Solution Manual for Corporate Finance, Canadian Edition, 5th Edition By Berk

Certainly! Here'samore detailed explorationof"Contemporary Business Mathematicswith Canadian Applications"thatexpands toapproximately 3000 words:### Introduction toContemporary Business Mathematicswith Canadian ApplicationsContemporary Business Mathematicswith Canadian Applications" isatextbook designed tobridgethegap between mathematicaltheoryandpractical business applications, tailored specifically forCanadian studentsandbusinesses. This comprehensive text coversawide rangeofmathematical concepts essential forunderstandingandmanaging various aspectsofbusiness operations intheCanadian context.inthis detailed exploration, we will delve intothekey components,themes,andrelevanceofthis textbook.

The textbook startswith foundational mathematical conceptsthatformthebasis formore complex calculationsandanalysesinbusiness settings. This includes arithmetic operations, algebraic equations,andbasic statistical measures. Understandingthese fundamental principles is crucial asthey providetheframework upon which more advanced topicsarebuilt.Financial mathematics isacornerstoneofbusiness education, particularlyinunderstandingthetime valueofmoneyandfinancial decision-making. Topics covered often include interest calculations, compound interest, annuities, loans,andinvestments.these conceptsareessential forevaluatingtheprofitabilityofinvestments, determining loan payments,andunderstandingtheimplicationsofinterest ratesonbusiness finances.#### 3. Inventory ControlEffective inventory management is vital forbusinesses toensure adequate stock levels while minimizing holding costs.thetextbook covers mathematical techniquesandmodels used forinventory control, including inventory turnover rates, economic order quantity (EOQ),andreorder points. Canadian applications may include considerations specific toCanadian supply chain dynamicsandregulations.#### 4. Marketing MathematicsMathematical tools playacrucial roleinmarketing decisionsbyproviding quantitative insights into consumer behavior, pricing strategies,andsales forecasts. Topics covered often include break-even analysis, pricing models, market segmentation,andpromotional budgeting. Case studiesandexamples tailored toCanadian markets illustrate howthese mathematical conceptsareappliedinreal-world marketing scenarios.#### 5. Data AnalysisandStatisticsIn today's data-driven business environment, proficiencyindata analysisandstatistical methods is essential.thetextbook typically covers statistical techniques used toanalyze business data, suchasdescriptive statistics, probabilitytheory, hypothesis testing,andregression analysis.these tools enable businesses toextract meaningful insights from data tosupport decision-makingandstrategic planning.#### 6. Spreadsheet ApplicationsSpreadsheetsarepowerful tools extensively usedinbusiness forfinancial modeling, data analysis,anddecision support.thetextbook includes guidanceonutilizing spreadsheet software like Excel toperform calculations, create financial models,andvisualize data. Practical exercisesandexamples help students develop proficiencyinusing spreadsheets forbusiness applications.### RelevanceofCanadian ApplicationsThe inclusionof"Canadian Applications" inthetextbook's title emphasizes its focusonintegrating Canadian business practices, regulations,andeconomic conditions intothelearning experience. This ensuresthatstudents not only grasp mathematical concepts but also understand howthese conceptsareappliedwithintheunique contextofCanada.

Examplesandcase studies drawn from Canadian businesses provide practical insightsandrelevance, preparing students forthechallengesandopportunitiesthey may encounter intheCanadian business landscape.### Pedagogical ApproachandLearning Resources#### Pedagogical ApproachThe textbook typically adoptsapedagogical approachthatcombinestheoretical explanationswith practical examplesandapplications. Conceptsarepresented inaclearandstructured manner tofacilitate understanding,with step-by-step explanationsofmathematical calculationsandproblem-solving techniques. This approach helps students develop both mathematical proficiency andtheability toapplythese skills toreal-world business scenarios.#### Learning ResourcesIn addition tothetextbook itself, supplementary learning resources often accompany "Contemporary Business Mathematicswith Canadian Applications" toenhance student learningandengagement.these may include:- **Online Resources**: Access tointeractive tutorials, practice exercises,andsolutions.- **Case Studies**: Detailed analysesofreal-world business scenarios relevant toCanadian markets.- **Workbook Exercises**: Additional problemsandexercises toreinforce learningandmasteryofmathematical concepts.- **Instructor Support**: Guidanceandsupport forinstructors toeffectively teachandintegratethetextbook intotheir courses.### BenefitsandPractical Applications#### Benefits forStudents- **Practical Relevance**:byfocusingonCanadian applications,thetextbook prepares students toapply mathematical concepts inthespecific contextofCanadian businesses, enhancingtheir readiness forcareers intheCanadian business landscape.**Skill Development**: Students develop critical quantitative skills thatarehighly valuedinvarious business disciplines, including finance, marketing, operations,andstrategic management.**Problem-Solving Abilities**: Through exercisesandcase studies, students learn toanalyze business problems, formulate mathematical models,andderive solutions, improvingtheir analyticalandproblem-solving abilities.#### Practical ApplicationsinBusiness- **Financial Decision-Making**: Businesses use financial mathematics toevaluate investment opportunities, manage cash flow,andmake informed financial decisions.- **Marketing Nursing is a profession deeply rooted in caring for individuals, families, and communities to promote

health and well-being, prevent illness and injury, and provide support to those experiencing health challenges. It encompasses a wide range of roles, from direct patient care to leadership in healthcare policy and education. In this essay, we will explore the multifaceted nature of nursing, its historical evolution, educational pathways, professional responsibilities, ethical considerations, and its crucial role in the healthcare system.### Historical Evolution of NursingThe history of nursing dates back centuries and has evolved significantly over time. Nursing as a formalized profession began to take shape in the 19th century with the pioneering efforts of figures like Florence Nightingale, who is often regarded as the founder of modern nursing. Nightingale's work during the Crimean War demonstrated the critical importance of sanitation and hygiene in healthcare settings, laying the foundation for evidence-based nursing practice.Throughout the 20th century, nursing continued to professionalize and expand its scope. World War I and World War II highlighted the essential role of nurses in caring for wounded soldiers and civilians, leading to advancements in nursing education and practice. The establishment of nursing schools and regulatory bodies further solidified nursing as a respected profession with defined standards of practice.### Educational Pathways in NursingToday, nursing education encompasses a variety of pathways, from diploma programs to bachelor's and master's degrees in nursing (BSN, MSN) and doctoral programs (DNP, PhD). Each level of education prepares nurses for different roles within the profession, ranging from bedside care to advanced clinical practice, research, and leadership positions.Diploma programs, typically offered by hospitals, provide a more hands-on, clinical-focused education. Associate degree programs (ADN) and BSN programs offered by colleges and universities combine clinical training with a broader education in sciences, humanities, and nursing theory. BSN programs are increasingly becoming the preferred entry-level education for nurses due to their emphasis on critical thinking, leadership, and evidence-based practice.Advanced practice roles such as nurse practitioners (NPs), nurse anesthetists (CRNAs), nurse midwives (CNMs), and clinical nurse specialists (CNSs) require additional education at the master's or doctoral level. These nurses are trained to provide specialized care, prescribe medications, and diagnose health conditions, expanding access to healthcare services, particularly in underserved areas.### Professional Responsibilities and RolesThe responsibilities of nurses vary depending on their level of education, specialization, and practice setting. At the core of nursing practice is the provision of compassionate, patient-centered care. Nurses assess patients' health needs, develop and implement care plans, monitor patient progress, and advocate for their wellbeing. They collaborate with other healthcare professionals, such as physicians, therapists, and social workers, to ensure comprehensive care.In addition to direct patient careStrategy**: Mathematical models aidindetermining optimal pricing strategies, forecasting sales volumes,andassessingtheeffectivenessofmarketing campaigns.- **Operations Management**: Inventory control techniques help businesses optimize inventory levels, minimize costs,andensure timely availabilityofproducts.### Conclusion"Contemporary Business Mathematicswith Canadian Applications" isavaluable resourcethatequips studentswiththemathematical skillsandknowledge essential forsuccessintoday's competitive business environment.byintegrating Canadian-specific examplesandapplications,thetextbook enhances its relevanceandpractical utility forCanadian studentsandbusinesses alike. Throughastructured pedagogical approachandcomprehensive coverageofkey mathematical concepts,thetextbook prepares students tobecome adept problem solversanddecision makers capableofapplying mathematical principles toaddress real-world business challenges.In summary, this textbook serves asafoundational resource foranyone seeking tounderstandtheintersectionofmathematicsandbusinesswithintheCanadian context, providingasolid framework foracademic learningandprofessional development inthefieldofbusiness mathematics.---This expanded exploration should provideathorough overviewof"Contemporary Business Mathematicswith Canadian Applications"andits relevance intherealmofCanadian business education.Certainly, let's continuewithadeeper exploration intothetopics coveredin"Contemporary Business Mathematicswith Canadian Applications"andhowtheyareappliedwithintheCanadian business context.---### Detailed ExplorationofKey Topics#### 1. Basic Mathematical ConceptsThe textbook beginsbylayingasolid foundationinbasic mathematical concepts essential forbusiness applications. This includes arithmetic operations suchasaddition, subtraction, multiplication,anddivision, which formthebasis forfinancial calculationsandstatistical analyses. Algebraic concepts suchaslinear equationsandinequalitiesareintroduced, providing tools formodelingandsolving business problems mathematically. Basic statistical measures suchasmean, median, mode,andstandard deviationarecovered, enabling students toanalyzeandinterpret data effectively.#### 2. Financial MathematicsFinancial mathematics playsapivotal roleinbusiness decision-makingbyquantifyingthefinancial implicationsofvarious choices. Key topics covered include:- **Interest Calculations**: Understanding simpleandcompound interest formulas, andtheir applicationsindetermining interest earnedorpaidoninvestmentsorloans. - **Time ValueofMoney**: Concepts suchaspresent value, future value,andannuitiesareexplored toevaluatetheprofitabilityofinvestments andthecostoffinancing over time. - **Loan Repayment**: Techniques forcalculating loan payments, understanding amortization schedules,andassessingthetotal costofborrowing.- **Investment Analysis**: Toolsandmethods foranalyzing investment opportunities, including net present value (NPV), internal rateofreturn (IRR),andpayback period calculations.- **Risk Assessment**: Techniques forassessingandmanaging financial risk through probability analysisandsensitivity analysis.IntheCanadian context,these conceptsareapplied tounderstandthefinancial implicationsofdecisionswithintheframeworkofCanadian financial regulationsandmarket conditions.#### 3.

Inventory ControlEffective inventory management is crucial forbusinesses tobalance stock levelswith customer demand while minimizing costs.thetextbook covers mathematical modelsandtechniques such as:- **Economic Order Quantity (EOQ)**: Calculatingtheoptimal order quantitythatminimizes total inventory costs, considering factors suchasordering costs, holding costs,anddemand variability.- **Reorder Point**: Determiningtheinventory level at which an order should be placed toavoid stockouts while accounting forlead timeanddemand variability.**Inventory Turnover**: Calculatingandinterpreting inventory turnover ratios toassess how efficiently inventory is being managedandutilized.In Canada, specific considerations suchassupply chain dynamics, seasonal variations,andregulatory requirements may influence inventory management strategies, makingthese mathematical tools particularly relevant.#### 4. Marketing MathematicsMathematical modelsandanalysesareessential formaking informed marketing decisionsandoptimizing marketing strategies. Key topics covered include:- **Break-Even Analysis**: Determiningthesales volume needed tocover costsandachieve profitability, considering factors suchasfixed costs, variable costs,andselling price.- **Pricing Strategies**: Using pricing models toset optimal pricesthatmaximize profitability while considering consumer demand elasticityandcompetitive pricing.- **Market Segmentation**: Analyzing customer data toidentifyandtarget specific market segments effectively.- **Sales Forecasting**: Using quantitative techniques suchastrend analysisandregression analysis toforecast future sales volumes basedonhistorical dataandmarket trends.Canadian-specific examplesandcase studies illustrate howthese mathematical conceptsareapplied inthecontextofCanadian consumer behavior, market dynamics,andregulatory environment.#### 5. Data AnalysisandStatisticsIntheeraofbig data, proficiencyindata analysisandstatistical methods is essential forextracting actionable insights from business data.thetextbook covers:- **Descriptive Statistics**: Summarizingandinterpreting data using measures suchasmean, median, mode, variance,andstandard deviation. **Probabilitytheory**: Understanding basic probability conceptsanddistributions toassess riskanduncertaintyinbusiness decisions. - **Hypothesis Testing**: Using statistical tests tomake inferencesanddraw conclusions from sample data, suchastesting hypotheses about population parameters.- **Regression Analysis**: Building regression models toanalyze relationships between variablesandmake predictions basedondata trends.These statistical tools enable businesses toanalyze market trends, customer preferences,andoperational performance, providingaquantitative basis forstrategic decision-making.#### 6. Spreadsheet ApplicationsSpreadsheets like Excelareindispensable tools forperforming mathematical calculations, creating financial

models,andanalyzing datainbusiness contexts.thetextbook provides guidance on:- **Excel Functions**: Using built-in functionsandformulas toautomate calculationsandperform complex analyses. - **Financial Modeling**: Creating financial models toevaluate investment scenarios, conduct sensitivity analysis,andgenerate forecasts.- **Data Visualization**: Using charts, graphs,andpivot tables topresent data visuallyandcommunicate insights effectively.Practical exercisesandhands-on examples help students develop proficiencyinusing spreadsheetsaspowerful tools forbusiness analysisanddecision support.### Practical ApplicationsinCanadian Business#### Financial Decision-MakingIn Canada, businesses use financial mathematics toevaluate investment opportunities, manage cash flow,andmake strategic decisionsthatimpact profitabilityandgrowth. forexample, Canadian companies may apply financial models

Chapter 8 Investment Decision Rules 8-1.

NPV = 12,000/1.1 − 10,000 = 909.09. So take theinvestment opportunity!

IRR = 12,000/10,000 − 1 = 20% The cost of capital can increase by up to10% without changing thedecision.

8-2.  1000000  NPV     200000   6193 9  1.2  1/9

 1000000  IRR     200000 

 1  19.58%

Do not take theproject. A drop of just 20 − 19.58 = 0.42% in thecost of capital would change thedecision.

8-3.

Since theplant will take a year tobuild, it will not generate profit until thesecond year.

Timeline (amounts in $ millions): 0

1

–100

2

3

4

30

30

30

 1  30  100 = $247.22 million , so make theinvestment.  1.08 0.08

NPV = 

Solving forIRR,

 1  30    100  0  r  24.16% 1  r  r So, thecost of capital can be underestimated by 16.16% without changing thedecision.

8-4.

Excel Solution

See thetable below forthe NPVs given different discount rates: Rate 0% 5%

NPV $5,000,000 $2,988,430

10% 15% 20% 25% 30%

$1,372,360 $56,465 ($1,028,164) ($1,932,160) ($2,693,291)

The project should be accepted as long as thediscount rate is below theIRR of 15.238237%.

8-5.

Excel Solution and Excel Project a. Timeline (amounts in $ millions), assuming thepublisher paid Mr. Clinton up front: 0

1

2

3

10

–8

–8

–8

8 

1  1    $9.895 million 0.1  1.13 

NPV  10 

b. Timeline: 0

1

2

3

4

10

–8

–8

–8

5

5

6

5(1 − 0.3) 5(1 − 0.3)2

First calculate thePV of theroyalties at year 3. theroyalties are a declining perpetuity: PV3 

5 0.1   0.3



5 0.4

 12.5 million .

So thevalue today is PVroyalties 

12.5

1.13

 9.391 .

Now add this tothe NPV from part (a), .

8-6.

Excel Solution and Excel Project

Timeline: 0

1

2

3

6

–200,000 –200,000 –200,000

7

–200,000 300,000

16

300,000

a.

NPV = 

200,000 

=

r

1 1   1  300,000   1  +  r 1    1  r    1  r    1  r   6

6

10

200,000  0.1

1   1  300,000  1  1  +  0.1  1     1.1    1.1    1.1  6

6

10

= $169,482.24

Since NPV > 0, thecompany should take theproject. b.

Setting NPV = 0 and solving forr (using a spreadsheet), theanswer is IRR = 12.66%. So, if theestimate is lowered by more than 2.66%, thedecision will change from accept toreject.

c.

200,000  1   1  300,000  1  1  +  1   6 6  1  r    1  r    1  r 10  r r            200, 000 1 1 300, 000 1  1         1  0.14  1.14 6   1.14 6  0.14  1.14 10         $64, 815.87

NPV = 

Since NPV < 0, thecompany should not make theinvestment.

8-7.

Excel Solution a.

8-8.

b.

The IRR is thepoint at which theline crosses thex-axis. In this case, it falls very close to13%. Using Excel, theIRR is 12.724192%.

c.

Yes, thepurchase is attractive because theNPV is positive at thediscount rate of 12%.

d.

The discount rate could be underestimated by 0.724192% before theinvestment decision would change.

IRR = 120,000/100,000 − 1 = 20%. You are indifferent tothe investment.

8-9.

1/ 40

 100000  IRR     1  12.2%  1000  100000 NPV   1000   986.71 1.2540 Both rules agree—do not undertake theinvestment.

8-10. Timeline (amounts in $ millions): 0

1

–100

2

3

4

30

30

30

 1  30  100  $247.22 million   1.08  0.08

NPV  

Solving forIRR,

 1  30    100  0  r  24.16% . 1  r  r Since theIRR exceeds the8% discount rate, theIRR rule gives thesame answer as theNPV rule.

8-11. Excel Solution Timeline forpart (a) of problem 5 (amounts in $ millions).

0

1

2

3

10

–8

–8

–8

IRR is ther that solves NPV  0  10 

8

 1 . 1  3  r 1  r  

To determine how many solutions this equation has, plot theNPV as a function of r:

From theplot, there is one IRR of 60.74%.

Since theIRR is much greater than thediscount rate, thenormal IRR rule says write thebook. But thecash flows of this project make it an unconventional project; it has a negative NPV, so thenormal IRR rule gives thewrong answer. Once we note that theproject has unconventional (or borrowing-type) cash flows, then we know we should reverse theIRR rule and accept theproject only when theIRR is less than thediscount rate. When we use IRR correctly, noting theunconventional cash flows, we get a decision consistent with NPV. Timeline forpart (b) of Problem 5 (amounts in $ millions): 0

1

2

3

4

10

–8

–8

–8

5

5

6

5(1 − 0.3) 5(1.03)2

From 6.5(b), theNPV of these cash flows is

8 1  1  5   NPV  10  1   .

3 r  1  r   1  r 3  r  0.3  Plotting theNPV as a function of thediscount rate gives

The plot shows that there are two IRRs: using Excel’s solver we determine they are 7.165% and 41.568%.

When there are multiple IRRs, you cannot rely on theIRR rule tomake a decision—so, no, theIRR rule does not work in this case.

8-12. The timeline of this investment opportunity is 0

1

2

12

50,000

–4400

–4400

–4400

Computing theNPV of thecash flow stream, NPV  50, 000 

4, 400  r

1  . 1  12   (1  r) 

To compute theIRR, we set theNPV equal tozero and solve forr. Using theannuity spreadsheet, N

I

PV

PMT

FV

12

0.8484%

50,000

–4400

0

The monthly IRR is 0.8484%, so since 12

(1.008484)

 1.106696 ,

0.8484% monthly corresponds toan EAR of 10.67%. Smith’s cost of capital is 15%, so according tothe IRR rule, she should turn down this opportunity if we ignore thefact that thecash flows are unconventional. If we adjust theIRR rule forthe unconventional cash flows (i.e., accept a project when IRR < cost of capital) then it indicates she should take theopportunity.

Let’s see what theNPV rule says. If you invest at an EAR of 15%, then after one month you will have 1

(1.15) 12  1.011715 ,

so themonthly discount rate is 1.1715%. Computing theNPV using this discount rate gives NPV  50, 000 

4, 400 

1   $1010.06 , 1  12  0.011715  (1.011715) 

which is positive, so thecorrect decision is toaccept thedeal. Smith can also be relatively confident in this decision. Based on thedifference between theIRR and thecost of capital, her cost of capital would have tobe 15 − 10.67 = 4.33% lower toreverse thedecision.

8-13. a.

Timeline (amounts in $ millions): 0

1

2

10

11

12

–5

1

1

The PV of theprofits is PVprofits 

1

 1 . 1  10  r 1  r  

1

–0.1

–0.1

The PV of thesupport costs cash flows is PVsup port 

0.1 r



1

1  r 

10

NPV  5  PVprofits  PVsup port  5 

1

 1  0.1 1   1   10  10 r  1  r   r 1  r 

r = 6% then NPV = $1,429,429.09 r = 2% then NPV = –$119,156.49 r = 15% then NPV = –$146,021.18 . b.

From theanswer topart (a), there are at least two IRRs: one between 2% and 6% and theother between 6% and 15%. You can use Excel and Solver todetermine theprecise IRRs: 2.06243162% and 14.13593788%.

c.

The IRR rule says nothing in this case because it cannot be used when there are two IRRs.

8-14. The timeline of this investment opportunity is (amounts in $ millions): 0

1

2

10

11

12

–120

20

20

20

–2

–2

Computing theNPV of thecash flow stream: NPV  120 

20 

1 1  2   . 1  10  10 r  (1  r)  r (1  r)

You can verify that r = 0.02924 or 0.08723 gives an NPV of zero. There are two IRRs, so you cannot apply theIRR rule. Let’s see what theNPV rule says. Using thecost of capital of 8% gives NPV  120 

20 

1 1  2    2.621791 . 1  10  10 r  (1  r)  r (1  r)

So theinvestment has a positive NPV of $2,621,791. In this case theNPV as a function of thediscount rate is n shaped.

If theopportunity cost of capital is between 2.93% and 8.72%, theinvestment should be undertaken.

8-15. a.

IRR of thedecision toforgo maintenance = 15.091%.

b.

No, thenormal IRR rule does not work forthis decision, but if you adjust it forthe unconventional cash flows, then you should reject theproposal when IRR > cost of capital.

c.

If thecost of capital > IRR = 15.091%, then forgoing themaintenance is a good decision.

8-16. Excel Solution Timeline (amounts in $ millions): 0

1

2

3

20

21

22

–250

20

20

20

20

–5

–5

PVoperating profits 

20 

 1 1  20  r  1  r  

In year 20, thePV of thestabilizations costs are PV20  So thePV today is PVstabilization cos ts  NPV  250 

5

.

r

5  . 1  r   r  1

20

20 

 1 1 5.  1  20  20   r  1  r   1  r   r 

Plotting this out gives

So no IRR exists. If you use Excel and try touse Solver, no solution toIRR will be found either.

8-17. Excel Solution Timeline (amounts in $ millions): 0

1

2

3

21

Cash Flow

–100

15

15

15 + 10

The NPV is NPV  100 million 

15 million 

 1 1 10 million    $962, 787.51 . 1  20  1  .12  1  .12 21  1  .12   

.12

The IRR is 12.132192%. So thetwo rules agree that theproject should be accepted.

8-18. Compute theIRR(s) by plotting theNPV as a function of thediscount rate and marking theIRR(s) where thegraph crosses thex-axis. Give your boss this plot along with theIRR number(s) so as toensure your boss makes theright decision, no matter whether theproject has conventional or unconventional cash flows or multiple or no IRRs. 8-19.

a.

IRR = IRR = NPV at 10% =

b.

0 500 8.53% 31.16%

1 -450

2 -450

4 900

$ (4.37)

No, theNPV < 0 at a 10% cost of capital, so theopportunity is not attractive.

0 1 2 500 -450 -450 IRR = #NUM! (does not exist) IRR = #NUM! NPV at 10% = $ 63.93

8-20.

3 -450

3 -450

4 1000

c.

No IRR exists.

d.

Yes, theopportunity is attractive as theNPV > 0 (at 10% and, in fact, forall costs of capital).

Excel Solution Timeline (amounts in $ millions): 0

Cash Flow

a.

–100

1

2

3

21

15

15

15 + –200

1 1  15 1   r  (1  r )20  200 with r = 12%, NPV = –$18.5 million.

NPV  100   (1  r ) (1  r )21

b.

No, theIRR rule is not reliable, because theproject has a negative cash flow that comes after thepositive ones.

c.

Because thetotal cash flows are equal tozero (–100 + 15 × 20 − 200 = 0), one IRR must be 0%.

Because thecash flows change sign more than once, we can have a second IRR. This IRR is is therate that would make theNPV = 0 and so is r in thefollowing equation: 1 1  15 1   r (1  r )20  200   100  0 . This cannot be solved algebraically but can be determined (1  r ) (1  r )21

using most financial calculators or Excel. By plotting theNPV profile, we can see thesecond IRR is around 7% and when we input that as theguess rate into our financial calculator or Excel, we find thesecond IRR of 7.06%. Because there are two IRRs, therule does not apply.

8-21.

The payback period can be computed as 5000/500 = 10 months.

8-22

Timeline (amounts in $ millions): 0

1

2

3

4

5

6

–10

0

5

2

2

2

2

It will take five years topay back theinitial investment, so thepayback period is five years. Since theproject’s payback period is longer than therequired payback cut-off period of two years, you will not make themovie.

NPV  10 

 2 1  1 5 2 1   1   10   1   $628,322 4 2 2 2  1  r  r  1  r   1  r  1.1 0.11.1  1.14  5

2

So theNPV agrees with thepayback rule in this case.

8-23. a.

Timeline (amounts in $ millions):

A B

0

1

2

3

–10 –10

2 1.5

2 1.5(1.02)

2 1.5(1.02)2

NPVA 

2

 10

r

Setting NPVA = 0 and solving forr: IRRA = 20% NPVB 

1.5 r  0.02

 10 .

Setting NPVB = 0 and solving forr: 1.5 r  0.02

 10  r  0.02  0.15  r  17%. So, IRR B  17% .

Based on theIRR, you always pick project A since it has higher IRR. b.

Substituting r = 0.07 into theNPV formulas derived in part (a) gives NPVA = $18.5714 million NPVB = $20 million.

So theNPV favours project B, since it has a higher NPV.

c.

Here is a plot of NPV of both projects as a function of thediscount rate. theNPV rule selects A (and so agrees with theIRR rule) forall discount rates tothe right of thepoint where thecurves cross.

NPVA  NPVB 2



r r



1.5 r  0.02 r  0.02

2

1.5

1.5r  2r  0.04 0.5r  0.04 r  0.08 So theIRR rule will give thecorrect answer fordiscount rates greater than 8% 8-24.

Excel Solution a.

Using theIRRs and theexisting cash flows, we can solve forthe present values of themissing cash flows as follows: Project B, Year 0 Outflow: ($111.26).

Project C, PV(salvage) = $14.89, therefore amount in year 3 is $50.25.

b.

The NPVs are as follows: Project A: NPV(A)  100  30 /1.10  153 /1.10  88 /1.10  $119.83 2

3

Project B: NPV(B)  111.26  0 /1.10  206 /1.10  95 /1.10  $130.37 2

3

Project C: NPV(C)  100  37 /1.10  0 /1.10  254.25 /1.10  $124.66 2

3

All projects have positive NPV, but since they are mutually exclusive only one can be chosen—the one with thehighest NPV, that is, project B. c.

Ranking theprojects by their IRR is not valid in this situation because theprojects have different scale and different pattern of cash flows over time.

8-25. To calculate theincremental IRR, subtract A from B. Timeline (amounts in $ millions): 0

1

2

3 ∞

A –10 B –10 B−A 0 NPVB A 

2



r r 2



1.5 r  0.02

2 1.5 1.5 − 2 

2

2 2 1.5(1.02) 1.5(1.02)2 1.5(1.02)–2 1.5(1.02)2–2

0

r

1.5 r  0.02 r  0.02 1.5

1.5r  2r  0.04 0.5r  0.04 r  0.08

So theincremental IRRB-A is 8%. These are conventional cash flows and this rate is above thecost of capital, so we should take B instead of A (note: if we did not know already, we would want toverify that B is better than thedo-nothing alternative).

8-26. We need tosubtract thePlayhouse cash flows from theFort.

Timeline (amounts in $ thousands): 0

Playhouse –30 Fort –80 Fort − Playhouse –50

NPVFort  Playhouse  50 

24 1 r



2  50   24 

 7.522%

15 39 24

20 52 32

1  r 

24  4  50  32  2

2  50 

2

32

Solving forr, r

1

.

2

Since theincremental IRRFort-Playhouse of 7.522% is less than thecost of capital of 8%, you should undertake thePlayhouse project.

NPV ($10 millions)

Playhouse

Fort 8-27. Excel Solution First, consider A–C incremental cash flows: Incremental Analysis

Year 0

Year 1

Year 2

Year 3

IRR

A–C

$0.00

($7.00)

$153.00

($166.25)

14.68% 1971.04%

You can see at 0%, thesum of thecash flows is negative, so fordiscount rates between 14.68% and 1971.04% (the two IRRs) theNPV will be positive and A will be better than C. Next, consider B–A incremental cash flows: Incremental Analysis

Year 0

1

2

3

IRR

B–A

($11.26)

($30.00)

$53.00

$7.00

30.57%

B is better than A as long as thediscount rate is < IRR. Note: B–A has conventional cash flows.

Next, consider B–C incremental cash flows: Incremental Analysis

Year 0

1

2

3

IRR

B–C

($11.26)

($37.00)

$206.00

($159.25)

1.55% 115.69%

B is better than C as long as thediscount rate is between thetwo IRRs, that is, greater than 1.55% and less than 115.69%.

See theNPV profile below toconfirm.

8-28. The two projects have thesame scale and thesame timing (10-year annuities). Thus, as long as they have thesame risk (and therefore, thesame cost of capital), we can compare them based on their IRRs. 8-29. a.

The IRR of theinvestment is 6.16%.

b.

Yes, you can decide by simply comparing theEAR with theIRR because thetwo projects have thesame timing, scale, and risk (measure of safety). You can choose theinvestment with thehigher IRR.

a.

IRR forHuawei = 83.9%, IRR forCisco = 36.3%.

b.

NPV forHuawei bid = $28.0 million, NPV forCisco bid = $44.1 million.

c.

Net Cash Flows = –20, 25, 25, 25

8-30.

The IRR of theCisco bid is now 111.9%. d.

No! Despite a higher IRR, thenew bid actually involves borrowing $80 million up front (instead of paying $100 million upfront, only $20 million is paid upfront − so implicitly leasing instead of buying is equivalent toa loan of $80 million) and requiring payments of $35 million per year forthree years. theIRR of theimplied loan from doing thelease instead of buying is 14.9% which can be thought of as a borrowing cost of 14.9%— this would certainly be higher than Facebook’s borrowing cost.

8-31.

Roses

NPV per bunch $3

Cost per bunch $20

Max . bunches 25

Profitability index Max. (per bunch) investment 0.150

$500

Lilies

$8

$30

10

0.267

$300

Pansies

$4

$30

10

0.133

$300

Orchids

$20

$80

5

0.250

$400

Buy $300 of lilies, $400 of orchids, and $300 of roses

8-32.

Space requirement (sq. ft.)

NPV/sqft

Model

NPV

MB345

$3000

200

MC237

$5000

250

MY456

$4000

240

MG231

$1000

150

MT347

$6000

450

MF302

$4000

200

MG201

$1500

150

$15.0 $20.0 $16.7 $6.7 $13.3 $20.0 $10.0

Take the MC237, MF302, MY456, and MB345 (890 sqft) Use remaining 1110 sqft for office space.

8-33. a.

We can compute theIRR foreach as IRR = (Sale Price/Cost)1/5 − 1. See spreadsheet below.

b.

We can compute theNPV foreach as NPV = Sale Price/(1+r)5 − Cost. See spreadsheet below. (Note, thePI numbers use unrounded NPVs calculated within thespreadsheet.)

Project Mountain Ridge Ocean Park Estates Lakeview Seabreeze Green Hills West Ranch

Cost Today Discount Rate $ 3,000,000 15% 15,000,000 15% 9,000,000 15% 6,000,000 8% 3,000,000 8% 9,000,000 8%

Expected Sale Price in Year 5 $ 18,000,000 $ 75,500,000 $ 50,000,000 $ 35,500,000 $ 10,000,000 $ 46,500,000

IRR 43.1% 38.2% 40.9% 42.7% 27.2% 38.9%

NPV $ 5,949,181 22,536,844 15,858,837 18,160,703 3,805,832 22,647,119

Profitability Index 1.98306041 1.50245623 1.76209297 3.02678392 1.26861066 2.51634652

c.

We can rank projects according totheir profitability index = NPV/Cost, as shown above. Thus, KP should invest in Seabreeze, West Ranch, and Mountain Ridge. (Note that ranking projects according totheir IRR would not maximize KP’s total NPV, and so would not lead tothe correct selection.)

d.

The profitability index fails because thetop-ranked projects do not completely use up thebudget. In this case, you should take Mountain Ridge and West Ranch.

8-34. Project I II III IV

PI 1.01 1.27 1.47 1.25

V

2.01

NPV/Headcount 5.1 6.3 5.5 8.3 6.0

a. The PI rule selects projects V, III, and II. These are also theoptimal projects toundertake (as thebudget is used up fully taking theprojects in order). b. The PI rule (calculated with NPV/Headcount) selects IV and II alone, because theproject with thenext highest PI (that is NPV/Headcount), V, cannot be undertaken without violating theresource constraint. However, this choice of projects does not maximize NPV. Orchid should also take on III and I. This solution is better than

taking V and I (which is also affordable), and shows that it may be optimal toskip some projects in thePI ranking if they will not fit within thebudget (and there is unused budget remaining). c.

We cannot use thePI rankings because they are so bad at hitting theconstraint exactly. Instead, look tothe NPVs of all feasible combinations of projects that use up thebudget of scientists. thefeasible combination that maximizes NPV is V and IV. Thus, do projects V and IV.

Chapter 9 Fundamentals of Capital Budgeting

Sales of new pizza – lost sales of original = 20 – (0.40 × 20) = $12 million.

Sales of new pizza – lost sales of original pizza from customers who would not have switched brands = 20 – (0.50 × 0.40 × 20) = $16 million.

Year Incremental Earnings Forecast ($000s) 1 Sales of Mini Mochi Munch 2 Other Sales 3 Cost of Goods Sold 4 Gross Profit 5 Selling, General & Admin.

6 Depreciation 7 EBIT 8 Income tax at 35% 9 Unlevered Net Income

1

2

9000 2000 (7,350) 3,650 (5,000) (1,350) 473 (878)

7,000 2,000 (6,050) 2,950 2,950 (1,033) 1,918

No, this is a sunk cost and will not be included directly. (But see (f) below.)

Yes, this is a cost of opening thenew store.

Yes, this loss of sales at theexisting store should be deducted from thesales at thenew store todetermine theincremental increase in sales that opening thenew store will generate forHBS. No, this is a sunk cost.

This is a capital expenditure associated with opening thenew store, so it will not affect thereported earnings other than through depreciation expenses (recall that depreciation is usually used forreporting purposes but not fortax purposes). This capital expenditure affects thecash flows of theproject by causing a large initial

outflow (the construction costs) followed by a series of inflows as theCCA tax shields are realized.

Yes, this is an opportunity cost of opening thenew store. (By opening thenew store, HBS forgoes theafter-tax proceeds it could have earned by selling theland. This loss is equal tothe sale price less thetaxes owed on thecapital gain from thesale, which is thedifference between thesale price and theinitial cost of theland.)

While these financing costs will affect HBS’s reported earnings, forcapital budgeting purposes we calculate theincremental earnings without including financing costs todetermine theproject’s unlevered net income.

Change in EBIT = Gross profit with price drop – Gross profit without price drop = 25,000 × (300 – 200) – 20,000 × (350 – 200) = –$500,000.

Change in EBIT from ink cartridge sales = 25,000 × $75 × 0.70 – 20,000 × $75 × 0.70 = $262,500 Therefore, incremental change in EBIT forthe next three years is Year 1: $262,500 – 500,000 = –$237,500 Year 2: $262,500 Year 3: $262,500.

Excel Solution Year 0

Year 1 Year 2 Year 3 Year 4 Year 5 Cash 6 12 15 15 15 Accounts Receivable 21 22 24 24 24 Inventory 5 7 10 12 13 Accounts Payable 18 22 24 25 30 Net working capital (1 + 2 + 3 – 4) 0 14 19 25 26 22 Increase in NWC 14 5 6 1 –4 Since an increase in NWC is equivalent toa negative cash flow, thecash flow effects are as follows:

Cash Flow

Year 0

Year 1

Year 2

Year 3

Year 4

Year 5

0

–14

–5

–6

–1

+4

In addition, if NWC reverts back to0 in year 6, then the―increase‖ in NWC foryear 6 is –22 and thecash flow caused by thechange in NWC is +22. (Note that thesum of theNWC cash flows from years 0 to6 is then 0.)

Note: we have assumed any incremental cost of goods sold is included as part of operating expenses.

Incremental earnings are as follows: Year Incremental Earnings Forecast ($000s)

1

2

1 2 3 4 5 6

Sales Operating Expenses Capital Cost Allowance EBIT Income tax at 35% Unlevered Net Income

125.0 (40.0) (25.0) 60.0 (21.0) 39.0

160.0 (60.0) (36.0) 64.0 (22.4) 41.6

Free Cash Flows Free Cash Flow ($000s) 7 Plus: CCA Less: Capital 8 Expenditures 9 Less: Increases in NWC 1 0 Free Cash Flow

1 25.0

2 36.0

(30.0) (5.0)

(40.0) (8.0)

29.0

29.6

Excel Solution

Before thefree cash flows can be calculated, theCCA deductions must be determined and used instead of depreciation. theCapEx amount is $25 million. Using Eq. 7.1 and 7.2, theUCC and CCA amounts are (in $ thousands): Year 1 2 3 4 5 6 7 8 9 10 UCC 12,500 21,250 14,875 10,413 7289 5102 3571 2500 1750 1225 CCA 3750 6375 4463 3124 2187 1531 1071 750 525 368 The free cash flows are (in $ thousands): Year Cost of machine cash flow Cash flow from change in net working capital Sales revenue

0

1

2

3

4

5

6

7

8

9

10

–25,000 –10,000

10,000 30,000

30,000

30,000

30,000

30,000

30,000

30,000

30,000

30,000

30,000

Minus cost of goods sold

18,000

18,000

18,000

18,000

18,000

18,000

18,000

18,000

18,000

18,000

Equals gross profit Minus general, sales, and administrative expense Plus overhead that would have occurred anyway Minus CCA Equals net operating income Minus income tax @ 35% Equals Net income Plus CCA Plus machine and change in net working capital cash flows Equals free cash flow

12,000

12,000

12,000

12,000

12,000

12,000

12,000

12,000

12,000

12,000

2000

2000

2000

2000

2000

2000

2000

2000

2000

2000

1000

1000

1000

1000

1000

1000

1000

1000

1000

1000

3750

6375

4463

3124

2187

1531

1071

750

525

368

7250

4625

6538

7876

8813

9469

9929

10,250

10,475

10,632

2538

1619

2288

2757

3085

3314

3475

3587

3666

3721

0

4713 3750

3006 6375

4249 4463

5120 3124

5729 2187

6155 1531

6454 1071

6662 750

6809 525

6911 368

–35,000

0

0

0

0

0

0

0

0

0

10,000

–35,000

8463

9381

8712

8243

7915

7686

7525

7413

7334

17,279

The free cash flows in theprevious table should not be used tocalculate theNPV as there are CCA tax shields in years 11 and onward. If only thefree cash flows shown in (a) are used forthe NPV, theresult will underestimate theproject’s actual NPV. We can calculate thefree cash flows excluding CCA effects and calculate thePV of CCA tax shields separately. This is shown below.

Year Cost of machine cash flow Cash flow from change in net working capital Sales revenue

0

1

2

3

4

5

6

7

8

9

10

–25,000 –10,000

10,000 30,000

30,000

30,000

30,000

30,000

30,000

30,000

30,000

30,000

30,000

Minus cost of goods sold

18,000

18,000

18,000

18,000

18,000

18,000

18,000

18,000

18,000

18,000

Equals gross profit Minus general, sales, and administrative expense Plus overhead that would have occurred anyway Minus CCA Equals net operating income Minus income tax @ 35%

12,000

12,000

12,000

12,000

12,000

12,000

12,000

12,000

12,000

12,000

2000

2000

2000

2000

2000

2000

2000

2000

2000

2000

1000

1000

1000

1000

1000

1000

1000

1000

1000

1000

11,000

11,000

11,000

not included as analyzed separately 11,000 11,000 11,000 11,000

11,000

11,000

11,000

3850

3850

3850

3850

3850

3850

7150

7150

715 0

715 0

7150

7150

Equals Net income Plus CCA Plus cost of machine and change in net working capital cash flows Equals free cash flow excluding CCA tax shields

3850 7150

3850 7150

3850

7150

3850

7150

not included as analyzed separately –35,000

0

0

0

0

0

0

0

0

0

10,000

–35,000

7150

7150

7150

7150

7150

7150

7150

7150

7150

17,150

(Note: Consistent with thetables above, all dollar values are expressed in $thousands.)

NPV of FCFexcluding CCA tax shields = 35,000 

7,150  1  17,150 1   $4,992.66 .14  1.149  1.1410

 .14  1  25, 000  0.30  .35  2  PV of CCA tax shields =   $5,599.58 .14  .30 1  .14

NPV = 4992.66 + 5599.58 = $10,592.25 Assuming thereported net income is calculated with depreciation equal tothe CCA, thefree cash flow is FCF = Unlevered Net Income + CCA – CapEx – Increase in NWC = 250 + 100 – 200 – 10 = $140 million.

Excel Solution We can use Eq. 7.9 toevaluate thefree cash flows associated with each alternative. Note that we need toinclude only thecomponents of free cash flows that vary across each alternative. forexample, since NWC is thesame foreach alternative, we can ignore it.

The spreadsheet below computes therelevant FCF from each alternative. Note that each alternative has a negative NPV—this represents thePV of thecosts of each alternative. We should choose theone with thehighest NPV (lowest cost), which, in this case, is topurchase thecurrent machine.

Notes tocalculations in spreadsheet: After-tax amounts forrent, maintenance, and other costs are calculated as follows: After-tax amount = Before-tax amount × (1 – τC).

For example, After-tax rent cash flow = –50,000 × (1 – 0.35) = –32,500.

PV CCA tax shields are calculated as follows:  r  0.08  1 1  CapEx  d   C  2  150, 000  0.25  0.35  2  Option A: PV      $38, 299.66 rd 0.08  0.25 1  r  1  0.08

.

r   0.08  1 1  CapEx  d   C  2  250, 000  0.25  0.35  2  Option B: PV      $63,832.77 . rd 0.08  0.25 1  r  1  0.08

NPVs of free cash flows excluding CCA tax shields (FCF) are calculated as follows: 10

FCFt t t 0 1  r  .

NPV  

Using Eq. 7.1 and 7.2, theCCA deductions are as follows: Year CapE x UCC

0 15,000,000.0 0

CCA

1

2

3

4

5

7,500,000.0 0 1,875,000.0 0

13,125,000.0 0 3,281,250.00

9,843,750.0 0 2,460,937.5 0

7,382,812.5 0 1,845,703.1 3

5,537,109.3 8 1,384,277.3 4

4

5

The CCA tax shield in year t is equal toτC × CCAt .

Year τC CCA tax

0 35%

1

2

3

656,250.00 1,148,437.50 861,328.13 645,996.09 484,497.07

shield t The CCA tax shield in year t has present value equal to  C  CCA t

1  r  .

Year r CCA tax shield PV of CCA tax shield NPV of first five CCA tax shields

0 10%

1

2

3

4

5

656,250.0 0 596,590.9 1

1,148,437.5 0

861,328.1 3 647,128.5 7

645,996.0 9 441,224.0 2

484,497.0 7 300,834.5 6

949,121.90

2,934,899.9 6

The PV of theperpetuity of CCA tax shields is as follows:  r  .10  1  1     CapEx  d   C 15, 000, 000  .25  .35  2 2 PV      $3,579,545.45. rd .10  .25 1  r  1  .10

If themarginal corporate tax rate increases over thenext five years, thePV of CCA tax shields will be higher than what is calculated in part (d).

In addition, if thetax rate will increase substantially, then Spherical may be better off todelay claiming CCA amounts until later years. theincreased CCA tax shield generated in thefuture may more than make up forthe extra discounting required tocompensate forthe time value of money.

Note that, in this case, it may be easier tomake a spreadsheet where thetax rates can be adjusted on a year-to-year basis as we cannot use theformula in part (d) because it assumes that a constant tax rate holds.

The expected cash flow in year 5 is 240,000 × 1.03 = $247,200. We can value thecash flows in year 5 and beyond as a growing perpetuity: Continuation Value in Year 4 = 247,200 / (0.14 – 0.03) = $2,247,273.

We can then compute thevalue of thedivision by discounting theFCF in years 1 through 4, together with thecontinuation value:

NPV 

185,000 12,000 99,000 240,000  2, 247, 273     $1,367,973.

1.14 1.142 1.143 1.144

FCF in year 6 = 110 × 1.02 = 112.2.

Continuation value in year 5 = 112.2 / (12% – 2%) = $1122, that is, $1,122,000,000.

We can estimate thecontinuation value as follows: Continuation value in year 5 = (earnings in year 5) × (P/E ratio in year 5) = $50 × 30 = $1500, that is, $1,500,000,000.

We can estimate thecontinuation value as follows:

Continuation value in year 5 = (book value in year 5) × (M/B ratio in year 5) = $400 × 4 = $1600, that is, $1,600,000,000.

Excel Solution Replacing themachine increases EBITDA by 40,000 – 20,000 = 20,000. Therefore, FCF excluding CCA tax shields will increase by (20,000) × (1 – 0.45) = $11,000 in years 1 through 10.

PV 

11,000  1  1  67,590.24  .10  1.1010 

In year 0, theinitial cost of themachine is $150,000. Selling thecurrent machine for$50,000 generates an incremental CapEx of 150,000 – 50,000 = 100,000. Thus theyear 0 cash flow is –100,000.

In addition, theincremental CapEx generates CCA tax shields. thePV of CCA tax shields is

 .10  1  100, 000  .40  .45  2 PV    34,363.64. .10  .40 1  .10 The NPV of replacing themachine is thus NPV = –100,000 + 67,590.24 + 34,363.64 = $1953.87.

Since theNPV ≥ 0, yes, your company should replace theold machine.

Excel Solution It is best tocalculate theFCF excluding CCA tax shields, take theNPV of these FCFs, and then add thePV of CCA tax shields toget theNPV.

The PV CCA tax shields are as follows:  r  0.12  1  1   CapEx  d   C  2  150, 000  0.10  0.35  2  PV      $22,585.23. rd 0.12  0.10 1  r  1  0.12

The FCFs excluding CCA tax shields, their NPV, and thetotal NPV including CCA tax shields are calculated in thetable below (in $ millions):

Revenue

10% higher

10% lower

NPV

$86.91971

$13.46681

Growth Rate NPV

0%

2%

5%

$50.19326

$65.38491

$91.06262

NPV is positive fordiscount rates below theIRR of 19.41%.

Excel Solution See spreadsheet below for(a)–(f).

Certainly! Here'samore detailed explorationof"Contemporary Business Mathematicswith Canadian Applications"thatexpands toapproximately 3000 words:### Introduction toContemporary Business Mathematicswith Canadian ApplicationsContemporary Business Mathematicswith Canadian Applications" isatextbook designed tobridgethegap between mathematicaltheoryandpractical business applications, tailored specifically forCanadian studentsandbusinesses. This comprehensive text coversawide rangeofmathematical concepts essential forunderstandingandmanaging various aspectsofbusiness operations intheCanadian context.inthis detailed exploration, we will delve intothekey components,themes,andrelevanceofthis textbook.

The textbook startswith foundational mathematical conceptsthatformthebasis formore complex calculationsandanalysesinbusiness settings. This includes arithmetic operations, algebraic equations,andbasic statistical measures. Understandingthese fundamental principles is crucial asthey providetheframework upon which more advanced topicsarebuilt.Financial mathematics isacornerstoneofbusiness education, particularlyinunderstandingthetime valueofmoneyandfinancial decision-making. Topics covered often include interest calculations, compound interest, annuities, loans,andinvestments.these conceptsareessential forevaluatingtheprofitabilityofinvestments, determining loan payments,andunderstandingtheimplicationsofinterest ratesonbusiness finances.#### 3. Inventory ControlEffective inventory management is vital forbusinesses toensure adequate stock levels while minimizing holding costs.thetextbook covers mathematical techniquesandmodels used forinventory control, including inventory turnover rates, economic order quantity (EOQ),andreorder points. Canadian applications may include considerations specific toCanadian supply chain dynamicsandregulations.#### 4. Marketing MathematicsMathematical tools playacrucial roleinmarketing decisionsbyproviding quantitative insights into consumer behavior, pricing strategies,andsales forecasts. Topics covered often include break-even analysis, pricing models, market segmentation,andpromotional budgeting. Case studiesandexamples tailored toCanadian markets illustrate howthese mathematical conceptsareappliedinreal-world marketing scenarios.#### 5. Data AnalysisandStatisticsIn today's data-driven business environment, proficiencyindata analysisandstatistical methods is essential.thetextbook typically covers statistical techniques used toanalyze business data, suchasdescriptive statistics, probabilitytheory, hypothesis testing,andregression analysis.these tools enable businesses toextract meaningful insights from data tosupport decision-makingandstrategic planning.#### 6. Spreadsheet ApplicationsSpreadsheetsarepowerful tools extensively usedinbusiness forfinancial modeling, data analysis,anddecision support.thetextbook includes guidanceonutilizing spreadsheet software like Excel toperform calculations, create financial models,andvisualize data. Practical exercisesandexamples help students develop proficiencyinusing spreadsheets forbusiness applications.### RelevanceofCanadian ApplicationsThe inclusionof"Canadian Applications" inthetextbook's title emphasizes its focusonintegrating Canadian business practices, regulations,andeconomic conditions intothelearning experience. This ensuresthatstudents not only grasp mathematical concepts but also understand howthese conceptsareappliedwithintheunique contextofCanada.

