditors of public corporations. Consistency in financial reporting and auditing practices/procedures promotes investor confidence in the financial information firms release to the public.

3-2

The four major financial statements are the following:

3-3



Income Statement, which summarizes firm operating results over a specified time period. It is a ―flow‖ document—demonstrating whether revenues over a month, quarter, or year exceed costs with sufficient detail to explain profits or losses.



Balance Sheet, which summarizes firm financial condition at a given point in time. It is a ―stock‖ document—noting assets, liabilities and net financial position (owner’s stake) on a specific date in detail.



Statement of Retained Earnings, which reconciles net income earned during the year (and any cash dividends paid) with the change in retained earnings from the beginning to the end of the year. It is a condensed version of the statement of stockholders’ equity.



The Statement of Cash Flows summarizes cash inflows and outflows experienced by a firm over a specific period (like a month, quarter, or year). In general, inflows and outflows are grouped under three headings: operating, investment, and financing. This statement is important to investors because cash flows, unlike profits, can be used to meet ongoing firm obligations.

Notes to the Financial Statements offer important background details for firm financial statements. Specifically, these notes explain how a firm’s accounting policies, procedures, calculations, and transactions have affected specific line items, thereby making financial statements easier to interpret. © 2022 Pearson Education, Inc.

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Interest Rates and Bond Valuation

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3-4

FASB Statement No. 52 governs rules for consolidating a firm’s foreign and domestic financial statements. The statement requires U.S.-based companies to translate foreign-currencydenominated assets and liabilities into U.S. dollars using the exchange rate on the last day of the fiscal year (current rate). Income-statement items are treated similarly. Equity accounts, in contrast, are translated into dollars using the exchange rate at the time of the parent’s equity investment (historical rate).

3-5

Current and prospective shareholders care about ratios bearing on expected cash flows and uncertainty about those flows because risk and return drive stock price. Creditors, on the other hand, focus on ratios gauging the firm’s short-term liquidity and ability to make scheduled interest and principal payments. Management needs to track ratios related to both risk/return and debt service. Managers should focus on maximizing shareholder wealth over the long term, but missing scheduled interest and principal payments could cause bankruptcy and prevent the firm from living past the short term.

3-6

Cross-sectional analysis involves comparing performance ratios for different firms at a specific point in time. Benchmarking is cross-sectional comparison of one firm’s performance ratios with those of a key competitor, group of competitors, or the industry average. Time-series analysis, in contrast, looks at the same firm’s performance over time (such as quarter-toquarter or year-over-year).

3-7

An analyst should focus on significant differences between firm ratios and those of a designated peer (competitor, group of competitors, or industry average), irrespective of whether the ratio is above or below the benchmark. For example, above-normal inventoryturnover ratio could indicate highly efficient inventory management or critically low inventory (and lost sales). When benchmarking, an analyst should also examine multiple ratios for a complete picture of each aspect of firm condition.

3-8

Analyzing financial data from different points in the year could lead to inaccurate conclusions because of seasonality. For example, many retailers post more sales in the fourth quarter than in the other three combined because of Christmas. So, comparing sales in the second and fourth quarters for such firms would make the second quarter look extraordinarily weak or the fourth quarter extraordinarily strong.

3-9

The current ratio is a better metric when current assets are all reasonably liquid while the quick ratio is preferred if the firm operates with high levels of illiquid inventory.

3-10 Most firms listed in Table 3.5 are large players in their industries; such firms typically rely on credit lines with banks for emergency cash. Put another way, small firms ―self-insure‖ against liquidity risk with a high current ratio while large firms insure through bank credit. Dillard’s is one of the smaller firms listed in the table. It’s roughly one-fourth the size of Nordstrom, and Target and Walmart are both more than 100 times larger than Dillard’s. Dillard’s also has a somewhat more upscale clientele and focuses less on ―necessities‖ compared with Target and Walmart. Because of that, Dillard’s is probably more sensitive to the business cycle and requires more liquidity to get through economic slowdowns.

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3-11 Average collection period, or average age of accounts receivable, is useful in evaluating a firms credit and collection policies. It equals accounts receivable divided by average daily sales.

Interpreting the ratio (in cross-section or time series analysis) requires context—specifically, what are the firm’s credit policies, how do they compare with other firms, and have they changed over time? Average payment period is accounts payable divided by average purchases per day. The difficulty in calculating this ratio is that the denominator—average daily purchases—is not available in firm financial statements. This too is context dependent and difficult to judge without knowing the credit terms offered to a firm by its suppliers.

3-12 Financial leverage refers to a firm’s reliance on debt (or other types of fixed-cost financing such as preferred stock) to fund ongoing operations. Financial leverage is important because greater reliance on debt can improve returns to shareholders but at the cost of higher risk.

3-13 The debt and debt-to-equity ratios gauge firm indebtedness (leverage). Specifically, the debt ratio is the percentage of firm assets financed by debt, while the debt-to-equity ratio is the relative proportion of debt and equity in the firm’s funding mix. Higher debt and debt-to-equity ratios correspond to greater financial leverage. Coverage ratios measure ability to service debts and other fixed obligations. Specifically, the times-interest-earned ratio captures the firm’s ability to pay interest on its debts, while the fixed-payment-coverage ratio shows its capacity to meet a broader set of fixed obligations (such as lease payments, principal payments on firm debt, and preferred stock dividends). For both coverage ratios, higher values indicate the firm is better able to pay fixed obligations.

3-14 The three profitability ratios found on a common-size income statement are (1) gross profit margin, (2) operating profit margin, and (3) net profit margin. Gross margin is the percentage of each sales dollar remaining after the cost of goods sold is covered. Operating margin is percentage of each sales dollar remaining after deducting all firm costs/expenses except interest, taxes, and preferred stock dividends. Net profit margin deducts all firm costs and expenses. For all three ratios, higher values are preferred.

3-15 Firms with high gross profit but low net profit margins have high operating expenses and other expenses that appear in the income statement after cost of goods sold. For example, a firm with significant financial leverage will have high interest expense, which will reduce its net profit margin but not its gross profit margin. Firms with higher-than-average leverage may report lower net profit margins relative to industry competitors that use little or no debt.

3-16 Return on assets (ROA) equals earnings available to common stockholders divided by total assets; return on equity (ROE) is earnings divided by common stock equity. ROA and ROE have the same numerator but different denominators. Firms with positive earnings and debt will have ROEs above their ROAs. Only when assets are entirely financed by common stock will ROE equal ROA. 3-17 The price-earnings ratio (P/E) captures what investors will pay for a dollar of earnings while the market/book (M/B) ratio shows market perception of firm value relative to the historical cost of assets. Both ratios embody a forward-looking perspective in that their numerators reflect investor expectations about future cash flows and the riskiness of those flows.

Interpreting these ratios for a specific firm is complicated by ―backward-looking‖ denominators (i.e., earnings already posted for the P/E ratio and historical cost of assets for the M/B ratio). Another issue with P/E ratios is the tendency of earnings to plummet during recessions, which can boost the ratio to eye-popping levels.

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3-18 Liquidity ratios measure firm capacity to meet current (short-term) financial commitments while activity ratios capture how rapidly a firm can convert various accounts into cash or sales.

Debt ratios gauge a firm’s dependence on creditors to finance ongoing operations and ability to service these obligations, and profitability ratios note a firm’s return with respect to sales, assets, or equity. Finally, market ratios provide insight into investor perceptions of firm risk and return and risk. Creditors will be more concerned with liquidity and debt ratios as these bear on the firm’s ability to meet its fixed commitments.

3-19 The analyst should use a ―level, peer, trend‖ approach for each of the five perspectives on firm condition (liquidity, activity, debt, profitability, and market). ―Level‖ means starting with computation of multiple ratios for each of the five perspectives and then determining whether the ratios tell a consistent story. If, for example, the current ratio is high (indicating strong liquidity), but the quick ratio low, then the firm is carrying significant inventory. The next step should be ascertaining whether that inventory can be sold with minimal losses in a cash crunch. ―Peer‖ means comparing firm ratios with key competitors or the industry average. ―Trend‖ means extending those comparisons over time to see longer-term patterns in each of the five areas and how these patterns compare with peer firms.

3-20 The DuPont system of analysis breaks firm return on equity (ROE) into three components: profitability (net profit margin), asset efficiency (total-asset turnover), and leverage (the debt ratio). This breakdown allows an analyst to isolate the impact of each factor on shareholder returns. For example, suppose firm A posts a significantly higher ROE than firm B. The DuPont system will highlight the ―big picture‖ reasons for the difference. If the firms have similar profit margins and asset efficiency, but firm A has higher leverage, then its higher ROE comes with greater risk of bankruptcy (and may not be a good thing). If, however, the difference stems from firm A’s higher net profit margin, then higher ROE reflects customer perception of firm A products as distinctive (and worth a significant mark-up).

 Suggested Answer to Focus on Practice Box: More Countries Adopt International Financial Reporting Standards What costs and benefits might be associated with a switch to IFRS in the United States?

The benefit is global standardization of accounting standards, which would allow American companies with foreign operations (or foreign companies with American operations) to reduce financial-reporting costs by using only one set of accounting policies/procedures—at least for operations in the 80+ countries now adhering to IFRS. In theory, investors would also benefit from greater ease of comparing firm performance across countries. The cost is a less rigorous standard.

Many consider GAAP ―the gold standard‖ and argue movement to IFRS would lower the quality of financial reporting by U.S. firms. One reason U.S. financial markets attract a large share of global savings is the greater transparency of American firms due to GAAP. If U.S. reporting standards were weakened, American firms could face greater difficulty selling stocks and bonds. A better approach to global standardization might be to bring the world in line with GAAP.

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 Suggested Answer to Focus on Ethics Box: Earnings Shenanigans Logitech understated potential warranty expenses by assuming customers would submit defectiveproduct claims within one quarter—even though warranties extended for many years. Suppose instinct tells you the assumption is reasonable and ethical because problems with electronic devices occur soon after purchase or not at all. What evidence might you compile to challenge your instincts and satisfy auditors?

If assumptions about warranty expense are material (i.e., significantly affect earnings), the firm should develop a warranty-expense policy grounded in historical experience and then apply that policy consistently over time to a range of company products. (Put another way, auditors and SEC lawyers should be satisfied the company did not just create a policy ―on the fly‖ to inflate earnings artificially.) Specifically, the firm should start by conducting extensive research on defective-product claims for a variety of its electronic devices. The goal would be documenting patterns in the number and dollar volume of claims for all major products. Once the database is built, differences across products can be investigated. If the number/dollar value of claims differs significantly across devices, the next step is selecting products similar (and dissimilar) to the one for which a warranty-expense policy is being developed. The firm should build a case by noting exactly why one product is a good match but another product is not and then document defective-claim experience for benchmark products. Ultimately, the firm will want a paper trail demonstrating a good-faith effort to develop expense policies using an extensive historical database, not cherry-picked data. Auditors and SEC lawyers will want to see evidence ―peer‖ products were selected using the same reasonable, a consistent approach over time, and that policies were developed based on consistent analysis of a historical database of firm products.

 Answers to Warm-Up Exercises E3-1

Preparing income statements (LG 1)

Answer: a.

Income Statement ($ Millions) Sales revenue Less: Cost of goods sold Gross profits Less: Operating expenses Sales expense General and administrative expenses Lease expense Depreciation expense Total operating expense Operating profits (EBIT) Less: Interest expense Net profit before taxes

$345.0 255.0 $90.0 $18.0 22.0 4.0 25.0 $69.0 $21.0 3.0 $18.0

b. Taxes = (Corporate tax rate)  (Net profits before taxes) = 0.21  $18 million = $3.78 million, so net profit after taxes = $18 – $3.78 = $14.22 million. c. EPS = Net profits after taxes / Shares outstanding = $14.22 million / 4.25 million = $3.34 Addition to retained earnings = (Net profit after taxes) – (Total dividends paid) Total dividends = (4.25 million)  ($1.10) = $4.675 million, so Addition to retained earnings = $14.22 – $4.675 = $9.545 million. © 2022 Pearson Education, Inc.

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E3-2

Interest Rates and Bond Valuation

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Income statements and balance sheets (LG 1)

Answer: On income statements, calculations begin with sales revenue and end with net profits after taxes. There is no guarantee a firm will be profitable. If net profits after taxes is positive, the firm earns a profit; if negative, the firm suffers a loss. As for balance sheets, the fundamental equation of accounting is: Total Assets = Total Liabilities + Owners’ Equity.

This simply means creditors or owners must supply the funds to purchase firm assets. The balance sheet demonstrates this relationship by showing the value of assets must equal debt and equity claims on those assets.

E3-3 Answer:

Statement of retained earnings (LG 1) Cooper Industries, Inc. Statement of Retained Earnings ($000) Year Ending December 31

Retained earnings balance (January 1) Plus: Net profits after taxes Less: Cash dividends Preferred stock Common stock Total dividends paid Retained earnings balance (December 31) E3-4

$25,320 5,150 750 3,850 4,600 $25,870

Current ratios and quick ratios (LG 3)

Answer: Between 2017 and 2022, Bluestone’s current ratio rose while the quick ratio fell.

Inventory appears in the numerator of the current ratio (as part of current assets), but not in the quick ratio. So the differing trends indicate Bluestone built up inventory over the six-year period— inconsistent with the CEO’s claims about the firm’s lean manufacturing model.

E3-5

The DuPont System (LG 6)

Answer: Return on equity (ROE) = 4.5%  0.72  1.43 = 4.63%. The DuPont system allows decomposition of firm ROE into a profit-on-sales component (net profit margin), an efficiency-of-asset-use component (total asset turnover), and a use-of-financial-leverage component (financial leverage multiplier). Such a decomposition highlights each component’s role in determining firm ROE. For example, the numbers above show how debt finance ―levers up‖ returns to shareholders. If the firm financed assets entirely with common equity, the financial-leverage multiplier would be 1.0 (rather than 1.43), and ROE would be only would be 3.24% (rather than 4.63%). Decomposing ROE with the DuPont system also allows comparison of component with key competitors or the industry average to see where the firm under- and outperforms.

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 Solutions to Problems P3-1

Financial statement account identification (Basic)

a., b.

Account Name Accounts payable Accounts receivable Accruals

Column 1 for (a) Statement BS BS BS

Column 2 for (b) Type of Account CL CA CL

Account Name Accumulated depreciation Administrative expense Buildings Cash Common stock (at par) Cost of goods sold Depreciation Equipment General expense Interest expense Inventories Land Long-term debt Machinery Marketable securities Notes payable Operating expense Paid-in capital in excess of par Preferred stock Preferred stock dividends Retained earnings Sales revenue Selling expense Taxes Vehicles

Column 1 for (a) Statement BS IS BS BS BS IS IS BS IS IS BS BS BS BS BS BS IS BS BS IS BS IS IS IS BS

Column 2 for (b) Type of Account FA* E FA CA SE E E FA E E CA FA LTD FA CA CL E SE SE E SE R E E FA

*

Not a fixed asset, but a charge against a fixed asset (and better known as a contra-asset).

P3-2

Income statement preparation (LG 1; Intermediate) a.

Cathy Chen, CPA – Income Statement Year End

Sales revenue Less: Operating expenses Salaries Employment taxes and benefits Supplies Travel and entertainment Lease payment

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$360,000 $180,000 34,600 10,400 17,000 32,400

Chapter 6

Depreciation expense Total operating expense Operating profits Less: Interest expense Net profits before taxes Less: Taxes (30%) Net profits after taxes

Interest Rates and Bond Valuation

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15,600 290,000 $70,000 15,000 $55,000 16,500 $38,500

b. In her first year of business, Ms. Chen covered operating expenses and earned a net profit of $38,500 (net profit margin of $38.500/$360,000 or 10.7%). Assuming the salary she paid herself ($96,000) is the going rate for accountants with comparable credentials and experience, she made what she would have working elsewhere and reaped an additional $38,500— a good year, indeed.

P3-3

Personal finance problem: Income statement preparation (LG 1; Intermediate) a. Personal Income and Expense Statement—Adam and Arin Adams Income Adam’s salary Arin’s salary Interest received Dividends received Total Income

$45,000 30,000 500 150 $75,650

Expenses Mortgage payments Utility expense Groceries Auto loan payment Home insurance Auto insurance Medical expenses Property taxes Income tax and social security Clothes and accessories Gas and auto repair Entertainment Total Expenses

$14,000 3,200 2,200 3,300 750 600 1,500 1,659 13,000 2,000 2,100 2,000 $46,309

Cash Surplus (Deficit)

$29,341

b. Income exceeds expenses, so the Adams have a cash surplus. c. The cash surplus can be used for a variety of purposes. In the short term, the Adams could replace their car, buy better furniture, or reduce their mortgage debt. Alternatively, they could put the surplus in a bank deposit or money-market account as source of emergency liquidity or purchase stocks and bonds as a long-term investment.

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P3-4

Calculation of EPS and retained earnings (LG 1; Intermediate) a.

Earnings per share: Net profit before taxes Less: Taxes at 21% Net profit after tax Less: Preferred stock dividends Earnings available to common stockholders

Earnings per share =

$436,000 91,560 $344,440 64,000 $280,440

Earning available to common stockholders $280,440 = = $1.65 Total shares outstanding 170,000

b. Addition to retained earnings: 170,000 shares  $0.80 = $136,000 common stock dividends Earnings available to common shareholders Less: Common stock dividends Retained earnings P3-5

$280,440 136,000 $144,440

Balance sheet preparation (LG 1; Basic) Mellark’s Baked Goods Balance Sheet Year End Assets Current assets Cash Marketable securities Accounts receivable Inventories Total current assets Gross fixed assets Land and buildings Machinery and equipment Furniture and fixtures Vehicles Total gross fixed assets Less: Accumulated depreciation Net fixed assets Total assets

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$215,000 75,000 450,000 375,000 $1,115,000 $325,000 560,000 170,000 25,000 $1,080,000 265,000 $ 815,000 $1,930,000

Chapter 6

Liabilities and stockholders’ equity Current liabilities Accounts payable Notes payable Accruals Total current liabilities Long-term debt Total liabilities

$220,000 475,000 55,000 $750,000 420,000 $1,170,000

Stockholders’ equity Preferred stock Common stock (at par) Paid-in capital in excess of par Retained earnings Total stockholders’ equity

$100,000 90,000 360,000 210,000 $760,000

Total liabilities and stockholders’ equity P3-6

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$1,930,000

Effect of net income on a firm’s balance sheet (Basic) Account a.

b. c.

P3-7

Interest Rates and Bond Valuation

Marketable securities Total assets Retained earnings Total liability and equity Long-term debt Retained earnings Buildings Total assets Retained earnings Total liability and equity

Initial Value

Change

Revised Value

$35,000 $4,900,000 $1,575,000 $4,900,000 $2,700,000 $1,575,000 $1,600,000 $4,900,000 $1,575,000 $4,900,000

+$1,365,000 +$1,365,000 +$1,365,000 +$1,365,000 –$865,000 +$865,000 +$865,000 +$865,000 +$865,000 +$865,000

$1,400,000 $6,265,000 $2,940,000 $6,265,000 $1,835,000 $2,440,000 $2,465,000 $5,765,000 $2,440,000 $5,765,000

Initial sale price of common stock (LG 1; Basic) Total proceeds from original sale of common stock = Par value of common stock + Paid-in capital in excess of par on common stock = $140,000 + $19,460,000 = $19,600,000 Original issue price of common stock = Proceeds from original sale / Total shares = $19,600,000/1,400,000 = $14

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P3-8

Statement of retained earnings (LG 1; Basic) a. Common stock dividends = Net profits after taxes – Preferred dividends – ∆ Retained earnings = $377,000 – $47,000 – ($1,048,000 – $928,000) = $210,000 Hayes Enterprises – Statement of Retained Earnings Year End

Retained earnings balance (January 1) Plus: Net profits after taxes Less: Cash dividends Preferred stock Common stock Retained earnings

$928,000 377,000 (47,000) (210,000) $1,048,000

b. Earnings available for common stockholders = Net profits after taxes – Preferred stock dividends = $377,000 – $47,000 = $330,000 Earnings per share = Earnings available for common stockholders / Common shares outstanding = $330,000 / 140,000 = $2.36 c. Cash dividend per share of common stock = Total cash dividends / common shares outstanding = $210,000 (from part a) / 140,000 = $1.50 P3-9

Changes in stockholders’ equity (LG 1; Intermediate) a. 2022 Net income = ∆ Retained earnings + Dividends paid = ($1,500,000 − $1,000,000) + $200,000 = $700,000 b. New shares issued = Outstanding shares in 2022 – Outstanding shares in 2021 = 1,500,000 − 500,000 = 1,000,000 c. Total proceeds from new issue = Par value of newly issued stock + Paid-in capital in excess of par on newly issued stock = ($1 Par value  1,000,000 shares) + $4,000,000 = $5,000,000 Average price of newly issued common stock = Total Proceeds from New Issue / Shares Issues = $5,000,000 / 1,000,000 = $5.00 d. Original issue price = Total Par Value (2021) + Paid-in Capital (2021) / Shares Outstanding (2021) = [$500,000 + $500,000] / 500,000 = $2.00

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P3-10 Ratio comparisons (LG 2, LG 3, LG 4, and LG 5; Basic) a. The companies operate in dissimilar industries, with wide-ranging differences in the nature of the product delivered, amount of plant/equipment needed for production, age of the industry, and degree of government regulation. So ratio comparisons will be apples to oranges. b. The electric utility and fast-food restaurant operate with lower liquidity ratios than the other firms, implying the two firms probably need less liquidity. Utilities tend to be relatively large, well-established firms, so Island Electric probably has a line of credit with a commercial bank to meet unexpected liquidity needs. Both Island Electric and Burger Heaven operate in relative stable industries—another reason for needing less liquidity. Finally, both firms operate primarily on a cash basis, so accounts receivable balances are going to lower than the other firms. c. Firms can operate with high levels of debt if cash flows are large, steady, and predictable.

Demand for power does vary by season, but in fairly predictable ways. Moreover, power is a necessity, not a luxury, so consumers cannot ―dump‖ the service in a recession. The software firm, in contrast, will have uncertain and changing cash flows because of rapidly changing technology, intense competition, and consumer fickleness, which can combine to make today’s ―hot‖ software tomorrow’s white elephant. In addition, a software company’s assets are mostly intangible and cannot be pledged as collateral for a loan, which means if the software company wanted to borrow money, it would likely have to issue unsecured debt which is more costly than secured debt. d. Although the software industry offers potentially high profits and returns, it also carries significant risk. As noted, rapidly changing technology, intense competition, and consumer fickleness combine to cause large, unexpected changes in the cash flows of software firms. Also, by investing in only one stock, investors would lose the riskreduction benefits of diversification.

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P3-11 Liquidity management (LG 3; Challenging) a. Both Bauman’s liquidity ratios are falling over time as shown below.

Ratio

2019

2020

2021

2022

Current ratio Quick ratio

1.88 1.22

1.74 1.19

1.79 1.24

1.55 1.14

b. Both ratios fall over the four-year period, indicating deterioration in Bauman’s liquidity position. Because peer data are not given, it is not clear if this deterioration is industrywide or only at Bauman. The slide is more pronounced for the current ratio—between 2019 and 2022, the current ratio tumbled 17.6% while the quick ratio slipped 6.5%.

Inventory figures in calculation of the current ratio but not the quick ratio. So the faster decline in the current ratio suggests inventories – though rising in dollar terms—are declining as a percentage of Bauman’s current assets over time. This trend may also be seen in the narrowing gap between the current and quick ratios. In short, inventories are becoming a less important part of the liquidity picture. c. Since 2019, Bauman’s inventory turns have generally increased (though not dramatically when comparing 2022 with 2019). This pattern is consistent with a declining inventory investment over time, at least relative to cost of goods sold. On the other hand, Bauman’s inventory turnover ratio is only about 60% of the industry average, suggesting the firm does a relatively poor job managing inventory. Put another way, Bauman carries more inventory (on average) than its peers, relative to cost of goods sold. This is at odds with the finding in part b that Bauman’s inventory is a shrinking part of current assets (and shrinking relative to current liabilities).

P3-12 Personal finance problem: Liquidity ratio (LG 3; Basic) a. Liquidity ratio = Total liquid assets / Total current debts = ($3,200 + $1,000 + $800) / ($1,200 + $900) = $5,000 / $2,100 = 2.38 b. Josh’s liquidity ratio exceeds 1.8, so he has more liquidity than financial advisors recommend for someone his age.

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P3-13 Inventory management (LG 3; Intermediate) Inventory-turnover ratio = Cost of goods sold / Inventories. So: Firm

2020

2019 2018 2017

Campbell Soup

7.5

6.3

4.9

5.4

Hormel

7.4

7.2

7.3

7.8

Tyson

9.6

8.8

9.1

9.9

Campbell Soup tends to have a slower rate of turnover than does Hormel or Tyson. Hormel and Tyson focus on meat products, which can be frozen but must be processed relatively quickly. Campbell’s main product is canned soup, which can last much longer than meat before spoiling. From 2017 to 2019, Campbell’s turnover averaged 6.0 compared with 7.4 for Hormel and 9.3 for Tyson. All three companies show increased turnover in 2020 versus 2019, but that comparison is problematic. The 2020 turnover figures could be distorted by the COVID outbreak in 2020, which shut down many food processing facilities for a time. Those shutdowns might have cut inventories, resulting in higher turnover temporarily. In addition, the figures for these ratios are based on each firm’s fiscal year, which means that they do not necessarily line up in terms of calendar time (if firms operate on different fiscal years). Some of the differences in turnover ratios might reflect seasonal patterns that show up because firms are reporting their financial results at different times of the year.

P3-14. Accounts receivable management (LG 3; Basic) a. A good ratio for evaluating a firm’s collection system is average collection period (= Accounts receivable  Average sales per day).

Average collection period 

$300,000 $300,000   45.62 days $2,400,000 6,575.34 365

Blair Supply normally extends 30-day credit to customers, so an average collection period over 15 days above 30 days suggests management should pay greater attention to accounts receivable. b. Seasonality could distort the average collection period and make it appear worse than it really is. In part a., we see that average sales per day figure is $6,575.34, but that average is taken over the whole year. If 70% of the firm’s sales occur in the last half of the year, then during that time the average sales per day figure is

Recalculating the average collection period using $9,205.48 as the average sales per day gives a collection period of just 32.6 days, which doesn’t seem out of line with the firm’s credit policy.

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P3-15 Interpreting liquidity and activity ratios (LG 3; Intermediate) a. Current ratio = Current assets / Current liabilities; Quick ratio = [Current assets – Inventory] / Current liabilities; Inventory-turnover ratio = Cost of goods sold / Inventories; Average collection period = Accounts receivable / Average sales per day; Average sales per day = Total sales / 365; and Total asset turnover = Sales / Total assets.

So, Firm Proctor & Gamble

Current Ratio 0.89

Quick Ratio 0.72

Inventory Turnover 6.89

Average Collection Period 26.46

Total Asset Turnover 0.56

Colgate-Palmolive

1.31

0.96

5.19

33.89

1.25

Clorox

0.76

0.51

6.45

31.93

1.29

b. Colgate-Palmolive boasts the highest current and quick ratios, so they have the most liquidity. Clorox’s relatively low liquidity ratios are surprising because it is considerably smaller than P&G and Colgate. Usually smaller companies have greater liquidity on their balance sheets because of (i) difficulty obtaining external finance in times of crisis and (ii) less predictable revenues than larger companies. The explanation might be that Clorox sells fewer products than P&G and Colgate, and consumer demand for those products are relatively stable over the business cycle. c. All three firms collect on sales in about 30 days, with the differences in average collection periods between the shortest (P&G) and longest (Colgate) collection periods only seven days. The most likely explanation is that companies compete with each other, selling similar products to most of the same customers, so probably offer similar credit terms. d. Procter & Gamble turned inventory over a bit faster than the other firms but assets much slower. This is surprising because both ratios measure asset efficiency—how could P&G excel at managing inventories (receivables, too, given its average collection period) but not overall assets? The most likely explanation is P&G employs more fixed assets than its competitors.

P3-16 Debt analysis (LG 4; Basic) Debt ratio = Total liabilities ÷ Total assets = $36,500,000 ÷ $50,000,000 = 0.73 Compared with an industry average of 0.51 Times interest earned ratio = EBIT ÷ Interest expense = $3,000,000 ÷ $1,000,000 = 3.00 Compared with an industry average of 7.30 Fixed payment coverage ratio =

(

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)

Compared with an industry average of 1.85.

Creek Enterprises finances a much larger percentage of assets with debt and has less ability to service additional debt than the average firm in the industry, so Springfield Bank should either reject the loan request or conduct more analysis to be sure that Creek will be able to repay the loan.

P3-17 Profitability analysis (LG 4 and LG 5; Intermediate) Gross profit margin = [Sales – Cost of goods sold] / Sales; Net profit margin = Earnings available for common stockholders / Sales; Return on Assets (ROA) = Earnings available for common stockholders / Total assets; Return on Equity (ROE) = Earnings available for common stockholders / Common stock equity So,

Firm Coca-Cola

Gross Profit Margin Net Profit Margin ROA ROE 60.7% 15.6% 7.5% 28.3%

Pepsico

55.1%

10.1%

8.5%

56.3%

Keurig Dr Pepper

59.9%

13.2%

8.7%

39.7%

It is difficult to say which company is most profitable because the four measures provide different answers. Coca-Cola posted the highest gross and net profit margins; at the same time, Keurig Dr Pepper generated the highest ROA, while Pepsico did the most for shareholders (in percentage terms). b. ROE exceeds ROA because—while both ratios have the same numerator (earnings)— ROE has a smaller denominator for all firms with some debt. (If a firm has no debt, assets equal equity, so ROE equals ROA.) Pepsico’s ROA lies between the other two firms, but its ROE is much larger, suggesting Pepsico uses more leverage than the other two firms.

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P3-18. Common-size statement analysis (LG 5) Creek Enterprises Common-Size Income Statement Years Ending December 31, 2021 and 2022

2021 Sales revenue Less: Cost of goods sold Gross profits Less: Operating expenses: Selling General Lease expense Depreciation Operating profits Less: Interest expense Net Profits before taxes Less: Taxes Net profits after taxes Less: Preferred stock dividends Earnings available for common stockholders

2022

100.0% 65.9% 34.1% 12.7% 6.3% 0.6% 3.6%

23.2% 10.9% 1.5% 9.4% 2.0% 7.4% 0.1% 7.3%

100.0% 70.0% 30.0% 10.0% 6.0% 0.7% 3.3%

20.0% 10.0% 3.3% 6.7% 1.4% 5.3% 0.3% 4.9%

In dollar terms, sales declined from $35 to $30 million. Meanwhile, as a percentage of sales, cost of goods sold increased, probably reflecting a loss of productive efficiency.

Operating expenses (as a percentage of sales) fell— a favorable development unless the decline contributed to drop in sales (say by reducing product quality). Interest expense as a percentage of sales more than doubled, suggesting the firm has too much debt. Further analysis should focus on the rise in cost of goods sold and the high debt level.

P3-19

The relationship between financial leverage and profitability (LG 4 and LG 5; Challenge) a.

(1) Debt ratio 

total liabilities total assets

$1,000,000  0.10  10% $10,000,000 $5,000,000 Debt ratio Timberland   0.50  50% $10,000,000 Debt ratio Pelican 

(2) Times interest earned 

earning before interest and taxes interest

$6,250,000  62.5 $100,000 $6,250,000 Times interest earned Timberland   12.5 $500,000 Times interest earned Pelican 

Timberland has much more financial leverage and, consequently, greater debt service than does Pelican, which will make Timberland’s earnings more volatile and expose its commonstock owners to greater risk. © 2022 Pearson Education, Inc.

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b. (1) Operating profit margin 

Interest Rates and Bond Valuation

li

operating profit sales

$6,250,000  0.25  25% $25,000,000 $6,250,000 Operating profit margin Timberland   0.25  25% $25,000,000 Operating profit margin Pelican 

Earnings available for common stockholders sales $3,690,000 Net profit margin Pelican   0.1476  14.76% $25,000,000 $3,450,000 Net profit margin Timberland   0.138  13.80% $25,000,000

(2) Net profit margin 

Earnings available for common stockholders total assets $3,690,000 Return on total assets Pelican   0.369  36.9% $10,000,000 $3,450,000 Return on total assets Timberland   0.345  34.5% $10,000,000

(3) Return on total assets 

(4) Return on common equity 

Earnings available for common stockholders Common stock equity

$3,690,000  0.41  41.0% $9,000,000 $3,450,000 Return on common equity Timberland   0.69  69.0% $5,000,000 Return on common equity Pelican 

Pelican is more profitable than Timberland, as shown by the higher operating profit margin, net profit margin, and return on assets. However, Timberland’s return on common equity exceeds Pelican’s because of Timberland’s greater leverage. c.

Pelican has a higher net profit margin because it uses less debt and, hence, has lower interest expense. But Timberland has a higher ROE because of greater financial leverage. Put another way, Timberland can spread its somewhat lower profits over a much smaller number of shareholders. That said, those shareholders face higher risk. Specifically, Timberland’s greater reliance on debt means a higher probability an unexpected fall in cash flows will cause bankruptcy.

P3-20 Analysis of debt ratios (LG 4; Intermediate) a. Debt Ratio = Total Liabilities / Total Assets. For Estée Lauder, the debt ratio is $5,636,000 / $9,223,300 = 0.61, and for e.l.f. Beauty, the ratio is $273,867 / $414,729 = 0.66.

Times Interest Earned Ratio = Earnings before Interest and Taxes / Interest. For Estée Lauder, the Times Interest Earned Ratio is $1,625,900 / $70,700 = 23.0, and for e.l.f.

Beauty, the ratio is $26,095 / $16,283 = 1.6. © 2019 Pearson Education, Inc.

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Debt ratios for the two companies are relatively similar, but Estée Lauder has a much larger ability to cover interest expense with current cash flow, so lenders would probably view its position as much less risky. b. Interest expense divided by total liabilities is conceptually similar to an interest rate—the ratio indicates how many dollars of interest a company must pay for each dollar of liabilities on its books. For Estée Lauder, this ratio is $70,700 / $5,636,000 = 0.0125 or about 1.25%, whereas for e.l.f. Beauty the ratio is $16,283 / 273,867 = 0.0594 or 5.94%.

Viewed this way, e.l.f. Beauty appears to pay a much higher interest rate on its debts.

There are two possible explanations for this pattern. The most important is the companies have different mixes of interest-bearing and non-interest-bearing liabilities. In fact, Estée Lauder’s long-term debt as a percentage of its liabilities is much lower than for e.l.f.

Beauty. Put another way, a sizable portion of Estée Lauder’s liabilities are current liabilities, and many of those do not bear interest at all. So the 1.25% ―interest rate‖ calculated above partly reflects the fact Estée Lauder pays no interest on many of its liabilities. A second explanation is e.l.f. Beauty is a much smaller company than Estée Lauder, so its interest-bearing debts probably carry a higher interest rate than Estée Lauder’s interest-bearing debts. In the eyes of lenders, smaller companies generally have fewer funding options in a cash squeeze, so lending to e.l.f. Beauty is riskier than lending to Estée Lauder.

P3-21 Ratio proficiency (LG 6; Basic) a.

Gross profit = sales × gross profit margin Gross profit = $37,500,000 × 0.745 = $27,937,500

b. Cost of goods sold = sales – gross profit Cost of goods sold = $37,500,000 – 27,937,500 = $9,562,500 c.

Operating profit = sales × operating profit margin Operating profit = $37,500,000 × 0.33 = $12,375,000

d. Operating expenses = gross profit – operating profit Operating expenses = $27,937,500 – 12,375,000 = $15,562,500 e.

Earnings available for common shareholders = sales × net profit margin = $37,500,000 × 0.076 = $2,850,000 sales $37,500,000   $11,718,750 total asset turnover 3.2

f.

Total assets 

g.

Total common equity 

h.

Accounts receivable  average collection period 

earnings available for common shareholders ROE $2,850,000 Total common equity   $15,079,365 0.189

Accounts receivable  58.7 days 

sales 365

$37,500,000  58.7  $102,740.73  $6,030,821.92 365

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P3-22 Cross-sectional ratio analysis (LG 6; Intermediate) a.

Ratio Analysis—Fox Manufacturing Company Debt Ratios Current ratio Quick ratio Activity Ratios Inventory turnover Average collection period Total asset turnover Debt Ratios Debt ratio Times interest earned Profitability Ratios Gross profit margin Operating profit margin Net profit margin Return on total assets Return on common equity Earnings per share

Industry Average

Fox

2.35 0.87

1.84 0.75

4.55 times 35.8 days 1.09

5.61 times 20.7 days 1.47

0.30 12.3

0.55 8.0

0.202 0.135 0.091 0.099 0.167 $3.10

0.233 0.133 0.072 0.105 0.234 $2.15

Liquidity: By both the current and quick ratios, Fox has a weaker liquidity position than the industry.

Activity: Inventory and asset turnover ratios compare favorably with the industry. Further analysis is necessary to determine whether Fox is truly in a weaker or stronger position than the industry. Higher inventory turnover, for example, may be the product of excessively low inventory levels (and resulting lost sales). Similarly, Fox’s shorter average collection period could be explained by extremely efficient receivables management, an overly zealous credit department, or excessively tight credit terms that reduce sales growth.

Debt: Fox uses more debt than the industry average, resulting in lower ability to cover interest with current cash flow.

Profitability: Fox posted a higher gross profit margin than the industry, suggesting a higher sales price or a lower cost of goods sold. Operating profit margin is in line with the industry, but net profit margin is lower— indicating relatively high expenses other than cost of goods sold. The likely explanation is interest expenses from Fox’s relatively high level of debt. On the bright side, Fox’s use of leverage produced a superior return on equity (ROE). b. Fox Manufacturing Company needs to improve its liquidity ratios and possibly reduce its debt load. Fox uses more leverage than the industry and, therefore, has more financial risk. At the same time, the firm’s leverage transforms relatively weak profitability into superior ROE.

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P3-23. Financial statement analysis (LG 6; Intermediate) a.

Zach Industries—Ratio Analysis Industry Actual Average 2021 Liquidity Ratios Current ratio Quick ratio Activity Ratios Inventory turnover Average collection period Debt Ratios Debt ratio Times interest earned Profitability Ratios Gross profit margin Net profit margin Return on total assets Return on common equity Market Ratio Market/book ratio

Actual 2022

1.80 0.70

1.84 0.78

1.04 0.38

2.50 37.5 days

2.59 36.5 days

2.33 57 days

65% 3.8

67% 4.0

61.3% 2.8

38% 3.5% 4.0% 9.5%

40% 3.6% 4.0% 8.0%

34% 4.1% 4.4% 11.3%

1.1

1.2

1.3

b. Liquidity: Zach Industries’ liquidity position deteriorated from 2021 to 2022 and now compares unfavorably with the industry average. The firm could have problems satisfying short-term obligations as they come due.

Activity: Zach’s activity ratios deteriorated from 2021 to 2022. Indeed, the firm now turns inventory over more slowly than peer firms. Similarly, average collection period lengthened by nearly three weeks, and now also compares unfavorably with the industry average.

Debt: Zach Industries’ debt position has improved since 2021 and now compares favorably with industry. That said, Zach Industries’ ability to service interest payments deteriorated – perhaps due to the higher rates creditors demanded in the past because of the firm’s above average debt load. As a result, Zach’s debt and debt service compare unfavorably with the industry average.

Profitability: Although Zach Industries’ gross margin is below the industry average (suggesting a high cost of goods sold), net profit margin rose from 2021 to 2022 and now compares favorably with industry. The high net margin translated into returns on assets and equity exceeding the average for peer firms—indeed, ROE is all the more impressive given that Bartlett reduced leverage absolutely and relative to the industry norm.

Market: Zach Industries’ market-to-book value rose from 2021 to 2022 and now compares even more favorably to industry norm. This trend indicates investors have recently come to view firm performance even more favorably than that of peer firms.

Overall, the firm maintains superior profitability at the risk of illiquidity. Investigation into the management of accounts receivable and inventory is warranted.

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P3-24. Integrative: Complete ratio analysis (LG 6; Challenge) a.

Sterling Company—Ratio Analysis

Ratio

(Given) Actual 2020

(Given) Actual 2021

Actual 2022

(Given) Industry 2022

Liquidity Current ratio

1.40

1.55

1.67

1.85

Quick ratio

1.00

0.92

0.88

1.05

Activity Inventory turnover

9.52

9.21

7.89

8.60

45.6 days

36.9 days 29.2 days

35.5 days

59.3 days 0.74

61.6 days 53.0 days 0.80 0.83

46.4 days 0.74

Debt Debt ratio

0.20

0.20

0.35

0.30

Times interest earned

8.2

7.3

6.5

8.0

4.5

4.2

4.051

4.2

0.30

0.27

0.25

0.25

0.12 0.062

0.12 0.062

0.13 0.082

0.10 0.053

0.045

0.050

0.068

0.040

0.061

0.067

0.120

0.066

$1.75

$2.20

$4.10

$1.50

12.0

10.5

9.63

11.2

1.20

1.05

1.16

1.10

Average collection period Average payment period Total asset turnover

Fixed-payment coverage Profitability Gross profit margin Operating profit margin Net profit margin Return on total assets Return on common Equity Earnings per share (EPS) Market Price/earnings (P/E) Market/book ratio (M/B)

TS: Time Series CS: Cross Sectional TS: Improving CS: Below Peer TS: Deteriorating CS: Below Peer TS: CS: TS: CS: TS: CS: TS: CS:

Deteriorating Below Peer Improving Above Peer Deteriorating Above Peer Improving Above Peer

TS: Deteriorating CS: Above Peer TS: Deteriorating CS: Poor TS: Deteriorating CS: Below Peer TS: Deteriorating CS: Equal to Peer TS: Improving CS: Above Peer TS: Improving CS: Above Peer TS: Improving CS: Above Peer TS: Improving CS: Above Peer TS: Improving CS: Above Peer TS: CS: TS: CS:

Falling Below Peer Steady Above Peer

Note: 1. Principal payments are not given in the problem; this figure assumes such payments are zero.

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Liquidity: Sterling’s overall liquidity position is weak compared with peer firms. The significant (and growing) difference in the current and quick ratios suggests the firm is holding a significant level of relatively illiquid inventory.

Activity: Sterling’s inventory turnover has deteriorated and currently lags the industry.

At the same time, the firm has made more efficient use of assets and now outperforms its peers. Sterling has also significantly improved collections, both absolutely and relative to peer.

Debt: Sterling’s debt ratio has risen from 2017 and now exceeds the industry average. As a consequence, times interest has deteriorated and now compares unfavorably with peer firms.

Profitability: Sterling’s gross margin, while in line with industry norms, has declined, probably because of increases in the cost of goods sold. Operating and net profit margins have been stable (or improving) at levels exceeding industry averages. Both ROA and the ROE have improved and compare favorably with the industry. EPS has risen considerably since 2017, but the P/E ratio has fallen, suggesting a lack of confidence in Sterling’s ability to match or exceed those earnings in the future.

Market: The firm’s P/E ratio has been falling. Its market-to-book ratio—though somewhat volatile—compare favorably with industry norms.