Examplesandcase studies drawn from Canadian businesses provide practical insightsandrelevance, preparing students forthechallengesandopportunitiesthey may encounter intheCanadian business landscape.### Pedagogical ApproachandLearning Resources#### Pedagogical ApproachThe textbook typically adoptsapedagogical approachthatcombinestheoretical explanationswith practical examplesandapplications. Conceptsarepresented inaclearandstructured manner tofacilitate understanding,with step-by-step explanationsofmathematical calculationsandproblem-solving techniques. This approach helps students develop both mathematical proficiency andtheability toapplythese skills toreal-world business scenarios.#### Learning ResourcesIn addition tothetextbook itself, supplementary learning resources often accompany "Contemporary Business Mathematicswith Canadian Applications" toenhance student learningandengagement.these may include:- **Online Resources**: Access tointeractive tutorials, practice exercises,andsolutions.- **Case Studies**: Detailed analysesofreal-world business scenarios relevant toCanadian

markets.- **Workbook Exercises**: Additional problemsandexercises toreinforce learningandmasteryofmathematical concepts.- **Instructor Support**: Guidanceandsupport forinstructors toeffectively teachandintegratethetextbook intotheir courses.### BenefitsandPractical Applications#### Benefits forStudents- **Practical Relevance**:byfocusingonCanadian applications,thetextbook prepares students toapply mathematical concepts inthespecific contextofCanadian businesses, enhancingtheir readiness forcareers intheCanadian business landscape.**Skill Development**: Students develop critical quantitative skills thatarehighly valuedinvarious business disciplines, including finance, marketing, operations,andstrategic management.**Problem-Solving Abilities**: Through exercisesandcase studies, students learn toanalyze business problems, formulate mathematical models,andderive solutions, improvingtheir analyticalandproblem-solving abilities.#### Practical ApplicationsinBusiness- **Financial Decision-Making**: Businesses use financial mathematics toevaluate investment opportunities, manage cash flow,andmake informed financial decisions.- **Marketing Nursing is a profession deeply rooted in caring for individuals, families, and communities to promote

health and well-being, prevent illness and injury, and provide support to those experiencing health challenges. It encompasses a wide range of roles, from direct patient care to leadership in healthcare policy and education. In this essay, we will explore the multifaceted nature of nursing, its historical evolution, educational pathways, professional responsibilities, ethical considerations, and its crucial role in the healthcare system.### Historical Evolution of NursingThe history of nursing dates back centuries and has evolved significantly over time. Nursing as a formalized profession began to take shape in the 19th century with the pioneering efforts of figures like Florence Nightingale, who is often regarded as the founder of modern nursing. Nightingale's work during the Crimean War demonstrated the critical importance of sanitation and hygiene in healthcare settings, laying the foundation for evidence-based nursing practice.Throughout the 20th century, nursing continued to professionalize and expand its scope. World War I and World War II highlighted the essential role of nurses in caring for wounded soldiers and civilians, leading to advancements in nursing education and practice. The establishment of nursing schools and regulatory bodies further solidified nursing as a respected profession with defined standards of practice.### Educational Pathways in NursingToday, nursing education encompasses a variety of pathways, from diploma programs to bachelor's and master's degrees in nursing (BSN, MSN) and doctoral programs (DNP, PhD). Each level of education prepares nurses for different roles within the profession, ranging from bedside care to advanced clinical practice, research, and leadership positions.Diploma programs, typically offered by hospitals, provide a more hands-on, clinical-focused education. Associate degree programs (ADN) and BSN programs offered by colleges and universities combine clinical training with a broader education in sciences, humanities, and nursing theory. BSN programs are increasingly becoming the preferred entry-level education for nurses due to their emphasis on critical thinking, leadership, and evidence-based practice.Advanced practice roles such as nurse practitioners (NPs), nurse anesthetists (CRNAs), nurse midwives (CNMs), and clinical nurse specialists (CNSs) require additional education at the master's or doctoral level. These nurses are trained to provide specialized care, prescribe medications, and diagnose health conditions, expanding access to healthcare services, particularly in underserved areas.### Professional Responsibilities and RolesThe responsibilities of nurses vary depending on their level of education, specialization, and practice setting. At the core of nursing practice is the provision of compassionate, patient-centered care. Nurses assess patients' health needs, develop and implement care plans, monitor patient progress, and advocate for their wellbeing. They collaborate with other healthcare professionals, such as physicians, therapists, and social workers, to ensure comprehensive care.In addition to direct patient careStrategy**: Mathematical models aidindetermining optimal pricing strategies, forecasting sales volumes,andassessingtheeffectivenessofmarketing campaigns.- **Operations Management**: Inventory control techniques help businesses optimize inventory levels, minimize costs,andensure timely availabilityofproducts.### Conclusion"Contemporary Business Mathematicswith Canadian Applications" isavaluable resourcethatequips studentswiththemathematical skillsandknowledge essential forsuccessintoday's competitive business environment.byintegrating Canadian-specific examplesandapplications,thetextbook enhances its relevanceandpractical utility forCanadian studentsandbusinesses alike. Throughastructured pedagogical approachandcomprehensive coverageofkey mathematical concepts,thetextbook prepares students tobecome adept problem solversanddecision makers capableofapplying mathematical principles toaddress real-world business challenges.In summary, this textbook serves asafoundational resource foranyone seeking tounderstandtheintersectionofmathematicsandbusinesswithintheCanadian context, providingasolid framework foracademic learningandprofessional development inthefieldofbusiness mathematics.---This expanded exploration should provideathorough overviewof"Contemporary Business Mathematicswith Canadian Applications"andits relevance intherealmofCanadian business education.Certainly, let's continuewithadeeper exploration intothetopics coveredin"Contemporary Business Mathematicswith Canadian Applications"andhowtheyareappliedwithintheCanadian business context.---### Detailed ExplorationofKey Topics#### 1. Basic Mathematical ConceptsThe textbook beginsbylayingasolid foundationinbasic mathematical concepts essential forbusiness applications. This includes arithmetic operations suchasaddition, subtraction, multiplication,anddivision, which formthebasis forfinancial calculationsandstatistical analyses. Algebraic concepts suchaslinear equationsandinequalitiesareintroduced, providing tools formodelingandsolving business problems mathematically. Basic statistical measures suchasmean, median, mode,andstandard deviationarecovered, enabling students toanalyzeandinterpret data effectively.#### 2. Financial MathematicsFinancial mathematics playsapivotal roleinbusiness decision-makingbyquantifyingthefinancial implicationsofvarious choices. Key topics covered include:- **Interest Calculations**: Understanding simpleandcompound interest formulas, andtheir applicationsindetermining interest earnedorpaidoninvestmentsorloans. - **Time ValueofMoney**: Concepts suchaspresent value, future value,andannuitiesareexplored toevaluatetheprofitabilityofinvestments andthecostoffinancing over time. - **Loan Repayment**: Techniques forcalculating loan payments, understanding amortization schedules,andassessingthetotal costofborrowing.- **Investment Analysis**: Toolsandmethods foranalyzing investment opportunities, including net present value (NPV), internal rateofreturn (IRR),andpayback period calculations.- **Risk Assessment**: Techniques forassessingandmanaging financial risk through probability analysisandsensitivity analysis.IntheCanadian context,these conceptsareapplied tounderstandthefinancial implicationsofdecisionswithintheframeworkofCanadian financial regulationsandmarket conditions.#### 3.

Inventory ControlEffective inventory management is crucial forbusinesses tobalance stock levelswith customer demand while minimizing costs.thetextbook covers mathematical modelsandtechniques such as:- **Economic Order Quantity (EOQ)**: Calculatingtheoptimal order quantitythatminimizes total inventory costs, considering factors suchasordering costs, holding costs,anddemand variability.- **Reorder Point**: Determiningtheinventory level at which an order should be placed toavoid stockouts while accounting forlead timeanddemand variability.**Inventory Turnover**: Calculatingandinterpreting inventory turnover ratios toassess how efficiently inventory is being managedandutilized.In Canada, specific considerations suchassupply chain dynamics, seasonal variations,andregulatory requirements may influence inventory management strategies, makingthese mathematical tools particularly relevant.#### 4. Marketing MathematicsMathematical modelsandanalysesareessential formaking informed marketing decisionsandoptimizing marketing strategies. Key topics covered include:- **Break-Even Analysis**: Determiningthesales volume needed tocover costsandachieve profitability, considering factors suchasfixed costs, variable costs,andselling price.- **Pricing Strategies**: Using pricing models toset optimal pricesthatmaximize profitability while considering consumer demand elasticityandcompetitive pricing.- **Market Segmentation**: Analyzing customer data toidentifyandtarget specific market segments effectively.- **Sales Forecasting**: Using quantitative techniques suchastrend analysisandregression analysis toforecast future sales volumes basedonhistorical dataandmarket trends.Canadian-specific examplesandcase studies illustrate howthese mathematical conceptsareapplied inthecontextofCanadian consumer behavior, market dynamics,andregulatory environment.#### 5. Data AnalysisandStatisticsIntheeraofbig data, proficiencyindata analysisandstatistical methods is essential forextracting actionable insights from business data.thetextbook covers:- **Descriptive Statistics**: Summarizingandinterpreting data using measures suchasmean, median, mode, variance,andstandard deviation. **Probabilitytheory**: Understanding basic probability conceptsanddistributions toassess riskanduncertaintyinbusiness decisions. - **Hypothesis Testing**: Using statistical tests tomake inferencesanddraw conclusions from sample data, suchastesting hypotheses about population parameters.- **Regression Analysis**: Building regression models toanalyze relationships between variablesandmake predictions basedondata trends.These statistical tools enable businesses toanalyze market trends, customer preferences,andoperational performance, providingaquantitative basis forstrategic decision-making.#### 6. Spreadsheet ApplicationsSpreadsheets like Excelareindispensable tools forperforming mathematical calculations, creating financial models,andanalyzing datainbusiness contexts.thetextbook provides guidance on:- **Excel Functions**: Using built-in functionsandformulas toautomate calculationsandperform complex analyses. - **Financial Modeling**: Creating financial models toevaluate investment scenarios, conduct sensitivity analysis,andgenerate forecasts.- **Data Visualization**: Using charts, graphs,andpivot tables topresent data visuallyandcommunicate insights effectively.Practical exercisesandhands-on examples help students develop proficiencyinusing spreadsheetsaspowerful tools forbusiness analysisanddecision support.### Practical ApplicationsinCanadian Business#### Financial Decision-MakingIn Canada, businesses use financial mathematics toevaluate investment opportunities, manage cash flow,andmake strategic decisionsthatimpact profitabilityandgrowth. forexample, Canadian companies may apply financial models

Chapter 10 Capital Markets and thePricing of Risk 10-1.

Excel Solution and Excel Project a.

E  R   0.25(0.1)  0.1(0.2)  0.1(0.25)  0.25(0.3)  5.5%

b.

Variance  R    0.25  0.055   0.1   0.1  0.055   0.2 2

2

  0  0.055   0.15   0.1  0.055   0.25   0.25  0.055   0.3 2

2

2

 2.648%

Standard Deviation  0.02648  16.27% 10-2.

Excel Solution a.

E [R] = - 1(0.4) - 0.75 (0.2) - 0.5 (0.2) - 0.25 (0.1) +10 (0.1) = 32.5%

b.

Variance [ R] = (- 1 - 0.325) 0.4 + (- 0.75 - 0.325) 0.2 + (- 0.5 - 0.325) 0.2

2

2

2

2

2

+ (- 0.25 - 0.325) 0.1 + (10 - 0.325) 0.1 = 10.46 Standard Deviation  10.46  3.235  323.5%

10-3.

Startup Inc. has a higher expected return, but is riskier. It is impossible tosay which stock would be preferable. It depends on risk performances and what other stocks are being held.

10-4.

Compute therealized return and dividend yield on this equity investment.

1  (55  50)  0.12  12% 50

a.

R

b.

Rdiv 

1  2% 50

Rcapital gain 

55  50  10% 50

The realized return on theequity investment is 12%. thedividend yield is 10%.

10-5.

Compute thecapital gain and dividend yield under theassumption thestock price has fallen to$45. a.

Rcapital gain  45  50 / 50  10%. Yes, thecapital gain is different, because thedifference between thecurrent price and thepurchase price is different than in Problem 4.

b.

The dividend yield does not change, because thedividend is thesame as in Problem 4.

The capital gain changes with thenew lower price; thedividend yield does not change.

10-6.

Excel Solution

Date 02/01/03 05/02/03 14/05/03 13/08/03 12/11/03 02/01/04

Date 02/01/08 06/02/08 07/05/08 06/08/08 05/11/08 02/01/09

10-7.

Price 33.88 30.67 29.49 32.38 39.07 41.99

Price 86.62 79.91 84.55 65.4 49.55 45.25

Dividend

R

1+R

0.17 0.17 0.17 0.17

–8.97% –3.29% 10.38% 21.19% 7.47%

0.910272 0.967069 1.103764 1.211859 1.074738

26.55%

1.265491

Dividend

R

1+R

0.4 0.4 0.4 0.4

–7.28% 6.31% –22.18% –23.62% –8.68%

0.927153 1.063071 0.778238 0.763761 0.913219

–46.5%

0.535006

Excel Solution

Date

End of Month Price

Dividend (end of month)

Monthly Return (1+Rt)

Monthly Return % (Rt)

Aug-94

$29.25

$0.00

Sep-94

$27.75

$0.00

0.949

–5.13%

Oct-94

$29.50

$0.26

1.072

7.24%

Nov-94

$27.12

$0.00

0.919

–8.07%

Dec-94

$27.87

$0.00

1.028

2.77%

Jan-95

$25.25

$0.26

0.915

–8.47%

Feb-95

$26.12

$0.00

1.034

3.45%

Mar-95

$26.87

$0.00

1.029

2.87%

Apr-95

$27.12

$0.31

1.021

2.08%

May-95

$29.25

$0.00

1.079

7.85%

Jun-95

$29.75

$0.00

1.017

1.71%

Jul-95

$29.00

$0.31

0.985

–1.48%

Aug-95

$30.75

$0.00

1.060

6.03%

Sep-95

$31.12

$0.00

1.012

1.20%

Oct-95

$28.75

$0.35

0.935

–6.49%

Nov-95

$28.25

$0.00

0.983

–1.74%

Dec-95

$28.87

$0.00

1.022

2.19%

Jan-96

$29.50

$0.35

1.034

3.39%

Feb-96

$31.25

$0.00

1.059

5.93%

Mar-96

$34.38

$0.00

1.100

10.02%

Apr-96

$35.88

$0.35

1.054

5.38%

May-96

$36.50

$0.00

1.017

1.73%

Jun-96

$32.38

$0.00

0.887

–11.29%

Jul-96

$32.38

$0.39

1.012

1.19%

Aug-96

$33.50

$0.00

1.035

3.46%

Sep-96

$31.25

$0.00

0.933

–6.72%

Oct-96

$31.25

$0.39

1.012

1.23%

Nov-96

$32.75

$0.00

1.048

4.80%

Dec-96

$32.25

$0.00

0.985

–1.53%

Jan-97

$32.13

$0.39

1.008

0.82%

Feb-97

$32.88

$0.00

1.023

2.33%

Mar-97

$31.37

$0.00

0.954

–4.59%

Apr-97

$34.75

$0.42

1.121

12.11%

May-97

$37.50

$0.00

1.079

7.91%

Jun-97

$38.00

$0.00

1.013

1.33%

Jul-97

$40.88

$0.42

1.087

8.68%

Aug-97

$43.00

$0.00

1.052

5.19%

Sep-97

$45.13

$0.00

1.050

4.95%

Oct-97

$43.69

$0.24

0.973

–2.66%

Nov-97

$43.00

$0.00

0.984

–1.58%

Dec-97

$48.56

$0.00

1.129

12.93%

Jan-98

$51.00

$0.24

1.055

5.52%

Feb-98

$56.56

$0.00

1.109

10.90%

Mar-98

$64.81

$0.00

1.146

14.59%

Apr-98

$45.81

$12.31

0.897

–10.33%

May-98

$51.88

$0.00

1.133

13.25%

Jun-98

$59.00

$0.00

1.137

13.72%

Jul-98

$57.00

$0.24

0.970

–2.98%

Aug-98

$44.63

$0.00

0.783

–21.70%

Total Return (product of 1 + Rs) = 2.2428.

Equivalent monthly return = (Total return)(1/48) – 1 = 1.697%.

10-8.

Excel Solution Using thesame data as Problem 7: a.

average return over this period = 1.959%.

b.

standard deviation over theperiod = 7.241%.

10-9.

Both numbers are useful. therealized return (in problem 10-7) tells you what you actually made by holding thestock over this period. theaverage return (problem 10-8a) over theperiod can be used as an estimate of themonthly expected return. If you use this estimate, then this is what you can expect tomake on thestock in thenext month.

10-10.

Excel Solution Using thesame data as Problem 7: standard error of theestimate = 1.045%.

95% confidence interval high value = 4.050%.

95% confidence interval low value = –0.131%.

10-11.

For large portfolios, there is a relationship between returns and volatility—portfolios with higher returns have higher volatilities. forstocks, no clear relationship exists.

10-12.

The expected payoffs are thesame, but bank A is less risky because therisk is diversified across several loans that are not correlated with each other.

10-13. a.

Expected payoff is thesame forboth banks.

Bank B  $100 million  0.95  $95 million.

Bank A  $1 million  0.95  100  $95 million.

b.

Bank B

Variance  100  95 0.95  0  95 0.05  475.

2

2

Standard Deviation  475  21.79.

Bank A

Variance of each loan  1  0.95 0.95  0  0.95 0.05  0.0475.

2

2

Standard Deviation of each loan  0.0475  0.2179.

Now thebank has 100 loans that are all independent of each other, so thestandard deviation of theaverage loan is

0.2179 100

 0.02179.

But thebank has 100 such loans so thestandard deviation of theportfolio is 100  0.02179  2.179,

which is much lower than thedeviation forBank B. 10-14.

A risk-averse investor would choose theeconomy in which stock returns are independent because there is only unique or idiosyncratic risk and all such risk can be diversified away in a large enough portfolio.

10-15.

E R  0.15 0.6  0.10.4  0.05

a.

Standard Deviation 

0.15  0.052 0.6    0.1  0.052 0.4  0.12247.

Because all S firms in theportfolio move together, there is no diversification benefit. So thestandard deviation of theportfolio is thesame as thestandard deviation of thestocks: 12.25%

E [R] = 0.15 (0.6) - 0.1(0.4) = 0.05

b.

Standard Deviation =

(0.15 - 0.05)2 0.6 + (- 0.1 - 0.05)2 0.4 = 0.12247

Type I stocks move independently. Hence thestandard deviation of theportfolio is 0.12247 SD Portfolio of 20 Type I stocks   2.74%.

20 10-16.

Volatility of Portfolio

14.00% 12.00% 10.00% 8.00% Type S

6.00%

Type I

4.00% 2.00% 0.00% 1

21

41

61

81

Number of Stocks

Expected return of a stock Standard Deviation of a stock Number of Stocks 1 2 3 4 5 6 7 8 9 10 11

0.05 0.122474

Type S 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25%

Type I 12.25% 8.66% 7.07% 6.12% 5.48% 5.00% 4.63% 4.33% 4.08% 3.87% 3.69%

(Continues)

(Continues) Number of Stocks 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56

Type S 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25%

Type I 3.54% 3.40% 3.27% 3.16% 3.06% 2.97% 2.89% 2.81% 2.74% 2.67% 2.61% 2.55% 2.50% 2.45% 2.40% 2.36% 2.31% 2.27% 2.24% 2.20% 2.17% 2.13% 2.10% 2.07% 2.04% 2.01% 1.99% 1.96% 1.94% 1.91% 1.89% 1.87% 1.85% 1.83% 1.81% 1.79% 1.77% 1.75% 1.73% 1.71% 1.70% 1.68% 1.67% 1.65% 1.64%

57

12.25%

1.62%

58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99

Type S 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25% 12.25%

Type I 1.61% 1.59% 1.58% 1.57% 1.56% 1.54% 1.53% 1.52% 1.51% 1.50% 1.49% 1.47% 1.46% 1.45% 1.44% 1.43% 1.42% 1.41% 1.40% 1.40% 1.39% 1.38% 1.37% 1.36% 1.35% 1.34% 1.34% 1.33% 1.32% 1.31% 1.31% 1.30% 1.29% 1.28% 1.28% 1.27% 1.26% 1.26% 1.25% 1.24% 1.24% 1.23%

(Continues) (Continues) Number of Stocks

10-17.

Investors can costlessly remove diversifiable risk from their portfolio by diversifying. They therefore do not demand a risk premium forit.

10-18.

10-19.

a.

Diversifiable risk

b.

Systematic risk

c.

Diversifiable risk

d.

Diversifiable risk

Excel Solution R(i) : (1.05)(1.40) – 1 = 47% or (1.05)(0.80) –1 = –16% R(ii) : 1.42 – 1 = 96%, 1.4 × 0.8 – 1 = 12%, 0.8 × 1.4 – 1 = 12%, 0.8 × 0.8 – 1 = –36% a.

ER(i) = (47% – 16%) / 2 = 15.5% ER(ii) = (96% + 12% + 12% – 36%) / 4 = 21%

b.

Vol(i) = (1/2 (47% – 15.5%)2 + 1/2 (–16% – 15.5%)2)1/2 = 31.5% Vol(ii) = (1/4 (96% – 21%)2 + 1/2 (12% – 21%)2 + 1/4 (–36% – 21%)2) 1/2 = 47.5%

c.

No, holding stocks fora longer period does not decrease therisk.

10-20.

An efficient portfolio is any portfolio that contains only systematic risk—it contains no diversifiable risk.

10-21.

Beta measures theamount of systematic risk in a stock.

10-22.  Stock

b.

Beta 

c.

A firm that moves independently has no systematic risk, so Beta = 0

 Stock  Market



30   10

18  22 30   10



60

Beta 

 Market



43   17 

a.



40

 1.5

40 40

 1

10-23.

a. E[RM] = ½ (30%) + ½ (–10%) = 10% i. E[R] = 4% + 1.5 (10% – 4%) = 13% ii. Actual Expected Return = (43% – 17%) / 2 = 13% b.

i. E[R] = 4% –1(10% – 4%) = –2% ii. Actual Expected Return = (–22% + 18%) / 2 = –2%

10-24.

10-25.

Excel Solution a.

E[RSU] = 0.04 + 1.2688  0.06 = 11.6128%

b.

E[RBBB/B] = 0.04 + 2.7674  0.06 = 20.6044%

c.

E[RMFI] = 0.04 + 0.3824  0.06 = 6.2944%

Cost of Capital  rf    E  R m   rf   5  1.2  6.5  12.8%

10-26. a.

This statement is inconsistent with both.

b.

This statement is consistent with both.

c.

This statement is inconsistent with theCAPM but not necessarily with efficient capital markets.

Chapter 11 Optimal Portfolio Choice and theCapital Asset Pricing Model 11-1. a.

Let ni be thenumber of shares in stock i, then nG 

200, 000  0.5

 4, 000 25 200, 000  0.25 nM   625 80 200, 000  0.25 nV   25, 000 2

The new value of theportfolio is

p  30nG  60nM  3nV  $232, 500 .

b.

Return 

232, 500

 1  16.25% .

200, 000

c.

The portfolio weight is thefraction of value invested in each stock: n  30 GoldFinger: G  51.61% 232, 500 Moosehead:

nM  60

 16.13%

232, 500 Venture:

nV  3 232, 500

 32.26%.

11-2.

Apple 1000 Cisco 10000 Goldman 5000

125 19 120 Total

Value a.

12 125000 0.136612022 10 190000 0.207650273 10.5 600000 0.655737705 915000

b.

1.639344262 2.076502732 6.885245902 10.6010929

New Price 130 24 110

New Value 130000 240000 550000

c. d.

0.141304348 1.6956522% 0.260869565 2.6086957% 0.597826087 6.2771739% 10.5815217%

11-3.

Total Number of Shares Outstanding 100.00

Current Price per Share $100

Expected Return 18%

Value $10,000.00

b.

0.454545455

8.18%

Fast Mover

50.00

$120

12%

$6,000.00

0.272727273

3.27%

Funny Bone

200.00

$30

15%

$6,000.00

0.272727273

4.09%

Stock First Bank

a. (in millions) $22,000.00

c. 15.55%

11-4.

Both calculations of expected return of a portfolio give thesame answer.

11-5.

If theprice of one stock goes up, theother stock price always goes up proportionally as well. If theprice of one stock goes down, theother stock price always goes down proportionally as well.

11-6.

Excel Solution a.

RA 

10  20  5  5  2  9

 3.5%

6 RB 

21  30  7  3  8  25 6

 12%

 0.1  0.035 2     2 2 1  0.2  0.035    0.05  0.035     Variance of A   5  0.05  0.035 2   0.02  0.035 2       0.09  0.035 2   0.01123 Volatility of A = SD( RA )  Variance of A  0.01123  10.60%  0.21  0.12 2   0.3  0.12 2    1 2 2 Variance of B   0.07  0.12    0.03  0.12     5  0.08  0.12 2   0.25  0.12 2     0.02448 Volatility of B = SD( RB )  Variance of B  0.02448  15.65%

b.

 0.1  0.035  0.21  0.12     0.2  0.035  0.3  0.12        0.05  0.035 0.07  0.12     1 Covariance    5  0.05  0.035  0.03  0.12     0.02  0.035  0.08  0.12       0.09  0.035  0.25  0.12    0.00794

c.

Correlation



Covariance SD(RA ) SD (RB )

 0.479 11-7.

Excel Solution Parts a, b, and c:

Year

2015

2016

2017

2018

2019

2020

A&B Ave Vol

5.5% 7.75% 11.36%

25.0%

6.0%

-4.0%

-3.0%

17.0%

d.

11-8.

The portfolio has a lower volatility than theaverage volatility of thetwo stocks because some of theidiosyncratic risk of thestocks in theportfolio is diversified away.

Excel Solution Variance = 0.7 0.106  0.3 0.1565  2(0.7)(0.3)(0.4789)(0.106)(0.1565).5 2

2

Standard Deviation = 11-9.

=10.51%

Excel Solution Covariance  corr  S /B 0.40  0.36  0.31  0.04464



11-10.

Because thecorrelation is perfect, they move together (always) and so theprobability is 1.

11-11. stock vol

40%

corr

50-50 Portfolio Vol 1

40.0%

0.5

34.6%

0

28.3%

-0.5

20.0%

-1

0.0%

Volatility of portfolio is less if thecorrelation is < 1.

11-12.

Vol Corr Wesley 60% 25% Addison 30% Portfolio x_A x_W Vol 100% 0% 30.00% 75% 25% 30.00% 50% 50% 36.74% 11-13.

Avon has twice therisk, so theportfolio needs twice as much weight on Nova => 1/3 Avon, 2/3 Nova.

11-14.

Excel Solution

Tex Mex Portfolio x_tex 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

Vol 40% 20%

Corr 0%

x_mex 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0%

Vol 20.00% 18.44% 17.89% 18.44% 20.00% 22.36% 25.30% 28.64% 32.25% 36.06% 40.00%

(a) 20% weight in Tex is thelowest volatility portfolio. (b) 40% weight in Tex gives thesame volatility as Mex. 11-15.

Volatility = standard deviation = 27.72%

11-16.

Vol Var Corr Covar N

a. b. c.

1 30 1000

50% 0.25 20% 0.05 Vol 50.0% 23.8% 22.4%

11-17.

Volatility: 11-18. a.

ave cov = (0.5  0.5  0.4)0.5 = 31.62%

Ave Covar = 40%  40%  20% = 0.032 Limit Vol = (0.032)0.5 = 17.89%

b.

From Eq. 11.13 Corr = SD(RP)/SD(Ri) = 17.89% / 40% = 44.72%

11-19.

From Eq. 11.13, marginal contribution torisk is SD(Ri)  Corr(Ri,RP): For A: 65%  10% = 6.5%; For B: 30%  25% = 7.5%.

So, volatility increases if we sell A and add B. 11-20. a.

Max vol = weighted average = 0.5(60%) + 0.25(30%) + 0.25(20%) = 42.5%

b.

Correlation = 1 (otherwise there would be some diversification)

11-21. a.

A ER Vol XA

B 20% 40%

XB 50%

Corr 10% 30%

Vol 50%

0% ER

25.0%

b.

No, dominated by 50–50 portfolio.

c.

Yes, not dominated.

15.0%

11-22. a.

If thetwo stocks are perfectly negatively correlated, they fluctuate due tothe same risks, but in opposite directions. Because Intel is twice as volatile as Coke, we will need tohold twice as much Coke stock as Intel in order tooffset Intel’s risk. That is, our portfolio should be 2/3 Coke and 1/3 Intel.

We can check this using Eq. 11.9:

Var ( RP )  (2 / 3) 2 SD( RCoke ) 2  (1 / 3) 2 SD( RIntel ) 2  2(2 / 3)(1 / 3)Corr ( RCoke , RIntel ) SD( RCoke ) SD( RIntel )  (2 / 3) 2 (0.252 )  (1 / 3) 2 (0.502 )  2(2 / 3)(1 / 3)( 1)(0.25)(0.50) 0 b.

From Eq. 11.3, theexpected return of theportfolio is

E[ RP ]  (2 / 3) E[ RCoke ]  (1 / 3) E[ RIntel ]  (2 / 3)6%  (1 / 3)26%  12.67%.

Because this portfolio has no risk, therisk-free interest rate must also be 12.67%.

11-23. a.

In this case, theportfolio weights are xB = xS = 0.50. From Eq. 11.3,

E[ RP ]  xB E[ RB ]  xS E[ RS ]  0.50(7%)  0.50(10%) .  8.5% b.

We can use Eq. 11.9: SD( RP ) 

xB 2 SD( RB ) 2  xS 2 SD( RS ) 2  2 xB xS Corr ( RB , RS ) SD( RB ) SD( RS )

 0.502 (0.162 )  0.502 (0.20) 2  2(0.50)(0.50)(0.22)(0.16)(0.20)  14.1%.

11-24. a. b.

The expected return would remain constant, assuming only thecorrelation changes: 0.5  0.07 + 0.5  0.10 = 0.085.

The volatility of theportfolio would increase (due tothe correlation term in theequation forthe volatility of a portfolio).

11-25. a.

In this case, thetotal investment is $10,000 – 2,000 = $8,000, so theportfolio weights are xB = 10,000 / 8,000 = 1.25, xS = –2,000 / 8,000 = –0.25. From Eq. 11.3:

E[ RP ]  xB E[ RB ]  xS E[ RS ]  1.25(7%)  0.25(10%) .  6.25% We can use Eq. 11.9: SD( RP ) 

xB 2 SD( RB ) 2  xS 2 SD( RS ) 2  2 xB xS Corr ( RB , RS ) SD( RB ) SD( RS )

 1.252 (.162 )  (0.25)2 (0.20)2  2(1.25)(0.25)(0.22)(0.16)(0.20)  19.5%.

11-26.

Excel Solution The set of efficient portfolios is those portfolios with no more than about 65% invested in Biovail (this is theportfolio with thelowest possible volatility, as shown in thechart and was solved forusing Excel’s Solver). See thespreadsheet file forthe details of thesolution.

Weight in Biovail 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1

Weight in Shoppers Drug Mart 1 0.95 0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

E [R p ] 10.00% 9.85% 9.70% 9.55% 9.40% 9.25% 9.10% 8.95% 8.80% 8.65% 8.50% 8.35% 8.20% 8.05% 7.90% 7.75% 7.60% 7.45% 7.30% 7.15% 7.00%

SD [R p ] 20.00% 19.19% 18.42% 17.68% 16.99% 16.35% 15.77% 15.24% 14.79% 14.41% 14.11% 13.90% 13.78% 13.75% 13.82% 13.98% 14.23% 14.56% 14.97% 15.45% 16.00%

11-27. a.

The total value of theportfolio is $50m (= –$35 + $85). This means that theweight on Oracle is –70% and theweight on Intel is 170%. theexpected return is

Expected return  0.7  12%  1.7  14.5%  16.25%. b.

Variance   0.7  0.45  1.7  0.40 2

2

2

2

 2   0.7  1.7  0.65  0.45  0.40  0.283165.

Std dev = (0.283165)^0.5 = 53.2%.

11-28.

In this case, with short-selling, an increase in correlation will decrease thevolatility and risk of theportfolio.

11-29.

From Eq. 11.13, fora small transaction size, short selling A and investing in P changes risk according toSD(RP) – SD(RA)Corr(RA,RP).

We gain therisk of theportfolio and lose therisk A has in common with theportfolio. forthis tobe negative, we must have SD(RP) / SD(RA) < Corr(RA,RP), or Corr > 30%/40% = 75%.

11-30. a.

A ER Vol XA

XB 50%

11-31.

B 20% 40%

Corr 12% 30%

Vol 50%

0% ER

25.0%

16.0%

b.

No, it has thesame expected return with higher volatility.

c.

Yes, adding this stock and reducing weight on theothers will reduce risk while leaving expected return unchanged.

Expected return = 18% Volatility= x2 0.32  y 2 0.252  2 xy0.9  0.3  0.25  39.05%

11-32.

Either strategy has expected return of 2(20%) – 1(10%) = 30%.

But theportfolio has lower volatility if correlation is +0.5; because you are shorting a positive correlation, it leads tolower risk. +2 HGH – KBH volatility = 52.9%

+2 HGH – LWI volatility = 72.1%

11-33. a.

x

115, 000  1.15 100, 000





E  R   rf  x E  R j   r  4%  1.15 11%   16.65% Volatility  x SD  R j   1.15(25%)  28.75%

b.

R

115, 000(1.25)  15, 000(1.04)

 1  28.15%

100, 000

c.

R

115, 000(0.80)  15, 000(1.04)

 1  23.6%

100, 000

11-34.

11-35.

a.

Er = 5% + 1.5  (10% – 5%) = 12.5%

b.

Vol = 1.5  15% = 22.5%

Invest an amount x in theother portfolio and theexpected return and volatility are

E[ Rx ]  rf  x( E[ RO ]  rf )  5%  x(20%  5%) SD( Rx )  x SD( RO )  x(12%). a.

So tomaintain thevolatility at 8%, x  8% /12%  2 / 3, you should invest $66,667 in theother portfolio and theremaining $33,333 in therisk-free investment. Your expected return will then be 15%.

b.

Alternatively, tokeep theexpected return equal tothe current value of 12%, x must satisfy 5% + x(15%) = 12%, so x = 46.667%. Now you should invest $46,667 in theother portfolio and $53,333 in theriskfree investment, lowering your volatility to5.6%.

11-36.

Sharpe ratios of A, B, and C are 0.6, 0.5, and 1, so you would choose C; it is thebest choice no matter what your clients’ risk preferences.

11-37.

Investors who want tomaximize their expected return fora given level of volatility will pick portfolios which maximize their Sharpe ratio. theset of portfolios that do this is a combination of therisk-free asset and thetangent portfolio.

11-38.

Required Return = 0.04 

11-39.

Your current portfolio is not efficient.

0.8  0.2  0.12  0.04  0.0912  9.12% 0.25 You should add some of theventure fund toyour portfolio because it has an expected return of 20% and this exceeds therequired return.

11-40.

Initial Portfolio 60–40 Portfolio 85–15 Portfolio Natasha fund Expected return Volatility

0.14 0.2

0.14 0.2

0.14 0.2

Hannah stock Expected return Volatility

0.2 0.6

0.2 0.6

0.2 0.6

Risk-free rate

0.038

0.038

0.038

Portfolio weight in Hannah Expected return of Portfolio Volatility of portfolio Beta Required return

0 0.14 0.2 0 0.038

0.4 0.164 0.268328157 2 0.29

0.15 0.149 0.192353841 1.459459459 0.2

a. Yes, because theexpected return of Hannah stock exceeds therequired return. b. Yes, because theexpected return of Hannah stock is less than therequired return. c. Yes, because now therequired and expected return are thesame.

11-41.

Initial Portfolio

60–40 Portfolio

85–15 Portfolio

Natasha Fund Expected Return Volatility

0.14 0.2

0.14 0.2

0.14 0.2

Hannah Stock Expected Return Volatilty

0.2 0.6

0.2 0.6

0.2 0.6

Risk Free Rate

0.038

0.038

0.038

Portfolio weight in Hannah Expected Return of Portfolio Volatility of Portfolio Sharpe Ratio

0 0.4 0.14 0.164 0.2 0.268328157 0.51 0.469574275 0.09002 0.097166359 The Sharpe Ratio is maximized at 15% in Hannah Stock.

0.15 0.149 0.192353841 0.577061522 0.102053829

11-42. a.

0.32

b.

0.271

c.

13% Initial Portfolio

50-50 Split

Tanglewood fund Expected return Volatility

0.12 0.25

0.12 0.25

Venture fund Expected return Volatilty

0.2 0.8

0.2 0.8

Risk-free rate

0.04

0.04

0.25

Portfolio weight in Hannah Expected return of portfolio Volatility of portfolio Sharpe ratio

0 0.5 0.12 0.16 0.25 0.44229515 0.32 0.271312041 0.0656 0.072557445 The Sharpe Ratio is maximized at 12% in theventure fund.

Weight in Venture Fund 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0.11 0.12 0.13 0.14 0.15 0.16 0.17 0.18 0.19 0.2 0.21 0.22

Expected Return 0.12 0.1208 0.1216 0.1224 0.1232 0.124 0.1248 0.1256 0.1264 0.1272 0.128 0.1288 0.1296 0.1304 0.1312 0.132 0.1328 0.1336 0.1344 0.1352 0.136 0.1368 0.1376

Volatility 0.25 0.249223293 0.248694592 0.248415479 0.248386795 0.248608628 0.249080308 0.24980042 0.250766824 0.251976685 0.253426518 0.25511223 0.257029181 0.25917224 0.261535848 0.264114085 0.266900731 0.269889329 0.27307325 0.276445745 0.28 0.283729184 0.287626494

Sharpe Ratio 0.32 0.324207256 0.328113287 0.331702358 0.334961446 0.337880469 0.340452445 0.342673563 0.344543184 0.346063763 0.347240694 0.348082097 0.34859855 0.348802788 0.348709366 0.348334319 0.347694814 0.346808821 0.34569479 0.34437137 0.342857143 0.341170403 0.339328963

0.25

Part a

Part c

Weight in Venture Fund 0.23 0.24 0.25 0.26 0.27 0.28 0.29 0.3 0.31 0.32 0.33 0.34 0.35 0.36 0.37 0.38 0.39 0.4 0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.5 0.51 0.52 0.53 0.54 0.55 0.56 0.57 0.58 0.59 0.6 0.61 0.62 0.63 0.64 0.65 0.66 0.67 0.68 0.69

Expected Return 0.1384 0.1392 0.14 0.1408 0.1416 0.1424 0.1432 0.144 0.1448 0.1456 0.1464 0.1472 0.148 0.1488 0.1496 0.1504 0.1512 0.152 0.1528 0.1536 0.1544 0.1552 0.156 0.1568 0.1576 0.1584 0.1592 0.16 0.1608 0.1616 0.1624 0.1632 0.164 0.1648 0.1656 0.1664 0.1672 0.168 0.1688 0.1696 0.1704 0.1712 0.172 0.1728 0.1736 0.1744 0.1752

Volatility 0.29168519 0.295898631 0.300260304 0.304763843 0.309403054 0.314171927 0.319064649 0.324075608 0.329199408 0.33443086 0.339764992 0.345197045 0.350722469 0.356336919 0.362036255 0.36781653 0.373673989 0.379605058 0.385606341 0.39167461 0.3978068 0.404 0.410251447 0.416558519 0.422918727 0.42932971 0.435789227 0.44229515 0.448845463 0.45543825 0.462071694 0.468744067 0.475453731 0.482199129 0.488978783 0.495791287 0.502635305 0.509509568 0.516412868 0.523344055 0.530302037 0.537285771 0.544294268 0.551326582 0.558381814 0.565459106 0.572557639

Sharpe Ratio 0.337350004 0.335249945 0.333044358 0.330747896 0.328374263 0.325936187 0.323445422 0.320912766 0.318348082 0.315760334 0.313157631 0.31054727 0.307935789 0.305329013 0.302732112 0.300149642 0.297585605 0.295043487 0.29252631 0.290036671 0.287576784 0.285148515 0.282753421 0.280392777 0.278067611 0.275778725 0.273526725 0.271312041 0.269134947 0.26699558 0.264893958 0.262829994 0.260803506 0.258814238 0.256861861 0.254945989 0.253066187 0.251221975 0.24941284 0.247638239 0.245897604 0.244190349 0.242515874 0.240873566 0.239262807 0.237682971 0.236133431

Part b

11-43.

All investors will want tomaximize their Sharpe ratios by picking efficient portfolios. When a riskless asset exists, this means that all investors will pick thesame efficient portfolio, and because thesum of all investors’ portfolios is themarket portfolio, this efficient portfolio must be themarket portfolio.

11-44. a.

26.16% is in themarket; therest in therisk-free asset.

b.

52.53% is in themarket; therest in therisk-free asset.

a.

Under theCAPM assumptions, themarket is efficient; that is, a leveraged position in themarket has thehighest expected return of any portfolio fora given volatility and thelowest volatility fora given expected return. By holding a leveraged position in themarket portfolio, you can achieve an expected return of

11-45.

E R p   rf  x  E  Rm   rf   5%  x  5% .

Setting this equal to12% gives 12  5  5x  x  1.4.

So theportfolio with thelowest volatility and that has thesame return as Microsoft has $15,000 1.4  $21,000 in themarket portfolio and borrows $21,000  $15,000  $6,000 ; that is, – $6,000 in theforce asset.

SD  Rp   xSD  Rm   1.4 18  25.2% Note that this is considerably lower than Microsoft’s volatility. b.

A leveraged portion in themarket has volatility 

SD  Rp   xSD  Rm   x 18%.

Setting this equal tothe volatility of Microsoft gives

40%  x 18% x

40  2.222.

18

So theportfolio with thehighest expected return that has thesame volatility as Microsoft has $15,000  2.2  $33,000 in themarket portfolio and borrows 33,000  15,000  $18,333.33 , that is – $18,333.33 in thein force asset.





E  Rp   rf  x E  Rm   rf  5%  2.222  5%  16.11% Note that this is considerably higher than Microsoft’s expected return.

11-46. a.

Erm = 15%, vol = 16.3%

b.

Value stocks have a higher Sharpe ratio than themarket, so market is not efficient.

11-47.

11-48.

0.2  0.075 0.16

a.

 JJ  0.06 

b.

E  RJJ   0.04  0.075  0.1  0.04  4.45%

Excel Solution and Excel Project Portfolio Weight

HEC Corp 0.25 Green Midget 0.35 AliveAndWell 0.4

Volatility

12% 25% 13%

Correlation Beta (Part a Expected with the answer) Return (Part a Market answer) Portfolio 0.4 0.48 5.4 0.6 1.5 10.5 0.5 0.65 6.25

Part c answer: Part b answer 0.905 7.525 Part d answer Expected Return calculated from porfolio beta 7.525 Porfolio

11-49.

  0.62.16  0.40.69  1.572 E  R  4  1.57210  4  13.432%

11-50.

Risk premium = 0. It is uncorrelated with themarket, so there is no incremental risk from adding it toyour portfolio.

Note also that since thestock is positively correlated with itself (which is part of themarket), tohave zero beta it must be negatively correlated with theother stocks. Thus, it offsets risk that other stocks have. Thus, taking it out will not reduce risk.

Certainly! Here'samore detailed explorationof"Contemporary Business Mathematicswith Canadian Applications"thatexpands toapproximately 3000 words:### Introduction toContemporary Business Mathematicswith Canadian ApplicationsContemporary Business Mathematicswith Canadian Applications" isatextbook designed tobridgethegap between mathematicaltheoryandpractical business applications, tailored specifically forCanadian studentsandbusinesses. This comprehensive text coversawide rangeofmathematical concepts essential forunderstandingandmanaging various aspectsofbusiness operations intheCanadian context.inthis detailed exploration, we will delve intothekey components,themes,andrelevanceofthis textbook.

The textbook startswith foundational mathematical conceptsthatformthebasis formore complex calculationsandanalysesinbusiness settings. This includes arithmetic operations, algebraic equations,andbasic statistical measures. Understandingthese fundamental principles is crucial asthey providetheframework upon which more advanced topicsarebuilt.Financial mathematics isacornerstoneofbusiness education, particularlyinunderstandingthetime valueofmoneyandfinancial decision-making. Topics covered often include interest calculations, compound interest, annuities, loans,andinvestments.these conceptsareessential forevaluatingtheprofitabilityofinvestments, determining loan payments,andunderstandingtheimplicationsofinterest ratesonbusiness finances.#### 3. Inventory ControlEffective inventory management is vital forbusinesses toensure adequate stock levels while minimizing holding costs.thetextbook covers mathematical techniquesandmodels used forinventory control, including inventory turnover rates, economic order quantity (EOQ),andreorder points. Canadian applications may include considerations specific toCanadian supply chain dynamicsandregulations.#### 4. Marketing MathematicsMathematical tools playacrucial roleinmarketing decisionsbyproviding quantitative insights into consumer behavior, pricing strategies,andsales forecasts. Topics covered often include break-even analysis, pricing models, market segmentation,andpromotional budgeting. Case studiesandexamples tailored toCanadian markets illustrate howthese mathematical conceptsareappliedinreal-world marketing scenarios.#### 5. Data AnalysisandStatisticsIn today's data-driven business environment, proficiencyindata analysisandstatistical methods is essential.thetextbook typically covers statistical techniques used toanalyze business data, suchasdescriptive statistics, probabilitytheory, hypothesis testing,andregression analysis.these tools enable businesses toextract meaningful insights from data tosupport decision-makingandstrategic planning.#### 6.