Overall, Sterling should work to improve inventory management and reduce debt load and service. Other than these issues, the firm appears to be doing reasonably wellespecially in generating returns on sales. Investors have generally endorsed this assessment, but somewhat volatile market ratios do hint at some uncertainty about the firm’s ability to take needed action.

P3-25 DuPont system of analysis (LG 6; Intermediate) a. Net profit margin = Earnings Available to Common Stockholders / Sales = $13,900 / $181,193 = 7.7% for AT & T = $19,265 / $131,868 = 14.6% for Verizon Asset turnover = Sales / Total assets = $181,193 / $551,669 = 0.33 for AT & T = $131,868 / $291,727 = 0.45 for Verizon Verizon has a higher net profit margin and a higher asset turnover. So Verizon is doing betting than AT & T in both categories. b. Return on assets (ROA) = Earnings Available to Common Stockholders / Total assets = $13,900 / $551,669 = 2.5% for AT & T = $19,265 / $291,727 = 6.6% for Verizon Return on equity (ROE) = Earnings Available to Common Stockholders / Common equity = $13,900 / $201,934 = 6.9% for AT & T = $19,265 / $62,835 = 30.7% for Verizon Verizon is outperforming AT & T in both ROA and ROE. c. Financial leverage multiplier (FLM) = ROE / ROA = 0.069 / 0.025 = 2.73 for AT & T = 0.307 /0.066 = 4.64 for Verizon

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Verizon’s FLM is higher than AT & T’s, which means Verizon is utilizing relatively more debt financing to leverage its ROE.

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P3-26 Complete ratio analysis, recognizing significant differences (LG 6; Intermediate) a. Note: the textbook contains an error in that it instructs students to use 2022 as the base year in the percentage change calculation rather than using 2021. Below we report the percentage difference using either 2021 or 2022 as the base year.

Home Health, Inc.

Ratio Current ratio Quick ratio Inventory turnover Average collection period Total asset turnover Debt ratio Times interest earned Gross profit margin Operating profit margin Net profit margin Return on total assets Return on common equity Price/earnings ratio Market/book ratio

2021 3.25 2.50 12.80 42.6 days 1.40 0.45 4.00 68% 14% 8.3% 11.6% 21.1% 10.7 1.40

2022 Difference 3.00 – 0.25 2.20 – 0.30 10.30 – 2.50 31.4 days – 11.2 days 2.00 + 0.60 0.62 + 0.17 3.85 – 0.15 65% – 3% 16% + 2% 8.1% – 0.2% 16.2% + 4.6% 42.6% +21.5% 9.8 – 0.9 1.25 – 0.15

Percentage Difference (2021 base year) – 7.69% – 12.00% – 19.53% – 26.29% +42.86% + 37.78% – 3.75% – 4.41% +14.29% – 2.41% + 39.65% +101.90% – 8.41% – 10.71%

Percentage Difference (2022 base year) –8.33% –13.64% –24.27% –35.67% +30.00% +27.42% –3.90% –4.62% +12.50% –2.47% +28.40% +50.47% –9.18% –12.00%

b.

Ratio Quick ratio Inventory turnover Average collection period Total asset turnover Debt ratio Operating profit margin Return on total assets Return on equity Market/book ratio

Proportional Difference 2022 base year

Home Health’s Favor

– 12.00% – 19.53% – 26.29% + 42.86% +37.78% + 14.29% + 39.65% +101.90% – 10.71%

No No Yes Yes No Yes Yes Yes No

c. The most obvious relationship is associated with the increase in the ROE value. The increase in this ratio is connected with the increase in the ROA. The higher ROA is partially attributed to the higher total asset turnover (as reflected in the DuPont model).

The ROE increase is also associated with the slightly higher level of debt as captured by the higher debt ratio.

P3-27 ETHICS PROBLEM (LG 1; Intermediate) Answers will vary by article chosen, but in general, students will report that financial statements are more trustworthy if company financial executives implement the provisions of Sarbanes-Oxley.

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 Case Case studies are available on www.pearson.com/mylab/finance.

Assessing Martin Manufacturing’s Current Financial Position Martin Manufacturing Company is an integrative case study employing financial-analysis techniques.

The company is a capital-intensive firm that manages accounts receivable and inventory poorly. The industry average inventory turnover can fluctuate from 10 to 100 depending on the market. a.

Ratio calculations Financial Ratio

2019

Current ratio Quick ratio Inventory turnover (times) Average collection period (days) Total asset turnover (times) Debt ratio Times interest earned (times) Gross profit margin Net profit margin Return on total assets Return on equity

Ratio

Historical Ratios—Martin Manufacturing Company Actual Actual Actual 2017 2018 2019

Current ratio Quick ratio Inventory turnover (times) Average collection period (days) Total asset turnover (times) Debt ratio Times interest earned Gross profit margin Net profit margin Return on total assets Return on equity Price/earnings ratio Market/book b.

$1,531,181  $616,000 = 2.5% ($1,531,181 – $700,625)  $616,000 = 1.3% $3,704,000  $700,625 = 5.3% $805,556  ($5,075,000  365) = 58.0 $5,075,000  $3,125,000 = 1.6 $1,781,250  $3,125,000 = 57% $153,000  $93,000 = 1.6 $1,371,000  $5,075,000 = 27% $33,000  $5,075,000 = 0.65% $33,000  $3,125,000 = 1.1% $33,000  $1,293,000 = 2.6%

1.7 1.0 5.2 50.7 1.5 45.8% 2.2 27.5% 1.04% 1.6% 3.0% 33.5 1.0

1.8 0.9 5.0 55.8 1.5 54.3% 1.9 28.0% 0.94% 1.4% 3.2% 38.7 1.1

2.5 1.3 5.3 58.0 1.6 57.0% 1.6 27.0% 0.65% 1.1% 2.6% 34.48 0.88

Industry Average 1.5 1.2 10.2 46.0 2.0 24.5% 2.5 26.0% 1.14% 2.3% 3.1% 43.4 1.2

Liquidity: Martin’s liquidity has improved in recent years and now compares favorably with industry. One potential concern is the significant difference between the current and quick ratio, implying the firm has much of its liquidity tied up in inventory.

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Activity: Inventory turnover has been relatively stable but at a much lower level than the industry norm, suggesting Martin is carrying too much inventory. The firm’s average collection period has risen and now exceeds the industry average by nearly two weeks—both indications of collection problems. Total asset turnover ratio has been stable, but again, at a lower level than the industry average, suggesting inefficient use of existing assets.

Debt: The debt ratio has increased and now substantially exceeds the industry average. The times interest ratio has also deteriorated and now falls considerably short of the industry norm.

Both ratios point to significant financial risk.

Profitability: Gross margin has been stable at a level just exceeding the industry average. Net profit margin, however, has deteriorated markedly and now trails the industry average. Martin’s debt level and service are the likely explanation for a strong gross but weak net margin.

Market: The market values Martin’s common stock at less than book value; in addition, the firm’s market-to-book ratio compares unfavorably with peer firms. The market has also put a price on Martin stock that is a lower multiple of earnings than the industry norm. c.

Martin Manufacturing clearly has a problem managing inventory and generating sales appropriate for its capital investment. The firm also carries substantial debt, which is further depressing profitability. Investors are aware of these weaknesses and priced them accordingly, hence Martin’s weak market ratios.

 Spreadsheet Exercise Answer to Chapter 3’s Dayton, Inc., financial-statement spreadsheet problem are available on www.pearson.com/mylab/finance.

 Group Exercise Group exercises are available on www.pearson.com/mylab/finance.

This chapter’s exercise focuses on each group’s shadow firm. Groups are asked to obtain their firm’s latest 10-K from the Securities and Exchange website (www.sec.gov) and then calculate/interpret basic performance ratios as done in the text. The number of years for time-series analysis is up to the instructor. That said, fewer years is advised the analysis can be performed using a single 10-K filing.

The conclusion of this assignment is calculation of the DuPont analysis for the shadow firm. This exercise shouldn’t require much assistance, particularly if students made a good choice of firm in Chapter 1.

Modifications could include dropping intertemporal analysis and focusing solely on the most recent year. Alternatively, groups could be asked to compare the ratios from their shadow firm with the ratios from another firm within the same industry.

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Chapter 4 Long- and Short-Term Financial Planning  Instructor’s Resources Chapter Overview This chapter introduces the financial-planning process, starting with an overview of long-term or strategic planning and moving to a detailed exploration of short-term (operating) financial planning and its two key components: cash and profit planning. Cash planning involves preparation of a cash budget, while profit planning involves preparation of a pro forma income statement and balance sheet. Step-by-step examples of cash budget and pro forma statement development are used to illustrate nuances students might find challenging—such as depreciation expenses as a cash inflow and the distinction between operating and free cash flow. The chapter ends by highlighting weaknesses of pro forma statements while still emphasizing the importance of these statements (along with the cash budget) as tools for disciplining management thinking about the range of possible cash flow and profitability outcomes and responses to those outcomes.

 Suggested Answer to Opener-in-Review Rapidly expanding firms—like Netflix—often must make significant investments in inventory, receivables, and fixed assets to sustain their growth. And the related cash outlays will not always appear immediately in calculation of profit. For example, purchase of a new capital equipment for $1 million dollars requires an outlay of $1 million dollars now, but that equipment may be expensed piecemeal through depreciation over several years (although under the Tax Cuts and Jobs Act of 2017, firms are allowed to fully expense many capital investments). Although not so much the case for Netflix, growing firms may also need to offer generous credit terms to attract customers from rivals. Sales on credit are typically recognized in the current year’s profit/loss calculation, even though full payment of the outstanding balance may take years. Accordingly, a firm could post enviable profits, yet at the same time, lack the cash flow to meet ongoing obligations without external finance.

 Answers to Review Questions 4-1

The financial-planning process is a two-step, highly collaborative endeavor to track the financial implications of the firm’s specific plan for creating value for shareholders. Step one is long-term or strategic planning, which involves detailing firm financial initiatives and the expected consequences of those initiatives over a two-to-ten-year horizon. Step two is developing a short-term (operating) financial plan, consistent with the strategic plan, which lays out the firm’s financial actions and likely results over a one- to two-year period.

4-2 The three key outputs of the short-term (operating) financial planning process are the (i) cash budget, (ii) pro forma income statement, and (iii) pro forma balance sheet.

4-3

Property classes under the Modified Accelerated Cost Recovery System (MACRS) are categorized by the length of the depreciation (recovery) period. The first four classes are 3-, 5-, 7-, and 10-years: © 2019 Pearson Education, Inc.

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Recovery Period 3 years

Definition Research and experiment equipment and certain special tools

5 years

Computers, printers, copiers, duplicating equipment, cars, light-duty trucks, qualified technological equipment, and similar assets

7 years

Office furniture, fixtures, most manufacturing equipment, railroad track, and single-purpose agricultural and horticultural structures

10 years

Equipment used in petroleum refining or in the manufacture of tobacco products and certain food products

For tax purposes, firms usually depreciate assets in the first four MARCS property classes via the double-declining-balance method, using a half-year convention (i.e., taking one-half year’s depreciation in the purchase year) and switching to straight-line depreciation when advantageous.

4-4

Cash flow from operating activities captures cash inflows/outflows related to the firm’s production cycle, beginning with the purchase of raw materials and ending with the finished product. Cash flow from investment activities tracks cash inflows/outflows from the purchases and sales of fixed assets and equity investments in other firms. Finally, cash flow from financing activities highlights cash inflows/outflows from transactions related to debt and equity financing—such as incurrence/repayment of debt, sales/repurchases of stock, and dividend payments.

4-5

A decrease in the cash balance is a source of cash flow because funds will be used for some other purpose, such as investment in inventory. Similarly, an increase in the cash balance is a use of cash flow because the funds, which the firm obtained by some other action such as an asset sale, are now being held in cash.

4-6

In compiling a cash-flow budget, it is important to recognize depreciation is a noncash expenditure on the firm’s income statement. Put another way, depreciation expense reduces taxable income but does not involve an actual cash outlay. [In contrast, there was a cash outlay when the depreciating asset was purchased.] So, depreciation must be added back to net income to find operating cash flows. The same logic holds for other non-cash expenses.

4-7

The statement of cash flows traces cash inflows/outflows from three different activities: (1) operating, (2) investing, and (3) financing. Traditionally, cash outflows are shown as negative numbers and cash inflows as positive numbers.

4-8

Operating cash flow isolates cash inflows/outflows from routine operations. Interest expense and taxes are excluded to keep the focus on cash flow from firm operations, independent of how the firm finances or government taxes those operations.

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Operating cash flow is cash flow generated from the firm’s regular production/sales of goods and services. Free cash flow is the remainder available to providers of debt (creditors) and equity finance (owners) after the firm has met operating needs and paid for net investment in fixed/and current assets. Formally, FCF = OCF – Net Fixed Asset Investment (NFAI) – Net Current Assets Investment (NCAI).

4-10 The cash budget is a statement of the firm’s planned cash inflows and outflows. Management uses this budget to estimate short-term cash needs and identify future periods with likely cash surpluses and shortages. The sales forecast is key to preparing the cash budget because most of the assumptions about cash inflows and outflows depend on the sales forecast.

4-11 The basic format of the cash budget appears below.

Total cash receipts Less: Total cash disbursements Net cash flow Add: Beginning cash Ending cash Less: Minimum cash balance Required total financing Excess cash balance

Jan.

Feb.

$XX XX XX XX XX XX $XX

$XX XX XX XX XX XX

   

Nov.

Dec.

$XX XX XX XX XX XX

$XX XX XX XX XX XX

$XX

The cash budget includes the following: 

Cash receipts: Total cash inflows in a given period (a month in the example above). The most common receipts are cash sales, collections of accounts receivable, and cash received from sources other than sales (dividends and interest received, asset sales, etc.).



Cash disbursements: Total cash outlays in a given period (again, a month above). The most common disbursements are cash purchases, payments of accounts payable, rent/lease payments, wages/salaries, tax payments, fixed-asset outlays, interest payments, payments of cash dividends, principal payments (loans), and repurchases/retirement of stock.



Net cash flow: Difference between cash receipts and disbursements in a given period (a month above).



Ending cash: Sum of beginning cash and net cash flow for a given period (a month above).



Required total financing: Amount of funds the firm needs if ending cash for the period is less than the desired minimum cash balance. Any short-term borrowing necessary to plug the gap will appear on the balance sheet as ―notes payable.‖



Excess cash: Amount of surplus funds if the firm’s ending cash exceeds its desired minimum cash balance. Managers usually invest the surplus in liquid, short-term, interest-paying securities.

4-12 The ending cash balance, along with the required minimum cash balance, indicate whether the firm will need additional cash or enjoy a surplus for the given period. If the ending cash balance falls short of the desired minimum cash balance, the firm must borrow short term to plug the gap. If ending cash exceeds minimum cash, the firm should invest the surplus short term in marketable securities.

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4-13 Uncertainty is the result of forecast error. For example, the linchpin of the cash budget is the sales forecast. No matter how complex the model used—that is, how many company-specific, market-specific, and macroeconomic variables are employed—the sales forecast is ultimately an extrapolation based on past relationships. Periodically, unexpected shocks will sever historical links between company/market/macroeconomic factors and sales, causing actual outcomes to differ significantly from predictions. One technique for coping with uncertainty is scenario analysis, which involves preparing different cash budgets for a range of specific sales levels—such as a pessimistic forecast, a most likely forecast, and an optimistic forecast. When uncertainty surrounding the cash budget is high, firms may also deal with that uncertainty by constructing cash budgets on a more frequent basis.

4-14 Pro forma statements provide the firm with a framework for analyzing future profitability.

They depend on two key inputs: the sales forecast and financial statements from the preceding year. Specifically, the sales forecast is applied to the latest balance sheet and income statement to obtain projected values for next period’s balance sheet and income statements.

4-15 In the percent-of-sales method of generating pro forma income statements, the financial manager estimates dollar values for individual expense items (such as cost of goods sold, operating expenses, and interest expense) as a fixed percentage of projected sales.

4-16 The percent-of-sales method assumes all costs are variable—a weakness because most firms have some fixed costs. For firms with significant fixed costs, the percent-of-sales method understates estimated profit when sales are projected to rise and overstates estimated profit when sales are projected to fall. An analyst can minimize this problem by dividing the expense portion of the pro forma income statement into fixed and variable components.

4-17 The judgmental approach to the pro forma balance sheet involves estimation of some balancesheet items, with external finance used as a balancing or ―plug‖ factor. This method assumes variables such as cash, accounts receivable, and inventory can be forced to take certain values, rather than occurring as a natural product of ongoing firm transactions.

4-18 The ―plug‖ figure in the judgmental approach is the external financing necessary to bring the pro forma balance sheet into balance. A positive value means the firm must raise funds externally to meet operating needs—by incurring debt, issuing equity, or reducing dividends. A negative plug figure implies the firm expects more than sufficient internal finance to support asset growth. Surplus funds can be used to repay debt, repurchase stock, or increase dividends.

4-19 Pro forma statements have two weaknesses, each arising from a problematic assumption— namely, (1) the firm’s past financial condition is an accurate predictor of its future and (2) the values of certain variables in the statements can be forced to take desired values.

4-20 The financial manager uses pro forma statements to provide a baseline for evaluating ongoing performance as the year begins. For example, she might perform ratio analysis and prepare source/use statements. She could also treat the pro forma statements as actual statements to assess various aspects of firm condition—liquidity, activity, debt, and profitability—and then adjust planned operations to achieve short-term financial goals.

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 Suggested Answer to Focus on Ethics Box: “Is Excess Cash Always a Good Thing?”

Suppose unexpected events in the market for your product leave you with significant free cash flow. What benchmark should you use in determining the best (and most ethical) use of those funds?

Managers should invest free cash flow in only those projects likely to earn returns above what firm shareholders could obtain elsewhere in financial markets by investing in other firms with comparable risk profiles (assuming shareholders could invest the free cash flow themselves). Otherwise, managers should return the firm’s excess cash to shareholders, perhaps through a special dividend.

 Suggested Answer to Focus on Practice Box: “Free Cash Flow at Abercrombie & Fitch” Free cash flow is often considered a more reliable measure of a company’s income than reported earnings. In what possible ways might corporate accountants change earnings to present a more favorable earnings statement?

Reported earnings for a given year can be boosted many ways. For example, a firm could offer exceedingly generous credit terms near the end of the fiscal year to pump up sales. The firm could also push some expenditures planned for the last quarter of the year to the next quarter so the expenses would show up on next year’s income statement. Although shady, such practices are not illegal. The Focus on Ethics box in Chapter 3 described how Logitech fraudulently boosted earnings by overstating the value of unsold units of a disappointing product and understating likely expenses associated with honoring warranties.

 Answers to Warm-Up Exercises E4-1

Depreciation schedule (LG 2)

Answer: Recovery Year 1 2 3 4 5 6 Total Depreciation

Recovery

by Year 20% 32% 19% 12% 12% 5% 100%

Depreciation $13,000 $20,800 $12,350 $ 7,800 $ 7,800 $ 3,250 $65,000

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E4-2

Cash inflows and outflows (LG 3)

Answer: a. b. c. d. e. f.

E4-3

Marketable securities increased Land and buildings decreased Accounts payable increased Vehicles decreased Accounts receivable increased Dividends paid

Cash Outflow Cash Inflow Cash Inflow Cash Inflow Cash Outflow Cash Outflow

Operating cash flow (LG 3)

Answer: OCF  [EBIT  (1  T)]  depreciation, and EBIT Sales – Cost of goods sold − Operating expenses − Depreciation. So, OCF = [($2,500  $1,800  $300 $200)  (1  0.35)] + $200 OCF  [$200  (0.65)]  $200  $330 E4-4

Free cash flow (LG 3)

Answer: FCF  OCF  Net fixed asset investment (NFAI)  Net current asset investment (NCAI), and NFAI  Change in net fixed assets  depreciation; NCAI  Change in current assets  change in (accounts payable  accruals) NFAI  $300,000  $200,000  $500,000 NCAI  $150,000  $75,000  $75,000. Hence, FCF  $700,000  $500,000  $75,000  $125,000 E4-5

Estimating net profits before taxes (LG 5)

Answer:

Rimier Corp—Pro Forma Income Statement Sales revenue $650,000 Less: Cost of goods sold Fixed cost 250,000 Variable cost (0.35 × sales) 227,500 Gross profits $172,500 Less: Operating expenses Fixed expense 28,000 Variable expenses (0.075 × sales) 48,750 Operating profits $ 95,750 Less: Interest expense (all fixed) 20,000 Net profits before taxes $ 75,750

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 Solutions to Problems P4-1

Depreciation (LG 2; Basic) Depreciation Schedule Percentages Cost (1) from Table 4.2 (2)

Year Asset A Research Equipment 1 2 3 4 Asset B Duplicating Equipment 1 2 3 4 5 6

Depreciation (3) = [(1)  (2)]

$17,000 $17,000 $17,000 $17,000

33% 45% 15% 7%

$5,610 $7,650 $2,550 $1,190

$45,000 $45,000 $45,000 $45,000 $45,000 $45,000

20% 32% 19% 12% 12% 5%

$9,000 $14,400 $8,550 $5,400 $5,400 $2,250

P4-2 Depreciation (LG 2; Basic) Over the next four years, Sosa will show the same total earnings and cash flows under each method, but the timing of earnings and cash flows will be different. MACRS will result in higher reported earnings this year and lower reported earnings in subsequent years compared to what Sosa will report if they use bonus depreciation. From a cash flow perspective, bonus depreciation means that the initial year will have the lowest earnings and therefore the lowest tax bill, so cash flow in the initial year is higher using bonus depreciation. In subsequent years, using bonus depreciation means reporting higher earnings and paying higher tax bills again compared to MACRS. From a valuation perspective, it is better for Sosa to use bonus depreciation. Their total tax bill over time will be the same no matter which method they choose, but they will defer more of the tax bill to later years if they use bonus depreciation.

Due to the time value of money, paying taxes later is better than paying them now.

P4-3

MACRS depreciation expense and accounting cash flow (LG 2 and LG 3; Challenge) a. Depreciation expense  $80,000  0.20  $16,000 (MACRS depreciation percentages can be found in Table 4.2 in the text.) b. The tax savings equal the depreciation deduction times the marginal tax rate. For example, if the firm’s tax rate is 21%, then the tax savings is 0.21 $16,000 = $3,360.

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P4-4

Depreciation and accounting cash flow (LG 2 and LG 3; Intermediate) a. Operating cash flow (OCF) = [Earnings before interest and taxes Tax rate)] Depreciation Sales revenue Less: Total costs before depreciation, interest, and taxes Depreciation expense (0.19 × $180,000) Earnings before interest and taxes (EBIT)

$400,000 290,000 34,200 $ 75,800

So, OCF = [$75,800 × (1 $34,200 = $94,082. b. Depreciation expense is an accounting entry to smooth the cost of an asset over time; there is no cash outlay. So, when a firm deducts depreciation on its income statement, net income will be less than cash flow, but the depreciation charge reduces a firm’s tax bill.

To find cash flow, depreciation must be added back to after-tax earnings.

P4-5

Classifying inflows and outflows of cash (LG 3; Basic) Notes: (i) Any reduction in an asset is a cash inflow because that cash has been released for another purpose. Hence, the $300 decline in cash below is an inflow. (ii) Depreciation does not involve a cash outlay but is deducted from income to obtain profit; it must be added back to after-tax earnings to determine cash flow (i.e., treated as a cash inflow).

Item

Change ($) Cash 300 Accounts payable 1,200 Notes payable 500 Long-term debt 1,000 Inventory 200 Fixed assets 400 P4-6

I/O I O I I O O

Item Accounts receivable Net profits Depreciation Repurchase of stock Cash dividends Sale of stock

Change ($) 700 900 100 900 800 1,000

I/O O I I O O I

Finding operating and free cash flow (LG 3; Intermediate) NOTE: The original version of this problem said that Nike earned EBIT of $1.19 billion with depreciation expenses of $0.51 million. There are two errors here. First, it is NOPAT, not EBIT, that equals $1.19 billion. Because the problem gives EBIT, we need a tax rate to find NOPAT, and below we assume a 21% tax rate. Second, the depreciation expense should be $0.51 billion rather than $0.51 million, though in our answer below we use $0.51 million as stated in the text. Beneath the solution below, we show a revised solution for the corrected problem as it will appear in reprints of the text.

All values are in $ billions. a. NOPAT = EBIT × (1 – T) = $1.19(1 – 0.21) = $0.94 OCF  NOPAT  Depreciation  $0.94  $0.00051  $0.94051 b. Net fixed asset investment (NFAI) = ∆Net fixed assets Depreciation = ($4.78  $4.67)  $0.00051 = $0.11051 Note that this represents additional investment for Nike and therefore a cash outflow.

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c. Net current asset investment (NCAI) = ∆Current Assets ∆(Accounts payable + Accruals) = ($15.74  $16.37) ($8.28  $8.26) = $0.65 Note that by reducing their current asset investment Nike experienced a cash inflow. d. FCF = OCF  Net fixed asset investment (NFAI)  Net current asset investment (NCAI) = $0.94051  $0.11051 + $0.65  $1.48 Here is a solution if the text you are using is a reprint with the errors described above corrected.

NOPAT = $1.19 billion OCF = NOPAT + Depreciation = $1.19 + $0.51 = $1.7 Net fixed asset investment (NFAI) = ∆Net fixed assets Depreciation = ($4.78  $4.67) $0.51 = $0.62 Net current asset investment (NCAI) = ∆Current Assets ∆(Accounts payable + Accruals) = ($15.74  $16.37) ($8.28  $8.26) = $0.65 Note that by reducing their current asset investment Nike experienced a cash inflow.

FCF = OCF  Net fixed asset investment (NFAI)  Net current asset investment (NCAI) = $1.7  $0.62 + $0.65  $1.73 P4-7

Finding operating and free cash flow (LG 3; Intermediate) a. Net operating profit after taxes (NOPAT)  Earnings before interest and taxes (EBIT)  [1  Tax rate (T)]  $2,700  (1  0.21)  $2,133. b. OCF  NOPAT  Depreciation  $2,133  $1,600  $3,733 c. FCF  OCF  Net fixed asset investment (NFAI)  Net current asset investment (NCAI) NFAI = ∆Net fixed assets Depreciation = ($14,800  $15,000)  $1,600 = $1,400 NCAI = ∆Current Assets ∆Accounts payable ∆Accruals) = ($8,200  $6,800)  ($1,600  $1,500)  ($200  $300) = $1,400. So, FCF  $3,733  $1,400  $1,400  $933 d. Keith Corporation has positive cash flows from operating activities. Operating cash flow (OCF) is more than twice NOPAT. FCF is positive, meaning cash flows from operations are adequate to cover both operating expense plus investment in fixed and current assets.

P4-8

Statement of cash flows (LG 4; Intermediate) a. The change in stockholders equity of $157 on the balance sheet is entirely traceable to a $157 increase in retained earnings. No other equity accounts changed in the current year.

From Tables 4.4, 4.5 and 4.6, the increase in retained earnings may be broken down as follows: Net profits after taxes $237 Less preferred dividends $10 Less common dividends $70 © 2019 Pearson Education, Inc.

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Net change in retained earnings

$157

On the cash-flow statement, the entry for net profits after taxes appears as a positive $237 (cash inflow) under ―cash flow from operations.‖ The entry common/preferred dividends paid appears as a negative $80 (cash outflow) under ―cash flow from financing.‖ The net effect of those two entries is an increase of cash of $157, so including the $157 change in retained earnings separately in the cash flow statement would be double counting. b. Other values are possible. For, example, the company could sell new stock for cash payment.

P4-9

Cash receipts (LG 4; Basic) Sales (St) Cash sales (0.50 St) Collections: Lag 1 month (0.25 St-1) Lag 2 months (0.25 St-2) Total cash receipts

April $65,000 $32,500

May $60,000 $30,000

June $70,000 $35,000

July $100,000 $50,000

August $100,000 $50,000

$16,250

$15,000 $16,250 $66,250

$17,500 $15,000 $ 82,500

$25,000 $17,500 $92,500

P4-10 Cash disbursements schedule (LG 4; Basic) Sales (St) Purchases (Pt = 0.6 St+1) Disbursements Cash purchases (0.1 Pt) Payments of A/P: 1-month delay (0.5 Pt-1) 2-month delay (0.4 Pt-2) Rent payments Wages & salary Fixed Variable (0.07 St) Taxes payments Fixed assets outlays Interest payments Cash dividend payment Total cash disbursement

February $500,000

March $500,000

April $560,000

May $610,000

June July $650,000 $650,000

$300,000

$336,000

$366,000

$390,000

$390,000

36,600

39,000

39,000

168,000

183,000

195,000

120,000 8,000

134,400 8,000

146,400 8,000

6,000

6,000

6,000

39,200

42,700

45,500 54,500

75,000 30,000 12,500 $465,300

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$413,100

$524,400

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P4-11 Cash budget: Basic (LG 4; Basic) Sales Cash sales (0.2 St)) Lag 1 month (0.6 St-1) Lag 2 months (0.2 St-2) Other income Total cash receipts Disbursements Purchases Rent Wages & salaries (0.1 St-1) Dividends Principal & interest Purchase of new equipment Taxes due Total cash disbursements

March $50,000 $10,000

April $60,000 $12,000

May $70,000 $14,000 36,000 10,000 2,000 $62,000

June $80,000 $16,000 42,000 12,000 2,000 $72,000

July $100,000 $20,000 48,000 14,000 2,000 $ 84,000

$50,000 3,000 6,000

$70,000 3,000 7,000 3,000 4,000

$80,000 3,000 8,000

6,000

March

April

Total cash receipts Total cash disbursements Net cash flow Add: Beginning cash Ending cash Minimum cash Required total financing (notes payable) Excess cash balance (marketable securities)

$59,000

6,000 $93,000

$97,000

May $62,000 59,000 $ 3,000 5,000 $ 8,000 5,000

June $72,000 93,000 ($21,000) 8,000 ($13,000) 5,000

July $84,000 97,000 ($13,000) (13,000) ($26,000) 5,000

0

$18,000

$31,000

$ 3,000

0

0

The firm should establish a credit line of at least $31,000 but may need to secure three to four times this amount based on scenario analysis.

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P4-12 Personal finance problem: Preparation of cash budget (LG 4; Basic) a.

Sam and Suzy Sizeman—Personal Budget This Year October November Income Take-home pay

December

$4,900

$4,900

$4,900

$1,470 245 490 343 25 147

$1,470 245 490 343 25 440

49 98 294 368 245 221 $3,995

$1,470 245 490 343 25 147 564 49 98 294 368 245 221 $4,559

Cash surplus or (deficit)

$ 905

$ 341

$ (594)

Cumulative cash surplus or (deficit) Note: Amounts are rounded.

$ 905

$1,246

$ 652

Expenses Housing Utilities Food Transportation Medical/Dental Clothing Property taxes Appliances Personal care Entertainment Savings Other Excess cash Total expenses

Percent 30.0% 5.0% 10.0% 7.0% 0.5% 3.0% 11.5% 1.0% 2.0% 6.0% 7.5% 5.0% 4.5%

b. December is a deficit month. c. The cumulative cash surplus is at the end of December is $652.

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49 98 1,500 368 245 221 $5,494

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P4-13 Cash budget: Advanced (LG 4; Challenge) a. and b.

Xenocore, Inc.($000) Apr

May

Jun

Jul

Aug

Sep

Oct

Nov

Forecast of Sales St) Cash sales (0.20 St) Collections Lag 1 month (0.40 St-1) Lag 2 months (0.40 St-2) Other cash receipts Total cash receipts

$210 $250

$170 $ 34

$160 $ 32

$140 $ 28

$180 $ 36

$200 $ 40

$250 $ 50

100 84

68 100

$218

$200

64 68 15 $175

56 64 27 $183

72 56 15 $183

80 72 12 $214

Purchases Forecast (Pt) Cash purchases (0.10 Pt) Payments Lag 1 month (0.50 Pt-1) Lag 2 months (0.40 Pt-2) Salaries & wages (0.20 St-1) Rent Interest payments Principal payments Dividends Taxes Purchases of fixed assets Total cash disbursements

$120 $150

$140 $ 14

$100 $ 10

$ 80 $ 8

$110 $ 11

$100 $ 10

$ 90 $ 9

75 48 50 20

70 60 34 20

50 56 32 20 10

40 40 28 20

55 32 36 20

50 44 40 20 10 30 20 80

Net cash flow (Receipts – disbursements) Add: Beginning cash Ending cash Less: Minimum cash balance Required total financing (notes payable) Excess cash balance (marketable securities)

20

$207

25 $219

$196

$139

$153

$303

11 22 33 15

(19) 33 14 15

(21) 14 (7) 15

44 (7) 37 15

30 37 67 15

(89) 67 (22) 15

1

22

18

37 22

52

c. The line of credit should be at least $37,000 to cover the maximum borrowing needs (for November).

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P4-14 Cash flow concepts (LG 4; Basic) Note to instructor: There are a variety of possible answers to this problem, depending on student assumptions. The question is designed to provoke discussion of differences among cash flows, income, and assets.

Cash Pro Forma Pro Forma Budget Income Statement Balance Sheet Transaction Cash sale Credit sale Accounts receivable are collected Asset with a five-year life is purchased Depreciation is taken Amortization of goodwill is taken Sale of common stock Retirement of outstanding bonds Fire insurance premium is paid for the next three years

X X X X

X X

X X

X X X X X X X X

X

X

X X

P4-15 Cash budget: Scenario analysis (LG 4; Intermediate) a.

Trotter Enterprises, Inc.—Multiple Cash Budgets ($000)

Total cash receipts Total cash disbursements Net cash flow Add: Beginning cash Ending cash: Less minimum cash balance Required total financing Excess cash balance

OCTOBER PessiMost Optimistic Likely mistic

NOVEMBER Pessi- Most Optimistic Likely mistic

DECEMBER Pessi- Most Optimistic Likely mistic

$260

$342

$462

$200

$287

$366

$191

$294

$353

285 25

326 16

421 41

203 (3)

261 26

313 53

287 (96)

332 (38)

315 38

(20) 45 18

(20) (4) 18

(20) 21 18

(45) (48) 18

(4) 22 18

21 74 18

(48) (144) 18

22 (16) 18

74 112 18

$ 63

$ 22

$ 162

$ 34

$ 66 $3

$4

$56

$ 94

b. Under the pessimistic scenario, Trotter will need a credit line of at least $162,000 to cover a cash shortfall of that amount in December. In the most likely scenario, Trotter will need a credit cline of at least $34,000 to cover a cash shortfall of that magnitude in December. In the optimistic scenario, Trotter will not have a cash shortfall in the last three months of the calendar year.

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P4-16 Multiple cash budgets: Scenario analysis (LG 4; Intermediate) a. and b.

Brownstein, Inc. Multiple Cash Budgets ($000) 1st Month

Sales Asset sale Purchases Wages Taxes Purchase of fixed asset Net cash flow Add: Beginning cash Ending cash:

2nd Month

3rd Month

Pessimistic

Most Likely

Opti- Pessi- Most Opti- Pessi- Most Optimistic mistic Likely mistic mistic Likely mistic

$ 80

$100

$120

$ 80

$100

$120

(60) (14) (20)

(60) (15) (20)

(60) (16) (20)

(60) (14)

(60) (15)

$(14)

$ 5

$ 24

(15) $ (9)

0 $(14)

0 $ 5

0 $ 24

(14) $(23)

(60) (16)

$80 $100 8 8 (60) (60) (14) (15)

$120 8 (60) (16)

(15) $ 10

(15) $ 29

$14

$ 33

$ 52

5 $ 15

24 $ 53

(23) $ (9)

15 $ 48

53 $105

c. Considering the extreme values reflected in the pessimistic and optimistic outcomes allows Brownstein to plan borrowing or short-term investments carefully. For example, the firm knows the worst-case scenario is the need for a credit line of at least $23,000 to cover a cash shortfall in the second month (in the pessimistic scenario).

P4-17 Pro forma income statement (LG 5; Intermediate) a.

Pro Forma Income Statement—Metroline Manufacturing, Inc. (Percent-of-Sales Method) Sales Less: Cost of goods sold (0.65  sales) Gross profits Less: Operating expenses (0.086  sales) Operating profits Less: Interest expense Net profits before taxes Less: Taxes (0.21  NPBT) Net profits after taxes Less: Cash dividends To retained earnings

$1,500,000 975,000 $ 525,000 129,000 $ 396,000 35,000 $ 361,000 75,810 $ 285,190 70,000 $ 215,190

Note: Operating expense percentage was found by dividing operating expenses ($) by sales ($) for the year just ended, and rounding to third decimal place.

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b.

Pro Forma Income Statement - Metroline Manufacturing, Inc (Fixed and Variable Data) Sales Less:

$1,500,000 Cost of goods sold - fixed 210,000 Cost of goods sold - variable (0.5  sales) 750,000 Gross profits $ 540,000 Less: Fixed expense 36,000 Variable expense (0.06  sales) 90,000 Operating profits $ 414,000 Less: Interest expense 35,000 Net profits before taxes (NPBT) $ 379,000 Less: Taxes (0.21  NPBT) 79,590 Net profits after taxes $ 299,410 Less: Cash dividends 70,000 To retained earnings $ 229,410 Note: Variable cost and expense percentages were found by dividing variable cost or expense ($) by sales ($) for the year just ended, and rounding to third decimal place. c. When sales are projected to rise, and the firm has fixed costs, pro forma income statements based on percentages of sales will overstate costs and understate profits. Both conditions are met here, so a pro forma income statement decomposing fixed and variable cost would be more accurate.

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P4-18 Pro forma income statement: Scenario analysis (LG 5; Challenge) a.

Pro Forma Income Statement—Allen Products, LP for the Coming Year Sales Less cost of goods sold (45% of sales) Gross profits Less operating expense (25% of sales) Operating profits Less interest expense (3.2% of sales) Net profit before taxes Less Taxes (24%) Net profits after taxes

Pessimistic

Most Likely

Optimistic

$900,000 405,000 $495,000 225,000 $270,000 28,800 $241,200 57,888 $183,312

$1,125,000 506,250 $ 618,750 281,250 $ 337,500 36,000 $ 301,500 72,360 $229,140

$1,280,000 576,000 $ 704,000 320,000 $ 384,000 40,960 $ 343,040 82,330 $260,710

b. The simple percent-of-sales method assumes all costs/expenses are variable. In the pessimistic case, this assumption causes all costs/expenses to fall by a given percentage of the decline in sales from the ―most likely‖ baseline. In reality, however, some costs/expenses are fixed, so total costs/expenses will not decline as much as projected. In short, the percent-of-sales method understates costs/expenses and overstates profit relative to the fixed-variable method when sales are falling. The opposite occurs in the optimistic case—that is, the percent-of-sales method assumes all costs/expenses rise by a given percentage of sales when in reality only the variable portion increases. So, when sales rise, percent-of-sates estimates overstate costs/expenses and understate profits relative to the fixed-variable method. c.

Pro Forma Income Statement—Allen Products, LP for the Coming Year Pessimistic

Most Likely

Optimistic

Sales $900,000 $1,125,000 $1,280,000 Less cost of goods sold: Fixed 250,000 250,000 250,000 a Variable (18.3%) 164,700 205,875 234,240 Gross profits $485,300 $ 669,125 $ 795,760 Less operating expense Fixed 180,000 180,000 180,000 Variable (5.8%)b 52,200 65,250 74,240 Operating profits $253,100 $ 423,875 $ 541,520 Less interest expense 30,000 30,000 30,000 Net profit before taxes $223,100 $ 393,875 $ 511,520 Less Taxes (24%) 53,544 94,530 122,765 Net profits after taxes $169,556 $299,345 $388,755 a Cost of goods sold variable percentage  ($421,875  $250,000) / $937,500 b Operating expense variable percentage  ($234,375  $180,000) / $937,500 d. Profits for the pessimistic case are larger in (a) than in (c). For the optimistic case, profits are lower in (a) than in (c). This pattern confirms the relationship stated (b).

P4-19 Pro forma balance sheet: Basic (LG 5; Intermediate) a.

Pro Forma Balance Sheet—Leonard Industries © 2019 Pearson Education, Inc.

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Assets Current assets Cash Marketable securities Accounts receivable (10% of sales) Inventories (12% of sales) Total current assets Net fixed assets1 Total assets

1

2

3

$ 50,000 15,000 300,000 360,000 $725,000 658,0001 $1,383,000

Liabilities & stockholders’ equity Current liabilities Accounts payable (14% of sales) Accruals Other current liabilities Total current liabilities Long-term debts Total liabilities Common stock Retained earnings Total stockholders’ equity External funds required3 Total liabilities & stockholders’ equity

Beginning gross fixed assets

$ 420,000 60,000 30,000 $ 510,000 350,000 $ 860,000 200,000 270,0002 $ 470,000 53,0003 $1,383,000

$ 600,000

Plus: Fixed asset outlays

90,000

Less: Depreciation expense

(32,000)

Ending net fixed assets

$ 658,000

Beginning of year retained earnings

$ 220,000

Plus: Net profit after taxes (4% of $3,000,000) 0.04)

120,000

Less: Dividends paid

(70,000)

End of year retained earnings

$ 270,000

Total assets

$1,383,000

Less: Total liabilities and equity

1,330,000

External funds required

$ 53,000

b. The finance manager should arrange a credit line of at least $53,000. Of course, if financing cannot be obtained, one or more of the constraints may be changed. c. If planned dividends for next year were reduced to $17,000 or less, Leonard would not need additional financing. Reducing the dividend would allow the firm to retain more cash to cover the growth in other asset accounts.

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P4-20 Pro forma balance sheet (LG 5; Intermediate) a.

Pro Forma Balance Sheet—Peabody & Peabody (as of December 31, 2023)

1.

Assets Current assets Cash Marketable securities Accounts receivable (12% of 2023 sales) Inventories (18% of 2023 sales) Total current assets Net fixed assets1 Total assets

$ 480,000 200,000 1,440,000 2,160,000 $4,280,000 4,820,000 $9,100,000

Liabilities and stockholders’ equity Current liabilities Accounts payable (14% of 2023 sales) Accruals Other current liabilities Total current liabilities Long-term debts Total liabilities Common equity2 External funds required

$1,680,000 500,000 80,000 $2,260,000 2,000,000 $4,260,000 4,065,000 775,000

Total liabilities and stockholders’ equity

$9,100,000

Beginning net fixed assets (1/1/2022)

$4,000,000

Plus: Fixed asset outlays

1,500,000

Less: Depreciation expense

(680,000)

Ending net fixed assets (12/31/2023) 2.

$4,820,000

Common equity is the sum of common stock and retained earnings.

Beginning common equity (January 1, 2022)

$3,720,000

Plus: Net profits after taxes (2022 = 3% of 2022 sales)

330,000

Net profits after taxes (2022 = 3% of 2023 sales)

360,000

Less: Dividends paid (2022 = 50% of 2022 NPAT) Dividends paid (2023 = 50% of 2022 NPAT) Ending common equity (December 31, 2023)

(165,000) (180,000) $4,065,000

b. Total assets at year-end 2023 equal $9.1 million while total liabilities and common equity equal $8.235 million—implying Peabody & Peabody needs additional financing of at least $775,000 over the next two years (to bring the pro forma balance sheet into balance).

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P4-21 Integrative: Pro forma statements (LG 5; Challenge) a.

Red Queen Restaurants Pro Forma Income Statement for Next Year (Percent-of-Sales Method) Sales $900,000 675,000 Less: Cost of goods sold (0.75  sales) Gross profits $225,000 112,500 Less: Operating expenses (0.125  sales) Net profits before taxes $112,500 23,635 Less: Taxes (0.21  NPBT) Net profits after taxes $ 88,875 Less: Cash dividends 35,000 To Retained earnings $ 53,875

b. and c.

Red Queen Restaurants Pro Forma Balance Sheet for Next Year (Judgmental Method) Assets Liabilities and Equity Cash $ 30,000 Accounts payable $112,500 Marketable securities 18,000 Taxes payable4 5,906 1 Accounts receivable 162,000 Other current liabilities 5,000 Inventories2 112,500 Current liabilities $123,406 Current assets $322,500 Long-term debt 200,000 3 Net fixed assets 375,000 Common stock 150,000 Retained earnings5 228,875 6 External funds surplus -4,781 Total assets $697,500 Total liabilities & stockholders’ equity $697,500 Notes: 1. Accounts receivable = 0.18  next year’s sales ($900,000) = $162,000 2. Last year’s inventories as a % of last year’s sales is 0.125, so 0.125  next year’s sales of $900,000 = $112,500 3. Net fixed assets (beginning of next year) $350,000 Plus: New machine 42,000 Less: Depreciation (17,000) Ending net fixed assets (next year) $375,000 4. Taxes payable = $23,635 × 0.25 = $5,906 5. Beginning retained earnings (next year) Plus: Net profit after taxes Less: Dividends paid Ending retained earnings (next year)

$175,000 88,875 (35,000) $228,875

6. Total assets equal $697,500 while total liabilities and stockholders’ equity equal $702,281. The difference—the amount needed to balance the pro forma balance sheet is –$4,781. Red Queen will have a funding surplus equal to this amount next year.

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P4-22 Integrative: Pro forma statements (LG 5; Challenge) a.

b.

Pro Forma Income Statement—Provincial Imports, Inc. for Next Year (Fixed and Variable Cost Method) Sales $6,000,000 3,100,000 Less: Cost of goods sold (0.35  sales  $1,000,000) Gross profits $2,900,000 970,000 Less: Operating expenses (0.12  sales  $250,000) Operating profits $1,930,000 Less: Interest expense 200,000 Net profits before taxes (NPBT) $1,730,000 363,300 Less: Taxes (0.21  NPBT) Net profits after taxes (NPAT) $1,366,700 546,680 Less: Cash dividends (0.40  NPAT) To Retained earnings $ 820,020 Pro Forma Balance Sheet—Provincial Imports, Inc. for Next Year (Judgmental Method) Assets Liabilities and Equity Cash $ 400,000 Accounts payable Marketable securities 225,000 Taxes payable Accounts receivable 750,000 Notes payable Inventories 1,000,000 Other current liabilities Current assets $2,325,000 Current liabilities 2 Net fixed assets 1,646,000 Long-term debt Common stock Retained earnings External funds required Total liabilities & Total assets $4,021,000 stockholders’ equity 1

$ 840,000 138,0541 200,000 6,000 $1,184,400 500,000 75,000 2,195,0203 66,926 $4,021,000

Taxes payable for last year were nearly 38% of last year’s taxes on the income statement. The pro forma value is obtained by taking 38% of next year’s taxes (0.38  $363,300  $138,054).

2

Net fixed assets (beginning of next year) Plus: New computer Less: Depreciation Net fixed assets (end of next year) 3 Beginning retained earnings (next year)

$1,400,000 356,000 (110,000) $1,646,000 $1,375,000

Plus: Net profit after taxes

1,366,700

Less: Dividends paid

(546,680)

Ending retained earnings (next year)

$2,195,020

c. Total assets equal $4,021,000, total liabilities equal $1,684,054, and stockholders’ equity equals $2,270,020. The difference between total assets and total liabilities plus stockholders’ equity—the amount needed to balance the pro forma balance sheet—is $66,926. Provincial Imports will need this amount of external financing next year.

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P4-23 Ethics problem (LG 1; Intermediate) Investors welcome increased transparency, accountability, and integrity. Speedy dissemination of negative information will cause a firm’s stock price to fall sooner than it otherwise would have. But a consistent policy of releasing negative information quickly (rather than trying to bury it) would signal the firm’s strong commitment to ethical business practices. And over the long run, investors should reward the company for its commitment to ethics.

 Case Case studies are available on www.pearson.com/mylab/finance.

Preparing Martin Manufacturing’s Pro Forma Financial Statements In this case, the student will prepare pro forma financial statements and use them to determine whether Martin Manufacturing will require external funding to embark on a major expansion program. a.

Martin Manufacturing Company—Pro Forma Income Statement (Year Ending 2020) Sales revenue $6,500,000 (100%) Less: Cost of goods sold 4,745,000 (0.73  sales) Gross profits $1,755,000 (0.27  sales) Less: Operating expenses Selling expense and general and administrative expense $1,365,000 (0.21  sales) Depreciation expense 185,000 Total operating expenses $1,550,000 Operating profits $ 205,000 Less: Interest expense 97,000 Net profits before taxes $ 108,000 Less: Taxes (40%) 43,200 Total profits after taxes $ 64,800 Note: Calculations were ―driven‖ by cost of goods sold and operating expense percentages (excluding depreciation, which is given).

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Martin Manufacturing Company—Pro Forma Balance Sheet (as of December 31, 2020) Assets Current assets Cash Accounts receivable Inventories Total current assets Gross fixed assets Less: Accumulated depreciation Net fixed assets Total assets Liabilities and stockholders’ equity Current liabilities Accounts payable Notes payable Accruals Total current liabilities Long-term debts Total liabilities Stockholders’ equity Preferred stock Common stock (at par) Paid-in capital in excess of par Retained earnings Total stockholders’ equity Total External funds required Total liabilities and stockholders’ equity 1 2

$ 25,000 890,4111 677,8572 $1,593,268 $2,493,819 685,000 $1,808,819 $3,402,087

$ 276,000 311,000 75,000 $ 662,000 1,165,250 $1,827,250 $ 50,000 400,000 593,750 344,8003 $1,388,550 $3,215,800 186,287 $3,402,087

$6,500,000/365  50 days  $890,411 Inventory turnover = cost of goods sold / inventory Inventory = 4,745,000 / 7 = 677,857

3

Beginning retained earnings (January 1, 2020) Plus: Net profits

c.

$300,000 64,800

Less: Dividends paid

(20,000)

Ending retained earnings (December 31, 2020)

$344,800

Based on the pro forma financial statements prepared above, Martin Manufacturing will need to raise about $200,000 ($186,287) in external financing to undertake its construction program.

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

Spreadsheet Exercise Answers to Chapter 4’s ACME Company spreadsheet problem are available on www.pearson.com/mylab/finance.

 Group Exercise Group exercises are available on www.pearson.com/mylab/finance.

This chapter’s exercise focuses on each group’s fictitious firm, with particular attention to asset depreciation and cash flows. Students are directed to the IRS’s website to retrieve depreciation information from publication number 946. Using this information, each group will then provide examples of property depreciation for its firm. Next, students will turn to financial planning—with each group evaluating recent statements of cash flows for its shadow firms and accounting for any changes. Similar evaluation/analysis should then be performed on each group’s fictitious firm. Students should also take short- and long-term planning information from the strategy section of the shadow firm’s annual report and apply it their fictitious firms. Cash budgeting for the fictitious firm and pro forma statements from the shadow firm conclude the assignment.

The best advice here is for students to keep it simple. Impress upon them the rapidly increasing complexity of the budgeting process as the number of accounts is increased. Encourage students to following the text’s examples and use information from the shadow firm.

Chapter 5 Time Value of Money  Instructor’s Resources Chapter Overview This chapter introduces a key—indeed, perhaps the most important—concept in finance: the time value of money. The present and future value of a sum and an annuity are explained. Special applications include intra-year compounding, mixed cash-flow streams, mixed cash flows with an embedded annuity, perpetuities, loan amortization, and deposits necessary to accumulate a future sum. The discussion employs numerous business and personal examples to stress all applications as variations on the same theme—sums received at different points in time are worth different amounts to the recipient, with differences traceable to when the sums are received and how frequently interest is compounded. The chapter drives home the need to understand the time value of money to analyze project profitability as a finance professional and achieve personal-finance goals as an individual.

 Suggested Answer to Opener-in-Review A lottery winner can choose between two options: (i) a lump-sum $477 million now or (ii) a mixed stream of 30 payments, with an immediate payment of $11.56 million and 29 additional annual payments growing at 5% per year. [So the second payment will be $11.56 million  (1.05), the third $11.56 million  (1.05)2, …, and the 30th $11.56 million  (1.05)29.] If the winner could earn 2% on © 2022 Pearson Education, Inc.

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cash invested today, should he take the lump sum or mixed stream? What if the rate of return is 3%?

What general principle do those calculations illustrate?

The lottery winner should choose the payment option with the higher present value. With a 2% discount rate, the present value of the mixed stream is $544.8 million—significantly higher than immediate payment of $480 million—so the winner should opt for the mixed-stream payment.

But with a 3% discount rate, the present value of the mixed stream is only $464.7 million—less than the immediate payment option of $477 million. Now, the lottery winner should take the lump sum.

In general, other things equal, a rise in the interest rate reduces the present value of an annuity due.

 Answers to Review Questions 5-1

Future value (FV) is the sum an amount today will grow to equal by a future date with compound interest. Present value (PV) is the dollar value today of an amount promised at a specific future date for a given interest rate. Viewed another way, it is also the amount, if invested today at the given interest rate, would grow to equal the future amount. Financial managers prefer evaluating projects with present value because decisions are typically made before a project starts (i.e., time period zero).

5-2

A single amount cash flow refers to a single payment or cash receipt, occurring at one point in time. An annuity is an unbroken series of equal cash flows occurring over multiple periods. A mixed stream is an unequal series of cash flows that take place over multiple periods.

5-3

Compounding occurs when interest earned in the initial period is added to the original principal and that sum grows by the interest rate in the second period, and so on. With compounding, interest is earned not only on original principal but also on interest earned. Assuming interest compounds once per period, the equation showing how much a sum today (present value, or PV0) will grow to equal (FVn) if compounded over (n) periods at a given interest rate per period (r) is: FVn  PV0(1  r)n

5-4

A fall in the interest rate reduces the future value of a deposit for a given holding period because of the decline in the amount of interest on which additional interest is subsequently paid (i.e., the deposit compounds at a lower rate). A rise in the holding period for a given interest rate increases future value because interest is paid on interest over a longer period of time (i.e., the deposit compounds longer).

5-5.

Present value (PV0) is the current dollar value of a future amount (FVn)—that is, the amount today that would grow to equal the future amount if compounded at a given interest rate (r) for a specific number of compounding periods (n). The equation showing the present value of a future sum (FVn) is: PV0  FVn  (1  r)n

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5-6.

Increasing required rate of return (discount rate) reduces the present value of a sum promised in the future. Mathematically, a higher interest rate (r) increases the denominator in the presentvalue equation in question 5.2—which for a given number of periods (n) and future value (FV) implies a smaller value. More intuitively, with a higher interest rate (compounding periods and future value held constant), a smaller sum would grow to the same future value.

5-7

The same equation links present and future value for a given interest rate and compounding period; the only difference is the given information. Consider, for example, the present-value equation in question 5.2. Now, assume present value is given, and the goal is solving for future value. All that is needed is to multiply both sides of the equation by (1  r)n—which yields the future-value equation in question 5.1.

5-8 The value at retirement of a sum invested today may be obtained with the future-value equation for a simple sum, FVn  PV0(1  r)n, assuming FVn is value at retirement, PV0 the sum invested today, n the number of periods to retirement, and r the interest rate per period. The Excel function for future value is FV(rate,nper,pmt,pv,type). To solve this problem in Excel, input the given values as shown below and enter =FV(B3,B4,0,B2,0) in cell B5. Completed Excel worksheets are available to students in MyFinanceLab.

5-9 The amount needed for investment today to cover future college expenses may be obtained with present-value equation for a simple sum. Assuming, for simplicity, expenses for all four years must be paid at the beginning of the child’s freshmen year, PV0  FVn  (1  r)n, FVn is the amount needed n periods in the future, n the number of periods in the future, and r the interest rate per period. The Excel function for present value is PV(rate,nper,pmt,fv,type). To solve this problem in Excel input the given values as shown below and enter =PV(B3,B4,0,B2,0) in cell B5. Completed Excel worksheets are available to students in MyFinanceLab.

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5-10 Annuities offer equal per-period payments for a given number of periods. Ordinary annuities make payments at the end of each period while payments for annuities due occur at the beginning of the period. For the same interest rate and number of payments, annuities due are more valuable—that is, have higher present values—because each payment of an annuity due comes one period sooner than the equivalent payment on an ordinary annuity. For example, the first payment of an annuity due is immediate, so the future value of that payment equals its present value. In contrast, the first payment on an ordinary annuity takes place one period in the future, so its present value equals CF1  (1  r).

5-11 The most efficient ways to calculate the present value of an ordinary annuity are with an algebraic equation, a financial calculator, or a spreadsheet program like Excel.

5-12 The future value (FV) of an ordinary annuity, where CF1 is the first payment made at end of period 1, n the number of payments, and r the interest rate, is given by: ì é 1+ r n - 1ù ü ) úû ï ï ê( FVn = CF1 ´ í ë ý r ï ï î þ The future value of an annuity due, where CF0 is the first payment made immediately (period 0), n is number of payments, and r the interest rate, is given by: ì é 1+ r n - 1ù ü ) ûú ï ï ê( FVn = CF0 ´ í ë ý ´ (1+ r ) r ï ï î þ Assuming the same cash flows (i.e., CF0 = CF1), interest rate and number of payments, the only difference is that each payment on the annuity due arrives one period sooner than the payment on an ordinary annuity. In other words, each annuity-due payment earns interest one more period than payments on an ordinary annuity. Multiplying the first two terms in the future-value equation for an ordinary annuity (which are identical in the annuity due and ordinary annuity equations) by (1 r) corrects for the additional year of compounding.

5-14 A perpetuity is an infinite-lived annuity; the present value of an ordinary annuity is given by: æ CF ö é 1 ùú PV0 = ç 1 ÷ ´ ê1è r ø ê (1+ r )n ú ë û where CF1 is the first payment received at the end of the first period, n is the number of periods, and r the interest rate per period. For a perpetuity, n is infinity, so raising to the infinite power makes equal to zero, and the formula for present value simplifies to: (

)

[

]

(

)

If the number of payments (n) goes to infinity in the equations for the future value of ordinary annuities and annuities due (meaning the instrument is a perpetuity rather than a regular annuity), then approaches infinity and future value is infinity. Intuitively, future value of a perpetuity is infinite because cash flows never end and never stop earning interest. The present value of a perpetuity, in contrast, is finite because present value of cash flows far into the future is effectively zero.

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5-15 The future value (at retirement) of equal annual IRA contributions at the end of every year can be determined using the formula for the future value of an ordinary annuity: ì é 1+ r n - 1ù ü ) ûú ï ï ê( FVn = CF1 ´ í ë ý r ï ï î þ where CF1 is the equal contribution made at the end of each year, r the interest rate, and n the number of contributions until retirement. The Excel function for future value is FV(rate,nper,pmt,pv,type). To solve this problem in Excel, input the given values as shown below and enter =FV(B3,B4,B2,0,0) in cell B5. Completed Excel worksheets are available to students in MyFinanceLab.

5-16 The future value (at retirement) of equal annual IRA contributions at the beginning of every year can be determined using the formula for the future value of an annuity due: ì é 1+ r n - 1ù ü ) úû ï ï ê( FVn = CF0 ´ í ë ý ´ (1+ r ) r ï ï î þ where CF0 is the equal contribution made at the beginning of each year, r the interest rate, and n the number of contributions until retirement. The Excel function for future value is FV(rate,nper,pmt,pv,type). Type is set to 1 for beginning of period payments. To solve this problem in Excel, input the given values as shown below and enter =FV(B3,B4,B2,0,1) in cell B5. Completed Excel worksheets are available to students in MyFinanceLab.

5-17 The amount you can spend on your new dream home can be found using the formula for the present value of an ordinary annuity, assuming that payments are due at the end of each month: æ CF ö é 1 ùú PV0 = ç 1 ÷ ´ ê1è r ø ê (1+ r )n ú ë û

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where CF1 is the monthly mortgage payment, r the monthly interest rate, and n the number of monthly mortgage payment over the life of the loan. The Excel function for present value is PV(rate,nper,pmt,fv,type). To solve this problem in Excel, input the given values as shown below and enter =PV(B3/12,B4*12,B2,0,0) in cell B5. Completed Excel worksheets are available to students in MyFinanceLab. Note that if you assume that mortgage payments are due at the start of each month, the correct answer is $171,025.50.

5-18

The future value of a mixed stream of cash flows equals the sum of the future values of the individual cash flows—that is, each cash flow should be treated like a single payment in the ―future value of a single payment‖ equation [FVn  PV0(1  r)n] and individual future values summed. Similarly, the present value of a mixed stream of cash flows equals the sum of the present value of each individual cash flow—that is, cash flow should be treated like a single payment in the ―present value of a single payment equation‖ [PV0  FVn  (1  r)n] and individual present values summed.

5-19 The value of your financial head start four years after your first annual investment is equal to the sum of the future value for each of your four investment amounts. Future values may be obtained with the future-value equation for a simple sum, FVn  PV0(1  r)n, assuming FVn is value at retirement, PV0 the sum invested today, n the number of periods to retirement, and r the interest rate per period. The Excel function for future value is FV(rate,nper,pmt,pv,type). To solve this problem in Excel, input the given values as shown below and enter =FV(B7,4,0,B3,0)+FV(B7,3,0,B4,0)+FV(B7,2,0,B5,0)+FV(B7,1,0,B6,0) in cell B8. Because this problem has a mixed stream of cash flows it is necessary to find the future value for each cash flow. Completed Excel worksheets are available to students in MyFinanceLab.

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5-20 For a given interest rate and holding period, as the number of compounding periods per year rises, both (a) future value and (b) the effective annual rate of interest (EAR) increase. Future value and EAR rise because more frequent compounding means more periods in which interest can be earned on interest.

5-21 Continuous compounding means interest is compounded an infinite number of times per year— that is, every nanosecond. Continuous compounding at a given rate of interest produces the largest future value compared with any other compounding period.

5-22 The nominal annual rate is the contractual annual rate of interest charged by the lender or promised to the borrower, while the effective annual rate is the rate actually charged or paid after accounting for the number of compounding periods per year. When interest is compounded annually, the two rates are the same, but when compounding occurs more frequently, the effective annual rate exceeds the nominal annual rate. ―Truth in lending‖ laws require disclosure of the annual percentage rate (APR) to consumers; this rate equals the rate of interest charged on loans in each compounding period multiplied by the number of compounding periods each year. The APY, or annual percentage yield, is the effective rate of interest depository institutions pay on savings products after accounting for the number of compounding periods per year; ―truth-in-savings laws‖ mandate disclosure of APYs. 5-23 The future value of your company’s short-term investments in four years for each of the compounding frequencies may be obtained with the future-value equation for a simple sum, FVn  PV0(1  r)n, assuming FVn is value at the end of n periods, PV0 the sum invested today, n the number of periods, and r the interest rate per period. The Excel function for future value is FV(rate,nper,pmt,pv,type). To solve this problem in Excel, input the given values as shown below and enter =FV(B3/B4,B5*B4,0,B2,0) in cell B6, =FV(B8/B9,B10*B9,0,B7,0) in cell B11, =FV(B13/B14,B15*B14,0,B12,0) in cell B16, and =FV(B18/B19,B20*B19,0,B17,0) in cell B21. The general principle is, other things equal, the greater the compounding frequency, the higher the future value of investment opportunities. Completed Excel worksheets are available to students in MyFinanceLab.

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5-24 The future value of your company’s short-term investments in four years for each of the compounding frequencies may be obtained with the future-value equation for a simple sum, FVn  PVe r×n, assuming FVn is value at retirement, PV the sum invested today, n the number of periods to retirement, and r the interest rate per period. To solve this problem in Excel, input the given values as shown below and enter =-B2*EXP(B3*B4) in cell B5. The general principle is, other things equal, the greater the compounding frequency, the higher the future value of investment opportunities. Completed Excel worksheets are available to students in MyFinanceLab.

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5-25 The effective annual rate (EAR) can be found using the following equation: EAR = (1  r/m)m – 1. The Excel function for the effective annual interest rate is EFFECT(nominal_rate, npery). To solve this problem in Excel, input the given values as shown below and enter =EFFECT(B2,B3) in cell B4, =EFFECT(B5,B6) in cell B7, =EFFECT(B8,B9) in cell B10, and =EFFECT(B11,B12) in cell B13. The effective annual rate will always be greater than the nominal rate when the compounding frequency is greater than annual or 1 per year, and it will be equal to the nominal rate with the compounding frequency is annual. Completed Excel worksheets are available to students in MyFinanceLab.

5-26 The equal annual end-of-year deposits (CF1) needed to accumulate a given amount over a certain number of periods (n) for a specific rate of interest per period (r) can be determined using the equation for the future value of an ordinary annuity: ì é 1+ r n - 1ù ü ) ûú ï ï ê( FVn = CF1 ´ í ë ý r ï ï î þ Here, FVn, n, and r are given; the answer may be obtained by solving for CF1.

5-27 Amortizing a loan over equal annual payments involves finding the sequence of payments with a present value at the loan interest rate equal to the initial principal borrowed. In other words, start with the formula for the present value of an ordinary annuity: æ CF ö é 1 ùú PV0 = ç 1 ÷ ´ ê1è r ø ê (1+ r )n ú ë û

Insert loan amount for PV0, n for number of payments, r for interest rate r; and solve for CF1.

5-28 If the present value, future value, and interest rate are given, and the goal is finding the number of periods needed for a sum today to grow to the future sum, then insert given information in the equation for future value of simple sum [FVn  PV(1  r)n] and solve for n: © 2022 Pearson Education, Inc.

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)(1  r)n (

)

( [

) (

]

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)

The same approach will yield the number of periods it will take an ordinary annuity to grow to a specific future value; the algebra is just a bit more complicated: *(

)

+

An alternative approach is to use trial and error, plugging in various values for the number of periods in a calculator or spreadsheet program (like Excel) with the given present value (or cash flows), future value, and interest rate until an approximate value of n is obtained.

 1  CF0  PV0  r  1   1 r  n   1  r   5-29 The general approach is to treat the car payment as an annuity due (because such payments are made at the beginning of the period) and use the present-value formula to solve for CF0. The payment amount for a present value of an annuity due is given by: Plug in given values for present value (car price), interest rate, and number of periods; then solve for the cash flow (CF0). The Excel function for payment amount is PMT(rate,nper,pmt,pv,fv,type). To solve this problem in Excel, input the given values as shown below and enter =PMT(B3/12,B4*12,B2,0,1) in cell B5. Completed Excel worksheets are available to students in MyFinanceLab.

5-30 The approach is to treat retirement age as n months from now in the formula for future-value of an ordinary annuity. That is, plug in given values for future value of the ordinary annuity (FVn), monthly contribution (CF1), interest rate (r), and solve for number of months. The number of periods for a future value of an ordinary annuity is given by:

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 FVn  r  m  1 ln   CF1   nmonths   ln 1  r m





Plug in given values for future value (retirement amount), monthly interest rate, and cash flow (monthly contribution); then solve for the cash flow (CF0). The Excel function for n is NPER(rate,pmt,pv,fv,type). To solve this problem in Excel, input the given values as shown below and enter =NPER(B4/12,B3,0,B2,0)/12 in cell B5. After finding the number of years until you retire, add your current age to find your age at retirement. In cell B7 enter =B6+B5.

Completed Excel worksheets are available to students in MyFinanceLab.

 Suggested Answer to Focus on People/Planet/Profits Box: “The Time Value of Money Heats Up Climate Change Debate” The cost/benefit analysis of climate-change policies often uses perpetuities in the present value calculations because the costs and benefits of these policies are spread out over very long horizons.

Think of a stream of costs that stretches out indefinitely into the future. How much does the present value of those costs change if the discount rate moves from 3% to 0.1%?

To think about this question, imagine a perpetuity that costs $1 per year. At a discount rate of 3%, the present value of costs is $1  0.03 = $33.33. If climate-change policies generate more benefits than they cost, then the amount spent on such policies should not exceed $33.33 (in today’s dollars) for each $1 of annual harm abated. But if the discount rate is 0.1%, the present value of the climate change cost stream is $1  0.001 = $1,000, potentially justifying policies that cost 30 times more.

 Suggested Answer to Focus on Ethics Box: “Was the Deal for Manhattan a Swindle?

How much responsibility do lenders have to educate borrowers? Does the fact that the government requires disclosure statements with a few standardized examples illustrating the time value of money change your answer?

This question is an excellent springboard to class discussion because there is no ―correct‖ answer.

Conservative-leaning students may argue the borrower has a responsibility to educate herself—that is, ―let the buyer beware.‖ Progressive-leaning students might counter that poorer borrowers lack the numeracy to offer informed consent to complex loan contracts. Irrespective of politics, recent © 2022 Pearson Education, Inc.

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experience and research suggests two factors to keep in mind when thinking about lender responsibility. First, failure to educate borrowers could come back to haunt lenders through bad publicity and the resulting loss of customers. Second, many patrons of payday lenders understand the high cost of their loans yet remain satisfied customers. A recent Freakonomics podcast summarizing research on payday loans, along with interviews with payday-loan customers, may be accessed here: http://freakonomics.com/podcast/payday-loans/.

 Suggested Answer to Focus on Practice Box: “New Century Brings Trouble for Subprime Mortgages” As a reaction to problems in the subprime area, lenders tightened lending standards. What effect do you think this change had on the housing market?

When mortgage lenders tightened underwriting standards, fewer people qualified for home loans, and demand for homes declined. This decline contributed the unprecedented nationwide slide in home prices, which did not bottom out until early 2012.

 Answers to Warm-Up Exercises E5-1

Future value of a lump-sum investment (LG 2)

Answer: FV  $2,500  (1  0.007)  $2,517.50, or in Excel, write the bracketed formula [=fv(0.007,1,0,-2500,0)] in a cell to obtain $2,517.50] in a cell.

E5-2

Finding future value (LG 2 and LG 5)

Answer: Because interest is compounded monthly, the number of periods is 4 (years)  12 (months)  48 and the monthly interest rate is 1/12th of the annual rate = 0.00166667: FV48  PV  (1  r)48 where r is the monthly interest rate FV48 ($1,260  $975)  (1  0.00166667)48 ($2,235)  1.083215  $2,420.99 Or in Excel insert the bracketed formula [= fv(0.02/12,48,0,-2235,0)] in a cell.

E5-3

Comparing a lump sum with an annuity (LG 3)

Answer: Note the 25-year payout option is an ordinary annuity. So the answer turns on whether the present value of an ordinary annuity offering $100,000 annual cash flows for 25 years (given a 5% interest rate) is greater or less than the sum available immediately ($1.3 million). Recall, the formula for the present value of an ordinary annuity is: æ CF ö é 1 ùú PV0 = ç 1 ÷ ´ ê1è r ø ê (1+ r )n ú ë û

(

)*

+= $1,409,394.06

This problem can also be solved with a financial calculator or spreadsheet program like Excel. In Excel, the command for the present value of an ordinary is: =PV(interest rate, periods, -[cash flow], 0 if ordinary annuity [1 if annuity due]) =PV(0.05,25, 100000,0) = $1,409,394.06.

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Gabrielle should take the 25-year payout because the present value ($1.41 million) exceeds the lump-sum payment of $1.3 million.

E5-4

Comparing the present value of two alternatives (LG 4)

Answer: You have the option of investing a sum today in software that offers the company a mixed stream of savings over the next five years. This option will be profitable if the present value of the savings (marginal benefit of the investment) exceeds the current cost (marginal cost). To find the marginal benefit, the present value of expected savings over the 5-year life of the software must be calculated. Present value (PV)  FVn  (1  r)n, where FVn is the cost savings in year n, n is the number of years in the future, and r the interest rate per period (9%).

Year 1 2 3 4 5

Savings Estimate $35,00 0 50,000 45,000 25,000 15,000

Discount Factor (1+r)n (1.09)1

Present Value of Savings $32,110

(1.09)2 (1.09)3 (1.09)4 (1.09)5

42,084 34,748 17,711 9,749 $136,402

Because the $136,402 present value of the savings (marginal benefit) exceeds the $130,000 cost of the software (marginal cost), the firm should invest in the new software.

E5-5

Compounding more frequently than annually (LG 5)

Answer: The future value of $12,000 invested for one year in Partners’ Savings Bank at 3%, compounded semi-annually (where m is compounding periods per year, and n is years, so r/m is the interest rate per compounding period, and m r is the number of total compounding periods) is: FV = PV0

( ) $12,362.70

= $12,000

(1 + 0.03/2)2 = $12,000 + $1.030225 =

In Excel, the bracketed command [=FV(0.015,2,0,-12000,0)] will yield the answer.

FV of $12,000 invested one year in Selwyn’s Bank at 2.75%, compounded continuously, is: FV = PV0

, where the value of e is approximately 2.7183

FV = $12,000 $12,334.58

= $12,000

2.71860.0275 = $12,000

1.027882 =

In Excel, the command for continuous compounding is the exp() function; the bracketed formula [=$12,000*exp(0.0275) = $12,334.58] in a cell will yield the correct answer.

Joseph should choose Partners’ Savings Bank because the 3% rate with semiannual compounding will earn him $28.12 more over the course of a year.

E5-6

Determining deposits needed to accumulate a future sum (LG 6)

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Answer: Jack and Jill want to make equal, end-of-year contributions for 18 years to accumulate $150,000 for their child’s college education. Equal, end-of-year contributions means ordinary annuity. Specifically, plug future value ($150,000), interest rate (6%), and period (years = 18) into the formula for the FV of an ordinary annuity and solve for CF1: ì é 1+ r n - 1ù ü ) ûú ï ï ê( FVn = CF1 ´ í ë ý r ï ï î þ 18 $150,000 = CF1  [(1.06) – 1]/0.06 → CF1 = $4,853.48

This problem can also be solved with a financial calculator or spreadsheet program like Excel. In Excel, annual contributions may be found using the PMT command.

Specifically, = PMT(rate,nper,pv,fv,type) = PMT(0.06,18,0, 150000,0) = $4,853.48 Jack and Jill should put aside $4,853.48 each year.

 Solutions to Problems P5-1 Using a timeline (LG 1; Basic) a, b, and c

d. Financial managers rely more on present value than future value because their decisions are typically made before a project starts (i.e., at time zero).

P5-2 Future value calculation (LG 2; Basic) To find future value in each case, plug $1 for PV0 as well as the given interest rates (r) and compounding periods (n) into FVn  PV0  (1  r)n.

Case A: FV2  $1  (1  0.12)2 = $1.2544 or in Excel: =fv(0.12,2,0,-1,0) = $1.2544 B: FV3  $1  (1  0.06)3 = $1.1910 or in Excel: =fv(0.06,3,0,-1,0) = $1.1910 C: FV2  $1  (1  0.09)2 = $1.1881 or in Excel: =fv(0.09,2,0,-1,0) = $1.1881 D: FV4  $1  (1  0.03)4 = $1.1255 or in Excel: =fv(0.03,4,0,-1,0) = $1.1255

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P5-3

Future value (LG 1; Basic) With compound interest: FVn  PV0  (1  r)n = $100  (1  0.05)10 =$162.89, or in Excel the bracketed formula [=fv(0.05,10,0,-100,0) = $162.89]. With simple interest, 5% would be earned on the original principal each year; no interest would be earned on prior interest earned. So, FV (simple) = $100 + ($5  10 years) = $150.

P5-4

P5-5

Future values (LG 2; Intermediate) Case A: FV20  $200  (1  0.05)20 = $530.66; in Excel: =fv(0.05,20,0,-200,0) = $530.66 B: FV7  $4,500  (1  0.08)7 = $7,712.21; in Excel: =fv(0.08,7,0,-4500,0) = $7,712.21 C: FV10  $10,000  (1  0.09)10 = $23,673.64; in Excel: =fv(0.09,10,0,-10000,0) = $23,673.64 D: FV12  $25,000  (1  0.10)12 = $78,460.71; in Excel: =fv(0.10,12,0,-25000,0) = $78,460.71 E: FV5  $37,000  (1  0.11)5 = $62,347.15. In Excel: =fv(0.11,5,0,-37000,0) = $62,347.15 F: FV9  $40,000  (1  0.12)9 = $110,923.15. In Excel: =fv(0.12,9,0,-40000,0) = $110,923.15 Personal finance problem: Time value (LG 2; Intermediate) a. (1) FV3 = $1,500  (1 + 0.07)3 = $1,837.56. In Excel: =fv(0.07,3,0,-1500,0) = $1,837.56 (2) FV6 = $1,500  (1 + 0.07)6 = $2,251.10. In Excel: =fv(0.07,6,0,-1500,0) = $2,251.10 (3) FV9 = $1,500  (1 + 0.07)9 = $2,757.69. In Excel: =fv(0.07,9,0,-1500,0) = $2,757.69 b. (1) Interest earned, years 1 through 3 = FV3 PV0 = $1,837.56 $1,500 = $337.56 (2) Interest earned, years 4 through 6 = FV6 – FV3= $2,251.10 $1,837.56 = $413.53 (3) Interest earned, years 7 through 9 = FV9 –FV6 = $2,757.69

$2,251.10 = $506.59

c. The amount of interest earned in the second three-year period ($413.53) exceeds the amount earned in the first ($337.56), and interest earned in the third three-year period ($506.59) exceeds interest earned in the second. Interest earned in each subsequent threeyear period rises because of compounding. That is, in each three-year period, interest is earned on prior interest paid, and the greater the interest earned in prior periods, the greater the impact of compounding.

P5-6

Personal finance problem: Time value (LG 2; Challenge) a. To find car price in five years with 2% inflation, use future-value equation with PV0 = $14,000, r = 2% and solve for FV5 [$14,000  (1 + 0.02)5 = $15,457.13]. To find price with 4% inflation, change r to 4% and re-solve [$14,000  (1 + 0.04)5 = $17,033.14]. b. The car will cost $1,576.01 more with a 4% inflation rate than an inflation rate of 2% – an increase of 10.2% ($1,576  $15,457). c. Again, use the future-value equation for both steps. Specifically, for the first two years: FV2 = $14,000  (1 + 0.02)2 = $14,565.60 Now, use FV2 as present value: FV5 = $14,565.60  (1 + 0.04)3 = $16,384.32

P5-7 Personal finance problem: Time value (LG 2; Challenge) To find the future value of $10,000 today, invested at 9% annual interest for 40 years: FV40 = $10,000  (1 + 0.09)40 = $314,094.20 or =fv(0.09,40,0,-10000,0) = $314,094.20 With 10 fewer years to compound:

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FV30 = $10,000  (1 + 0.09)30 = $132,676.78 or =fv(0.09,30,0,-10000,0) = $132,676.78 Investing now rather than waiting 10 years will yield $181,417 more ($314,094  $132,677).

The difference is the magic of compounding (i.e., earning interest on interest).

P5-8

Personal finance problem: Time value (LG 2; Challenge) The problem asks students to use a financial calculator or Excel spreadsheet to obtain approximations. Exact answers are offered below. Using the future-value framework (5 years, $15,000 = FV5), solve for r with the various starting present values. a. FVn = PV0  (1 + r)n (i) $15,000 = $10,200  (1 + r)5 (ii) 1.4706 = (1 + r)5

P5-9

(iii) √ r (iv) r = 8.02%

b. FVn = PV0  (1 + r)n (i) $15,000 = $8,150  (1 + r)5 (ii) 1.8405 = (1 + r)5

(iii) √ r (iv) r = 12.98%

c. FVn = PV0  (1 + r)n (i) $15,000 = $7,150  (1 + r)5 (ii) 2.0979 = (1 + r)5

(iii) √ r (iv) r = 15.97%

Personal finance problem: Single-payment loan repayment (LG 2; Intermediate) To find the amount that must be repaid, use the future-value framework with $200 as the present value, 8.5% the interest rate, and various values for n. a. In one year: FVn = PV0  (1 + r)n = $200 (1 + 0.085) = $217.00 b. In four years: FVn = PV0  (1 + r)n = $200 (1 + 0.085)4 = $277.17 c. In eight years: FVn = PV0  (1 + r)n = $200 (1 + 0.085)8 = $384.12

P5-10 Present value calculation (LG 2; Basic) In all cases, solve for present value using the present-value equation: PV0 = FVn (1 + r)n Case A: PV0 = $1 (1 + 0.02)4 $0.9238 or in Excel: =pv(0.02,4,0,-1,0) = $0.9238 B: PV0 = $1 (1 + 0.10)2 = $0.8264 or in Excel: =pv(0.10,2,0,-1,0) = $0.8264 C: PV0 = $1 (1 + 0.05)3 = $0.8638 or in Excel: =pv(0.05,3,0,-1,0) = $0.8638 D: PV0 = $1 (1 + 0.13)2 = $0.7831 or in Excel: =pv(0.13,2,0,-1,0) = $0.7831 P5-11 Present values (LG 2; Basic) Case A: PV0 = $7,000 (1 + 0.12)4  $4,448.63; in Excel: =pv(0.12,4,0,-7000,0) = $4,448.63 B: PV0 = $28,000 (1 + 0.08)20 = $6,007.35; in Excel: =pv(0.08,20,0,-28000,0) = $6,007.35 C: PV0 = $10,000 (1 + 0.14)12 = $2,075.59; in Excel: =pv(0.14,12,0,-10000,0) = $2,075.59 D: PV0 = $150,000 (1 + 0.11)6 = $80,196.13; in Excel: =pv(0.11,6,0,-150000,0) = $80,196.13 E: PV0 = $45,000 (1 + 0.20)8 = $10,465.56; in Excel: =pv(0.20,8,0,-45000,0) = $10,465.56

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P5-12 Present value concept (LG 2; Intermediate) a. PV0 = FVn b. PV0 = FVn

(1 + r)n = $6,000 (1 + r)n = $6,000

(1 + 0.12)6  $3,039.79 (1 + 0.12)6  $3,039.79

c. PV0 = FVn (1 + r)n = $6,000 (1 + 0.12)6  $3,039.79 If you can buy the investment for less than $3,039.79 then your implied rate of return will be greater than 12%. d. The same question is asked in three different ways, so the answer is the same each time.

P5-13 Personal finance problem: Time value (LG 2; Basic) a. PV0 = FVn

(1 + r)n = $500

(1 + 0.07)3  $408.15

b. Jim should pay no more than $408.15 for $500 in three years if his discount rate is 7%. c. If Jim pays less than $408.15, his rate of return will exceed 7%.

P5-14 Time value (LG 2; Intermediate) The state will sell bonds at a price equal to present value of cash flows. The bond can be converted to $100 in 6 years, and comparable bonds offer 3% compounded annually. So: PV0 = FVn

(1 + r)n = $100

(1 + 0.03)6  $83.75; in Excel: =pv(0.03,6,0,-100,0)

P5-15 Personal finance problem: Time value and discount rates (LG 2; Intermediate) a. (1) PV0 = FVn (1 + r)n = $1,000,000 (1 + 0.06)10  $558,394.78 (2) PV0 = $1,000,000 (1 + 0.09)10  $422,410.81 (3) PV0 = $1,000,000 (1 + 0.12)10 $321,973.24 The values represent the lowest you would be willing to accept because they are based on required rates of return. b. (1) PV0 = FVn (1 + r)n = $1,000,000 (1 + 0.06)15  $417,265.06 (2) PV0 = $1,000,000 (1 + 0.09)15  $274,538.04 (3) PV0 = $1,000,000 (1 + 0.12)15 $182,696.26 c. As the discount rate increases, present value decreases because of the higher opportunity cost associated with the higher rate. Moreover, the longer the time until the lottery payment is collected, the lower the present value because of the longer opportunity for compounding. More generally, the larger the discount rate and number of periods until the money is received, the lower the present value of a future payment.

P5-16 Personal finance problem: Calculating missing cash flow (LG 2; Challenge) Step 1: Determine lump sum future value at time 3 of the cash flows at time 1 and 3.

FVn = PV0  (1 + r)n $100  (1 + 0.05)2 = $110.25 $300  (1 + 0.05)0 = $300 Lump Sum FV = $110.25 + $300 = $410.25 Step 2: Determine future value at time 3 of the cash flow at time 2. $567.75  $410.25  $157.50 © 2022 Pearson Education, Inc.

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Step 3: Calculate the present value at time 2 of future value of the time 2 cash flow at time 3 (i.e., the cash flow at time 2): PV0 = $157.50  (1 + 0.05)1  $150.00 It might help to see the underlying algebra for this solution. The three payments have a future value of $567.75. The cash flow at time 1 earns 5% interest for 2 years, the missing cash flow at time 2 earns 5% for 1 year, and the third cash flow at time 3 earns 5% for 0 years. To solve for the missing payment:

In the last equation, X is the present value of the missing cash flow at time 2 and $157.50 is its future value at time 3.

P5-17 Time value comparisons of single amounts (LG 2; Intermediate) a. A: PV0 = FVn (1 + r)n = $28,500 (1 + 0.09)3  $22,007.22 B: PV0 = $54,000 (1 + 0.09)9  $24,863.10 C: PV0 = $160,000 (1 + 0.09)20 $28,548.94 b. The benchmark for an acceptable alternative is $23,000, what you would have to pay today to undertake it. Alternatives (B) and (C) have present values exceeding $23,000 (i.e., marginal benefits exceeding marginal cost) and should, therefore, be undertaken. c. Although alternatives (B) and (C) are both attractive, (C) is the most attractive if only one can be chosen because it has the highest present value. Put another way, for the same marginal cost, alternative (C) offers the greater marginal benefit.

P5-18 Personal finance problem: Cash flow investment decision (LG 2; Intermediate) The general approach is to determine the present value of each investment and then compare that present value to the purchase price. All investments with present values exceeding price should be purchased. As always, the formula for present value is PV0 = FVn (1 + r)n.

A: PV0 = $30,000 (1 + 0.10)5  $18,627.64. Price = $18,000.

Decision: Purchase B: PV0 = $3,000 (1 + 0.10)20  $445.93. Price = $600.

Decision: Do not purchase C: PV0 = $10,000 (1 + 0.10)10  $3,855.43. Price = $3,500.

Decision: Purchase D: PV0 = $15,000 (1 + 0.10)40  $331.42. Price $1,000.

Decision: Do not purchase If limited to one investment, Tom should purchase option A because marginal benefits exceed price by $627.644 while marginal benefits for option C exceed price by $355.43.

P5-19 Calculating deposit needed (LG 2; Challenge) Step 1: Determine future value of initial deposit at the end of the 7 years.