Spreadsheet ApplicationsSpreadsheetsarepowerful tools extensively usedinbusiness forfinancial modeling, data analysis,anddecision support.thetextbook includes guidanceonutilizing spreadsheet software like Excel toperform calculations, create financial models,andvisualize data. Practical exercisesandexamples help students develop proficiencyinusing spreadsheets forbusiness applications.### RelevanceofCanadian ApplicationsThe inclusionof"Canadian Applications" inthetextbook's title emphasizes its focusonintegrating Canadian business practices, regulations,andeconomic conditions intothelearning experience. This ensuresthatstudents not only grasp mathematical concepts but also understand howthese conceptsareappliedwithintheunique contextofCanada. Examplesandcase studies drawn from Canadian businesses provide practical insightsandrelevance, preparing students forthechallengesandopportunitiesthey may encounter intheCanadian business landscape.### Pedagogical ApproachandLearning Resources#### Pedagogical ApproachThe textbook typically adoptsapedagogical approachthatcombinestheoretical explanationswith practical examplesandapplications. Conceptsarepresented inaclearandstructured manner tofacilitate understanding,with step-by-step explanationsofmathematical calculationsandproblem-solving techniques. This approach helps students develop both mathematical proficiency andtheability toapplythese skills toreal-world business scenarios.#### Learning ResourcesIn addition tothetextbook itself, supplementary learning resources often accompany "Contemporary Business Mathematicswith Canadian Applications" toenhance student learningandengagement.these may include:- **Online Resources**: Access tointeractive tutorials, practice exercises,andsolutions.- **Case Studies**: Detailed analysesofreal-world business scenarios relevant toCanadian markets.**Workbook Exercises**: Additional problemsandexercises toreinforce learningandmasteryofmathematical concepts.- **Instructor Support**: Guidanceandsupport forinstructors toeffectively teachandintegratethetextbook intotheir courses.### BenefitsandPractical Applications#### Benefits forStudents- **Practical Relevance**:byfocusingonCanadian applications,thetextbook prepares students toapply mathematical concepts inthespecific contextofCanadian businesses, enhancingtheir readiness forcareers intheCanadian business

landscape.- **Skill Development**: Students develop critical quantitative skills thatarehighly valuedinvarious business disciplines, including finance, marketing, operations,andstrategic management.- **Problem-Solving Abilities**: Through exercisesandcase studies, students learn toanalyze business problems, formulate mathematical models,andderive solutions, improvingtheir analyticalandproblem-solving abilities.#### Practical ApplicationsinBusiness- **Financial Decision-Making**: Businesses use financial mathematics toevaluate investment opportunities, manage cash flow,andmake informed financial decisions.- **Marketing Nursing is a profession deeply rooted in

caring for individuals, families, and communities to promote health and well-being, prevent illness and injury, and provide support to those experiencing health challenges. It encompasses a wide range of roles, from direct patient care to leadership in healthcare policy and education. In this essay, we will explore the multifaceted nature of nursing, its historical evolution, educational pathways, professional responsibilities, ethical considerations, and its crucial role in the healthcare system.### Historical Evolution of NursingThe history of nursing dates back centuries and has evolved significantly over time. Nursing as a formalized profession began to take shape in the 19th century with the pioneering efforts of figures like Florence Nightingale, who is often regarded as the founder of modern nursing. Nightingale's work during the Crimean War demonstrated the critical importance of sanitation and hygiene in healthcare settings, laying the foundation for evidence-based nursing practice.Throughout the 20th century, nursing continued to professionalize and expand its scope. World War I and World War II highlighted the essential role of nurses in caring for wounded soldiers and civilians, leading to advancements in nursing education and practice. The establishment of nursing schools and regulatory bodies further solidified nursing as a respected profession with defined standards of practice.### Educational Pathways in NursingToday, nursing education encompasses a variety of pathways, from diploma programs to bachelor's and master's degrees in nursing (BSN, MSN) and doctoral programs (DNP, PhD). Each level of education prepares nurses for different roles within the profession, ranging from bedside care to advanced clinical practice, research, and leadership positions.Diploma programs, typically offered by hospitals, provide a more hands-on, clinical-focused education. Associate degree programs (ADN) and BSN programs offered by colleges and universities combine clinical training with a broader education in sciences, humanities, and nursing theory. BSN programs are increasingly becoming the preferred entry-level education for nurses due to their emphasis on critical thinking, leadership, and evidencebased practice.Advanced practice roles such as nurse practitioners (NPs), nurse anesthetists (CRNAs), nurse midwives (CNMs), and clinical nurse specialists (CNSs) require additional education at the master's or doctoral level. These nurses are trained to provide specialized care, prescribe medications, and diagnose health conditions, expanding access to healthcare services, particularly in underserved areas.### Professional Responsibilities and RolesThe responsibilities of nurses vary depending on their level of education, specialization, and practice setting. At the core of nursing practice is the provision of compassionate, patient-centered care. Nurses assess patients' health needs, develop and implement care plans, monitor patient progress, and advocate for their well-being. They collaborate with other healthcare professionals, such as physicians, therapists, and social workers, to ensure comprehensive care.In addition to direct patient care Strategy**: Mathematical models aidindetermining optimal pricing strategies, forecasting sales volumes,andassessingtheeffectivenessofmarketing campaigns.- **Operations Management**: Inventory control techniques help businesses optimize inventory levels, minimize costs,andensure timely availabilityofproducts.### Conclusion"Contemporary Business Mathematicswith Canadian Applications" isavaluable resourcethatequips studentswiththemathematical skillsandknowledge essential forsuccessintoday's competitive business environment.byintegrating Canadian-specific examplesandapplications,thetextbook enhances its relevanceandpractical utility forCanadian studentsandbusinesses alike. Throughastructured pedagogical approachandcomprehensive coverageofkey mathematical concepts,thetextbook prepares students tobecome adept problem solversanddecision makers capableofapplying mathematical principles toaddress real-world business challenges.In summary, this textbook serves asafoundational resource foranyone seeking tounderstandtheintersectionofmathematicsandbusinesswithintheCanadian context, providingasolid framework foracademic learningandprofessional development inthefieldofbusiness mathematics.---This expanded exploration should provideathorough overviewof"Contemporary Business Mathematicswith Canadian Applications"andits relevance intherealmofCanadian business education.Certainly, let's continuewithadeeper exploration intothetopics coveredin"Contemporary Business Mathematicswith Canadian Applications"andhowtheyareappliedwithintheCanadian business context.---### Detailed ExplorationofKey Topics#### 1. Basic Mathematical ConceptsThe textbook beginsbylayingasolid foundationinbasic mathematical concepts essential forbusiness applications. This includes arithmetic operations suchasaddition, subtraction, multiplication,anddivision, which formthebasis forfinancial calculationsandstatistical analyses. Algebraic concepts suchaslinear equationsandinequalitiesareintroduced, providing tools formodelingandsolving business problems mathematically. Basic statistical measures suchasmean, median, mode,andstandard deviationarecovered, enabling students toanalyzeandinterpret data effectively.#### 2. Financial MathematicsFinancial mathematics playsapivotal roleinbusiness decision-makingbyquantifyingthefinancial implicationsofvarious choices. Key topics covered include:- **Interest Calculations**: Understanding simpleandcompound interest formulas, andtheir applicationsindetermining interest earnedorpaidoninvestmentsorloans. - **Time ValueofMoney**: Concepts suchaspresent value, future value,andannuitiesareexplored toevaluatetheprofitabilityofinvestments andthecostoffinancing over time. - **Loan Repayment**: Techniques forcalculating loan payments, understanding amortization schedules,andassessingthetotal costofborrowing.- **Investment Analysis**: Toolsandmethods foranalyzing investment opportunities, including net present value (NPV), internal rateofreturn (IRR),andpayback period calculations.- **Risk Assessment**: Techniques forassessingandmanaging financial risk through probability analysisandsensitivity analysis.IntheCanadian context,these conceptsareapplied tounderstandthefinancial implicationsofdecisionswithintheframeworkofCanadian financial regulationsandmarket conditions.#### 3. Inventory ControlEffective inventory management is crucial forbusinesses tobalance stock levelswith customer demand while minimizing costs.thetextbook covers mathematical modelsandtechniques such as:- **Economic Order Quantity (EOQ)**: Calculatingtheoptimal order quantitythatminimizes total inventory costs, considering factors suchasordering costs, holding costs,anddemand variability.- **Reorder Point**: Determiningtheinventory level at which an order should be placed toavoid stockouts while accounting forlead timeanddemand variability.- **Inventory Turnover**: Calculatingandinterpreting inventory turnover ratios toassess how efficiently inventory is being managedandutilized.In Canada, specific considerations suchassupply chain dynamics, seasonal variations,andregulatory requirements may influence inventory management strategies, makingthese mathematical tools particularly relevant.#### 4. Marketing MathematicsMathematical modelsandanalysesareessential formaking informed marketing decisionsandoptimizing marketing strategies. Key topics covered include:- **Break-Even Analysis**: Determiningthesales volume needed tocover costsandachieve profitability, considering factors suchasfixed costs, variable costs,andselling price.- **Pricing Strategies**: Using pricing models toset optimal pricesthatmaximize profitability while considering consumer demand elasticityandcompetitive pricing.- **Market Segmentation**: Analyzing customer data toidentifyandtarget specific market segments effectively.**Sales Forecasting**: Using quantitative techniques suchastrend analysisandregression analysis toforecast future sales volumes basedonhistorical dataandmarket trends.Canadian-specific examplesandcase studies illustrate howthese mathematical conceptsareapplied inthecontextofCanadian consumer behavior, market dynamics,andregulatory environment.#### 5. Data AnalysisandStatisticsIntheeraofbig data, proficiencyindata analysisandstatistical methods is essential forextracting actionable insights from business data.thetextbook covers:- **Descriptive Statistics**: Summarizingandinterpreting data using measures suchasmean, median, mode, variance,andstandard deviation. - **Probabilitytheory**: Understanding basic probability conceptsanddistributions toassess riskanduncertaintyinbusiness decisions. - **Hypothesis Testing**: Using statistical tests tomake inferencesanddraw conclusions from sample data, suchastesting hypotheses about population parameters.- **Regression Analysis**: Building regression models toanalyze relationships between variablesandmake predictions basedondata trends.These statistical tools enable businesses toanalyze market trends, customer preferences,andoperational performance, providingaquantitative basis forstrategic decision-making.#### 6. Spreadsheet ApplicationsSpreadsheets like Excelareindispensable tools forperforming mathematical calculations, creating financial models,andanalyzing datainbusiness contexts.thetextbook provides guidance on:- **Excel Functions**: Using built-in functionsandformulas toautomate calculationsandperform complex analyses. - **Financial Modeling**: Creating financial models toevaluate investment scenarios, conduct sensitivity analysis,andgenerate forecasts.- **Data Visualization**: Using charts, graphs,andpivot tables topresent data visuallyandcommunicate insights effectively.Practical exercisesandhands-on examples help students develop proficiencyinusing spreadsheetsaspowerful tools forbusiness analysisanddecision support.### Practical ApplicationsinCanadian Business#### Financial Decision-MakingIn Canada, businesses use financial mathematics toevaluate investment opportunities, manage cash flow,andmake strategic decisionsthatimpact profitabilityandgrowth. forexample, Canadian companies may apply financial models

Chapter 12 Estimating theCost of Capital 12-1.

12-2.

3% + 0.57  (8% – 3%) = 5.85%

a.

No, volatility includes diversifiable risk, and so it cannot be used toassess theequity cost of capital.

b.

Microsoft stock would need tohave a beta of 1.

12-3.

Alcoa is 5%  (2 – 0.45) = 7.75% higher.

12-4.

Excel Solution and Excel Project Total value of themarket  10  10  20  12  8  3  50  1  45  20  $1.314 billion Stock

Portfolio Weight 10 10  7.61% 1314 20 12  18.26% 1314 8 3  1.83% 1314 50  3.81% 1314 45  20  68.49% 1314

A B C D E 12-5.

Excel Solution Total value of all four stocks  13  1000  22  1.25  43  30  5  10  $14, 367.5 billion

Stock Grizzly Ltd. DAS Polar Beer PDJB

Portfolio Weight 13 1000  90.48% 14367.5 22 1.25  0.19% 14367.5 43  30  8.98% 14367.5 5 10  0.35% 14367.5

12-6. a.

12,000  (MC A / MC C) = 12,000  (10  10)/(8  3) = $50,000

b.

12,000/8 = 1,500 shares of B, 1500  (shares of B/shares of C) = 1500  12/3 = 6000 shares of B

c.

No trades are needed; it is a passive portfolio.

12-7.

Best Buy has 16/40 = 0.4 billion shares outstanding.

Therefore, you hold 100  (1.8/.4) = 450 shares of Disney.

12-8. a.

Sell winners, buy losers, tomaintain an equal investment in each.

b.

The market portfolio should represent theaggregate portfolio of all investors. However, in aggregate, investors must hold more of larger market cap stocks; theaggregate portfolio is value weighted, not equally weighted.

12-9.

No. theexpected return of this portfolio would be lower due tobonds. Compute thehistorical excess return of this new index.

12-10.

No. Investors were not expecting a negative return. toestimate an expected return, we need much more data.

12-11.

No trades are required. In a value-weighted portfolio, an equal fraction of thetotal number of shares of each security is held. There has been only a price change, not a change in thenumber of shares outstanding.

12-12. a.

(10% – 45%)/2 = –17.5%

b.

Excess returns: MKT 3%, –38% XYZ 7%, –46% Beta = (7 – (–46))/(3 – (–38)) = 1.29

c.

Alpha = intercept = E[RS – rf] – beta  (E[RM – rf]) = (7% – 46%)/2 – 1.29  (3% – 38%)/2 = 3.1%

d.

E[R] = 3% + 1.29  (8% – 3%) = 9.45%

e.

Use (d). CAPM is more reliable than average past returns, which would imply a negative cost of capital in this case!

Ignore (c), as alpha is not persistent.

12-13.

Excel Solution Nike beta = 0.625 HP Inc. beta = 1.57

12-14.

Excel Solution Nike alpha = 1.44% per month HP Inc. alpha = –1.74% per month

12-15. Excel Solution

12-16. a.

Risk-free => y = 3.75%

b.

y – d  l = 3.75% – 1%(0.40) = 3.35%

12-17.

Rd = 3% + 0.31(5%) = 4.55% = y – pL = 17.3% – p(0.60) p = (17.3% – 4.55%)/.60 = 21.25% 12-18. a.

Use CAPM toestimate expected return, using AAA rate as rf rate: r + 0.1  rp = 4% + 0.10(5%) = 4.5% So, y – p  l = 4.5% y = 4.5% + p(60%) = 4.5% + 0.4%(60%) = 4.74% Spread = 0.74%

b.

Use CAPM toestimate expected return, using AAA rate as rf rate: r + 0.1  rp = 4% + 0.10(5%) = 4.5% y = 4.5% + 3%(80%) = 6.90% Spread = 2.9%

c.

Use CAPM toestimate expected return, using AAA rate as rf rate: r + 0.1  1.2  rp  1.2 = 4% + 0.10(5%)1.22 = 4.72% So, y – p  l = 4.5% y = 4.72% + p(80%) = 4.72% + 3%(80%) = 7.12% Spread = 3.12%

12-19.

Project beta = 0.85 (using all equity comp) Thus, rp = 4% + 0.85(5%) = 8.25% 12-20. a.

E = 20  15 = 300 E + D = 400 BU = 300/400  1.00 + 100/400  0 = 0.75 RU = 4% + 0.75(5%) = 7.75%

b.

Re = 4% + 1.0  5% = 9% RU = 300/400  9% + 100/400  4.5% = 7.875%

c.

In thefirst case, we assumed thedebt had a beta of zero, so rd = rf = 4%.

In thesecond case, we assumed rd = ytm = 4.5%.

d.

Thurbinar Ru = (7.75 + 7.875)/2 = 7.8125%.

Harburtin Ru = 8.25%.

Estimate = (8.25% + 7.8125%)/2 = 8.03%.

12-21 Assume debt is risk-free and market value = book value. Assume comparable assets have same risk as project.

Be = 1.20, Bd = 0 E = 15  80 = 1200 D = 400 BU = (1200/1600)  1.20 + (400/1600)  0 = 0.90 12-22. a.

EV = E + D – C = 115 + 10 – 34 = $91 billion

b.

Net Debt = 10 – 34 = –24 RU = (115/91)  1.27 + (–24/91)  0 = 1.60

12-23.

Market Capitalization Total Enterprise

2-Year

Debt

Beta

Ratings

Company Name

($mm)

Value ($mm)

Delta Air Lines (DAL)

4,938.5

17,026.5

2.04 BB

Southw est Airlines (LUV)

4,896.8

6,372.8

0.966 A/BBB

JetBlue Airw ays (JBLU)

1,245.5

3,833.5

1.91 B/CCC

Continental Airlines (CAL)

1,124.0

4,414.0

1.99 B

Debt Beta Asset Beta

0.17 0.075 0.285 0.26

0.712 0.760 0.813 0.701

Average

0.746

12-24. a.

Soft drink RU = 4% + 0.6  5% = 7% V = 50/(7% – 3%) = 1250 Chemical RU = 4% + 1.20  5% = 10% V = 70/(10% – 2%) = 875 Total = 1250 + 875 = $2.125 billion

b.

Weston Beta (portfolio) 1250/2125  0.6 + 875/2125  1.2 = 0.85 Re = 4% + 0.85  5% = 8.25% Not useful. Individual divisions are either less risky or more risky. Over time, Weston’s equity beta will decline toward 0.6, as thesoft drink division has a higher growth rate and so will represent a larger fraction of thefirm.

12-25.

HHI Equity = 32  20 = $640 HHI debt = $64 HHI asset beta = (640 / (640 + 64)) 1.33 + (64 / (640 + 64)) 0 = 1.21 Holdings of Hotdogs = 850 / 2 = 425 Value of Hockey Team = (640 + 64) – 425 = $279 Hotdog equity beta = (850 / 1050)  Be + (200 / 1050)  0 = 0.75 Be = 0.75  1050 / 850 = 0.93 forhotdog equity So, if B = hockey team beta, (425 / (425+279)) 0.93 + (279 / (425+279))  B = 1.21 B = 1.64 Beta of hockey team = 1.64.

12-26. a.

FCF = (30 – 0.8(30))(1 – 0.40) = 3.6 million RU = 4% + 1.25  5% = 10.25% V = 3.6 / 0.1025 = 35.12 million

b.

FCF without energy = (30 – 18)(1 – 0.40) = 7.2 Cost of capital = 10.25% Energy cost after tax = 3(1 – 0.40) = 1.8 Cost of capital = 4% V = 7.2/.1025 – 1.8/0.04 = 70.24 – 45 = 25.24 million

c.

FCF = 7.2 – 1.8 = 5.4 5.4/25.24 = 21.4% Risk is increased because now energy costs are fixed. Thus a higher cost of capital is appropriate.

12-27. a.

E = 40  $10 = $400 D = $100 RU = 400 / 500  15% + 100 / 500  8% = 13.6%

b.

Rd = 8%  (1 – 40%) = 4.8%

c.

rwacc = 400 / 500  15% + 100 / 500  4.8% = 12.96%

12-28.

RU = 9% = 75% Re + 25% Rd = 75% Re + 25%(6%) Re = (9% – 25%(6%))/75% = 10% rwacc = 75%(10%) – 25%(6%)(1 – 40%) = 8.4%

Chapter 13 Investor Behaviour and Capital Market Efficiency 13-1.

Yes. When thenew information arrives, it will change theattractiveness of this stock. If other stock prices do not change, then investors would want toincrease their weight in this stock, implying they would not be holding themarket portfolio.

13-2.

Excel Solution and Excel Project According tothe CAPM, we should hold themarket portfolio. But once new news arrives and we update our expectations, we may find profitable trading opportunities if we can trade before prices fully adjust tothe news. Assuming we initially hold themarket portfolio, we can improve gain by investing more in stocks with positive alphas and less in stocks with negative alphas.

Expected Return 12%

Volatility

Beta

20%

NatSam

10%

HanBel Rebecca Automobile

Green Leaf

Alpha

1.5

Required Return (CAPM) 9.0%

40%

1.8

10.2%

-0.2%

9%

30%

0.75

6.0%

3.0%

6%

35%

1.2

7.8%

-1.8%

3.0%

a.

Green Leaf and HanBel represent buying opportunities.

b.

A sell order would make sense forRebecca Automobile and possibly NatSam (although its alpha is close enough tozero that we might regard it as insignificant).

a.

No, it will no longer be thetangency point if therisk-free rate has changed.

13-3.

b.

13-4.

Stocks with betas (calculated using themarket portfolio prior tothe interest rate change) greater than one will have positive alphas, and so would be buying opportunities. Similarly, stocks with betas less than one will be selling opportunities.

Invest in themarket portfolio. Because theaverage investor must hold themarket, by investing in themarket you guarantee theaverage investor return. If theinformed traders make higher returns than theaverage investor, somebody must make lower returns, so by holding themarket you can guarantee that it is not you.

13-5.

The market portfolio can be inefficient (so it is possible tobeat themarket) only if a significant number of investors either

13-6.

1.

do not have rational expectations, so that they misinterpret information and believe they are earning a positive alpha when they are actually earning a negative alpha, or

2.

care about aspects of their portfolios other than expected return and volatility, and so are willing tohold inefficient portfolios of securities.

By investing in themarket portfolio, investors can protect themselves from being exploited by investors with better information than they have themselves. By choosing not toinvest in themarket portfolio, investors expose themselves tobeing exploited. If they do this because of overconfidence, they will lose money.

13-7.

13-8.

a.

You will underperform because you will lose every time you trade with an informed investor.

b.

This time theonly source of losses is transaction costs (which are assumed tobe zero). In this case, your trades should break even, so you should earn thesame return.

Investors should trade very rarely because they should hold themarket portfolio, which is a value-weighted portfolio and thus requires no re-trading when prices change tomaintain thevalue weights.

13-9. a.

In this case theinformed traders and your brother will both submit buy orders forany price less than $20, so theonly market-clearing price is $20, and nobody trades. Zero profits.

b.

In this case theinformed traders will submit sell orders forany price above $15, and your brother will submit his buy order forany price below $20. Trade will occur at some price in between, and your brother will make negative profits.

c.

Your brother’s expected profits are negative.

13-10.

46 / 523 = 8.8%.

13-11.

The disposition effect causes investors tosell stocks that have appreciated and hold on tostocks that have depreciated. Thus investors are paying capital gains taxes that they could defer and deferring tax deductions they could take immediately. Because of thetime value of money, these investors are therefore increasing their required tax obligations.

13-12. a.

Sell 1, hold 2.

b.

Sell both.

c.

Sell both.

d.

Sell 2, hold 1.

a.

$55

b.

Buy if theprice goes up by 10% or more.

a.

Zero

b.

$200 million

c.

$2 million

13-13.

13-14.

13-15. a.

Bill: 0% Allison 1%

b.

Because investors in Allison’s fund earn a positive alpha, it will experience an inflow of funds.

c.

Capital will flow into Allison’s fund until investors earn a zero alpha. If thetotal amount of new capital 100(2%)  x(0%)  1%  0 is x, this means that 100  x x  100

d.

Since no new capital flowed into Bill’s fund, his before- and after-fee alphas remain thesame.

100(2%)  x(0%) 100(2%) forAllison thealpha after fees is zero. Before fees it is   1%. So, both 100  x 200 managers now have thesame alpha before and after fees.

e.

Bill’s compensation is 100 × 1% = $1 million/year. Allison’s compensation is 200 × 1% = $2 million/year. Allison, thehigher skilled manager, is paid more.

a.

15% – (5% + 1.5(11% – 6%) = 1%

b.

0

c.

(11% – (0.45(11%) + 0.05(15%))/0.5 = 10.6%

d.

A 1% alpha forthe informed traders (representing 5% of theeconomy) means that thefad followers (representing 50% of theeconomy, and 10 times more than theinformed traders) will make –0.1% alpha.

13-16.

13-17.

The size effect is theempirical observation that firms with lower market capitalizations on average have higher average returns.

13-18.

Firms with higher expected returns will have lower market values, and firms with high dividend yields will have high expected returns.

13-19.

Excel Solution a.

Firm Dividend Cost of Capital Market value S1 10 8% $125.00 S2 10 12% $83.33 S3 10 14% $71.43 B1 100 8% $1250.00 B2 100 12% $833.33 B3 100 14% $714.29 b.

Firm Market Value Cost of Capital Self-financing weights S1 $125.00 8% 1 S2 $83.33 12% S3 $71.43 14% (1.00) B1 $1250.00 8% 1 B2 $833.33 12% B3 $714.29 14% (1.00) E[R] (S firms) –6.00% E[R] (B firms) –6.00%

c.

Firm B1 B2 B3 S1 S2 S3

Market Value $1250.00 $833.33 $714.29 $125.00 $83.33 $71.43 E[R] (All firms)

Cost of Capital 8% 12% 14% 8% 12% 14% –6.00%

Self-financing weights 1

(1.00)

Firms will lower costs of capital tend tobe higher in this sort, but theranking is not perfect. d.

Firm S1 B1 S2 B2 S3 B3

Dividend Yield/Cost of Capital 8% 8% 12% 12% 14% 14% 6.00%

Market Value $125.00 $1250.00 $83.33 $833.33 $71.43 $714.29 E[R] (All firms)

Self-financing weights (1.00)

1.00

Because thedividend yield equals thecost of capital, thesort ranks firms perfectly (in contrast toparts b and c)—firms with higher dividend yields have higher costs of capital.

13-20.

Excel Solution and Excel Project a.

Expected Return Stock A

25.0%

Stock B

33.3%

Stock C

5.3%

Stock D

11.1%

b.

The ranking is theopposite, so thecorrelation is negative.

Correlation coefficient 13-21.

–0.99842

Excel Solution a.

Market Capitalization ($ million)

Expected Liquidating Dividend ($ million)

Beta

Stock A

800

1000

0.77

25.00%

8.39%

Stock B

750

1000

1.46

33.33%

13.22%

Stock C

950

1000

1.25

5.26%

11.75%

Stock D

900

1000

1.07

11.11%

10.49%

CAPM Expected Expected Return from Return Problem 20 Prediction

b.

Slope

0.78

Intercept

0.101

c.

Expected Return from Problem 19

Expected Return from Regression

Residual

Stock A

25.00%

16.66%

8.34%

Stock B

33.33%

20.44%

12.89%

Stock C

5.26%

19.29%

–14.03%

Stock D

11.11%

18.31%

–7.20%

d.

Expected Return from Problem 19

Residual

Stock A

25.00%

8.34%

800

Stock B

33.33%

12.89%

750

Stock C

5.26%

–14.03%

950

Stock D

11.11%

–7.20%

900

Market Capitalization Correlation ($ million) Coefficient

–0.9984

negative

e.

Negative correlation is consistent with theevidence regarding small versus large stock returns.

There is a negative relation between market capitalization and expected returns.

13-22.

You buy stocks that have done well in thepast, and sell stocks that have done poorly.

13-23.

If themarket portfolio is efficient, then all stocks have zero alphas, and you could not construct any strategy that has a positive alpha.

13-24.

Because theproxy portfolio is not highly correlated with themarket portfolio, it will not capture some components of systematic risk. thealphas reflect therisk components that theproxy portfolio is not capturing.

13-25.

The market portfolio consists of thecombination of all investors’ portfolios. Because some investors hold inefficient portfolios that depart from efficient in systematic ways, thesum of all these investors’ portfolios is not efficient. Since therest of investors hold efficient portfolios, thesum of all investors’ portfolios will not be efficient.

13-26.

Employees are already partially invested in their company due totheir human capital. Their optimal diversification strategy should take this into account, and thus should underweight their own company’s stock.

For Problems 27–29, refer tothe following table of estimated factor betas based on data from 2007–2017.

Factor

MSFT

XOM

GE

MKT

1.10

0.66

1.22

SMB

–0.71

–0.38

–0.29

HML

–0.01

0.17

0.78

PR1YR

0.02

0.21

–0.17

13-27.

Excel Solution Risk Premium (monthly)

0.93%

RP annual

11.11%

The Annual RP produced by theCAPM beta = 1.47 × 0.66% × 12 = 11.64%

13-28.

Excel Solution MKT SMB HML PR1YR

Factor 0.66 0.22 0.38 0.66

Risk Premium (monthly) RP annual Rf Cost of capital

13-29.

XOM 0.66 -0.38 0.17 0.21 0.56% 6.66% 3.00% 9.66%

Excel Solution MKT SMB HML PR1YR

Factor 0.66 0.22 0.38 0.66

Risk Premium (monthly) RP annual Rf Cost of capital CAPM cost of capital

MSFT 1.10 -0.71 -0.01 0.02 0.58% 6.95% 3.00% 9.95% 10.84%

Certainly! Here'samore detailed explorationof"Contemporary Business Mathematicswith Canadian Applications"thatexpands toapproximately 3000 words:### Introduction toContemporary Business Mathematicswith Canadian ApplicationsContemporary Business Mathematicswith Canadian Applications" isatextbook designed tobridgethegap between mathematicaltheoryandpractical business applications, tailored specifically forCanadian studentsandbusinesses. This comprehensive text coversawide rangeofmathematical concepts essential forunderstandingandmanaging various aspectsofbusiness operations intheCanadian context.inthis detailed exploration, we will delve intothekey components,themes,andrelevanceofthis textbook.

The textbook startswith foundational mathematical conceptsthatformthebasis formore complex calculationsandanalysesinbusiness settings. This includes arithmetic operations, algebraic equations,andbasic statistical measures. Understandingthese fundamental principles is crucial asthey providetheframework upon which more advanced topicsarebuilt.Financial mathematics isacornerstoneofbusiness education, particularlyinunderstandingthetime valueofmoneyandfinancial decision-making. Topics covered often include interest calculations, compound interest, annuities, loans,andinvestments.these conceptsareessential forevaluatingtheprofitabilityofinvestments, determining loan payments,andunderstandingtheimplicationsofinterest ratesonbusiness finances.#### 3. Inventory ControlEffective inventory management is vital forbusinesses toensure adequate stock levels while minimizing holding costs.thetextbook covers mathematical techniquesandmodels used forinventory control, including inventory turnover rates, economic order quantity (EOQ),andreorder points. Canadian applications may include considerations specific toCanadian supply chain dynamicsandregulations.#### 4. Marketing MathematicsMathematical tools playacrucial roleinmarketing decisionsbyproviding quantitative insights into consumer behavior, pricing strategies,andsales forecasts. Topics covered often include break-even analysis, pricing models, market segmentation,andpromotional budgeting. Case studiesandexamples tailored toCanadian markets illustrate howthese mathematical conceptsareappliedinreal-world marketing scenarios.#### 5. Data AnalysisandStatisticsIn today's data-driven business environment, proficiencyindata analysisandstatistical methods is essential.thetextbook typically covers statistical techniques used toanalyze business data, suchasdescriptive statistics, probabilitytheory, hypothesis testing,andregression analysis.these tools enable businesses toextract meaningful insights from data tosupport decision-makingandstrategic planning.#### 6. Spreadsheet ApplicationsSpreadsheetsarepowerful tools extensively usedinbusiness forfinancial modeling, data analysis,anddecision support.thetextbook includes guidanceonutilizing spreadsheet software like Excel toperform calculations, create financial models,andvisualize data. Practical exercisesandexamples help students develop proficiencyinusing spreadsheets forbusiness applications.### RelevanceofCanadian ApplicationsThe inclusionof"Canadian Applications" inthetextbook's title emphasizes its focusonintegrating Canadian business practices, regulations,andeconomic conditions intothelearning experience. This ensuresthatstudents not only grasp mathematical concepts but also understand howthese conceptsareappliedwithintheunique contextofCanada.

Examplesandcase studies drawn from Canadian businesses provide practical insightsandrelevance, preparing students forthechallengesandopportunitiesthey may encounter intheCanadian business landscape.### Pedagogical ApproachandLearning Resources#### Pedagogical ApproachThe textbook typically adoptsapedagogical approachthatcombinestheoretical explanationswith practical examplesandapplications. Conceptsarepresented inaclearandstructured manner tofacilitate understanding,with step-by-step explanationsofmathematical calculationsandproblem-solving techniques. This approach helps students develop both mathematical proficiency andtheability toapplythese skills toreal-world business scenarios.#### Learning ResourcesIn addition tothetextbook itself, supplementary learning resources often accompany "Contemporary Business Mathematicswith Canadian Applications" toenhance student learningandengagement.these may include:- **Online Resources**: Access tointeractive tutorials, practice exercises,andsolutions.- **Case Studies**: Detailed analysesofreal-world business scenarios relevant toCanadian markets.- **Workbook Exercises**: Additional problemsandexercises toreinforce learningandmasteryofmathematical concepts.- **Instructor Support**: Guidanceandsupport forinstructors toeffectively teachandintegratethetextbook intotheir courses.### BenefitsandPractical Applications#### Benefits forStudents- **Practical Relevance**:byfocusingonCanadian applications,thetextbook prepares students toapply mathematical concepts inthespecific contextofCanadian businesses, enhancingtheir readiness forcareers intheCanadian business landscape.**Skill Development**: Students develop critical quantitative skills thatarehighly valuedinvarious business disciplines, including finance, marketing, operations,andstrategic management.**Problem-Solving Abilities**: Through exercisesandcase studies, students learn toanalyze business problems, formulate mathematical models,andderive solutions, improvingtheir analyticalandproblem-solving abilities.#### Practical ApplicationsinBusiness- **Financial Decision-Making**: Businesses use financial mathematics toevaluate investment opportunities, manage cash flow,andmake informed financial decisions.- **Marketing Nursing is a profession deeply rooted in caring for individuals, families, and communities to promote

health and well-being, prevent illness and injury, and provide support to those experiencing health challenges. It encompasses a wide range of roles, from direct patient care to leadership in healthcare policy and education. In this essay, we will explore the multifaceted nature of nursing, its historical evolution, educational pathways, professional responsibilities, ethical considerations, and its crucial role in the healthcare system.### Historical Evolution of NursingThe history of nursing dates back centuries and has evolved significantly over time. Nursing as a formalized profession began to take shape in the 19th century with the pioneering efforts of figures like Florence Nightingale, who is often regarded as the founder of modern nursing. Nightingale's work during the Crimean War demonstrated the critical importance of sanitation and hygiene in healthcare settings, laying the foundation for evidence-based nursing practice.Throughout the 20th century, nursing continued to professionalize and expand its scope. World War I and World War II highlighted the essential role of nurses in caring for wounded soldiers and civilians, leading to advancements in nursing education and practice. The establishment of nursing schools and regulatory bodies further solidified nursing as a respected profession with defined standards of practice.### Educational Pathways in NursingToday, nursing education encompasses a variety of pathways, from diploma programs to bachelor's and master's degrees in nursing (BSN, MSN) and doctoral programs (DNP, PhD). Each level of education prepares nurses for different roles within the profession, ranging from bedside care to advanced clinical practice, research, and leadership positions.Diploma programs, typically offered by hospitals, provide a more hands-on, clinical-focused education. Associate degree programs (ADN) and BSN programs offered by colleges and universities combine clinical training with a broader education in sciences, humanities, and nursing theory. BSN programs are increasingly becoming the preferred entry-level education for nurses due to their emphasis on critical thinking, leadership, and evidence-based practice.Advanced practice roles such as nurse practitioners (NPs), nurse anesthetists (CRNAs), nurse midwives (CNMs), and clinical nurse specialists (CNSs) require additional education at the master's or doctoral level. These nurses are trained to provide specialized care, prescribe medications, and diagnose health conditions, expanding access to healthcare services, particularly in underserved areas.### Professional Responsibilities and RolesThe responsibilities of nurses vary depending on their level of education, specialization, and practice setting. At the core of nursing practice is the provision of compassionate, patient-centered care. Nurses assess patients' health needs, develop and implement care plans, monitor patient progress, and advocate for their wellbeing. They collaborate with other healthcare professionals, such as physicians, therapists, and social workers, to ensure comprehensive care.In addition to direct patient careStrategy**: Mathematical models aidindetermining optimal pricing strategies, forecasting sales volumes,andassessingtheeffectivenessofmarketing campaigns.- **Operations Management**: Inventory control techniques help businesses optimize inventory levels, minimize costs,andensure timely availabilityofproducts.### Conclusion"Contemporary Business Mathematicswith Canadian Applications" isavaluable resourcethatequips studentswiththemathematical skillsandknowledge essential forsuccessintoday's competitive business environment.byintegrating Canadian-specific examplesandapplications,thetextbook enhances its relevanceandpractical utility forCanadian studentsandbusinesses alike. Throughastructured pedagogical approachandcomprehensive coverageofkey mathematical concepts,thetextbook prepares students tobecome adept problem solversanddecision makers capableofapplying mathematical principles toaddress real-world business challenges.In summary, this textbook serves asafoundational resource foranyone seeking tounderstandtheintersectionofmathematicsandbusinesswithintheCanadian context, providingasolid framework foracademic learningandprofessional development inthefieldofbusiness mathematics.---This expanded exploration should provideathorough overviewof"Contemporary Business Mathematicswith Canadian Applications"andits relevance intherealmofCanadian business education.Certainly, let's continuewithadeeper exploration intothetopics coveredin"Contemporary Business Mathematicswith Canadian Applications"andhowtheyareappliedwithintheCanadian business context.---### Detailed ExplorationofKey Topics#### 1. Basic Mathematical ConceptsThe textbook beginsbylayingasolid foundationinbasic mathematical concepts essential forbusiness applications. This includes arithmetic operations suchasaddition, subtraction, multiplication,anddivision, which formthebasis forfinancial calculationsandstatistical analyses. Algebraic concepts suchaslinear equationsandinequalitiesareintroduced, providing tools formodelingandsolving business problems mathematically. Basic statistical measures suchasmean, median, mode,andstandard deviationarecovered, enabling students toanalyzeandinterpret data effectively.#### 2. Financial MathematicsFinancial mathematics playsapivotal roleinbusiness decision-makingbyquantifyingthefinancial implicationsofvarious choices. Key topics covered include:- **Interest Calculations**: Understanding simpleandcompound interest formulas, andtheir applicationsindetermining interest earnedorpaidoninvestmentsorloans. - **Time ValueofMoney**: Concepts suchaspresent value, future value,andannuitiesareexplored toevaluatetheprofitabilityofinvestments andthecostoffinancing over time. - **Loan Repayment**: Techniques forcalculating loan payments, understanding amortization schedules,andassessingthetotal costofborrowing.- **Investment Analysis**: Toolsandmethods foranalyzing investment opportunities, including net present value (NPV), internal rateofreturn (IRR),andpayback period calculations.- **Risk Assessment**: Techniques forassessingandmanaging financial risk through probability analysisandsensitivity analysis.IntheCanadian context,these conceptsareapplied tounderstandthefinancial implicationsofdecisionswithintheframeworkofCanadian financial regulationsandmarket conditions.#### 3.

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tools forbusiness analysisanddecision support.### Practical ApplicationsinCanadian Business#### Financial Decision-MakingIn Canada, businesses use financial mathematics toevaluate investment opportunities, manage cash flow,andmake strategic decisionsthatimpact profitabilityandgrowth. forexample, Canadian companies may apply financial models

Chapter 14 Financial Options 1-1.

1-2.

Definitions: a.

Option: A contract that gives one party theright, but not theobligation, tobuy or sell an asset at some point in thefuture.

b.

Expiration date: thelast date on which theholder still has theright toexercise theoption. If theoption is American, theright can be exercised until theexercise date; if it is European, theoption can be exercised only on theexercise date.

c.

Strike price: theprice at which theholder of theoption has theright tobuy or sell theasset.

d.

Call: An option that gives its holder theright tobuy an asset.

e.

Put: An option that gives its holder theright tosell an asset.

European options can be exercised only on theexercise date, while American options can be exercised on any date prior tothe exercise date. Both types of options are traded in Europe and in theAmericas.

1-3. a.

15 Oct 145.00 put

b.

15 Oct 155.00 call

c.

$1.54  100 = $154

d.

Last sale may have happened earlier in theday, whereas bid/ask are current quotes.

e.

$2.24  100 = $224

f.

Calls: 140.00, 145.00 Puts: 150.00, 155.00

1-4.

g.

Identical except that thesecond expires one month later than thefirst.

h.

This expires theSaturday following thethird Friday of October 2015, or October 17 th, 2015. tobe valuable IBM’s stock price must be less than $140 at expiration.

When a party has a long position in a put, it has theright tosell theunderlying asset at thestrike price; when it has a short position in a call, it has theobligation tosell theunderlying asset at thestrike price if exercised.

These are clearly different positions.

1-5.

The positions that benefit if a stock price increases are (i) long call and (iv) short put.

1-6.

Excel Solution and Excel Project Long-call option: value at expiration:

1-7.

a.

$15

b.

$0

c.

Payoff diagram:

Excel Solution and Excel Project Short-call option: value at expiration date: a.

You owe $15.

b.

You owe nothing.

c.

Payoff diagram:

1-8.

Excel Solution Long-put value at expiration:

1-9.

a.

$2

b.

$0

c.

Payoff diagram:

Excel Solution Short-put value at expiration:

1-10.

a.

You owe $2.

b.

You owe nothing.

c.

Payoff diagram:

Downside exposure is larger with a short call (the downside is unlimited) than with a short put (the downside cannot be larger than thestrike price).

1-11.

a.

For calls, strike + ask. forputs, strike – ask.

b.

155 call option is worthless if IBM is below 155 at expiration.

c.

155 call option: it will have a return of 1 / 0.19 – 1 = 426%.

1-12.



1-13.

The red curve is the$40 call; theblue curve is the$60 call; thegreen curve is theshort position in two $50 calls; and theblack curve is thecombination.

$40 Call

$60 Call

combination

Short position 

This is called a butterfly spread.

1-14.

A forward with price p can be constructed by longing a call and shorting a put with strike p.

1-15.

To protect against a fall in theprice of Costco, you can buy a put with Costco as theunderlying asset. By doing this, over thelife of theoption you are guaranteed toget at least thestrike price from selling thestock you already have.

1-16. a.

To ensure that thevalue of your IBM does not fall significantly, you would purchase a protective put. thecurrent ask price fora protective put with a strike price of $140 that expires thethird Friday of October is $0.36 per share. As a percentage of your portfolio this cost toinsure is $0.36/$149.04 = 0.0024 or 0.24% of thevalue of your portfolio.

b.

The ask price of a protective put with a strike price of $140 that expires on thethird Friday of November is $2.00. thecost toinsure thevalue of your holdings will not fall is $2.00/$149.04 = 0.0134 or 1.34%.

c.

The ask price of a protective put with a strike price of $145 that expires on thethird Friday of November is $3.35. thecost toinsure thevalue of your holdings will not fall is $3.35/$149.04 = 0.0225 or 2.25%.

1-17.

Put-call parity:

C  PS 

K 35  2.10  33   3.282 1 r 1.1

1-18. The arbitrage opportunity exists because $7  $3.33  $20 

$18  $6.66 . 1  0.08

So thecall is overpriced compared tothe portfolio of a put, thestock, and risk-free borrowing.

As a result, thestrategy would be tosell thecall option, buy theput, buy thestock and borrow $16.67 (the present value of $18).

The net amount left after doing this is $0.34, with no cash flows when theoptions expire.

1-19. a.

Consider: Buy call and T-Bills, sell stock & put –9.50 – 140 + 148.72 + 0.35 = –0.43.

b.

Sell call and T-Bills, buy stock & put +9.15 + 140 – 149.04 – 0.36 = –0.25.

c.

Both are negative due totransactions costs: call spread (0.35) + put spread (0.01) + stock spread (0.32) = 0.68 in total loss in (a) & (b).

d.

In this case, there appears tobe a small profit if we buy thestock and theput, and sell thecall and a riskfree bond with a face value of 150: –2.47 – 149.04 + 1.53 + 150 = 0.02. Of course, most traders could not actually borrow at a 0% interest rate, and so would receive less than $150 forthe bond, eliminating this profit. And even if this were possible, after executing a few trades prices would quickly move toeliminate this arbitrage.

a.

From put–call parity:

1-20.

S  PV ( Div )  C  P  PV ( K )

From the$860 calls and puts:

S  PV ( Div)  76.30  102.21 

860  827.82 1.00422/12

Therefore, forthe $840 put:

P  C  PV ( K )  (S  PV ( Div))  88.00  b.

840  827.82  94.05 1.00422/12

And forthe $880 call:

C  P  PV ( K )  (S  PV ( Div))  111.56 

1-21. a.

$70 (stock price)

b.

$100 (strike price)

c.

Intrinsic value = $20

880  827.82  65.80 1.00422/12

d.

Intrinsic value = $30

a.

No one will pay more than theprice of theNovember option with thesame exercise price (which expires later) that can easily be purchased at its Ask price of $5.55.

b.

No one would sell forless than theprice of theOctober option with thesame exercise price (which expires earlier) that can easily be sold at its Bid price of $2.24.

a.

Good news about thestock, which raises its stock price

b.

Bad news, which lowers thestock’s price

c.

News that increases thevolatility of thestock

1-22.

1-23.

1-24.

It is never optimal in this situation because you can always sell theAmerican option formore money.

1-25.

It always has thesame price because theoption toexercise early is worthless; theAmerican option provides no more benefits than its European counterpart.

14-26. a.

It is optimal toexercise early puts with higher strikes, so value = intrinsic value of 60 – 10 = $50.

b.

Because put has no time value, call value must be less than dis(55) = 55 – 55 / 1.10 = $5.

14-27.

0  dis( K )  P  PV ( Div)  dis( K )  PV ( Div) , Time value

so PV ( Div)  dis ( K ).

I.e., thediscount on strike must be smaller than dividend. Given 6%/12 = 0.5% interest over themonth, K – K / 1.005 < 0.30, so K < 0.30  1.005 / 0.005 = $60.30.

So, strikes at or below $60 per share could be exercised early.

14-28.

Call has negative time value implies dis ( K )  P  PV ( Div )  0 ,

which means that

dis(K) - PV (Div) < 0 or PV(Div) > dis(K), or FV(Divs) > interest on K = 5%  400 = 20. So dividend yield must be at least 20 / 900 = 2.22%.

14-29.

- dis(K) + C + PV (div) < 0 implies that

- dis(K) + PV (div) < 0 or

dis(K) > PV (div) Kr div  1 r 1 r Kr  div, so interest on strike must exceed thedividend: (0.02  K) > $30, so K > 1500.

14-30. a.

Maturity of thecall = 5 years Market price of underlying assets = E + D = $25  20 + 0.5($25  20) = $500 + 250 = $750 million.

Strike price = face value of debt when it comes due

 $250 million  1  .10  $402,627,500.

5

b.

Long thefirm’s assets and short theequity call option above.

c.

Long risk-free debt and short a put option on StangWay’s assets with a five-year maturity and $402,627,500 exercise price.

14-31.

An equity holder can be thought toown (or be long in) thefirm’s assets, be short in a risk-free debt obligation with thesame maturity value as thefirm’s risky debt, and be long in a put option on thefirm’s assets with exercise price equal tothe maturity value of thefirm’s debt.

14-32.

Following themethod in Example 14.11, $105 billion face value of thedebt is equivalent toa claim of $105 billion / (300 million shares) = $350 per share on Buffin’s current assets. thevalue of a call option with a strike price of $350 is approximately $128.8, implying an equity value of $38.6 billion after therecap. Thus, we estimate a debt value of 121.8 – 38.6 = $83.2 billion, fora yield tomaturity of 12

 105  25    1  11.8% .

83.2  Thus, Buffin’s credit spread with this higher level of debt would be about 11.8% – 4.5% = 7.3%. This rate is higher than in Example 14.11 because of thegreater risk of default with this higher level of debt.

14-33.

To start, ignore thejunior debt as it is irrelevant tothe senior debt claim. We can compute thevalue of thesenior debt as in Example 14.11. the$93 billion face value of senior debt is equivalent toa claim of $93 billion/300 million shares = $310/share on Buffin’s assets. Ignoring thejunior debt, Buffin’s shareholders will receive only thevalue of Buffin in excess of this debt claim; thevalue of Buffin’s equity after thesenior debt financing is equivalent tothe current value of a call option with a strike price of $310.

From thequotes of Figure 14.10, thevalue of thecall is themidpoint between thebid and ask, and equals $152.50. Multiplying by Buffin’s total number of shares, we can estimate thetotal value of Buffin’s equity (ignoring thejunior debt) as $152.50 × 300 million shares = $45.75 billion. To estimate thevalue of

thesenior debt, we can subtract theestimated equity value (ignoring thejunior debt) from Buffin’s total value of $121.8 billion; thus, theestimated senior debt value is 121.8 – 45.75 = $76.05 billion.

We can now redo thecalculation of theequity value by taking into account both thesenior and junior debt. thetotal debt claim is $105 billion, which works out to$105 billion / 300 million shares = $350 per share.

Buffin’s shareholders will receive only thevalue of Buffin in excess of this debt claim, and this is equivalent toa call option on Buffin’s assets with an exercise price of $350/share. thevalue of this call option (from Figure 14.10 using themidpoint of thebid and ask) is $128.80. Multiplying by Buffin’s total number of shares, we can estimate theactual value of Buffin’s equity tobe $128.80 × 300 million shares = $38.64 billion.

Because Senior Debt + Junior Debt + Equity must equal thetotal value of $121.8 billion, Junior Debt has a value of 121.8 – 76.05 – 38.64 = $7.11 billion.

Therefore, theyield on thejunior debt is  12    7.11

12/25

 1  28.56%.

Chapter 15 Option Valuation 15-1.

In this case, thestock price either rises toSu = 25 × 1.20 = 30 or falls toSd = 25 × 0.80 = 20. theoption payoff is therefore either Cu = 5 or Cd = 0. thereplicating portfolio is  = (5 − 0) / (30 − 20) = 0.5 and B = (0 − 20 × 0.5) / 1.06 = − 9.43.

Therefore, C = 0.5 × 25 − 9.43 = $3.07.

15-2.

The parameters are thesame as in 15-1, but thepayoff of theput is 0 if thestock goes up and 5 if thestock goes down. Therefore, thereplicating portfolio is

 = (0 − 5) / (30 − 20) = –0.5, B = (5 − 20 × [–0.5]) / 1.06 = 14.15.

Therefore, P = –0.5 × 25 + 14.15 = $1.65.

15-3.

Up state at time 1:  = (4 − 0) / (11 − 6.50) = 0.889, B = (0 − 6.50 × 0.889) / 1.03 = −5.61, therefore Cu = 0.889 × 8.50 − 5.61 = $1.95 In thedown state at time 1, theoption is worth nothing. thecall option at time 0 is therefore equivalent tothe replicating portfolio: Δ = (1.95 − 0) / (8.50 − 4) = 0.433, B = (0 − 4 × 0.433) / 1.03 = −1.68.

And so, by theLaw of One Price, theinitial option price is 0.433×6 − 1.68 = $0.92.

15-4.

If thestock goes up twice, theput is worth zero. If thestock ends up at $6.50, theput is worth $0.50, if thestock goes down twice, theput is worth $5. Given these final values, we can calculate thevalue of theput at earlier dates using thebinomial model.

Up state at time 1: Δ = (0 − 0.50) / (11 − 6.50) = − 0.111, B = (0.50 − 6.50 × (− 0.111)) / 1.03 = 1.19.

Therefore, Pu = − 0.111 × 8.50 + 1.19 = $0.25.

Down state at time 1: Δ = (0.50 − 5) / (6.50 − 2) = –1, B = (5 − 2 × (–1)) / 1.03 = 6.80.

Therefore, Pd = − 1 × 4 + 6.80 = $2.80.

Time 0: Δ = (0.25 − 2.80) / (8.50 − 4) = − 0.567, B = (2.80 − 4 × (− 0.567)) / 1.03 = 4.92.

Therefore, Pd = − 0.567 × 6 + 4.92 = $1.52.

15-5.

In Example 15.1, thetheoretical put price is $3.30. If it actually sells fora higher price, it is overvalued, and we can sell it and buy thereplicating portfolio toearn an arbitrage profit. This means that at t = 0, you will sell theput, short 0.3333 shares of stock, and invest $23.30 in Treasury Bills.

Following this strategy, you will earn theput price less (60 × (–0.3333) + 23.30) = $3.30 up front.

Then, if thestock goes up, our portfolio of theput, shares, and borrowing is worth $0 − 0.3333 × 72 + 23.30 × 1.03 = $0. If it goes down, our portfolio is worth –$6 − 0.3333 × $54 + 23.30 × 1.03 = $0, so that at maturity, thepayoff of theoption and thevalue of thereplicating portfolio cancel out.

(Note that this is not theonly arbitrage strategy one can follow—e.g., we could invest more in T-Bills initially—but it is theone that generates themost cash up front without any risk of loss in thefuture.)

15-6.

In Example 15.2, thetheoretical put price is $8.68. If it actually sells for$5, it is underpriced, which means we can buy it and sell thereplicating portfolio toearn an arbitrage profit. This means that at t = 0, you will: buy the$5 put, buy 0.6633 shares of stock, and borrow $41.84.

Following this strategy, you will end up with $41.84 − 0.6633 × 50 − 5 = $3.68 up front.