FVn = PV0  (1 + r)n = $10,000  (1 + 0.05)7 = $14,071 Step 2: Determine future value of second deposit

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$20,000  $14,071  $5,929 Step 3: Calculate the present value of second deposit at the end of year 3 (i.e., actual amount deposited at that time): PV0 = $5,929 (1 + 0.05)4  $4,877.80 It might help to see the underlying algebra for this solution. The two payments have a future value of $20,000. The first earns 5% interest for 7 years, and the second earns 5% for 4 years. To solve for the missing payment:

In the last equation, $5,929 is future value of the missing payment, and X is the present value of the missing payment.

P5-20 Present value and discount rates (LG 2; Challenge) If discount rates are negative, the relation between present value and the number of periods will be upward sloping. This means that the present value will be greater than the future value. In other words, if an investment pays $1 in the future, it will cost more than $1 to buy the investment today.

P5-21 Future value of an annuity (LG 3; Intermediate) a. (1) The future value of an ordinary annuity is given by: ì é 1+ r n - 1ù ü ) úû ï ï ê( FVn = CF1 ´ í ë ý r ï ï î þ where CF1 is the equal end-of-period payments, r the interest rate, and n the number of periods.

Case A: B: C: D: E:

FV10 = $2,500  {[(1 + 0.08)10  $36,216.41 FV6 = $500  {[(1 + 0.12)6  $4,057.59 FV5 = $30,000  {[(1 + 0.20)5  $223,248.00 FV8 = $11,500  {[(1 + 0.09)8  $126,827.45 FV30 = $6,000  {[(1 + 0.14)30  $2,140,721.08

Future value of an ordinary annuity may also be found with a financial calculator or spreadsheet program like Excel. In Excel, command format is FV(r, n, -CF1,0,0) where the final ―0‖ inside the parentheses indicates ordinary annuity (1 = annuity due). For case A above, the specific Excel entry is: =FV(0.08,10,-2500,0,0) (2)The future value of an annuity due is given by: ì é 1+ r n - 1ù ü ) úû ï ï ê( FVn = CF0 ´ í ë ý ´ (1+ r ) r ï ï î þ where CF0 is the equal beginning-of-period payments, r the interest rate, and n the number of periods.

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FV10 = $2,500  {[(1 + 0.08)10   (1 + 0.08) $39,113.72 FV6 = $500  {[(1 + 0.12)6  $4,544.51 FV5 = $30,000  {[(1 + 0.20)5  $267,897.60 FV8 = $11,500  {[(1 + 0.09)8  $138,241.92 FV30 = $6,000  {[(1 + 0.14)30  $2,440,422.03

Again, future value of an annuity due may be found with a financial calculator or spreadsheet program like Excel. In Excel, command format is FV(r, n, -CF1,0,1) where the final ―1‖ inside the parentheses indicates annuity due. For case A above, the following would be entered in a cell: =FV(0.08,10,-2500,0,1) b. The annuity due has a greater future value in each case. By making deposits at the beginning rather than the end of the year, each cash flow enjoys one additional year of compounding.

P5-22 Present value of an annuity (LG 3; Intermediate) a. Using the formula for present value of an ordinary annuity: æ CF1 ö é 1 ùú ê PV0 = ç ´ 1è r ÷ø ê (1+ r )n ú ë û

where CF1 is the equal end-of-period payments, n the number of periods, and r the interest rate per period.

Case A B C D E

PV0 = ($12,000   [1  [1 ÷ (1 + 0.07)3 $31,491.79 PV0 = ($55,000   [1 ÷ 1 + 0.12)15$374,597.55 PV0 = ($700  [1  [1 ÷ (1 + 0.20)9$2,821.68 PV0 = ($140,000   [1  [1 ÷ (1 + 0.05)7$810,092.28 PV0 = ($22,500   [1  [1 ÷ (1 + 0.10)5$85,292.70

Present value of an ordinary annuity may also be in Excel using the bracketed formula [=PV(r, n, -CF1,0,0)], where the final ―0‖ inside the parentheses denotes ordinary annuity (1 = annuity due). For case A above, the specific cell for the present value of the ordinary annuity is: =PV(0.07, 3, -12000,0,0) The present value of an annuity due is given by: æ CF ö é 1 ùú PV0 = ç 0 ÷ ´ ê1´ (1+ r ) è r ø ê (1+ r )n ú ë û where CF0 is the equal beginning-of-period payments, r the interest rate, and n the number of periods.

Case A B C D E

PV0 = ($12,000   [1  [1 ÷ (1 + 0.07)3 (1 + 0.07)  $33,696.22 PV0 = ($55,000   [1 ÷ 1 + 0.12)15 (1 + 0.12) $419,549.25 PV0= ($700  [1  [1 ÷ (1 + 0.20)9 (1 + 0.20) $3,386.01 PV0 = ($140,000   [1  [1 ÷ (1 + 0.05)7 (1 + 0.05) $850,596.89 PV0 = ($22,500   [1  [1 ÷ (1 + 0.10)5 (1 + 0.10)$93,821.97

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Present value of an annuity due may be in Excel with the bracketed formula [=PV(r, n, -CF0,0,1)], where the final ―1‖ inside the parentheses denotes annuity due. For case A above, the specific cell entry to obtain PV is: =PV(0.07, 3, -12000,0,1) b. The annuity due has the greater present value in each case. By making deposits at the beginning rather than the end of the year, each cash flow is discounted one less year.

P5-23 Personal finance problem: Time value: Annuities (LG 3; Challenge) a. The future value of the ordinary annuity is $32,951.99 when the interest rate is 6% and $39,843.56 when the interest rate is 10%. The future value of the annuity due is $32,134.78 when the interest rate is 6% and $40,321.68 when the interest rate is 10%. b. When the interest rate is 6%, the ordinary annuity has a higher future value, but when the interest rate is 10%, the annuity due has the higher future value. c. The present value of the ordinary annuity is $18,400.22 when the interest rate is 6% and $15,361.42 when the interest rate is 10%. The present value of the annuity due is $17,943.89 when the interest rate is 6% and $15,545.75 when the interest rate is 10%. d. At 6% the ordinary annuity has a higher present value, but at 10% the annuity due has the higher present value. e. Ignoring the time value of money, the ordinary annuity pays out more cash ($25,000 vs. $23,000 for the annuity due), but Marian receives the cash at the end of the year. When interest rates are low, the penalty for waiting longer is not very high, so the ordinary annuity is more desirable. As interest rates rise, however, collecting cash faster becomes more important until at some point the annuity due becomes more valuable. Even though it offers less total cash, it pays that cash faster.

P5-24 Personal finance problem: Future and present value of an annuity (LG 3; Intermediate) Using the formula for future value of an ordinary annuity: ì é 1+ r n - 1ù ü ) úû ï ï ê( FVn = CF1 ´ í ë ý r ï ï î þ and the formula for present value of an ordinary annuity:

æ CF ö é 1 ùú PV0 = ç 1 ÷ ´ ê1è r ø ê (1+ r )n ú ë û

where CF1 is the equal end-of-period payments, n the number of periods, and r the interest rate per period. a. FV5 = $100  {[(1 + 0.20)5 $744.16 PV0 = ($100    [1  (1 + 0.20)–5$299.06 b. FV5 = $100  {[(1 + 0.10)5 $610.51 PV0 = ($100    [1  (1 + 0.10)–5$379.08 c. FV5 = $100  {[(1 + 0.01)5 $510.10 PV0 = ($100    [1  (1 + 0.01)–5$485.34 d. FV5 = $100  {[(1 + 0.001)5 $501.00 © 2022 Pearson Education, Inc.

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PV0 = ($100    [1  (1 + 0.001)–5$498.50 e. The general principle is that future values are directly related to interest rates, while present values are inversely rated to interest rates. As the interest rate gets closer to zero, the future and present values get closer to each other and closer to the simple sum of the annuity payments of $500. If the interest rate were actually zero, we could just add up annuity payments over time because money would have no time value (i.e., a dollar today is worth a dollar in the future).

P5-25 Retirement planning (LG 3; Challenge) a. The problem asks students to find the future value of an ordinary annuity. Specifically, FV with end-of-year $2,000 contributions, an interest rate of 10% and 40 years to retirement, is $885,185.11. b. If contributions begin 10 years later (i.e., are made for 30 years), but all other information remains the same, future value falls to $328,988.05. c. The opportunity cost of delaying deposits for 10 years is the difference in future values in parts (a) and (b), or $556,197. This large difference is traceable to two factors: (i) 10 years of lost deposits and (ii) 10 years of lost compound interest. d. The problem asks students to find the future value of an annuity due. Specifically, future value of an annuity due—given annual end-of-year contributions of $2,000, an interest rate of 10% and 40 years to retirement, is $973,703.62. If contributions begin 10 years later (i.e., are made for 30 years), but all other information remains the same, future value falls to $361,886.85. The opportunity cost of delaying deposits for 10 years is the difference in future values, $611,816.77.

P5-26 Personal finance problem: Value of a retirement annuity (LG 3; Intermediate) The correct framework is present value of an ordinary annuity, with $12,000 payments for 25 years and an interest rate of 9%. Based on your 9% required rate of return the PV ($117,870.96) is the maximum you would pay for the annuity. æ CF ö é 1 ùú PV0 = ç 1 ÷ ´ ê1è r ø ê (1+ r )n ú ë û

PV0 = ($12,000    [1  (1 + 0.09)-25$117,870.96 P5-27 Personal finance problem: Funding your retirement (LG 2, 3; Challenge) a. At age 65, Emily will be one year away from making the first of 25, $50,000 annual withdrawals (i.e., she will take out $50,000 each birthday from age 66 to age 90).

Imagine today is Emily’s 65th birthday, and she expects the 25-year annuity to begin in one year. The present value of that ordinary annuity (on Emily’s 65th birthday) is $421,087.23. This value can be obtained using Equation 5.4 in the text where CF1 = $50,000, n = 25, and r = 11%, or with the PV function in Excel with the syntax PV(0.11,25,-50000,0,0). Now, to find out how much Emily should invest now at an interest rate of 11%, so she will have $421,087.23 in 20 years, plug FV20 = $421,087.23, r = 11%, and n = 20 into Equation 5.2 in the text (i.e. PV0 = $421,087.23 (1.11)20) or use the PV function in Excel with the syntax PV(0.11,20,0,421087.23,0). Emily must invest $52,229.09 to accumulate the funds necessary to pay the $50,000 annuity. b. The approach is the same as in part (a) except the present value of the 25-year, $50,000 annuity on Emily’s 65th birthday is calculated with a discount rate of 8% rather than © 2019 Pearson Education, Inc.

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11%. Now, present value is $533,738.81—a much larger than in part (a) because Emily will need more money at age 65 if the rate of return after retirement is lower. To accumulate this higher sum over the next 20 years given a discount rate of 11%, Emily must invest $66,201.71 today [PV = $533,738.81 (1.11)20]—nearly $14,000 more today than if she earned 11% to age 90. c. In part (b) Emily needs to invest $66,201.71 today to reach her retirement-income goals.

If she deposits $75,000 instead, the difference ($8,798.29) will compound at 11% for 20 years then at 8% for 25 more years. Over the first 20 years, the $8,798.42 will reach a value of $70,934.56 [FV20 = $8,798.29 (1.11)20]. Over the next 25 years, $70,934.56 will earn 8% to reach a value of $485,793.57 [FV20 = $70,934.56 (1.08)25].

Correct answers may be obtained another way. Consider part (b). Suppose Emily deposits $75,000 today at 11% to reach a value of $604,673.36 in 20 years [FV = $75,000 (1.11)20]. Now, at age 65 she must purchase an ordinary annuity paying $50,000 per year for 25 years given an interest rate of 8%. At that point, the price of the annuity (which equals the present value) will equal $533,781.81. After purchasing the annuity, Emily will have $70,934.55 remaining ($604,673.36 $533,781.81). Now, she can invest this remainder at 8% for 25 years to reach a final value at age 90 of $485,793.51.

P5-28 Personal finance problem: Value of an annuity versus a single amount (LG 2 and LG 3; Intermediate) a. n  25, r  5%, PMT  $40,000; solve for PV  $563,757.78. Take the annuity because its present value exceeds the lump sum by $63,757.58. b. n  25, r  7%, PMT  $40,000; solve for PV  $466,143.33. Take the lump sum because it exceeds the present value of the annuity by $33,856.67. c. View this problem as a $500,000 investment offering a 25-year annuity of $40,000, and determine the discount rate necessary to make the present value of the annuity equal $500,000. The discount rate equating the two sums is 6.24%. This may be obtained by solving for r, given n  25, PV  $500,000, PMT  $40,000. In Excel, use RATE function with the bracketed syntax [=rate(periods, -(annuity),present value,0,0)].

Specifically, [= rate (25,-40000,500000,0,0)].

P5-29 Perpetuities (LG 3; Basic) Case A

Equation $20,000  0.08 $100,000  0.10 $3,000  0.06 $60,000  0.05

B C D

Present Value $250,000 $1,000,000 $50,000 $1,200,000

P5-30 Perpetuities (Intermediate) a. The present value of the perpetuity is $100 0.07 = $1,428.57. To see this is the correct answer, suppose you invested $1,428.57 invested in an account right now paying 7% interest. If you withdrew the interest each year, you would have $1,428.57 7% = $100.

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b. This problem is identical to part (a) except here you also get $100 immediately. If the stream in part (a) is worth $1,428.57, then the part (b) stream is worth $100 more or $1,528.57. c. This situation is the same as part (a) except here you must wait 2 additional years before the $100 payments begin. The perpetuity in part (a) is worth $1,428.57; to obtain the present value of this amount in two years, then simply discount this amount by 2 years.

PV = $1,428.57 (1.07)2 = $1,247.77. Again it is easy to verify this answer is correct.

Suppose you have $1,247.77 in an account right now paying 7%. If that grows at 7% for two years, then on January 1 two years later you have FV = $1,247.77 1.072 = $1,428.57.

P5-31 Perpetuities (Intermediate) a. Present value is $75

1.10 = $68.18.

b. In 100 years, the payment will be $75 1.0499 = $3,642.18. The present value of this payment will be $3,642.18 1.10100 = $0.26. c. Using Equation 5.8 in the text, the present value of a perpetuity initially paying $75 but growing by 4% per year thereafter, and a discount rate of 10% is PV = $75 (0.10 0.04) = $1,250. d. Payments further in the future have a lower present value, with present value going to zero as the number of periods goes to infinity. Because the present value of payments after a point way into the future is zero, the sum of the present values of all future payments is finite.

P5-32 Personal finance problem: Creating an endowment (Intermediate) a.

Cost next year = $600

(1.02) = $612.

b.

The present value of an annuity paying $612 initially but growing at 2% per year with a discount rate of 6% is $612 (0.06 0.02) = $15,300.

c. The present value of an annuity paying $612 initially but growing at 2% per year with a discount rate of 9% is $612 (0.09 0.02) = $8,742.86.

P5-33 Value of a mixed stream (LG 4; Challenge) a.

Stream A

Y e a r 1

Compounding Years 2

2 3

1 0

Cash flow $ 900 1,000 1,200

Interest Rate    Sum =

Stream

Y e ar 1

Future Value $ 1,128.96 1,120.00 1,200.0 0 $ 3,448.96

Compounding Years

Cash flow

Interest Rate

Future Value

4

$30,00



$

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2 3 4 5

B

0 25,000 20,000 10,000 5,000

3 2 1 0

    Sum =

Stream

C

Y e ar 1 2 3 4

Compounding Years

Cash flow

Interest Rate

3 2 1 0

$ 1,200 1,200 1,000 1,900

    Sum =

47,205.58 35,123.20 25,088.00 11,200.00 5,000.0 0 $123,616.7 8 Future Value $1,685.91 1,505.28 1,120.00 1,900.0 0 $ 6,211.19

b. If payments are made at the beginning of each period, the present value of each end-ofperiod cash flow should be multiplied by (1  r) to obtain present values for beginning-ofperiod cash flows. So, A: $3,448.96 (1  0.12)  $3,862.84, B: $123,616.78 (1  0.12)  $138,450.79, and C: $6,211.19 (1  0.12)  $6,956.53.

P5-34 Personal finance problem: Value of a single amount versus a mixed stream (LG 4; Challenge) If Gina takes $24,000 and leaves it in account earning 7% for five years, she will have $33,661.24 (PV = $24,000, n = 5, and r = 0.07) for her home. The future value of the mixed stream is: Time 0

Compounding Years 5

Cash Flow $ 2,000 $ 4,000 $ 6,000 $ 8,000 $10,00 0

Interest Rate 7%

Future Value $ 2,805.10 1 4 $  5,243.18 2 3 $  7,350.26 3 2 7% $ 9,159.20 4 1 $10,700.0  0 Total =  $35,257 .75 Gina should take the mixed-stream the stream of payments because its future value will be $1,596.51 than the lump-sum payment.

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P5-35 Value of mixed streams (LG 4; Basic) Project A: Interpret the negative cash flow as a payment made rather than one received; discount it as you would a positive value. Cash flows are received/paid at year end, so the $2,000 payment made should be discounted back one period, the $3,000 payment made discounted back two periods, and so on. Given cash flows of CF1 = $2,000, CF2  $3,000, CF3  $4,000, CF4  $6,000, CF5  $8,000 and an interest rate (r) of 12%, present value = $11,805.51.

Project B: Given end-of-year cash flows of CF1  $10,000, CF2  $5,000, F2  4, CF3  $7,000 and an interest rate of 12%, present value = $26,034.58.

Project C: Treat this mixed stream as two ordinary annuities—the first paying $10,000 for 5 years with an interest rate of 12% and present value of $36,047.76 and the second paying $8,000 for 5 years with an interest rate of 12%, and present value of $28,838.21. The second annuity does not begin until the end of year six, so $28,838.21 is present value in six years. To obtain the total present value of the mixed stream, $28,838.21 must first be discounted to current dollars before adding the present value of the first annuity. Discounting the second annuity s yields $16,363.57. So, present value of the two annuities together –the present value of the 10-year mixed stream—is $52,411.34.

P5-36 Present value: Mixed streams (LG 4; Intermediate) a. The present value of stream A with cash flows of CF0  $50,000 (i.e., a $50,000 payment is made immediately), CF1  $40,000 (i.e., a $40,000 payment is received at the end of year one), CF2  $30,000, CF3  $20,000, CF4  $10,000 and an interest rate of 5% = $40,809.90. The present value of stream B with cash flows of CF0  $10,000 (i.e., a $10,000 payment is received immediately), CF1  $20,000, CF2  $30,000, CF3  $40,000, CF4  $50,000 (i.e., a $50,000 payment is made at the end of year 4) and an interest rate of 5% = $49,676.88. b. Both streams pay $50,000, but stream A has a large negative cash outflow right away whereas stream B’s large negative cash outflow occurs 4 years later. Because money today is more valuable than money in the future, the cash-flow stream that delays the large outflow (i.e., stream B) will generally be more valuable. The two streams will have the same present value if the discount rate is 0%, which means money today and money tomorrow have the same value. With a 0% discount rate, it does not matter when the $50,000 outflow occurs.

P5-37 Value of a mixed stream (LG 1 and LG 4; Intermediate) a.

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b.

Year Cash Flow Interest Rate Present Value 1 $30,000 12% $ 26,785.71 2 $25,000 12% $ 19,929.85 3 $15,000 12% $ 10,676.70 4 $15,000 12% $ 9,532.77 5 $15,000 12% $ 8,511.40 6 $15,000 12% $ 7,599.47 7 $15,000 12% $ 6,785.24 8 $15,000 12% $ 6,058.25 9 $15,000 12% $ 5,409.15 10 $10,000 12% $ 3,219.73 Total = $ 104,508.28

c. Harte should still accept the offer of a 10-year mixed stream because its present value of that mixed stream exceeds the $100,000 immediate payment.

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P5-38 Value of a mixed stream (LG 4; Intermediate) a.

b. Total undiscounted cash flow is $3 million. At first glance, this project seems unattractive, but the large cash outflow of $22 million comes six years in the future, and a high discount rate could make its present value relatively low. In other words, total present value for the project might be positive if the cost of capital is sufficiently. c. Project present value with a 5% discount rate is –$98,832, which means the project is unattractive. However, at 10% opportunity cost, the PV is $1,744,721, which is quite attractive. These numbers illustrate the key idea: future outflows have small present values if payment is sufficiently distant or the discount rate sufficiently high.

P5-39 Relationship between future value and present value: Mixed stream (LG 4; Intermediate) a. The present value of end-of-year cash flows CF1  $800, CF2  $900, CF3  $1,000, CF4  $1,500, CF5  $2,000 with a 5% discount rate of 5% is $5,243.17. b. FV = $5,243.17

(1 + 0.05)5 = $6,691.76.

c. Future value is $6,691.77, apart for rounding error the same as in part (b). The point here is the three ways of obtaining equivalent value: (i) taking the mixed stream as received, (ii) taking $5,243.17 today, or (iii) taking $6,691.76 in 5 years. d. The appropriate price for the mixed stream is its present value ($5,243.17).

P5-40 Relationship between future value and present value: Mixed Stream (LG 4; Intermediate) Step 1: Calculate present value of known cash flows: Year CFt PV @ 4% 1 $10 $9,615. ,00 38 0 2 $5, $4,622.

000 78 3 ?

4 $20 $17,09 ,00 6.08 0 5 $3, $ 000 2,465.7 8 Total = $33,80 0.02 Step 2: Subtract present values for years 1, 2, 4, and 5 from present value of entire stream: $32,911.03  $33,800.02  $888.99.

Step 3: Calculate value in 3 years of Step 2 value today: The future value of $888.99, compounding for 3 years at 4% is $999.99

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P5-41 Changing compounding frequency (LG 5; Intermediate) Future value with different compounding frequencies: (1) Annual Semiannual n  5, r  12%, PV  $5,000 n  5  2  10, r  12%  2  6%, PV  $5,000 Solve for FV  $8,811.71 Solve for FV  $8,954.24 Quarterly n  54  20 periods, r  12%  4  3%, PV  $5,000. Solve for FV  $9,030.56 (2) Annual Semiannual n  6, r  16%, PV  $5,000 n  6  2  12, r  16%  2  8%, PV  $5,000 Solve for FV  $12,181.98 Solve for FV  $12,590.85 Quarterly n  6  4  24 periods, r  16%  4  4%, PV  $5,000. Solve for FV  $12,816.52 (3) Annual Semiannual n  10, r  20%, PV  $5,000 n  10  2  20, r  20%  2  10%, PV  $5,000 Solve for FV  $30,958.68 Solve for FV  $33,637.50 Quarterly n  10  4  40 periods, r  20%  4  5%, PV  $5,000. Solve for FV  $35,199.94 Effective interest rate: reff  (1  r/m)m – 1 (1) Annual reff  (1  0.12/1)1 – 1 reff  (1.12)1 – 1 reff  (1.12) – 1 reff  0.12  12%

Semiannual reff  (1  12/2)2 – 1 reff  (1.06)2 – 1 reff  (1.124) – 1 reff  0.124  12.4%

Quarterly reff  (1  12/4)4 – 1 reff  (1.03)4 – 1 reff  (1.126) – 1 reff  0.126  12.6%

(2) Annual reff  (1  0.16/1)1 – 1 reff  (1.16)1 – 1 reff  (1.16) – 1 reff  0.16  16%

Semiannual reff  (1  0.16/2)2 – 1 reff  (1.08)2 – 1 reff  (1.166) – 1 reff  0.166  16.6%

Quarterly reff  (1  0.16/4)4 – 1 reff  (1.04)4  1 reff  (1.170)  1 reff  0.170  17%

(3) Annual reff  (1  0.20/1)1 – 1 reff  (1.20)1 – 1 reff  (1.20) – 1 reff  0.20  20%

Semiannual reff  (1  0.20/2)2 – 1 reff  (1.10)2 – 1 reff  (1.210) – 1 reff  0.210  21%

Quarterly reff  (1  0.20/4)4 – 1 reff  (1.05)4 – 1 reff  (1.216) – 1 reff  0.216  21.6%

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P5-42 Finding nominal rate of return (LG 5; Challenge) a. (EAR + 1)1/m – 1 = r/m (0.0938 + 1)1/12 – 1 = 0.00750 or 0.75% b. 40 year total spending = monthly spending × 12 × 40 = $125 × 12 × 40 = $60,000 c. Using the formula for future value of an ordinary annuity: ì é 1+ r n - 1ù ü ) ûú ï ï ê( FVn = CF1 ´ í ë ý r ï ï î þ where CF1 is the end-of-month payments of $125, n the total number of months over 40 years, and r/m is the nominal monthly interest rate.

FV40 yrs = $125  {[(1 + 0.0075)480 $585,165 P5-43 Compounding frequency, time value, and effective annual rates (LG 5; Intermediate) a.

Different compounding frequencies: A: n  10, r  3%, PV  $2,500 Solve for FV5  $3,359.79

B: n  18, r  2%, PV  $50,000 Solve for FV3  $71,412.31

C: n  10, r  5%, PV  $1,000 Solve for FV10  $1,628.89

D: n  24, r  4%, PV  $20,000 Solve for FV6  $51,226.08

b. Effective interest rate: reff  (1  r/m)m – 1 A: reff  (1  0.06/2)2  1 reff  (1  0.03)2  1 reff  (1.061)  1 = 0.061  06.1%

B: reff  (1  0.12/6)6  1 reff  (1  0.02)6  1 reff  (1.126)  1 = 0.126  12.6%

C: reff  (1  0.05/1)1  1 D: reff  (1  0.16/4)4 – 1 1 reff  (1  0.05)  1 reff  (1  0.04)4  1 reff  (1.05)  1 = reff  0.05  5% reff  (1.170)  1 = reff  0.17  17% c. Effective interest rates rise relative to stated rates as compounding frequency rises.

P5-44 Continuous compounding (LG 5; Intermediate) FVcont.  PVer n (where e  2.718282, r = annual interest rate, and n = number of years) A: FVcont.  $1,000e0.18  $1,197.22 B: FVcont.  $ 600e1  $1,630.97

C: FVcont.  $4,000e0.56  $7,002.69 D: FVcont.  $2,500e0.48  $4,040.19

The Excel command [in brackets] is [=PV*exp(r*n)].

P5-45 Personal finance problem: Compounding frequency and time value (LG 5; Challenge) a. (1) Annually: n  10; r  8%, PV  $2,000. FV  $4,317.85.

(2) Semiannually: n  20, r  4%, PV  $2,000. FV $4,382.25.

(2) Daily: n  3650; r  8%  365  0.022, PV  $2,000. FV  $4,450.69 (4) Continuously: FV10  $2,000(e0.8). FV = $4,451.08 b. (1) reff  (1  0.08/1)1  1

(2) reff  (1  0.08/2)2  1

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reff  (1  0.08)1  1 reff  (1.08) – 1= 0.08  8% (3) reff  (1  0.08/365)365  1 reff  (1  0.00022)365  1 reff  (1.0833) – 1 = 0.0833  8.33%

reff  (1  0.08)2  1 reff  (1.0816)  1 =0.0816  8.16% (4) reff   1 reff  e0.08 1 reff  1.0833  1 = 0.0833  8.33%

c. Continuous compounding yields $133.23 more than annual compounding over 10 years. d. The more frequent the compounding, the larger the future value. Part (a) demonstrates this idea with larger future values as compounding increases from annual to continuous.

Because future value is larger for a given amount invested, effective return also rises with compounding frequency. Part (b) demonstrates this idea with the effective rate rising from 8% to 8.33% when compounding changed from annual to continuous.

P5-46 Personal finance problem: Annuities and compounding (LG 3 and LG 5; Intermediate) a. For the ordinary annuity with annual compounding: n  10, r  8%, PMT  $300, and solve for FV  $4,345.97. For the ordinary annuity with semiannual compounding: n  10  2  20; r  8  2  4%, PMT  $150, and solve for FV  $4,466.71. For the ordinary annuity with, quarterly compounding: n  10  4  40; r  8  4  2%; PMT  $75, and solve for FV  $4,530.15. b. The sooner a deposit is made, the sooner the funds can earn interest. Thus, the sooner the deposit and more frequent the compounding, the larger the future sum.

P5-47 Deposits to accumulate future sums (LG 6; Basic) Using the framework for future value of an annuity: Case A B C D

Terms 12%, 3 years 7%, 20 years 10%, 8 years 8%, 12 years

Given Information n  3, r  12%, FV  $5,000 n  20, r  7%, FV  100,000 n  8, r  10%, FV  $30,000 n  12, r  8%, FV  $15,000

Payment $1,481 .74 $2,439 .29 $2,623 .32 $ 790.43

P5-48 Personal finance problem: Creating a retirement fund (LG 6; Intermediate) a. Given n  42, r  8%, and FV  $220,000, solve for PMT  $723.10. b. Given n  42, r 8%, and PMT  $600, solve for FV  $182,546.11.

P5-49 Personal finance problem: Accumulating a growing future sum (LG 6: Intermediate) Step 1: Determining cost of home in 20 years: Given n  20, r  6%, and PV  $185,000, solve for FV20  $593,320.06.

Step 2: Determining how much to save annually to afford home: Given n  20, r  10%, and FV  $593,320.06, solve for PMT  $10,359.15.

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P5-50 Personal finance problem: Inflation, time value, and annual deposits (LG 2, LG 3, and LG 6; Challenge) a. n  25, r  5%, PV = $200,000. Solve for FV25  $677,270.99. b. n  25, r  9%, FV25  $677,270.99. Solve for PMT  $7,996.03. c. Because each of John’s deposits will earn interest an additional year’s worth of interest, he can deposit a smaller sum each year and still hit his target of $677,270.99 in 25 years.

To determine how much smaller, let n = 25, r = 9%, and FV25 = $677,270.99, and solve for the annuity due PMT (= $7,335.81).

P5-51 Loan payment (LG 6; Basic) A: Given n  3, r  8%, and PV  $12,000, solve for PMT  $4,656.40.

B: Given n  10, r  12%, and PV  $60,000, solve for PMT  $10,619.05.

C: Given n  30, r  10%, and PV  $75,000, solve for PMT  $7,955.94.

D: Given n  5, r  15%, PV  $4,000, solve for PMT  $1,193.26.

P5-52 Personal finance problem: Loan-amortization schedule (LG 6; Intermediate) a. Treat the problem like an ordinary annuity with n= 3, r = 4%, PV = $45,000, and solve for PMT. Loan payment is $16,215.68. b.

End of Year 1 2 3

Loan Payment $16,215.68 16,215.68 16,215.68

Beginning-ofYear Principal

Interest

Payments Principal

$45,000.00 30,584.32 15,592.00

$1,800.00 1,223.37 623.68 Total =

$14,415.68 14,992.31 15,592.00 $45,000.00

c. The interest portion falls each period because some principal is being repaid (so the same interest rate is applied to smaller principal).

P5-53 Loan interest deductions (LG 6; Challenge) a. Use the ordinary annuity framework with n  3, r  13%, PV (loan amount)  $10,000, and solve for PMT. Annual end-of-year loan payments = $4,235.22. b. and c.

End of Year 1 2 3

Loan Payment

Beginning-ofYear Principal

Interest

$4,235.22 4,235.22 4,235.22

$10,000.00 $7,064.78 $3,747.98

$1,300.00 918.42 487.24

$2,935.22 3,316.80 3,747.98

$2,705.66

$10,000.00

Totals =

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P5-54 Personal finance problem: Monthly loan payments (LG 6; Challenge) a. Use the ordinary annuity framework with n  12 3 = 36, r  6% 12 = 0.005, PV (loan amount)  $25,000, and solve for PMT. End-of-month payments = $760.55. b.

Use the ordinary annuity framework with n  36, r  4% 12 = 0.003333, PV (loan amount)  $25,000, and solve for PMT. End-of-month payments = $738.10.

P5-55 Growth rates (LG 6; Basic) To find average annual growth rate, use the future-value framework and solve for the interest rate that makes stock price grow from the purchase to the sales price over the holding period. Specifically: Amazon: ($1,875/$134)1/10 – 1 = 30.2% Chipotle: ($291/$88)1/8 – 1 = 16.1% Netflix: ($259/$8)1/9 – 1 = 47.2%

P5-56 Personal finance problem: Rate of return (LG 6, Intermediate) a.

Use the future-value framework, and solve for the interest rate (r) that makes the investment grow from initial to final value over the holding period, where n  3, PV  $1,500, FV  $2,000. Average annual growth rate (r) = 10.06%. In Excel, the answer may be obtained with the RATE function, but PV must be entered as a negative number.

b. Mr. Singh should make the investment that returns $2,000 because it offers a higher return for the same amount of risk.

P5-57 Personal finance problem: Rate of return and investment choice (LG 6; Intermediate) a. For each investment, use the equation (FVn/PV0)1/n – 1 = r to find the average annual rate of return for each investment. Note that the present value for each investment is the $5,000 purchase price.

Purchase Future Average Annual Investment Price Cash Inflow Years Rate of Return A $ 5,000 $ 8,400 6 9.03% B $ 5,000 $ 15,900 15 8.02% C $ 5,000 $ 7,600 4 11.04% D $ 5,000 $ 13,000 10 10.03% In Excel, use RATE function, but PV must be entered as a negative number. b. Investment C provides the highest return of the four alternatives. Assuming all investments have equal risk, Clare should choose C. P5-58 Rate of return: Annuity (LG 6; Basic) Use the framework for an ordinary annuity, which is given by æ CF ö é 1 ùú PV0 = ç 1 ÷ ´ ê1è r ø ê (1+ r )n ú ë û

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where CF1 is the first payment received at the end of the first period, n is the number of periods, and r the interest rate per period. Because you cannot algebraically rearrange the ordinary annuity equation to isolate r, you have to use trial and error. Solve the annuity equation using the given 10 annuity payments of $2,000 and a guess for r. When the result equals $10,606, you will have found the correct rate of return (i.e., the one that equates the discounted present value of the 10 $2,000 payments to $10,606). Maybe try an initial guess of 10%, in which case you find:  1  $2, 000    $12, 289.13    1  10   0.10   1  0.10  

Because $12,289.13 is greater than $10,606, a discount rate of 10% is too low, so guess something higher. Maybe try 12%, in which case you would find a present value of $11,300.45 that is still larger than $10,606. Keep refining your guess through repeated trial and error, and you will find that 13.58% is the correct discount rate and, therefore, the rate of return on the investment.  1  $2, 000    1   $10, 605.51    10   0.1358   1  0.1358 

Alternately, you can enter the given information into your financial calculator. Enter n  10, PV  $10,606, PMT  $2,000, and solve for I/Y. Using the same trial and error, but doing it faster, the financial calculator finds the same rate of return of 13.58%.

P5-59 Personal finance problem: Choosing the best annuity (LG 6; Intermediate) Note: In Excel, use RATE function, but PV must be entered as a negative number. a. Annuity A n  20, PV  $30,000, PMT  $3,100 Solve for r  8.19%

Annuity B n  10, PV  $25,000, PMT  $3,900 Solve for r  9.03%

Annuity C n  15, PV  $40,000, PMT  $4,200 Solve for r  6.30%

Annuity D n  12, PV  $35,000, PMT 4,000 Solve for r  5.23%

b. Annuity B offers the highest return at 9.03%. Because Raina considers all four annuities equally risky and is indifferent about their differing lives, she should choose Annuity B. P5-60 Personal finance problem: Interest rate for an annuity (LG 6; Challenge) Note: In Excel, use RATE function, but PV must be entered as a negative number. a. Defendant’s interest rate assumption: n  25, PV  $2,000,000, and PMT  $156,000 Solve for r  5.97% (or 6% when rounded to the nearest whole percent). b. Anna’s interest rate assumption: n  25, PV  $2,000,000, and PMT  $255,000.

Solve for r  12.0%. c. n  25, r  9%, PV  $2,000,000. Solve for PMT  $203,612.50.

P5-61 Personal finance problem: Loan rates of interest (LG 6; Intermediate) Note: In Excel, use RATE function, but PV must be entered as a negative number. a. Loan A: n  5, PV  $5,000, PMT  $1,352.81. Solve for r  11.0% © 2019 Pearson Education, Inc.

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Loan B: nPV  $5,000 PMT  $1,543.21. Solve for r  9.0% Loan C: n  3, PV  $5,000, PMT  $2,010.45. Solve for r  10.0% b. Mr. Fleming should choose Loan B, which has the lowest interest rate.

P5-62 Number of years to equal future amount (LG 6; Intermediate) Note: In Excel, use NPER function, but PV must be entered as a negative number.

A Given r  7%, PV  $300, and FV  $1,000, solve for n  17.79 years.

B: Given r  5%, PV  $12,000, FV  $15,000, solve for n  4.57 years.

C: Given r  10%, PV  $9,000, and FV  $20,000, solve for n  8.38 years.

D: Given r  9%, PV  $100, and FV  $500, solve for n  18.68 years.

E: Given r  15%, PV  $7,500, and FV  $30,000, solve for n  9.92 years.

P5-63 Personal finance problem: Time to accumulate a given sum (LG 6; Intermediate) Note: In Excel, use NPER function, but PV must be entered as a negative number. a. r  10%, PV  $10,000, FV  $20,000. Solve for n  7.27 years. b. r  7%, PV  $10,000, FV  $20,000. Solve for n  10.24 years. c. r  12%, PV  $10,000, FV  $20,000. Solve for n  6.12 years. d. The higher the rate of interest, the less time is required to accumulate a given future sum.

P5-64 Number of years to provide a given return (Intermediate) Note: In Excel, use the NPER function, but PV must be entered as a negative number.

A: Given r  11%, PV  $1,000, and PMT  $250, solve for n  5.56 years.

B: Given r  15%, PV  $150,000, and PMT  $30,000, solve for n  9.92 years.

C: Given r  10%, PV  $80,000, and PMT  $10,000, solve for n  16.89 years.

D: Given r  9%, PV  $600, and PMT  275, solve for n  2.54 years E: Given r  6%, PV  $17,000, and PMT  $3,500, solve for n  5.91 years.

P5-65 Personal finance problem: Time to repay installment loan (LG 6; Intermediate) Note: In Excel, use the NPER function, but PV must be entered as a negative number. a. r  12%, PV  $14,000, PMT  $2,450. Solve for n  10.21 years. b. r  9%, PV  $14,000, PMT  $2,450. Solve for n  8.38 years. c. r  15%, PV  $14,000, PMT  $2,450. Solve for n  13.92 years. d. The higher the interest rate, the longer it will take Mia to repay the loan.

P5-66 Ethics problem (LG 6; Intermediate) To find average annual return, use the future-value equation – plugging in purchase price ($711,000) for PV, sales price ($1,340,000) for FV17, and 17 for the number of periods (n): (i) FVn = PV (1 + r)n (ii) $1,340,000 = $711,000 (iii) 1.8847 = (1 + r)17

(1 + r)

17

(iv) √ –1=r 1/17 (v) 1.8847 – 1 = r (vi) r = 0.038 = 3.80%

As to Samantha ―swindling‖ Michael by selling the house 17 years later for nearly double the price she paid him for it, Samantha’s return on home-ownership does not appear excessive © 2022 Pearson Education, Inc.

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compared to the return she might have earned on an investment in stocks. At a broader level, at the time Samantha purchased the home, neither she nor Michael knew with certainty what the market for San Francisco real estate would look like decades later. If both parties were reasonable informed about housing-market fundamentals (i.e., what could have been known and reasonable predictions about future prices based on that knowledge) and neither had undue power in the real-estate market, then many economists would argue whatever price they agreed upon was fair.

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 Case: “Finding Jill Moran’s Retirement Annuity” Case studies are available on www.pearson.com/mylab/finance.

Chapter 5’s case challenges the student to apply present and future value techniques to a real-world situation. The first step is determining the total amount Sunrise Industries needs to accumulate until Ms. Moran retires, remembering to take into account interest that will be earned during the 20-year payout period. Then, needed annual deposits can be determined. a.

b. Total amount needed at the end of year 12 is the present value of future annuity payments to Ms. Moran. Those payments ($42,000) will be made at year end (ordinary annuity) for 20 years, and the interest rate is 12%. The future value of this ordinary annuity is $313,716.63. c. End-of-year deposits necessary over 12 years to fund Mr. Moran’s annuity may be found with the formula for future value of an ordinary annuity, given a future value of $313,716.63 and an interest rate of 9%. Necessary deposits = $15,576.24—that is, Sunrise must deposit $15,576.24 at year end for the next 12 years to accumulate the funds needed to pay $42,000 for 20 years. d. If the interest rate rises to 10%, needed deposits fall to $14,670.43—Sunrise must deposit $14,670.43 at the end of years 1–12 to provide Ms. Moran a $42,000 in years 13 to 32. e. Step one is determining the present value of the $42,000 perpetuity to Ms. Moran; the present value of this perpetuity equals annual cash flows ($42,000) divided by the interest rate (12%) or $350,000. Annual deposits needed to fund this perpetuity may be found using the formula for future value of an ordinary annuity—given a future value of $350,000 and an interest rate of 9%.

Necessary deposits equal $17,377.73—that is, Sunrise must deposit $17,377.73 at year end for the next 12 years to accumulate the funds needed to pay $42,000 annuity in perpetuity. 

Spreadsheet Exercise Answers to Chapter 5’s Uma Corporation spreadsheet problem are available on www.pearson.com/mylab/finance.

 Group Exercise Group exercises are available on www.pearson.com/mylab/finance.

This chapter’s exercises provide each group with opportunities to use time value of money techniques on their fictitious firm. In part (a), students analyze options for leasing a new copy machine to replace the current unreliable one. In part (b), students analyze options for buying a replacement copier outright. Students are asked to furnish a discount rate; instructors should discuss various market rates as candidates. [A good source for interest-rate data is the Federal Reserve Economic Data (FRED), the data website of the Federal Reserve Bank of St. Louis (https://fred.stlouisfed.org/).] In part (c), students are asked to create an amortization schedule for a loan to upgrade the firm’s computer © 2022 Pearson Education, Inc.

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systems. Finally, in part (d), students are asked to compute the present value of a four-year stream of settlement payments, given a 6% discount rate.

 Integrative Case 2: Track Software Inc. Integrative Case 2, Track Software Inc., places the student in the role of financial manager to introduce basic concepts like setting financial goals, measuring firm performance, and analyzing firm condition.

This seven-year-old company has cash-flow problems, so the student must prepare/analyze the statement of cash flows. Interest expense is increasing, and the firm’s financing strategy should be evaluated in view of current yields on loans of different maturities. Ratio analysis of Track’s financial statements provides additional insight into firm condition. The student must then confront a cost/benefit tradeoff: Is the additional expense of a new software developer (which depresses short-term profitability) a good long-term investment? Wrestling with such decisions highlight the importance of financial decisions to day-to-day firm operations and long-term profitability. a.

1.

Stanley has focused on maximizing profit, as suggested by the rise in net profits from 2016 to 2022. His concern about adding a software designer, which would depress near-term earnings, also underscores his focus on profits. Stanley should maximize wealth, a goal which considers risk and cash flows over time. Profit maximization does not integrate these variables (cash flow, timing, risk) into decision-making.

2. An agency problem exists when managers place personal goals ahead of corporate goals.

Stanley owns 40% of outstanding equity, so agency problems are not a major concern. b.