At t = 1, you rebalance that portfolio according tothe new  and B. If thestock goes up at date 1: we reduce our stock holdings to0.3333 shares, using theproceeds toreduce our debt to41.84 × 1.03 − (0.6633 − 0.3333) × $60 = $23.30. Then, if thestock goes up again, our portfolio of theput, shares, and borrowing is worth $0 + 0.3333 × 72 − 23.30 × 1.03 = $0. If it goes down, our portfolio is worth $6 + 0.3333 × $54 − 23.30 × 1.03 = $0.

If thestock goes down at date 1: we increase our stock holdings to1 share, and increase our debt to 41.84 × 1.03 + (1 − 0.6633) × $45 = $58.25. Then, if thestock goes up, our portfolio of theput, shares, and borrowing is worth $6 + 1 × 54 − 58.25 × 1.03 = $0. If it goes down, our portfolio is worth $19.50 + 1 × 40.50 − 58.25 × 1.03 = $0.

Thus, we have made a profit of $3.68 up front, with a zero payoff no matter what happens in thefuture.

15-7. a.

Pu = 0, Pd = 5, ∆ = (0 − 5) / (20 − 5) = − 1 / 3, B = (5 − 5(–1 / 3)) / 1.25 = 5.33 P = − 1 / 3 (10) + 5.33 = $2.00

b.

The payoffs are Pu = 0, Pd = 15. Because these payoffs are three times thepayoffs in (a), theput must be worth 3 × $2 = $6.

c.

We can exercise theput immediately and get its intrinsic value. In (a), intrinsic value = 0, so not relevant, value is still $2 In (b) intrinsic value = 20 − 10 = $10, so it is better toexercise now => value is $10 (not $6).

15-8.

Delta <= 1, Delta >= 0

15-9. a.

Either $1050 million (if thefirm does well) or $900 million (if thefirm does poorly and defaults).

b.

We can use thebinomial model (expressing all $ amounts in millions): ∆ = (1050 − 900) / (1400 − 900) = 0.3, B = (900 − 900(.3)) / (1.05) = 600, Debt value = 0.3(1000) + 600 = 900. So thedebt value today is $900 million.

c.

1050 / 900 = 16.67%

d.

The initial value should not change as thevalue of theassets and liabilities have both increased by thesame amount, so thevalue of theequity = $1000 million. Once thedividend is paid, subtract dividend amount of $900 million ( this was thevalue of debt issued) todetermine ex-dividend value = $100 million.

e.

Equity payoffs are 1400 − 1050 = 350 or 0 (all in $ millions) ∆ = (350 − 0) (1400 − 900) = 0.70 B = (0 − 900(.7)) / 1.05 = –600 Equity Value = 0.7(1000) − 600 = 100. So theequity value is $100 million.

a.

Bankruptcy costs are 0 or 90.

15-10. ∆ = (0 − 90) / (1400 − 900) = − 0.18 B = (90 − 900(–.18)) / 1.05 = 240 BC value = − 0.18(1000) + 240 = $60 million b.

Debt value = $900 less BC = 900 − 60 = $840 Yield = 1050 / 840 − 1 = 25%

c.

15-11.

On announcement, equity value declines by BC = $1000 − 60 = $940. Subtract dividend of $840 (debt value) todetermine ex-div value = $100. (Note this ex-div value is thesame as in Problem 9 because equity holders have thesame final payoffs.)

Black-Scholes value = $8.50

15-12. a.

Using theBlack-Scholes formula:

PV(K) = 100 /(1.0638)

90 365

= 98.487, d1 

 120  ln   98.487 



0.4 90 / 365

0.4 90 / 365  1.094 2

d 2  1.094  0.4 90 / 365  0.895

C = S × N(d1) − PV(K)N(d2) = 120 × 0.863 − 98.487 × 0.815 = $23.29 b.

Using put-call parity: P = C + PV(K) − S = 23.29 + 98.487 – 120 = $1.78

a.

December 2009 call option with a $5 strike price:

15-13.

The December contracts expire on thethird Friday in December, 2009, so there are 147 days until expiry.

PV ( K )  5 1  0.01

147 365

S

d1 

ln( PV (t K ) )

 T



 T



2

 4.980 , ln( 5.035 4.98 )

0.65 147 / 365



0.65 147 / 365  0.23288 , 2

d2  d1   T  0.23288  0.65 147 / 365  0.17962 .

Substituting d1 and d2 into theBlack-Scholes formula gives

Ct (St , K , T ,  , r )  St N (d1 )  PV ( K ) N (d 2 )  5.035  0.59207  4.98  0.42873  0.846.

Note, this is between thebid and ask prices forthis option. b.

December 2009 put option with a $6 strike price: The December contracts expire on thethird Friday in December, 2009, so there are 147 days until expiry.

PV ( K )  6 1  0.01

147 365

 5.976 ,

 St   5.035  ln  ln    PV ( K )  T   5.976   0.65 147 / 365  0.20911 , d1    2 2  T 0.65 147 / 365

d2 = d1 - s T = -0.20911- 0.65 147 / 365 = -0.62162 .

Substituting d1 and d2 into theBlack-Scholes put formula gives

Pt  St , K , T ,  , r   PV ( K )[1  N (d 2 )]  S[1  N (d1 )] .  5.976  [1  (0.26711)]  5.035  [1  (0.417202)]  1.45 Note, this is between thebid and ask prices forthis option.

c.

March 2010 put option with a $7 strike price: The March contracts expire on thethird Friday in March, 2010, so there are 238 days until expiry.

238

PV ( K )  7 /(1  0.01) 365  6.9547 ,



St   5.035  ln    PV ( K )  T    6.9547   0.65 238 / 365  0.352964  , 2 2  T 0.65 238 / 365

ln  d1 

d 2  d1   T  0.352964  0.65 238 / 365  0.877838 .

Substituting d1 and d2 into theBlack-Scholes formula gives:

Pt  St , K , T ,  , r   PV ( K )[1  N (d 2 )]  S[1  N (d1 )] 6.9547  [1  (0.19)]  5.035  [1  (0.362088)]  2.42 15-14.

The bid and ask prices of thecall are $151.10 and $153.90; thetime toexpiration is approximately 2.12 years (Dec. 5, 2014 toJan. 18, 2017 = 26 + 365 +365 + 18 = 774 days), and theprice of thestock is $405.85.

With PV(K) = 310 / (1 + 0.045)2.12 = 282.38,  405.85  ln   282.38   2.12 d1    , d 2  d1   2.12 .

2  2.12

The Black-Scholes formula, Eq. 15.7, implies: C = 405.85 × N(d1) − 282.38 × N(d2).

You can verify C is between thebid and ask prices forthe call option when σ = 38% or 39%. (You can find theσ by guessing or using a calculator or spreadsheet program.) Thus, theimplied volatility forBuffin derived from this call option is about 38.5%.

15-15.

Implied volatility is 38.5% from Problem 15.14. thestrike price = 340; current stock price is $405.85; and therisk-free rate is 4.5%, with a time tomaturity of 2.12 years.

The Black-Scholes formula gives d1 

ln( 405.85 ) 0.385 2.12 309.71   0.763, d 2  0.763  0.385 2.12  0.202.

2 0.385 2.12

Therefore, C  S  N  d1   PV ( K ) N  d 2   405.85  0.777  309.71 0.580  $135.71

P  C  PV ( K )  S  135.71  309.71  405.85  $39.57.

Note that thecall price is within therange of thequotes provided in Table 15.5.

15-16.

The price of theput is given by thestandard Black-Scholes equation forputs, but with thestock price replaced everywhere with S / (1 + q)T = S / (1 + 0.05)2 = S / 1.1025. PV(K) = 20 / 1.042 = 18.491.

Therefore,  Sx   S / 1.1025  ln  ln    PV ( K )   T 18.491    d1  ln    0.1 2 2  T 0.2 2

 S / 1.1025  ln   18.491  d 2  d1   T    0.1 2 0.2 2 So thevalue of theput option given current stock price S is

P( S )  PV ( K )(1  N (d 2 ))  ( S x )(1  N (d1 ))    S / 1.1025   S / 1.1025     ln 18.491   ln 18.491   18.491  1  N   0.1 2    S / 1.1025  1  N   0.1 2      0.2 2    0.2 2           Plotting this function (the curved line below) gives Option Value

Intrinsic Value

20

Value ($)

15

10

5

0 0

5

10

15

20

25

30

35

40

Stock Price ($)

Notice that when theput is deep in themoney it is worth less than its intrinsic value. In this case thetime value is negative because thesize of thediscount on a two-year, zero-coupon bond is larger than thevalue of thedividends and thecall option, implying a negative time value forthe option.

15-17. a.

∆ = N(d1) = N(0.312634) = 0.6227, so purchase 623 shares.

B = –28.859, so borrow $28,859.

Cost = 622.27 × 60 − 28859 = $8,504

b.

Portfolio = 622.27 × 62 − 28859 = 9749.

Call = 9790.

Difference = (9749 − 9790) / 9790 = –0.42%.

c.

∆ = 0.663.

Increase shares to663.

Borrow additional (663 − 623) × 62 = $2480.

15-18.

Original BS price = $8.50. a.

BS price = $9.14.

b.

BS price = $8.73.

c.

BS price = $8.78.

d.

BS price = $8.09.

e.

Ex-div stock price = $59, BS price = $7.89.

a.

0.5(30 + 18) / 20 − 1 = 20%

b.

25%

a.

25% × (30 − 25) / 1.05 = $1.19

b.

50% × (5) / 1.19 − 1 = 110%

c.

75% × (25 − 18) / 1.05 = $5

d.

50% × (7) / 5 − 1 = –30%

15-19.

15-20.

15-21.

The risk-neutral probabilities can be calculated using 

(1  rf )S  Sd (1.06)25  20   0.65 Su  Sd 30  20 .

Using these probabilities theprice of theoption is 5  0.65   0  0.35  1.06

 $3.066

.

15-22.

The risk neutral probabilities are

p1 

(1.03)6  4  48.44% 8.5  4

p2u 

(1.03)8.5  6.5  50.11% 11  6.5

p2u 

(1.03)4  2  47.11% 6.5  2

The value of thecall is therefore

1 (4 p1 p2u  0( p1 (1  p2u )  (1  p1 ( p2d ))  0(1  p1 )(1  p2d )) 1.032 

4(0.4844)(0.5011) = $0.9153.

1.032

15-23.

Actual probabilities are theprobabilities that an event will happen. Risk-neutral probabilities are theprobabilities of an event happening in a world where investors are risk-neutral. Thus, risk-neutral probabilities are a construction and do not reflect reality. Assuming that investors are risk-averse, riskneutral probabilities are lower in good states and higher in bad states. This has tobe thecase because if investors were risk-neutral, theexpected returns are therisk-free rate, whereas risk-averse investors demand higher returns. And given thesame payoffs, a higher expected return implies good states are more likely.

15-24.

Risk-neutral probabilities can be used toprice derivative securities because thepricing of derivatives only depends on thecharacteristics of theunderlying asset. By construction, thevalue of theunderlying asset can be calculated using risk-neutral probabilities and therefore thevalue of thecall will depend on these probabilities. Risk-neutral probabilities are theeasiest probabilities towork with, given that they simplify thecalculations, and that is why we use them.

15-25.  call 

N (d1 ) S S S  S S  B C

Call option: 175 days tomaturity, S = 5.035, K = 9,  = 65%, rf  1%  N (d1 )  0.146, C  $0.141.  call 

N (d1 ) S 0.146  5.035  0.85  4.4, and Leverage ratio   5.2 0.141 C

 put 

(1  N (d1 ) S S S  S S  B P

15-26.

Put option: 238 days tomaturity, S = 5.035, K = 5,  = 65%, rf  1%  N (d1 )  0.613, P  $1.002 (1  N (d1 )) S  1.9 P

a.

Leverage ratio 

b.

 put 

c.

E ( R)  rf   put  E  RMkt   rf   0.01   1.65  0.06   0.089

d.

(1  0.613)  5.035  0.85  1.65 1.002

An investor would buy a put option given a negative expected return toact as a hedge against losses. thenegative beta implies thereturn will move inverse tothe market, giving good returns when times are bad (when positive returns are themost valuable).

15-27.

E   a.

A  D U   1   U E  E

We want the  E following thedebt issuance:  

 E   (1 

D  D   )  U   N (d1 )(1  )  U E  E  . 

We have S = 122, K = 90, r = 4.5%, T = 2 years, implied = 38.5% from Problem 15.14 ⇒ N (d1) = 0.840, C = 47.01  122  47.01  So Buffin’s new equity beta would be 0.84  1    1.45  3.16 .

47.01  

b.

 D  (1  )

A E  U  (1  ) 1   U D D 

So Buffin’s debt beta would be 1  0.84   1  

47.01    1.45  0.3774.

122  47.01 

Note: you can verify that (a) and (b) are correct by taking theweighted average of thedebt and equity betas and you get theunlevered equity beta of 1.45.

15-28.

The equity can be interpreted as a four-year call option on thefirm’s assets with a strike price of $100 million. thecurrent market value of assets is 400 + 75 = $475 million.

Using theBlack-Scholes formula, we can get theimplied volatility of assets: Call option value = 400, K = 100, S = 475, T = 4, r = 0.0513    60.6% Using thevolatility:     475  ln   4  100 / (1  0.0513)  0.606 4    N (d1 )  N     0.980   2 0.606 4    

U 

E

 D  1    E



1.2  1.03.

75   0.980  1    400 

15-29. a.

Miles’ equity can be viewed as a call option on Miles’ assets with a strike price of $900, five years tomaturity, and a market value of $20 / share × 25 million shares = $500 million. themarket value of Miles’ debt is D = $900 / 1.095 = $585. Using Black-Scholes with parameters S = D + E = 585 + 500 = 1085, rf = 5%, K = 900, T = 5, we find an implied volatility of Mile’s assets of 35%.

b.

We can use Black-Scholes tocalculate thedelta of theequity call option in part a tobe  = 0.827. Then, theprofitability index must exceed (see Eq. 15.24 and Eq. 15.23) D / E × d / e = (1 –) /  = (1 − .827) / .827 = 0.209. (To understand this, note that a $1 investment by equity holders with an NPV of $0.209 will increase thevalue of theassets by $1.209, and so increase thevalue of equity by approximately 0.827 × 1.209 = $1.)

Chapter 16 Real Options 16-1.

Excel Solution Decision Tree

16-2.

Decision Tree

16-3. a.

Without rights, expected profit from first film only is 400 – 300 = 100. Therefore, sequel rights add $50 million toUnited’s expected payoff.

b.

United can pay $30 million now, or buy thesequel rights only if Film I is a blockbuster, which has expected cost $10 million + 50%(30 million) = $25 million. So, it should pay $10 million forthe option.

16-4.

Decision Tree

16-5.

By delaying, you delay thebenefits of taking on theproject and your competitors might take advantage of this delay. However, by delaying, uncertainty can be resolved, so you can become better informed and make better decisions.

16-6. a.

S x = S – PV(Div) = 40(1 – 0.15) = 40(0.85) PV(K) = 35 / (1.04) = 33.6538 T=1

 = 0.25 ln( S x / PV ( K )  T   0.1659 2  T d 2  0.0841 d1 

C  S x N (d1 )  PV ( K ) N (d 2 )  $3.54 So, thevalue of waiting is $3.54 million. thevalue of investing today is $40 – $35 = $5 million.

So, they should enter thebusiness now. b.

S x = S – PV(Div) = 36(1.0 – 0.15) = 36(0.85) PV(K) = 35 / (1.04) = 33.6538 T=1

 = 0.25 ln( S x / PV ( K )  T   0.2555 2  T d 2  0.5055 d1 

C  S x N (d1 )  PV ( K ) N (d 2 )  $1.90

So, thevalue of waiting is $1.90 million. thevalue of investing today is $36 – $35 = $1 million.

So, they should not enter thebusiness now. c.

Decision Tree

d.

S x  N (d1 )  shoe C The beta of theinvestment opportunity is when theshoe business is worth less then $37.4 million and is equal to1 (the beta of a shoe business) when it is worth more:



($ millions) 16-7.

Decision Tree

If you book now, your expected benefit from skiing in 4 months is: $6000(0.60)  0(0.40)  $3600

The NPV of booking today is therefore: NPV 

$3600 4

 $2500  $1008.82

1.0812

If you wait tobook, theexpected benefit in 4 months is:

$2000  0.60  0  0.40  $1200

The PV of this today is:

$1200 4 1.0812

 $1169.61

So, you should wait.

16-8.

Excel Solution a.

Decision Tree

0

1

20%

–100 80%

2

3

10(1.1)2

10(1.1)3

10(1.05)2 10(1.05)3

5

1 6

10(1.1)5 10(1.1)5 × (1 – 0.02)

2 7

12 17

10(1.1)5 × (1 – 0.02)2

10(1.1)5 × (1 – 0.02)12

10(1.05)5 10(1.05)5 10(1.05)5 × (1 – 0.02) × (1 – 0.02)2

10(1.05)5 × (1 – 0.02)12

If thehigh growth rate state occurs, then theNPV is:  10 1.15 1  0.02     0.98 12    1   NPVhigh  4 10      100 5 1.1  0.1  0.02    1.1    $1, 247,147.

1

Note: Since thefirst four cash flows grow out of thesame rate as thediscount rate, their present value is just thesum of thecash flows.

If thelow growth rate state occurs, then theNPV is:

NPVlow 

2 5 4 12 1  10 1.05   1.05    1  10 1.05  1  0.02     0.98       1   1       100      1.1   1.1  0.1  0.05   1.1    1.15  0.1  0.02   

 17.428 million.

So theexpected value is: NPV = 1.25(0.20) + –17.428(0.80) = –13.6927 million. b.

If thehigh growth rate state occurs, then theNPV at time 1 is:

 10 1.15 1  0.02     0.98 12    1      100 4 1.1  0.1  0.02    1.1    0.372 million.

NPVhigh  3 10 1.1 

1

Note: Since thefirst three cash flows grow at thesame rate as thediscount rate, their present value is just thesum of thecash flows, so thevalue at time 1 is just thePV compounded.

If thelow growth rate state occurs, then theNPV at time 1 is:

NPVlow 

3 5 3 12 1  10 1.05   1.05    1  10 1.05 1  0.02     0.98       1   1       100      1.1   1.1  0.1  0.05   1.1    1.14  0.1  0.02   

 19.19 million.

So investment will only occur in thehigh growth state. thevalue of this today is:

 0.372  NPV0   0.2     $67.6 million  1.1  c.

Since theNPV of investing today is negative, you should wait and only invest if thehigh state occurs.

16-9.

First, you must compute thecurrent value of theasset without thedividends that will be missed:

S x  S  PV ( Div)  $5 million 

$0.5 million  $4.55 million 1.12

Next, using theinformation from Table 16.1 calculate thevalue of thecall option toopen thedealership:

d1 

I n[S x / PV ( K )]  T I n(4.55/4.76)    0.20  0.087 2 0.40  T

d2  d1   T  0.087  0.40  0.313 C  S x N (d1 )  PV ( K ) N (d 2 )  $4.55  0.535  $4.76  0.377  $0.64

 corporation 

S x  N (d1 ) $4.55  0.535  dealership   2  7.6 C $0.64

When thevalue of thefirm is $7 million, it is optimal toinvest immediately. We can see this by computing thevalue of thecall similar toabove: Sx = $6.38, d1 = 0.932, d2 = 0.532, C = $1.91. thevalue exercising immediately ($7 – $5 = $2 million) is greater than thevalue of thecall ($1.91 million) so thebeta of thecorporation is equal tothe beta of thedealership: 2.

16-10. a.

From section 16.3, thevalue of theoperating dealership is $6 million and thevalue of a one-year call option is $1.2 million. Additionally from section 16.3, thebeta of theoperating dealership is 2, while thebeta of theone-year option is 6.4.

The value of thefirm is therefore Vc = ($6 + 5 × $1.2) = $12 million.

corporation  b.

Vd 5  Vo $6 5  $1.2  dealership   option  2  6.4  4.2 Vc Vc $12 $12

For expected cash flows of $300,000, thevalue of theoperating dealership is $3 million. thevalue of a one-year call is:

S x  S  PV ( Div)  million  d1 

In(2.73/4.76)  0.20  1.190 0.40

$0.3 million  $2.73 million 1.12

d2  d1   T  0.404  0.40  1.590

C  $2.73  0.117  $4.76  0.056  $0.05 Vc = ($3 + 5 × $0.05) = $3.25 million.

 option 

S x  N (d1 ) $2.73  0.117  Dealership   2  12.8 C $0.05

corporation  c.

Vd 5  Vo $3 5  $0.05  dealership   option  2 12.8  2.8 Vc Vc $3.25 $3.25

The options have a higher beta when cash flows are lower (in part b). thefirm has a higher beta when cash flows are higher (in part a). With higher cash flows, thebeta of theoption is lower because theoption is closer toin-the-money. This causes thevalue of theoption tobe higher and toaffect thevalue and beta of thefirm. With higher cash flows, thebeta of thefirm will be higher because a larger portion of its value comes from thevalue of theoptions. Intuitively, a firm with higher future expected cash flows in this setting has larger growth options.

16-11.

Notice that if thefirm makes theinvestment, thecurrent dividend will be 10(1 – 0.1) = $9 million. It will have two possible growth rates in one period. If thereturn on investment turns out tobe 14%, then thegrowth rate will be g = retention ratio × return on new investment = 0.1 × 0.14 = 1.4%. If thereturn on new investment is 10%, then thegrowth rate will be g = 0.1 × 0.1 = 1%. Assume thefirm decides towait tofind out what thereturn on investment will be before making a decision. In this case, we need todecide whether tomake theinvestment decision at time 1. If we make theinvestment and thereturn on investment is 10%, so thegrowth rate is 1% (the low growth state), thetimeline will be as shown.

1

Low Growth No Growth

9 10

2

3

4

9 × 1.011 9 × 1.012 9 × 1.013 10 10 10

From thetimeline we can see that thevalue of thefirm at time 1 (before thedividend is paid) if thefirm decides toinvest and grow is: l Pinvest  9

9(1.01)  108.89.

0.101  0.01

When thefirm chooses not toinvest, thevalue of thefirm is: Pnoinvest  10 

10  109.01.

0.101

Clearly, thefirm is better off not investing, so its value in this state is $109.01 million. (One way tosee this immediately is tonote that in this state thereturn on investment is less than theopportunity cost of capital.)

The return on investment is 14%, so thegrowth rate is 1.4% (the high growth state); thesame logic shows that if theinvestment is undertaken, thevalue of thefirm is:

h Pinvest  9

9(1.014)  113.897.

0.101  0.014

This value exceeds $109.01, thevalue if theinvestment is not undertaken, so thefirm should undertake theinvestment. (Here thereturn on investment exceeds theopportunity cost of capital.)

What is thevalue of this firm today? tosolve this problem, first write down thedecision tree: 0

1

50% 10

113.89

50% 109.01

Since each state is equally likely, theexpected value of thefirm in one period is 113.897 (0.5) + 109.01 (0.5). Computing thepresent value and adding today’s dividend (remember, we are computing thevalue before thedividend is paid) gives thefirm value today of: 1

P 2

(109.01  113.897)  10  111.229.

1.101

Consider what would have happened if instead of waiting one period tomake theinvestment decision, Western’s managers decided tomake thedecision today (i.e., at time 0 on thetimeline). In this case, thedecision tree looks like this: 0

1

2

3

4

High Growth

9 × 1.014

9 × 1.0142

9 × 1.0143

9 × 1.0144

Low Growth

1

2

3

4

9 × 1.01

9 × 1.012

9 × 1.013

9 × 1.014

We can derive thevalue of thefirm using thesame logic as above. thepresent value today (time 0) of thefuture dividends if thegrowth rate turns out tobe high is: h Pinvest 

9(1.014)  104.897.

0.101  0.014

When thegrowth rate is low, thepresent value of future dividends at time 0 is: l Pinvest 

9(1.01)  99.89.

0.101  0.01

The value of thefirm today is thepresent value of theexpected future dividends plus thedividend today.

Since both states are equally likely, we have:

Pinvest  9  12 (99.89)  12 (104.897)  111.39.

Note that this value is higher than thevalue of thefirm if management waits and makes thedecision at time 1 when thereturn on investment becomes certain. So, managers should give up theoption towait and invest today. Hence thevalue of thecompany is $111.39 million.

16-12.

Decision Tree

First let’s calculate theNPV of investing in 3 possible states: 1.

$100 Million State: Timeline: 1

2

3

–500

100

100

100  500 r NPV 11%   $409 million NPV  r  

NPV  9%   611.11 million

So theexpected value if this state occurs is

EV100  409  0.8   611.11 0.2   450.

2.

$50 Million State: Timeline: 1

2

3

–500

50

50

100  500 r NPV 11%    45.45 NPV  r  

NPV  9%   55.56

EV100  55.56  0.2   11.11 3.

NPV is zero in theworst state, since profits are zero so theproject will not be undertaken.

So thepresent value at time 0 of theexpected value at time 1 is: 0.6  450  0.2 11.11  $235.54 million.

1.1544

If theinvestment is made at time 0, then theNPV is thePV of theexpected cash flows minus theinitial investment. theexpected cash flows are: 0.6  100  0.2  50  70 per year.

Timeline: 0

NPV 

1

2

3

70

70

70

70  500  $136.36 million 0.11

Alternatively, you can calculate thevalue of theproject one year from now and discount that toget thesame answer: 1 70 1 70  70   70  NPV  0.2      500   0.8      500   0.09 1.1544 1.1544   1.1 1.1544 1.1544   136.39 million.

So you are better off waiting.

16-13.

Decision Tree 0

1

2

4

5

6

7

–1000

100

100

0

0

0

Successful 25% 75% –10

–10

–10

–10 Unsuccessful

If development is successful, then theNPV (at time 5) of producing thewire is: NPV5  r  

100  1, 000. r

This is negative forr > 10%, so thewire will only be produced if therates are 8% or 5%: 100  1000  250 0.08 NPV5  0.05   1000.

NPV5  0.08  

So theexpected value of thegrowth opportunity at time 5 if development is successful is:

EVS  250  0.25   1000  0.25   312.5.

There is a 25% chance of success so theexpected value at time 5 of theinvestment opportunity is: EV  312.5  0.25  $78.125.

The NPV of thedevelopment opportunity at time 0 is therefore:

NPV  10 

4 10   1   78.125 1      0.1   1.1   1.15

 $6.811 million.

16-14.

Take a project that arrives in year n. thetimeline is as follows.

N

n+1

n+2

n+3

–10

1

(1 + g)

(1 + g)2

The NPV of this investment opportunity is:

1  10.

0.12  g

NPVn  g   Now:

1  10  $1.11 million 0.12  0.03 1 NPVn  g  0%    10   $1.67 million 0.12  0 1 NPVn  g   3%    10   $3.33 million 0.12  0.03 NPVn  g  3%  

Only theprojects with positive growth rates will be taken on. Thus, theexpected value of any given investment opportunity is EVn  1.11

1  $370, 370.

3

The probability that a project will arrive in any given year is 2/3, and so theexpected value of thegrowth opportunity that will arrive in year n is: Gn  370, 370 

2  $246, 914.

3

Putting this on a timeline: 0

1

2

3

246,914

246,914

246,914

So thePV of all these opportunities today is: PV 

246,914  $2.058 million.

0.12

16-15.

Take a project that arrives in year n. thetimeline is as follows.

N

n+1

n+2

n+3

–10

1

(1 + g)

(1 + g)2

The NPV of this investment opportunity is:

NPVn  g , r  

1  10. rg

If therisk free rate is 10%:

1  10  $4.286 million 0.1  0.03 1 NPV  0%,10%    10  0 0.1 1 NPV   3%,10%    10   $2.308 million.

0.1  0.03 NPV  3%,10%  

Only theprojects with positive rates will be taken on. Thus, theexpected value of any given investment opportunity is: EVn  4.286 

1  $1.4286 million.

3

The probability that a project will arrive in any given year is 2/3, so theexpected value of thegrowth opportunity that will arrive in year n is: Gn  1.4286 

2  $952, 381.

3

Putting this on a timeline: 0

1

2

3

952,381

952,381

952,381

So thePV at time 1 of all these opportunities is:

952,381 0.1  $10.4762 million.

PV  952,381 

Now, if therisk free rate is 6%:

1  10  23.333 0.06  0.03 1 NPV  0%, 6%    10  6.667 0.06 1 NPV   3%, 6%    10  1.111.

0.06  0.03 NPV  3%, 6%  

Therefore, regardless of thegrowth rate, all projects will be taken on. Thus, theexpected value of any given investment opportunity is: 1 1 1 EVn  23.33   6.67   1.111  $10.37 million.

3 3 3

The probability that a project will arrive in any given year is 2/3, and so theexpected value of thegrowth opportunity that will arrive in year n is

Gn  10.37 

2  $6.914 million.

3

Putting this on a timeline: 0

1

2

3

6.914 m

6.914 m

6.914 m

So thePV at time 1 of all these opportunities is: PV  6.914 

6.914  $122.14 million.

0.06

There is a 64.375% chance of rates going to10% and a 35.625% chance of rates going to6%. Putting this on a decision tree: 10.4762

64.375%

35.625%

122.14

So theexpected value is: 10.4762 × 0.64375 + 122.14 × 0.35625 = 50.2564.

Computing thePV gives theanswer:

50.2564  $46.97 million.

1.07 16-16. a. b.

The offer is worth more than $25 million because of theput option.

The value of theoffer is thecurrent value of theshares plus thevalue of theput option.

To calculate thevalue of theput: S = 25 PV(K) = 25/(1.0618) T=1  = 0.30 ln( S / PV ( K )  T   0.3499 2  K d 2  0.04989 d1 

P  PV ( K )(1  N (d 2 )  S (1  N (d 2 ))  2.224

So, thevalue of theoffer is 25 + 2.224 = $27.224 million.

(Note that theactual value will be slightly higher because this uses thevalue of a European put.)

16-17.

Decision Tree

The value of thestore is themaximum of thePV of all future profits and $500,000, thevalue of selling thestore.

If therevenues decrease, then future profits are zero, so thestore should be shut down. Hence thevalue in this state is $500,000.

If therevenues increase, then thePV of future profits at time 1 are:

1.05  0.9 

1.05  0.9 0.15  0.15   $1.65 million.

0.1 0.1

So it is optimal tokeep thestore running. Putting this on a decision tree:

So, thePV of thetime 1 expected value is:

1 1 1.65  0.5 2 2  $977, 273.

1.1 Since this is greater than $500,000, thevalue of thestore is $977,273 with theoption toabandon.

16-18.

Excel Solution Decision Tree: Copper Price:

Cash Flows

First, determine whether themine is operating or shut down in each possible state at time 2.

When thecopper price is $2.344, themine’s profits are $0.344 million/year for98 years and then it will cost $5 million toshut themine down. thePV at time 2 of these cash flows is: PV2,0.344 

 0.344  1 5 1    $2.293 million.

98 0.15  1.15  1.1598

Since it will cost $5 million toshut themine down, it is better toleave it operating.

Similarly, when thecopper price is $1.406, PV1.406 

  0.594  1 5 1    3.96 million.

9.8 015  1.15  1.1598  

Again, since thevalue is greater than –$5 million, it is better tooperate themine.

Finally, when thecopper price is $0.844: PV0.844 

  1.156  1 5 1     $7.7 million.

98 98   0.15  1.15  1.15

Since it is smaller than –$5 million, themine should be shut down.

Next, calculate thevalue of themine if it is operating at time 1 when thecopper price is $1.85. Putting thevalue of themine plus thecash flow at time 2 on a decision tree:

The present value at time 1 of theexpected value at time 2 is

 2.293  0.344

1 1   3.96  0.594  2 2   $0.8337 million.

1.15

So it is optimal torun themine in this state.

Next, calculate thevalue of themine if it is operating at time 1 when thecopper price is $1.125. Putting thevalue of themine plus thecash flow at time 2 on a decision tree:

The present value at time 1 of theexpected value at time 2 is

1 1  5     3.96  1.594  2 2   $4.154 million.

1.15 So it is optimal torun themine in this state.

Finally, we calculate thevalue of themine at time 0 if it is operating.

Putting thevalue of themine plus thecash flow at time 1 on a decision tree:

The present value at time 0 of theexpected value at time 1 is:

  0.8337  0.125

1 1    4.154  0.875 2 2   $2.603 million.

1.15

So themine is worth –$2.603 million. Even though it is worth a negative amount because in most states it will lose money forthe next 98 years, it is not optimal toshut down themine at time 0 because thecosts of shutting down are even greater.

16-19.

The silver dollar is actually a real option because you always have theoption touse it as a dollar coin.

Although at thecurrent price of silver this does not make sense, if theprice of silver dropped below 4.1¢/gram ($1.28/troy ounce) thevalue of thesilver in thecoin would drop below $1. thecoin can always be used a dollar coin, so its price cannot fall below a dollar (by theLaw of One Price). Since theprice of 24 grams of silver can drop below $1, thecoin must be worth more than 24 grams of silver or $4.50. So thecoin must be worth $4.50.

16-20.

a.

Converting thecost of capital toa monthly discount rate gives: 1

1.12 12  1  .00949.

So 12% per year is equivalent to0.949% per month. If rents rise then a building of size q will be worth: 200q .

0.00949

This building will cost 0.1q2 tobuild, so theNPV is: 200q  0.1q 2 .

0.00949

To find themaximum value of this function, take thederivative and set theresults equal tozero: 200  0.2q  0.

0.00949

Solving forq gives: q = 105,387 sq. ft.

Doing thesame thing in thestate with $100/sq. ft. rents gives: q = 52,694. b.

If you wait, theNPV of building in 200 sq. ft. state is 200 105, 387 200q 2  0.1q 2   0.1 105, 387   $1.11 billion.

0.00949 0.00949

Repeating forthe $100 sq. ft. state gives $278 million.

So thePV of this today is:

PV 

1110  0.5  278  0.5

1.00949

60

 $394 million.

If we build today theexpected cash flows are: 0 1 2

100q NPV 

100q

60

61

62

100q

150q

150q

60  100q  1 150q  1  2 1       0.1q .

0.00949  1.00949 60  0.00949  1.00949 

Taking thederivative, setting theresult equal tozero and solving forq gives theoptimal size of thebuilding today: 60  100  1 150  1  1       0.2q  0.

0.00949  1.00949 60  0.00949  1.00949 

q = 67,644 sq. ft.

So theNPV is: NPV = $458 million.

So we should build a building of 67,644 sq. ft. today.

16-21.

The equivalent annual cost method implicitly assumes that, at theend of thelife of theshorter length project, you can replace theshorter length project on theoriginal terms.

16-22. a.

Timeline: 0

Lease –30,000

Buy

1

2

60

500

500

500

900

900

900

61

96

900

900

Converting thecost of capital toa monthly discount rate gives: 1

1.12 12  1  .00949 , so themonthly discount rate is 0.949%.  500  1 1   60 0.00949  1.00949      $22, 794

NPVLease 

NPVBuy   30, 000 

 900  1 1   96 0.00949  1.00949    

 $26, 541 b.

Timeline: 0

1

2

60

Lease – $22,794

X

X

X

– $26,541

Y

Y

Y

Buy

Solving forthe EAB of leasing: 22, 794 

 X  1 1    X  $500.

60 0.00949  1.00949  

Solving forthe EAB of buying: 26,541 

 Y  1 1    Y  $422.

60 0.00949  1.00949  

61

96

Y

Y

c.

Decision Tree

In month 60, theNPV of buying a taxi is either NPV  10, 000 

 500  1 1    $5,188.

36  0.00949  1.00949  

or NPV  16, 000 

 500  1 1     $813.

0.00949  1.00949 36   

The expected value of replacing thetaxis in year 5 is:

EV  5188  0.5  $2594.

The value today of this: PV 

2594

1.00949 

60

 $1472.

Adding this tothe NPV of leasing from part a gives: NPVLease  1472  22, 794  $24, 266.

Since theNPV of buying as not changed

NPVBuy  $26,541.

So you should buy thecab.

16-23. a.

− 10 − 30 + 0.05 × 20 × 2000/1.064 = − 24 million

b.

Potency: (1 − 0.05/1.062)/10 = 0.096 Toxicity: (1 − 0.20/1.064)/30 = 0.028 So, work on potency, then toxicity—higher risk and smaller investment.

c.

If potency works, then theNPV of continuing is (− 30 + 0.2 2000/1.064) So theNPV of starting is − 10 + 0.05 (NPV of starting toxicity)/1.062 = − 10 + 0.05 (− 30 + 0.2 2000/1.064)/1.062 = $2.76 million.

16-24. a.

Software: (1 − 0.80/1.02)/5 = 0.043 Hardware: (1 − 0.50/1.02)/10 = 0.051 So, hardware should go first

b.

Hardware: (1 − 0.75/1.02)/10 = 0.026 In this case, software should go first.

16-25. a.

V0 = ( 0.45 × (150) + 0.55 × (50)+10)/1.05 = 100 NPV(expand) = 100 − 80 = 20 NPV(wait) = 0.45 × (150 – 80)/1.05 = 30 So, it is optimal towait.

b.

NPV(expand) = 100 − C = NPV(wait) = 0.45 × (150 − C)/1.05 So, C = 62.5 Prof Index = NPV/C = (100 − 62.5)/62.5 = 0.6

16-26. a.

100k/.054 − 1m = 851.852k

b.

NPV(up) = 0, NPV(down) = 100k/0.05 − 1m = 1m PV = 0.9 × 1m/1.08 = $833,333k

c.

Hurdle rate rule: NPV = 100k/0.09 − 1m = 111.11k > 0, so invest now, correct.

a.

80k/0.054 − 1m = 481.48k

b.

NPV(down) = 80k/0.05 − 1m = 600k

16-27.

PV = 0.9 × 600/1.08 = 500k c.

80/0.09 − 1m = − 111k, so wait, correct toassessthefeasibilityofexpanding operations, acquiring new equipment,orentering new markets.#### Marketing StrategyMathematical models playacrucial roleinCanadian marketing strategiesbyhelping businesses optimize pricing, forecast sales,andallocate marketing budgets effectively. Canadian businesses leveragethese tools totailor marketing campaigns tospecific demographics, analyze consumer behavior trends,andmeasurethereturnoninvestment (ROI)ofmarketing initiatives.#### Operations ManagementInventory control techniquesarevital forCanadian businesses tomanage supply chain operations efficiently, minimize carrying costs,andensure product availability.byapplying mathematical models suchasEOQandreorder point analysis, Canadian companies can optimize inventory levels, reduce stockouts,andenhance overall operational efficiency.### Conclusion"Contemporary Business Mathematicswith Canadian Applications" servesasan invaluable resource forCanadian studentsandprofessionals seeking todevelop proficiencyinapplying mathematical principles toreal-world business scenarios.byintegrating Canadian-specific examples, case studies,andapplications,thetextbook enhances its relevanceandpractical utilitywithintheCanadian business landscape. From financial decision-makingandmarketing strategy tooperations managementanddata analysis,thetextbook equips learnerswiththequantitative skillsandanalytical tools necessary forsuccessindiverse business environments.In summary, this comprehensive textbook not only providesathorough understandingofbusiness mathematics but also prepares students totackle complex business challengeswith confidenceandcompetence, making itacornerstoneofbusiness educationinCanada.---This expanded exploration should provideacomprehensive overviewof"Contemporary Business Mathematicswith Canadian Applications"andits applicationwithintheCanadian business context, coveringtheessential topicsandpractical implicationsindetail.Certainly, let's continue exploring "Contemporary Business Mathematicswith Canadian Applications"inmore depth, focusingonits pedagogical approach, benefits forstudents,andits broader impactonCanadian business education.--### Pedagogical ApproachandLearning Methods#### Structured Learning Pathway"Contemporary Business Mathematicswith Canadian Applications" typically followsastructured learning pathway designed tofacilitate understandingandmasteryofmathematical conceptsinbusiness contexts.thepedagogical approach often includes:**Conceptual Foundations**: Each chapter beginswith clear explanationsoffundamental mathematical principles relevant tobusiness applications. This ensures students haveasolid understanding ofthetheoretical framework before movingontopractical applications.- **Real-World Examples**:thetextbook integrates real-world examplesandcase studies drawn from Canadian businesses.these examples illustrate how mathematical conceptsareappliedinvarious industriesandscenarios, helping students bridgetheorywith practice.- **Step-byStep Solutions**: Detailed step-by-step solutions toproblemsandexercisesareprovided, enabling students tofollowthelogical progressionofmathematical reasoningandproblemsolving techniques.- **Application Exercises**: Practical exercisesandproblems reinforce learningbychallenging students toapply mathematical concepts tosolve business problems.these exercises often require students toanalyze data, make decisions basedonquantitative analysis,andinterpret results.#### Interactive Learning ResourcesTo complementthetextbook, interactive learning resourcesareoften available toenhance student engagementandunderstanding.these resources may include:- **Online Tutorials**: Interactive tutorialsthatprovide additional explanationsandexamples toclarify difficult concepts. - **Practice QuizzesandTests**: Self-assessment toolsthatallow students togaugetheir understandingandtracktheir progress. - **Case StudiesandSimulations**: Virtual simulationsordetailed case studiesthatsimulate real-world business scenarios, requiring students toapply mathematical concepts tomake strategic decisions.- **Discussion Forums**: Online forumsordiscussion boards where students can collaborate, ask questions,anddiscuss challenges related tothecourse material.### Benefits forStudents#### Skill Development"Contemporary Business Mathematicswith Canadian Applications" equips studentswith essential quantitative skillsandanalytical abilities thatarehighly valuedintoday's business environment. Key benefits include:- **Critical Thinking**: Through

problem-solving exercisesandcase studies, students develop critical thinking skills toanalyze complex business problemsandformulate effective solutions. - **Quantitative Literacy**: Masteryofmathematical concepts enables students tointerpret numerical data, assess risks,andmake informed decisions basedonquantitative analysis. - **Technical Proficiency**: Proficiencyinusing spreadsheet software (e.g., Excel) forfinancial modeling, data analysis,andpresentationofresults enhances students' technical skillsandproficiencyinbusiness applications.#### Career ReadinessBy providingacomprehensive understandingofbusiness mathematicswithintheCanadian context,thetextbook prepares students forvarious careers across different industries. Whether pursuing rolesinfinance, marketing, operations management,orstrategic planning, students gainthefoundational knowledgeandpractical skills necessary tosucceed intheir chosen field.#### AcademicandProfessional Development"Contemporary Business Mathematicswith Canadian Applications" supports both academic learningandprofessional development by:- **Meeting Curriculum Standards**: Aligningwith curriculum standards setbyCanadian educational institutions ensuresthatstudents receivearigorousandrelevant educationinbusiness mathematics. - **Professional Certification**: Preparing students forprofessional certification exams (e.g., CPA, CFA)thatrequire proficiencyinfinancial mathematicsandquantitative analysis. - **Continued Learning**: Serving asareference tool forlifelong learning, enabling professionals tostay updatedwith mathematical techniquesandbusiness trends throughouttheir careers.### Broader ImpactonCanadian Business Education#### Integration into CurriculumThe textbook's focusonCanadian applicationsandbusiness practices enhances its integration into Canadian business education programs.byaddressing local regulatory frameworks, economic conditions,andindustry dynamics,thetextbook providesacontextualized learning experiencethatresonateswith Canadian studentsandeducators alike.#### Industry-Relevant SkillsBy emphasizing practical applicationsandreal-world examples, "Contemporary Business Mathematicswith Canadian Applications" helps bridgethegap between academic learningandindustry needs. Graduates equippedwith strong quantitative skillsarebetter prepared tocontribute effectively toCanadian businesses, driving innovation, efficiency,andgrowth.#### Enhancing CompetitivenessCanadian businesses benefit fromatalent pool equippedwith advanced quantitative skills andadeep understandingofbusiness mathematics. This enhancestheir competitiveness inaglobalized economy, enablingthem tomake data-driven decisions, manage financial resources effectively,andcapitalizeonmarket opportunities.### Conclusion"Contemporary Business Mathematicswith Canadian Applications" playsacrucial roleinshapingthenext generationofbusiness leadersandprofessionalsinCanada.bycombiningtheoretical foundationswith practical applications tailored totheCanadian business landscape,thetextbook equips studentswiththeknowledge, skills,andconfidence tonavigate complex business challengesandcontribute meaningfully toorganizational success.In summary, this comprehensive textbook not only fosters academic excellence but also cultivatesamindsetofinnovationandstrategic thinking essential forthrivingintoday's dynamic business environment. AsacornerstoneofCanadian business education, "Contemporary Business Mathematicswith Canadian Applications" continues toempower students, educators,andbusinesses acrossthecountry, driving economic growthandprosperity.---This expanded discussion providesadeeper insight intothepedagogical approach, benefits forstudents,andbroader impactof"Contemporary Business Mathematicswith Canadian Applications"withintherealmofCanadian business education.Business,inits broadest sense, refers totheactivities involvedinproducing, buying,andselling goodsandservices forprofit. It encompasses all formsofcommercial activities conductedbyindividuals, organizations,orentitieswiththeprimary goalofgenerating revenueandsustaining economic growth.### Key CharacteristicsofBusiness:1. **Profit Motive**:thefundamental objectiveofmost businesses is toearnaprofitbyselling goodsorservices ataprice higher thanthecostofproduction. Profitability ensures sustainabilityandgrowth.2. **GoodsandServices**: Businesses engage intheproductionorprocurementofgoods (tangible products)orservices (intangible offerings)thatmeettheneedsanddesiresofconsumers.3. **Exchange**: Business involves transactions where goodsorservicesareexchanged formoneyorother goodsandservices.these exchanges create value forboththebusinessandits customers.4. **RiskandUncertainty**: Business operationsareinherently exposed torisks suchasmarket fluctuations, competition, regulatory changes,andeconomic downturns.

Managingthese risks is crucial forlong-term success.5. **InnovationandAdaptation**: Successful businesses innovate tocreate competitive advantages, adapt tochanging market conditions,andmeet evolving consumer preferences.6. **LegalandEthical Considerations**: Businesses operatewithin legal frameworksthatgovern commercial activities, ensuring compliancewith lawsandregulations. Ethical considerations involve conducting business inasocially responsible manner.### TypesofBusiness Entities:- **Sole Proprietorship**: Ownedandoperatedbya single individual who assumes all risksandresponsibilities. - **Partnership**: Formedbytwoormore individuals who share ownership, responsibilities, profits,andlosses. - **Corporation**:alegal entity separate from its owners (shareholders), offering limited liabilityandperpetual existence. - **Limited Liability Company (LLC)**: Combines aspectsofpartnershipsandcorporations, providing limited liability toits members (owners).### FunctionsofBusiness:1. **Production**: Manufacturingoracquiring goodsandservices tomeet consumer demand.2. **Marketing**: Promotingandselling goodsorservices totarget consumers through advertising, sales,anddistribution channels.3. **Finance**: Managing financial resources, including budgeting, investments,andsecuring funding forbusiness operations.4. **Human Resources**: Recruiting, training,andmanaging employees tooptimize workforce performanceandproductivity.5. **Operations**: Overseeing day-to-day activitiesandprocessesthatcontribute totheproductionanddeliveryofgoodsandservices.### ImportanceofBusiness:- **Economic Growth**: Businesses drive economic developmentbycreating jobs, generating income,andcontributing toGDP growth.

- **Innovation**: Businesses innovate todevelop new products, services,andtechnologiesthatimprove qualityoflifeanddrive societal progress.

- **Wealth Creation**: Successful businesses create wealth forowners, employees, shareholders,andcommunities through profitable operationsandinvestments.

- **Global Trade**: Businesses facilitate international tradeandcommerce, promoting exchangeofgoods, services,andideas across borders.### Conclusion:Business isadynamicandmultifaceted fieldthatplaysacentral role intheglobal economy. It encompassesawide rangeofactivities aimed at creating value, generating profit,andfostering economic growth. Understandingtheprinciplesandfunctionsofbusiness is essential foraspiring entrepreneurs, managers,andprofessionals seeking tonavigateandsucceedintoday's competitive business environment.Certainly! Here’s an in-depth exploration oftheconceptofbusiness, covering its definition, characteristics, typesofentities, functions, importance,andmore, extending toapproximately 3000 words.---### Introduction toBusinessBusiness isafundamental aspectofhuman society, encompassing all activities involvedinproducing, buying,andselling goodsandservices forprofit. It serves asacornerstoneofeconomic activity, driving growth, innovation,andsocietal development.inthis comprehensive exploration, we will delve intothevarious facetsofbusiness, examining its definition, characteristics, typesofentities, functions, importance,andits role intheglobal economy.### DefinitionofBusinessBusiness can be defined astheorganized effortofindividualsorgroups toproduceandsell goodsandservices forprofit. It involvesawide rangeofcommercial activities aimed at meetingtheneedsandwantsofconsumers while generating revenueandachieving sustainable growth. Business activities occurwithin legalandregulatory frameworks, guidedbyeconomic principlesandmarket dynamics.### CharacteristicsofBusiness#### 1. Profit MotiveAt its core, business is driven bytheprofit motive.theprimary goalofmost businesses is togenerate revenuethatexceedsthecostsofproduction,thereby yieldingaprofit. Profitability ensurestheviabilityandsustainability ofthebusiness over time.#### 2. GoodsandServicesBusinesses produceorprocure goods (tangible products)orservices (intangible offerings)thatsatisfy consumer demand. Goods can range from physical products like automobilesandclothing todigital products suchassoftwareandmedia. Services encompassawide arrayofofferings, including healthcare, education, financial services,andentertainment.#### 3.