Earnings per share (EPS) calculation: Year

Net Profits After Taxes

EPS (NPAT  50,000 shares)

2016 2017 2018 2019 2020 2021 2022

($50,000) (20,000) 15,000 35,000 40,000 43,000 48,000

$ 0 0 0.30 0.70 0.80 0.86 0.96

EPS has increased steadily, suggesting Stanley has been focused on profit maximization. c.

Operating and Free Cash Flows OCF  EBIT (1  T)  depreciation = $89 (1  0.2) 11  $82.2 FCF  OCF  net fixed asset investment*  net current asset investment**  $82.2  15  47  $20.2 * NFAI ∆ net fixed assets  depreciation  ($132 – 128)  11  $15. ** NCAI  ∆ current assets  ∆ accounts payable ∆ accruals) = ($421

362)  (136

126)  (27

25)  47.

Track Software is generating good cash flow from operating activities. OCF is sufficient to provide needed cash for investment in fixed assets and net working capital, with $20,200 left over for investors (creditors and equity holders). d.

Ratio Analysis—Track Software Inc. Actual

Industry Avg.

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Ratio

2021

2022

2022

Net working capital

$21,000

$58,000

$96,000

TS: Improving CS: Poor

Current ratio

1.06

1.16

1.82

TS: Improving CS: Poor

Quick ratio

0.63

0.63

1.10

Inventory turnover

10.40

5.39

12.45

TS: Stable CS: Poor TS: Deteriorating CS: Poor

Avg. collection period (days) Total asset turnover

29.6

35.8

20.2

TS: Deteriorating CS: Poor

2.66

2.80

3.92

TS: Improving CS: Poor

Actual

CS:

Cross Section

Ratio Debt ratio

2021 0.78

2022 0.73

Industry Avg. 2022 0.55

TS: Time Series CS: Cross Section TS: Decreasing CS: Poor

Times interest earned

3.0

3.1

5.6

TS: Stable CS: Poor

Gross profit margin

32.1%

33.5%

42.3%

TS: Improving CS: Fair

Operating profit margin

5.5%

5.7%

12.4%

TS: Improving CS: Poor

Net profit margin

3.0%

3.1%

4.0%

TS: Stable CS: Fair

Return on assets (ROA) Return on equity (ROE)

8.0%

8.7%

15.6%

TS: Improving CS: Poor

36.4%

31.6%

34.7%

TS: Deteriorating CS: Fair

Ratio analysis of Track Software: 1.

Liquidity: Track’s liquidity (based on its current ratio, net working capital, and quick ratio) has been stable or improved slightly but remains well below peer (industry average).

2.

Activity: Inventory turnover has deteriorated considerably and is now much worse than peer. Average collection period has also deteriorated and is now substantially worse than peer. Total asset turnover improved slightly but remains well below the industry norm.

3.

Debt: Track’s debt ratio improved slightly from 2021 but remains higher than the industry average. The times interest earned ratio is stable and, though it suggests a reasonable cushion for the firm, is still below the industry average.

4.

Profitability: Track’s gross, operating, and net profit margins improved slightly in 2022 but remain low by industry standards. ROA ticked up but is still only half the industry average.

ROE dipped below the industry average.

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Track Software, while showing improvement in most liquidity, debt, and profitability ratios, still compares unfavorably with its peers. The firm should also take steps to improve activity ratios, particularly inventory turnover and accounts-receivable collection.

5.

Stanley should find the cash to hire the software developer. Adding a new product would increase sales and lead to greater earnings for Track Software over the long term.

e.

Stanley should seek to maximize the value of Track Software, not earnings in any one period.

Accordingly, he should focus less on the initial negative impact of hiring the software developer and more on the potential for a significant long-term rise in sales/earnings.

f.

The investor should view a $5,000 annual payment as a perpetuity, with a present value equal to expected cash flows ($5,000) divided by required rate of return (10%), or $50,000.

g.

You should view the $20,200 annual free cash flow as a perpetuity with a present value equal to expected cash flows ($20,200) divided by required rate of return (10%) or $202,000.

Part Three Valuation of Securities Chapters in This Part Chapter 6

Interest Rates and Bond Valuation

Chapter 7

Stock Valuation

Integrative Case 3: Encore International

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Chapter 6 Interest Rates and Bond Valuation  Instructor’s Resources Chapter Overview This chapter introduces interest-rate and bond-market fundamentals, beginning with a simple supplyand-demand framework for understanding how the market determines an equilibrium interest rate. A brief section discusses the widespread current phenomenon of negative interest rate, and then the chapter explores the distinction between nominal and real interest rates and the role of expected inflation in linking the two. Risk premia are added to highlight components of the nominal return on a risky security, namely the (i) real risk-free rate, (ii) expected inflation rate, and (iii) risk premium on the security. Next, the discussion turns to the relationship between the nominal interest rate on a bond and its term to maturity—formally referred to as the term structure of interest rates and represented pictorially by the yield curve. The exposition notes the general upward slope of the yield curve—that is, that long-term interest rates tend to exceed short-term rates—and offers three explanations: (a) expectations about future short-term rates, (ii) general investor preference for short-term, liquid debt, and (iii) segmentation of short- and long-term debt markets. The focus then moves to bond-market institutions with a catalogue of the major types of issues along with their legal issues, risk characteristics, and indenture provisions. The role of rating agencies is also emphasized. The chapter concludes by presenting the basic model for bond valuation (with annual or semiannual coupons) as a special case of the general model for valuing assets (i.e., value is simply the present value of expected cash flows from the asset). Examples are provided of the impact of variation in coupon/principal payments, timing of coupon/principal payments, and required rates of return on the market price of a bond. The final topic is yield to maturity—explained as nothing more than the interest rate equating the present value of a bond’s remaining coupons and principal payments with its market price.

 Suggested Answers to Opener-in-Review Assume the bonds Aston Martin issued in 2019 pay interest semiannually. a.

How much cash would a bondholder receive every 6 months if the bonds have a coupon rate of 12%?

The bonds pay 12%  2 or 6% semiannually. Although not stated explicitly in the problem, par value of the typical corporate bond is $1,000, implying a semiannual interest payment is $60 or ($1,000 × 0.06).

b.

When Austin Martin issued the bonds, maturity was three years, and the required return was 11.75% per year. What was the market price? Did the bond sell at par, at a premium, or at a discount? Why? The market price of the bond (given 6 semiannual coupon payments of $60, a $1,000 principal payment at the end of the 6th period, and a semiannual discount rate of 11.75%  2 = 0.05875) is $1,006.17. The solution may be obtained in Excel using the present-value command with this syntax:

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=pv(market interest rate, # of periods, coupon payments, par value,0) or =pv(0.05875,6,60,1000,0) The bond sells at a premium because the current market rate (11.75%) is less than the coupon rate (12%). When the market rate falls below the coupon rate, the bond price rises until yield-tomaturity equals the market rate. c.

Given your answer to part b, what was the current yield of Aston Martin bonds when they were first issued? Suppose the bonds are now worth $1,075 each. What is the current yield on the bonds now?

Current yield = Bond’s annual interest payment  Current price As established in part b, the bond price at issuance was $1,006.17, and the annual interest is $120, so the current yield at issue was $120 ÷ $1,006.17 = 11.93%. If the bond now sells for $1,075, the current yield is $120 ÷ $1,075 = 11.16%.

 Answers to Review Questions 6-1

The real rate of interest measures the return on an investment, not in dollars, but in terms of how much the investment increases one’s purchasing power. The nominal rate of interest is the actual rate of interest charged by suppliers and paid by demanders of funds; it differs (approximately) from the real rate of interest by expected inflation. Specifically, let r* be the real rate of interest, r the nominal interest rate, and i the expected rate of inflation. Then, (1 + r) = (1 + r*)  (1 + i) → r = r* + i + (r*  i) r*  r – i (or r  r* + i) when the inflation rate and real rate are relatively low because (r*  i) is a very small number.

In words, demanders and suppliers of loanable funds care about the real interest rate –the cost of borrowing or return to saving expressed in terms of command of goods and services—but rates are negotiated and paid in nominal terms. To ensure the nominal interest rate delivers the desired real rate, demanders and suppliers add a premium equal to expected inflation.

6-2

The term structure of interest rates is the relationship between nominal rate of return and time to maturity for bonds with similar risk. The yield curve is a graphical representation of this relationship.

6-3

Under the expectations theory of the term structure, the slope of the yield curve for bonds with similar risk should reflect expectations about future short-term interest rates on those securities. a. Downward sloping: Investors expect short-term interest rates to fall. b. Upward sloping: Investors expect short-term interest rates to rise. c. Flat: Investors expect future short-term interest rates to roughly equal current short-term rates.

6-4

a. Under the expectations theory of the term structure of interest rates, the shape of the yield curve solely reflects investor expectations about future short-term interest rates. So, an upward sloping curve means investors expect short-term rates to rise.

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b. The liquidity-preference theory assumes investors prefer short- to long-term debt instruments because short-term debt is more liquid and less risky (i.e., suffer lower capital losses when interest rates rise). The general preference for short maturities means investors will demand a premium to hold longer-term debt instruments. This risk premium will produce an upward sloping yield curve even if investors expect no changes in interest rates. c. The market-segmentation theory assumes the market for short- and long-term debt instruments is distinct, with demand and supply determining the interest rate in each market.

Under the market-segmentation theory, long-term interest rates exceed short-term rates when demand for long-term instruments is stronger than for demand for short-term instruments (and/or supply of long-term debt is relatively weaker than demand for short-term debt).

6-5

Prior discussion noted the nominal rate of interest (r) approximately equals the real rate (r*) plus expected inflation (i)—ignoring risk. Broadening to encompass risky debt instruments: RF = r* + i, where RF is the risk-free rate, and rj  r*  i  RPj where RPj, is a risk premium on debt instrument j reflecting:

6-6

 Default risk: The risk the issuer will not pay contractual interest or principal as scheduled.  Interest rate risk: The risk interest rates will rise and cause the price of the debt instrument to fall; price volatility increases with instrument’s maturity.

Most corporate bonds are issued in $1,000 denominations with maturities of 10 to 30 years.

The stated interest rate represents the percentage of the bond’s par value to be paid annually, though most bonds pay interest semiannually. Most corporate bonds are also callable, meaning the issuer can—under conditions detailed in the indenture—repurchase outstanding bonds from their current holders. Mechanisms that protect bondholders are in the bond indenture and may include restrictive covenants, sinking fund provisions, and collateral provisions.

6-7

A bond indenture is a complex legal document specifying rights of bondholders and duties of the issuing firm. Indentures generally contain standard debt provisions and restrictive covenants. Standard debt provisions specify generally acceptable record-keeping and business practices for the issuer and typically do not burden a financially sound business. Restrictive covenants, in contrast, place specific operating and financial constraints on the issuer.

Violation of standard or restrictive provisions may give bondholders the right to demand immediate repayment. Violations could also trigger other adverse consequences for the issuing firm, such as a rating downgrade or renegotiation of the indenture.

6-8

Borrowing long term usually costs more than borrowing short term. On top of the base risk-free Treasury rate, the cost of long-term debt reflects term-to-maturity, offering size, and default risk.

Other things equal, the interest rate on long-term debt rises when (i) term-to-maturity is longer, (ii) offering size is smaller (flotation/administrative costs can be spread over more bonds), and (iii) default risk is higher.

6-9

A bond with a conversion feature gives holders the option of converting the bond into a certain number of shares of stock within a certain period of time. A call feature gives the issuer the opportunity to repurchase, or call, bonds at a stated price prior to maturity. Stock purchase warrants give bondholders the right to purchase a certain number of shares of common stock at a specified price.

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6-10 The current yield equals a bond’s annual interest payment divided by its current market price.

Bonds prices are quoted as a percentage of par value; for example, a quote of 98.621 means the bond price is 98.621 percent of par. Independent private agencies such as Moody’s, Fitch, and Standard & Poor’s rate bonds based on the likelihood interest and principal payments will be made. Ratings reflect detailed financial ratio and cash-flow analyses of the issuing firm.

Prospective bond holders and issuers value ratings as third-party certification—like a ―Yelp‖ rating in the market for goods and services. Bond ratings affect rates of return; other things equal, a higher rating implies lower default risk (and coupon rate).

6-11 Eurobonds are issued by international borrowers in countries with currencies other than that in which the bond is denominated—such as a dollar denominated bond sold by an American firm in Japan. Foreign bonds, in contrast, are issued by a foreign borrower in a host country’s capital market and denominated in the host currency—such as a yen-denominated bond issued in Japan by a French firm.

6-12 A financial manager should understand valuation because, on her firm’s behalf, she will (i) issue bonds and stocks and must, as a consequence, understand how investors value those securities and (ii) determine whether investment projects generate cash flows with present value exceeding cost.

6-13 The three inputs in asset valuation are (i) Cash flows—cash received from asset ownership, (ii) Timing—time period(s) when cash is received; and (iii) Required return—risk-adjusted interest rate used to discount cash flows (i.e., higher risk implies a discount rate).

6-14 The valuation process applies to assets providing cash flows of any size (constant, mixed stream, and lump sum) over any time period (intermittent, annual, semiannual, etc.).

6-15 The value of any asset is the present value of all cash flows expected from the asset; value depends on the cash flows, their timing, and required rate of return. Formally, CF3 CFn CF1 CF2 V0 =     1 2 3 n 1  r  1  r  1  r  1  r  where: V0  asset value at time zero CFt  cash flow expected in period t

r  required return (discount rate) n  time period

6-16 The basic valuation equation for a bond paying annual interest is B0

=

C

+ 1

1  r 

C

1  r 

+ 2

C

1  r 

+…+ 3

 M    n  1  r   where: V0  Value of bond at t = 0 C  Coupon payment each period n  periods to maturity

C

1  r 

+ n

 n C  =   +  n n 1  r   t 1 1  r   M

M = Par value (in dollars) r = Required rate of return on bond

For annual interest, n is the number of years and r the annual rate of return. For semiannual interest, n is the number of years × 2, and r is annual required rate of return  2.

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6-17 A bond sells at a discount when required return exceeds the coupon rate and at a premium when the coupon rate exceeds required return. A bond sells at par value when required return equals the coupon rate. Coupon is generally fixed, whereas required return fluctuates as market conditions change or as the issuer’s credit quality changes.

6-18 Interest rate risk is the chance that interest rates will change and thereby alter the required return and bond value. Other things equal, interest rate risk increases with maturity, meaning that long-term bond prices move more than short-term bond prices do when rates change.

Likewise, other things being equal, a bond’s interest rate risk is higher if its coupon rate is lower.

6-19 If a bond sells at a premium (discount) because the market’s required return is less than (greater than) the bond’s coupon rate, then over time the bond price will gradually fall (increase) and approach par value as the bond approaches its maturity date.

6-20 Yield-to-maturity (YTM) is the compound annual rate of return investors would earn by purchasing a bond at the current market price (which, after issue, does not necessary equal par value) and holding to maturity. YTM is the interest rate equating the present value of the bond’s remaining coupon and principal payments with its market price. YTM can be found using timevalue functions on a financial calculator—treating current market price (B0) as present value (PV), dollar coupon per period (C) as remaining annuity-type payments, principal (M) as lumpsum payment at maturity, and n as number of remaining periods, then solving for yield (r). In Excel, YTM can be found with the RATE function: =rate(remaining periods,coupon payments per period,-current market price,principal,0) Note: Bond price must be entered as a negative number. Also, the command produces interest rate per period and requires annualizing if coupon payments are more frequent than once per year.

6-21 The Excel present value function can be used to find the bond’s value. The present value is PV(rate,nper,pmt,fv,type). To solve this problem in Excel, input the given values as shown below, enter =B2*B3 in cell B4, and enter =PV(B5,B6,B4,B2,0) in cell B7. Completed Excel worksheets are available to students in MyFinanceLab.

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6-22 The Excel present value function can be used to find the bond’s value with different coupon payment frequencies. The present value is PV(rate,nper,pmt,fv,type). To solve for the bond’s value with semiannual coupon payments in Excel, input the given values as shown below in column B, enter =B2*B3 in cell B4, and enter =PV(B6/B5,B7*B5,B4/B5,B2,0) in cell B8. To solve for the bond’s value with monthly coupon payments in Excel, input the given values as shown below in column C, enter =C2*C3 in cell C4, and enter =PV(C6/C5,C7*C5,C4/C5,C2,0) in cell C8. Completed Excel worksheets are available to students in MyFinanceLab.

6-23 The Excel RATE function can be used to find the bond’s yield to maturity. The RATE function is =RATE(nper,pmt,pv,fv,type,guess). To solve this problem in Excel, input the given values as shown below, enter =B2*B3 in cell B4, and enter =RATE(B6*B5,B4/B5,B7,B2,0)*B5 in cell B8. Completed Excel worksheets are available to students in MyFinanceLab.

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 Suggested Answer to Focus on Practice Box: “I-Bonds Adjust for Inflation” What effect do you think the inflation-adjusted interest rate has on the price of an I-bond in comparison with similar bonds having no allowance for inflation?

I-bonds offer inflation insurance. Specifically, the I-bond interest rate includes (i) a fixed rate that remains constant over the life of the bond and (ii) an adjustable rate that equals the inflation rate (at a short lag). If inflation rises, the adjustable component of the I-bond rate rises to ensure the actual real return roughly equals the promised real return. Because I-bonds carry inflation protection, the U.S. Treasury can offer a slightly lower interest rate (and, therefore obtain, a slightly higher price) than on comparable Treasuries without such protection.

 Suggested Answer to Focus on Ethics Box: “Can Bond Ratings Be Trusted?”

What ethical issues could arise because companies or governments issuing debt (rather than investors) pay NRSROs to rate those instruments?

Rating agencies (National Recognized Statistical Ratings Organizations or NRSROs) want to keep issuers happy to secure future business. Knowing this, an issuer could apply pressure to ―inflate‖ the rating on a debt issue—that is, award a rating indicating lower default risk than justified by the issuer’s financial information.—to keep interest costs down.

Why do you think NRSROs inflated ratings for new complex MBSs but not traditional corporate bonds in the run-up to the Great Recession?

An NRSRO cannot ignore an issuer’s financial condition and award any rating it wishes.

Investors/firms value ratings because of the rater’s reputation for sound, independent analysis. In the run-up to the Great Recession, NRSROs were asked to rate new, complex mortgage-backed securities (MBSs). These securities had not been ―stress tested‖ by a credit cycle, so little hard evidence existed about how they might fare in a serious recession. Put another way, NRSROs had more latitude to accept an issuer’s rosy claims about minimal default risk. In contrast, NRSROs have rated traditional corporate debt for over a century—long enough to include many boom and bust cycles. Any attempt to inflate ratings on traditional corporate bonds –instruments with historical data to benchmark default risk—would seriously tarnishing the rater’s reputation.

 Suggested Answer to Focus on People/Planet/Profit Box: “How Do I Love Thee? Let Me Count the Yield” The average corporate bond offering in the United States raises about $1 billion. For a bond offering this size, how much interest would a company save each year by issuing bonds with the CBI certification if that reduced the coupon rate the company had to pay by 10 basis points?

Ten basis points is 0.1%. If the interest rate that the company has to pay on certified bonds is 0.1% lower, this saves interest costs of $1 billion × 0.001 = $1 million per year.

 Answers to Warm-Up Exercises © 2022 Pearson Education, Inc.

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E6-1

Finding the real rate of interest (LG 1)

Answer: If r* real interest rate, r the nominal interest rate (risk free), and i expected inflation, then: r*  r – i, or i  r – r* when the inflation rate is relatively low.

Expected inflation is approximately equal to 0.0123  0.008 = 0.0043 or 0.43%. More precisely, (1 + r) = (1 + r*)  (1 + i) → i = [(1 + r)  (1 + r*)] – 1 i = [(1.0123)  (1.008)] – 1 = 0.0042659 Note: Expected inflation is well under 1%, so the approximation is close.

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Yield curve (LG 1)

Answer: a.

b.

Consider two options for a 10-year investment, purchasing: (i) a 10-year bond paying 4.51% or (ii) a 5-year bond paying 3.7% and another 5-year bond in 5 years. Under the expectations hypothesis, these options should have the same expected return.

Assuming the two investing options offer the same return and denoting expected return on a 5-year bond in 5 years E(r): (1.0451)10 = (1.037)5 × (1+E(r))5 → (1.0451)10  (1.037)5 = (1+E(r))5 E(r) = 5 1.296238 – 1 = 0.053263 = 5.326% Simple, intuitive math yields a close approximation. Ignoring compounding, the 10year return on the 10-year, 4.51% bond is 45.1% (10 × 4.51%), and the 5-year return on the 5-year, 3.7% bond is 18.5% (3.7% × 5). So, the 10-year bond and two 5-year bonds will provide the same return if the second 5-year bond offers 26.6% (45.1%  18.5%) or 5.32% per year (26.6% ÷ 5 years)—a figure close to the one including compound interest.

c.

Using the logic from part (b), the return on purchasing a 3-year bond today must equal the return on purchasing a 2-year bond today and a 1-year bond in two years.

So, (1.0301)3 = (1.0268)2(1+E(r))1 → (1.0301)3  (1.0268)2 = (1+E(r))1 E(r) = 1.036732—1 = 3.673%

d.

Yield curves may have an upward slope for several reasons other than expectations of rising interest rates. According to the liquidity-preference theory, investors prefer short- to long-term debt because short-term debt is more liquid and less subject to capital losses when interest rates rise. Accordingly, long-term rates must contain a premium to overcome investor preference for the short term. In addition to the expectations and the liquidity preference theories, the market-segmentation theory holds that markets for short- and long-term debt instruments are distinct and relatively higher long-term rates indicate strong demand for and/or weak supply of long-term debt (relative to short-term debt).

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E6-3

Calculating inflation expectation (LG 1)

Answer: The inflation expectation for a specific maturity is approximately the difference between the nominal yield and real interest rate at that maturity (i.e., in the Table below we use Equation 6.1a): Maturity 3 months 6 months 2 years 3 years 5 years 10 years 30 years E6-4

Yield 1.41% 1.71 2.68 3.01 3.70 4.51 5.25

Real Interest Rate 0.80% 0.80 0.80 0.80 0.80 0.80 0.80

Inflation Expectation 0.61% 0.91 1.88 2.21 2.90 3.71 4.45

Real returns (LG 1)

Answer: A T-bill will yield a negative real return if its nominal return falls below the inflation rate.

More precisely, let r* be the real interest rate, r the nominal interest rate, and i the expected inflation rate, so (1 + r) = (1 + r*)  (1 + i) → r = [(1 + r*)  (1 + i)] – 1 Now, plugging in values for the real return (r* = 0%) and expected inflation (r = 3.3%) yields: r = [(1 + 0.00)  (1 + 0.033)] – 1 = 3.3% In words, if the nominal rate on T bills falls below 3.3%, the real return will be negative.

To earn a 2% real return, the nominal rate (r) must be: [(1 + 0.02)  (1 + 0.033)] – 1 = 5.37% E6-5

Calculating risk premium (LG 1)

Answer: Let rj be the nominal rate of interest on security j, RF the risk-free rate, and RPj the risk premium on security j; then rj = RF + RPj. Or alternatively, RPj= rj – RF. So: Security AAA BBB B E6-6

Nominal interest rate 5.12% 5.78 7.82

Risk premium 5.12%  4.51%  0.61% 5.78%  4.51%  1.27% 7.82%  4.51%  3.31%

The basic valuation model (LG 4)

Answer: Find the present value of the cash-flow stream for each asset by discounting expected cash flows with the required return: Asset 1: PV  $300  0.09  $3,333.33 Asset 2: PV =

$1,400

1.07 

1



$1,300

1.07 

2



$850

1.07 

3

= $1,308.41 + 1,135.47 + $693.85 = $3,137.73

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Calculating present value of a bond when required return exceeds the coupon rate (LG 4)

Answer: The value of a bond is the present value of its future cash flows, discounted at the required rate of return. Let C represent dollar coupon payments per period, M the dollar face value or principal to be returned on maturity, n the number of periods until maturity, and r the required rate of return.

Value of Bond (B0) =

C

1  r 

1



C

1  r 

2

 

C

1  r 

n



M

1  r 

n

Face value (or principal) of the bond is $20,000, interest is paid annually, and the coupon rate is 6%. This information implies a $1,200 coupon payment is made annually. Plugging this value and the other given information yields: Value of Bond (B0) =

$1,200

1.08 

1



$1,200

1.08 

2



$1,200

1. 08 

5



$20,000

1.08 

5

= $18,402.92

Bond value (also market price) is below face value because the coupon rate (6%) is less than the required rate of return. Bond value may also be obtained in Excel using the PV command and the following syntax: =pv(rate,nper,pmt,fv,type), where the bond’s coupon plays the role of payment in Excel syntax, and par value plays the role of future value].

Specifically: =PV(0.08,5,-1200,-20000) Note: Dollar coupon payment must be a negative number. Also, the final entry before the right parenthesis indicates whether payments occur at beginning (1) or end of period (0).

E6-8

Bond valuations using required rates of return (LG 5)

Answer: a. Specific student answers will vary but the following general relationships will hold:  When required rate of return exceeds coupon rate, bond sells at a discount.  When required rate of return equals coupon rate, bond sells at par value.  When required rate of return is below coupon rate, bond sells at a premium. b. Student answers will vary but should be consistent with their answers to part (a).

 Solutions to Problems P6-1

Interest rate fundamentals: The real rate of return (LG 1; Basic) Real rate of return (r*)  nominal interest rate (r) – expected inflation (i) = 1.5%  0.5%  1.0% More precisely: (1 + r) = (1 + r*)  (1 + i) → [(1 + r)  (1 + i) – 1] = r* r* = [(1.015)  (1.005)] – 1 = 0.00995 = 0.995%  1%

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Equilibrium rate of interest (LG 1; Intermediate) a,b and c.

The intersection of the demand and supply curves for funds determines the equilibrium rate of interest. Initial equilibrium is given by point O where r  4%. d. The change in tax law shifts the demand curve up and to the right, raising the equilibrium interest rate to 6% (i.e., the intersection of demand and supply after change in tax law).

P6-3

Real and nominal rates of interest (LG 1; Basic) a. The term (1 + r) represents the growth in money that an investment provides. b. The term (1 + r*) represents the growth in an investor’s purchasing power. c. The term (1 + i) represents the growth in prices in the economy.

Point N: New Equilibrium

P6-4

Personal finance problem: Real and nominal rates of interest (LG 1; Intermediate) a. $100 budget  $2.5 per pair of socks = 40 pair of socks. b. Nominal return on $100 invested at 9% for one year =$100  (1.09)1  $109. c. Price of a pair of socks in one year with a 5% inflation rate = $2.50  (1.05)1  $2.625.

Pointcan O: Initial Equilibrium d. Number of pairs Zane purchase in one year = $109  $2.625 = 41.524. In percentage terms, he can buy 3.81% more socks [($41.524  $40) – 1 = 0.0381).

e. Real rate of return is: (1 + r) = (1 + r*)  (1 + i) → [(1 + r)  (1 + i) – 1] = r* r* = [(1.09)  (1.05)] – 1 = 0.038095 = 3.81% where r is the nominal rate of return on risk-free Treasuries (9%), and i is expected inflation (5%). Conceptually, real return is the additional command over goods and services an investment offers. Here, Zane can purchase 3.81% more socks by investing in the Treasury and buying socks in one year (as opposed to buying them now). Notice 3.81% is close to real rate obtained with Equation 6.1a (9% - 5% = 4%).

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Yield curve (LG 1; Intermediate) a. b. The yield curve is slightly downward sloping, which under the expectations theory of the term structure means investors expect short–term interest rates to fall slightly in the coming years. In addition, a downward sloping yield curve usually signals a recession is likely, because interest rates typically fall during a recession (as demand for funds weakens).

P6-6

Nominal interest rates and yield curves (LG 1; Challenge) a. For individual security j, nominal rate of return rj equals real rate of interest (r*) plus expected inflation rate (i) plus risk premium on security j (RPj) or rj = r*+ i + RPj. Treasuries are risk free (RP = 0), and expected inflation will remain 2% per year, so the real return on each Treasury is: rj = r* + i + RPj → rj – 2.0 – 0.0 3-month bill: r*  5.0%  2%  3.0% 6.5% 2-year note: r*  6.0%  2%  4.0% 7.0% 5-year bond: r*  8.0%  2%  6.0%

10-year bond: r*  8.5%  2%  20-year bond: r*  9.0%  2%

b. If nominal rates of interest at every maturity fall 1.5%, but expected inflation remains at 2%, and Treasuries remain risk free, then real return at every maturity in part (a) will decline by 1.5%. c.

The yield curve for Treasuries is upward sloping, which, under the expectations theory of the term structure, means investors expect short-term rates to be higher in the future than they are today. d. Followers of the liquidity-preference theory attribute the upward-sloping shape of the curve to a strong preference by investors for short-term debt (because it is more liquid and less likely to suffer a large capital loss than long-term debt). Put another way, the equilibrium rate offered on longer term bonds must be high enough to overcome investor preference for short-term debt. e. Believers in market segmentation would argue the upward slope is attributable to greater demand (and/or weaker supply) for long-term debt.

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Nominal and real rates (LG 1; Challenge) a. The approximate real rate of interest is the nominal rate minus expected inflation: r* ≈ r – i ≈ 5% - 1% ≈ 4% If Tyra spends $200 at Dollar Barrel today where everything costs $1, she can purchase 200 items. If, however, she invests the $200, at year end she will have $210. In one year, inflation will cause Dollar Barrel prices to rise to $1.01 per item. So, Tyra will be able to purchase $210  1.01 = 207.92 items. By investing, her real purchasing power increases by 3.96% [(207.92  200) – 1]. The actual increase in Tyra’s purchasing power (3.96%) is roughly equal to the approximate real return from investing (4%). b. The approximate real rate of interest is the nominal rate minus expected inflation: r* ≈ r – i ≈ 20% - 10% ≈ 10% If Tyra spends $200 at Dollar Barrel today where everything costs $1, she can purchase 200 items. If, however, she invests the $200, at year end she will have $240. In one year, inflation will cause Dollar Barrel prices to rise to $1.10 per item. So, Tyra will be able to purchase $240  1.10 = 218.18 items. By investing, her real purchasing power increases by 9.09%. The actual increase in Tyra’s purchasing power (9.09%) is roughly the same as the approximate real return from investing (10%). But notice the approximation is much worse. In part (a), the error was 0.04%; here the error is 0.91%. In words, the approximation deteriorates (i.e., error rises) as expected inflation rises.

P6-8

Term structure of interest rates (LG 1; Intermediate) a. b. and c.

Five years ago, the yield curve was slightly upward sloping, suggesting (under the expectations theory) investors expected future short-term interest rates to be only slightly higher than current short-term rates. Two years ago, the yield curve had a sharp downward slope, suggesting investors expected a dramatic decline in short-term interest rates (perhaps due to a coming recession). Today, the yield curve is upward sloping, suggesting investors expect short-term rates to rise. d. Consider two 10-year investment options five years ago: (i) a 10-year bond offering 9.5% or (ii) a 5-year bond offering 9.3% and another 5-year bond in 5 years. Under the expectations theory of the term structure, the options should offer the same return.

Assuming the options offered the same return and denoting expected return on a 5-year bond 5 years ago E(r): (1.095)10 = (1.093)5 ( 1+E(r))5 → (1.095)10  (1.093)5 = (1+E(r))5 E(r) = 5 1.58869 – 1 = 0.097 = 9.7% Alternatively, return on 5-year bond in five years = [(10  9.5%) – (5  9.3%)]  5 = 9.7%.

P6-9

Term structure (LG 1; Basic) Consider two 2-year investment options: (i) a 2-year Treasury note offering 5.5% or (ii) a 1year Treasury bill offering 5% and another 1-year bond in 1 year. Under the expectations theory of the term structure, the options should offer the same return. Denoting expected return on a 1-year bill in one year E(r): © 2022 Pearson Education, Inc.

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(1.055)2 = (1.05)  (1+E(r)) → (1.055)2  (1.05) = (1+E(r)) E(r) = 1.06002 – 1 = 0.060 = 6.0% Alternatively, return on 1-year bill in one year = (2  5.5%) – 5% = 11% – 5% = 6%.

P6-10 Risk premiums (LG 1; Intermediate) The coupon rate (3.3%) on the Anheuser-Busch (AB) bond exceeds yield to maturity (2.82%, also the current market interest rate on bonds of equivalent risk), so the AB bond sells at a premium. The coupon rate on the Santander Holdings (SH) bond (3.571%) also exceeds yield to maturity (3.341%), so the SH bond also sells at a premium The bonds mature at the same time, so any difference in yield to maturity (YTM) likely reflects differences in perceived risk. The SH bond has the higher YTM, so it probably has the higher risk and lower rating.

As noted, the bonds mature at the same time, so comparing their current yields reveals nothing about the shape of the yield curve; the difference in yields is likely traceable to a difference in risk. Students might be tempted to say the yield curve sloped down when these bonds were issued because the SH bond has a shorter maturity, higher yield, and higher coupon rate than the AB bond. This is incorrect because the bonds (i) were issued at different times [so comparing their yields or coupons at issuance is not apples-to-apples] and (ii) do not have the same risk.

P6-11 Bond interest payments before and after taxes (LG 2; Intermediate)

Yearly interest = ($1,000  0.07) = $70.00 Total interest expense  $70.00 per bond  2,500 bonds  $175,000 Net after-tax interest (given a 21% tax rate) = $175,000 $138,250 P6-12 Bond types and characteristics (LG 3; Basic) The call premium is always the difference between par and the call price, so $60 here.

P6-13 Bond prices and yields (LG 3 LG 4; Intermediate)

Because the bond is selling at par, its required return must equal its coupon rate.

Because the bond pays a $40 coupon each year, its coupon rate is 4% ($40  $1,000) so its required return is also 4%.

Intuitively, you can think of this like a 4-year bond that sells at par but has an extra $40 payment immediately. In other words, price will be the par value plus one interest payment or $1,040. You could also calculate the price of the bond as follows, noting that the first payment is not discounted because it comes immediately after you purchase the bond: B0  $40 

$40 $40 $40 $40     $1,040 2 3 1.04  104  104 1.044

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P6-14 Bond prices and yields (LG 4; Basic)

0.97708  $1,000  $977.08 (0.057  $1,000)  $977.08  $57.00  $977.08  0.0583  5.83% The bond sells at a discount because its coupon rate (5.7%) is below the current market rate on bonds of equivalent risk (i.e., required rate of return of 6.034%).

The yield to maturity (YTM) is higher than the current yield because YTM includes $22.92 in price appreciation from now to maturity on May 15, 2027.

P6-15 Personal finance problem: Valuation fundamentals (LG 4; Basic) a. In years 1 – 4, $6,000 is paid on property taxes and maintenance, but $10,000 is saved on rent. Also, in year 4, the condo will sell for $125,000. So, the timeline is: Year 0

1

2

3

4

?

$4,000

$4,000

$4,000

$129,000

b. Value of Condo Purchase =

$4,000

1.04 

1



$4,000

1.04 

2



$4,000

1. 04 

3



$129,000

1.04 

4

= $121,370.10

The value of any asset is the present value of its cash flows. At a 4% discount rate, the present value of cash flows from the condo purchase is $121.370.10, so this is the maximum price to pay.

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P6-16

Risk and Return

Valuation of assets (LG 4; Basic) Asset

A

End of Year 1 2 3

B

1–

C

1 2 3 4 5

D

1–5 6 1 2

E

3 4 5 6

Cash Flows $ 3,00 0 $ 3,00 0 $ 3,00 0 $ 500 0 0 0 0 $45, 000 $ 1,50 0 8,50 0 $ 2,00 0 3,00 0 5,00 0 7,00 0 4,00 0 1,00 0

Discount Rate

Present Value

0.08

$7,731.29

0.05

$10,000.0 0

0.06

$33,626.6 2

0.04

$13.395.4 1

0.07

$17,429.5 2

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P6-17 Personal finance problem: Asset valuation and risk (LG 4; Intermediate) a. The value of an asset is the present value of its cash flows. So: Discount Rates n

CFn 14 5

$3, 00 0 15, 00 0 Present Value =

Low Risk (r =4%)

Avg. Risk (r = 7%)

$23,218.59

$20,856.43

High Risk (r =14%)

$16,531.67

b. To be sure of a good deal, Laura must take care to not understate risk. A conservative approach is to pay no more than $16,531.67—the present value of the asset assuming the highest possible risk (and discount rate). c. Higher risk means discounting cash flows at a higher rate, which reduces present value (asset value), holding cash flows and their timing constant.

P6-18 Bond types and valuation (LG 3 LG 5; Intermediate)

Because this bond doesn’t pay interest, its value is simply the discounted present value of its par value to be received in 10 years. The bond’s value is $1,000  (1 + 0.05)10 = $613.91.

Because this bond doesn’t pay interest, its value is simply the discounted present value of its par value to be received in 9 years. The bond’s value is $1,000  (1 + 0.05)9 = $644.61.

You earned 5% return during the year. The 5% return is found by calculating the change in the bond’s price over the year [($644.61  $613.91) – 1].

You must first calculate the bond’s price with 9 years to maturity and a 4% required rate of return. The bond’s value is $1,000  (1 + 0.04)9 = $702.59. So your return for the year is ($702.59  $613.91) – 1 = 0.144 or 14.4%. Because the market interest rate fell during the year, the bond’s price rose considerably so there is a larger percentage change in the bond’s price.

P6-19 Basic bond valuation (LG 5; Intermediate) a. The answer may be obtained in Excel using the PV function with r  10%, n  16, C (PMT)  $120, and M (FV)  $1,000. Solving for PV  $1,156.47. b. Complex Systems bonds sell at a premium, meaning required return has fallen since issuance. One possible reason is a decline in the risk-free rate because of changes in macroeconomic conditions; another is a decline in default risk because the firm’s financial condition has improved.

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c. The answer may be found in Excel using the PV function with r  12%, n  16, C (PMT)  $120, and M (FV)  $1,000. PV  $1,000. Alternatively, since required return equals the coupon rate, bond price will equal par value ($1,000). When required return falls below the coupon rate, traders race to buy the bond, causing price to rise until yield to maturity equals the required rate of return.

P6-20

Bond valuation: Annual interest (LG 5; Basic)

B

16

Required Return (r) 12 % 8%

C

8

9%

D

13

6%

0.06  $500  $30

$500

E

10

7%

0.07  $1,000  $70

$1,000

Bond A

Years to Maturity (n) 20

Coupon (C) 0.11  $1,000  $110 0.08  $1,000  $80 0.09  $100  $9

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Par Value (M) $1,000 $1,000 $100

Bond Value $925.

31 $1,00 0.00 $ 111.9 4 $ 420.9 6 $1,15 4.43

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P6-21 Bond value and changing required returns (LG 5; Intermediate) a.

Required Return (r) 11%

Years to Maturity (n) 12

15%

12

8%

12

Coupon (C) 0.11  $1,000  $110 0.11  $1,000  $110 0.11  $1,000  $110

Par Value (M) $1,000 $1,000 $1,000

Bond Value $1,00 0.00 $783.

18 $1,22 6.08

b. c. When the coupon rate exceeds required return, market value exceeds par value (i.e., the bond sells at a premium). When required return exceeds the coupon rate, par value exceeds market value (i.e., the bond sells at a discount). When required return equals the coupon rate, market value equals par. d. The required return on the bond could differ from the coupon rate because of changes in the: (i) risk-free rate (perhaps because of changes in macroeconomic conditions), or (ii) risk of the issuing firm.

P6-22 Interest rate risk and bond price changes (LG 5; Intermediate) a. Use Equation 6.5a to find the bond prices. The initial price of the annual coupon bond is  1 $1,000  $70    B0    1 10 10  0.07   1  0.07   1  0.07 

B0  $1,000  0.49165  $508.35  $491.65  $508.35  $1,000

The Excel PV(rate,nper,pmt,fv,type) function can also be used with =PV(0.07,10,70,1000,0). The new price of the annual coupon bond is  1 $1,000  $70    B0    1 10 10  0.08   1  0.08   1  0.08 

B0  $875 0.53681  $463.19  $469.71  $463.19  $932.90

The Excel PV(rate,nper,pmt,fv,type) function can also be used with =PV(0.08,10,70,1000,0). The percentage change in annual coupon bond price is ($932.90  $1,000) – 1 = -0.0671 or -6.71%.

The initial price of the zero coupon bond is  1 $1,000  $0     B0   1   10 10  0.07   1  0.07   1  0.07 

B0  $0  0.49165  $508.35  $0  $508.35  $508.35

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The Excel PV(rate,nper,pmt,fv,type) function can also be used with =PV(0.07,10,0,1000,0). The new price of the zero coupon bond is  1 $1,000  $0     B0   1   10 10  0.08   1  0.08   1  0.08 

B0  $0  0.53681  $463.19  $0  $463.19  $463.19

The Excel PV(rate,nper,pmt,fv,type) function can also be used with =PV(0.08,10,0,1000,0). The percentage change in annual coupon bond price is ($463.19  $508.35) – 1 = –0.0888 or –8.88%. b. Use Equation 6.5a to find the bond prices. The initial price of the annual coupon bond is  1 $1,000  $70    B0    1 20 20 0.07    1  0.07   1  0.07 

B0  $1,000  0.74158  $258.42  $741.58  $258.42  $1,000

The Excel PV(rate,nper,pmt,fv,type) function can also be used with =PV(0.07,20,70,1000,0). The new price of the annual coupon bond is:  1 $1,000  $70    B0    1 20 20  0.08   1  0.08   1  0.08 

B0  $875 0.78545  $214.55  $687.27  $214.55  $901.82

The Excel PV(rate,nper,pmt,fv,type) function can also be used with =PV(0.08,20,70,1000,0). The percentage change in annual coupon bond price is ($901.82  $1,000) – 1 = –0.0982 or –9.82%.

The initial price of the zero coupon bond is  1 $1,000  $0     B0   1   20 20  0.07   1  0.07   1  0.07 

B0  $0  0.74158  $258.42  $0  $258.42  $258.42

The Excel PV(rate,nper,pmt,fv,type) function can also be used with =PV(0.07,20,0,1000,0). The new price of the zero coupon bond is  1 $1,000  $0    B0    1 20 20  0.08   1  0.08  1  0.08 

B0  $0  0.78545  $214.55  $0  $214.55  $214.55

The Excel PV(rate,nper,pmt,fv,type) function can also be used with =PV(0.08,20,0,1000,0). The percentage change in annual coupon bond price is ($214.55  $258.42) – 1 = –0.1698 or –16.98%. c. Use Equation 6.5a to find the bond prices. The initial price of the annual coupon bond is

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 1 $1,000  $70    B0    1 10 10  0.07   1  0.07   1  0.07 

B0  $1,000  0.49165  $508.35  $491.65  $508.35  $1,000

The Excel PV(rate,nper,pmt,fv,type) function can also be used with =PV(0.07,10,70,1000,0). The new price of the annual coupon bond is  1 $1,000  $70    B0    1 10 10  0.06   1  0.06   1  0.06 

B0  $1,166.67  0.44161  $558.39  $515.21  $558.39  $1,073.60

The Excel PV(rate,nper,pmt,fv,type) function can also be used with =PV(0.06,10,70,1000,0). The percentage change in annual coupon bond price is ($1,073.60  $1,000) – 1 = 0.0736 or 7.36%.