ExchangeBusiness involves transactions where goodsorservicesareexchanged formoneyorother goodsandservices.these exchanges create value forboththebusinessandits customers, facilitating economic activityandwealth creation.#### 4. RiskandUncertaintyBusiness operationsareinherently exposed torisksanduncertainties. Factors suchasmarket fluctuations, competitive pressures, technological changes, regulatory shifts,andeconomic downturns can impact business performance. Effective risk management strategiesareessential formitigatingthese uncertaintiesandsafeguarding business continuity.#### 5. InnovationandAdaptationSuccessful businesses innovate tocreate competitive advantages, differentiatethemselves inthemarketplace,andmeet evolving consumer preferences. Innovation can involve technological advancements, product development, process improvements,andmarketing strategies. Adaptation tochanging market conditionsandconsumer behaviors is crucial formaintaining relevanceandsustainability.#### 6.

LegalandEthical ConsiderationsBusinesses operatewithin legal frameworksthatregulate commercial activities, ensuring compliancewith laws, regulations,andindustry standards.

Ethical considerations guide business conduct, encompassing principlesoffairness, transparency, accountability,andsocial responsibility. Adhering toethical practices enhances corporate reputation, fosters trust among stakeholders,andcontributes tolong-term success.### TypesofBusiness EntitiesBusiness entities varyinlegal structure, ownership, liability,andtaxation.thechoiceofentity dependsonfactors suchasbusiness objectives, organizational size, ownership preferences,andregulatory requirements.themain typesofbusiness entities include:#### 1. Sole Proprietorship- **Definition**:asole proprietorship isabusiness ownedandoperatedbya single individual.theowner retains full control over business decisionsandassumes all risksandresponsibilities. - **Characteristics**: Simplest formofbusiness entity,with minimal regulatory requirements. Owner reports business incomeandlossesonpersonal tax returnsandhas unlimited liability forbusiness debts. - **Examples**: Freelancers, independent consultants, small retail shops.#### 2. Partnership**Definition**:apartnership is formedbytwoormore individuals who share ownership, management responsibilities, profits,andlosses ofthebusiness. - **Characteristics**: Partners enter intoapartnership agreement outlining rights, duties,andprofit-sharing arrangements. Partnerships can be general (equal partnership)orlimited (with limited partners having restricted liability). - **Examples**: Law firms, accounting firms, medical practices.#### 3. Corporation- **Definition**:acorporation isalegal entity separate from its owners (shareholders)with rightsandliabilities distinct from thoseofits shareholders. - **Characteristics**: Corporations offer limited liability protection toshareholders, meaningtheir personal assetsareshielded from business debtsandliabilities. Corporations have perpetual existenceandcan raise capital throughthesaleofstock. - **Examples**: Large multinational companies (MNCs), publicly traded corporations.#### 4. Limited Liability Company (LLC)- **Definition**:alimited liability company (LLC) combines elementsofpartnershipsandcorporations, offering limited liability protection toits members (owners).

- **Characteristics**: LLCs provide flexibilityinmanagement structureandprofit distribution while shielding members from personal liability forbusiness debts. LLCsaretaxed similarly topartnerships (pass-through taxation). - **Examples**: Small businesses, real estate ventures, professional services firms.### FunctionsofBusinessBusinesses perform various functions essential fortheir operation, growth,andsuccess inthemarketplace.these functions encompass core activitiesthatcontribute

Chapter 17 Capital Structure in a Perfect Market

7-2. 17-1. a.

E C 1  NPV 

1 2

130, 000  180, 000  155, 000,

155, 000

 100, 000  129,167  100, 000  $29,167

1.20 b. c.

Equity value  PV C1 

155, 000

 129,167

1.20

Debt payments  100, 000, equity receives $20,000 or $70,000.

Initial value, by MM, is 129,167  100, 000  $29,167 .

7-3. 17-2. a.

Total value of equity  2  $2m  $4m .

b.

MM says total value of firm is still $4 million. $1 million of debt implies total value of equity is $3 million. Therefore, 33.3% of equity must be sold toraise $1 million.

c.

In (a), 50% × $4m = $2m. In (b), 2 / 3 × $3m = $2m. Thus, in a perfect market thechoice of capital structure does not affect thevalue tothe entrepreneur.

7-4. 17-3. a.

E[Value in one year]  0.8  50  0.2  20  44 . E 

b.

D=

c.

Without leverage, r =

44 44  20  1  10% , with leverage, r =  1  14.55% 40 20.952

d.

Without leverage, r =

20 0  1   50% , with leverage, r =  1  100% 40 20.952

7-5. 17-4. a.

44  $40m 1.10

20  19.048 . Therefore, E  40  19.048  $20.952m 1.05

(0.5 × 450 + 0.5 × 200) / 250 = 1.30 => 30%

b.

E + D = 250, D = 100 => E = 150

c.

(0.5 × (450 − 105) + 0.5 × (200 − 105)) / 150 = 1.4667 => 46.67%

7-6. 17-5. a.

Excel Solution and Excel Project

FCF

b.

ABC Debt Equity Payments Dividends

XYZ Debt Equity Payments Dividends

$800

0

800

500

300

$1000

0

1000

500

500

Unlevered equity = debt + levered equity. Buy 10% of XYZ debt and 10% of XYZ equity, get 50 + (30, 50) = (80, 100).

c.

Levered equity = unlevered equity + borrowing. Borrow $500, buy 10% of ABC, receive (80, 100) − 50 = (30, 50).

7-7. 17-6. a. b.

7-8. 17-7. a.

V(Alpha) = 10  22 = 220m = V(Omega) = D + E  E = 220 − 60 = 160m  p = $8 per share.

Omega is overpriced. Sell 20 Omega, buy 10 Alpha, and borrow $60. Initial = 220 − 220 + 60 = $60.

This assumes we can trade shares at current prices and assumes we can borrow at same terms as Omega (or own Omega debt and can sell at same price).

Assets = cash + non-cash.

Liabilities = equity + options.

Non-cash assets = equity + options − cash = 12 × 5 + 8 − 5 = $63 billion.

b.

Equity = 60 − 5 = 55.

Repurchase

5b  0.417b shares  4.583 billion shares remain.

12

Per share value 

7-9. 17-8. a. b.

55  $12 .

4.583

Share price = 4b / 100m = $40, Issue 2b / 40 = 50 million shares You can undo theeffect of thedecision by borrowing tobuy additional shares, in thesame proportion as thefirm’s actions, thus re-levering your own portfolio. In this case, you should buy 50 new shares and borrow $2000.

7-10.

17-9. a. i)

Excel Solution

A = 50 cash + 700 non-cash L = 750 equity

ii)

A = 350 cash + 700 non-cash L = 750 equity + 100 short-term debt + 100 long-term debt + 100 preferred stock

iii) A = 700 non-cash

L = 400 equity + 100 short-term debt + 100 long-term debt + 100 preferred stock b.

Repurchase

350

 46.67 shares  53.33 remain. Value is

7.50

400

 $7.50

53.33

7-11.

17-10. Any leverage raises theequity cost of capital. In fact, risk-free leverage raises it themost (because it does not share any of therisk). Think of therisk of thefirm’s assets as theaverage riskiness of thecash flows theassets generate. If thefirm has no debt, then equity has a claim toall thecash flows and so therisk of theequity is equal tothe risk of theaverage cash flows of thefirm. If risk-free debt is now used, then risk-free cash flows must be removed from thepool of cash flows that would have previously been claimed by equity. Thus theaverage risk of thecash flows left forequity goes up as you remove therisk-free cash flows that now go todebt.

7-12.

17-11. a. E = 1000 − 750 = 250.

CF = (1400, 900) − 750 (1.05) = (612.5, 112.5) b.

RE = (145%, −55%), E[RE] = 45%

c.

Risk premium = 45% − 5% = 40%.

Return sensitivity = 145% − (−55%) = 200%. This sensitivity is four times thesensitivity of unlevered equity (50%). Its risk premium is also four times that of unlevered equity (40% vs. 10%).

d.

750

3

250 e.

7-13.

25%(45%) + 75%(5%) = 15%

17-12. a. rE = rU + D / E(rU − rD) = 12% + 0.50(12% − 6%) = 15% b.

rE = 12% + 1.50(12% − 8%) = 18%

c.

Returns are higher because risk is higher—the return fairly compensates forthe risk. There is no free lunch.

7-14.

17-13.

At a cost of debt of 6%:

D (rU  rD ) E 0.13 rE  0.092  (0.092  0.06) 0.87  0.0968  9.68%. rE  rU 

7-15.

17-14 a.

rwacc 

2 15%



6%

3

3

 12%  rU .

b. i. ii.

7-16.

7-17.

17-15.

17-16.



150 12  6 150

rwacc  rU  10% 

rU  rwacc 

1 2

1

13% 

1.6

0.6 1.6

11  5  8% 1

2

200 150

8%  5%  12%

Initial enterprise value = 75 × 10 + 100 = 850 million New debt = 350 million E = 850 − 350 = 500 million Share price = 500m / 10m = $50

c.

RU = (750 / 850) × 8.5% + (100 / 850) × 4.25% = 8% RE = 8% + 350 / 500 (8% − 5%) = 10.1%

7-20.

x 

x  1.6 10  13  3  0.6 x  x  5%

17-17. a. MM => no change, $75 b.

 18% lim

If rD is higher, rE is lower. thedebt will share some of therisk.

rE  8% 

7-18.

7-19.



rE  rU  D / E rU  rD  12 

7-21.

17-18. a.

128b − 25b = $103billion

b.

Because thedebt is risk free, U 

c.

rwacc  rf    E[ RMkt ]  rf   5  2.11 4  13.4%

E E .

ED 128  (1.7) 103  2.11

Alternatively,

rE  rf   E  E[ RMkt ]  rf   5  1.7  4  11.8% E D $128 $25 rE  rD  (11.8%)  (5%)  13.4% .

ED ED $103 $103

rwacc  7-22.

7-23.

17-19. Indell increases its net debt by $40 million ($30 million in new debt + $10 million in cash paid out). Therefore, thevalue of its equity decreases to120 − 40 = $80 million. o

If thedebt is risk free,

o

 e  u 1 

 

D

u  E  D 

E

E



 u 

EV

,

E

o where D is net debt, and EV is enterprise value. theonly change in theequation is thevalue of equity. Therefore '

E

120

  e  1.50  2.25 . e E' 80

o 7-24.

7-25.

17-20. a.

The market value of the80 million shares Campbell does not own is $80 million.

b.

The value of equity that could be repurchased is thevalue of thedebt: $75 million / 1.02 = $73.53 million. Jim already owns $20 million in equity. Because theunlevered value of equity is $100 million, Jim would still not own $100 − $20 − $73.53 = $6.47 million.

c.

The payoff is $75 million if thesoftware is not a hit. If thesoftware is a hit, thepayoff is $75 + (6.47 / 26.47) × ($160 − $75) = $95.77 million. thevalue of this portfolio is themarket value of thedebt plus thevalue of theremaining equity: $73.53 + $6.47 = $80 million.

d.

Risky debt with a face value of thepayoff in thegood state ($95.77 million) would yield thesame payoffs.

e.

With a face value of $95.77 million and a market value of $80 million, theyield is 95.77 / 80 − 1 = 19.71%.

f.

Without leverage, thereturn on equity is 160 / 100 − 1 = 60% or 75 / 100 − 1 = −25%, foran expected return of (60% − 25%) / 2 = 17.5%. As thecompany is currently all equity, this is its initial WACC.

g.

OpenStart’s debt return is 19.71% in thegood state, and 75 / 80 − 1 = − 6.25% in thebad state, foran expected return of (19.71% − 6.25%) / 2 = 6.73%. OpenStart’s remaining equity is worth $20 million, and has a payoff of $160 − $95.77 = $64.23 million or zero, or an expected payoff of $64.23 / 2 = $32.11 million. This corresponds toan expected return of 32.11 / 20 − 1 = 60.55%. theWACC is therefore (20 / 100)(60.55%) + (80 / 100)(6.73%) = 17.5%, just as in part f.

7-26.

7-27.

17-21.

Excel Solution

 

a.

 e  u 1  D /E   1.2 1 

b.

rE  r f  b rm  r f



40 

2

60 

  r  r  12.51.2 5  6.25  r  5  2 6.25  17.5% from theCAPM, m

f

E

or





rE  rU  D /E rU  rD  12.5  c.

40 12.5  5

 17.5

60

p  14 1.50   $21 .

Borrow 40%(21) = 8.4, interest = 5%(8.4) = 0.42.

Earnings = 1.50 − 0.42 = 1.08, per share 

1.08

 1.80 .

0.60 No benefit; risk is higher. thestock price does not change. d.

P /E 

21

 11.67 . It falls due tohigher risk.

1.80

7-28.

17-22. a.

Issue

180

 2 million new shares  12 million shares outstanding.

90 New EPS 

24  $2.00 per share.

12

b.

Interest on new debt  180  5%  $9 million. theinterest expense will reduce earnings to24 − 9 = $15 15 million. With 10 million shares outstanding, EPS   $1.50 per share.

10

c.

By MM, share price is $90 in either case. PE ratio with equity issue is P / E ratio with debt is

90  45 .

2

$90  60 .

1.50

The higher P / E ratio is justified because with leverage, EPS will grow at a faster rate.

7-29.

7-30.

17-23.

850  $7.73 110

a.

Assets = $850m. New shares = 110m.  price 

b.

Cost = 100(8.50 − 7.73) = 10(7.73) = $77m. Issuing equity at below market price is costly.

17-24. a.

Assets = 10 / 0.02 = 500 500 × 4.22% − 490 × 4% = 1.5 1.5 / 10 = 15% ROE

b.

Equity ⇒ $20 billion Earnings = 15% × 10 + 4% × 10 = 19% × 10 ROE = 19% × 10 / 20 = 9.5%

c.

11% versus 5.5%—premium drops in half

d.

Vol = 500 × 0.25% / 10 = 12.5% before, versus 500 × 0.25% / 20 = 6.25% after.

e.

No, premium and risk both fall by half.

Chapter 18 Debt and Taxes

7-31.

18-1. a.

Net Income  EBIT  Interest  Taxes   325  125   1  0.40   $120 million.

b.

Net income + Interest = 120 + 125 = $245 million.

c.

Net income  EBIT  Taxes  325  1  0.40  $195 million. This is 245  195  $50 million





lower than in part b. d.

7-32.

7-33.

Interest tax shield  125  40%  $50 million.

18-2.





a.

Net income will fall by theafter-tax interest expense to $20.750  1  1  0.35  $20.10 million.

b.

Free cash flow is not affected by interest expenses.

18-3.

a)





Net income  1000  1  40%  $600 . Thus, equity holders receive dividends of $600 per year with no risk. E 

600

 $12, 000 .

5% b) Net income  1000  500   1  0.40   $300  E  of $500 per year  D = $10,000. c)

300

 $6000 . Debt holders receive interest

5%

Total value of firm with leverage = 6000 + 10,000 = $16,000.



Total value of firm without leverage = $12,000.



Difference = 16,000 – 12,000 = $4000.

4000

d)

 40%  corporate tax rate.

10, 000

7-34.

7-35.

18-4.

Excel Solution and Excel Project (in $ millions) Year 0 1 2 3 Debt 35 28 21 14 Interest 2.8 2.24 1.68 Tax shield 1.12 0.896 0.672

18-5.

5 0 0.56 0.224

4 0 2.5 1

5 0 0 0

$0.175

 $3.5 million.

Excel Solution

Year Debt Interest Tax shield PV 7-36.

4 7 1.12 0.448

0 100

1 75 10 4

(in $ millions) 2 3 50 25 7.5 5 3 2

$8.30

18-6. a) Interest tax shield  $10  6%  35%  $0.21 million. b) PV(Interest tax shield) 

$0.21

 $3.5 million.

0.06 c)

Interest tax shield = $10 × 5% × 35% = $0.175 million. PV 

0.05 7-37.

18-7.

Excel Solution and Excel Project

a.

Interest tax shield  $10  6%  35%  $0.21 million.

b.

PV(interest tax shield) 

$0.21  $5.25 million.

0.04

Alternatively, new market value of debt is D = (10  0.06) / 0.04 = $15 million. Tc  D = 35%  15 = $5.25 million.

7-38.

7-39.

18-8.

7-40.

Interest tax shield in year 1 = $30 × 6.5% × 40% = $0.78 million. As theoutstanding balance declines, so will theinterest tax shield. Therefore, we can value theinterest tax shield as a growing perpetuity with a growth rate of g = –5% and r = 6.5%: 7-41.

PV 

$0.78 6.5%  5%

 $6.78 million.

7-42.

7-43.

18-9.

7-44.

D = –$10 million (negative debt).

So PV(interest tax shield) = Tc  D = –$3.5 million.

This is thepresent value of thefuture taxes Safeco will pay on theinterest earned on its reserves.

7-45.

7-46.

18-10. a.

rwacc 

E D 2 1 rE  rD (1   c )  12  7  10.33% ED ED 3 3

b.

rwacc 

E D 2 1 rE  rD (1   c )  12  7(0.65)  9.52% ED ED 3 3

7-47.

7-48.

18-11. a. E  $15  30  $450 million. D  $150 million.

Pre-tax WACC 

450

10% 

600 b.

WACC 

450

10% 

600

18-12.

D ED



0.65

150

5%  8.75% .

600

150 600

5% 1  35%   8.3125% .

 0.394

1.65

7-49.

Therefore, WACC = pre-tax WACC – 0.394 × 7% × 0.40 = pre-tax WACC – 1.10%

7-50.

So, it lowers Summit’s WACC by 1.1%.

7-51.

18-13.

Pre-tax WACC 

D rD  15%  0.5  0.09  0.35  13.425% ED

7-52.

18-14. a.

b.

1 0.85 12%  7% 1  0.40   8.42% 1.85 1.85 FCF 10 V L  E  D  220  1.85  407   WACC  g 0.0842  g 10 g  0.0842   5.96% 407 WACC 

1 0.85 12%  7%  9.70% 1.85 1.85 FCF 10 VU    $267 million Pre-tax WACC  g 0.0970  0.0596 Pre-tax WACC 

PV  Interest Tax Shield   407  267  $140 million 18-15.

FCF 7  . Therefore WACC = 8%.

WACC  g WACC  3%

a.

V L  E  D  140 

b.

Pretax WACC  WACC 

D 40 rD C  8%   7.5% 0.35  8.75% ED 140

VU 

FCF 7   $122 million pre-tax WACC  g 0.0875  0.03

PV  Interest Tax Shield   V L  V U  140  122  $18 million

18-16.

18-17.

5  $33.33 million 0.15

a.

VU 

b.

V L  V U  C D  33.33  0.35  19.05  $40 million

Reducing cash is equivalent toincreasing leverage by $50 billion. PV of tax savings = 35%  50 = $17.5 billion, or 17.5 / 8.75 = $2.00 per share price increase.

7-53.

18-18. a.

Assets = Equity = $7.50 × 20 = $150 million

b.

Assets = 150 (existing) + 50 (cash) + 40% × 50 (tax shield) = $220 million

c.

E = Assets – Debt = 220 – 50 = $170 million. Share price 

$170 million

 $8.50 .

20 million Kurz will repurchase

50 million

 5.882 million shares.

8.50 d.

Assets = 150 (existing) + 40% × 50 (tax shield) = $170 million Debt = $50 million E = A – D = 170 – 50 = $120 million Share price 

$120 20  5.882

 $8.50/share .

18-19. a.

Share price 

25

 $2.50 per share

10 b.

Just before theshare repurchase:

Assets  25  existing   10  cash   35%  10  tax shield   $38.5 billion E  38.5  10  28.5 Share price 

28.5

 $2.85/share 10 Therefore, shareholders will not sell for$2.75 per share.

c.

Assets = 25 (existing) + 35% × 10 (tax shield) = $28.5 billion E = 28.5 – 10 = 18.5 billion

Shares  10 

10

 6.667 billion. Share price 

3

18.5

 $2.775 / share.

6.667

d.

From part b, fair value of theshares prior torepurchase is $2.85. At this price, Rally will have 10 18.5 10   6.49 million shares outstanding, which will be worth  $2.85 per share after 2.85 6.49 therepurchase. Therefore, shareholders will be willing tosell at this price.

a.

$15 × (1 – 0 .333) = $10 million each year

b.

Given a corporate tax rate of 40%, an interest expense of $15 million per year reduces net income by 15(1 – 0.4) = $9 million after corporate taxes.

c.

$9 million dividend cut  $9 × (1 – 0.15) = $7.65 million per year.

d.

Interest taxes = 0.333 × 15 = $5 million

18-20.

Less corporate taxes = 0.40 × 15 = $6 million Less dividend taxes = 0 .15 × 9 = $1.35 million  Govt tax revenues change by 5 – 6 – 1.35 = $2.35 million (Note this equals (a) – (c)).

1  0.40 1  0.15

e.

*  1 

a

$1 billion / 7.5% = $13.33 billion.

b.

To reduce taxes by $1 billion, Apple will need tomake interest payments of 1 / 0.35 = $2.857 billion, or issue 2.857 / 0.075 = $38.1 billion in debt.

1  0.333

 23.5%

18-21.

 = 1 – (1 – c)(1 – e)/(1 – i) = 1 – (0.65)(0.85) / 0.65 = 15%   D = 15%  $38.1 = $5.71 billion 7-54.

18-22. a.

PV = C D = 35% × 100 = $35 million

b.

*  1 

1  0.351  0.20  1  0.40

 13.33%

PV = C D = 13.33% × 100 = $13.33 million

18-23. a.

Investors receive 6% × (1 – 0.35) = 3.9% after tax from risk-free debt. They must earn thesame aftertax return from risk-free preferred stock. Therefore, thecost of capital forpreferred stock is

3.9% 1  0.15%

 4.59% .

b.

After-tax debt cost of capital = 6% × (1 – 0 .40) = 3.60% which is cheaper than the4.59% cost of capital forpreferred stock.

c.

*  1 

1  0.40 1  0.15 1  0.35

 21.54%

4.59% × (1 – 0.2154) = 3.60%

18-24.

*  1 

7-55.

1  C 1  e  1  i

 0 if and only if 1  C 

or equivalently: C  1 

1  i 1  e

1  i 0.65 1  27.8% .

1  e 0.90

Thus, there is a tax advantage of debt as long as themarginal corporate tax rate is above 27.8%.

7-56.

18-25.

Net income of $4.5 million 

4.5 1  0.35

 $6.923 million in taxable income.

Therefore, Impi can increase its interest expenses by $6.923 million, which corresponds todebt of

6.923

 $86.5 million.

0.08 7-57.

18-26. a.

FCF  EBIT  1     Dep  CapEx  NWC  15  1  0.35   3  6  6.75 E 

b.

6.75 10%  8.5%

 $450 million

Interest expense of $15 million  debt of

15

 $187.5 million.

0.08 c.

No. themost they should borrow is $187.5 million—there is no interest tax shield from borrowing more.

18-27.

Excel Solution 1  C 1  e  1  0.351  0.15 a. *  1  1  15% 1  i 1  0.35 b.

For interest expenses over $20 million, net income is negative so τC = 0..

Therefore, *  1 

c.

1  C 1  e  1  i

1

1  01  0.15 1  0.35

 31% .

For interest expenses between $10 million and $15 million, there is a

2

chance that net income will be

3 positive. Therefore, theexpected corporate tax savings is

2

 35%  23.3% . Thus,

3 *  1  d.

1  C 1  e  1  i

1

1  0.23  1  0.15  0.26% .

1  0.35

There is thegreatest incremental tax advantage up toan interest expense of $10 million.

toassessthefeasibilityofexpanding operations, acquiring new equipment,orentering new markets.#### Marketing StrategyMathematical models playacrucial roleinCanadian marketing strategiesbyhelping businesses optimize pricing, forecast sales,andallocate marketing budgets effectively. Canadian businesses leveragethese tools totailor marketing campaigns tospecific demographics, analyze consumer behavior trends,andmeasurethereturnoninvestment (ROI)ofmarketing initiatives.#### Operations ManagementInventory control techniquesarevital forCanadian businesses tomanage supply chain operations efficiently, minimize carrying costs,andensure product availability.byapplying mathematical models suchasEOQandreorder point analysis, Canadian companies can optimize inventory levels, reduce stockouts,andenhance overall operational efficiency.### Conclusion"Contemporary Business Mathematicswith Canadian Applications" servesasan invaluable resource forCanadian studentsandprofessionals seeking todevelop proficiencyinapplying mathematical principles toreal-world business scenarios.byintegrating Canadian-specific examples, case studies,andapplications,thetextbook enhances its relevanceandpractical utilitywithintheCanadian business landscape. From financial decision-makingandmarketing strategy tooperations managementanddata analysis,thetextbook equips learnerswiththequantitative skillsandanalytical tools necessary forsuccessindiverse business environments.In summary, this comprehensive textbook not only providesathorough understandingofbusiness mathematics but also prepares students totackle complex business challengeswith confidenceandcompetence, making itacornerstoneofbusiness educationinCanada.---This expanded exploration should provideacomprehensive overviewof"Contemporary Business Mathematicswith Canadian Applications"andits applicationwithintheCanadian business context, coveringtheessential topicsandpractical implicationsindetail.Certainly, let's continue exploring "Contemporary Business Mathematicswith Canadian Applications"inmore depth, focusingonits pedagogical approach, benefits forstudents,andits broader impactonCanadian business education.---### Pedagogical ApproachandLearning Methods#### Structured Learning Pathway"Contemporary Business Mathematicswith Canadian Applications" typically followsastructured learning pathway designed tofacilitate understandingandmasteryofmathematical conceptsinbusiness contexts.thepedagogical approach often includes:- **Conceptual Foundations**: Each chapter beginswith clear explanationsoffundamental mathematical principles relevant tobusiness applications. This ensures students haveasolid understanding ofthetheoretical framework before movingontopractical applications.- **Real-World Examples**:thetextbook integrates real-world examplesandcase studies drawn from Canadian businesses.these examples illustrate how mathematical conceptsareappliedinvarious industriesandscenarios, helping students bridgetheorywith practice.- **Step-by-Step Solutions**: Detailed step-by-step solutions toproblemsandexercisesareprovided, enabling students tofollowthelogical progressionofmathematical reasoningandproblem-solving techniques.- **Application Exercises**: Practical exercisesandproblems reinforce learningbychallenging students toapply mathematical concepts tosolve business problems.these exercises often require students toanalyze data, make decisions basedonquantitative analysis,andinterpret results.#### Interactive Learning ResourcesTo complementthetextbook, interactive learning resourcesareoften available toenhance student engagementandunderstanding.these resources may include:- **Online Tutorials**: Interactive tutorialsthatprovide additional explanationsandexamples toclarify difficult concepts. - **Practice QuizzesandTests**: Self-assessment toolsthatallow students togaugetheir understandingandtracktheir progress. - **Case StudiesandSimulations**: Virtual simulationsordetailed case studiesthatsimulate realworld business scenarios, requiring students toapply mathematical concepts tomake strategic decisions.- **Discussion Forums**: Online forumsordiscussion boards where students can collaborate, ask questions,anddiscuss challenges related tothecourse material.### Benefits forStudents#### Skill Development"Contemporary Business Mathematicswith Canadian Applications" equips studentswith essential quantitative skillsandanalytical abilities thatarehighly valuedintoday's business environment. Key benefits include:- **Critical Thinking**: Through problem-solving exercisesandcase studies, students develop critical thinking skills toanalyze complex business problemsandformulate effective solutions. - **Quantitative Literacy**: Masteryofmathematical concepts enables students tointerpret numerical data, assess risks,andmake informed decisions basedonquantitative analysis. - **Technical Proficiency**: Proficiencyinusing spreadsheet software (e.g., Excel) forfinancial modeling, data analysis,andpresentationofresults enhances students' technical skillsandproficiencyinbusiness applications.#### Career ReadinessBy providingacomprehensive understandingofbusiness mathematicswithintheCanadian context,thetextbook prepares students forvarious careers across different industries. Whether pursuing rolesinfinance, marketing, operations management,orstrategic planning, students gainthefoundational knowledgeandpractical skills necessary tosucceed intheir chosen field.#### AcademicandProfessional Development"Contemporary Business Mathematicswith Canadian Applications" supports both academic learningandprofessional development by:- **Meeting Curriculum Standards**: Aligningwith curriculum standards setbyCanadian educational institutions ensuresthatstudents receivearigorousandrelevant educationinbusiness mathematics. - **Professional Certification**: Preparing students forprofessional certification exams (e.g., CPA, CFA)thatrequire proficiencyinfinancial mathematicsandquantitative analysis. - **Continued Learning**: Serving asareference tool forlifelong learning, enabling professionals tostay updatedwith mathematical techniquesandbusiness trends throughouttheir careers.### Broader ImpactonCanadian Business Education#### Integration into CurriculumThe textbook's focusonCanadian applicationsandbusiness practices enhances its integration into Canadian business education programs.byaddressing local regulatory frameworks, economic conditions,andindustry dynamics,thetextbook providesacontextualized learning experiencethatresonateswith Canadian studentsandeducators alike.#### Industry-Relevant SkillsBy emphasizing practical applicationsandreal-world examples, "Contemporary Business Mathematicswith Canadian Applications" helps bridgethegap between academic learningandindustry needs. Graduates equippedwith strong quantitative skillsarebetter prepared tocontribute effectively toCanadian businesses, driving innovation, efficiency,andgrowth.#### Enhancing CompetitivenessCanadian businesses benefit fromatalent pool equippedwith advanced quantitative skills andadeep understandingofbusiness mathematics. This enhancestheir competitiveness inaglobalized economy, enablingthem tomake data-driven decisions, manage financial resources effectively,andcapitalizeonmarket opportunities.### Conclusion"Contemporary Business Mathematicswith Canadian Applications" playsacrucial roleinshapingthenext generationofbusiness leadersandprofessionalsinCanada.bycombiningtheoretical foundationswith practical applications tailored totheCanadian business landscape,thetextbook equips studentswiththeknowledge, skills,andconfidence tonavigate complex business challengesandcontribute meaningfully toorganizational success.In summary, this comprehensive textbook not only fosters academic excellence but also cultivatesamindsetofinnovationandstrategic thinking essential forthrivingintoday's dynamic business environment. AsacornerstoneofCanadian business education, "Contemporary Business Mathematicswith Canadian Applications" continues toempower students, educators,andbusinesses acrossthecountry, driving economic growthandprosperity.---This expanded discussion providesadeeper insight intothepedagogical approach, benefits forstudents,andbroader impactof"Contemporary Business Mathematicswith Canadian Applications"withintherealmofCanadian business education.Business,inits broadest sense, refers totheactivities involvedinproducing, buying,andselling goodsandservices forprofit. It encompasses all formsofcommercial activities conductedbyindividuals, organizations,orentitieswiththeprimary goalofgenerating revenueandsustaining economic growth.### Key CharacteristicsofBusiness:1. **Profit Motive**:thefundamental objectiveofmost businesses is toearnaprofitbyselling goodsorservices ataprice higher thanthecostofproduction. Profitability ensures sustainabilityandgrowth.2. **GoodsandServices**: Businesses engage intheproductionorprocurementofgoods (tangible products)orservices (intangible offerings)thatmeettheneedsanddesiresofconsumers.3. **Exchange**: Business involves transactions where goodsorservicesareexchanged formoneyorother goodsandservices.these exchanges create value forboththebusinessandits customers.4. **RiskandUncertainty**: Business operationsareinherently exposed torisks suchasmarket fluctuations, competition, regulatory changes,andeconomic downturns. Managingthese risks is crucial forlong-term success.5. **InnovationandAdaptation**: Successful businesses innovate tocreate competitive advantages, adapt tochanging market conditions,andmeet evolving consumer preferences.6. **LegalandEthical Considerations**: Businesses operatewithin legal frameworksthatgovern commercial activities, ensuring compliancewith lawsandregulations. Ethical considerations involve conducting business inasocially responsible manner.### TypesofBusiness Entities:- **Sole Proprietorship**: Ownedandoperatedbya single individual who assumes all risksandresponsibilities. - **Partnership**: Formedbytwoormore individuals who share ownership, responsibilities, profits,andlosses. - **Corporation**:alegal entity separate from its owners (shareholders), offering limited liabilityandperpetual existence. - **Limited Liability Company (LLC)**: Combines aspectsofpartnershipsandcorporations, providing limited liability toits members (owners).### FunctionsofBusiness:1. **Production**: Manufacturingoracquiring goodsandservices tomeet consumer demand.2. **Marketing**: Promotingandselling goodsorservices totarget consumers through advertising, sales,anddistribution channels.3. **Finance**: Managing financial resources, including budgeting, investments,andsecuring funding forbusiness operations.4. **Human Resources**: Recruiting, training,andmanaging employees tooptimize workforce performanceandproductivity.5. **Operations**: Overseeing dayto-day activitiesandprocessesthatcontribute totheproductionanddeliveryofgoodsandservices.### ImportanceofBusiness:- **Economic Growth**: Businesses drive economic developmentbycreating jobs, generating income,andcontributing toGDP growth.

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InnovationandAdaptationSuccessful businesses innovate tocreate competitive advantages, differentiatethemselves inthemarketplace,andmeet evolving consumer preferences. Innovation can involve technological advancements, product development, process improvements,andmarketing strategies. Adaptation tochanging market conditionsandconsumer behaviors is crucial formaintaining relevanceandsustainability.#### 6. LegalandEthical ConsiderationsBusinesses operatewithin legal frameworksthatregulate commercial activities, ensuring compliancewith laws, regulations,andindustry standards. Ethical considerations guide business conduct,

encompassing principlesoffairness, transparency, accountability,andsocial responsibility. Adhering toethical practices enhances corporate reputation, fosters trust among stakeholders,andcontributes tolong-term success.### TypesofBusiness EntitiesBusiness entities varyinlegal structure, ownership, liability,andtaxation.thechoiceofentity dependsonfactors suchasbusiness objectives, organizational size, ownership preferences,andregulatory requirements.themain typesofbusiness entities include:#### 1. Sole Proprietorship- **Definition**:asole proprietorship isabusiness ownedandoperatedbya single individual.theowner retains full control over business decisionsandassumes all risksandresponsibilities. - **Characteristics**: Simplest formofbusiness entity,with minimal regulatory requirements. Owner reports business incomeandlossesonpersonal tax returnsandhas unlimited liability forbusiness debts.

- **Examples**: Freelancers, independent consultants, small retail shops.#### 2. Partnership**Definition**:apartnership is formedbytwoormore individuals who share ownership, management responsibilities, profits,andlosses ofthebusiness. - **Characteristics**: Partners enter intoapartnership agreement outlining rights, duties,andprofit-sharing arrangements. Partnerships can be general (equal partnership)orlimited (with limited partners having restricted liability). - **Examples**: Law firms, accounting firms, medical practices.#### 3. Corporation- **Definition**:acorporation isalegal entity separate from its owners (shareholders)with rightsandliabilities distinct from thoseofits shareholders. - **Characteristics**: Corporations offer limited liability protection toshareholders, meaningtheir personal assetsareshielded from business debtsandliabilities. Corporations have perpetual existenceandcan raise capital throughthesaleofstock.

- **Examples**: Large multinational companies (MNCs), publicly traded corporations.#### 4. Limited Liability Company (LLC)- **Definition**:alimited liability company (LLC) combines elementsofpartnershipsandcorporations, offering limited liability protection toits members (owners). - **Characteristics**: LLCs provide flexibilityinmanagement structureandprofit distribution while shielding members from personal liability forbusiness debts. LLCsaretaxed similarly topartnerships (passthrough taxation). - **Examples**: Small businesses, real estate ventures, professional services firms.### FunctionsofBusinessBusinesses perform various functions essential fortheir operation, growth,andsuccess inthemarketplace.these functions encompass core activitiesthatcontribute

Chapter 19 Financial Distress, Managerial Incentives, and Information

7-58.

Excel Solution and Excel Project a)

0.25 

150  135  95  80  $109.52 million 1.05

b)

0.25 

100  100  95  80  $89.29 million 1.05

c)

YTM =

100 − 1= 12% 89.29

expected return = 5% d) equity = 0.25 

50  35  0  0  $20.24 million 1.05

total value = 89.28 + 20.24 = $109.52 million (rounding is a small factor here)

7-59.

7-60.

a)

81  36  $4.50 / share 10

b)

36  8 million shares 4.5

c)

81  $4.50 / share 18

No. Some of these losses are due todeclines in thevalue of theassets that would have occurred whether or not thefirm defaulted. Only theincremental losses that arise from thebankruptcy process are bankruptcy costs.

7-61. a)

Intuit Inc.—its customers will care about their ability toreceive upgrades totheir software.

b) Allstate Corporation—its customers rely on thefirm being able topay future claims.

7-62. a)

Office building—there are many alternate users who would be likely tovalue theproperty similarly.

b) Raw materials—they are easier toreuse. c)

Patent rights—they would be easier tosell toanother firm.

7-63.

Creditors receive $80 million in bankruptcy, so they need toreceive at least this much forthe workout.

Therefore, Tefco should offer its creditors 80% of thefirm’s equity toensure a successful workout.

7-64.

a.

Firm B’s contract is more attractive as it offers thehigher salary forboth years.

b.

If you quit, you would earn $85,000 forthree-fourths of a year, or $63,750. Thus, Firm B would have tooffer at least this much foryou tostay.

c.

A = 85 + 85 / 1.05 = $165.95k B = 90 + ½ (90 + 63.75) / 1.05 = $163.21k

d.

7-65.

The risk of bankruptcy decreases theexpected wage an employee is set toreceive; therefore thefirm must pay a higher wage toincentivize theemployee totake a new position or tostay in (i.e., not quit from) a current position.

Excel Solution 150  135  95  80 a) 0.25   $109.52 million 1.05

100  100  95  0.75  80  0.75  $78.87 million 1.05

b)

0.25 

c)

YTM 

100  1  26.79% 78.87

Expected return = 5% d)

Equity  0.25 

50  35  0  0  $20.24 million 1.05

Total value  0.25 

150  135  95  0.75  80  0.75  $99.11 million 1.05

(or 78.87 + 20.24 = $99.11 million) e)

109.52  $10.95/share 10

f)

99.11  $9.91/share . Bankruptcy cost lowers share price.

10

Note that Gladstone will raise $78.87 million from thedebt, and repurchase 78.87  7.96 million shares. Its equity will be worth $20.24 million, fora share price of 9.91 20.24  $9.91 after thetransaction is completed.

10  7.96 7-66. a)

NPV = $10 million / 0.08 − $50 million = $75 million.

b) Unlevered firm value = PV of future free cash flows = $10 million / 0.08 = $125 million.

Share price = $125 million / 5 million shares = $25/share. c)

From part b, thevalue of theunlevered firm is $125 million. Therefore thevalue of thelevered firm is $125 million + 40% × $50 million = $125 million + $20 million = $145 million. thevalue of theequity is thelevered firm value minus thedebt value = $145 million − $50 million = $95 million. With only 3 million shares outstanding, theprice per share is $95 million / 3 million shares = $31.67/share.

d) The present value of financial distress costs is thePV of thedifference in FCF (which is $1 million per year in perpetuity). Therefore thevalue of thelevered firm is $125 million + $20 million − $1 million / 0.08 = $132.5 million. thevalue of theequity is thevalue of thelevered firm minus thevalue of thedebt = $132.5 million − $50 million = $82.5 million. theprice per share is $82.5 million / 3 million shares = $27.50 per share.

7-67.

No, not necessarily. He has neglected theeffect on customers. Customers will be less willing tobuy thecompany’s cars because thewarranty is not as solid as those provided by thecompany’s competitors.

Since thewarranty is presumably offered toentice customers tobuy more cars, theoverall effect could easily be a reduction in value.

7-68.

Apple has volatile cash flows, has a high beta (around 2), and is a human-capital intensive firm. All of these things imply that Apple has relatively high distress costs.

7-69. a)

The same price, $5.50/share, because financial transactions do not create value.

b)

0.3 

c)

(6.1 − 5.75) × 10 = $3.5 million

a.

80

b.

60

c.

40

a)

r = 5% + 1.1 × (15% − 5%) = 16%

20  5.5  $6.10 / share 10

7-70.

7-71.

V

16  $100 million 0.16

b) r = 5% + 1.1 × (15% − 5%) = 16%

V

7-72.

15  0.35  40  $107.75 million 0.16

According totradeoff theory, tax shield adds value while financial distress costs reduce a firm’s value. thefinancial distress costs fora real estate investment are likely tobe low, because theproperty can generally be easily resold forits full market value. In contrast, corporations generally face much higher costs of financial distress. As a result, corporations choose tohave lower leverage.

7-73.

7-74.

a.

Agency cost—cashing out

b.

By paying a dividend, executives increased theprobability of bankruptcy and therefore theprobability of receiving government funds. Since these government funds are funds that investors would not otherwise be entitled to, thepayment of a dividend could actually raise firm value in this case.

If Dynron has no debt, or if in all scenarios Dynron can pay thedebt in full, equity holders will consider only theproject’s NPV in making thedecision. If Dynron is heavily leveraged, equity holders will also gain from theincreased risk of thenew investment.

7-75. a)

equity = 0 debt =

10 = $9.09 million 1.1

b) NPV =

25 − 20 = $2.73 million 1.1

c)

15 = $13.64 million 1.1

debt =

equity 

35  15  $18.18 million 1.1

d) Equity holders will not be willing toaccept thedeal, because forthem it is a negative-NPV investment (18.18 − 20 < 0).

7-76.

Excel Solution and Excel Project a.

A+D+E

b.

Not taking B & C => loss of 6 + 10 = $16 million

d/e equity beta debt beta Cutoff

1.20 2.00 0.30 0.180

Project Investment NPV

A 100 20

B 50 6

C 85 10

D 30 15

E 75 18

0.200

0.120

0.118

0.500

0.240

NPV/I

7-77. a)

E(A) = $75 million E(B) = 0.5 × 140 = $70 million E(C) = 0.1 × 300 + 0.9 × 40 = $66 million Project A has thehighest expected payoff.

b) E(A) = 75 − 40 = $35 million E(B) = 0.5 × (140 − 40) = $50 million E(C) = 0.1 × (300 − 40) + 0.9 × (40 − 40) = $26 million Project B has thehighest expected payoff forequity holders. c)

E(A) = $0 million E(B) = 0.5 × (140 − 110) = $15 million E(C) = 0.1 × (300 − 110) = $19 million Project C has thehighest expected payoff forequity holders.

d) With $40 million in debt, management will choose project B, which has an expected payoff forthe firm that is 75 − 70 = $5 million less than project A. Thus, theexpected agency cost is $5 million.

With $110 million in debt, management will choose project C, resulting in an expected agency cost of 75 − 66 = $9 million.

7-78. a)

Market value of firm assets  30 / (2 / 3)  $45 million. With debt of $20 million, equity is worth 45 − 20 = $25 million, so you will need tosell

10  40% of theequity.

25

b) Given debt D, equity is worth $(45 − D) million. Selling 50% of equity, together with debt, must raise $30 million: 0.5   45  D   D  30 . Solving, we get D = $15 million.

7-79.

Excel Solution a)

In addition totax benefits of leverage, debt financing can benefit Empire by reducing wasteful investment.

b) Net income will fall by $1 × 0.65 = $0.65 Because 10% of net income will be wasted, dividends and share repurchases will fall by $0.65 × (1 − 0.10) = $0.585 c)

Paying $1 in interest, giving up $0.585 in dividends and share repurchases  increase of 1 − 0.585 = $0.415 per $1 of interest.

a)

Without personal spending, there is a 1% chance of bankruptcy.

7-80.

With $10 million personal spending, there is a 7% chance—so theprobability of bankruptcy increases by 6%.

b) Debt between $90 million and $100 million will provide theCEO with thebiggest incentive not toproceed with personal spending because by doing so thechance of bankruptcy would increase by 38%.

7-81.

Overinvestment: investing in negative-NPV projects.

Underinvestment: not investing in positive-NPV projects.

Cashing out: paying out dividends instead of investing in positive-NPV projects.

Employee job security: highly leveraged firms run therisk of bankruptcy and so cannot write long-term employment contracts and offer job security.

7-82. a)

50  100  150  $100 million 3

b) (i) Empire building: value = 100 − 5 = $95 million (ii) Value = $100 million (iii) Empire building and increased risk: value = 0.5(50) + 0.1(100) + 0.4(150) − 5 = $90 million (iv) Increased risk: value = $95 million c)

Because thetax benefits are paid as a dividend, themanagers will empire build or increase risk as determined in part (b). We can therefore determine theexpected value with leverage by adding theexpected tax benefit tothe value calculated in part (b): (i) $95 + 10%(44) = $99.4 million (ii) $100 + 10%(49) = $104.9 million (iii) $90  10%  0.5  45  0.5  90   $96.75 million (iv) $95 + 10%(0.5 × 50 + 0.5 × 99) = $102.45 million Therefore, $49 million in debt is optimal; even though there is a tax benefit, thefirm’s optimal leverage is limited due toagency costs.

7-83. a)

Tobacco firms: high optimal debt level—high free cash flow, low growth opportunities

b) Accounting firms: low optimal debt level—high distress costs c)

Mature restaurant chains: high optimal debt level—stable cash flows, low growth, low distress costs

d) Lumber companies: high optimal debt level—stable cash flows, low growth, low distress costs e)

Cell phone manufacturers: low optimal debt level—high growth opportunities, high distress costs

7-84.

Unlevered Value 

90  $900 million.

0.10

Levered Value with Raider = 900 + 40%(750) = $1.2 billion To prevent a successful raid, current management must have a levered value of at least $1.2 billion  $1 billion .

1.20 Thus, theminimum tax shield is $1 billion − 900 million = $100 million, which requires

100  $250 0.40

million in debt.

7-85. a) (i) Borrowing has a net cost of $20 million, or

$20  $0.20 per share.

100

500  37 million shares at a premium of $1 per share has a benefit of $37 million, 13.50    12.50 100  500  37 or  $0.27 per share, that is,   12.77  12.50  0.27.  .

137  100  500    13.50   Selling

Therefore, issue equity. (ii) Borrowing has a net cost of $20 million, or Selling

$20  $0.20 per share.

100

500  37 million shares at a discount of $1 per share has a cost of $37 million, or 13.50

37  $0.27 per share.

137 Therefore, issue debt. b) If IST issues equity, investors would conclude IST is overpriced, and theshare price would decline to$12.50. c)

If IST issues debt, investors would conclude IST is undervalued, and theshare price would rise to$14.50.

d) If there are no costs from issuing debt, then equity is issued only if it is overpriced. But knowing this, investors would buy equity only at thelowest possible value forthe firm. Because there would be no benefit toissuing equity, all firms would issue debt.

7-86.

If thefirm must pay 10% more than thetarget firm was worth, but can do thepurchase using shares that were overvalued by more than 10%, in thelong run thefirm will gain from theacquisition.

7-87. a)

NPV of expansion  20  Equity value 

500  80  $58 / share 10

b) NPV of expansion  4 

c)

0.65  50  $80 million 0.1

0.65  50  $24 million 0.1

Share price 

500  24  $47.60 / share 10

New shares 

50  1, 050, 420 shares 47.6

Share price 

500  50  80  $57.01/ share 11.05

The share price is now lower than theanswer from part a, because in part a, share price is fairly valued, while here shares issued in part b are undervalued. New shareholders’ gain of 57.01  47.60 1, 050, 420  $9.9 million = old shareholders’ loss of (58 − 57.01) × 10. d) Tax shield = 35%(50) = $17.5 million Share price 

500  50  80  17.50  50  $59.75 per share.

10

Gain of $2.75 per share compared tothe case in part c. $1 toavoid issuing undervalued equity, and $1.75 from interest tax shield.

Chapter 20 Payout Policy

20-1.

It can retain its free cash flows and use them tomake investments, or hold them in cash. It can pay them out toequity holders, either by issuing a dividend or by repurchasing shares.

20-2.

Excel Solution e) April 8 f)

20-3.

April 12 (Monday)

There are three mechanisms: 1) In an open market repurchase, thefirm repurchases theshares in theopen market. This is themost common mechanism in Canada. 2) In a tender offer, thefirm announces theintention toall shareholders torepurchase a fixed number of shares fora fixed price, conditional on shareholders agreeing totender their shares. If not enough shares are tendered, thedeal can be cancelled. 3) A targeted repurchase is similar toa tender offer, except it is not open toall shareholders; only specific shareholders can tender their shares in a targeted repurchase.

20-4.

Assuming perfect markets, thefirst ex-dividend price should drop by exactly thedividend payment. Thus, thefirst ex-dividend price should be $49 per share. In a perfect capital market, thefirst price of thestock on theex-dividend day should be theclosing price on theprevious day less theamount of thedividend.

20-5. a.

$1 billion / 20 million shares = $50 per share.

b.

$100 million / $50 per share = 2 million shares.

c.

If markets are perfect, then theprice right after therepurchase should be thesame as theprice immediately before therepurchase. Thus, theprice will be $50 per share.

a.

(500 − 200) / 10 = 30

b.

(450 − 200) / 10 = 25

c.

(450 − 200) / (10 − 1.667) = 30

d.

200 / 250 = 0.8

20-6.

20-7. g) The dividend payoff is $250 / $500 = $0.50 on a per share basis. In a perfect capital market, theprice of theshares will drop by this amount to$14.50. h) $15 i)

Both are thesame.

20-8.

If you sell 0.5 / 15 of your shares, you receive $0.50 per share, and your remaining shares will be worth $14.50 each, leaving you in thesame position as if thefirm had paid a dividend.

20-9.

Because thepayoff of theoption depends upon Oracle’s future stock price, you would prefer that Oracle use share repurchases, as it avoids theprice drop that occurs when thestock price goes ex-dividend.

20-10. j)

P = $1.60 / 0.12 = $13.33

k) P = $2.00 / 0.12 = $16.67 20-11.

Use Eq. 20.3 as follows:

  d   g   .4011  .2666    0.18339  18.339%  1     1  .2666  g  

Quebec:  d   *

    g   0.3171  0.24  Alberta:  d*   d   .10145  10.145%  1     1  0.24  g  

20-12.

20-13.

a.

t*_ d = (50% − 25%) / (1 − 25%) = 33.3%, P_ex = 30 − 6(1 − t*) = $26

b.

With dividend, tax would be 6  50% = $3 fordividend, with a tax savings of 4  25% = $1 forcapital loss, fora net tax from thedividend of $2 per share. This amount would be saved if Arbuckle does a share repurchase instead.

c.

Stock price rises by $2 to$32 toreflect thetax savings.

If theinvestor sells theshares today, theinvestor will receive $105 per share and no dividend. If theinvestor sells theshares tomorrow, theinvestor will receive theex-dividend price (expected tobe $100 per share) but will also receive the$5 dividend per share. Now taxes need tobe considered. According toTable 18.3, in Quebec

thedividend will be taxed at 40.11%, giving a dividend after tax of $3.00 per share, thecapital gain (of thesame $5 amount) will be taxed at 26.7% giving a capital gain after tax of $3.665 per share; therefore thecapital gain is preferred in Quebec, so thestock should be sold today. In Alberta thedividend will be taxed at 31.7% giving an after-tax dividend of $3.415 per share and thecapital gain (of thesame $5 amount) will be taxed at 24% giving an after-tax capital gain of $3.80 per share, therefore thecapital gain is preferred so thestock should only be sold before it goes ex-dividend. In either province, if thestock is held in a TFSA, then there is no tax on dividend or capital gain income, so it doesn’t matter if thestock is sold today (to get thegain) or tomorrow (to get thedividend), either way $5 per share is received. thetotal cash flows are as follows: Where Held

Quebec

Sell today (receive capital gain, forgo dividend)

Sell tomorrow (receive dividend, no capital gain)

After-tax gain per share

Net Proceeds (500 shares × $100 per share) + (500 shares × aftertax dividend or capital gain) $51,833.50

= $5 × (1 − 0.2666) = $3.667 Quebec

After-tax dividend per share

$51,497.25

= $5 × (1 − 0.4011) = $2.9945 Alberta

After-tax gain per share

$51,900.00

= $5 × (1 − 0.24) = $3.80 Alberta

After-tax dividend per share

$51,707.25

= $5 × (1 − 0.3171) = $3.4145 TFSA

$5

TFSA 20-14.

$52,500 $5

$52,500

Excel Solution l) The price drop was $2.63 / $3.08 = 85.39% of thedividend amount, implying an effective tax rate of 14.61%. m) With an effective dividend tax rate of about 15%, we can solve forτg from Eq. 20.3:    g   0.15   g   d*   d  0.15      g  0  1    1  g  g   

Thus, if investors are recognizing a zero tax on capital gains, they must be long-term individual investors. (The zero tax on gains is theeffective result because thecapital gains taxes are deferred fora very long time, thus thetax has PV of about 0.)

20-15.

d. Corporations

20-16.

Because thestock price drops by 80% of thedividend amount, shareholders are indifferent if τd* = 20%.

From Eq. 20.3, (τd − τg) / (1 − τg) = τd*, so τd = τg + τd* (1 − τg) = 36%. Investors who pay a lower tax rate than 36% could gain from a dividend capture strategy.

20-17.

Dividend-capture theory states that investors with high effective dividend tax rates sell toinvestors with low effective dividend tax rates just before thedividend payment. theprice drop therefore reflects thetax rate of thelow effective dividend tax rate individuals.

20-18.

20-19.

a.

Invest the$5 special dividend, and earn interest of $0.50 per year.

b.

Borrow $5 today, and use theincrease in theregular dividend topay theinterest of $0.50 per year on theloan.

Excel Solution n) The value of Kay will remain thesame. o) The value of Kay will fall by $100 million. c.

20-20.

It will neither benefit nor hurt investors.

Excel Solution p) The value of Kay will rise by $35 million. q) The value of Kay will fall by $100 million. r)

It will benefit investors.

20-21. Excel Solution a.

Assuming investors pay a 15% tax on dividends but no capital gains taxes nor taxes on interest income, and Kay does not pay corporate taxes: (a.) The value of Kay will remain thesame. (Dividend taxes do not affect thecost of retaining cash, as they will be paid either way.) (b.) The value of Kay will fall by $85 million (i.e., 100  (1 − 15%)) toreflect after-tax dividend value. (c.) It will neither benefit nor hurt investors.

b.

Assuming investors pay a 15% tax on dividends and capital gains, and a 35% tax on interest income, while Kay pays a 35% corporate tax rate: (a.) Effective tax disadvantage of retaining cash is 1 − (1 − τc)(1 − τg)/(1 − τi) = 1 − (1 − 35%)(1 − 15%)/(1 − 35%) = 15%, the equity value of Kay would go up by 15% × 100 = 15 million on announcement. (b.) The value of Kay will fall by $100 million on ex-dividend date (since τg = τd, τ*d = 0). (c.) Given these price reactions, this decision will benefit investors by $15 million.

20-22. Effective tax disadvantage of retention is τ* = 35%. (The reason is that Harris will pay 35% tax on theinterest income it earns.) Thus, stock price falls by 35% × $250m / 100m shares = $0.875 per share.

20-23. a.

100  10%  (1 − 40%) = $6 million

b.

$6  (1 − 0.20) = $4.8 million

c.

100 × 10%  (1 − 0.30) = $7 million

d.

$1 spent on fees = $1  (1 − 0.40)  (1 − 0.20) = $0.48 toinvestors after corporate and capital gains tax. tomake up theshortfall, fees = (7 − 4.8) / 0.48 = $4.583 million.

20-24.

Assuming firms are not small businesses, we get thefollowing: s)

Using Eq. 20.7, forAlberta thetax disadvantage forretained cash is

1-

(1 - 0.25)(1 - 0.24)

= - 9.6154% (i.e., there is a tax advantage forretained cash).

(1 - 0.48) t)

For Quebec, thetax disadvantage forretained cash

1-

(1 - 0.265)(1 - 0.2666)

= - 15.4528% (i.e., there is a tax advantage forretained cash).

(1 - 0.5331) 20-25. u) By increasing dividends, managers signal that they believe that future earnings will be high enough tomaintain thenew dividend payment. v) Raising dividends signals that thefirm does not have any positive NPV investment opportunities, which is bad news.

20-26.

By choosing todo a share repurchase, management credibly signals that they believe thestock is undervalued.

20-27.

Excel Solution and Excel Project w) Because enterprise value = equity + debt − cash, AMC’s equity value is Equity = EV + cash = $500 million.

Therefore, Share price = ($500 million) / (10 million shares) = $50 per share. x) AMC repurchases $100 million / ($50 per share) = 2 million shares. With 8 million remaining shares outstanding (and no excess cash) its share price if its EV goes up to$600 million is Share price = $600 / 8 = $75 per share.

And if EV goes down to$200 million: Share price = $200 / 8 = $25 per share.

y) If EV rises to$600 million prior torepurchase, given its $100 million in cash and 10 million shares outstanding, AMC’s share price will rise to Share price = (600 + 100) / 10 = $70 per share.

If EV falls to$200 million, Share price = (200 + 100) / 10 = $30 per share.

The share price after therepurchase will also be $70 or $30, since theshare repurchase itself does not change thestock price.

Note: thedifference in theoutcomes for(a) vs. (b) arises because by holding cash (a risk-free asset) AMC reduces thevolatility of its share price. z)

If management expects good news tocome out, they would prefer todo therepurchase first, so that thestock price would rise to$75 rather than $70. On theother hand, if they expect bad news tocome out, they would prefer todo therepurchase after thenews comes out, fora stock price of $30 rather than $25.

(Intuitively, management prefers todo a repurchase if thestock is undervalued—they expect good news tocome out—but not when it is overvalued because they expect bad news tocome out.)

aa) Based on (d), we expect managers todo a share repurchase before good news comes out and after any bad news has already come out. Therefore, if investors believe managers are better informed about thefirm’s future prospects, and that they are timing their share repurchases accordingly, a share repurchase announcement would lead toan increase in thestock price.

20-28. $120,000 per old share / $50 per new share = 2400 new shares / old share. A 2400:1 split would be required.

20-29.

Excel Solution and Excel Project bb) With a 20% stock dividend, an investor holding 100 shares receives 20 additional shares. However, since thetotal value of thefirm’s shares is unchanged, thestock price should fall to Share price = $20 × 100 / 120 = $20 / 1.20 = $16.67 per share. cc) A 3:2 stock split means forevery two shares currently held, theinvestor receives a third share. This split is therefore equivalent toa 50% stock dividend. theshare price will fall to Share price = $20 × 2/3 = $20 / 1.50 = $13.33 per share. dd) A 1:3 reverse split implies that every three shares will turn into one share. Therefore, thestock price will rise to Share price = $20 × 3 / 1 = $60 per share.

20-30.

Companies use stock splits tokeep their stock prices in a range that reduces investor transaction costs.

20-31.

It might be advantageous toundertake a reverse stock split toavoid being delisted from an exchange because theprice of thestock has fallen below theminimum required tostay listed.

20-32.

The value of thedividend paid per Adaptec share was (0.1646 shares of Roxio) × ($14.23 per share of Roxio) = $2.34 per share. Therefore, ignoring tax effects or other news that might come out, we would expect Adaptec’s stock price tofall to$10.55 − 2.34 = $8.21 per share once it goes ex-dividend. (Note: In fact, Adaptec stock opened on Monday, May 14, 2001—the next trading day—at a price of $8.45 per share.)

Chapter 21 Capital Budgeting and Valuation with Leverage 7-88.

7-89.

a.

While there may be some differences, themarket risk of thecash flows from this new product is likely tobe similar toClorox’s other household products. Therefore, assuming it has thesame risk as theaverage risk of thefirm is reasonable.

b.

A real estate investment likely has very different market risk than Google’s other investments in Internet search technology and advertising. It would not be appropriate toassume this investment has risk equal tothe average risk of thefirm.

c.

An expansion in thesame line of business is likely tohave risk equal tothe average risk of thebusiness.

d.

The theme park will likely be sensitive tothe growth of theChinese economy. Its market risk may be very different from GE’s other divisions, and from thecompany as a whole. It would not be appropriate toassume this investment has risk equal tothe average risk of thefirm.

E = 665 million × $74.77 = $49.7 billion, D = $25 billion, D / E = 25 / 49.722 = 0.503.

E = 700 million × $83.00 = $58.1 billion. Constant D / E implies D = 58.1 × 0.503 = $29.2 billion.

7-90.

7-91.

Intel’s debt is a tiny fraction of its total value. Indeed, Intel has more cash than debt, so its net debt is negative. Intel is also very profitable; at an interest rate of 6%, interest on Intel’s debt is only $792 million per year, which is around 4.95% of its EBIT. Thus, therisk that Intel will default on its debt is extremely small. This risk will remain extremely small even if Intel borrows an additional $1 billion. Thus, adding debt will not really change thelikelihood of financial distress forIntel (which is nearly zero), and thus will also not lead toagency conflicts. As a result, themost important financial friction forsuch a debt increase is thetax savings Intel would receive from theinterest tax shield. A secondary issue may be thesignalling impact of thetransaction—borrowing todo a share repurchase is usually interpreted as a positive signal that management may view theshares tobe underpriced.

a.

Given a D / E ratio of 0.25 = 0.25 / 1, we can determine theD/(D + E) ratio as 0.25/(0.25 + 1) = 0.2.

If thefirm’s value currently is 3.5 million, then theinitial amount of debt is 0.2 × $3.5 million = $0.7 million, and theequity is $3.5 million – $0.7 million = $2.8 million.

b.

Before recapitalization, thevalue of debt remains constant so in thegood state, thevalue of equity will be $5 million – $0.7 million = $4.3 million and in thebad state, thevalue of theequity will be $2.5 million – $0.7 million = $1.8 million. % change in Firm value

c.

% change in Equity value

% change in Debt value

Good state

(5 – 3.5) / 3.5 = 42.86%

(4.3 – 2.8) / 2.8 = 53.57%

0%

Bad state

(2.5 – 3.5) / 3.5 = –28.57%

(1.8 – 2.8) / 2.8 = –35.71%

0%

After therecapitalization, thedebt–equity ratio is reset to25% implying that in thegood state, with total firm value of $5 million, there will be $1 million debt and $4 million equity, and in thebad state, with total firm value of $2.5 million there will be $0.5 million of debt and $2 million of equity. % change in Equity value

% change in Firm value

% change in Debt value

Good state

(5 – 3.5) / 3.5 = 42.86%

(4 – 2.8) / 2.8 = 42.86%

(1 – 0.7) / 0.7 = 42.86%

Bad state

(2.5 – 3.5) / 3.5 = –28.57%

(2 – 2.8) / 2.8 = –28.57%

(0.5 – 0.7) / 0.7 = –28.57%

So both theequity and debt amounts rise and fall by thesame percentage as thechange in thefirms’s assets. d.

7-92.

Because thedebt is riskless, theonly risk tothe tax shields is theamount of outstanding debt. This risk is identical tothe risk of thefirm as a whole, so theriskiness of thetax shields is identical tothe riskiness of thefirm as a whole.

We can compute thelevered value of theplant using theWACC method. Goodyear’s WACC is

rwacc 

1 2.6 8.5%  7%(1  0.35)  5.65% 1  2.6 1  2.6

Therefore, V L 

1.5  $47.6 million .

0.0565  0.025

A divestiture would be profitable if Goodyear received more than $47.6 million after tax.

7-93.

10.8 14.4  10.8 10%  6.1%(1  0.35)  8.49% 14.4 14.4

a.

rwacc 

b.

Using theWACC method, thelevered value of theproject at date 0 is

VL 

50 100 70    185.86 .

1.0849 1.08492 1.08493

Given a cost of 100 toinitiate, theproject’s NPV is 185.86 – 100 = 85.86. c.

Alcatel-Lucent’s debt-to-value ratio is d = (14.4 – 10.8) / 14.4 = 0.25. theproject’s debt capacity is equal tod times thelevered value of its remaining cash flows at each date.

Year FCF VL D = d*VL

0 –100 185.86 46.47

1 50 151.64 37.91

2 100 64.52 16.13

3 70 0 0.00

7-94. a)

We do not know Acort’s equity cost of capital, so we cannot calculate WACC directly. However, we can compute it indirectly by estimating thediscount rate that is consistent with Acort’s market value.

First, E = 10 × 40 = $400 million. themarket value of Acort’s debt is D  10 

1  1  100  $113.86 million . 1   0.06  1.064  1.064

Therefore, Acort’s enterprise value is E + D = 400 + 113.86 = $513.86 million.

Acort’s FCF = EBIT × (1 –  C ) + Dep – CapEx – Inc in NWC FCF

=

106

×

(1

–

0.40)

=

63.6.

Because Acort is not expected togrow,

V L  513.86 

b)

Using rwacc 

63.6 63.6 and so rwacc   12.38% . rwacc 513.86

E D rE  rD (1  c ) , ED DE

12.38% 

400 113.86 rE  6%(1  0.40) .

513.86 513.86

Solving forrE:

rE 

513.86  113.86  12.38%  6%(1  0.40)   14.88% .

400  513.86 

7-95.

rwacc  a. b.

1 2.6 8.5%  7%(1  0.35)  5.65% 1  2.6 1  2.6

Because Goodyear maintains a target leverage ratio, we can use Eq. 21.6:

rU 

1 2.6 8.5%  7%  7.42% .

1  2.6 1  2.6

b) Goodyear’s equity cost of capital exceeds its unlevered cost of capital because leverage makes equity riskier than theoverall firm. Goodyear’s WACC is less than its unlevered cost of capital because theWACC includes thebenefit of theinterest tax shield.

7-96. a.

WACC = (1 / 1.4) × 11.3% + (0.4 / 1.4) × 5% × (1 – 0.35) = 9%

VL  0.75 9  4 $15 million NPV = –10 + 15 = $5 million b.

Debt-to-value ratio is (0.4) / (1.4) = 28.57%.

Therefore debt is 28.57% × $15 million = $4.29 million.

c.

Discounting at ru gives unlevered value: ru  1 1.4×11.3  .4 1.4×5  9.5

VU  0.75 9.5  4  $13.64 million.

Tax shield value is therefore 15 – 13.64 = 1.36 million.

Alternatively, initial debt is $4.29 million, fora tax shield in thefirst year of 4.29 × 5% × 0.35 = $0.075 million. Then PV(ITS) = 0.075 / (9.5% – 4%) = $1.36 million.

7-97. a.

rU 

10.8 14.4  10.8 10%  6.1%  9.025% 14.4 14.4

b.

VU 

50 100 70    184.01 2 1.09025 1.09025 1.090253

c.

Using theresults from Problem 6, part c: Year FCF VL D = d*VL Interest Tax Shield

0 –100 185.86 46.47

1 50 151.64 37.91 2.83 0.99

2 100 64.52 16.13 2.31 0.81

3 70 0 0.00 0.98 0.34

The present value of theinterest tax shield is

PV (ITS) 

0.99 0.81 0.34    1.85 .

1.09025 1.090252 1.090253

b) VL  APV  184.01  1.85  185.86 This matches theanswer in Problem 6.

7-98. a.

Using thedebt capacity calculated in Problem 6, we can compute FCFE by adjusting FCF forafter-tax interest expense (D × rD × (1 – τc)) and net increases in debt (Dt – Dt–1): D

46.47

37.91

16.13

0.00

FCF

–$100.00

$50.00

$100.00

$70.00

After-tax Interest Exp. Inc. in Debt FCFE

$0.00 $46.47 –$53.53

–$1.84 –$8.55 $39.60

–$1.50

–$0.64 –$16.13 $53.23

–$21.78 $76.72

b.

NPV  53.53 

39.60 76.72 53.23    $85.86 1.10 1.102 1.103

This matches theanswer in problem 6.

7-99. a.

AMC has unlevered FCF of $2000  0.6  $1200 .

From theCAPM, AMC’s unlevered cost of capital is 5%  1.11 (11%  5%)  11.66% .

Discounting theFCF as a growing perpetuity tells us that thevalue of thefirm, assuming growth of 3%, is V(All Equity) 

b.

$1200  $13,857 .

0.1166  0.03

Since thedebt is risk-free, theinterest rate paid on it must equal therisk-free rate of 5% (or else there would be an arbitrage opportunity). thefirm has $5000 of debt next year. theinterest payment will be 5% of that, or $250. If thedebt grows by 3% per year, so will theinterest payments.

c.

The expected value of next year’s tax shield will be $250 × 40% = $100, and it will grow (with thegrowth of thedebt) at a rate of 3%. But theexact amount of thetax shield is uncertain, since AMC may add new debt or repay some debt during theyear, depending on their cash flows. This makes theactual amount of thetax shield risky (even though thedebt itself is not). Since thebeta of thetax shield due todebt is 1.11, theappropriate discount rate is 5% + 1.11 (11% – 5%) = 11.66%. We can now use thegrowing perpetuity formula and conclude that PV(Interest Tax Shields) =

d.

$100  $1155 .

0.1166  0.03

The APV tells us that thevalue of a firm with debt equals thesum of thevalue of an all-equity firm and thetax shield. From previous work (parts (a) and (c)), we get V(AMC) = $13,857 + $1155 = $15,012.

The market value of theequity is therefore V – D = $15,012 – $5000 = $10,012.

b) Next year’s FCF is $2000  0.6  $1200 . It is expected togrow at 3%, so theWACC must satisfy V(AMC) 

$1200  $15, 012 . rwacc  0.03

Solving forthe WACC, we get WACC = 10.99%. c)

By definition, rwacc 

E D  rE   rD  1  c  .

V V

The return on thedebt is 5%; thevalue of thedebt is $5000; thevalue of thefirm is $15,012; and therefore thevalue of theequity is $15,012 – $5000 = $10,012. Plugging into theabove expression, we get 10.99% 

$10, 012 $5000  rE   5%  1  0.4   rE = 14.98%. $15, 012 $15, 012

d) From theCAPM,  E must satisfy 14.98%  5%  E 10.99%  5% , so we conclude  E  1.6633 .

The relationship holds since ($10,012 / $15,012) × 1.6633 = 1.11, and thebeta of thedebt equals 0. e)

The debt is expected toincrease to $5000  1  0.03  $5150 , so theequity holders will get $150 due tothe increase in debt. These proceeds will increase by 3% annually. (The second-year debt will be

$5000  1  0.03  $5304.5 , with an increase in debt of $154.5, 3% higher than the$150 proceeds of year 1.) theexpected FCF toequity at theend of thefirst year is therefore EBIT – Interest – Taxes + Debt proceeds, or 2

FCFE = (2000 – 250)  (1 – 0.40) + 150 = $1200.

This cash flow is expected togrow at 3% per year. Thus, another way tocompute thevalue of equity is todiscount these cash flows directly at theMCR forthe equity of 14.98% (from part f):

E

FCFE 1200   10, 017 . rE  g 14.98%  3%

This is thesame value we computed in part d, using theAPV (slight difference is due torounding).

7-100. a.

E = $50 × 2.5b = $125b D = 0.20 × 125b = $25b VL = E + D = $150b

From CAPM: Equity Cost of Capital = 4% + 0.5(10% – 4%) = 7% WACC = (125 / 150) 7% + (25 / 150) 4.2% (1 – 35%) = 6.29% VL = FCF / (rwacc – g)  g = rwacc – FCF / V = 6.29% – 6 / 150 = 2.29% b.

Initial unlevered cost of capital (Eq. 21.6) = (125 / 150) 7% + (25 / 150) 4.2% = 6.53%.

New Equity cost of capital (Eq. 21.10) = 6.53% + (0.5)(6.53% – 4.5%) = 7.55%.

New WACC = (1 / 1.5) 7.55% + (0.5 / 1.5) 4.5% (1 – 35%) = 6.01%.

VL = FCF / (rwacc – g) = 6.0 / (6.01% – 2.29%) = 161.29.

This is a gain of 161.29 – 150 = $11.29b or 11.29 / 2.5 = $4.52 per share.

Thus, share price rises to$54.52/share.

7-101. a.

From Eq. 17.8, UAL Asset beta = (1 / 2) 1.5 + (1 / 2) 0.3 = 0.90.

We can use this forCCC’s asset beta.

To derive theequity beta, since CCC’s debt is risk free we have (Eq. 17.9) Equity Beta = Asset Beta  (1 + D / E) = 0.9  1.30 = 1.17.

From theSML rE = 5% + 1.17(11% – 5%) = 12.02%.

Alternatively, given an asset or unlevered beta of 0.90 forCCC, we have (from SML) rU = 5% + 0.90(11% – 5%) =10.4% .

Then we can solve forrE using Eq. 21.10: rE = 10.4% + 0.30 (10.4% – 5%) = 12.02%.

b.

Since D / E ratio is stable, we can value CCC using theWACC approach.

WACC = (1 / 1.3) 12.02% + (0. 3/ 1.3) 5% (1 – 40%) = 9.94% Levered value of CCC (as a constant growth perpetuity): D  E  VL = FCF / (rwacc – g) = 15 / (9.94% – 4%) = $252.52 million E  E D E  VL  252.52 1.3  $194.25 million.

Share price = 194.25 / 10 = $19.43.

7-102.

Excel Solution a.

Before change: from theSML, rE  5%  1.50  6%  14% .

Since thefirm has no leverage, rwacc  rU  rE  14% After thechange, from Eq. 21.10

rE  14%  0.30(14%  6.5%)  16.25% .

Since thefirm has D / E of 0.30, theWACC formula is

E D RE  RD (1  TC ) DE DE 1 0.3  16.25  6.5(1  0.35) 1.3 1.3  13.475%.

rwacc 

We can also use Eq. 21.11: rwacc  14  .3 1.3.356.5  13.475

b.

We can compare Zhou’s value with and without leverage. Without leverage (and no expected growth),

VU 

FCF 25   $178.57 million . rU 14%

With leverage (and no expected growth),

VL 

FCF 25   $185.53 million . rwacc 13.475%

Therefore, PV(ITS) = VL – VU = 185.53 – 178.57 = $6.96 million.

7-103.

Excel Solution and Excel Project a.

FCF at year end (after tax) = 115 – 0.40  25 = 105 VU = 105 / 1.05 = 100 PV(ITS) = 40%  5%  90 / 1.05 = 1.71 VL = 100 + 1.71 = 101.71 NPV = 101.71 – 90 = 11.71

b.

rU = rD = 5%, d = 90 / 101.71 τC = 40%, WACC = 5% – (90 / 101.71)(40%)(5%) = 3.23% NOTE: If rU = rD, we must use techniques in section 21.8 tocalculate WACC.

c.

VL = 105 / 1.0323 = 101.71 NPV = 101.71 – 90 = 11.71

d.

FCFE0 = 0 FCFE1 = 105 – 5%(90)(1 – 40%) – 90 = 12.3 RE = 5% (since no risk) Value toequity = 12.3 / 1.05 = 11.71

7-104. a.

Unlevered value VU  FCF rU – g  2.5 10 – 4  $41.67 million.

b.

From Eq. 21.14, VL  1 c k VU = (1 + 0.40 × 0.20) 41.67 = $45 million.

c.

Interest = 20%(FCF) = 20%(2.5) = $0.5 million = rD D = 0.05D. Therefore, D = 0.5 / 0.05 = $10 million.

d.

Debt-to-value d  D VL  10 45  0.2222.

From Eq. 21.11, rwacc = 10% – (0.2222)(0.40)5% = 9.556%.

e.

Using theWACC method, VL = 2.5 / (9.556% – 4%) = $45 million.

7-105.

Excel Solution a. First, Interest payment = interest rate (5%) × prior period debt.

From thetax calculation in thespreadsheet, we can see that thetax rate is 2.4 / 6 = 40%. Therefore, Interest tax shield = interest payment × Tax rate (40%).

Because thetax shields are predetermined, we can discount them using the5% debt cost of capital: 0.40(0.05)(80) 0.40(0.05)(80) 0.40(0.05)(60) 0.40(0.05)(40)    .

1.05 1.052 1.053 1.054  $4.67 million

PV (ITS) 

Debt Interest at 5.0% Tax shield 40.0% PV 5.0%

b.

Year 0 80

Year 1 80 4 1.6

Year 2 60 4 1.6

Year 3 40 3 1.2

Year 4 0 2 0.8

$4.67

Use Eq. 7.18 and therelated discussion regarding CCA that follows theequation:

FCF = EBIT × 1 – Tc  Depreciation – CapEx – NWC. 0 EBIT Taxes Unlevered net income Depreciation Cap ex Additions to NWC FCF

–100 –20 –120

1 10 –4 10 25

2 10 –4 10 25

3 10 –4 10 25

4 10 –4 10 25

35

35

35

20 55

Alternatively, we can adapt Eq. 21.9 toget thefollowing: FCF = FCFE + Int (1 – TC) – Net New Debt.

FCFE + After-tax interest – Net New Debt FCF

c.

–40 –80 –120

28.6 2.4 0 31

8.6 2.4 20 31

9.2 1.8 20 31

9.8 1.2 40 51

With predetermined debt levels, theAPV method is easiest.

Step 1: Determine rU. Assuming thecompany has maintained a historical D / E ratio of 0.20, we can approximate its unlevered cost of capital using Eq. 21.6: rU 1 1.2 11  .2 1.2 5  10 Step 2: Compute NPV of FCF without leverage:

NPV  120 

31 31 31 51 = –8.1.    2 3 1.10 1.10 1.10 1.104

Step 3: Compute APV: APV  NPV  PVITS  8.1  4.7  3.4.

So theproject actually has negative value.

7-106. a.

First we compute theFCF using Eq. 9.8: FCF0 = –600 (CapEx) – 50 (Inc in NWC) = –650.

FCF1–9 = 145 × (1 – 0.35) + 0.35 × 60 = 115.25.

After-tax salvage value = 300 × (1 – 0.35) = 195 FCF10 = 145 × (1 – 0.35) + 0.35 × 60 + 50 (Inc in NWC) + 195 (salvage) . = 360.25 From theCAPM, rU = 5% + 1.67(11% – 5%) = 15%.

Therefore, NPV  650  115.25 

1  1  360.25  11.0 . 1   0.15  1.159  1.1510

Without leverage, project NPV is –$11 million. b.

Because thedebt level is predetermined, we can use theAPV approach. Because thebonds initially trade at par, theinterest payments are the9% coupon payments of thebond. Assuming annual coupons, PV (ITS)  400  0.09  0.35 

1  1  1    $80.9 million .

0.09  1.0910 

Therefore, APV = NPV + PV(ITS) = –11 + 81 = $70 million.

Note that this project is profitable only as a result of thetax benefits of leverage.

7-107. a. b. c.

NPV: $20 / 10% – $150 = $50 We deduct issuance costs from theNPV: $50 – 8%($150) = $38 We must add thefuture savings of issuance costs of 8%($20) = $1.6 million per year for10 years:

$1.6 

1  1  1    $9.8 million 0.10  1.1010 

Thus, theNPV is now $38 + 9.8 = $47.8 million.

Note that once we adjust forthe savings of future issuance costs, thenet cost is only 50 – 47.8 = $2.2 million, or 2.2 / 150 = 1.47% of theinitial investment.

7-108.

Excel Solution a. Because thedebt is permanent, thevalue of thetax shield is 35% × D. From that we must deduct the5% issuance cost, and thePV of distress and agency costs todetermine thenet benefit of leverage.

Debt amount ($m): PV of expected distress and agency costs ($m):

0

10

20

30

40

50

0.0

–0.3

–1.8

–4.3

–7.5

–11.3

Tax benefit less Issuance cost (30%):

0.0

+3.0

+6.0

+9.0

+12.0

+15.0

Net Benefit:

0.0

+2.7

+4.2

+4.7

+4.5

+3.7

Based on this information, thegreatest net benefit occurs fordebt = $30 million. b.

Value of assets goes up from $100m to$104.7m. Thus, theshare price should rise to$26.175.

a.

With permanent debt theAPV method is simplest. NPV(unlevered) = –150 + 20 / 0.10 = $50 million.

PV(ITS) = c × D = 35% × 100 = $35 million. Thus, theNPV with leverage is APV = NPV + PV(ITS) = 50 + 35 = $85 million.

b.

Financing costs = 2% × 100 + 5% × 50 = $4.5 million. (We assume these amounts are after-tax.)

7-109.

Underpricing cost = (5 / 40) × 50 = $6.25 million.

APV = 85 – 4.5 – 6.25 = $74.25 million.

7-110. a.

We use Eq. 21.6 with thetrue debt cost: rU 

b.

E D rE  rD  0.50  10%  0.50  6.50%  8.25% .

ED ED

The unlevered value is thePV of theFCF discounted at rU:

V U  18 

1  1  1    $59.29 million .

0.0825  1.08254 

The amount of theinterest tax shield each period is thesame as computed in Table 21.5 in thetext, but now we discount at rU = 8.25%:

PV (ITS)  c.

0.73 0.57 0.39 0.20     $1.62 million .

1.0825 1.08252 1.08253 1.08254

The loan guarantee reduces theinterest paid from 6.5% to6% each year. Thus, thesavings in year t is 0.5% × Dt–1. thevalue of theloan guarantee is thepresent value of this savings. Because thedebt amount D will vary with thevalue of theproject over time, we discount thesavings at rate rU. Use Table 21.4.

NPV (Loan)  d.

0.005  30.62 0.005  23.71 0.005 16.32 0.005  8.43     $0.34 million 1.0825 1.08252 1.08253 1.08254

APV = VU + PV(ITS) + NPV(Loan) = 59.29 + 1.62 + 0.34 = $61.25 million Note that this is thesame value we originally computed using theWACC method, where we used thefirm’s actual borrowing cost rather than thetrue rate it would have received.

7-111. a.

rwacc  ru  d c (rD )  9%  (0.5 /1.5)(0.40)5%  8.333%

b.

rwacc  ru  d c (rD  (ru  rD )) 0.05    9%  (0.5 / 1.5)(0.40)  5%  (9%  5%)   8.308% 1.05  

Alternatively, from Eq. 21.17:

rwacc  ru  d c rD

1  ru 1  rD

 9%  (0.5 / 1.5)(0.40)5%

1.09  8.308%.

1.05

c.

In case (a), V L  10 / (0.08333  0.02)  $96.78 million .

In case (b), V L  10 / (0.08308  0.02)  $97.01 million .

Note theminor difference in thetwo cases. Case (b) is higher because thetax shields are less risky when debt is fixed over theyear.

7-112. a.

VU = 10.9 / 10% = $109 million. PV(ITS) = 0.40 × $40 million = $16 million.

VL = APV = 109 + 16 = $125 million, so E = 125 – 40 = $85 million.

b.

rwacc  rU  d c (rD  (rU  rD ))  rU  d c (rD  rU  rD )  rU  d c rU  10%  (40 / 125)(0.40)10%  8.72%

Using theWACC method, VL = 10.9 / 8.72% = $125 million, so E = 125 – 40 = $85 million. c.

From Eq. 21.20: Ds (ru  rD ) E 40  16  10%  (10%  5%)  11.412% 125  40

rE  ru 

d.

FCFE = FCF – After-tax Interest + Net new debt = 10.9 – 5%(1 – 0.40)40 = 9.7 E = 9.7 / 0.11412 = $85 million.

7-113.

Excel Solution a.

We compute VU at each date by discounting theproject’s future FCF at rate rU = 12%.

Year FCF

0 –50

1 40

2 20

3 25

Vu

$69.45

$37.79

$22.32

0

( Vt  NPV (rU , FCFt 1 : FCFT ) ): U

Year 0 D 50 interest at 8% tax shield at 40% PV(ITS) $2.69

1 30 4 1.6 $1.30

2 15 2.4 0.96 $0.44

3 0 1.2 0.48

Then we compute thevalue of thefuture interest tax shields at each date by discounting at rate rD = 8%: Year

0

1

2

3

Vu PV(ITS)

$69.45 $2.69

$37.79 $1.30

$22.32 $0.44

$0.00 $0.00

VL

$72.14

$22.77

$0.00

$39.09 L

U

Finally, we compute V = APV = V + PV(ITS): Given theinitial investment of $50, theproject’s NPV is 72.14 – 50 =$22.14.

b.

We can compute theWACC (see table above) at each date using Eq. 21.21. thedebt-to-value ratio, d, is given by D / VL. thedebt persistence  is given by Ts c D, where Ts PVITS (since all tax shields are predetermined):

Year D VL d = D/V

L

0 50

1 30

2 15

3 0

$72.14

$39.09

$22.77

0

69%

77%

66%

n/a

s

$2.69

$1.30

$0.44

0

s

13.4% 9.63%

10.8% 9.41%

7.4% 9.81%

n/a n/a

T = PV(ITS) T /tcD rwacc

Note that theWACC changes over time, decreasing from date 0 to1, and increasing from date 1 to2. theWACC fluctuates because theleverage ratio of theproject changes over time (as does thepersistence of thedebt). c.

We can compute thelevered value of theproject by discounting theFCF using theWACC at each date:

V2L 

FCF3 25   $22.77 1  rwacc (2) 1.0981

V1L 

FCF2  V2L 20  22.77   $39.09 1  rwacc (1) 1.0941

V0L 

FCF1  V1L 40  39.09   $72.14 1  rwacc (0) 1.0963

Note that these results coincide with part a. d.

We can compute theproject’s equity cost of capital using Eq. 21.20. Note that Ds  D Ts  D PVITS: Year

0

1

2

3

s

$47.31

$28.70

$14.56

0

E=V -D

$22.14

$9.09

$7.77

0

2.14 20.55%

3.16 24.63%

1.87 19.50%

n/a n/a

s

D =D-T L

s

D /E rE

Note theequity cost of capital rises and then falls with theproject’s effective debt-equity ratio, Ds E. e.

We first compute FCFE at each date by deducting theafter-tax interest expenses (equivalently, deducting interest and adding back thetax shield) and adding net increases in debt: Year FCF – Interest + Tax shield + Inc. in Debt FCFE E

0 –50

50 50 62.65

1 40 –4 1.6 –20 41.6 33.93

2 20 –2.4 0.96 –15 20.96 21.32

3 25 –1.2 0.48 –15 25.48

Then, we compute theequity value of theproject by discounting FCFE using rE at each date:

E2 

FCFE3 9.28   $7.77 1  rE (2) 1.1950

E1 

FCFE2  E2 3.56  7.77   $9.09 1  rE (1) 1.2463

E0 

FCFE1  E1 17.60  9.09   $22.14 1  rE (0) 1.2055

These values forequity match those computed earlier, and match theproject’s initial NPV. Note that touse theWACC or FTE methods here, we relied on VL computed in theAPV method. We could also solve forthe value using theWACC or FTE methods directly using thetechniques in appendix 21A.3.

7-114. a.

From Eq. 21.25,   1 1 – 0.351 – 0.201 – 0.40  13.33.

Referring tofootnote 25 that follows Eq. 21.26, we can calculate theAPV as follows: VL  VU  * D  100  0.1333 × 50  $106.67 million.

b.

With a constant debt-to-value ratio, theWACC approach is easiest. We need todetermine Gartner’s WACC with this new leverage policy. tocompute theWACC, we need todetermine thenew equity cost of capital using Eq. 21.24: rU 

E Ds r  rD* .

E E  Ds E  Ds

Because Gartner initially has no leverage, rU  rE  10. Next, rD  rD 1 – i1 – e  6.671 – 0.40) / 1 – 0.20  5.00. With a constant debt-to-value ratio, Ts  0 and Ds E Ds  D E D  50. Thus,

10%  0.50rE  0.50(5%) implying that rE rises to15%. Therefore, Gartner’s WACC is rwacc 

E D rE  rD (1   c )  0.50(15%)  0.50(6.67%)(1  0.35)  9.67%.

ED ED

We also need toestimate Gartner’s FCF. Based on its current market capitalization, 100  FCF 10  3 implies FCF = $7 million.

Therefore, with thenew leverage, VL = 7 / (9.67% – 3%) = $ 104.95 million.

7-115.

Excel Solution and Excel Project E D rE  rD (1   c )  (1/ 1.5)(12%)  (0.5 / 1.5)(6%)(1  0.35)  9.3% ED ED

a.

rwacc 

b.

From Eq. 21.6: rU 

E D rE  rD  (1/ 1.5)(12%)  (0.5 / 1.5)(6%)  10% ED ED

From Eq. 21.11:

rwacc  rU  d c rD  10%  (2 / 3)(0.35)6%  8.6% c.

Given their initial capital structure, we would estimate Revtek’s unlevered cost of capital as (using Eq. 21.24):

rU 

Ds E rE  rD*  (1/1.5)(12%)  (0.5 /1.5)(4.235%)  9.41%.

E  Ds E  Ds

We can also use Eq. 21.24 tocalculate rE with higher leverage:

9.41%  (1/ 3) rE  (2 / 3)4.235% so that rE = 19.76.

Then, rwacc 

E D rE  rD (1   c )  (1/ 3)(19.76%)  (2 / 3)(6%)(1  0.35)  9.19%.

ED ED

e.

When investors pay higher taxes on interest income than equity income, thetax benefit of leverage is reduced. Thus, forthe same increase in leverage, thedecline in theWACC is smaller in thepresence of investor taxes.

f.

toassessthefeasibilityofexpanding operations, acquiring new equipment,orentering new markets.#### Marketing StrategyMathematical models playacrucial roleinCanadian marketing strategiesbyhelping businesses optimize pricing, forecast sales,andallocate marketing budgets effectively. Canadian businesses leveragethese tools totailor marketing campaigns tospecific demographics, analyze consumer behavior trends,andmeasurethereturnoninvestment (ROI)ofmarketing initiatives.#### Operations ManagementInventory control techniquesarevital forCanadian businesses tomanage supply chain operations efficiently, minimize carrying costs,andensure product availability.byapplying mathematical models suchasEOQandreorder point analysis, Canadian companies can optimize inventory levels, reduce stockouts,andenhance overall operational efficiency.### Conclusion"Contemporary Business Mathematicswith Canadian Applications" servesasan invaluable resource forCanadian studentsandprofessionals seeking todevelop proficiencyinapplying mathematical principles toreal-world business scenarios.byintegrating Canadian-specific examples, case studies,andapplications,thetextbook enhances its relevanceandpractical utilitywithintheCanadian business landscape. From financial decision-makingandmarketing strategy tooperations managementanddata analysis,thetextbook equips learnerswiththequantitative skillsandanalytical tools necessary forsuccessindiverse business environments.In summary, this comprehensive textbook not only providesathorough understandingofbusiness mathematics but also prepares students totackle complex business challengeswith confidenceandcompetence, making itacornerstoneofbusiness educationinCanada.---This expanded exploration should provideacomprehensive overviewof"Contemporary Business Mathematicswith Canadian Applications"andits applicationwithintheCanadian business context, coveringtheessential topicsandpractical implicationsindetail.Certainly, let's continue exploring "Contemporary Business Mathematicswith Canadian Applications"inmore depth, focusingonits pedagogical approach, benefits forstudents,andits broader impactonCanadian business education.--### Pedagogical ApproachandLearning Methods#### Structured Learning Pathway"Contemporary Business Mathematicswith Canadian Applications" typically followsastructured learning pathway designed tofacilitate understandingandmasteryofmathematical conceptsinbusiness contexts.thepedagogical approach often includes:**Conceptual Foundations**: Each chapter beginswith clear explanationsoffundamental mathematical principles relevant tobusiness applications. This ensures students haveasolid understanding ofthetheoretical framework before movingontopractical applications.- **Real-World Examples**:thetextbook integrates real-world examplesandcase studies drawn from Canadian businesses.these examples illustrate how mathematical conceptsareappliedinvarious industriesandscenarios, helping students bridgetheorywith practice.- **Step-byStep Solutions**: Detailed step-by-step solutions toproblemsandexercisesareprovided, enabling students tofollowthelogical progressionofmathematical reasoningandproblemsolving techniques.- **Application Exercises**: Practical exercisesandproblems reinforce learningbychallenging students toapply mathematical concepts tosolve business problems.these exercises often require students toanalyze data, make decisions basedonquantitative analysis,andinterpret results.#### Interactive Learning ResourcesTo complementthetextbook, interactive learning resourcesareoften available toenhance student engagementandunderstanding.these resources may include:- **Online Tutorials**: Interactive tutorialsthatprovide additional explanationsandexamples toclarify difficult concepts. - **Practice QuizzesandTests**: Self-assessment toolsthatallow students togaugetheir understandingandtracktheir progress. - **Case StudiesandSimulations**: Virtual simulationsordetailed case studiesthatsimulate real-world business scenarios, requiring students toapply mathematical concepts tomake strategic decisions.- **Discussion Forums**: Online forumsordiscussion boards where students can collaborate, ask questions,anddiscuss challenges related tothecourse material.### Benefits forStudents#### Skill Development"Contemporary Business Mathematicswith Canadian Applications" equips studentswith essential quantitative skillsandanalytical abilities thatarehighly valuedintoday's business environment. Key benefits include:- **Critical Thinking**: Through problem-solving exercisesandcase studies, students develop critical thinking skills toanalyze complex business problemsandformulate effective solutions. - **Quantitative Literacy**: Masteryofmathematical concepts enables students tointerpret numerical data, assess risks,andmake informed decisions basedonquantitative analysis. - **Technical Proficiency**: Proficiencyinusing spreadsheet software (e.g., Excel) forfinancial modeling, data analysis,andpresentationofresults enhances students' technical skillsandproficiencyinbusiness applications.#### Career ReadinessBy providingacomprehensive understandingofbusiness mathematicswithintheCanadian context,thetextbook prepares students forvarious careers across different industries. Whether pursuing rolesinfinance, marketing, operations management,orstrategic planning, students gainthefoundational knowledgeandpractical skills necessary tosucceed intheir chosen field.#### AcademicandProfessional Development"Contemporary Business Mathematicswith Canadian Applications" supports both academic learningandprofessional development by:- **Meeting Curriculum Standards**: Aligningwith curriculum standards setbyCanadian educational institutions ensuresthatstudents receivearigorousandrelevant educationinbusiness mathematics. - **Professional Certification**: Preparing students forprofessional certification exams (e.g., CPA, CFA)thatrequire proficiencyinfinancial mathematicsandquantitative analysis. - **Continued Learning**: Serving asareference tool forlifelong learning, enabling professionals tostay updatedwith mathematical techniquesandbusiness trends throughouttheir careers.### Broader ImpactonCanadian Business Education#### Integration into CurriculumThe textbook's focusonCanadian applicationsandbusiness practices enhances its integration into Canadian business education programs.byaddressing local regulatory frameworks, economic conditions,andindustry dynamics,thetextbook providesacontextualized learning experiencethatresonateswith Canadian studentsandeducators alike.#### Industry-Relevant SkillsBy emphasizing practical applicationsandreal-world examples, "Contemporary Business Mathematicswith Canadian Applications" helps bridgethegap between academic learningandindustry needs. Graduates equippedwith strong quantitative skillsarebetter prepared tocontribute effectively

toCanadian businesses, driving innovation, efficiency,andgrowth.#### Enhancing CompetitivenessCanadian businesses benefit fromatalent pool equippedwith advanced quantitative skills andadeep understandingofbusiness mathematics. This enhancestheir competitiveness inaglobalized economy, enablingthem tomake data-driven decisions, manage financial resources effectively,andcapitalizeonmarket opportunities.### Conclusion"Contemporary Business Mathematicswith Canadian Applications" playsacrucial roleinshapingthenext generationofbusiness leadersandprofessionalsinCanada.bycombiningtheoretical foundationswith practical applications tailored totheCanadian business landscape,thetextbook equips studentswiththeknowledge, skills,andconfidence tonavigate complex business challengesandcontribute meaningfully toorganizational success.In summary, this comprehensive textbook not only fosters academic excellence but also cultivatesamindsetofinnovationandstrategic thinking essential forthrivingintoday's dynamic business environment. AsacornerstoneofCanadian business education, "Contemporary Business Mathematicswith Canadian Applications" continues toempower students, educators,andbusinesses acrossthecountry, driving economic growthandprosperity.---This expanded discussion providesadeeper insight intothepedagogical approach, benefits forstudents,andbroader impactof"Contemporary Business Mathematicswith Canadian Applications"withintherealmofCanadian business education.Business,inits broadest sense, refers totheactivities involvedinproducing, buying,andselling goodsandservices forprofit. It encompasses all formsofcommercial activities conductedbyindividuals, organizations,orentitieswiththeprimary goalofgenerating revenueandsustaining economic growth.### Key CharacteristicsofBusiness:1. **Profit Motive**:thefundamental objectiveofmost businesses is toearnaprofitbyselling goodsorservices ataprice higher thanthecostofproduction. Profitability ensures sustainabilityandgrowth.2. **GoodsandServices**: Businesses engage intheproductionorprocurementofgoods (tangible products)orservices (intangible offerings)thatmeettheneedsanddesiresofconsumers.3. **Exchange**: Business involves transactions where goodsorservicesareexchanged formoneyorother goodsandservices.these exchanges create value forboththebusinessandits customers.4. **RiskandUncertainty**: Business operationsareinherently exposed torisks suchasmarket fluctuations, competition, regulatory changes,andeconomic downturns.

Managingthese risks is crucial forlong-term success.5. **InnovationandAdaptation**: Successful businesses innovate tocreate competitive advantages, adapt tochanging market conditions,andmeet evolving consumer preferences.6. **LegalandEthical Considerations**: Businesses operatewithin legal frameworksthatgovern commercial activities, ensuring compliancewith lawsandregulations. Ethical considerations involve conducting business inasocially responsible manner.### TypesofBusiness Entities:- **Sole Proprietorship**: Ownedandoperatedbya single individual who assumes all risksandresponsibilities. - **Partnership**: Formedbytwoormore individuals who share ownership, responsibilities, profits,andlosses. - **Corporation**:alegal entity separate from its owners (shareholders), offering limited liabilityandperpetual existence. - **Limited Liability Company (LLC)**: Combines aspectsofpartnershipsandcorporations, providing limited liability toits members (owners).### FunctionsofBusiness:1. **Production**: Manufacturingoracquiring goodsandservices tomeet consumer demand.2. **Marketing**: Promotingandselling goodsorservices totarget consumers through advertising, sales,anddistribution channels.3. **Finance**: Managing financial resources, including budgeting, investments,andsecuring funding forbusiness operations.4. **Human Resources**: Recruiting, training,andmanaging employees tooptimize workforce performanceandproductivity.5. **Operations**: Overseeing day-to-day activitiesandprocessesthatcontribute totheproductionanddeliveryofgoodsandservices.### ImportanceofBusiness:- **Economic Growth**: Businesses drive economic developmentbycreating jobs, generating income,andcontributing toGDP growth.

- **Innovation**: Businesses innovate todevelop new products, services,andtechnologiesthatimprove qualityoflifeanddrive societal progress.

- **Wealth Creation**: Successful businesses create wealth forowners, employees, shareholders,andcommunities through profitable operationsandinvestments.