The initial price of the zero coupon bond is  1 $1,000  $0     B0   1   10 10  0.07   1  0.07   1  0.07 

B0  $0  0.49165  $508.35  $0  $508.35  $508.35

The Excel PV(rate,nper,pmt,fv,type) function can also be used with =PV(0.07,10,0,1000,0). The new price of the zero coupon bond is  1 $1,000  $0    B0    1 10 10  0.06   1  0.06   1  0.06 

B0  $0  0.44161  $558.39  $0  $558.39  $558.39

The Excel PV(rate,nper,pmt,fv,type) function can also be used with =PV(0.06,10,0,1000,0). The percentage change in annual coupon bond price is ($558.39  $508.35) – 1 = 0.0984 or 9.84%. d. Use Equation 6.5a to find the bond prices. The initial price of the annual coupon bond is  1 $1,000  $70    B0    1 20 20  0.07   1  0.07   1  0.07 

B0  $1,000  0.74158  $258.42  $741.58  $258.42  $1,000

The Excel PV(rate,nper,pmt,fv,type) function can also be used with =PV(0.07,20,70,1000,0). The new price of the annual coupon bond is  1 $1,000  $70     B0   1   20 20  0.06   1  0.06   1  0.06 

B0  $1,166.67  0.68820  $311.80  $802.89  $311.80  $1,114.70

The Excel PV(rate,nper,pmt,fv,type) function can also be used with =PV(0.06,20,70,1000,0). The percentage change in annual coupon bond price is ($1,114.70  $1,000) – 1 = 0.1147 or 11.47%. © 2022 Pearson Education, Inc.

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The initial price of the zero coupon bond is  1 $1,000  $0     B0   1   20 20  0.07   1  0.07   1  0.07 

B0  $0  0.74158  $258.42  $0  $258.42  $258.42

The Excel PV(rate,nper,pmt,fv,type) function can also be used with =PV(0.07,20,0,1000,0). The new price of the zero coupon bond is  1 $1,000  $0     B0   1   20 20  0.06   1  0.06   1  0.06 

B0  $0  0.68820  $311.80  $0  $311.80  $311.80

The Excel PV(rate,nper,pmt,fv,type) function can also be used with =PV(0.06,20,0,1000,0). The percentage change in annual coupon bond price is ($311.80  $258.42) – 1 = 0.2066 or 20.66%. e. The zero coupon bond’s price is more sensitive to interest rate risk because it has a lower coupon rate. However, regardless of the level of interest rate risk for given bond, the magnitude of the price change that results from an increase in underlying interest rates is less than the price change that results from a decrease in interest rates.

P6-23 Bond value and time: Constant required returns (LG 5; Intermediate) a. Using the PV function in Excel with the syntax: =pv(required return, years to maturity, coupon, par value) Required Return (r) 14 % 14 % 14 % 14 % 14 % 14 %

Years to Maturity (n) 1

ParValue (M) $1,000

3

Coupon (C) 0.12  $1,000  $120 $120

6

$120

$1,000

9

$120

$1,000

12

$120

$1,000

15

$120

$1,000

$1,000

Bond Value $982.

46 $953.

57 $922.

23 $901.

07 $886.

79 $877.

16

b. c. As can be seen in part (b), other things equal, when required return differs from the coupon rate and remains constant to maturity, bond value will approach par value as time to maturity declines.

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P6-24 Personal finance problem: Bond value and interest rate risk (LG 5; Basic) a. Bonds A and B have the same 5% annual coupon and 20-year maturity, but their required rates of return are different. Because bond A is selling at a premium, its required rate of return is less than the 5% coupon rate, and because bond B is selling at a discount, its required rate of return is greater than the 5% coupon rate. Bond A has more interest rate risk because its required rate of return is lower than that of bond B. b. Bonds M and N have the same 15 years to maturity and 8% required rate of return, but their coupon payments are different. Because bond N has a zero coupon payment, it has more interest rate risk than bond M, which has an annual coupon payment. c. Bonds Y and Z have the same 9% annual coupon and 8% required rate of return, but their times to maturity are different. Because bond Y has a longer remaining time to maturity, it has more interest rate risk than bond Z, which has a shorter remaining time to maturity.

P6-25 Bond value and time: Changing required returns (LG 5; Challenge) a. and b.

Bond

A

Years to Maturity (n) 5

Required Return (r) 8% 11 % 14 % 8%

5 5 15

B

Coupon (C) 0.11  $1,000  $110 $110

11 % 14 %

15 15

Par Value (M) $1,000 $1,000

$110

$1,000

$110

$1,000

$110

$1,000

$110

$1,000

Bond Value $1,11 9.78 $1,00 0.00 $897.

01 $1,25 6.78 $1,00 0.00 $815.

73

c.

Value Required Return 8% 11% 14%

Bond A

Bond B

$1,119.78 1,000.00 897.01

$1,256.75 1,000.00 815.73

The longer the time to maturity, the more responsive bond price is to changes in required return. d. Lynn could minimize interest-rate risk by choosing Bond A with the shorter maturity.

The price of bond A will move less with any given change in required return than the price of bond B.

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P6-26 Interest rate risk and bond price changes (LG 5; Intermediate) a. Use Equation 6.5a to find the initial price of both bonds. The initial price of the high yield bond is  1 $1, 000  $110    B0    1 9  0.14   1  0.14   1  0.14 9

B0  $785.71 0.69249  $307.51  $544.10  $307.51  $851.61

The Excel PV(rate,nper,pmt,fv,type) function can also be used with =PV(0.14,9,110,1000,0). The initial price of the high yield bond is  1 $1, 000  $65    B0    1 22 0.09    1  0.09   1  0.09 22

B0  $722.22 0.84982  $150.18  $613.76  $150.18  $763.94

The Excel PV(rate,nper,pmt,fv,type) function can also be used with =PV(0.09,22,65,1000,0). b. First, use Equation 6.5a to find the new price of both bonds. The new price of the high yield bond is  1 $1, 000  $110    B0    1 9 0.16    1  0.16   1  0.16 9

B0  $687.50 0.73705  $262.95  $506.72  $262.95  $769.67

The Excel PV(rate,nper,pmt,fv,type) function can also be used with =PV(0.16,9,110,1000,0). The new price of the high yield bond is  1 $1, 000  $65    B0    1 22  0.11   1  0.11  1  0.1122

B0  $590.910.89933  $100.67  $531.42  $100.67  $632.09

The Excel PV(rate,nper,pmt,fv,type) function can also be used with =PV(0.11,22,65,1000,0).

Now calculate the percentage change in price for both bonds. For the high yield bond, it’s ($769.67  $851.61) – 1 = –0.096 or –9.6%, and for the investment grade bond, it’s ($632.09  $763.94) – 1 = –0.173 or –17.3%. The investment grade bond has more price sensitivity to interest rate risk because its percentage price decrease was greater. c. First, use Equation 6.5a to find the new price of both bonds. The new price of the high yield bond is  1 $1, 000  $110    B0    1 9 0.12    1  0.12   1  0.12 9

B0  $916.67  0.63939  $360.61  $586.11  $360.61  $946.72

The Excel PV(rate,nper,pmt,fv,type) function can also be used with =PV(0.12,9,110,1000,0). The new price of the high yield bond is © 2022 Pearson Education, Inc.

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 1 $1, 000  $65    B0    1 22  0.07   1  0.07   1  0.07 22

B0  $928.57 0.77429   $225.71  $718.98  $225.71  $944.69

The Excel PV(rate,nper,pmt,fv,type) function can also be used with =PV(0.07,22,65,1000,0).

Now calculate the percentage change in price for both bonds. For the high yield bond, it’s ($946.72  $851.61) – 1 = 0.112 or 11.2%, and for the investment grade bond, it’s ($944.69  $763.94) – 1 = 0.237 or 23.7%. The investment grade bond has more price sensitivity to interest rate risk because its percentage price increase was greater. d. The investment grade bond’s price is more sensitive to interest rate risk because it has a longer time to maturity, lower coupon rate, and lower initial YTM. However, regardless of the level of interest rate risk for given bond, the magnitude of the price change that results from an increase in underlying interest rates is less than the price change that results from a decrease in interest rates. e. No, because even though both bonds would have the same maturity and coupon rate, the investment grade bond’s cash flows are less risky so it has a lower initial required rate of return, which will cause its price to be more sensitive to changes in interest rates.

P6-27 Bond value, interest rate changes, and return (LG 5; Intermediate) a. Use Equation 6.5a to find the bond’s purchase price.  1 $1, 000  $60    B0    1 6  0.07   1  0.07   1  0.07 6

B0  $857.14 0.33366  $666.34  $285.99  $666.34  $952.33

The Excel PV(rate,nper,pmt,fv,type) function can also be used with =PV(0.07,6,60,1000,0). b. Use Equation 6.5a to find the bond’s selling price.  1 $1, 000  $60    B0    1 5 0.07    1  0.07   1  0.07 5

B0  $857.14 0.28701  $712.99  $246.01  $712.99  $959

The Excel PV(rate,nper,pmt,fv,type) function can also be used with =PV(0.07,5,60,1000,0). c. The idea here is for the student to calculate the total return, which is formally introduced in Chapter 8. The total return can be found by summing the investment’s cash flows and dividing by the investment cost: Total Return  ($959  $60  $952.33)  $952.33  0.07 or 7.0%

d. First, use Equation 6.5a to find the bond’s selling price:  1 $1, 000  $60    B0    1 5  0.06   1  0.06   1  0.06 5

B0  $1,000  0.25274  $747.26  $252.74  $747.26  $1,000

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The Excel PV(rate,nper,pmt,fv,type) function can also be used with =PV(0.06,5,60,1000,0). Next calculate the total return: Total Return  ($1,000  $60  $952.33)  $952.33  0.113 or 11.3%

P6-28 Yield to maturity (LG 6; Basic) Bond A will sell at a discount to par; Bond B will sell at par value; Bond C will sell at a premium to par; Bond D will sell at a discount to par, Bond E will sell at a premium to par.

P6-29 Yield to maturity (LG 6; Intermediate) a. Yield to maturity (YTM) may be found in Excel using the RATE function with the following syntax: =rate(n,C,V0,M) = rate(15,60,$867.59,1000) = 7.50% where n is years to maturity, C the coupon payment, V0 the bond price (entered as a negative number), and M the par value. b. Note in part (a), the bond sells at a discount ($867.59) from par ($1,000), and the YTM (7.5%) exceeds the coupon rate (6%). The required return has risen since the bond was issued. For this bond to offer the same return available on bonds of similar risk now, the price must full until the YTM equals the current required return.

Similarly, if the bond sells at a premium, the YTM must be less than the coupon rate. The required return has fallen since the bond was issued, so bond traders raced to get the attractive coupon rate, pushing up the bond price until the YTM equaled the current required return.

P6-30 Yield to maturity (LG 6; Intermediate) a. In Excel, the RATE function will generate a bond’s yield to maturity (YTM). For example, for bond A in the table below, the proper syntax is =rate(periods, payment,present value, future value) where periods (n) is number of periods to maturity, payment (C) is coupon payment, present value is bond price (V0), and future value is principal or par value (M). Note: Values for coupon and principal must be entered as negative numbers. Specifically, for bond A =rate(8,90,820,1000) which gives a rate of 12.71% b. If YTM exceeds coupon rate, the bond sells at a discount, and if the coupon rate exceeds YTM, the bond sells at a premium. When YTM equals the coupon rate, the bond sells at par value.

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P6-31

Personal finance problem: Bond valuation and yield to maturity (LG 2, LG 5, and LG 6; Challenge) a. Value of the Crabbe Waste bond may be found in Excel using the PV function and n  5, YTM = r  7.5%, C  0.06324  $1,000  $63.24, and M  $1,000: =pv(0.075,5, 63.24,1000) = $952.42 Value of the Malfoy bond may be found in Excel using the PV function and n  5, YTM = r  7.5%, C  0.088  $1,000  $88.00, and M  $1,000: =pv(0.075,5, 63.24,1000) = $1,052.60 b. The number of Crabbe Waste bonds  $20,000  $952.42  21, and the number of Malfoy bonds  $20,000  $1052.60  19. c. Annual interest income on Crabbe Waste bonds  21 bonds$63.24 per year  $1,328.04, and annual interest income on Malfoy bonds  19 bonds$88  $1,672.06. d. By purchasing the Crabbe Waste bonds, Mark will receive $1,328.04 in interest at the end of years 1-5 and $21,000 (21 bonds$1,000) in principal at the end of year 5. The future value of reinvested interest may be found using the FV function in Excel with the following syntax: =fv(r,n,) = rate(0.10,5,1328.04) = $8,107.82 where r is the required rate of return, n is years interest will be earned, and CF is total interest earned at the end of each year. Adding the $21,000 in principal brings the total to $29,107.82.

By purchasing the Malfoy bonds, Mark will receive $1,672 in interest at the end of years 1-5 and $19,000 (19 bonds$1,000) in principal at the end of year 5. The future value of reinvested interest may be found using the FV function in Excel with the following syntax: =fv(r,n,) = rate(0.10,5,1672) = $10,207.73 Adding the $19,000 in principal brings the total to $29,207.73. e. The opportunity to reinvest coupon payments at 10% is attractive because both bonds offer yield to maturities of only 7.5%. Although both bonds offer the same YTM, Malfoy has the higher coupon rate. So buying Malfoy bonds would generate $88 per bond each year to reinvest at 10% while buying Crabbe Waste bonds would produce only $63.24.

Malfoy is, therefore, the better investment.

6-32

Bond valuation: Semiannual interest (LG 6; Intermediate) To adjust the bond-valuation framework for semiannual interest, let n be the number of semiannual periods (2  number of years = 12), r the semiannual interest rate (annual interest rate  2 = 7%)C the semiannual coupon payment (annual coupon payment  2 = $50), and M the par value ($1,000). Using the PV function in Excel: =pv(r,n,C,M) = pv(0.07,12,50,1000) = $841.15

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P6-33 Bond valuation: Semiannual interest (LG 6; Challenge)

P6-34 Bond valuation: Quarterly interest (LG 6; Challenge) To adjust the bond-valuation framework for quarterly interest, let n be the number of quarterly periods (4  number of years = 40), r the quarterly interest rate (annual interest rate  4 = 3%)C the quarterly coupon payment (annual coupon payment  4 = $1250), and M the par value ($5,000). Using the PV function in Excel: =pv(r,n,C,M) = pv(0.03,40,125,5000) = $4,422.13 P6-35 ETHICS PROBLEM (LG 1; Intermediate) This is a good question for class discussion. On the one hand, bundling ratings with other services could reduce welfare by: (i) giving ratings agencies some market power and (ii) tempting agencies to ―tweak‖ ratings based on issuer willingness to buy other services. The counterargument recognizes the information ratings agencies produce about the quality of private and public debt instruments improves financial-market efficiency. Put another way, a better informed market means more accurate prices/interest rates which, in turn, implies scarce loanable funds are more likely to find their way to the highest valued uses. Bundling could also allow ratings agencies to earn higher profits, which in turn, would spur internal research on ways to award more informative ratings at lower cost. Finally, provided debt instruments have been ―stress tested‖ by a credit cycle, ratings agencies have a strong incentive to award the correct rating—one justified by the issuer’s financial condition— because of the reputational cost for fudging to sell add-on services.

 Case: “Evaluating Annie’s Proposed Investment in Atilier Industries Bonds” Case studies are available on www.pearson.com/mylab/finance. a.

Annie should convert the bonds. The value of the stock if the bond is converted is 50 shares$30 per share  $1,500. If the bond is called, Annie would receive only $1,080.

b.

Current value of bond under different required returns:

c.

Under all three required returns for both annual and semiannual interest payments, the relationship between required return (relative to coupon rate) and bond price (relative to par value) is the same.

When required return is above (below) the coupon rate, the bond sells at a discount (premium).

When required return equals the coupon rate, the bond sells at par. With semiannual payments, the premium and discount are slightly larger because interest and principal payments compound more often. d.

If expected inflation increases by 1%, required return will increase from 8% to 9%, and bond price will drop to $901.77 (i.e., with n  25, r  9%, C  $80, and M  $1,000, present value  $901.77). This amount is the maximum Annie should pay for the bond after the increase in expected inflation. © 2022 Pearson Education, Inc.

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e.

If the ratings downgrade raises expected return from 8% to 8.75%, bond price will fall to $924.81 (i.e., with n  25, r  9%, C  $80, and M  $1,000, present value  $924.81). This amount is the maximum Annie should pay for the bond after the ratings downgrade.

f.

In three years, the bond will be worth $1,110.61 (i.e., with n  22, r  7%, C  $80, and M  $1,000, present value  price = $1,110.61)—the present value of remaining interest and principal payments. If Annie purchased the bond at par and sells it at the new price, her capital gain would be $110.61.

g.

In ten years, the bond will be worth $1,091.08 (i.e., with n  15, r  7%, C  $80, and M  $1,000, price  $1,091.08)—the present value of remaining interest and principal payments. If Annie purchased the bond at par and sells it at the new price, her capital gain would be $91.08.

Notice bond price is more sensitive to changes in interest rates for the longer time to maturity (22 years) than the shorter (15 years). Other things equal, maturity (interest-rate) risk decreases as the bond approaches maturity.

h.

Yield to maturity, given n  25, r  7%, C  $80, and a bond price (V0)  $983.80, is 8.154%.

Current yield = C V0 = 8.132%.

Annie should probably not invest in the Atilier bond because:  The potential for a rating downgrade means significant risk of capital loss from a rise in the default premium.  The threat of higher inflation means, coupled with the bond’s long time to maturity, means a significant risk of capital loss from a higher base (risk free) interest rate.

i.

 Spreadsheet Exercise Answers to Chapter 6’s CSM Corporation spreadsheet problem are available on www.pearson.com/mylab/finance.

 Group Exercise Group exercises are available on www.pearson.com/mylab/finance.

This chapter’s exercise focuses on debt. Each group will conduct Internet research to identify debt recently issued by its shadow firm—specifically the issue’s rating and interest rate. Then, armed with this information, each group will compare that interest rate with the rate on Treasuries with similar maturities to infer the risk premium on the shadow firm’s debt. An important lesson from this exercise is the lack of transparency in the bond market, particularly compared with the stock market.

Updated information is not as easily compared across multiple websites, and details are often sketchy.

(Because a recent debt issue is needed, groups may need to look at the recent SEC filings of their shadow firm.) Students may find their shadow firms have multiple recent debt offerings, and these offerings have different ratings because of different covenants. In the final part of the exercise, each group will explore a potential capital investment for its fictitious firm. The interest rate used will reflect research on the shadow firm, but other project details are entirely up to the group. Encourage creativity, as students will live with their fictitious firm another two months.

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Chapter 7 Stock Valuation  Instructor’s Resources Chapter Overview This chapter focuses on equity—distinctions between equity and debt, different forms of equity, and approaches to valuing equity instruments. The basic model for valuing equity is presented as an example of the asset-valuation framework introduced in Chapter 5 and applied to bonds in Chapter 6.

Specifically, the value of a share or common of preferred stock is the present value of expected future cash flows from that share, where the cash flows here are dividends, or in some cases, free cash flows. When capital markets are efficient, the stock price should equal the present value of expected dividends, and news about changes in risk or expected cash flows will be priced immediately. The discussion then expands the common-stock valuation framework to accommodate different assumptions about expected dividend growth. Other approaches to equity-valuation—ranging from variations on dividend-discounting like the free-cash-flow model to models based on market benchmarks like price/earnings multiples—are also compared and contrasted with the expecteddividend model. The chapter ends with discussion of interrelationships among financial decisions, expected return, risk, and firm value.

 Suggested Answer to Opener-in-Review In the chapter opener you learned that the stock of Darden Restaurants fell from $120.68 to $39 in just 22 days. Prior to the coronavirus outbreak, Darden was on pace to pay a dividend in 2020 of $3.52 per share. Given that dividend, and assuming that prior to the outbreak investors required a 9% return on Darden stock, how fast would it appear that investors expected dividends to grow in the very long run? (Hint: According to the constant growth dividend model, what growth rate would justify a $120.68 stock price if the next dividend is $3.52 and the required return is 9%?) The virus outbreak probably decreased the dividend growth rate that investors expected and increased their required return at the same time. Suppose the growth rate fell 1% (relative to your answer to the previous question), and the required return rose from 9% to 10%. What would the new stock price be under those assumptions? How did that compare to Darden’s stock price at its low point? What lessons do you learn by comparing the model’s estimate to the actual market price ?

The assumed growth rate is found by solving the constant-growth dividend model, Equation 7.4, for g: g = r – (D1  Po) = 0.09 – ($3.52  $120.68) = 0.060831 or 6.083%. The new stock price is found using Equation 7.4: P0 = D1  (r – g) = $3.52  (0.10 – 0.05083) = $71.59. The actual low for the stock price was $39, so the increase in the required return and/or the decline in the growth rate must have been even larger. Of course, a more realistic analysis here could take into account that, in the short run, dividends might actually decline before eventually going back to a new steady state growth rate. Modeling a scenario like that means using the variable growth rate model.

 Answers to Review Questions 7-1

Equity capital is permanent capital representing ownership, while debt capital is a loan that must be repaid. Holders of equity capital receive a claim on firm income and assets subordinate to creditor claims—that is, debt holders must receive all interest and principal owed prior to © 2022 Pearson Education, Inc.

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distributions of firm income or assets to equity holders. Equity capital is perpetual, while debt has a specified maturity date. To investors, interest on debt is currently taxed as ordinary income, but dividends and capital gains on common stock are taxed but at a lower rate. To the corporation, interest on debt is tax deductible while dividends are not.

7-2

A corporation’s owners are the common stockholders. As residual claimants, these stockholders are not guaranteed a return, only what is left after other claims on firm income and assets have been satisfied. Any funds invested are at risk; the only guarantee is that losses are capped at the purchase price of the common stock. Given the significant uncertainty about earnings, common stockholders expect relatively high returns in the form of dividends and capital gains.

7-3

Rights offerings are financial instruments that allow existing stockholders to purchase additional shares of new stock issues below the market price, in direct proportion to the number of shares they own. Rights offerings protect current shareholders against dilution of their voting power.

7-4

Authorized shares are the maximum number of shares a firm can sell without approval from existing shareholders; this limit is stated in the corporate charter. Authorized shares sold to and held by the public are called outstanding shares. Shares repurchased by the firm from the public are classified as treasury stock; this stock is not considered outstanding because it is not held by the public. Issued shares are shares of common stock that have been put into circulation and include both outstanding shares and treasury stock.

7-5

Issuing stock outside their domestic markets can benefit corporations by broadening the investor base and facilitating integration into the local economy. Specifically, a local stock listing increases community press coverage, thereby raising awareness about the firm. Locally traded stock also facilitates acquisitions. American depository shares (ADSs) are dollardenominated receipts for stocks of foreign companies held by U.S. financial institutions overseas. American depository receipts (ADRs) are securities that permit U.S. investors to hold shares of non-U.S. companies and trade them in U.S. markets. ADRs are issued in dollars and subject to U.S. securities laws; they offer U.S. investors an opportunity to diversify internationally.

7-6

Preferred stockholders have a fixed claim on firm income and assets behind creditors but ahead of common stockholders.

7-7

Cumulative preferred stock gives the holder the right to receive any dividends in arrears prior to dividend payment to common stockholders. A call feature allows the issuer to retire outstanding preferred stock within a certain time period at a pre-specified price. The call price is normally set at or above the initial issuance price but may decrease according to a predefined schedule. Firms use the call feature to escape the fixed-payment commitment of preferred stock.

7-8

According to the efficient market hypothesis (EMH): a. Securities prices are in equilibrium (fairly priced with expected returns equal to required returns); b. Securities prices reflect all public information and react quickly to new information; so c. Investors should not waste time searching for mispriced (over or undervalued) securities.

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The EMH is generally accepted as holding for securities traded on major exchanges and as framework for thinking about security pricing. But proponents of behavioral finance, or behaviorists, have challenged the EMH, arguing the key underlying assumption—investor rationality—is false. They point to a growing body of research showing markets can be driven by reluctance to recognize losses, desire to follow the herd, tendency to compartmentalize investments, and overweighting of recent performance.

7-9

a.

The zero growth model of common-stock valuation assumes constant dividends through time, so a stock is valued as a perpetuity with today’s value (price), P0, depending on the perpetual dividend, D1 and required return r as follows:  1 1 D P0  D1    D1   1 t r r t 1 1  r 

b. The constant growth model assumes dividends will grow at a constant rate, g, from D1.

Again, the required return is denoted by r: D P0  1 rg c. The variable growth model assumes dividends grow at a variable rate. The stock with a single change in the growth rate is valued as the present value of dividends in during the initial growth phase plus the present value of the price of stock at the end of the initial growth phase. Specifically, n

D0  1  g1 

t 1

1  r 

P0  

t

t

 1 D    n 1  n  1  r  r  g 2 

Present value of dividends Present during initial value of growth period

dividends during initial growth period

Present value of price of stock at Present end of initial value of stock growth period

price at end of initial growth period

where Dn is the expected dividend in year n, g1 is the dividend growth rate in the initial period, g2, is the dividend growth rate in the second period, and r is the required rate of return.

7-10 The free-cash-flow valuation model discounts future free cash flows rather than expected dividends. Because this discounted value represents total firm value, the market value of total debt and preferred stock must be subtracted to obtain the value of the firm’s common stock.

Dividing the value of common stock by outstanding shares gives the stock price. The freecash-flow model differs from the dividend-valuation model in two ways (i) total firm cash flows are discounted, not just dividends, and (ii) the discount rate is the firm’s cost of capital, not the required return on stock. This approach is appealing when valuing startups, firms with no dividend history, or an operating unit or division of a larger public company.

7-11

a. Book value per share is the hypothetical amount common shareholders would receive if firm assets were sold at book (accounting) value, liabilities (including preferred stock) were paid off at book value, and the remainder divided by shares of common stock outstanding. b. Liquidation value is the amount each common stockholder would expect to receive if firm assets were actually sold, creditors and preferred stockholders were actually paid, and any remainder divided among the common stockholders. Here, market rather than book values of assets and liabilities are used.

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c. Under the price-earnings-multiples approach, share value is estimated by multiplying expected earnings per share by the average price/earnings ratio for the industry. Of the three approaches to valuation, the price/earnings multiples approach is considered the best because it considers expected earnings.

7-12 A useful way to think about the impact of financial decisions on the firm is with the constantgrowth stock-valuation model: P0 = D1  (r – g). Actions of financial managers affect the stock price (and hence firm value) through their impact on expected dividends (either the next expected, D1, or the expected growth of dividends, g) or risk (which shows up in required return). Any action that increases expected dividends will boost the stock price, and any action that increases risk will depress the stock price.

7-13 A useful way to analyze the impact of events on stock price is to start with the constant-growth stock-valuation model and assign hypothetical initial values. Accordingly, let the next expected dividend (D1) be $5, expected rate of dividend growth (g) be 3%, and required rate of return (r) be 8%: P0 = D1  (r – g) = $5  (0.08 – 0.03) = $100 a. Now, required return (r) = risk-free rate (RF) + risk premium (RP). So, an increase in RP means an increase in r. Suppose r rises from 8% to 9%, but expected dividends (D1 and g) do not change: $5  (0.09 – 0.03) = $83.33. In words, stock price will fall from $100 to $83.33. b. Suppose r falls from 8% to 7%, but nothing else changes: $5  (0.07 – 0.03) = $125.00.

Stock price will rise from $100 to $125. c. Suppose the dividend expected next year (D1) rises from $5 to $6, but nothing else changes: $6  (0.08 – 0.03) = $120.00. Stock price will rise from $100 to $120. d. Suppose expected dividend growth (g) rises from 3% to 4%, but nothing else changes: $5  (0.08 – 0.04) = $125.00. Stock price will rise from $100 to $125.

 Suggested Answer to Focus on People/Planet/Profit Box: “Shrinking and Growing at the Same Time” Why do you think that profit margins are highest in the industries that have become very concentrated?

One possibility is that industries (product markets) are becoming less competitive. In a perfectly competitive market, price equals marginal cost and pure economic profits (profits above the cost of capital) are zero. As markets become less competitive, profits can rise above marginal cost.

Another possibility is that larger firms could be more efficient than smaller ones, benefitting from economies of scale and/or scope, which could make them more profitable.

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 Suggested Answer to Focus on Practice Box: “Understanding Human Behavior Helps Us Understand Investor Behavior” Theories of behavioral finance can apply to other areas of human behavior as well as investing. Think of a situation in which you may have demonstrated one of these behaviors. Share your situation with a classmate.

Student answers will vary. Examples for discussion: (i) regret theory may hold true for social situations in which a person makes a mistake and subsequently focuses on avoiding embarrassment at all costs; (ii) fear of regret can sometimes be rationalized away with ―everyone else is doing it‖ (herding theory), thereby explaining why some people do silly things at parties; and (iii) students may react to grades as investors react to news, placing more importance on recent events without recognizing the overall trend (anchoring).

 Suggested Answer to Focus on Ethics Box: “Index Funds and Corporate Governance” If you were the CEO of a publicly traded company, would you want a large bloc of your shares held by index funds? Why or why not?

A CEO’s chief responsibility is to take all actions that increase shareholder wealth. That said, some CEOs look for opportunities to pursue personal interests at the expense of shareholders. At first, it might seem an unethical CEO would like to see the bulk of her firm’s shares in an index fund because, to the extent such funds are poor monitors, she will have some room to use firm resources to advance her personal agenda. But recent evidence suggests firms largely owned by index funds have excellent corporate governance practices—most likely because the costs of organizing to oust management is smaller when the bulk of a firm’s shares are held by large shareholders. In short, an ethical CEO would see a large ownership stake in the hands of an index fund as a mechanism for committing to (and signaling that commitment to) shareholder welfare.

Now, suppose you manage a large index fund, what responsibility (if any) do you have for ensuring the companies in your portfolio maximize shareholder wealth?

As fund manager, your fiduciary duty is to your investors; they gave you money believing you will construct a portfolio to mimic a market index at the lowest possible fees. So monitoring and disciplining the management of firms in your portfolio is not your first concern. How much effort you spend monitoring depends on the availability of good substitutes and expected net benefits of monitoring. Devoting few resources to monitoring a company makes sense if monitoring costs are high, expected benefits are low (because firm management is entrenched), and investing in a similar company is easy. Moreover, other shareholders (or more likely blocks of shareholders) can do the necessary monitoring and alert you to serious governance issues. As a large shareholder, the threat you might ally with disgruntled shareholders might be sufficient to keep management focused on stockholder wealth.

 Answers to Warm-Up Exercises E7-1

Using debt ratio to calculate a firm’s total liabilities (LG 1)

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Answer: Debt ratio = Total liabilities  Total assets → Total liabilities = Debt ratio  Total assets = 0.75  $5,200,000 = $3,900,000.

E7-2

Determining net proceeds from the sale of stock (LG 2)

Answer: Net proceeds = (1,000,000  $20  0.95)   $20  0.90) = $23,500,000.

E7-3

Preferred and common stock dividends (LG 2)

Answer: Common-stock dividend  (Cash available  Preferred dividends owed)  Common shares  [$12,000,000  (4  $2.50  750,000)]  3,000,000  $1.50 per share.

E7-4

Price-earnings ratios (LG 3)

Answer: Earnings per share (EPS)  $11,200,000  4,600,000  $2.43 per share. So, today’s P/E ratio is $24.60  $2.43  10.12, and yesterday’s P/E ratio is $24.95  $2.43  10.27.

E7-5

Valuing stock with zero dividend growth (LG 4)

Answer: P0  D1  r, where D1 = $1.20  (1.05) = $1.26. and r = 8%. So, P0 = $1.26  0.08  $15.75.

E7-6

Valuing stock with zero dividend growth (LG 6)

Answer: Calculate required return, r  RF + RP = 4.5%  10.8%  15.3%. Then, calculate value of stock with zero-growth model, P0  D1  r, where D1 = $2.25 and r = 15.3%: $2.25  0.153  $14.71.

 Solutions to Problems P7-1

Authorized and available shares (LG 2; Basic) a. Maximum shares = Authorized shares – Shares outstanding = 2,000,000 – 1,400,000 = 600,000. b. Total shares needed = $48,000,000  $60 = 800,000 shares, meaning 200,000 additional shares must be authorized to raise needed funds. c. Aspin must seek approval through a vote of current shareholders.

P7-2

Preferred dividends (LG 2; Intermediate) a. Annual dividend in dollars = Price of preferred stock annual dividend rate = $40 $3.20 per year or $0.80 per quarter. b. $0.80—For noncumulative preferred, only the current dividend must be paid before dividends on common stock. c. $3.20—For cumulative preferred, all dividends in arrears must be paid before paying dividends on common stock. Here, the board must pay the three dividends missed plus the current dividend.

P7-3

Preferred dividends (LG 2; Intermediate) Case

Answer

Explanation

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P7-4

A

$16

B

$2.20

C

$3

D

$4.50

E

$2.10

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Three quarters of passed dividends plus current quarter (4 quarters × $4 per quarter) Dividend is 2% of $110 per quarter, or $2.20 per quarter; only current quarter must be paid because stock is noncumulative.

Only current dividend of $3 must be paid because stock is noncumulative.

Quarterly dividend is 1.5% of $60 or $0.90 per quarter. Total dividends owed equal four quarters passed plus current dividend (5 × $0.90).

Quarterly dividend is 3% of $70 or $2.10 per quarter. No dividends have been passed, so only current $2.10 dividend is due.

Convertible preferred stock (LG 2; Challenge) a. Conversion value or preferred stock  Conversion ratio  Common stock price  5  $20  $100. b. The investor should convert because they would obtain shares worth$100 while only giving up preferred shares worth $96. c. This question is trickier than it first appears. Students might note conversion of one share of preferred to five shares of common stock will reward an investor with $5 in common dividends annually ($1.00 per share 5) while retaining the preferred stock will yield $10.00 per year in dividends. This is true but fails to recognize that an investor converting preferred into common shares could immediately sell the common shares for $100, repurchase the preferred shares for $96, and pocket a $4 profit. Moreover, there is no limit on how often this trade can be repeated as long as the markets for preferred and common are liquid, and prices do not change. [In general, however, such trading will bid up the price of preferred stock (because investors are buying it) and depress the price of the common stock (because investors are selling it) until the profit opportunity vanishes.]

Students might also say that investors could choose not to convert because they prefer a less risky preferred stock investment over a more risky common stock one. Again, this ignores the opportunity that a preferred stockholder has to convert into common, sell the common for $100, and repurchase the preferred stock. This preserves the riskiness of their position while bringing in a profit of $4. If the transactions costs associated with these trades are high enough to more than wipe out the $4 profit, then preferred investors might choose to hold their stock rather than convert it. There might also be tax implications of converting that could make the conversion transaction unattractive.

P7-5

Voting rights and dual-class stock (LG 2; Challenge) a. Co-founders equity investment = (Co-founders Class A shares + Co-founders Class B shares)  All Class A and B shares = (1,369,182 + 12,779,709)  285,877,300 = 0.049 or 4.9%. b. Co-founders control rights = (Co-founders Class A votes + Co-founders Class B votes)  All Class A and B votes = (1,369,182 + 12,779,709 × 20)  (273,097,591 + 12,779,709 × 20) = 0.486 or 48.6%. c. Co-founders equity investment = (Co-founders Class A shares + Co-founders Class B shares)  All Class A and B shares = (1,369,182 + 12,779,709)  385,877,300 = 0.037 or 3.7%.

Co-founders control rights = (Co-founders Class A votes + Co-founders Class B votes)  All Class A and B votes = (1,369,182 + 12,779,709 × 20)  (373,097,591 + 12,779,709 × 20) = 0.409 or 40.9%. © 2022 Pearson Education, Inc.

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d. Co-founders equity investment = (Co-founders Class A shares + Co-founders Class B shares)  All Class A and B shares = 12,779,709  385,877,300 = 0.033 or 3.3%.

Co-founders control rights = (Co-founders Class A votes + Co-founders Class B votes)  All Class A and B votes = (12,779,709 × 20)  (373,097,591 + 12,779,709 × 20) = 0.407 or 40.7%. e. Co-founders equity investment = (Co-founders Class A shares + Co-founders Class B shares)  All Class A and B shares = 6,779,709  385,877,300 = 0.018 or 1.8%.

Co-founders control rights = (Co-founders Class A votes + Co-founders Class B votes)  All Class A and B votes = (6,779,709 × 20)  (379,097,591 + 6,779,709 × 20) = 0.263 or 26.3%.

P7-6

Preferred stock valuation (LG 4; Basic) a. Annual dividend = Price of preferred stock × annual dividend rate = $65 × 10% or $6.50. b. Because the dividend stream is a perpetuity, value of preferred stock is just annual dividend divided by required return, P0  D1  r, where D1 = $6.50 and r = 8%.: $6.50  0.08 = $81.25. c. To find share value, recognize the dividend stream is identical to part (b) except that in one year, investors will receive an extra $13 for two years of passed dividends.

Therefore, value of preferred stock equals value calculated in part (b) plus the present value of the passed dividends: $81.25 + ($13  1.08) = $81.25 + $12.04 = $93.29.

P7-7

Personal finance problem: Common-stock valuation: Zero growth (LG 4; Intermediate) Using the perpetuity formula (P0  D1  r, where D1 = $2.80 and r = 9.25%), Kelsey Drums common stock is worth $32.27 today. Ten years ago, it was worth $36.84 ($2.80  0.076), implying Kim would lose $6.57 per share ($36.84 – $32.27), and $1,314.37 in total (200 shares × $6.57).

P7-8

Preferred stock valuation (LG 4; Intermediate) a. Preferred stock price = Expected perpetual dividend (D1)  Required rate of return (r)  $6.40  0.093 = $68.82. b. New value of preferred stock is $6.40 $60.95. The investor would lose $7.87 per share ($68.82  $60.95) because the rise in required return caused the price of preferred stock to fall. Now, the same perpetual dividend is discounted at a higher rate.

P7-9 Common stock value: Constant growth (LG 4; Basic) Let P0 be the current price of the common stock, D1 the next expected dividend, r the required return, and g the expected constant growth rate of dividends: Firm A B C D E

P0  D1  (r  g) P0  $1.20  (0.13  0.08)  P0  $4.00  (0.15  0.05)  P0  $0.65  (0.14  0.10)  P0  $6.00  (0.09  0.08)  P0  $2.25  (0.20  0.08) 

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Share Price $ 24.00 $ 40.00 $ 16.25 $600.00 $ 18.75

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P7-10 Common stock value: Constant growth (LG 4; Intermediate) a. Let P0 be current price of common stock, D1 the next expected dividend, r the required return, and g the expected dividend growth rate; stock price is given by P0  D1  (r  g).

The next dividend (D1) = $1.20 × 1.05 = $1.26. So, plugging given information in the stock-price equation and solving for r: $28 = $1.26  (r – 0.05) → r = 9.50%. b. The next dividend (D1) = $1.20 × 1.10 = $1.32. So, $28 = $1.32  (r – 0.10) → r = 14.7%.

P7-11 Common stock value: Constant growth (LG 4; Intermediate) Let P0 be current price of common stock, D1 the next expected dividend, r the required return, and g the expected dividend growth; stock price is given by P0  D1  (r  g). Historical growth in dividends from 2018 to 2022 (also expected dividend growth) is 4

 $2.52  $1.52  – 1 = 13.47%. The next expected dividend (D2023) is $2.52 × 1.1347 =

$2.86. The required return is given as 20%, so P0 = $2.86  (0.20 – 0.1347) = $43.80.

P7-12 Common stock value: Constant growth (LG 4; Intermediate) Let P0 be current price of common stock, D1 the next expected dividend, r the required return, and g the expected dividend growth rate; stock price is given by P0  D1  (r  g). In the problem, D1 is $1.35, today’s share price is $114, and required return is 15.8%. Plugging these values in the stock-price equation and solving for expected dividend growth (g): $114 = $1.35  (0.158 – g) = $43.80 → g = 0.158 – ($1.35  $114) = 0.1462 = 14.62%.

P7-13 Personal finance problem: Common stock value: Constant growth (LG 4; Challenge) a. Let P0 be current stock price, D1 the next expected dividend, r the required return, and g the expected dividend growth rate; stock price is given by P0  D1  (r  g). Historical growth in dividends from 2017 to 2022 (also expected dividend growth) is 5

 $2.87  $2.25  – 1 = 4.99%. The next expected dividend (D2023) is $2.87 × 1.0499 =

$3.01, and the required return is given as 13%, so P0 = $3.01 (0.13 – 0.0499) = $37.61. b. If the required return falls to 10%, P0 = $3.01 (0.10 – 0.0499) = $60.12. c. A fall in the required return means future dividends are discounted at a lower rate, so the stock price rises.

P7-14 Common stock value: Variable growth (LG 4; Challenge) P0  Present value of dividends during initial growth period  Present value of stock price at end of initial growth period.

Step 1: Present value of cash dividends during initial growth period, given last dividend of $2.55, expected dividend growth for 3 years of 25%, and required return of 15%: n 1 2 3

D0 1.25n = Dn 1/(1.15)n = Present Value of Dividends $2.55 1.2500 $3.19 0.8696 $2.77 $2.55 1.5625 $3.98 0.7561 $3.01 $2.55 1.9531 $4.98 0.6575 $3.27 Total = $9.05

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Step 2: Present value of price of stock at end of initial growth period—given dividend at the end of third period of $4.98, expected dividend growth of 10%, and required return of 15%: At end of year 3, next expected dividend, D4 $4.98  (1  0.10) = $5.48. So, stock price at the end of year 3, P3 [D4  (r  g)] $5.48  (0.15  0.10) = $109.56.

Finally, the present value of stock price at end of year 3 is $109.56 $72.04.

Step 3: Add present value of dividends during initial growth period to present value of stock price at end of growth period: P0 $9.05  $72.04 = $81.09.

P7-15 Personal finance problem: Common stock value: Variable growth (LG 4; Challenge) P0  Present value of dividends during initial growth period  Present value of stock price at end of initial growth period.

Step 1: Present value of dividends during initial growth period—given last dividend (D0) of $3.40, variable dividend growth for 4 years, and required return of 14%: D1  $3.40  (1.00)  $3.40 D2  $3.40  (1.05)  $3.57 n 1 2 3 4

Dn $3.40 $3.57 $3.75 $4.31

(1.14)n 0.8772 0.7695 0.6750 0.5921

D3  $3.57  (1.05)  $3.75 D4  $3.75  (1.15)  $4.31 = Present Value of Dividends $2.98 $2.75 $2.53 $2.55 Total = $10.81

Step 2: Present value of price of stock at end of initial growth period—given dividend at the end of year 4 period of $4.31, expected dividend growth of 5%, and required return of 14%: At end of year 4, next expected dividend, D5, is $4.31  (1.05)  $4.53. So, stock price at the end of year 4, P4 [D5  (r  g)] $4.53  (0.14  0.05) = $50.33.

Finally, present value of stock price at end of year 4 is $50.33 $29.80.

Step 3: Add present value of dividends during initial growth period to present value of stock price at end of growth period: P0 $10.81  $29.80 = $40.61.

P7-16 Common stock value: Variable growth (LG 4; Challenge) a. P0 Present value of dividends during initial growth period  Present value of stock price at end of initial growth period.