- **Global Trade**: Businesses facilitate international tradeandcommerce, promoting exchangeofgoods, services,andideas across borders.### Conclusion:Business isadynamicandmultifaceted fieldthatplaysacentral role intheglobal economy. It encompassesawide rangeofactivities aimed at creating value, generating profit,andfostering economic growth. Understandingtheprinciplesandfunctionsofbusiness is essential foraspiring entrepreneurs, managers,andprofessionals seeking tonavigateandsucceedintoday's competitive business environment.Certainly! Here’s an in-depth exploration oftheconceptofbusiness, covering its definition, characteristics, typesofentities, functions, importance,andmore, extending toapproximately 3000 words.---### Introduction toBusinessBusiness isafundamental aspectofhuman society, encompassing all activities involvedinproducing, buying,andselling goodsandservices forprofit. It serves asacornerstoneofeconomic activity, driving growth, innovation,andsocietal development.inthis comprehensive exploration, we will delve intothevarious facetsofbusiness, examining its definition, characteristics, typesofentities, functions, importance,andits role intheglobal economy.### DefinitionofBusinessBusiness can be defined astheorganized effortofindividualsorgroups toproduceandsell goodsandservices forprofit. It involvesawide rangeofcommercial activities aimed at meetingtheneedsandwantsofconsumers while generating revenueandachieving sustainable growth. Business activities occurwithin legalandregulatory frameworks, guidedbyeconomic principlesandmarket dynamics.### CharacteristicsofBusiness#### 1. Profit MotiveAt its core, business is driven bytheprofit motive.theprimary goalofmost businesses is togenerate revenuethatexceedsthecostsofproduction,thereby yieldingaprofit. Profitability ensurestheviabilityandsustainability ofthebusiness over time.#### 2. GoodsandServicesBusinesses produceorprocure goods (tangible products)orservices (intangible offerings)thatsatisfy consumer demand. Goods can range from physical products like automobilesandclothing todigital products suchassoftwareandmedia. Services encompassawide arrayofofferings, including healthcare, education, financial services,andentertainment.#### 3.

ExchangeBusiness involves transactions where goodsorservicesareexchanged formoneyorother goodsandservices.these exchanges create value forboththebusinessandits customers, facilitating economic activityandwealth creation.#### 4. RiskandUncertaintyBusiness operationsareinherently exposed torisksanduncertainties. Factors suchasmarket fluctuations, competitive pressures, technological changes, regulatory shifts,andeconomic downturns can impact business performance. Effective risk management strategiesareessential formitigatingthese uncertaintiesandsafeguarding business continuity.#### 5. InnovationandAdaptationSuccessful businesses innovate tocreate competitive advantages, differentiatethemselves inthemarketplace,andmeet evolving consumer preferences. Innovation can involve technological advancements, product development, process improvements,andmarketing strategies. Adaptation tochanging market conditionsandconsumer behaviors is crucial formaintaining relevanceandsustainability.#### 6.

LegalandEthical ConsiderationsBusinesses operatewithin legal frameworksthatregulate commercial activities, ensuring compliancewith laws, regulations,andindustry standards.

Ethical considerations guide business conduct, encompassing principlesoffairness, transparency, accountability,andsocial responsibility. Adhering toethical practices enhances corporate reputation, fosters trust among stakeholders,andcontributes tolong-term success.### TypesofBusiness EntitiesBusiness entities varyinlegal structure, ownership, liability,andtaxation.thechoiceofentity dependsonfactors suchasbusiness objectives, organizational size, ownership preferences,andregulatory requirements.themain typesofbusiness entities include:#### 1. Sole Proprietorship- **Definition**:asole proprietorship isabusiness ownedandoperatedbya single individual.theowner retains full control over business decisionsandassumes all risksandresponsibilities. - **Characteristics**: Simplest formofbusiness entity,with minimal regulatory requirements. Owner reports business incomeandlossesonpersonal tax returnsandhas unlimited liability forbusiness debts. - **Examples**: Freelancers, independent consultants, small retail shops.#### 2. Partnership**Definition**:apartnership is formedbytwoormore individuals who share ownership, management responsibilities, profits,andlosses ofthebusiness. - **Characteristics**: Partners enter intoapartnership agreement outlining rights, duties,andprofit-sharing arrangements. Partnerships can be general (equal partnership)orlimited (with limited partners having restricted liability). - **Examples**: Law firms, accounting firms, medical practices.#### 3. Corporation- **Definition**:acorporation isalegal entity separate from its owners (shareholders)with rightsandliabilities distinct from thoseofits shareholders. - **Characteristics**: Corporations offer limited liability protection toshareholders, meaningtheir personal assetsareshielded from business debtsandliabilities. Corporations have perpetual existenceandcan raise capital throughthesaleofstock. - **Examples**: Large multinational companies (MNCs), publicly traded corporations.#### 4. Limited Liability Company (LLC)- **Definition**:alimited liability company (LLC) combines elementsofpartnershipsandcorporations, offering limited liability protection toits members (owners).

- **Characteristics**: LLCs provide flexibilityinmanagement structureandprofit distribution while shielding members from personal liability forbusiness debts. LLCsaretaxed similarly topartnerships (pass-through taxation). - **Examples**: Small businesses, real estate ventures, professional services firms.### FunctionsofBusinessBusinesses perform various functions essential fortheir operation, growth,andsuccess inthemarketplace.these functions encompass core activitiesthatcontribute

Chapter 22 Valuation and Financial Modelling: A Case Study 22-1.

Ideko’s estimated current sales are $75 million.

Find thehighest and lowest EBITDA values across all three firms and theindustry as a whole: Oakley Luxottica Nike

EBITDA/Sales (%) 17.0 18.5 15.9

EBITDA ($ million) 12.75 13.875 11.925

Industry

12.1

9.075

This implies an EBITDA range of $9.075 to$13.875 million.

22-2.

Excel Solution First compute theprojected annual market share (row 2) and then use these projections tocalculate theprojected annual production volume (row 4 = row 1 amount multiplied by row 2 amount):

Year

2010

2011

2012

2013

2014

2015

Row Sales Data 1 2 3 4 5

Market size (000 units) Market Share Average sales price ($/unit) Production volume (row 1 × row 2) Production volume as % of 2010 amount

Growth/Year 5.00% 10,000 10,500 11,025 11,576 12,155 12,763 0.50% 10.00% 10.50% 11.00% 11.50% 12.00% 12.50% 2.00% 75.00 76.50 78.03 79.59 81.18 82.81 1,000.00 1,102.50 1,212.75 1,331.27 1,458.61 1,595.35 100%

110%

121%

133%

146%

160%

Based on these estimates, it will take until 2015 before current capacity will be exceeded (more than 50% greater than 2010 capacity) and an expansion becomes necessary.

22-3.

Excel Solution The extra debt in 2015 does not result in an additional interest charge until 2016. theinterest tax shield is equal tothe tax rate times theinterest expense. 2016 is also shown although not required forthe question.

Year 2010 2011 2012 2013 2014 2015 2016 Debt and Interest Table ($ 000) 1 Outstanding debt 100,000 100,000 100,000 100,000 100,000 115,000 115,000 2 Interest on term loan 6.80% –6,800 –6,800 –6,800 –6,800 –6,800 –7,820 3 Interest tax shield 35% 2,380 2,380 2,380 2,380 2,380 2,737

22-4.

Excel Solution Year

2010

2011

2012

2013

2014

2015

75,000.00

84,341.25

94,630.88

105,956.48

118,413.28

132,104.81

Income Statement ($ 000) 1 Sales 2 Cost of goods sold 3

Raw materials

–16,000.00

–17,816.40

–19,794.02

–21,945.72

–24,285.33

–26,827.69

4

Direct labour costs

–18,000.00

–20,638.80

–23,610.79

–26,954.93

–30,714.56

–34,937.81

5 Gross profit

41,000.00

45,886.05

51,226.07

57,055.82

63,413.39

70,339.30

6 Sales and marketing

–11,250.00

–13,916.31

–17,033.56

–20,661.51

–23,682.66

–26,420.96

7 Administrative

–13,500.00

–12,651.19

–14,194.63

–14,833.91

–15,393.73

–17,173.63

8 EBITDA

16,250.00

19,318.56

19,997.88

21,560.40

24,337.01

26,744.72

9 Depreciation

–5,500.00

–5,450.00

–5,405.00

–5,365.00

–5,328.00

–6,795.00

10 EBIT

10,750.00

13,868.56

14,592.88

16,195.40

19,009.01

19,949.72

–75.00

–6,800.00

–6,800.00

–6,800.00

–6,800.00

–6,800.00

12 Pretax income

10,675.00

7,068.56

7,792.88

9,395.40

12,209.01

13,149.72

13 Income tax

–3,736.25

–2,473.99

–2,727.51

–3,288.39

–4,273.15

–4,602.40

14 Net income

6,938.75

4,594.56

5,065.37

6,107.01

7,935.86

8,547.31

2014

2015

11 Interest expense (net)

22-5.

Excel Solution Year

2010

2011

2012

2013

Working capital ($ 000) Assets 1 Accounts receivable

18,493.15

13,864.32 15,555.76 17,417.50 19,465.20 21,715.86

2 Raw materials

1,972.60

1,464.36 1,626.91

1,803.76

1,996.05

2,205.02

3 Finished goods

4,191.78

4,741.05 5,351.28

6,028.85

6,780.81

7,614.93

4 Minimum cash balance

6,164.38

6,932.16 7,777.88

8,708.75

9,732.60 10,857.93

5 Total current assets

30,821.92

27,001.89 30,311.83 33,958.86 37,974.66 42,393.73

Liabilities 6 Wages payable

1,294.52

1,368.08 1,553.65

1,717.35

1,894.86

2,141.57

7 Other accounts payable

3,359.59

3,912.25 4,540.39

5,252.95

5,913.86

6,564.90

8 Total current liabilities

4,654.11

5,280.33 6,094.03

6,970.30

7,808.72

8,706.47

Net working capital 9 Working capital (5 – 8) 10 Increase in net working capital

26,167.81

21,721.55 24,217.79 26,988.56 30,165.93 33,687.26 –4,446.26 2,496.24

2,770.77

3,177.37

3,521.33

22-6.

Excel Solution Year

2010

2011

2012

2013

2014

2015

Working capital ($ 000) Assets 1

Accounts receivable

18,493.15

20,796.47 23,333.64

26,126.25

29,197.79

32,573.79

2

Raw materials

1,972.60

2,196.54

2,440.36

2,705.64

2,994.08

3,307.52

3

Finished Goods

4,191.78

4,741.05

5,351.28

6,028.85

6,780.81

7,614.93

4

Minimum cash balance

6,164.38

6,932.16

7,777.88

8,708.75

9,732.60

10,857.93

5

Total current assets

30,821.92

34,666.22 38,903.16

43,569.49

48,705.28

54,354.17

Liabilities 6

Wages payable

1,294.52

1,368.08

1,553.65

1,717.35

1,894.86

2,141.57

7

Other accounts payable

3,359.59

3,912.25

4,540.39

5,252.95

5,913.86

6,564.90

8

Total current liabilities

4,654.11

5,280.33

6,094.03

6,970.30

7,808.72

8,706.47

26,167.81

29,385.89 32,809.13

36,599.19

40,896.56

45,647.70

3,218.08 3,423.23

3,790.07

4,297.36

4,751.14

Net Working Capital 9

Working capital (5-8)

10

Increase in net working capital

22-7.

Excel Solution Year 2010

2011

2012

2013

2014

2015

1 Net income

4,594.56

5,065.37

6,107.01

7,935.86

8,547.31

2 Plus: after-tax interest expense

4,420.00

4,420.00

4,420.00

4,420.00

4,420.00

3 Unlevered net income

9,014.56

9,485.37

10,527.01

12,355.86

12,967.31

4 Plus: depreciation

5,450.00

5,405.00

5,365.00

5,328.00

6,795.00

5 Less: increases in NWC

4,446.26

–2,496.24

–2,770.77

–3,177.37

–3,521.33

6 Less: capital expenditures

–5,000.00

–5,000.00

–5,000.00

–5,000.00 –20,000.00

7 Free cash flow of firm

13,910.82

7,394.14

8,121.24

9,506.49

–3,759.01

0.00

0.00

0.00

0.00

15,000.00

9 Less: after-tax interest expense

–4,420.00

–4,420.00

–4,420.00

–4,420.00

–4,420.00

10 Free cash flow toequity

9,490.82

2,974.14

3,701.24

5,086.49

6,820.99

Free cash flow ($ 000)

8 Plus: net borrowing

22-8.

Excel Solution Year

2010

2011

2012

2013

2014

2015

1 Net Income

4,594.56

5,065.37

6,107.01

7,935.86

8,547.31

2 Plus: after-tax interest expense

4,420.00

4,420.00

4,420.00

4,420.00

4,420.00

3 Unlevered net income

9,014.56

9,485.37

10,527.01

12,355.86

12,967.31

4 Plus: depreciation

5,450.00

5,405.00

5,365.00

5,328.00

6,795.00

5 Less: increases in NWC

–3,218.08

–3,423.23

–3,790.07

–4,297.36

–4,751.14

6 Less: capital expenditures

–5,000.00

–5,000.00

–5,000.00

–5,000.00 –20,000.00

7 Free cash flow of firm

6,246.48

6,467.14

7,101.94

8,386.49

–4,988.83

0.00

0.00

0.00

0.00

15,000.00

9 Less: after-tax interest expense

–4,420.00

–4,420.00

–4,420.00

–4,420.00

–4,420.00

10 Free cash flow toequity

1,826.48

2,047.14

2,681.94

3,966.49

5,591.17

2010

2011

2012

2013

2014

2015

6,164.38

6,932.16

7,777.88

8,708.75

9,732.60

10,857.93

Free cash flow ($ 000)

8 Plus: net borrowing

22-9.

Excel Solution

Balance Sheet ($ 000) Assets 1 Cash and cash equivalents

Year

2

Accounts receivable

18,493.15

13,864.32

15,555.76 17,417.50 19,465.20

21,715.86

3

Inventories

6,164.38

6,205.41

6,978.18

8,776.86

9,819.94

4

Total current assets

30,821.92

27,001.89

30,311.83 33,958.86 37,974.66

42,393.73

5

Property, plant, and equipment

49,500.00

49,050.00

48,645.00 48,280.00 47,952.00

61,157.00

6

Goodwill

72,332.00

72,332.00

72,332.00 72,332.00 72,332.00

72,332.00

7 Total assets Liabilities 8 Accounts payable

152,653.92 148,383.89 151,288.83 154,570.86 158,258.66

175,882.73

7,808.72

8,706.47

9

100,000.00 100,000.00 100,000.00 100,000.00 100,000.00

115,000.00

104,654.11 105,280.33 106,094.03 106,970.30 107,808.72

123,706.47

Debt

10 Total liabilities Stockholders’ equity 11 Starting Stockholders’ equity

4,654.11

5,280.33

6,094.03

7,832.61

6,970.30

48,000.00

43,103.74 45,194.98 47,600.76

50,450.13

4,594.56

5,065.37

7,935.86

8,547.31

–2,974.14 –3,701.24 –5,086.49

–6,820.99

12

Net income

13

Dividends

–2,000.00

–9,490.82

14

Capital contributions

50,000.00

0.00

15

Stockholders’ equity

48,000.00

43,103.74

16

Total Liabilities and equity

0.00

6,107.01

0.00

0.00

0.00

45,194.98 47,600.76 50,450.13

52,176.45

152,654.11 148,384.08 151,289.02 154,571.05 158,258.85

175,882.92

2011

2012

2013

2014

2015

1 Net income

4,594.56

5,065.37

6,107.01

7,935.86

8,547.31

2 Depreciation

5,450.00

5,405.00

5,365.00

5,328.00

6,795.00

4,628.84

–1,691.45

–1,861.74

–2,047.69

–2,250.66

5 Inventory

–41.03

–772.77

–854.42

–944.26

–1,043.08

6 Accounts payable

626.22

813.70

876.26

838.43

897.75

7 Cash from operating activities

15,258.59

8,819.86

9,632.11

11,110.33

12,946.32

8 Capital expenditures 9 Other investment 10 Cash from investing activities

–5,000.00

–5,000.00

–5,000.00

–5,000.00

–20,000.00

–5,000.00

–5,000.00

–5,000.00

–5,000.00

–20,000.00

0.00

0.00

0.00

0.00

15,000.00

–9,490.82

–2,974.14

–3,701.24

–5,086.49

–6,820.99

0.00

0.00

0.00

0.00

0.00

–9,490.82

–2,974.14

–3,701.24

–5,086.49

8,179.01

767.77

845.72

930.87

1,023.85

1,125.33

767.77

845.72

930.87

1,023.85

1,125.33

2010

2011

2012

2013

2014

2015

6,164.38

6,932.16

7,777.88

8,708.75

9,732.60

10,857.93

2 Accounts receivable

18,493.15

20,796.47

23,333.64

26,126.25

29,197.79

32,573.79

3 Inventories

6,164.38

6,937.59

7,791.64

8,734.48

9,774.89

10,922.45

4 Total current assets

30,821.92

34,666.22

38,903.16

43,569.49

48,705.28

54,354.17

5 Property, plant, and equipment

49,500.00

49,050.00

48,645.00

48,280.00

47,952.00

61,157.00

Statement of cash flows

Year

2010

3 Changes in working capital 4 Accounts receivable

11 Net borrowing 12 Dividends 13 Capital contributions 14 Cash from financing activities 15 Net change in cash Check figure: Change in cash from balance sheet

22-10.

Excel Solution

Balance Sheet ($ 000) Assets 1 Cash and Cash equivalents

Year

6 Goodwill

72,332.00

72,332.00

72,332.00

72,332.00

72,332.00

72,332.00

7 Total assets Liabilities

152,653.92

156,048.22

159,880.16

164,181.49

168,989.28

187,843.17

4,654.11

5,280.33

6,094.03

6,970.30

7,808.72

8,706.47

9 Debt

100,000.00

100,000.00

100,000.00

100,000.00

100,000.00

115,000.00

10 Total liabilities

104,654.11

105,280.33

106,094.03

106,970.30

107,808.72

123,706.47

48,000.00

50,768.08

53,786.32

57,211.39

61,180.75

4,594.56

5,065.37

6,107.01

7,935.86

8,547.31

8 Accounts Payable

Stockholders’ equity 11 Starting stockholders’ equity 12 Net income 13 Dividends

–2,000.00

–1,826.48

–2,047.14

–2,681.94

–3,966.49

–5,591.17

14 Capital contributions

50,000.00

0.00

0.00

0.00

0.00

0.00

15 Stockholders’ equity

48,000.00

50,768.08

53,786.32

57,211.39

61,180.75

64,136.89

16 Total liabilities and equity

152,654.11

156,048.42

159,880.35

164,181.68

168,989.47

187,843.36

Year

2011

2012

2013

2014

2015

1 Net income

4,594.56

5,065.37

6,107.01

7,935.86

8,547.31

2 Depreciation

5,450.00

5,405.00

5,365.00

5,328.00

6,795.00

–2,303.32

–2,537.17

–2,792.61

–3,071.54

–3,375.99

5 Inventory

–773.21

–854.04

–942.85

–1,040.40

–1,147.56

6 Accounts payable

626.22

813.70

876.26

838.43

897.75

7 Cash from operating activities

7,594.25

7,892.86

8,612.81

9,990.34

11,716.51

8 Capital expenditures

–5,000.00

–5,000.00

–5,000.00

–5,000.00

–20,000.00

–5,000.00

–5,000.00

–5,000.00

–5,000.00

–20,000.00

Statement of cash flows

2010

3 Changes in working capital 4 Accounts receivable

9 Other investment 10 Cash from Investing Activities 11 Net Borrowing 12 Dividends

0.00

0.00

0.00

0.00

15,000.00

–1,826.48

–2,047.14

–2,681.94

–3,966.49

–5,591.17

0.00

0.00

0.00

0.00

0.00

–1,826.48

–2,047.14

–2,681.94

–3,966.49

9,408.83

767.77

845.72

930.87

1,023.85

1,125.33

767.77

845.72

930.87

1,023.85

1,125.33

Common multiples EV/Sales P/E (levered) P/E (unlevered)

1.842 15.020 18.768

13 Capital contributions 14 Cash from financing activities 15 Change in cash Check figure: Change in cash from balance sheet









22-11.

ru  rf  u E  Rmkt   rf  4%  1.1 5%   9.5%

22-12.

ru  rf  u E  Rmkt   r f  5%  1.2  6%   12.2%

22-13.

Excel Solution Continuation value: multiples approach ($000) 1 EBITDA in 2015 26,744.72 9.1× 2 EBITDA multiple 3 Continuation enterprise value 243,377 4 Debt 115,000 5 Continuation equity value 128,377

22-14.

It does not affect theanswer because theworking capital savings do not affect EBITDA or debt levels.

Continuation value: multiples approach ($000) 1 EBITDA in 2015 26,744.72 Common multiples 9.1× EV/Sales 2 EBITDA multiple 3 Continuation enterprise value 243,377 P/E (levered) 4 Debt 115,000 P/E (unlevered) 5 Continuation equity value 128,377

22-15.

1.842 15.020 18.768

Excel Solution Approximately 5.6%. Note: This growth rate determines all amounts in thelower half of thetable that depend on g.

Enterprise value calculation using EBITDA multiple 1 EBITDA 2015 26,744.72 2 EBITDA multiple 9.1 3 Enterprise Value 243,376.91 4

Implied growth rate, g = 5.6058% (Used solver so PV of FCFs = enterprise value)

Free cash flow calculation (see Eq. 22.16) 5 Net Income 6 Add back interest after tax 7 Unlevered net income2015 × (1+g) 8 –NWC2015 × g 9 –Fixed Assets2015 × g 10 Free cash flow 22-16.

2015 2016 8,547.31 4,420 12,967.31× (1 + g) = 13,694.24 –33,687.26 × g = (1,888.45) –61,157 × g = (3,428.36) 8,377.43

Weighted average cost of capital 1 rU = 10% 2 rD = 6.80% 3 τC = 35% 4 D = 40% Using Eq. 22.11 5 rWACC = 9.0480% DCF Continuation Value PV2015 = FCF2016 / (rWACC–g) 8,377.43 PV2015 = 0.09048 -0.05606 6 PV2015 = 243,376.91 7 PV of FCFs – EV =

0.0000

Excel Solution Approximately 6.05%. Note: This growth rate determines all amounts in thelower half of thetable that depend on g.

Enterprise value calculation using EBITDA multiple 1 EBITDA 2015 26,744.72 2 EBITDA multiple 9.1 3 Enterprise value 243,376.91 4

Implied growth rate, g = 6.0542% (Used solver so PV of FCFs = enterprise value)

Free cash flow calculation (see Eq. 22.16) 5 Net Income 6 Add back interest after tax 7 Unlevered net income2015 × (1+g) 8 –NWC2015 × g 9 –Fixed Assets2015 × g 10 Free cash flow

2015 2016 8,547.31 4,420 12,967.31× (1+g) = 13,752.38 –45,647.70 × g = (2,763.60) –61,157 × g = (3,702.57) 7,286.21

Weighted average cost of capital 1 rU = 10% 2 rD = 6.80% 3 τC = 35% 4 d = 40% Using Eq. 22.11 5 rWACC = 9.0480% DCF continuation value PV2015 = FCF2016 / (rWACC-g) 7,286.21 PV2015 = 0.09048 -0.06054 6 PV2015 = 243,376.91 7 PV of FCFs – EV =

0.0000

22-17.

Excel Solution The equity value is $90 million so theNPV of thedeal is about 89.946 – 53 = $36.946 million.

22-18.

Excel Solution The equity value is $80 million so theNPV of thedeal is 79.918 – 53 = $26.918 million.

22-19.

Excel Solution The value of thesavings of cash flows due tobetter working capital management is thedifference between thevalue with and without thesavings (take valuation results in 22-17 and subtract equivalents from 22-18; you can look at any of APV, Equity, or NPV)—approximately $10.028 million.

Chapter 23 Raising Equity Capital

23-1. Private companies can raise equity capital from angel investors, venture capitalists, institutional investors, or corporate investors.

23-2. The advantage of raising money from a corporate investor is that thelarge corporate partner may provide benefits such as capital, expertise, or access todistribution channels. thecorporate partner may become an important customer or supplier forthe start-up firm, and thewillingness of an established company toinvest may be an important endorsement of thenew company.

The disadvantage is that not all corporate investments are successful. thecorporate partner may gain access toproprietary technology or eventually even become a competitor. Once a young firm has aligned itself with one corporate partner, thecompetitors of this partner may be unwilling todo business.

23-3. a. After thefunding round, thefounder’s 8 million shares will represent 80% ownership of thefirm. tosolve forthe new total number of shares (TOTAL): 8,000,000 = 0.80 × TOTAL So TOTAL = 10,000,000 shares. If thenew total is 10 million shares, and theventure capitalist will end up with 20%, then theventure capitalist must buy 2 million shares.

Given theinvestment of $1 million for2 million shares, theimplied price per share is $0.50. b. After this investment, there will be 10 million shares outstanding, with a price of $0.50 per share, so thepost-money valuation is $5 million.

23-4. Excel Solution a.

IRR solves NPV(Total invested) =

400  80  0 (1  r )10

1/10

 400  So r =    80 

b.

 1  17.46%

Timeline (values in $million): 0

1

2

–100

10

340

$100 million invested Profit = 400 – 100 = $300 million Carried interest = 20%  300 = $60 million LP payoff = 400 – 60 = $340 million 1/10

 340  So IRR =    100 

 1  13.02%

23-5. Excel Solution a.

Before theSeries D funding round, there are 5,000,000 + 1,000,000 + 500,000 = 6,500,000 shares outstanding. Given a Series D funding price of $4.00 per share, thepre-money valuation is (6,500,000) × $4.00/share = $26 million.

b.

After thefunding round, there will be 6,500,000 + 500,000 = 7,000,000 shares outstanding, so thepostmoney valuation is (7,000,000) × $4.00/share = $28 million.

c.

You will own 5,000,000 / 7,000,000 = 71.4% of thefirm after thelast funding round.

a.

12 – 8 = $4 million in Series A, 40 – 25 = $15 million in Series B, 150 – 100 = $50 million in Series C.

b.

Fraction after Series A = 8/12 = 66.7%, after Series B = 25/40 × 66.7% = 41.7%, after Series C = 100/150 × 41.7% = 27.8%.

c.

Series C = 50/150 = 33.3%, Series B = 15/40 × 100/150 = 25%, Series A = 4/12 × 25/40 × 100/150 = 13.9%, and 27.8% held by common (from (b)).

d.

Founders and employees get 27.8% × 500 = $139 million. Series A gets 13.9% × 500/4 = 17.4x.

Series B gets 25% × 500/15 = 8.3x, and Series C gets 33.3% × 500/50 = 3.3x.

23-6.

23-7. Prior round post-money value = 5 + 9 = $14 million. Beru.com’s pre-money value in thenew round must exceed this toavoid a down round. So thepost-money value in a new financing must be at least 6 + 14 = $20 million. So thelargest fraction new investors can receive is 6/20 = 30%.

23-8. Series A liquidation preference = 10 × 1.5 = $15m, ownership = 10/40 × 50/75 = 16.67% Series B liquidation preference = 25 × 3 = $75m, ownership = 25/75 = 33.33%.

Common owns 50%. a.

$75m toB, $10m toA.

b.

$75m toB, $15m toA, $10m tocommon.

c.

$75m toB, A converts and gets $125 × (16.67 / (16.67 + 50)) = $31.25m, common gets $93.75m.

d.

A and B convert, so A gets 16.67% × 300 = $50m, B gets 33.3% × 300 = $100m, and common gets $150m.

23-9. The two main advantages of going public are liquidity and access tocapital. One of themajor disadvantages of an IPO is that once a company becomes a public company, it must satisfy all of therequirements of being a public company, such as filings with thesecurities commissions and listing requirements of thesecurities commissions.

23-10. Underwriters face themost risk from a firm commitment IPO. With this method, they guarantee that they will sell all of theshares at theoffer price. If theentire issue does not sell at theIPO price, theremaining shares must be sold at a lower price and theunderwriter must take theloss.

With a best-efforts IPO, theunderwriter does not guarantee that theshares will be sold, but instead tries tosell theshares forthe best possible price. In an auction IPO, theunderwriters let themarket determine theprice by auctioning off thecompany.

23-11. First, compute thecumulative total number of shares demanded at or above any given price: Price

Cumulative Demand

14.00

100,000

13.80

300,000

13.60

800,000

13.40

1,800,000

13.20

3,000,000

13.00

3,800,000

12.80

4,200,000

The winning price should be $13.40 because investors have placed orders fora total of 1.8 million shares at a price of $13.40 or higher.

23-12. Excel Solution a. With a P/E ratio of 20.0x and 2021 earnings of $7.5 million, thetotal value of thefirm at theIPO should be P  20.0x  P  $150 million .

7.5

There are currently (500,000 + 1,000,000 + 2,000,000) = 3,500,000 shares outstanding (before theIPO). At theIPO, thefirm will issue an additional 6.5 million shares, so there will be 10 million shares outstanding immediately after theIPO. With a total market value of $150 million, each share should be worth $150 / 10 = $15 per share. b. After theIPO, you will own 500,000 of the10 million shares outstanding, or 5% of thefirm.

23-13. Underpricing refers tothe fact that, on average, underwriters pick theIPO issue price so that theaverage first-day return is positive. If you followed a strategy of placing an order fora fixed number of shares on every IPO, your order will be completely filled when theshare price goes down, but you will be rationed when it goes up. In effect, you only get substantial amounts of shares when you do not want them. thewinners’ curse is substantial enough so that thestrategy of investing in every IPO does not yield abovemarket returns.

23-14. The initial return on Margoles Publishing shares is ($19.00 – $14.00) / ($14.00) = 35.7%.

Who gains from theprice increase? Investors who were able tobuy at theIPO price of $14/share see an immediate return of 35.7% on their investment. Owners of theother shares outstanding that were not sold as part of theIPO see thevalue of their shares

increase. tothe extent that theinvestors who were able toobtain shares in theIPO have other relationships with theinvestment banks, theinvestment banks may benefit indirectly from thedeal through their future business with these customers.

Who loses from theprice increase? theoriginal shareholders lose, because they sold their shares for$14.00 per share when themarket was willing topay $19.00 per share.

23-15. The total dollar value of theIPO was ($18.50) × (4 million) = $74 million. thespread equalled (0.07) × ($74 million) = $5.18 million.

23-16. a.

5 million  (20 – 7%  20) = $93 million

b.

15 million  50 = $750 million

c.

Market value of firm assets absent new cash raised = 750 – 93 = $657 million. $657 million / (10 million original shares) = $65.70 per share Check: 93 million / 65.70 = 1.4155 million new shares, $750 / 11.4155 = $65.7.

d.

(65.7 – 50)  10 million = $157 million

a.

0.10(100%) + 0.80(10%) + 0.10(–15%) = 16.5%

b.

Average investment = 0.10(50  15) + 0.80(200  15) + 0.10(1000  15) = $3975

23-17.

Average gain = 0.10(50  15  100%) + 0.80(200  15  10%) + 0.10(1000  15  –15%) = $90 Return = 90 / 3975 = 2.3%

23-18. a.

The company sold 5 million shares at $42.50 per share, so it raised ($42.50) × (5,000,000) = $212.5 million. After underwriting fees, it will keep 212.50 × (1 – 0.05) = $201.875 million.

b.

The venture capitalists raised ($42.50) × (3,000,000) = $127.5 million. After underwriting fees, they will keep 127.5 × (1 – 0.05) = $121.125 million.

So, in total, theSEO was worth $201.875 + $121.125 = ($42.50) × (8,000,000) × 0.95 = $323 million.

23-19. A cash offer is when a company offers thenew shares toinvestors at large. A rights offer is when thenew shares are offered only toexisting shareholders. Rights offers protect existing shareholders from underpricing. However, with a rights offer, only existing shareholders are offered shares topurchase. Demand may be lower, because existing shareholders are only a subset of all possible investors and because they may not want toincrease thepercentage weight of these shares in their portfolios. If demand is lower, firms may receive a lower price from rights offers.

23-20. a.

10m shares / 5  40 = $80 million

b.

12m total shares, Value = $400m + $80m in new capital = $480 million Share price = 480 / 12 = $40

c.

10m  $8 = $80 million

d.

$480 / 20 = $24 per share

e.

Shareholders are thesame either way. In thefirst case, each share is worth $40, and exercising theright has 0 NPV, so thetotal value of a share is $40.

In thesecond case, theshare is worth $24, but theright is worth (24 – 8) = $16, so thetotal value from owning a share is $24 + $16 = $40 per share.

However, thesecond plan is much more likely tobe fully subscribed, because exercising theright is a good deal. In thefirst case, shareholders are indifferent between exercising and not exercising.

Chapter 24 Debt Financing 24-1. In a public debt offering, a prospectus is created with details of theoffering and a formal contract between thebond issuer and thetrust company is signed. thetrust company makes sure theterms of thecontract are enforced. In a private offering, there is no need fora prospectus or a formal contract. Instead a promissory note can be enough. Moreover, thecontract in a private placement does not have tobe standard.

24-2. If a corporation goes bankrupt, then bonds with lower seniority are supposed tobe repaid only after higher seniority bonds have been paid in full. Thus, there is greater default risk forlower-seniority bonds. This results in higher promised yields forlower-seniority bonds.

Note: See theexplanation in Chapter 6 (Section 6.4) that relates thepromised bond yield toa bond’s expected return.

24-3. A secured corporate bond gives thebondholder theright over particular assets that serve as collateral in case of default. An unsecured corporate bond does not offer such protection tothe bondholder. Thus, with an unsecured corporate bond thebondholders are residual claimants in thecase of bankruptcy after thesecured assets have been given tothe corresponding bondholders.

24-4. A foreign bond is a bond issued by a foreign company in a local market. Eurobonds, on theother hand, are bonds denominated in a different currency from that of thecountry in which they are issued.

24-5. The Canadian government uses cash management bills, Treasury Bills, fixed-coupon marketable bonds (otherwise known as Government of Canada bonds), and real return bonds tofinance thefederal debt. Cash management bills are pure discount bonds with maturities from 1 day toless than 3 months. Treasury bills are pure discount bonds with maturities of 3 to12 months. Fixed-coupon marketable bonds (or Government of Canada bonds) are semi-annual coupon bonds with maturities from 2 to40 years. Finally, real return bonds are bonds with coupon payment and face value amounts that adjust with therate of inflation.

24-6. The CPI index appreciated by 300  1.2 .

250

Consequently, theprincipal amount of thebond increased by this amount, that is, theoriginal face value of $1000 increased to$1200.

Since thebond pays semi-annual coupons, thecoupon payment is  0.03  1.2     $1000  $18 .  2 

24-7. The CPI index depreciated by 300  0.75 400

Consequently, theprincipal amount of thebond decreased by this amount, that is, theoriginal face value of $1000 decreased to$750.

Since thebonds pay semi-annual coupons, thecoupon payment is  0.06  0.75     $1000  $22.5 .  2 

However, we assumed that thefinal payment of thematurity (i.e., theprincipal) is protected against deflation. So, since $750 is less than theoriginal face value of $1000, theoriginal amount is repaid (i.e., $1000). If this assumption did not hold, only $750 would be repaid.

24-8. Holders of theCMHC securities face payment risk because homeowners have theoption toprepay their debt whenever they decide todo so. In particular, they will prepay if interest rates fall and they can obtain new debt at a lower interest rate. This is precisely when theholders of CMHC securities would like toavoid payments, since they can reinvest only at a lower interest rate.

24-9. The distinguishing feature is that income from U.S. municipal bonds is not taxed in theU.S. at thefederal level. In Canada, interest income is taxable regardless of its source unless thesecurity is held in a tax-free account (e.g., RRSP, TFSA, RRIF, pension fund).

24-10. Bond issuers benefit from placing restricting covenants because by doing so, they can obtain a lower interest rate.

24-11. Excel Solution and Excel Project Timeline: Time

0

Cash Flows

1

2

9

10

$6

$6

$6

$100 + $6

The present value formula tobe solved is 102 

 6  1 100 1   .

10 YTM  1  YTM   1  YTM 10  

Using theannuity calculator: YTM = 5.73.

YTC: Timeline: Time

0

Cash Flows

1

$100 + $6

The present value formula tobe solved is 102 

106 1  YTC

 YTC 

106  1  3.92% .

102

24-12.

Excel Solution

Timeline: Years

0

Periods

0

Cash Flows

1

2

3

1

2

3

4

5

6

$2.5

$2.5

$2.5

$2.5

$2.5

$100 + $2.5

The present value formula tobe solved is 99 

2.5  1  100 1   .

6 i  1  i   1  i 6  

Using theannuity calculator: i  2.68% .

So since YTM are quoted as APRs, YTM  i  2  2.68%  2  5.36% .

YTC: Timeline: Years

0

Periods

0

Cash Flows

1

2

1

2

3

4

$2.5

$2.5

$2.5

$100 + $2.5

The present value formula tobe solved is 99 

2.5  1  100 1   .

4  4  i  1  i   1  i 

Using theannuity calculator: i  2.77% .

Since YTM (and therefore YTC) are quoted as APRs, YTC  i  2 .  5.54%

24-13. The option toconvert thebond into stock is valuable and hence its price will be higher and therefore its yield lower.

24-14. Excel Solution

The conversion price is theface value of thebond divided by theconversion ratio. In this case, Face value $10, 000  Conversion ratio 450 P  $22.22.

P

Chapter 25 Leasing 25-1. a.

Since thelease is risk-free, thelease rate should be therisk-free rate which is 5% / 12, or 0.416667% per month. Using this todetermine thelease payment, from Eq. 25.1, fora five-year (60-month) lease, PV(Lease payments) = 200,000 – 60,000/(1 + 0.05/12)60 = $153,247.68.

Because thefirst lease payment is paid up front, thePV annuity formula discounts thepayments one too many periods; therefore, we have tofuture value theresult one period:

    1 1 1    1  0.05 12   153, 247.68  L   60  0.05 12  1  0.05 12           1 1 1   .

Note: This is thesame as 153, 247.68  L  1   0.05 12  1  0.05 12 59      Therefore, L = $2879.97. b.

From Eq. 25.2 (see also Example 25.2),

   1 1 1   .

200, 000  M    0.05 12  1  0.05 12 60      Therefore, M = $3774.25.

25-2.

From Eq. 25.1, PV(residual value) = purchase price – PV(lease payments) = $2 million –

 22, 000  1 1    1  0.05 12  84 0.05 12  1  0.05 12    

= $436,974.05.

The future residual value in 84 months is therefore Residual value = $436,974.05 × (1 + 0.05 / 12)84 = $619,644.96.

25-3. a.

From Eq. 25.1, fora five-year (60-month) lease with a monthly interest rate of 6% / 12 = 0.5%, PV(Lease payments) = 400,000 – 150,000 / (1.005)60 = $288,794.17.

Because thefirst lease payment is paid up front, and theremaining 59 payments are paid as an annuity,

 1  1   288, 794.17  L    1.005  . 1  60  0.005  1.005    Therefore, L = $5555.42. b.

In this case, thelessor will receive only $1 at theconclusion of thelease. Therefore, thepresent value of thelease payments should be PV(Lease payments) = 400,000 – 1 / (1.005)60 = $399,999.26:

 1  1   399, 999.26  L    1.005  . 1  60   0.005  1.005   Therefore, L = $7694.63. c.

In this case, thelessor will receive $80,000 at theconclusion of thelease. Thus, PV(Lease payments) = 400,000 – 80,000 / (1.005)60 = $340,690.22.

Because thefirst lease payment is paid up front, and theremaining 59 payments are paid as an annuity,

 1  1   340, 690.22  L    1.005  . 1  60  0.005  1.005    Therefore, L = $6553.73.

25-4.

Excel Solution and Excel Project (See Example 25.4) Capital lease: property added tobalance sheet, lease added todebt

Assets Cash Property, equipment

20

Liabilities Debt

150

255

Equity

125

plant,

Book D / E = 150 / 125 = 1.2.

Operating Lease: no change tobalance sheet. Book D / E = 70 / 125 = 0.56.

25-5. a.

PV(Lease payments) =

 1150  1 1    1  0.09 12   $46, 559.09 48 0.09 12  1  0.09 12    

This is 46,559.09 / 50,000 = 93.1% of thepurchase price. Because it exceeds 90% of thepurchase price, this is a capital lease. b.

c.

The lease term is 75% or more of theeconomic life of theasset (75% × 8 years = 6 years), and so this is a capital lease.  925  1 1    1  0.09 12   $44, 894.57 PV(Lease payments) = 0.09 12  1  0.09 12 60   

This is 44,895 / 50,000 = 89.8% of thepurchase price. Because it is less than 90% of thepurchase price, and theterm is less than six years, and it is a fair market value lease, this is an operating lease.

d.

Without thecancellation option, thePV of thelease payments would exceed 90% of thepurchase price.

With thecancellation option, PV(Minimum Lease payments) =

 1000  1 9000 1    1  0.09 12    $38, 560.00 36 0.09 12  1  0.09 12   1  0.09 12 36   As this is less than 90% of thepurchase price, thelease qualifies as an operating lease.

25-6. a.

FCF0 = –Capital expenditure = –$756,000 FCF1-7 = +Depreciation tax shield = +35% × 756,000 / 7 = +$37,800

b.

FCF0 = –Lease payment = –$130,000 FCF1-6 = –Lease payment + income tax saving = –130,000 + 130,000 × 35% = –$84,500 FCF7 = +Income tax saving = +130,000 × 35% = +$45,500

c.

FCF0 = –130,000 – (–756,000) = +$626,000 FCF1-6 = –84,500 – (37,800) = –$122,300 FCF7 = +45,500 – (37,800) = $7700

25-7.

Excel Solution (See theExcel Worksheet)

25-8.

25-8.

Excel Solution (See theExcel Worksheet)

25-9.

25-9.

Excel Solution (See theExcel Worksheet)

25-10.

Excel Solution (See theExcel Worksheet – Use solver todetermine thelease payment in part (a)) Lessor's Tax rate Borrowing Rate After tax borrowing Rate

40.00% 8.00% 4.80%

a. What is the lease rate for which the lessor will break even?

Because CCA is calculated using a 100% rate, the half-year rule implies half the CapEx may be claimed as CCA in the 1st year and the other half in the 2nd year .

Year 0 1 2 3 4 5 Buy 1 Capital Expenditure (48,000,000.00) 2 CCA tax shield 9,600,000.00 9,600,000.00 3 Free Cash Flow (Buy) (48,000,000.00) 9,600,000.00 9,600,000.00 0.00 0.00 0.00 Lease 4 Lease Payments 10,669,300.80 10,669,300.80 10,669,300.80 10,669,300.80 10,669,300.80 0.00 5 Income Tax (4,267,720.32) (4,267,720.32) (4,267,720.32) (4,267,720.32) (4,267,720.32) 6 Free Cash Flow (Lease) 10,669,300.80 6,401,580.48 6,401,580.48 6,401,580.48 6,401,580.48 (4,267,720.32) Lessor Free Cash Flow 7 Buy and Lease (37,330,699.20) 16,001,580.48 16,001,580.48 6,401,580.48 6,401,580.48 (4,267,720.32) NPV of Lease NPV of Buy

30,098,945.28 NPV buy and lease: (30,098,945.28)

0.00

10.00% 8.00% 7.20%

Lessee's Tax Rate Borrowing Rate After tax borrowing Rate b. What is the gain to Netflix with this lease rate?

Because CCA is calculated using a 100% rate, the half-year rule implies half the CapEx may be claimed as CCA in the 1st year and the other half in the 2nd year .

Year 0 1 2 3 4 5 Buy 1 Capital Expenditure (48,000,000.00) 2 CCA tax shield 2,400,000.00 2,400,000.00 3 Free Cash Flow (Buy) (48,000,000.00) 2,400,000.00 2,400,000.00 0.00 0.00 0.00 Lease 4 Lease Payments (10,669,300.80) (10,669,300.80) (10,669,300.80) (10,669,300.80) (10,669,300.80) 0.00 5 Income Tax Savings 1,066,930.08 1,066,930.08 1,066,930.08 1,066,930.08 1,066,930.08 6 Free Cash Flow (Lease) (10,669,300.80) (9,602,370.72) (9,602,370.72) (9,602,370.72) (9,602,370.72) 1,066,930.08 FCF Lease - FCF Buy NPV of FCF Lease-Buy Decision: c.

37,330,699.20

(12,002,370.72)

(12,002,370.72)

(9,602,370.72)

(9,602,370.72) 1,066,930.08

1,378,147.76 The lease is better

What is the source of the gain in this transaction?

The source of the gain is the differ ence in tax rates. By leasing instead of buying, the government collects less tax r evenue; this benefit accrues to the lessee as the lessor breaks even (which would be the same as if the lessor did nothing).

25-11.

a. NPV(purchase) = (225 + 50) / 1.125 – 225 = $19.44 million NPV(lease) = 35 / 1.125 – 25 = $6.11 million b. There is a 10% chance of default: NPV(purchase) = 90%(225 + 50) / 1.125 – 225 = –$5 million NPV(lease) = 90%(35) / 1.125 – 25 = $3 million c. Indifference: (1 – p)(225 + 50) / 1.125 – 225 = (1 – p)(35) / 1.125 – 25 p = (50 – 35) / (225 + 50 – 35) = 6.25%

Chapter 26 Working Capital Management

26-1. a. A firm’s cash cycle is theaverage length of time from when a firm pays cash forits inventory towhen it receives cash from thesale of that inventory (or theend product that thefirm produced with theinventory). It is calculated as theaverage number of days between thepurchase of theinitial inventory and thesale of theend product plus theaverage number of days it takes thefirm’s customers topay cash forthe inventory they purchase minus theaverage number of days thefirm takes topay its suppliers forthe inventory.

A firm’s operating cycle is theaverage length of time between when a firm purchases its inventory and when thefirm receives cash from thesale of theinventory. It is calculated as theaverage number of days between thepurchase of theinitial inventory and thesale of theend product plus theaverage number of days it takes thefirm’s customers topay forthe inventory they purchase. If a firm were topay cash forits inventory, rather than buying theinventory on credit, thecash cycle and theoperating cycle of thefirm would be identical. In most cases, however, firms buy their inventory on credit, so thecash cycle is shorter than theoperating cycle of thefirm. b. If a firm increases its inventory, its inventory days will increase, all else equal. This will, therefore, increase thecash cycle of thefirm. c. If a firm begins totake discounts offered by its suppliers, its accounts payable days will decrease. All else equal, this will cause thecash cycle of thefirm toincrease.

26-2. No. An increase in a firm’s cash cycle does not necessarily mean that thefirm is managing its cash poorly. theincrease may be due toa conscious management decision. forexample, a firm may decide toincrease its inventory in order toavoid excessive stockouts that it has been experiencing. All else equal, this would result in an increase in its cash cycle. Or, a firm may decide toloosen its credit policy in order toattract customers from its competitors. This would result in an increase in accounts receivable days and, all else equal, result in an increase in thefirm’s cash cycle. And if a firm chooses totake thediscounts offered by its suppliers, theaccounts payable days will decrease, leading toan increase in thefirm’s cash cycle.

26-3. Ignoring revenues and other expenses associated with thenew plant, theNPV of the$2 million investment in net working capital is simply thepresent value of the$2 million that thefirm will recoup at theend of 10 years minus theinitial $2 million investment.

NPV = –$2,000,000 +

$2, 000, 000 = –$883,210 (1.06)10

26-4. Excel Solution and Excel Project a. Net working capital is current assets minus current liabilities. Using this definition, theGreek Connection’s net working capital is $7250 – $3720 = $3530. Some analysts calculate thenet operating working capital instead, which is thenon-interest-earning current assets minus thenon-interest-bearing current liabilities. In this case, thenotes payable would not be included in thecalculation since they are assumed tobe interest bearing. Net operating working capital forThe Greek Connection is $7250 – ($1500 + $1220) = $4530.

b. The cash conversion cycle (CCC) is equal tothe inventory days plus theaccounts receivable days minus theaccounts payable days. theGreek Connection’s cash conversion cycle for2018 was 41.4 days.

CCC =

Inventory Accounts Receivable Accounts Payable   Average Daily COGS Average Daily Sales Average Daily COGS $1300 $3950 $1500   $20, 000 $32, 000 $20, 000 365 365 365  23.7 days  45.1 days  27.4 days  41.4 days

CCC2018 

c. If theGreek Connection accounts receivable days had been 30 days, its cash conversion cycle would have been only 26.3 days: CCC = 23.7 days + 30 days – 27.4 days = 26.3 days.

26-5. In this instance, thecustomer will have theuse of $97 foran additional 25 days (30 – 5) if he chooses not totake thediscount. It will cost him $3 todo so since he must pay $100 forthe goods if he pays after thefive-day discount period. Thus, theinterest rate per period is $3  0.0309  3.09%. $97

The number of 25-day periods in a year is 365 / 25 = 14.6 periods. So, theeffective annual cost of thetrade credit is EAR = (1.0309)14.6 – 1 = 55.94%.

26-6. If you were topay within the10-day discount period, you would pay $99 for$100 worth of goods. If you wait until day 45, you will owe $100. Thus, you are paying $1 in interest fora 35-day (45 – 10) loan. theinterest rate per period is $1  0.0101  1.01%. $99

The number of 35-day periods in a year is 365 / 35 = 10.43 periods. So, theeffective annual cost of thetrade credit is EAR = (1.0101)10.43 – 1 = 11.05%.

26-7. The benefit of outsourcing thebilling and collection tothe other firm is equal towhat Fast Reader can earn on thefunds that are freed up. Since average daily collections are $1200 and float will be reduced by 20 days, Fast Reader will have an additional $24,000 ($1200 × 20). (Think about this as follows. Immediately after hiring thebilling firm, its collection float drops by 20 days, so all collections due within thenext 20 days are immediately available.) thebilling firm charges $250 per month. At an 8% annual rate, themonthly discount rate is 1.0811/12 – 1 = 0.643%, so thepresent value of these charges in perpetuity is 250 / 0.00643 = $38,856. Thus, thecosts ($38,856) exceed thebenefits ($24,000), and Fast Reader should not employ thebilling firm.

26-8. The electronic funds transfer system will free up $50,000 (= 5 × $10,000). (Think about this as follows. Immediately after switching banks its collections due within thenext five days are immediately available.) On theother hand, Saban will have topay a cost because it has tohold $30,000 in a non-interest-earning account, which means it has essentially given up these funds. Because thebenefits ($50,000) are larger than thecosts ($30,000), it should switch banks.