Step 1: Present value of dividends during initial growth period—given last dividend of $1.80, expected dividend growth for 3 years of 8%, and required return of 11%: n 1 2 3

D0 × 1.08n = Dn 1/(1.11)n = Present Value of Dividends $1.80 1.0800 $1.94 0.9009 $1.75 $1.80 1.1664 $2.10 0.8116 $1.70 $1.80 1.2597 $2.27 0.7312 $1.66 Total = $5.11

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Step 2: Present value of price of stock at end of initial growth period—given dividend at the end of year 3 period of $2.27, expected dividend growth of 5%, and required return of 11%: At end of year 3, next expected dividend, D4 $2.27  (1  0.05) = $2.38. So, stock price at the end of year 3, P3 [D4  (r  g)] $2.38  (0.11  0.05) = $39.67. Finally, the present value of stock price at end of year 3 = $39.67 $29.01.

Step 3: Add present value of dividends during initial growth period to present value of stock price at end of growth period: P0 $5.11  $29.01 = $34.12. b. The present value of year 1-3 dividends is the same as in part (a). For step 2, dividend growth rate is now zero after year 3, so D3  D4 $2.27. Moreover, now P3 may be found with the perpetuity formula: P3 D4  r $2.27  0.11 = $20.64; present value of stock price at the end of year 3 = $20.64 = $15.09. For step 3, P0 $5.11  $15.09 = $20.20. c. Present value of year 1-3 dividends is the same as in part (a). For step 2, dividend growth is now 10% after year 3, so D4 $2.27  (1.10) = $2.50, and P3 [D4  (r  g)] $2.50  0.01 = $250.00. Present value of stock price at end of year 3 is $250.00= $182.80. For step 3, P0 $5.11  $182.80 = $187.91.

P7-17 Personal finance problem: Free cash flow valuation (LG 4; Challenge) a. Firm has no debt or preferred stock, so firm value (Vc) is present value of expected free cash flow (FCF). If current FCF is not expected to change (FCF0 = FCF1), and required return (r) is 18%: VC  FCF1  r = $42,500  0.18 = $236,111 b. Free cash flow next year, FCF1 = $42,500  1.07 = $45,475. Now, because FCF is expected to grow at a constant rate, VC can be found using the stock-valuation framework for constant dividend growth, where FCF1 is substituted for D1, and g represents constant FCF growth: VC  FCF1  (r – g) = $46,750  (0.18 – 0.07) = $413,409.09 c. VC Present value of FCF in first growth period  Present value of FCF after first growth period.

Step 1: Present value of FCF during initial growth period—given FCF0 = $42,500, expected FCF growth of 12% for two years, and required return of 18%: n 0 1 2

FCF0 $42,500 $42,500 $42,500

× 1.12n = FCFn

1/(1.18)n

1.1200 1.2554

0.8475 0.7182 Total =

$47,600 $53,312

= Present Value of FCFn $40,338.98 $38,287.85 $78,626.83

So, stock price at end of year 2, P2 [FCF3  (r  g)] $53,312  (0.18  0.07) = $518,580.36. Finally, the present value of P2 = $518,580.36 $372,436.34.

Step 3: Add present value of free cash flow during initial growth period to present value of stock price at end of growth period: VC $78,626.83  $372,436.34 = $451,063.17.

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Note: Student answers may vary by a few cents because of rounding.

P7-18 Free cash flow valuation (LG 5; Challenge) a. Firm value (VC) may be found in three steps: Step 1: FCF in year 6: FCF6 = $390,000 (1.03) = $401,700. Now calculate the present value as of year 5 of all cash flows that arrive in year 6 and beyond: = $401,700  (0.11  0.03)  $5,021,250.

Step 2: Add step 1 to the free cash flow in year 5: $5,021,250 + $390,000 = $5,411,250.

Step 3: Discount FCF from year 1 through 5 back to today: Years from Now (n) FCFn 1/(1.11)n = Present Value 1 $200,000 0.900901 $ 180,180.18 2 250,000 0.811622 $ 202,905.61 3 310,000 0.731191 $ 226,669.33 4 350,000 0.658731 $ 230,555.84 5 $5,411,250 0.593451 $3,211,313.50 Total Firm Value (VC) $4,051,624.46 = b. Total value of common stock (VS) = VC – Total value of debt (VD)– Total value of preferred stock (VP) = $4,051,624.46  $1,500,000  $400,000 = $2,151,624.46. c. Value per share of common stock = VS Common shares = $2,151,624.46  200,000  $10.76.

P7-19 Personal finance problem: Using the free cash flow valuation model to price an IPO (LG 5; Challenge) a. The value of the firm’s common stock may be found in four steps: Step 1: Find present value in year 4 of free cash flow (FCF) from year 5 to infinity: FCF5 = $1,100,000 (1.02) = $1,122,000.

Present value of FCF5 to infinity in year 4 = $1,100,000  (0.08  0.02)  $18,700,000.

Step 2: Add step 1 to given FCF4: $18,700,000 + $1,100,000 = $19,800,000.

Step 3: Discount FCF from year 1 through 4 to today to obtain total value of the firm (VC): Years from Now (n)

FCFn

1/(1.08)n

1 2 3 4

= Present Value

$ 700,000 0.925926 $648,148.15 $ 800,000 0.857339 $685,871.06 $ 950,000 0.793832 $754,140.63 $19,800,000 0.735030 $14,553,591.09 Total Value of Firm = $16,641,750.92 Step 4: Find value of common stock per share: VS = VC – VD – VP = $16,641,750.92 – $2,700,000 – $1,000,000 = $12,941,750.92.

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Value per share of common stock = Total value of common stock Common shares = $12,941,750.92  $11.77. b. IPO is overvalued by $0.73 ($12.50  $11.77), so you should not buy the stock. c. New value of common stock may be found in four steps: Step 1: Present value in year 4 of FCF from year 5 to infinity: FCF5 = $1,100,000 (1.03) = $1,133,000.

Present value of FCF5 to infinity-in year 4= $1,133,000  (0.08  0.03)  $22,660,000.

Step 2: Add step 1 to given FCF4: $22,660,000 + $1,100,000 = $23,760,000.

Step 3: Discount FCF from years 1 through 4 back to today.

Years from Now (n) 1 2 3 4

FCFn

(1.08)n

= Present Value

$ 700,000 0.925926 $648,148.15 $ 800,000 0.857339 $685,871.06 $ 950,000 0.793832 $754,140.63 0.735030 $17,464,309.30 $23,760,000 Total Firm Value (VC) = $19,552,469.14

Step 4: Find value of common stock per share: VS = VC – VD – VP = $19,552,469.14  $2,700,000  $1,000,000 = $15,852,469.14.

Value per share of common stock =VS Common shares = $15,852,469.14  $14.41.

The IPO is undervalued by $1.91 ($14.41  $11.77), so you should buy the stock.

P7-20 Book and liquidation value (LG 5; Intermediate) a. Book value per share = Book value of assets—Book value of liabilities – Book value of preferred stock Outstanding Shares = ($780,000 – $340,000 – $80,000) $36 per share. b. Liquidation value: Assets Cash Marketable Securities Accounts Receivable (0.90  $120,000) Inventory (0.90  $160,000) Land and Buildings (1.30  $150,000) Machinery & Equipment (0.70  $250,000) Liquidation Value of Assets

$40,000 60,000 108,000 144,000 195,000 175,000 $722,000

Liquidation Value for Common Stock Liquidation Value–Assets $722,000 Less: Current Liabilities (160,000) Long-Term Debt (180,000) Preferred Stock (80,000) Available for Common Stockholders $302,000

Liquidation value per share = Value available for common stockholders common shares = $302,000  $30.20.

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c. Book values reflect historical prices/costs, not current market value, so it is no surprise liquidation and book value differ for Gallinas Industries. Here, book exceeds liquidation value (as is the norm), so estimated value of common shares based on book value exceeds the estimate based on liquidation value.

P7-21 Valuation with price/earnings multiples (LG 5; Basic) To estimate stock price given earnings per share (EPS) and the average industry priceearnings multiple (P/E): Firm EPS  P/E  Stock Price A $3.00  6.2  $18.60 B 4.50  10.0  $45.00 C 1.80  12.6  $22.68 D 2.40  8.9  $21.36 E 5.10  15.0  $76.50 P7-22 Management action and stock value (LG 6; Intermediate) a. Last dividend (D0) was $3, expected dividend growth (g) is 5% per year, and required return (r) is 15%. Hence, the next expected dividend (D1) is $3.15 and stock price, P0 D1 r – g $3.15  (0.15  0.05) $31.50. b. If g rises to 6% and r dips to 14%, P0 $3.18  (0.14  0.06) $39.75. c. If g rises to 7% and r rises to 17%, P0 $3.21  (0.17  0.07) $32.10. d. If g falls to 4% and r rises to 16%, P0 $3.12  (0.16  0.04) $26.00. e. If g rises to 8% and r rises to 17%, P0 $3.24  (0.17  0.08) $36.00.

The best alternative is the one producing the highest share price, namely (b).

P7-23 Integrative: Risk and valuation (LG 4 and LG 6; Intermediate) The risk premium on Foster stock may be found in two steps: Step 1: Given stock price (P0) of $50, next expected dividend (D1) of $3 per share, and expected dividend growth (g) of 6.5%, solve for required return on the stock (r): P0  D1  (r  g) $50 = $3.00  (rs  0.065) → r = 0.125 or 12.5% Step 2: Given a 12.5% required return (r) and a 4.5% risk free return (RF), solve for risk premium (RP) on stock: r = RF + RP RP → RP = 0.08 or 8.0% P7-24 Integrative: Risk and valuation (LG 4 and LG 6; Challenge) a. Given a 14.8% required return (r), and a 4% risk-free rate (RF), solve for risk premium (RP) on Giant Enterprise stock: r = RF + RP RP → RP = 0.108 or 10.8% b. Dividend growth rate from 2016 to 2022 (also expected future growth rate) is – 1 = 0.0597 = 5.97%. Given the next expected dividend (D1) is $2.60 per share, expected dividend growth (g) is 5.97% per year, and required return (r) is 14.8%, solve the for value of Giant stock: P0 D1 r – g$2.60  (0.148  0.0597) $29.45. © 2022 Pearson Education, Inc.

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c. A decline in the risk premium would reduce required return. If expected dividends did not change, that stream would now be discounted at a lower rate, which means a higher value of (and market price for) Giant Enterprise stock.

P7-25 Integrative: Risk and valuation (LG 4 and LG 6; Challenge) a. The maximum price to pay for Craft stock may be found in three steps: Step 1: Find the 2017-2022 dividend growth rate (which is expected dividend growth rate) for Craft stock: g = – 1 = 0.0702 = 7.02%.

Step 2: Find the required return (r) on Craft stock, given a risk-free rate (RF) of 5% and a risk premium (RP) of 9%: r = RF + RP = 5% + 9% = 14%.

Step 3: Given a next expected dividend (D2023) of $3.68 per share, expected dividend growth (g) of 7.02%, and a required return (r) of 14%, solve for the maximum price (value) of Craft stock: P0 D2023 r – g$3.68  (0.14  0.0702) $52.72 per share. b. Part (1): The new value of Craft stock with lower dividend growth may be found in two steps: Step 1: Find new expected dividend for 2020 with expected growth rate two percentage points lower: D2023 = D2022 (1.0502) = $3.44 (1.0502) = $3.61 .

Step 2: Given an expected 2023 dividend (D1) of $3.61 per share, expected dividend growth (g) of 5.02%, and required return (r) of 14%, solve for new value of Craft stock:P0 D2023 r – g$3.61  (0.14  0.0502) $40.20 per share. A two-percentage-point decline in dividend growth reduced share price by $12.52.

Part (2): To find the new share price, first recognize the smaller risk premium means a smaller expected return. Specifically, risk premium (RP) falls to 4%, so required return is: (r) = RF + RP = 5% + 4% = 9%. Given the next expected dividend (D2023) is $3.68 per share, expected dividend growth (g) is 7.02%, and required return (r) is now 9%, solve for new value of Craft stock: P0 D2023 r – g$3.68  (0.9  0.0702) $185.86. A five percentage point decline in the risk premium boosted share price by $133.14.

P7-26 ETHICS PROBLEM (LG 4; Intermediate) a. ―Clearly not growing‖ means valuing with the zero-dividend-growth model. Given a next expected dividend (D1 = D0) of $5 per share and required return (r) of 11%, solve for value of Generic Utilities stock: P0 D1 r$5  0.11 $45.45 per share. b. A one-percentage-point ―credibility‖ risk premium means raising required return from 11% to 12%, making the new value of Generic stock: P0 D1 r$5  0.12 $41.67 per share. c. The added risk premium reduces the value of Generic stock by $3.79. Uncertainty about the reliability of the firm’s financials means expected dividends must be discounted at a higher rate. Put another way, uncertainty makes future dividends worth less to investors.

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 Case: Assessing Impact of Proposed Risky Investment on Suarez Stock Case studies are available on www.pearson.com/mylab/finance.

In this case, students will assess the potential impact of risky project on a hypothetical firm’s stock. a.

To find the current value per share of Suarez common stock, first obtain the dividend growth rate (g), which is expected to equal recent historical experience: g = – 1 = – 1 = 0.0995 = 9.95% Now, given g is 10%, the next expected dividend (D1) is $2.09, and required return is 14%, solve for stock price: P0 = D1 r – g) $2.09  (0.14 – 0.0995) $51.63 per share.

Now, given g is 10%, the next expected dividend (D1) is $2.09, and required return is 14%, solve for stock price: P0 = D1 r – g) $2.09  (0.14 – 0.0995) $51.63 per share.

b. If Suarez makes the risky investment, next year’s dividend (D1) will rise to $2.15 per share, the dividend growth rate (g) to 13%., and required return to 16%. The new value of common stock is: P0 = D1 r – g) $2.15  (0.16 – 0.13) $71.67 per share. Stock price jumps $20.04 (38.8%). c. Suarez should undertake the proposed project. Higher dividend growth more than compensates for the impact of project risk on required return, thereby boosting stock price and shareholder wealth. d. Now, dividends will grow 13% per year for three years (from the last dividend of $1.90); then, growth will slow to the historical growth rate of 9.95%. Stock price will equal the present value of dividends during 13% growth period plus the present value of stock price at the end of that period.

Step 1: Find the present value of dividends during the 13% growth period – given the last dividend (D0) of $1.90, and required return of 16%: D1 = $1.90 (1.13) = $2.15 $2.43 n 0 1 2 3

Dn $1.90 $2.15 $2.43 $2.74

1/(1.16)n 0.8621 0.7432 0.6407

D3 = $2.43 (1.13) = $2.74

D2 = $2.15 (1.13) =

= Present Value of Dividends $1.85 $1.80 $1.76 Total = $5.41

Step 2: Now, find the present value of price of stock at end of 13% growth period (when 9.95% growth resumes). At the end of year 3, the next expected dividend (D4) is $2.74 1.0995 = $3.01, expected dividend growth is 9.95%, and required return is 14%,, so stock price is: P3 [D4  (r  g)] $3.01  (0.16  0.0995) = $49.84. Finally, present value of stock price at end of year 3 is $49.84 $31.93 Step 3: Add present value of dividends during 13% growth period to present value of stock price at end of 13% growth period: P0 $5.41  $31.93 = $37.34.

Suarez should not undertake the risky project because share price would fall $14.29 (from $51.63 to $37.34). Additional dividends do not compensate for the impact on of additional risk on required return.

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 Spreadsheet Exercise Answers to Chapter 7’s Azure Corporation spreadsheet problem are available on www.pearson.com/mylab/finance.

 Group Exercise Group exercises are available on www.pearson.com/mylab/finance.

The semester began with the fictitious firms about to become public corporations. Out of necessity, few details were given. Now groups will begin to fill in the blanks. Specifically, using details from recent IPOs, each group will write a detailed prospectus following the example in the text. Students should quickly see similar patterns. Most IPOs, for example, are priced between $10 and $30 with few shares available at the offer price, forcing the public to pay a premium on and around the issuance date. The final group task is obtaining the most recent information on its shadow firm, including current market numbers and any recent news/analyses. Students will find much of the news fairly innocuous. Instructors can note the tendency in recent regulation for public companies to disclose more and more information. Class discussion can then explore the costs and benefits of erring on the side of over-disclosure.

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 Integrative Case 3: Encore International In this case, students will explore different methods of valuing a hypothetical firm, including price/earnings multiples, book value, and traditional dividend-growth models (under varying assumptions about the patterns of that growth). They will compare stock values generated by the models, discuss the differences, and select the approach best capturing the firm’s true value. a.

Book value per share = Book value of common equity Common shares outstanding = $60,000,000 2,500,000 = $24.

b.

Current price/earnings ratio = Current stock price Earnings per share (EPS) = $40 $6.25 = 6.4.

c.

(1) Current required return on common stock (r) = Risk-free rate (RF) + Risk premium (RP) = 6% + 8.8% = 14.8%.

(2) New required return on common stock = 6% + 10% = 16%.

d.

Because no dividend growth is anticipated, the valuation formula for perpetuities will indicate stock price. Given a constant dividend of $4.00 (D1) and a required return of 16%, stock price, P0 = D1 r = $4 0.16 = $25.

e.

(1) Given a 6% constant dividend growth, the next dividend is $4 (1.06) = $4.24. Stock price (P0) with 6% constant dividend growth (g), 16% required return (r), and $4.24 next dividend (D1) is: P0 = D1 (r – g) = $4.24 (0.16 – 0.06) = $42.40.

(2) Stock price when dividends grow 8% for two years then 6% forever may be found in three steps: Step 1: Present value of dividends in the 8% growth period, given last dividend (D0) was $4, and required return is 16%: First note: D1 = $4.00 (1.08) = $4.32 and D2 = $4.32 (1.08) = $4.67. So, n 0 1 2

Dn $4.00 $4.32 $4.67

1/(1.16)n

= Present Value of Dividends

0.8621 0.7432

$3.72 $3.47 Total = $7.19

Step 2: Present value of price of stock at end of 8% growth period: At end of year 2, next expected dividend, D3 = $4.67 (1.06) = $4.95, expected growth is 6%, and required return is 16%, so stock price, P2 [D3  (r  g)] $4.95  (0.16  .06) = 49.50. Finally, present value of end-of-year-2 stock price is $49.50 $36.79.

Step 3: Add present value of dividends during 8% growth period to present value of stock price at end of 8% growth period: P0 $7.19  $36.79 = $43.98. f.

Comparing value of Suarez stock with different valuation methods: Valuation Method Market value Book value Zero growth Constant growth

Share Price $40.00 24.00 25.00 42.40

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43.98

Book value has no relevance to the true value of the firm. Of the remaining methods, the most conservative estimate is given by the zero-growth model. Based on this estimate of stock value, wary analysts may advise paying no more than $25 per share—a figure hardly more than book value. The most optimistic prediction, the variable-growth model, estimates at $43.98, not far from the market value. The market appears to be more optimistic about Encore International’s future than wary analysts.

Part Four Risk and the Required Rate of Return Chapters in this Part (a) Chapter 8

Risk and Return

(b) Chapter 9

The Cost of Capital

(c) Integrative Case 4: Eco Plastics Company

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Chapter 8 Risk and Return  Instructor’s Resources Chapter Overview This chapter focuses on the fundamentals of risk and return—beginning with simple definitions of risk, total return, and expected return, and then describing the different types of risk preferences: risk neutral, risk averse, and risk seeking. The discussion then moves to risk measurement by focusing on a single asset and measuring risk with statistics associated with a probability distribution—namely, mean, standard deviation, variance, and coefficient of variation. To demonstrate that insights about risk for a single asset do not necessarily carry through to a portfolio of assets, the discussion broadens to risk and return for a portfolio.

The chapter explores the concept of correlation in depth with real-world examples and illustrates that the benefits of diversification depend on the correlation between assets in the portfolio. In many cases, investors are better off (meaning that they can create an efficient portfolio) by diversifying rather than investing in just one asset. The key takeaway is that the volatility (variance of standard deviation) of a portfolio will be less than a weighted average of the volatilities of the assets in the portfolio as long as the correlation is less than 1.0.

These ideas are used to motivate the Capital Asset Pricing Model (CAPM). Diversifiable and nondiversifiable risk are distinguished, with the key idea that the market only rewards bearing nondiversifiable risk because firm-specific risk can so easily be eliminated through diversification. The chapter introduces beta, a measure of an asset’s nondiversifiable risk. Examples demonstrate how to estimate a stock’s beta by using historical data on the stock’s return and the return on a market index. Then, the CAPM equation and its graphical representation (the Security Market Line or SML) are introduced to show the link between return and nondiversifiable risk. The chapter concludes by illustrating the impact of changes in inflation expectations and investor risk aversion on the SML.

 Suggested Answer to Opener-in-Review In the chapter opener, you learned about the Fidelity Select Air Transport Fund (FSAIX). Below are 10 years of returns on that fund and the S&P 500 stock index.

Year

Return on FSAIX

Return on S&P 500

2010

33.4%

15%

2011

6.1

2.1

2012



16.0

2013

50.7

32.4

2014

28.1

13.7

2015

–8.6

1.4

2016

20.0

12.0

2017

24.3

21.8

2018

–12.5

–4.4

2019

21.0

31.5

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Calculate the average return and the standard deviation for both of those investments, then calculate the correlation coefficient between them. Do you think this was a particularly good decade for FSAIX and for the S&P? Why or why not? What other lessons emerge from your calculations?

Average annual return on FSAIX = (33.40% – 6.1% + 19.2% + 50.7% + 28.1% – 8.6% + 20% + 24.3% – 12.5% + 21%)  10 = 169.5%  10 = 16.95%.

Average annual return on S & P 500 = (15.1% + 2.1% + 16.0% + 32.4% + 13.7% + 1.4% + 12.0% + 21.8% – 4.4% 31.5%)  10 = 141.6%  10 = 14.16%.

The FSAIX Fund performed better. A $1,000 investment in FSAIX January 1, 2010, would have grown to $4,160.56 by year-end 2019 ($1,000 × 1.334 × 0.939 × 1.192 × 1.507 × 1.281 × 0.914 × 1.200 × 1.243 × 0.875 × 1.210) while a $1,000 investment in the S&P 500 would have grown to only $3,568.57 ($1,000 × 1.151 × 1.021 × 1.160 × 1.324 × 1.137 × 1.014 × 1.120 × 1.218 × 0.956 × 1.315).

To find standard deviation of returns (σ) for each fund, use the standard-deviation formula for n observations of that fund’s historical data:

r  r  n

j 1



2

j

n 1

, where rj is return in year j (running to year n) and ̅ is average return over n years.

For the FSAIX Fund, start by calculating the difference between the fund’s return each year and its average return. Then square that difference each year. Add up those squared differences, divide by n – 1, and then take the square root: FSAIX standard deviation =

√

√ S&P500 standard deviation =

√

√

Note: In both equations, the last term under the square root symbol has a number and then the % symbol squared. For example, the first equation shows that value 3670.2%2. This is not an error and reflects that the unit of measure here is ―percent squared.‖ That is, read the value as 3,670.2 ―percent squared.‖ When you take the square root of this term, the value 3,670.2 becomes 20.19 and the units go from %2 to %. The Excel command for standard deviation is STDEV.S. If historical returns for one of the funds above appeared in rows 1–10 of column A of a worksheet, proper syntax would be = STDEV.S(A1:A10).

For both investments, it was a good decade. The average return on the index was better than the long-run returns on stocks in the United States reported elsewhere in the chapter, and the fund’s performance outpaced the index on average. The correlation between the two was 0.85, suggesting a high degree of positive correlation. Measured by the standard deviation, the fund was riskier, which is not surprising because it focuses on a narrow set of companies and is therefore not as diversified as the index.

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 Answers to Review Questions 8-1.

Risk refers to the uncertainty about the return an investment will earn.

8-2

Total return (gain or loss) on an investment over a given time period is the change in value over that period plus any cash distributions, expressed as a percentage of beginning-of- period value.

Specifically, [(ending value  initial value)  cash distribution] Return  initial value

8-3

a. Risk-averse investors dislike risk and, therefore, expect higher returns on riskier investments. b. Risk-neutral investors select investments based on expected return—the higher the better—without regard to risk. Such investors require no compensation for bearing more risk. c. Risk-seekers like risk. Just as risk-averse investors will give up some return to avoid some risk, risk-seeking investors will give up some return to take more risk.

Most financial managers are risk averse—they expect compensation for bearing additional risk. Risk tolerance refers an investor’s degree of risk aversion, that is the specific compensation she requires for taking additional risk. Two investors may both refuse to accept more risk unless awarded a higher expected return—that is, both are risk averse. But the more risk tolerant of the two will require less additional compensation.

8-4.

Scenario analysis assesses asset risk using more than one possible set of returns to gauge the variability of outcomes. Range—a measure of variability—is found by subtracting the pessimistic outcome from the optimistic outcome, with larger ranges suggesting greater risk. Note, however, after getting deeper into the chapter, students will learn the range of outcomes suffers as a risk measure by not distinguishing diversifiable and undiversifiable components.

8-5

Decision makers can estimate risk with a plot of the probability distribution—which relates probabilities to potential returns by showing the dispersion in returns. The wider the distribution of potential returns, the greater the variability (risk) associated with returns. It is important to note, however, the plot offers a feel for an asset’s risk but does not distinguish between diversifiable and undiversifiable components.

8-6.

The standard deviation of a distribution of asset returns is an absolute measure of dispersion of risk around the mean or expected value. A higher standard deviation means more variable returns.

8-7

The coefficient of variation (CV)—another risk indicator—is the standard deviation on an asset’s returns divided by its average return. In other words, rather than measuring risk solely by the volatility of an asset’s returns, CV shows the volatility of returns relative to average or expected return.

8-8

An efficient portfolio offers the maximum return for a given risk level. Portfolio return is just the weighted average of returns on individual assets in the portfolio. Specifically: rp = (w1

r1) + (w2

where: rp = portfolio return wj = the portfolio weight of asset j (∑

r2) + … + (wn

= 1)

rn) = ∑

(

)

rj = the return on asset j n = number of assets in the portfolio

The standard deviation of a portfolio is not the weighted average of the standard deviations of component assets except in the special case when the correlation between the assets in the portfolio is

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1.0. Nearly always, the standard deviation of a portfolio will be less than the weight average of the standard deviations of the assets in the portfolio. One way to calculate the standard deviation of a portfolio is simply to take historical data on the portfolio’s returns and calculate the standard deviation of that series of numbers as follows:

σp = √∑ 8-9

(

)

where: σp = standard deviation of portfolio returns

No, this is not necessarily true. With perfect negative correlation, when one stock’s return is above its average, the other stock’s return is below its average. Only in the special and very unusual case that both stocks average return equals zero would the review question statement be true, namely that when one stock earns a negative return the other earns a positive return.

8-10 The correlation between asset returns is key to evaluating the effect of a new asset on portfolio risk.

Returns on different assets that move in the same direction are positively correlated, while those moving in opposite directions are negatively correlated. Unless the returns on assets in a portfolio are perfectly positively correlated, portfolio standard deviation will be less than the weighted average of the standard deviations of portfolio assets. If the correlation between assets in the portfolio is sufficiently low—it need not be negative—portfolio standard deviation may be less than the standard deviation of the least volatile asset in the portfolio. In short, the magic of diversification is that portfolio returns can be less volatile than the returns on any single portfolio asset.

8-11 The total risk of a security is measured by the standard deviation of returns; it has two components – nondiversifiable risk and diversifiable risk. Diversifiable risk refers to the portion of risk attributable to firm-specific, random events (such as strikes, litigation, and loss of key contracts) that can be eliminated by diversification. Nondiversifiable risk, in contrast, is attributable to market factors affecting all firms at the same time (such as war, inflation, and political events). Nondiversifiable risk is the only relevant risk in the sense that the market only rewards investors with higher returns for the nondiversifiable risk that they take. There is no risk premium for diversifiable risk because it can be easily eliminated by forming a portfolio of assets with less than perfect positive correlation. Because investors can easily eliminate this risk, the market will not offer compensation in the form of higher returns to those who bear it.

8-12 Beta measures the nondiversifiable risk of a specific asset or portfolio; it is an index of the comovement of an asset’s return with the market return. The beta coefficient for an asset can be found by plotting the asset’s historical returns relative to the returns for the market and using statistical techniques to fit the ―characteristic line‖ to the data points. The slope of this line is beta. Beta coefficients for actively traded stocks are widely published in print and online. The beta of a portfolio is simply the weighted average of the betas of component assets.

8-13 The capital asset pricing model (CAPM) is given by rj  RF [βj(rm  RF)] where rj  required (or expected) return on asset j RF rate of return required on a risk-free security (U.S. Treasury bill) βj beta coefficient or index of nondiversifiable (relevant) risk for asset j rm  required return on market portfolio of assets (market return)

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The security market line (SML) is a pictorial presentation of the relationship between an asset’s systematic risk and the required return. Systematic risk—measured by beta—is on the horizontal axis while required return is on the vertical axis.

8-14 a. An increase in inflationary expectations shifts the SML upwards by an amount equal to the increase. The shift is parallel—that is, SML slope remains the same—because the only change is that required return for a given level of risk rose to reflect the higher expected inflation rate. b. If investors become less risk averse, the slope of the SML (beta coefficient) will decline—that is the SML will rotate clockwise around the given fixed risk-free rate because a lower return is now required for each level of risk.

 Suggested Answer to Focus on Ethics Box: “If It Seems Too Good to Be True, It Probably Is” What are warning signs an investment advisor’s activities may be suspect? Both Ponzi and Madoff claimed later they knew what they were doing was wrong, but they expected to earn enough eventually to deliver on their promises. Do you believe them?

Academic research on the investment performance of actively managed funds has produced two consistent findings: (i) after adjusting for differences in fees, active funds do not—on average—outperform passive (index) funds, and (ii) funds that do outperform the market for a while cannot maintain that edge. These findings suggest a fund manager can consistently beat the market over a long period only by taking more risk or committing fraud. (And ―taking more risk‖ in this context means returns will be more volatile than the market return.) So investors should beware of fund managers who claim to earn ―abnormal‖ returns consistently over a long stretch, particularly if the explanation offered is a new, proprietary strategy. As for believing Ponzi and Madoff, of course, it is impossible to know what was in their hearts. But successful scam artists have to be good salesman, and all good salesman must—to some extent at least—believe their pitch.

That is what makes the pitch so believable to others (along with greed). The bottom line here is that scam artists seldom wear nametags identifying themselves as such. So it is best to look past the sales pitch and ask skeptical questions about returns that appear too good to be true.

 Suggested Answer to Focus on Global Finance Box: “An International Flavor to Diversification” International mutual funds do not include any domestic assets, whereas global mutual funds include both foreign and domestic assets. How might this difference affect their correlation with U.S. equity mutual funds?

Global funds differ from international funds by investing in stocks and bonds around the world, including U.S. securities. International funds, in contrast, invest in stocks and bonds around the world but not U.S securities. Global funds, therefore, are more likely to be correlated with U.S. mutual funds because U.S. securities (particularly equities) are typically a sizeable chunk of their portfolios. A U.S. investor seeking to further internationally diversify her portfolio should either add international funds or—if the portfolio already includes global funds—increase the portfolio weight of those funds.

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 Suggested Answer to Focus on People/Planet/Profit Box: “Happy Employees Signal High Stock Returns” Would it be correct to interpret the results of this study as saying that having happy employees causes higher stock returns?

The causality here is reversed. The study suggests that employees have access to early good news about how a firm is performing. That news shows up in Glassdoor ratings before it is made public through a firm’s financial reports. Thus, it is the underlying fundamental good news that is driving employee morale and higher stock returns.

 Answers to Warm-Up Exercises E8-1

Total annual return (LG 1)

Answer: Total return is given by: rt 

Ct  Pt  Pt 1 where: Pt 1

rt = actual rate of return during period t Ct = cash (flow) received asset in period t

Pt = price (value) of asset at time t Pt – 1 = price (value) of asset at time t – 1

Logistics paid no dividend during the year. Market capitalization was $10 million at the beginning of the year and $12 million at the end. So total return is: ($0  $12,000,000  $10,000,000)  $10,000,000  $2,000,000  $10,000,000  20% Logistics, Inc., doubled the annual rate of return predicted by the analyst. The negative net income is irrelevant to the problem.

E8-2

Expected return (LG 2)

Answer: Expected return on Flourine Chemical stock is the weighted-average return, meaning each analyst’s prediction is weighted by the probability assigned; then weighted values are summed. Specifically: Analyst Probability Return Weighted Value

E8-3

1 2

0.35 0.05

5% 5%

3 4 Total =

0.20 0.40 1.00

10% 3% Expected return

Comparing the risk of two investments (LG 2)

Answer: The coefficient of variation is given by: CV 

 r

1.75% 0.25 % 2.00% 1.20% = 4.70%

where:

σ = standard deviation of returns on the investment

= average return on the investment.

Investment 1 has a standard deviation of 10% and an average return of 15% while Investment 2 has a standard deviation of 5% and an average return of 12%, so:

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CV1  0.10  0.15  0.6667

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181

CV2  0.05  0.12  0.4167

Based on standard deviations and coefficients of variation, Investment 2 has lower risk if held in isolation. That said, neither standard deviation nor coefficient of variation decomposes measured volatility into nondiversifiable and diversifiable components. But coefficient of variation does at least measure an asset’s volatility relative to expected return and, therefore, offers a broader perspective on risk than standard deviation alone.

E8-4 Computing the expected return of a portfolio (LG 3) Answer: Expected return on the portfolio is the weighted-average of the expected returns on portfolio components. Put another way, expected return on each asset is weighted by its percentage of the portfolio and then weighted values are summed. Specifically: rp  (0.45  0.02)  (0.4  0.10)  (0.15  0.15)  (0.009)  (0.04)  (0.02252)  0.0715  7.15% E8-5 Calculating a portfolio beta (LG 5) Answer: Portfolio beta is just the weighted average of betas of individual portfolio components, meaning the beta on each asset is weighted by its share of the portfolio and weighted betas summed. Specifically: Beta  (0.20  1.17)  (0.10  0.82)  (0.15  1.24)  (0.20  0.95)  (0.35  1.38)  0.234  0.082  0.186  0.190  0.483  1.175 E8-6 Calculating the required rate of return (LG 6) Answer: The CAPM equation can be used to find required return, given an asset’s beta, the risk-free rate, and market return. Specifically: rj  RF    j   rm  RF   where: rj = required return on asset j RF = risk-free rate

j = Beta coefficient for asset j

rm = Expected return on market portfolio of assets

a. Required return  0.05  1.8 (0.10  0.05)  0.05  0.09  0.14. b. Required return  0.05  1.8 (0.13  0.05)  0.05  0.144  0.194.

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c. The risk-free rate does not change, but market return increases. The resulting rise in the market-risk premium will cause the Security Market Line (SML) to rotate counterclockwise about the fixed risk-free rate..

 Solutions to Problems P8-1

Total rate of return (LG 1; Basic) Total return on an investment is given by rt =

(Pt  Pt 1  Ct ) Pt1

where rt = total return on the bond Pt = Price of the bond at time t a. Total rate of return  P8-2

Pt = Price of the bond at time t-1 Ct = Coupons received between t-1 and t

($925.65  $1,000  $30)  0.04435 or  4.435% $1,000

Total rate of return (LG 1; Basic) Total return on an investment is given by: rt =

where:

rt = total return on asset Pt = Price of asset at time t

(Pt  Pt 1  Ct ) Pt1

Pt = Price of asset at time t-1 Ct = Cash received between t-1 and t

GE’s Total Return = $5.79 – $10+$0.16 = –$4.05 GE’s Total Return % 

($5.79  $10  $0.16)  40.50% $10

Netflix’s Total Return = $438.27 – $354.45 = $83.82 Netflix’s Total Return % 

($438.27  $354.45)  23.65% $354.45

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P8-3 Return calculations (LG 1; Basic) Total return on an investment is given by rt =

(Pt  Pt 1  Ct ) Pt1

where rt = total return on asset Pt = Price of asset at time t Investment A B C D E P8-4

Pt = Price of asset at time t-1 Ct = Cash received between t-1 and t

Calculation ($100  $1,100 $800)  $800 ($118,000  $120,000  $15,000)  $120,000 ($48,000  $45,000  $7,000)  $45,000 ($500  $600  $80)  $600 ($12,400  $12,500  $1,500)  $12,500

rt (%) −163.64 10.83 22.22 3.33 11.20

Risk preferences (LG 1; Intermediate) a. If Sharon were risk neutral, differences in risk would not enter into her decision; she would care only about expected return. Moreover, she would not have a preference among investment A, B, or C because all three provide a 14% expected return. b. If Sharon were risk averse, she would select investment X because it offers less volatile returns for the same expected return as Y and Z. c. If Sharon were risk seeking, she would prefer Z because it has the highest standard deviation while offering the same expected return as X and Y. d. If Sharon were risk averse, it is not clear whether she would prefer investment W or X . From part (b), Sharon prefers X to Y and Z, but investment W has a higher expected return and standard deviation. Thus, Sharon’s preference between W and X will depend on whether the extra return expected on W is sufficient compensation for the extra risk. In other words, Sharon’s choice will depend on her risk tolerance (i.e, her degree of risk aversion).

P8-5

Risk preferences (LG 2 LG 3; Intermediate) a. The expected payoff and return for each investment: Investment 1: Expected payoff = (0.5 × $100) + (0.5 × $200) = $150 Expected return = ($150  $130) – 1 = 0.154 or 15.4% The expected return in dollars is $20.

Investment 2: Expected payoff = (0.5 × $50) + (0.5 × $250) = $150 Expected return = ($150  $130) – 1 = 0.154 or 15.4% The expected return in dollars is $20. b. The risk-seeker prefers investment 2 because it has the same return as investment 1 but has higher risk (more extreme high and low outcomes). c. The risk-neutral investor is indifferent between them because they have the same expected return. d. The risk-averse investor prefers investment 1 because it has the same expected return as investment 2 but is less risky. However, we cannot say for sure whether investment 1 is © 2022 Pearson Education, Inc.

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acceptable to the risk-averse investor. It offers some compensation for risk (i.e., it has a positive expected return), but we do not know if that return is sufficient compensation for the risk in the eyes of the investor, even though we can say that investment 1 is preferable to investment 2.

P8-6

Risk analysis (LG 2; Intermediate) a. Range is the difference between return with the best outcome and return with the worst.

Expansion

Range

Wind Farm

36%  4%  32%

Solar Farm

46%  (-6)%  52%

b. Wind Farm seems less risky because the range of outcomes is smaller than the range for Solar Farm. c. Both investments have the same mean return (20%), but returns on Solar Farm are more volatile, so a risk-averse investor would prefer Wind Farm. d. The answer is no longer clear because it now involves a risk-return tradeoff. Solar Farm now offers a slightly higher average return, but more risk, while Wind Farm has both lower return and lower risk.

P8-7

Risk and probability (Intermediate) a. Range is the difference between return with the best outcome and return with the worst.

Camera Nikon Canon b.

Camera Nikon

Range 30%  20%  10% 35%  15% 20%

Possible Outcomes Pessimistic Most likely Optimistic

Probability Pri 0.25 0.50 0.25 1.00

Pessimistic Most likely Optimistic

0.20 0.55 0.25 1.00

Expected Return ri 20 25 30 Expected return =

Weighted Value (%) (ri  Pri) 5.00% 12.50% 7.50% 25.00%

15 3.00% 25 13.75% 35 8.75% Expected 25.50% return = c. Canon seems riskier than Nikon because it has a broader range of outcomes. Note the risk-return tradeoff because Canon has more risk but also provides a higher return.

Canon

P8-8

Bar charts and risk (LG 2; Intermediate) a.

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Bar Chart-Hug

Bar Chart-Stretch 0.80

Probability

Probability

0.80 0.60 0.40 0.20

0.40 0.20 0.00

0.00

b.

0.60

0.0075 0.0125 0.0850 0.1475 0.1625

0.0100 0.0250 0.0800 0.1350 0.1500

Expected Return (%)

Expected Return (%)

Jeans Line

Hug

Stretch

Market Acceptance Very Poor Poor Average Good Excellent

Probability Pri 0.05 0.15 0.60 0.15 0.05 1.00

Expected Return ri 0.0075 0.0125 0.0850 0.1475 0.1625 Expected return =

Weighted Value (ri  Pri) 0.000375 0.001875 0.051000 0.022125 0.008125 0.083500

Very Poor Poor Average Good Excellent

0.05 0.15 0.60 0.15 0.05 1.00

0.010 0.025 0.080 0.135 0.150 Expected return =

0.000500 0.003750 0.048000 0.020250 0.007500 0.080000

c. Line Stretch appears less risky because of a slightly tighter distribution of potential outcomes.

P8-9

Coefficient of variation (LG 2; Basic) Coefficient of variation is given by CV  where

r

, r is the standard deviation of returns, and

is expected return

a. The coefficients of variation for projects A, B, C, and D are 7% 6% A: CVA  C: CVC   0.3500  0.3158 20% 19% B: CVB 

9.5%  0.4318 22%

D: CVD 

5.5%  0.3438 16%

b. If the goal is simply to minimize risk (volatility), project D is attractive because it has the lowest standard deviation. If the goal is to minimize volatility relative to the expected return, however, then asset C is the best choice.

P8-10 Standard deviation versus coefficient of variation as measures of risk (LG 2; Basic) a. Project A is least risky based on range because it has the smallest value (4%).

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b. Project A has the lowest standard deviation. Standard deviation fails to take into account both the volatility and return of the investment and does not distinguish between a project’s diversifiable and nondiversifiable risk.  c. Coefficient of variation is given by CV  r , r where is the standard deviation of returns and is expected return. The coefficients of variation for projects A, B, C, and D are A: CVA 

0.029  0.2417 0.12

C: CVC 

0.035  0.2692 0.13

B: CVB 

0.032  0.2560 0.125

D: CVD 

0.030  0.2344 0.128

Project D may be the best alternative because it has the least amount of risk per percentage point of return. Coefficient of variation is probably the best measure here because it provides a standardized method of capturing the risk-return tradeoff for investments with differing returns.

That said, like the standard deviation, the coefficient of variation does not distinguish between an investment’s diversifiable and nondiversifiable risk.

P8-11 Personal finance problem: Rate of return, standard deviation, and coefficient of variation (LG 1, LG 2; Challenge) a.

Year 2017 2018 2019

Stock Price Beginning End Returns $31.45 $47.29 50.37% 47.29 46.02 -2.69% 46.02 50.57 9.89% b. Average return = 19.19%

r  r  n

c. Standard deviation is given by  

j 1

2

j

,

n 1 where rj is return in year j (running to year n) and is average return over n years. In this problem,

= 27.7%

√

d. Coefficient of variation is given by CV =  r where  is standard deviation of returns and r is expected return: 27.7%  19.19% = 1.44. e. Note how coefficient of variation provides additional perspective—for each unit of return, the stock carries 1.44 units of volatility. Dell stock exceeds Mike’s risk limit of a coefficient of variation of returns of 0.9.

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P8-12 Assessing return and risk (LG 2; Challenge) a.

Project 257 (1) Range: 1.00  (.10)  1.10 percentage points (2) Expected or average return: n

r   ri  Pri where ri = return for outcome i, and Pri is probability of outcome i: i 1

(3) Standard deviation:  

n

 (r  r )  P , where: 2

i 1

i

ri

ri is the return for outcome i, r is the average return across all outcomes, and Pri is the probability of outcome i.

Column →

(1) (2) Return Average (r i ) Return (r̅) -0.10 0.45 0.10 0.45 0.20 0.45 0.30 0.45 0.40 0.45 0.45 0.45 0.50 0.45 0.60 0.45 0.70 0.45 0.80 0.45 1.00 0.45

(3)

(4)

= (1) ─ (2)

= (3)

-0.550 -0.350 -0.250 -0.150 -0.050 0.000 0.050 0.150 0.250 0.350 0.550

0.3025 0.1225 0.0625 0.0225 0.0025 0.0000 0.0025 0.0225 0.0625 0.1225 0.3025

2

(5) (6) Probability = (4) x (5) (P ri )

0.010 0.003025 0.040 0.004900 0.050 0.003125 0.100 0.002250 0.150 0.000375 0.300 0.000000 0.150 0.000375 0.100 0.002250 0.050 0.003125 0.040 0.004900 0.010 0.003025 Sum = 0.027350 √ (Sum) = Standard Deviation (σ) = 0.16538 = 16.54%

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(4) Coefficient of variation (CV) is given by σ  r , where σ is the standard deviation of returns on the asset and r is the average return. So: CV 

0.165378  0.3675 0.450

Project 432 (1) Range: 0.50  0.10  0.40 percentage points (2) Expected or average return: n

r   ri  Pri where ri = return for outcome i, and Pri is probability of outcome i: i 1

Return

ri

Probability P ri

0.10 0.050 0.15 0.100 0.20 0.100 0.25 0.150 0.30 0.200 0.35 0.150 0.40 0.100 0.45 0.100 0.50 0.050 Sum = 1.000

(3) Standard deviation:  

Weighted Value

r i x P ri 0.0050 0.0150 0.0200 0.0375 0.0600 0.0525 0.0400 0.0450 0.0250 Sum = Average Return (r̅) = 0.300000 = 30.0% n

 (r  r )  P , where: 2

i 1

i

ri

ri is the return for outcome i, r is the average return across all outcomes, and Pri is the probability of outcome i.

Column →

(1) (2) Return Average (r i ) Return (r̅) 0.10 0.30 0.15 0.30 0.20 0.30 0.25 0.30 0.30 0.30 0.35 0.30 0.40 0.30 0.45 0.30 0.50 0.30

(3)

(4)

= (1) ─ (2)

= (3)2

-0.200 -0.150 -0.100 -0.050 0.000 0.050 0.100 0.150 0.200

0.0400 0.0225 0.0100 0.0025 0.0000 0.0025 0.0100 0.0225 0.0400

(5) (6) Probability = (4) x (5) (P ri )

0.050 0.002000 0.100 0.002250 0.100 0.001000 0.150 0.000375 0.200 0.000000 0.150 0.000375 0.100 0.001000 0.100 0.002250 0.050 0.002000 Sum = 0.011250 √ (Sum) = Standard Deviation (σ) = 0.106066 = 10.61%

(4) Coefficient of variation (CV) is given by σ  r , where σ is the standard deviation of returns on the asset and r is the average return. So: CV = 0.106066  0.300 = 0.3536.

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b. Bar charts

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Chapter 8

c. Summary statistics Range Expected return (r ) Standard deviation ( r ) Coefficient of variation (CV)

Project 257 1.100 0.450 0.165 0.368

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Project 432 0.400 0.300 0.106 0.354

Project 432 has a lower range, standard deviation, and coefficient of variation, so it appears the less risky option (though, again, the given information does not allow decomposition of return volatility into diversifiable and nondiversifiable components).

P8-13 Integrative: Expected return, standard deviation, and coefficient of variation (LG 2; Challenge) a. Expected or average return: n

r   ri  Pri where ri = return for outcome i, and Pri is probability of outcome i: i 1

Return Asset

F

Asset

G

Asset

H

ri

Probability P ri

0.40 0.100 0.10 0.200 0.00 0.400 -0.05 0.200 -0.10 0.100 Sum = 1.000

ri

r i x P ri 0.0400 0.0200 0.0000 -0.0100 -0.0100 Sum = Average Return (r̅) = 0.040000 = 4.0%

r i x P ri

P ri

0.35 0.400 0.10 0.300 -0.20 0.300 Sum = 1.000

ri

Weighted Value

0.1400 0.0300 -0.0600 Sum = Average Return (r̅) = 0.110000 = 11.0%

r i x P ri

P ri

0.40 0.100 0.20 0.200 0.10 0.400 0.00 0.200 -0.20 0.100 Sum = 1.000

0.0400 0.0400 0.0400 0.0000 -0.0200 Sum = Average Return (r̅) = 0.100000 = 10.0%

Asset G provides the largest average return.

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b. Standard deviation:  

n

 (r  r )  P , , where ri is the return for outcome i, r is the 2

i 1

i

ri

average return across all outcomes, and Pri is the probability of outcome i.

Asset F Column →

(1) (2) Return Average (r i ) Return (r̅) 0.40 0.04 0.10 0.04 0.00 0.04 -0.05 0.04 -0.10 0.04

(3)

(4)

= (1) ─ (2)

= (3)

0.360 0.060 -0.040 -0.090 -0.140

0.1296 0.0036 0.0016 0.0081 0.0196

2

(5) (6) Probability = (4) x (5) (P ri )

0.100 0.012960 0.200 0.000720 0.400 0.000640 0.200 0.001620 0.100 0.001960 Sum = 0.017900 √ (Sum) = Standard Deviation (σ) = 0.133791 = 13.38%

Asset F Column →

(1) (2) Return Average (r i ) Return (r̅) 0.40 0.04 0.10 0.04 0.00 0.04 -0.05 0.04 -0.10 0.04

(3)

(4)

= (1) ─ (2)

= (3)

0.360 0.060 -0.040 -0.090 -0.140

0.1296 0.0036 0.0016 0.0081 0.0196

2

(5) (6) Probability = (4) x (5) (P ri )

0.100 0.012960 0.200 0.000720 0.400 0.000640 0.200 0.001620 0.100 0.001960 Sum = 0.017900 √ (Sum) = Standard Deviation (σ) = 0.133791 = 13.38%

Asset G Column →

(1) (2) Return Average (r i ) Return (r̅) 0.35 0.11 0.10 0.11 -0.20 0.11

(3)

(5) (6) Probability = (1) ─ (2) = (3)2 = (4) x (5) (P ri ) 0.240 0.0576 0.400 0.023040 -0.010 0.0001 0.300 0.000030 -0.310 0.0961 0.300 0.028830 Sum = 0.051900 √ (Sum) = Standard Deviation (σ) = 0.227816 = 22.78%

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Asset H Column →

(1) (2) Return Average (r i ) Return (r̅) 0.40 0.10 0.20 0.10 0.10 0.10 0.00 0.10 -0.20 0.10

(3)

(4)

= (1) ─ (2)

= (3)

0.300 0.100 0.000 -0.100 -0.300

0.0900 0.0100 0.0000 0.0100 0.0900

2

(5) (6) Probability = (4) x (5) (P ri )

0.100 0.009000 0.200 0.002000 0.400 0.000000 0.200 0.002000 0.100 0.009000 Sum = 0.022000 √ (Sum) = Standard Deviation (σ) = 0.148324 = 14.83%

Based on standard deviation, Asset G appears to have the greatest risk. c. Coefficient of variation (CV) is given by σ  r , where so σ is the standard deviation of returns on the asset and r is the average return. So: Asset F: CV = 0.1338  0.04 = 3.345 Asset G: CV = 0.2278  2.071

Asset H: CV = 0.1483  0.10 = 1.483

As measured by the coefficient of variation, Asset F has the largest relative risk.

P8-14 Rate of return, standard deviation, and coefficient of variation (LG 1, LG 2 Intermediate) a.

Calculate the monthly rate of return for each stock.

b.

Calculate the average monthly return for each stock.

Month Monthly return BLK May-20 –3.23% Apr-20 14.97% Mar-20 –4.97% Feb-20 –12.20% Jan-20 5.62% Dec-19 1.57% Nov-19 7.19% Oct-19 4.43% Sep-19 5.46% Aug-19 –9.65% Jul-19 0.42% Jun-19 12.93% Average Return = 1.88%

c.

Monthly return KKR –3.33% 7.41% –17.62% –10.35% 9.36% –0.68% 2.31% 7.36% 4.43% –3.41% 5.85% 14.03% 1.28%

Calculate the standard deviation of monthly returns for each stock.

BLK Month

Returns

Average Return

May-20

–3.23%

1.88%

–5.11%

0.00261475

Apr-20

14.97%

1.88%

13.09%

0.01714551

Mar-20

–4.97%

1.88%

–6.85%

0.00469693

rF – rF

(rF – rF )2

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Feb-20

–12.20%

1.88%

–14.08%

0.01982498

Jan-20

5.62%

1.88%

3.74%

0.00139995

Dec-19

1.57%

1.88%

–0.30%

9.2663E-06

Nov-19

7.19%

1.88%

5.31%

0.00282264

Oct-19

4.43%

1.88%

2.55%

0.00065258

Sep-19

5.46%

1.88%

3.58%

0.00128392

Aug-19

–9.65%

1.88%

–11.53%

0.01328381

Jul-19

0.42%

1.88%

–1.46%

0.0002136

Jun-19

12.93%

1.88%

11.05%

0.01221502

Sum of Squared Differences =

7.6%

n – 1=

11.0

Sum of Squared Differences / (n – 1) = Standard Deviation =

0.7% 8.3%

KKR Month

Returns

Average Return

rF - rF

(rF - rF )2

May-20

–3.33%

1.28%

–4.61%

0.00212669

Apr-20

7.41%

1.28%

6.13%

0.00376275

Mar-20

–17.62%

1.28%

–18.90%

0.0357203

Feb-20

–10.35%

1.28%

–11.63%

0.01353039

Jan-20

9.36%

1.28%

8.08%

0.0065292

Dec-19

–0.68%

1.28%

–1.96%

0.00038539

Nov-19

2.31%

1.28%

1.03%

0.0001057

Oct-19

7.36%

1.28%

6.08%

0.00369392

Sep-19

4.43%

1.28%

3.15%

0.00099227

Aug-19

–3.41%

1.28%

–4.69%

0.00219717

Jul-19

5.85%

1.28%

4.57%

0.00209042

Jun-19

14.03%

1.28% 12.75% Sum of Squared Differences = n - 1= Sum of Squared Differences / (n-1) = Standard Deviation =

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0.01626003 8.7% 11.0 0.8% 8.9%

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d. Based on parts b and c, determine the coefficient of variation for each stock.

BLK Standard deviation, σF

8.3%

Avg. return, r̅F

1.88%

Coefficient of variation, CVF

4.43

KKR Standard deviation, σF

8.9%

Avg. return, r̅F

1.28%

Coefficient of variation, CVF

6.97

P8-15 Normal probability distribution (LG 2; Challenge) a. Coefficient of variation: CV = σ  r , where so σ is the standard deviation of returns on the asset and r is the expected return. So, given the CV (0.75) and the expected return (0.189), solve for standard deviation: 0.75    0.189 →  0.750.189  0.14175 b. (1) 68% of the outcomes will lie between 1 standard deviation of the expected value, so: +1σ = 0.0189 + 0.14175 = 0.33075

–1σ = 0.0189 – 0.14175 = 0.04725

(2) 95% of the outcomes will lie between  2 standard deviations of the expected value: +2σ = 0.0189 + (2 0.14175) = 0.4725 0.0945

–2σ = 0.0189 – (2 0.14175) = –

(3) 99% of the outcomes will lie between 3 standard deviations of the expected value: +3σ = 0.0189 + (3 0.14175) = 0.61425 –2σ = 0.0189 – (3 0.14175) = – 0.23625 c.

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P8-16 Personal finance problem: Portfolio weights (Basic) a.

How much money do you have invested in each stock?

Stock Price Shares Investment

c. Weights

Chipotle

$902.59

23

$20,759.57

0.81

McDonald's

$172.82

14

$2,419.48

0.09

Shake Shack

$47.67

35

$1,668.45

0.07

Wendy's b. Total Portfolio

$19.88

40

$795.20

0.03

112

$25,642.70

1.00

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P8-17 Portfolio return and standard deviation (LG 3; Challenge) a. Actual portfolio return for each year: rp  (wLrL)  (wMrM), where w is the portfolio weight of asset L or M, and r is the actual return in a given year on asset L or M. Year 201 4 201 5 201 6 201 7 201 8 201 9

Asset L (wLrL) (13.63%  0.40)



Asset M (wMrM)  (19.12%0.60)

Portfolio Return (rp)  16.92%

(1.35%  0.40)

 (9.54%0.60 )



6.26%

(11.93%  0.40)

 (7.01%0.60 )



8.98%

(21.78%  0.40)

 (32.70%0.60 )



28.33%

(-4.42%  0.40)

 (-0.14%0.60 )



-1.85%

(31.46%  0.40

 (39.12%0.60)



36.06%

b. Average return, ETF VOO (rVOO): (13.63% + 1.35% + 11.93% + 21.78% + –4.42% + 31.46%)  6 = 12.62% Average return, ETF QQQ (rQQQ): (19.12% + 9.54% + 7.01% + 32.70% + –0.14% + 39.12%)  6 = 17.89% Portfolio return (rp): (16.92% + 6.26% + 8.98% + 28.33% + –1.85% + 36.06%)  6 = 15.78% n

(ri  r ) 2

 (n  1) ,

c. Standard deviation for VOO and QQQ:  rp 

i 1

where ri is the asset return in year i (running to year n), and r is average return over n years: Standard deviation for VOO:

13.63%  12.62%   1.35%  12.62%     31.46%  12.62%  2



2

2

 13.10%

5

Standard deviation for QQQ:

19.12%  17.89%    9.54%  17.89%     39.12%  17.89%  2



2

2

 15.39%

5

Standard deviation of portfolio:

16.92%  15.78%    6.26%  15.78%      36.06%  15.78%  2



2

2

 14.27% 5 The standard deviation of the portfolio returns is less than the standard deviation of QQQ but more than for VOO by itself.

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d.

Year 2014 2015 2016 2017 2018 2019

r VOO 13.63% 1.35% 11.93% 21.78% -4.42% 31.46%

r QQQ 19.12% 9.54% 7.01% 32.70% -0.14% 39.12%

Correlation coefficient, ρ LM

0.94

The two ETFs VOO and QQQ are very strongly positively correlated. They are very close to perfect positive correlation of 1.00. e. The two assets VOO and QQQ are nearly perfectly positively correlated. The diversification benefit would be greater for Jamie if she selected assets with less positive correlation. As it is with this portfolio, her portfolio standard deviation is less than asset QQQ, but more than VOO. Ideally, Jamie would seek two assets such that the portfolio standard deviation is less than that of both assets.

P8-18 Portfolio analysis (LG 3; Challenge) a. Expected portfolio return: Alternative 1: 100% Asset F rp 

16%  17%  18%  19%  17.5% 4

Alternative 2: 50% Asset F  50% Asset G Asset F Year (wFrF) 2019 2020 2021 2022

(16%0.50  8.0%) (17%0.50  8.5%) (18%0.50  9.0%) (19%0.50  9.5%) rp 

   

Asset G (wGrG) (17%0.50  8.5%) (16%0.50  8.0%) (15%0.50  7.5%) (14%0.50  7.0%)

2020 2021



16.5%



16.5%



16.5%



16.5%

16.5%  16.5%  16.5%  16.5%  16.5% 4

Alternative 3: 50% Asset F  50% Asset H Asset F Year (wFrF) 2019

Portfolio Return rp

(16%0.50  8.0%) (17%0.50  8.5%) (18%0.50 

  

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Asset H (wHrH)

Portfolio Return rp

(14%0.50  7.0%) (15%0.50  7.5%) (16%0.50 



15.0%



16.0%



17.0%

Chapter 8

2022

9.0%) (19%0.50  9.5%) rp 



8.0%) (17%0.50  8.5%)

Risk and Return



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18.0%

15.0%  16.0%  17.0%  18.0%  16.5% 4

b. Standard deviation:  rp 

(ri  r ) 2 ,  i 1 ( n  1) n

where ri is asset return in year i (running to year n), and r is average return over n years: Alternative 1: Alternative 1: 100% Asset F Year 2019 2020 2021 2022

Return 16.0% 17.0% 18.0% 19.0%

Average Return Return - Avg. Return (Return - Avg. Return)2 17.5% -1.5% 0.000225 17.5% -0.5% 0.000025 17.5% 0.5% 0.000025 17.5% 1.5% 0.000225 Sum of Squared Differences = 0.000500 n-1 3 Sum of Squared Differences / (n-1) = 0.000167 Standard Deviation = σ = 1.291%

Alternative 2: Alternative 2: 50% Asset F + 50% Asset G Year 2019 2020 2021 2022

Alternative 3:

Return 16.5% 16.5% 16.5% 16.5%

Average Return Return - Avg. Return (Return - Avg. Return)2 16.5% 0.0% 0.000000 16.5% 0.0% 0.000000 16.5% 0.0% 0.000000 16.5% 0.0% 0.000000 Sum of Squared Differences = 0.000000 n-1 3 Sum of Squared Differences / (n-1) = 0.000000 Standard Deviation = σ = 0.000%

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Alternative 3: 50% Asset F + 50% Asset H Year 2019 2020 2021 2022

Return 15.0% 16.0% 17.0% 18.0%

Average Return Return - Avg. Return (Return - Avg. Return)2 16.5% -1.5% 0.000225 16.5% -0.5% 0.000025 16.5% 0.5% 0.000025 16.5% 1.5% 0.000225 Sum of Squared Differences = 0.000500 n-1 3 Sum of Squared Differences / (n-1) = 0.000167 Standard Deviation = σ = 1.291%

c. Coefficient of variation (CV)  r  r , where σr is standard deviation of the investment alternative and r is the average return of the investment alternative: 1.291% 0 1.291% CVF   0.0738 CVFG  0 CVFH   0.0782 17.5% 16.5% 16.5% d. Summary: r CV r Alternative 1 17.5% 1.291% 0.0738 (F) Alternative 2 16.5% 0 0.0 (FG) Alternative 3 16.5% 1.291% 0.0782 (FH) Alternative 1 posted the highest return but Alternative 2 the lowest volatility (risk). When thinking about performance, it is instructive to ask how a hypothetical investor might view these alternatives. She would first note Alternative 2 is clearly preferable to Alternative 3 because it offers the same expected return but no volatility in returns. Now, as between Alternatives 1 and 2, Alternative 1 offers a higher expected return but also has more volatile returns. Without knowing an investor’s risk tolerance, it is not possible to say whether Alternative 1 or 2 is ―best.‖ P8-19 Portfolio efficiency (LG 4; Challenge) Not necessarily. An efficient portfolio maximizes return for any standard deviation, which is not the same thing as minimizing standard deviation for any return. In Figure 8.5, for example, the portfolio consisting of 100% of Medtronic has the lowest possible standard deviation given a return of 2.1%, but that portfolio is not efficient. There is another portfolio with the same standard deviation that has a higher return.

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P8-20 Portfolio efficiency (LG 4; Intermediate)

P8-21 Correlation, risk, and return (LG 4; Intermediate) a. (1) Expected return on the portfolio will range from 8% (expected return on Asset V) to 13% (expected return on Asset W). The degree of correlation between Asset V and Asset W does not affect does not affect expected return on the portfolio.

(2) Standard deviation of the portfolio will range from 14% (standard deviation of Asset V) and 19% (standard deviation of Asset W), depending on the weights of the assets in the portfolio. b. (1) Expected return on the portfolio will range from 8% (expected return on Asset V) to 13% (expected return on Asset W). The degree of correlation between Asset V and Asset W does not affect expected return on the portfolio.

(2) Students do not have enough information to determine the precise minimum standard deviation obtainable through diversification, but it certainly exceeds 0% (because that is only achievable when the correlation coefficient is –1.0), and likely to be less than 14% (i.e., less than the standard deviation of V). The upper bound for risk is still 19%, which occurs if 100% of the portfolio is invested in asset W. c.

(1) Expected return on the portfolio will range from 8% (expected return on Asset V) to 13% (expected return on Asset W). The degree of correlation between Asset V and Asset W does not affect does not affect expected return on the portfolio.

(2) Standard deviation of the portfolio will range from 0% (standard deviation if assets are perfectly negatively correlated and ―correct‖ portfolio weights are chosen) to 19% (standard deviation of Asset W).

P8-22 Personal finance problem: Correlation, risk, and return (LG 1 and LG 4; Intermediate) a. Calculate the correlation between SNAP and TWTR monthly returns

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Month May-19 Jun-19 Jul-19 Aug-19 Sep-19 Oct-19 Nov-19 Dec-19 Jan-20 Feb-20 Mar-20 Apr-20 May-20

SNAP 6.7% 20.3% 17.5% -5.8% -0.2% -4.7% 1.3% 7.1% 12.6% -22.9% -16.1% 48.1% -4.8% Correlation

TWTR -8.7% -4.2% 21.2% 0.8% -3.4% -27.3% 3.1% 3.7% 1.3% 2.2% -26.0% 16.8% -1.9% 0.57

b. Calculate the correlation between UNH and GD monthly returns

Month May-19 Jun-19 Jul-19 Aug-19 Sep-19 Oct-19 Nov-19 Dec-19 Jan-20 Feb-20 Mar-20 Apr-20 May-20

UNH 5.2% -15.1% 2.2% 6.9% 7.5% -4.8% -9.7% -14.4% 7.7% 6.4% -2.4% -0.9% -3.6% Correlation

GD -9.5% 13.1% 2.3% 3.4% -4.5% -3.2% 3.4% -3.0% -0.5% -8.5% -17.1% -1.3% 1.5% -0.38

c. SNAP and TWTR are positively correlated while UNH and GD are negatively correlated.

P8-23 Total, nondiversifiable, and diversifiable risk (LG 5; Intermediate) a. and b. c. Only nondiversifiable risk is relevant because, as shown above, building a portfolio of at least 20 securities with imperfectly correlated returns substantially reduces diversifiable risk. When additional securities no longer reduce risk, the remaining standard deviation of David Talbot’s portfolio is non-diversifiable. The market would not reward David for taking the extra risk associated with a portfolio of a small number of investments because it is easy for David to eliminate the risk via diversification. The market will reward David for the nondiversifiable risk that he takes.

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P8-24 Graphic derivation of beta (LG 5; Intermediate) a.

b. The betas for assets A and B are the slopes of the characteristic lines above. Typically, the slopes of these lines are estimated with a statistical technique known as linear regression. But, slope can also be calculated given any two points on a line. Here, we can obtain slopes (i.e., betas) with the highest and lowest returns for each asset. Specifically: Beta = ∆ Asset Return  ∆ Market Return BetaAsset A = Highest Return on Asset A – Lowest Return on Asset A

Highest Market Return – Lowest Market Return

= [0.19 – (–0.04)]  [0.16 – (–0.13)] = 0.23  0.29 = 0.793 BetaAsset B = Highest Return on Asset B – Lowest Return on Asset B Highest Market Return – Lowest Market Return = [0.30 – (–0.10)]  [0.16 – (–0.13)] = 0.40  0.29 = 1.38 c. With a higher beta, Asset B has more nondiversifiable risk (i.e., the risk that really matters to investors). For each one percentage point movement in the market return, Asset B will move 1.38 percentage points (compared with 0.793 percentage points for Asset B).

P8-25 Graphic derivation of beta (LG 5; Intermediate) a. The returns for Biotech are more disperse, so it has a higher standard deviation. b. The slope of the line is greater for Cyclical, which means that Cyclical stock tends to move more when the market moves than the Biotech stock. Thus, Cyclical has the higher beta. c. Cyclical has the higher beta (more nondiversifiable risk), so it should offer the higher expected return. Biotech is more volatile, but diversification can eliminate most of that risk. Only if forced to hold Cyclical or Biotech in isolation would Biotech be riskier (and

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even then investors holding Biotech should not expect the market to reward them with a higher return).

P8-26 Interpreting beta (LG 5; Basic) The impact of a change in market return for an asset with a beta of 1.20 if market return rises by: a. 15%: 1.20  (15%) = 18.0% increase in the asset’s return. b. 8%: 1.20  (8%) =9.6% decrease in the asset’s return. c. 0%: 1.20  (0%) = no change in the asset’s return. d. With a beta of 1.2, this asset is riskier than the market portfolio (which has a beta of 1.0).

The return moves 1.2 percentage points for every percentage point movement in the market return.

P8-27 Betas (LG 5; Basic) a., b.

Asset A

B

C

D

Beta 0 .

5 0 1 .

6 0  0 .

2 0 0 .

9 0

∆ Market Return 0.10 

∆ Asset Return = 0.0 5

0.10

= 0.1 6

B

0.10

 0.

02

C

0.10

= 0.0 9

D







Asset A

Beta 0 .

5 0 1 .

6 0  0 .

2 0 0 .

9 0

∆ Market Return 0.1  0 

∆ Asset Return  0 .0 5

0.1 0

 0 .1 6

0.1 0

= 0.

02

0.1 0

 0 .0 9

   

 

c. Asset B should be chosen because it will have the highest increase in return. d. Asset C would be the appropriate choice because it is a defensive asset, moving in opposition to the market. In an economic downturn, Asset C’s return is increasing.

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P8-28 Personal finance plan: Betas and risk rankings (LG 5; Intermediate) a., b., c.

Asset B

Beta 1.40

A

0.80

C 0.30

Risk Level Most Risk

∆ Market Return 0.12

∆ Asset Return 0.16 8

∆ Market Return 0.05

0.12

0.09 6

0.05

0.12

0.

036

0.05

Least Risk

∆ Asset Return 0.

07 0 0.

04 0 0.0 15

d. In a declining market, Stock C is the most attractive. The negative beta means the return will rise if the market return declines. e. Stock B is the most attractive in a rising market because it has the largest beta. A one percentage point rise in the market return leads to a 1.4 percentage point increase in Stock B’s return.

P8-29 Personal finance problem: Portfolio betas (LG 5; Intermediate) The beta of a portfolio is simply the weighted average of the betas of the individual securities in the portfolio, with portfolio shares serving as weights. More formally: n

 p w j   j , j 1

where  p is portfolio beta,  j is beta of asset j, and w j = portfolio share of the asset j (running to n) a.

Asset 1 2 3 4 5

Portfolio A Beta wj wjbj 1.30 0.1 0.13 0 0 0.70 0.3 0.21 0 0 1.25 0.1 0.12 0 5 1.10 0.1 0.11 0 0 0.90 0.4 0.36 0 0  p of Asset A  0.

93 5

Portfolio B wj wjbj 0.3 0.39 0 0.1 0.07 0 0.2 0.25 0 0.2 0.22 0 0.2 0.18 0  p of Asset 1.

B  11

b. Portfolio A is less risky than the market (average risk, beta = 1.0), and Portfolio B is more risky. When the market return increases by one percentage point, the return on portfolio A will rise by 0.935 percentage points and the return on Portfolio B will rise 1.11 percentage points.

P8-30

Capital asset pricing model—CAPM (LG 6; Basic)

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The CAPM equation is rj  RF  [j(rm  RF)], where rj is the expected/required return on asset j, j is beta for asset j, RF is the risk-free rate, and rm is the expected return on the market portfolio: Case RF  [ j(rm   rj RF)] A 1%   10.1% [1.30(8%  1%)] B 2%   5.6% [0.90(13%  2%)] C 5%   3.4% [0.20(12%  5%)] D 6%   12.0% [1.00(15%  6%)] E 4%  = 7.6% [0.60(10%  4%)] P8-31 Personal finance problem: Beta coefficients and the capital asset pricing model (LG 5 and LG 6; Intermediate) The CAPM equation is rj  RF  [j(rm  RF)], where rj is the expected or required return on asset j, j is the beta coefficient for asset j, RF is the risk-free rate, and rm is the expected return on the market portfolio. The problem provides values for rj, RF and rm and asks for beta; using the CAPM equation: (i) rj  RF  [j(rm  RF)]

(iii) j(rj  RF)  (rm  RF)]

(ii) rj  RF  j(rm  RF) a.

Beta 

10%  5% 5%   0.4545 16%  5% 11%

c.

Beta 

18%  5% 13%   1.1818 16%  5% 11%

b.

Beta 

15%  5% 10%   0.9091 16%  5% 11%

d.

Beta 

20%  5% 15%   1.3636 16%  5% 11%

e. If Katherine is willing to take no more than the average amount of risk, her upper limit on a beta is 1.0. Given a beta of 1.0, her expected return is 16% [r  5%  1.0(16%  5%)  16%.].

P8-32 Manipulating CAPM (LG 6; Intermediate) Using the CAPM equation: rj  RF  [  j (rm  RF)], where rj is the required return on asset j, RF the risk-free rate,  j the beta on asset j and rm the return on the market portfolio: a. Solve for rj if RF is 8%,  j is 0.90, and rm is 12%: 8%  [0.90(12%  8%)] =11.6%. b. Solve for RF if rj is 15%,  j is 1.25, and rm is 14%: 15% = RF  [1.25(14%  RF)] →

RF = 10%.

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c. Solve for rm if RF is 9%, if rj is 16%, and  j is 1.10: 16% = 9%  [1.10(rm  9%)] → rm = 15.36%. d. Solve for  j if rj is 15%, RF is 10%, and rm is 12.5%: 15% = 10%  [  j (12.5%  10%)] →  j = 2.0 P8-33 Personal finance problem: Portfolio return and beta (LG 1, LG 3, LG 5, and LG 6; Challenge) a. The beta of a portfolio asset is the weighted average of the betas of the individual assets in the portfolio, with the weight on each asset being that asset’s share in the portfolio.

Specifically, n

n

j 1

j 1

 p = w j   j and w j = 1 →  p  ( wA   A )  ( wB   B )  ( wC  C )  ( wD   D )

b.

Return 

rA 

rB 

[(ending value  initial value)  cash distribution] initial value

($20,000  $20,000)  $1,600 $1,600   8% $20,000 $20,000 ($34,500  $30,000)  0 $4,500    15% $30,000 $30,000

rC

($36,000  $35,000)  $1, 400 $2, 400   6.86% rD $35,000 $35,000 ($16,500  $15,000)  $375 $1,875    12.5% $15,000 $15,000

c. Total value of the portfolio today is $107,500, and total annual income from Assets A–D during the year was $3,375, so portfolio return is given by: rP 

($107,000  $100,000)  $3,375 $10,375   10.375% $100,000 $100,000

d. Using the CAPM equation, rj  RF  [  j (rm  RF)], where rj is the required return on asset j, RF the risk-free rate (given as 4%),  j the beta on asset j and rm the return on the market portfolio (given as 10%), solve for required return on each project: rA  4%  [0.80(10%  4%)] = 8.8%

rC  4%  [1.50(10%  4%)] = 13.0%

rB  4%  [0.95(10%  4%)] = 9.7%

rD  4%  [1.25(10%  4%)] = 11.5%

e.

Security analysts typically use statistical techniques to estimate an asset’s beta by obtaining a line of best fit through historical asset and market returns. The slope of this line is beta. Data points—that is, actual returns on the asset and market for a given

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Zutter/Smart • Principles of Managerial Finance, Sixteenth Edition

period—will be randomly scattered around the line no matter how well it ―fits‖ the data.

The point here is asset betas are estimates. The CAPM return is, in a sense, a forecast and even good forecasts are subject to random error. Another possibility is beta does not fully capture all nondiversifiable or systemic factors that affect expected returns. Still another possibility is the firm behind the asset has changed, so the beta estimated with historical data does not reflect the asset’s current beta.

P8-34 Security market line (SML) (LG 6; Intermediate) a, b, d. c. Using the CAPM equation: rj  RF  [  j (rm  RF)], where rj is the required return on asset j, RF the risk-free rate,  j the beta on asset j and rm the return on the market portfolio: Asset A: rj  0.09  [0.80(0.13  0.09)] = 0.122 or 12.2% Asset B: rj  0.09  [1.30(0.13  0.09)] = 0.142 or 14.2% d. Asset A has the smaller beta, hence the smaller risk premium and required return.

Specifically, the risk premium is 3.2% (12.2%  9%)—compared with 5.2% for Asset B’s (14.2%  9%).

P8-35 Shifts in the security market line (LG 6; Challenge) a, b, c, d. b. Using the CAPM equation: rA  RF  [  A (rm  RF)], where RF is the risk-free rate (here 8%),  A the beta on asset A (here 1.1) and rm the return on the market portfolio (here 12%), solve for rA is the required return on asset A: rA  8%  [1.1(12%  8%)] = 8%  4.4% = 12.4%. c. Using the CAPM equation, rA = RF  [  A (rm  RF)], with RF = 6%,  A = 1.1 and rm = 10%, solve for rA  6%  [1.1(10%  6%)] = 6%  4.4% = 10.4%. d. Using the CAPM equation, rA  RF  [  A (rm  RF)], with RF = 8%,  A = 1.1 and rm = 13%, solve for rA = 8%  [1.1(13%  8%)] = 8%  5.5% = 13.5%. e. (1) A decline in inflationary expectations reduces required return by the same amount for every beta—that is, produces a parallel downward shift of the SML. (2) Increased risk aversion gives the SML a steeper slope because a higher return is now be required for each beta.

P8-36 Integrative: Risk, return, and CAPM (LG 6; Challenge) a.

Using the CAPM equation, rj  RF  [  j (rm  RF)], where RF is the risk-free rate,  j the project beta  and rm the return on the market portfolio, solve for rj is the required return on each project j: Project

RF  [ β j (rm  RF)]

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rj

Chapter 8

A B C D E

9%  [1.5(14%  9%)] 9%  [0.75(14%  9%)] 9%  [2.0(14%  9%)] 9%  [0.0(14%  9%)] 9%  [(0.5)(14%  9%)]

Risk and Return

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 16.5 0%  12.7 5%  19.0 0%  9.00 %  6.50 %

b., c., and d. See SML graph below. c. The premium for project j’s nondiversifiable risk is  j (rm  RF). In words, the spread between the return on the market portfolio and risk-free rate is the risk premium necessary to induce a manager to undertake a project of average risk (i.e., one with the same nondiversifiable risk as the market portfolio). Multiplying this spread by the project beta—the co-movement of project returns with the market return—yields the risk premium for the project’s nondiversifiable risk. Projects A and C have more nondiversifiable risk than the market because their betas exceed 1.0; they require risk premiums above the average risk premium. Projects B, D, and E have less nondiversifiable risk than the market because their betas are less than 1.0; they carry risk premiums below the average risk premium). Projects D and E are particularly noteworthy. Project D requires no premium for nondiversifiable risk because its beta is 0.

Project E has a negative beta, meaning its return moves in the opposite direction of the market. Required return for Project E will actually be less than the risk-free rate. d. rA  9%  [1.5(12%  9%)] 13.50% 9.00% rB  9%  [0.75(12%  9%)] 11.25% 7.50% rC  9%  [2.0(12%  9%)] 15.00% e.

rD  9%  [0.0(12%  9%)] rE  9%  [0.5(12%  9%)]

When investor risk aversion declines, investors require lower returns for any given risk level (beta). The SML will rotate clockwise about the fixed risk-free rate (because the risk premium for an zero-beta asset will remain zero).

P8-37 ETHICS PROBLEM (LG 1; Intermediate) Investors expect managers to take risks with their money, so it is not unethical for managers to make risky investments with other people’s money. Managers do, however, have a duty to communicate truthfully with investors about the risk they take. Portfolio managers should not take risks they do not expect to generate returns sufficient to compensate investors for systemic risk.

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Zutter/Smart • Principles of Managerial Finance, Sixteenth Edition

 Case: Analyzing Risk and Return on Chargers Products’ Investments Case studies are available on www.pearson.com/mylab/finance This case requires students to use standard deviation, coefficient of variation, and CAPM to assess the trade-off between risk and return for two possible investments. a.

( Pt  Pt 1  Ct ) , Pt 1 where Pt is the end-of-year value, Pt-1 is the beginning-of-year value, and Ct is cash flow during the year.

Actual annual return for period t is given by rt 

Asset X:

Cash Flow (Ct) $1, 00 0 1,5 00

Ending Value (Pt) $22, 000

1,4 00 1,7 00

24,0 00 22,0 00

1,9 00 1,6 00 1,7 00

23,0 00 26,0 00 25,0 00

8

2,0 00

24,0 00

9

2,1 00 2,2 00

27,0 00 30,0 00

Year 1 2 3 4 5 6 7

10

Asset Y:

Year 1 2 3 4 5 6

Cash Flow (Ct) $1,500 1,600 1,700 1,800 1,900 2,000

21,0 00

Ending Value (Pt) $20,000 20,000 21,000 21,000 22,000 23,000

Beginning Value (Pt – 1) $20,00 0

Gain/ Loss $2, 00 0 22,000 1, 00 0 21,000 3,0 00 24,000 2, 00 0 22,000 1,0 00 23,000 3,0 00 26,000 1, 00 0 25,000 1, 00 0 24,000 3,0 00 27,000 3,0 00 Ten-Year Average =

Beginning Value (Pt – 1) $20,000 20,000 20,000 21,000 21,000 22,000

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Gain/ Loss $ 0 0 1,000 0 1,000 1,000

Annual Rate of Return 15.00 % 2.27 20.95 1.25 13.18 20.00 2.69 4.00 21.25 19.26 11.7 4%

Annual Rate of Return (rt) 7.50% 8.00 13.50 8.57 13.81 13.64

Chapter 8

7 8 9 10

2,100 2,200 2,300 2,400

23,000 24,000 25,000 25,000

Risk and Return

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23,000 0 9.13 23,000 1,000 13.91 24,000 1,000 13.75 25,000 0 9.60 Ten-Year Average = 11.14%

Mr. Sayou anticipates expected annual return over the next ten years will match average return over the past 10 years, so expected return on AssetXis 11.74%, and expected return on Asset Y is 11.14%. b.

Standard deviation of returns is given by  

n

 (r  r )  (n  1) , 2

i 1

i

where rt is return in year t, r is average return over n years, and n is the number of years. For Asset X: Asset X: Year 1 2 3 4 5 6 7 8 9 10

x 

rt 15.00% 2.27 20.95 1.25 13.18 20.00 2.69 4.00 21.25 19.26

rt – r r 11.74% 3.26% 11.74 9.47 11.74 9.21 11.74 12.99 11.74 1.44 11.74 8.26 11.74 -9.05 11.74 7.74 11.74 9.51 11.74 7.52 Sum of Squared Differences =

(rt – )2 0.001063 0.008968 0.008482 0.016874 0.000207 0.006823 0.008190 0.005991 0.009044 0.005655 0.071297

0.071297  0.07922  0.0890  8.90% 10  1

Coefficient of Variation (CVx) = x x = 8.90% 11.74% = 0.76 Ass et Y:

Ye ar 1

rt –  3.6 4% 3.1 4 2.36

11.14%

2

rt 7.50 % 8.00

3

13.50

11.14

4

8.57

11.14

5

13.81

11.14

2.5 7 2.67

6

13.64

11.14

2.50

7

9.13

11.14

8

13.91

11.14

2.0 1 2.77

11.14

(rt – )2 0.0013 25 0.0009 86 0.0005 57 0.0006 60 0.0007 13 0.0006 25 0.0004 04 0.0007 67

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Zutter/Smart • Principles of Managerial Finance, Sixteenth Edition

Y  c.

9

13.75

10

9.60

11.14

2.61

1.5 4 Sum of Squared Differences = 11.14

0.0006 81 0.0002 37 0.00695 5

0.006955  0.0773  0.0278  2.78% , and CVy = y y = 2.78% 11.14% = 0.25.

10  1

Summary statistics Expected return Standard deviation Coefficient of variation

Asset X 11.74% 8.90% 0.76

Asset Y 11.14% 2.78% 0.25

Comparing expected returns calculated in part (a), Asset X provides a return only slightly above that from Asset Y. At the same time, both the standard deviation of returns and coefficient of variation for Asset X are roughly three times greater than that for Asset Y. So Asset X appears significantly riskier than Asset Y. But the problem notes Mr. Sayou is choosing between X and Y to add to a diversified portfolio. That means he should care only about the nondiversifiable risk of the two assets. Standard deviation and coefficient of variation are measures of the total volatility of returns from both diversifiable and nondiversifiable shocks. For this reason, the better asset cannot be determined. d.

Using the CAPM equation: rj  RF  [(rm  RF)], where rj is the required return on asset j, RF the risk-free rate, the beta on asset j and rm the return on the market portfolio, the required returns on Asset X and Y are: CAPM Required Expected Return RF  [j(rm  RF)] Asset = Return Part (a) X = 11.8 11.74% 7%  [1.6(10%  % 7%)] Y = 10.3 11.14% 7%  [1.1(10%  % 7%)] Of the two assets, only Asset Y offers an expected return above the required return determined

by CAPM. Moreover, required return is calculated with the correct risk measure—beta—which captures nondiversifiable risk.

e.

Mr. Sayou wants to add X or Y to a diversified portfolio, so he should care only about the nondiversifiable risk of the two assets. [Standard deviation and coefficient of variation measure total volatility of returns from both diversifiable and nondiversifiable shocks.] CAPM will indicate the required return for the two assets based on their nondiversifiable risk (x = 1.60 and y = 1.10). As noted, of the two assets only Y offers an expected return above required return (based on nondiversifiable risk). So Asset Y should be recommended.

f.

(1) A one percentage point rise in expected inflation will boost the risk-free rate to 8% and market return to 11% and have the following effect on rx and ry: Asset X Y

CAPM: RF  [bj(rm RF)] 8%  [1.6(11%  8%)] 8%  [1.1(11%  8%)]

Required = Return (rj) = 12.8% = 11.3%

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Expected Return Part (a)

11.74% 11.14%

Chapter 8

Risk and Return

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Now, neither asset offers an expected return greater than required return.

(2) If investors become less risk averse, causing market return to fall to 9%, rx and ry will change to:

Asset X Y

CAPM RF  [bj(rm RF)] 7%  [1.6(9%  7%)] 7%  [1.1(9%  7%)]

Requir ed  Return = (rj) = 10.2 % = 9.2%

Expected Return

Difference

Part (a)

(Percentage Points)

11.74%

1.54

11.14%

1.94

The drop in market return causes required return to fall below expected returns for both assets. If limited to one asset, Mr. Sayou should choose Y because it provides the larger return compared with required return (and does so with less nondiversifiable risk).

 Spreadsheet Exercise Answers to Chapter 8’s stock portfolio analysis spreadsheet problem are available on www.pearson.com/mylab/finance.

 Group Exercise Group exercises are available on www.pearson.com/mylab/finance.

This exercise will give students insight into the world of stock-market analysis. Each group is asked to obtain stock-market data from several websites on the recent performance of its shadow firm and compare the numbers with a relevant market index over one and five years. Students will calculate annual returns, investigate correlation between returns and the market return, and graph the results.

As always, the instructor can modify the exercise the meet class needs, perhaps by adding other corporations to the comparison(s) and dropping more complex calculations.