26-9. The three steps involved in establishing a credit policy are (1) Establish credit standards. In this step, thefirm must decide how much credit risk it is willing toaccept.

(2) Establish credit terms. Here, thefirm decides on thelength of time before payment must be made and whether or not it will offer a discount. If a discount is tobe offered, theamount of thediscount and thelength of thediscount period must also be established.

(3) Establish a collection policy. In this step, thefirm must decide how it will handle late payers.

26-10. If we assume all thesales were made on credit, theaverage length of time it takes Manana tocollect on its sales is 12.2 days: Accounts Receivable Days =

Accounts Receivable $2,000,000   12.2 days. $60,000,000 Average Daily Sales 365

26-11. Mighty Power Tool Company Aging Schedule Days Outstanding

Percent of Amount Owed Accounts Receivable

0–15

$ 68,000

19.3%

16–30

$ 75,000

21.2%

31–45

$ 92,000

26.1%

46–60

$ 82,000

23.2%

over 60

$ 36,000 $353,000

10.2% 100.0%

26-12. ―Stretching theaccounts payable‖ refers toa customer’s not paying by thepayment date specified.

26-13. Excel Solution and Excel Project If Simple Simon’s takes thediscount, it must pay $99 in 10 days forevery $100 of purchases. If it elects not totake thediscount, it will owe thefull $100 in 25 days. theinterest rate on theloan is $1  1.01%. $99

The loan period is 15 days (= 25 – 10). theeffective annual cost of thetrade credit is EAR = (1.0101)365/15 – 1 = 27.7%.

Since thebank loan is only 12%, Simple Simon should borrow thefunds from thebank in order totake advantage of thediscount.

26-14. a. Your firm is paying $3 toborrow $97 for25 days (= 40 – 15). theinterest rate per period is $3  0.0309  3.09%. $97

The effective annual rate is (1.0309)365/25 – 1 = 55.9%. b. In this case, your firm is stretching its accounts payable. You are still paying $3 toborrow $97, so theinterest rate per period is 3.09%. However, theloan period is now 35 days (= 50 – 15). theeffective annual rate is reduced to37.4% because your firm has use of themoney fora longer period of time: EAR = (1.0309)365/35 – 1 = 37.4%.

26-15. Excel Solution a. The cash conversion cycle (CCC) is equal tothe inventory days plus theaccounts receivable days minus theaccounts payable days. IMC’s cash conversion cycle for2017 was 35.2 days, and for2018, it was 45.6 days.

CCC =

inventory accounts receivable accounts payable   average daily COGS average daily sales average daily COGS $6200 $2800 $3600   $52, 000 $60, 000 $52, 000 365 365 365  43.5 days  17.0 days  25.3 days  35.2 days

CCC2017 

$6600 $6900 $4600   $61, 000 $75, 000 $61, 000 365 365 365  39.5 days  33.6 days  27.5 days  45.6 days

CCC2018 

IMC’s cash conversion cycle has lengthened in 2018, due toan increase in its accounts receivable days. thenumber of days goods are held in inventory has decreased and IMC is taking longer topay its suppliers, both of which would decrease thecash conversion cycle, all else equal. These changes were not enough tooffset theincrease in theamount of time it is taking IMC’s customers topay forpurchases made on credit. thelengthening of thecash conversion cycle means that IMC will require more cash. b. If IMC’s suppliers are offering terms of net 30, IMC should consider waiting longer topay forits purchases. In 2017, it paid nearly five days earlier than necessary, and in 2018 it paid 2.5 days earlier. IMC could, therefore, have kept themoney working forit longer since there was no discount offered forearly payment. theearly payment may give IMC a preferred position with its suppliers, however, which may have benefits that are not presented here. IMC’s decision on whether toextend its accounts payable days would have totake these benefits into consideration.

26-16. a. The inventory days ratio is equal tothe inventory divided by average daily cost of goods sold. This was 91.25 days forOntario Valley Homecare Suppliers in 2018: inventory days 

$2,000,000  91.25 days $8,000,000 365 .

b. If theaverage days of inventory forthe industry is applied toOVHS, its inventory would be 73/365 times $8,000,000 = $1,600,000.

This means OVHS could reduce its investment in inventory by $400,000 (= $2,000,000 – $1,600,000).

26-17. Commercial paper. Treasury Bills and certificates of deposit are considered tobe free of default risk, so they offer lower returns than a security that has default risk associated with it. Bearer deposit notes have default risk, but therisk of banks is usually considered tobe less than that of other corporations, so theinterest rates on bearer deposit notes are generally lower. Commercial paper exposes theinvestor todefault risk and theinterest earned is fully taxable. It would, therefore, have tooffer theinvestor a higher before-tax return than theother instruments.

Chapter 27 Short-Term Financial Planning

27-1. The professional sports team is likely tohave very high short-term financing needs because of theseasonality of its revenue stream. Several months before theseason actually begins forthe team, it may have large cash-flow requirements—equipment, tickets, advertising, etc.—that it will need tofinance because thebulk of its cash inflows will occur during theseason.

The clothing retailer may have high short-term financing needs because of theseasonality of its business. Sales are often highest in thefall and spring, so theretailer may need shortterm financing in order topurchase inventory prior tothe high seasons. It may need even more short-term financing due toa negative cash-flow shock. forexample, if thestore’s buyer misjudged thefashion trends and overbought a particular style that thestore was unable tosell, thestore might experience a large net cash outflow forthat season. In that case, it may require more short-term financing than normal in order topurchase inventory forthe upcoming season.

27-2. To determine Sailboats’ seasonal working capital needs, we calculate thechanges in net working capital forthe firm: Changes capital

in

working

Month

($ 000) Accounts receivable Inventory Accounts payable Change in net working Capital

1 $1 –$1 0 $0

2 $1 $2 0 $3

3 $1 $1 0 $2

4 $2 0 0 $2

5 $3 –$1 0 $2

6 –$4 –$2 0 –$6

From thetable above, it can be seen that Sailboats’ working capital needs are highest in month 2 because its investments in accounts receivable and in inventory increased themost in that month.

Sailboats, Etc. has a surplus cash position in every month as shown below: Month 1 2 3 4 5 6 $10 $12 $15 $25 $30 $18 2 3 3 4 5 4 0 3 2 2 2 –6

($ 000) Net income Plus depreciation Minus changes in net working capital Cash flow from $12 $12 $16 $27 $33 $28 operations Minus capital expenditures 1 0 0 1 0 0 Change in cash $11 $12 $16 $26 $33 $28 27-3. Permanent working capital is theamount that a firm has tokeep invested in its short-term assets in order tosupport its operations. forexample, a toy retailer may need tokeep a minimum amount invested in inventory at all times. Temporary working capital, on theother hand, is theresult of seasonal fluctuations and/or unanticipated cash flow shocks. thetoy retailer, forexample, may need tohave more invested in inventory just prior toits peak season. Temporary working capital is thedifference between theactual level of investment in short-term assets and thepermanent working capital investment.

27-4. Excel Solution and Excel Project The net working capital foreach quarter is calculated below: ($ 000) Cash Accounts receivable Inventory Accounts payable Net working capital

Quarter 1 2 $100 $100 $200 $100 $200 $500 $100 $100 $400 $600

3 $ 100 $ 100 $ 900 $ 100 $1000

4 $100 $600 $ 50 $100 $650

The minimum level of net working capital—$400,000 in quarter 1—represents thefirm’s permanent working capital. thedifference between thehigher net working capital levels in each quarter and thepermanent working capital needs represents thefirm’s temporary working capital needs. Thus, thefirm has temporary working capital needs of $200,000 in quarter 2, $600,000 in quarter 3, and $250,000 in quarter 4.

27-5. Financing permanent working capital with short-term debt is an aggressive financing policy and is considered risky. Although under a normal term structure, theinterest rates on short-term debt are lower than those on long-term debt, thefirm faces therisk that it will have topay more when it needs torefinance thedebt in thefuture. This increased risk will be reflected in a higher cost of equity forthe firm. Nevertheless, a firm may decide touse short-term debt tofinance permanent working capital if it believes that one or more market imperfections exist. forexample, short-term debt is less sensitive toa firm’s credit quality than is long-term debt and so will be less affected by management’s actions or information. Thus, short-term debt may have lower agency and lemons costs than longterm debt. Management may also believe it has superior knowledge regarding thefuture cash flows forthe firm, which is not yet reflected in thefirm’s credit rating. If management believes its ability toproduce future cash flows will have a positive impact on its credit rating in thefuture, management may elect touse lower-cost, short-term debt tofinance its permanent working capital forthe time being with theexpectation that it will refinance it with long-term debt in thenear future when themarket has recognized thefirm’s improved future prospects and rewarded it with a higher credit rating, which can result in a lower cost of long-term debt tothe firm.

27-6. Excel Solution Alternative A: Pay $10,000 in one month rather than $10,000 × (1% – 2%) = $9800 today.

Alternative B: Pay $10,000 × (1% – 2%) = $9800 today tothe supplier. Borrow $9800 + 5% × $9800 (for compensating balance) + $100 (origination fee) = $10,390 from Bank A today, and repay 10,390 × (1 + 12%/12) = $10,493.90 in one month, using 5% × $9800 = $490 from thecompensating balance account.

Thus, thenet amount owed in one month is $10,493.90 – $490 = $10,003.90.

Alternative C: Pay $9800 today tothe supplier. Borrow $9800 + 1% × $9800 (origination fee) = $9898 from Bank B, and repay $9898 × (1 + 15%/12) = $10,021.73 in one month.

Alternative A (trade credit) is thecheapest source of financing forHand-to-Mouth: paying $10,000 in one month rather than Alternative B paying $10,003.90 or Alternative C paying $10,021.73.

27-7. The loan with the1% loan origination fee would cost themost since theloan origination fee is just another form of interest, so on a $1000 loan, theborrower is paying $90 in interest and will have theuse of only $990 forthe period, making theeffective annual cost of theloan over 9% ($90 / $990 = 9.1%). thecompensating balance requirement of 5% on a $1000 loan reduces theusable proceeds of thefirm by 5% to$950, but theinterest rate is still 8%, so theeffective annual cost of that arrangement is 8.4% theeffective annual rate is not increased by a full percentage.

 $80  8.4%   .  $950 

27-8. The major difference between a revolving line of credit and evergreen credit is thecommitment period. A revolving line of credit is a committed line of credit that involves a solid commitment from thebank fora period that is longer than thetypical oneyear term of a regular line of credit. therevolving line of credit is usually good fortwo or three years before it must be renegotiated. Evergreen credit is a revolving line of credit that has no fixed maturity.

27-9. The effective annual rates of each of thealternatives are calculated as follows: a. Since theAPR is 6%, themonthly rate is 6% / 12 = 0.5%. This translates toan effective annual rate of (1.005)12 – 1 = 6.2%. b. The compensating balance is $1000 × 0.10 = $100. Therefore, theborrower will have use of only $900 of the$1000. theinterest is 0.06 × $1000 = $60. theinterest rate per period is $60 / $900 = 6.7%. Since this alternative assumes annual compounding, theeffective annual rate is 6.7% as well. c. The interest expense is 0.06 × $1000, and theloan origination fee is 0.01 × $1000 = $10. theloan origination fee reduces theusable proceeds of theloan to$990 since it is paid at thebeginning of theloan. theinterest rate per period is $70 / $990 = 7.1%. Since theloan is compounded annually in this case, 7.1% is theeffective annual rate.

Thus, alternative A offers thelowest effective annual cost.

27-10. In this problem, Needy must pay $400 every three months tohave theuse of $10,000.

Thus, theinterest rate per period is $400 / $10,000 = 4%. Since there are four three-month periods in a year, theeffective annual rate is (1.04)4 – 1  17%.

27-11. Treadwater is paying $15,000 (= $1,000,000 – $985,000) touse $985,000 forthree months, so thethree-month interest rate is $15,000 / $985,000 = 1.523%. There are four three-month periods in a year, making theeffective annual rate (1.01523)4 – 1 = 6.2%.

27-12. Magna is paying $26,290 (= $1,000,000 – $973,710) touse $973,710 forsix months. thesix-month interest rate is $26,290 / $973,710 = 2.7%. There are two six-month periods in one year, so theeffective annual rate is (1.027)2 – 1 = 5.5%.

27-13. Direct paper is a method by which a firm sells its commercial paper directly toinvestors.

Dealer paper refers tothe sale of commercial paper through dealers. thedealers get a fee fortheir services, thus reducing theproceeds that theissuing firm receives (and increasing theeffective cost tothe firm).

27-14. Signet’s interest expense on this loan is $129,150 (= $6,000,000 – $5,870,850), and theusable proceeds are $5,870,850. theinterest rate forthe four-month period is $129,150 / $5,870,850 = 2.2%. There are three four-month periods in a year, so theeffective annual rate is (1.022)3 – 1 = 6.7%.

27-15. When accounts receivable are pledged, theborrowing firm is simply using its accounts receivable as collateral fora loan. thelender reviews theinvoices forthe credit sales of theborrower and determines which accounts are acceptable collateral. thelender will then

lend some percentage of thedollar amount of theaccepted invoices. If one or more of theborrowing firm’s customers fail topay, thefirm is still responsible tothe lender forthe money it has borrowed. In a factoring arrangement, theaccounts receivable are sold tothe lender (i.e., factor), and thefirm’s customers typically make their payments directly tothe lender. thelender agrees topay theborrowing firm theamount due from thefirm’s customers at theend of thefirm’s payment period, but thefirm may borrow a certain percentage of theface value of its receivables in order toreceive themoney in advance. thefactoring arrangement may be ―with recourse,‖ which means that if any of theborrowing firm’s customers defaults on its bills, thefactor can require theborrowing firm tomake thepayment. It may also be ―without recourse,‖ in which case thelender bears therisk that one or more customers will default on their bills. theborrowing firm is not responsible forthe payments.

27-16. The monthly interest rate is 9% / 12 = 0.75%, so SteelCo must pay 0.0075 × $5,000,000 = $37,500 in interest on theloan. Combining this with the$5000 warehouser fee makes themonthly cost of theloan $42,500. Since thefee is paid at theend of themonth, SteelCo has use of thefull $5,000,000 forthe month. theinterest rate per period is $42,500 / $5,000,000 = 0.85%. There are 12 months in a year, so theeffective annual rate is (1.0085)12 – 1 = 10.7%.

27-17. The three different methods under which inventory is used as collateral fora loan are floating liens, a trust receipt, and warehouse arrangements.

With a floating lien (also called a general lien or a blanket lien), all of theborrower’s inventory serves as collateral fora loan. thevalue of thecollateral declines as thefirm sells its inventory. In times of financial distress, a firm may decide tosell its inventory without making payments on theloan and may not have enough money toreplenish theinventory it has sold. theloan is then under-collateralized. This is theriskiest arrangement from thelender’s standpoint, and theloan will carry a higher interest rate than if one of theother two methods is used. Additionally, thelender will lend a much smaller percentage of theinventory value under this arrangement.

In a trust receipt (or floor planning) arrangement, specific inventory items are identified as collateral forthe loan. As thespecified inventory is sold, thefirm uses thecash received torepay theloan. thelender will send someone tothe borrower’s premises periodically toensure that none of thespecified inventory has been sold without a repayment made.

In a warehouse arrangement, theinventory serving as collateral is stored in a warehouse.

One type of warehouse is a public warehouse, which is a business that exists forthe sole purpose of tracking theflow of theinventory. theinventory is delivered tothe public warehouse by theborrowing firm, and thelender extends a loan based on thevalue of that inventory. When theborrowing firm needs theinventory tosell, it must return tothe warehouse toretrieve it after receiving permission from thelender todo so. This arrangement is theleast risky from thestandpoint of thelender since it allows thelender tomaintain thetightest control over theinventory, but it is only feasible forsome types of inventory. This method would not be usable forinventory that is subject tospoilage or that is bulky and difficult totransport. In this latter case, a field warehouse might be a good alternative. A field warehouse is established on theborrower’s premises, but it is separated from theborrower’s main plant. It is operated by a third party. This type of arrangement is more convenient forthe borrower, but it still gives thelender theadded security of having theinventory that serves as collateral tracked by a third party.

27-18. Rasputin’s interest expense is 0.10 × $500,000 = $50,000. thewarehouse fee is 0.01 × $500,000 = $5000. Because thewarehouse fee must be paid at thebeginning of theyear, Rasputin’s usable proceeds from theloan are only $495,000 (= $500,000 – $5000). theeffective annual rate is $55,000 / $495,000 = 11.1%.

Chapter 28

Mergers and Acquisitions 28-1.

Either another corporation or group of individuals can acquire thetarget firm, or thetarget firm can merge with another firm.

28-2.

There are many competing theories as towhy mergers cluster in time. They generally fall into two camps: either stock market valuations drive merger activity or industry shocks accompanying economic expansions drive merger activity. It is clear that merger activity is much greater during economic expansions than during contractions and that merger activity strongly correlates with bull markets. Thus, there must be something about economic expansions in general and higher stock market valuations in particular that grease thewheels of themerger process. However, unless you are willing tobelieve that themajority of managers buy other companies simply because they can, without regard toeconomic reasoning, this cannot be thewhole story. There must be real economic impetus tothe activity. Many of thesame technological and economic conditions that lead tobull markets also motivate managers toreshuffle assets through mergers and acquisitions. Thus, it takes a combination of forces usually present only during strong economic expansions todrive peaks in merger activity.

28-3.

Horizontal mergers are more likely tocreate value foracquiring shareholders. Horizontal mergers combine two firms in thesame industry. This provides forgreater potential synergies in eliminating redundant functions within thetwo firms and potentially increases pricing power with both vendors and customers.

28-4.

The acquiring firm has tocompete against other firms, thus reducing thegains it can obtain from thetransaction. Target shareholders benefit from this competition, as they obtain higher bids forthe company.

28-5.

In cases where you are buying a lot of intangible assets, especially human capital, you have tobe particularly worried about how you are going tocreate incentives forthe target’s employees tostay on. Retention bonuses are very common forkey employees in these types of acquisitions. It is also hard tobe successful with a hostile acquisition when retention of target employees is critical. Keeping uncertainty low and moving quickly during theintegration phase are both critical toacquisitions of expertise.

28-6.

The argument can go either way on this. Some of thecritical factors toconsider are as follows: What is thesocial good created by antitrust regulation? Do U.S. and European regulators have a right toregulate firms doing business in theU.S. or Europe, regardless of where those firms are headquartered (such as in Canada)? What would be thealternatives?

28-7.

Carryforward and carryback provisions generally reduce theattractiveness of tax losses as a motivation tomerge. Since thetax loss motivation is based on theability of a larger firm tocapture thetax deduction from thelosses of thetarget, it requires that thetarget itself not be able tocapture thevalue of that deduction. Carryforward and carryback provisions give thetarget more opportunities tocapture thededuction either through recapture of

previously paid taxes or by applying thededuction in thefuture when it returns toprofitability.

28-8.

Yes, diversification is good forshareholders, and they can do it efficiently themselves by purchasing shares in different companies. There is no need formanagers todo this forthem by creating a conglomerate by purchasing other companies at a premium tomarket price.

Given thepremium paid in an acquisition and thediffering preferences of shareholders, it cannot be efficient formanagers todiversify thecompany rather than leaving it toshareholders todiversify their portfolio.

28-9.

Excel Solution and Excel Project TargetCo’s shares are worth $25 and your shares are worth $40. You will have toissue $25 / $40 (= 5/8) shares per share of Target Co tobuy it. That means that in aggregate, you have toissue (5 / 8) × 1 million = 625,000 new shares. After themerger, you will have a total of 1,625,000 shares outstanding (the original 1 million plus the625,000 new shares).

Your total earnings will be $6 million. This comes from the$4 per share × 1 million shares = $4 million you were earning before themerger and the$2 per share × 1 million shares = $2 million that TargetCo was earning. Thus, your new EPS will be $6 million / 1.625 million shares = $3.69. b. A 20% premium means that you will have topay $30 per share tobuy TargetCo (= $25 × 1.20). Thus, you will have toissue $30 / $40 = 0.75 of your shares per share of TargetCo, or a total of 750,000 new shares. With total earnings of $6 million and total shares outstanding after themerger of 1,750,000, you will have EPS of $6 million / 1.75 million shares = $3.43. c. In part (a), thechange in theEPS simply came from combining thetwo companies, one of which was earning $4 per share and theother earning $2 per share. However, you will notice that even though TargetCo has half your EPS, it is trading formore than half your value. That is possible if TargetCo’s earnings are less risky or if they are expected togrow more in thefuture. Thus, although your shareholders end up with lower EPS after thetransaction, they have paid a fair price, exchanging their $4 per share before thetransaction foreither lower, but safer, EPS after thetransaction, or lower EPS that are expected togrow more in thefuture. Either way, focusing on EPS alone cannot tell you whether shareholders are better or worse off. d. If you simply combine thetwo companies and there are no synergies as indicated, then thetotal value of thecompany will be $40 million + $25 million = $65 million. You will have earnings totalling $6 million, so that your P/E ratio is $65 / $6 = 10.83.

Your P/E ratio before themerger was $40 / $4 = 10 and TargetCo’s was $25 / $2 = 12.5. You can see that by buying TargetCo forits market price and creating no synergies, thetransaction simply ends up with a company whose P/E ratio is between theP/E ratios of thetwo companies going into thetransaction. Again, simply focusing on metrics like P/E does not tell you whether you are better or worse off (your P/E went up from 10 to10.83, but your shareholders are no better or worse off). a.

28-10.

Excel Solution TargetCo has $2 in earnings, so if other companies in its industry are trading at 14 times earnings, then a starting point fora valuation of TargetCo in this transaction might be $28 per share, implying a 12% premium ($28 / $25).

28-11.

The CEO will be better off. His portion of the$50 million loss in firm value is 3%, or $1.5 million. If his compensation increases by $5 million, even foronly one year, he will be better off by $3.5 million.

28-12.

The premium is 40%, so thecompensation toThor shareholders must be 1.4 × 40 = $56.

Loki’s shares are worth $50, so it will need tooffer $56 / $50 = 1.12 shares of Loki forevery share of Thor.

28-13.

First, calculate thenumber of shares of LE: Number of shares 

$4, 000, 000, 000  160, 000, 000 . $25

Including synergies, LE will be worth $4 billion + $1 billion = $5 billion, or $31.25 per share (= $5 billion / 160,000,000).

Hence, themaximum exchange ratio that NFF can offer is Exchange ratio 

$31.25  0.893 . $35

Thus, NFF can offer a maximum exchange ratio of 0.893 of its share in exchange foreach share of LE.

28-14.

Excel Solution a.

Since 0.75 million new shares will be issued, theshare price will be (40 + 25) / 1.75 = $37.143.

b.

Same as theprice after themerger, $37.143.

c.

Since TargetCo shareholders will receive 0.75 × 37.143 = $27.86 million, and there are 1 million shareholders, theshare price will be $27.86.

d.

27.86/25 – 1 = 11.43%

a.

Price of XYZ = $3. Premium = 20%.

28-15.

Price of ABC = 20 – premium × 2.5 = 20 – (20% × 2.5) = $19.50. b.

ABC price = price of combined entity = 22.50 / 1.15 = $19.5652.

XYZ price = amount shareholders will receive = 0.15 × $19.5652 = $2.9345 Premium = 2.9345 / 2.5 = 17.4% premium.

c.

28-16.

No, thepremium in thestock offer is lower because market prices change toreflect thefact that ABC shareholders are giving XYZ shareholders money because they are paying a premium. thepart (b) announcement means XYZ stock goes up and ABC stock goes down, which lowers thepremium relative tothe cash offer.

Excel Solution and Excel Project If you trigger thepoison pill, then you own 20% of thecompany, or 400,000 shares (= 20% × 2,000,000 shares). When you trigger thepoison pill, every other shareholder will buy a new share forevery share they hold, so 1,600,000 shares (= 2,000,000 – 400,000) will be issued. These shares will be issued at $10, which is 50% of theprice immediately before triggering thepoison pill (which we assume stays constant at $20). b. After thenew 1,600,000 shares are issued, there will be a total of 3,600,000 shares (= 2,000,000 + 1,600,000). You will own 400,000 of them, so your participation will be 11.11% (= 400,000 / 3,600,000). c. When thepoison pill is triggered, themarket value of thefirm will increase to$56 million [= ($20 × 2,000,000) + ($10 × 1,600,000)]. thenew stock price will be $15.56 (= $56 million / 3,600,000). d. You lose from triggering thepoison pill. (You bought shares at $20 that are now worth $15.56.) Every other shareholder in thetarget firm gains. (They end up with $31.12 (= $15.56 × 2) worth of shares forwhich they only paid $30 (= $20 + $10).) a.

28-17.

Since theacquirer gains thefull amount of thevalue improvement on theshares acquired as a toehold, a toehold provides an incentive toundertake theacquisition, even if theacquirer must pay a price equal tothe with-improvement value forthe rest of theshares.

28-18. a.

The value should reflect theexpected improvement that you will make by replacing themanagement, so thevalue of thecompany will be $40 million plus 40% = $56 million. If you buy 50% of theshares for$25 apiece, you will buy 1 million shares,

paying $25 million. However, you will borrow this money, pledging theshares as collateral, and then assign theloan tothe company once you have control. This means that thenew value of theequity will be $56 million – $25 million in debt = $31 million. With 2 million shares outstanding, theprice of theequity will drop to$15.50. b. Since theprice of theshares will drop from $20 to$15.50 after thetender offer, everyone will want totender their shares for$25. c. Assuming that everyone tenders their shares and you buy them all at $25 apiece, you will pay $50 million toacquire thecompany and it will be worth $56 million. You will own 100% of theequity, which will be $56 million – $50 million loan tobuy theshares = $6 million gain.

Chapter 29 Corporate Governance 29-1.

The corporation allows forthe separation of management and ownership. Thus, those who control theoperations of thecorporation and how its money is spent are not thesame ones who have invested in thecorporation. This creates a clear conflict of interest and this conflict between theinvestors and managers creates theneed forinvestors todevise a system of checks on managers—the system of corporate governance.

29-2.

Examples of agency problems are excessive perquisite consumption (more company jets/company jet travel than needed, nicer office than necessary, etc.). Others are valuedestroying acquisitions that nonetheless increase thepecuniary or non-pecuniary benefits tothe CEO on net.

29-3.

The corporate organizational form allows those who have thecapital tofund an enterprise tobe different from those who have theexpertise tomanage theenterprise. This critical separation allows a wide class of investors toshare therisk of theenterprise. However, as mentioned in theanswer toquestion 1, this separation comes with a cost—the managers will act in their best interests, not thebest interests of theshareholders who own thefirm.

29-4.

The board of directors is theprimary internal control mechanism and thefirst line of defense toprevent fraud, agency conflicts, and mismanagement. theboard is empowered tohire and fire managers, set compensation contracts, approve major investment decisions, etc.

29-5.

Over time, a long-standing CEO can manoeuvre thenomination process so that his or her associates and friends are nominated tothe board. Additionally, board members representing customers, suppliers, or others who have thepotential forbusiness relationships with thefirm will sometimes compromise their fiduciary duty in order tokeep themanagement of thefirm happy. This desire tokeep theCEO happy or reluctance

tochallenge him or her interferes with theboard’s primary function of monitoring themanagement.

29-6.

By knowing a company and its industry as well as possible, security analysts are in a position touncover irregularities. They also participate in earnings calls with theCEO and CFO, asking difficult and probing questions.

29-7.

Lenders are exposed tothe firm as creditors and so are motivated tocarefully monitor thefirm. They often include covenants in their loans that require thecompany tomaintain certain profitability and liquidity levels. Breaking these covenants can be a warning sign of deeper trouble.

29-8.

A whistleblower can be anyone, but is typically an employee, who uncovers outright wrongdoing and ―blows thewhistle,‖ on thefraud by reporting it tothe authorities.

29-9.

The advantages are that, since options increase in value when thefirm’s stock price increases, theCEO’s wealth and incentives will be more closely tied tothe shareholders’ wealth. thedisadvantage is that option grants can increase a CEO’s incentives togame thesystem by timing therelease of information tofit theoption granting schedule or toartificially smooth earnings.

29-10.

No. There are two counter arguments here. First, as Demsetz and Lehn (1985) argue, there is no reason toexpect a simple relationship between ownership and performance.

There are many dimensions tothe corporate governance system, and a one-size-fits-all approach is too simplistic; thecorrect ownership level forone firm may not be thecorrect level foranother. Second, some studies have shown a non-linear relation between firm valuation and ownership—specifically that increasing ownership is good at first but that, in a certain range, managers can use their ownership level topartially block efforts toconstrain them, even though they still own a minority of theshares. In this ―entrenching‖ range, increasing ownership could reduce performance.

29-11.

Proxy contests are simply contested elections fordirectors. In a proxy contest, there are two competing slates of directors rather than just one slate proposed by thecompany. If a board has become captured or unresponsive toshareholder demands, shareholders can put their own slate of new directors up forelection. If thedissident slate wins, then shareholders will have succeeded in placing new directors, presumably not beholden tothe CEO, on theboard.

29-12.

A say-on-pay vote is a non-binding vote whereby theshareholders indicate whether they approve of an executive’s pay package or not.

29-13.

When confronted with a dissident shareholder, a board can ●

ignore theshareholder, which will result in either theshareholder going away or launching a proxy fight, in which case theboard will need toexpend resources in an attempt toconvince shareholders not toside with thedissident, or

●

negotiate with thedissident shareholder tocome toa solution on which theboard and theshareholder can agree.

29-14.

An activist hedge fund is a hedge fund that identifies undervalued companies and takes a significant stake in thecompany toput pressure on management tochange strategy.

29-15.

The government should be trying tomaximize societal welfare. Thus, in designing regulation, it must trade off theeffects of direct and indirect enforcement, compliance, and other costs associated with regulation against theaggregate benefits that accrue toshareholders and theeconomy as a whole.

29-16.

Auditors are very important tocorporate governance. Auditors ensure that thefinancial picture of thefirm presented tooutside investors is clear and accurate. Part of therole of auditors is todetect financial fraud before it threatens theviability of thefirm. SarbanesOxley Act of 2002 included measures designed toreduce conflicts of interest among auditors and toincrease thepenalties forfraud.

29-17.

If employees and directors are permitted tohedge against declines in stock price, it may give them theincentive toact in an overly risky manner (because they face less downside risk) which would not align them with theshareholders of thecompany.

29-18.

Trading is how prices come toreflect all material information about a company’s prospects. By restricting a set of investors from trading, we decrease theefficiency of theprices because it will take longer forthe prices toreflect that private information. We rely on efficient prices tomake sure that capital is allocated toits best use. While thereduction in market efficiency is a cost of prohibiting insider trading, there is also a benefit. In order fora capital market tofulfill its function, uninformed investors must be willing toinvest their money—providing liquidity and lowering thecost of capital. If investors thought that thestock market was just a fools’ game where they lost toinsiders, they would be unwilling toinvest or would price their expected loss into their required return. This increases thecost of capital forcompanies and slows economic growth.

29-19.

The laws are much stricter formerger-related trading. Anyone who has information about a pending merger is restricted from trading. Non-merger restrictions depend on thesource of thematerial non-public information. If thesource violated a fiduciary duty tothe shareholders, then thetrading is prohibited.

29-20.

They are better protected in theU.S. While thelegal systems in both theU.S. and Canada are based on British common law, which offers considerably more protection tominority shareholders than does civil law, theU.S. regulatory authorities are known tobe more active in their enforcement of shareholder rights.

29-21.

Because pyramidal structures allow a controlling family tocontrol firms in which they have little actual cash-flow rights, thefamily can use their control tomove profits away from firms where they get a small percentage of thecash flows tofirms in which they can claim a larger fraction of thecash flows. forexample, they can have one firm sell toanother at a reduced price.

Chapter 30 Risk Management 30-1.

From theSML, therequired return fora beta of –0.25 is rL = 5% – 0.25(10% – 5%) = 3.75%. From Eq. 30.1:

Premium 

3%  $65 million  $1.88 million .

1.0375

30-2. a.

From theSML, therequired return fora beta of –0.5 is rL = 5% – 0.5(10% – 5%) = 2.5%. From Eq. 30.1:

Premium  b.

2%  $450 million  $8.78 million .

1.025

With 15% overhead costs, theinsurance premium will be $8.78 × 1.15 = $10.098 million. Buying insurance is positive NPV forGenentech if it experiences distress or issuance costs equal to15% of theamount of theloss. That is, it must experience distress or issuance costs of 15% × $450 million = $67.5 million in theevent of a loss. In that case, NPV(buy insurance) = –10.098 +

2%  $(450  67.5) million = 0.

1.025

30-3. a.

From theSML, therequired return fora beta of –1.5 is rL = 5% – 1.5(10% – 5%) = – 2.5%. From Eq. 30.1:

Premium 

10%  $500,000  $51, 282.

1  0.025

b.

If we consider after-tax cash flows, NPV = –51,282 × (1 – 0.40) +

10%  $500,000  (1  0.10)  $15,385.

1  0.025

The gain arises because thefirm pays forthe insurance when its tax rate is high but receives theinsurance payment when its tax rate is low.

30-4. a.

New policies reduce thechance of loss by 9% – 4% = 5%, foran expected savings of 5% × $10 million = $500,000. Therefore, theNPV is NPV = –100,000 + 500,000 / 1.05 = $376,190.

b.

If thefirm is fully insured, then it will not experience a loss. Thus, there is no benefit tothe firm from thenew policies. Therefore, NPV = –100,000.

c.

If thefirm is fully insured, it will not have an incentive toimplement thenew safety policies. Therefore, theinsurance company will expect a 9% chance of loss. Therefore, theactuarially fair premium would be Premium = 9% × $10 million / 1.05 = $857,143.

d.

If theinsurance policy has a deductible, then thefirm will benefit from thenew policies because it will avoid a loss, and therefore avoid paying thedeductible 5% of thetime. Let D be theamount of thedeductible. Then theNPV of thenew policies is NPV = –100,000 + 5%(D) / 1.05.

Setting theNPV to0 and solving forD, we get D = $2.1 million.

e.

With a deductible of $2.1 million, theinsurance company can expect thefirm toimplement thenew policies. Therefore, it can expect a 4% chance of loss. In theevent of a loss, theinsurance will pay (10 – 2.1) = $7.9 million. Therefore, Premium = 4% × $7.9 million / 1.05 = $300,952.

Note: With this policy, thefirm will pay $300,952 forinsurance, $100,000 toimplement thenew policies, and 4% × $2.1 million = $84,000 in expected deductibles. Thus, thefirm will pay 300,952 + 100,000 + 84,000 = $484,952 in total, which is much less than theamount it would pay forfull insurance in (c).

30-5. a.

Operating profit = 2 billion pounds × (price per pound – $0.90/lb).

Price ($/lb) Operating profit ($ billion)

b.

1.25 0.70

1.50 1.20

1.75 1.70

In this case, BHP will sell forthe contract price of $1.45/lb, no matter what thespot price of copper is next year.

That is, Operating profit = 2 × (1.45 – 0.90) = $1.10 billion.

c.

In this case, Operating profit = 1 × (1.45 – 0.90) + 1 × (Price – 0.90). Therefore, Contract price ($/lb) Contract amount S pot price ($/lb) Operating profit ($ billion)

d.

1.45 1.00 billion pounds 1.25 1.50 0.90 1.15

1.75 1.40

Strategy (a) could be optimal if thefirm is sufficiently profitable that it will not be distressed even if thecopper price next year is low. Equity holders will, in this case, bear therisk of copper price fluctuations, and there is no gain from hedging therisk. It could also be optimal if thefirm is currently in or near financial distress. Then by not hedging, thefirm increases its risk. Equity holders can benefit if theprice of copper is high, but debt holders suffer if theprice is low. (Recall thediscussion in Chapter 16 regarding equity holders’ incentive toincrease risk when thefirm is in or near financial distress.)

Strategy (b) could be optimal if thefirm is not in distress now but would be if theprice of copper next year is low and thefirm does not hedge. Then, by locking in theprice it will receive at $1.45/lb, thefirm can avoid financial distress costs next year.

30-6.

Strategy (c) could be optimal if thefirm would risk distress with operating profits of $0.7 billion from copper but would not with operating profits of $0.9 billion. In that case, thefirm can partially hedge and avoid any risk of financial distress.

Excel Solution a.

You have gone long 100 × 1000 = 100,000 barrels of oil. Therefore, themark-to-market profit or loss will equal 100,000 times thechange in thefutures price each day:

b.

Summing thedaily profit/loss amounts, thetotal is a gain of $250,000. This gain offsets your increase in cost from theoverall $2.50 increase in oil prices over the10 days, which increases your total cost of oil by 100,000 × $2.50 = $250,000.

c.

After thesecond day, you have lost a total of $250,000. This loss could be a problem if you do not have sufficient resources tocover theloss. In that case, your position would have been liquidated on day 2, and you would have been stuck with theloss and had topay thehigher cost of oil on day 10.

30-7.

If theprice of coffee goes up by $0.01 per pound, Starbucks’ cost of coffee will go up by $0.01 × 100 million = $1 million. But because it can charge higher prices, its revenues will go up by 60% × $1 million = $0.6 million. tohedge this risk, Starbucks should lock in theprice for40 million pounds of coffee, so that it will suffer an increase in cost only forthe remaining 60 million pounds of coffee.

30-8.

Hans, as a producer of barley, is concerned that when he sells it, theprice may have dropped. He can consider various hedging strategies but must also consider thecost of hedging and his production costs. a.

Hans can take a short position in futures on 3000 tonnes of barley fordelivery in October.

b.

i.

3000 tonnes  $200/tonne = $600,000

ii.

3000 tonnes  $200/tonne = $600,000

iii. 3000 tonnes  $200/tonne = $600,000 c.

Hans can purchase theput options on 3000 tonnes of barley.

d.

now i. He already paid forthe put options an amount of $15/tonne  3000 tonnes = $45,000. With the market price of barley below theput’s strike price, he will exercise theput toreceive $200/tonne  3000 tonnes = $600,000. His total net amount received is thus $600,000 – $45,000 = $555,000.

ii.

He already paid forthe put options an amount of $15/tonne  3000 tonnes = $45,000. With the market price of barley equal theput’s strike price, he can let theputs expire and sell at themarket price, or he can exercise theputs and sell at thestrike price (it does not matter, both give thesame result) toreceive $200/tonne  3000 tonnes = $600,000. His total net amount received is thus $600,000 – $45,000 = $555,000.

iii. He already paid forthe put options an amount of $15/tonne  3000 tonnes = $45,000. With the market price of barley above theput’s strike price, he will let theputs expire and sell thebarley at themarket price of $250 per tonne toreceive $250/tonne  3000 tonnes = $750,000. His total net amount received is thus $750,000 – $45,000 = $705,000 e.

i.

Since thefutures contracts would be marked-to-market each day, Hans may have a margin call if his account balance falls too low. If he won’t have funds available, he should avoid thepotential liquidity problem that might arise from using thefutures hedge.

ii. Given thepossibility he will not have a crop tosell, theoptions hedge may be preferable as with theoptions contracts, he is not obliged tosell thebarley, but with thefutures contract he is obliged tosell. Thus, thefutures may result in a large loss if he is forced tooffset his position at a higher futures price. iii. With thefutures hedge, Hans will be guaranteed toreceive $600,000 (as long as his crop can be sold) so he will cover his costs with certainty and will not bear any financial distress costs. With theoptions hedge, if barley prices are below $205/tonne, he will receive less than $570,000 in net revenues (when barley is below $200/tonne, he will receive only net revenues of $555,000) so he will have a shortfall, go bankrupt, and bear significant personal financial distress costs. Thus, in this case, thefutures hedge is preferred. iv. With thefutures hedge, Hans will lock in a certain loss, go bankrupt, and bear thefinancial distress costs with certainty. His revenues minus costs will be $600,000 – $700,000 = –$100,000 with certainty. Thus, he should not use thefutures hedge. With theoptions hedge, Hans will have a loss forany barley price below $248.33/tonne but he will be solvent if barley prices are higher than that. theoptions hedge is better than thefutures hedge as it gives him a chance of solvency whereas thefutures give him certain failure. However, thebest strategy forHans in this situation is tonot hedge at all. With costs of $700,000, he will break even with a barley price of $700,000 / 3000 tonnes = $233.33/tonne. Note that, without hedging, thebarley price can be $15 less than required with theput options hedge ($15 is thecost of theput option), so there is more possibility that thefinal barley price will allow Hans toremain solvent. Your advice in this scenario is not tohedge.

Depending on thecircumstances, theoptions hedge may be better, thefutures hedge may be better, or not hedging at all may be better. It is important toconsider all factors and outcomes before deciding what, if any, hedging strategy should be used.

30-9.

Plan: Alberta Oil Refinery (AOR) will need topurchase oil in thefuture, and management is concerned that oil prices might rise. AOR’s management can hedge, but they must consider which hedging strategy is best or whether hedging is necessary at all. a.

AOR can hedge by taking a long position in futures for500,000 barrels of oil forSeptember delivery.

b.

i.

500,000 barrels  $110/barrel = $55 million

ii.

500,000 barrels  $110/barrel = $55 million

iii. 500,000 barrels  $110/barrel = $55 million

c.

AOR can purchase thecall options on 500,000 barrels of oil.

d.

i.

To purchase thecall options, AOR must pay $20/barrel  500,000 barrels = $10 million. If oil is $60 per barrel, AOR will let thecalls expire worthless and purchase theoil at themarket price, costing $60/barrel  500,000 barrels = $30 million. thenet amount paid is thus $40 million.

ii.

To purchase thecall options, AOR must pay $20/barrel  500,000 barrels = $10 million. If oil is $110 per barrel, which is thesame as thecall’s strike price, AOR can purchase on themarket or exercise thecalls (it does not matter, either way thecost is thesame), costing $110/barrel  500,000 barrels = $55 million. thenet amount paid is thus $65 million.

iii. To purchase thecall options, AOR must pay $20/barrel  500,000 barrels = $10 million. If oil is $160 per barrel, AOR will exercise thecalls and pay $110/barrel  500,000 barrels = $55 million. thenet amount paid is thus $65 million. e.

i.

Without much cash available between April and August, AOR should avoid thefutures hedge as it generates a liquidity risk due tothe marking tomarket and potential margin calls.

ii.

In this case, there is thepossibility that AOR will not need all 500,000 barrels of oil. theoptions hedge is preferred as it does not commit AOR topurchase theoil. With thefutures contract, if AOR ends up purchasing only half of theoil at a lower price, this offsets only half of thelosses from theshort futures position, so AOR will have additional losses from thefutures on the250,000 barrels of oil it did not need topurchase.

iii. With net revenues of $60 million, thefutures guarantee a cost of only $55 million, so financial distress is avoided with certainty. However, with theoptions hedge, if oil is above $100 per barrel AOR’s costs will be above $60 million (and will rise toas much as $65 million foroil prices above theoptions strike price of $110). Thus, theoptions hedge still leaves a significant chance of financial distress and associated costs. thefutures hedge is preferred in this case. iv. If AOR will have net revenues of $50 million, thefutures hedge guarantees a certain loss and should be avoided. theoptions hedge allows fora profit as long as theoil price is below $80 per barrel, so it is better than thefutures hedge. However, not hedging is even better. With no hedge, as long as oil prices are below $100 per barrel, AOR will be profitable and avoid financial distress costs. Note that theoptions hedge requires oil tobe lower because there is also the$20 cost per call option on a barrel of oil. v.

In this case, AOR has a natural hedge and should not use either of thefutures or options hedges.

Depending on thecircumstances, theoptions hedge may be better, thefutures hedge may be better, or not hedging at all may be better. It is important toconsider all factors and outcomes before deciding what, if any, hedging strategy should be used.

30-10.

Excel Solution a.

You could lock in an exchange rate of 3.1712 zloty per dollar in three months time through a forward contract with thebank. (Note that when converting zloty todollars, you pay thehigher rate.) In order toreceive $100,000, you would therefore need towrite thecontract for100,000 × 3.1712 = 317,120 zloty.

b.

The forward rates show that fewer zloty per dollar are needed forlonger maturities. From Eq. 30.3, in terms of zloty per dollar,

FT  S 

1  rz T

1  r$ 

T

.

Thus, thezloty (PLN) interest rate is below thedollar (USD) interest rate.

Excel Solution a.

Unhedged profit = (100,000 EUR) × (S1 CAD/EUR) – 110,000. See figure below.

b.

Forward Hedged profit = (100,000 EUR) × (1.55 CAD/EUR) – 110,000 = 45,000 CAD. See figure below.

c.

You want tosell EUR in exchange forCAD. Thus, buying put option will protect theprice at which you can sell EUR.

d.

Buying put options for100,000 EUR has cost of 100,000 × 0.10 CAD = 10,000 CAD. The put allows you tosell theEUR fora minimum of 1.55 CAD/EUR. Therefore, All-in option hedged profit = (100,000 EUR) × (max[1.55,S1] CAD/EUR) – 110,000 – 10,000 = max[5000 CAD, (100,000 EUR) × (S1 CAD/EUR) – 120,000].

See figure below.

Profit from King Crab Sale

Unhedged Profits

Profit

30-11.

Forward Hedge Put option hedge

CAD/EUR

e.

For theforward hedge, you receive a payoff from theforward if thevalue of theEUR declines. If theEUR appreciates, you have a loss on your forward position. theprofit is (100,000 EUR) × (1.55 – S1 CAD/EUR).

If theEUR appreciates significantly, theloss from theforward hedge can be very large. (See figure below.)

For theoption hedge, you receive a payoff from theput if thevalue of theEUR declines. If theEUR appreciates, theput is worthless (and you are out theoriginal purchase price of theputs). theprofit is max[0, (100,000 EUR) × (1.55 – S1 CAD/EUR)] – 10,000.

With theputs, themaximum loss is their initial cost of $10,000 CAD. (See figure below.)

Put Option

From thepicture, an advantage of hedging with options is thelimited downside risk in theevent of cancellation.

30-12.

The inputs are S = spot exchange rate = 1.80, K = strike price = 1.80, T = 0.5, r$ = 5.25%, r£ = 4.0%,  = volatility = 10%. From Eq. 30.3, FT = S(1 + r$)T / (1 + r£)T = 1.80(1.0525)0.5 / (1.04)0.5 = $1.8108/£.

Therefore, from Eq. 30.5, d1 

ln(1.8108 / 1.80) 10% 0.5



10% 0.5  0.120 , and d2  d1  10% 0.5  0.049 .

2

And so N(d1) = 0.548 and N(d2) = 0.520.

From Eq. 30.4, C = (1.80 / (1.04)0.5) × (0.548) – (1.80 / (1.0525)0.5) × (0.520) = $0.0549.

Thus, thecall option price is 0.0549 USD/GBP. 30-13.

The duration of a security is equal tothe weighted-average maturity of its cash flows (Eq. 30.6). Thus, theduration of a five-year zero-coupon bond is five years, and theduration of thenine-year zero-coupon bond is nine years (see Eq. 30.11).

We cannot determine thedurations of theannuities exactly without knowing thecurrent interest rate. But because thecash flows of an annuity are equal and at regular intervals, theduration of an annuity must be less than its average maturity (because weighting by present values will put less weight on later cash flows). Thus, thefive-year annuity has a duration of less than (1 + 2 + 3 + 4 + 5) / 5 = 3 years, and thenineyear annuity has a duration of less than (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9) / 9 = 5 years. theranking is therefore five-year annuity, nine-year annuity, five-year zero, nine-year zero.

30-14.

Excel Solution and Excel Project a.

From Eq. 30.8, 50 100 150 (0 yrs)  (2 yrs)  (7 yrs)  4.17 yrs .

300 300 300

Asset duration =

Liability duration =

80 100 100 (0 yrs)  (2 yrs)  (10 yrs)  4.29 yrs .

280 280 280

From Eq. 30.9,

b.

Equity Duration =

300 280 (4.17 yrs)  (4.29 yrs)  2.49 yrs .

20 20

Asset Duration =

100 100 100 (0 yrs)  (2 yrs)  (7 yrs)  3.0 yrs .

300 300 300

Equity Duration =

300 280 (3.0 yrs)  (4.29 yrs)  15.0 yrs .

20 20

Therefore, if interest rates drop by 1%, we would expect thevalue of Northland’s equity todrop by about 15% (or more precisely from Eq. 30.7, 15 × 0.01 / 1.04 = 0.1442 or 14.4%). c.

Northland would like toincrease theduration of its assets, so it should use cash tobuy long-term bonds.

Because 10-year STRIPS (zero-coupon bonds) have a 10-year duration, we can use Eq. 30.10: change in equity duration

Amount 

(15yrs)  10yrs

equity value

20

 30 .

duration of STRIPS (vs. cash)

That is, we should buy $30 million worth of 10-year STRIPS. 30-15.

Excel Solution a.

The duration of theassets is 13.5 years. toestimate theeffect of a parallel interest rate of 0.5%, we use theduration formula:  0.50% %change  Duration   13.5   6.40% , 1 r 1.055 or a drop in value of 6.40% × $44.8 million = $2.87 million.

Similarly, forthe liabilities:

%change   Duration 

 0.50%  4.0   1.90% , 1 r 1.055

or a drop in value of 1.90% × $39.2 million = $0.74 million.

As a result, thevalue of equity will decline by about 2.87 – 0.74 = $2.13 million (a loss of 38% of its value!). b.

From Eq. 30.9: Equity duration =

44.8 39.2 (13.5yrs)  (4.0yrs)  80yrs .

5.6 5.6

This explains theextreme sensitivity of theequity value tochanges in interest rates.

c.

Liquidating a portion of theassets and investing in T-Bills and notes will reduce theduration of these assets by 13.5 – 2 = 11.5 years. From Eq. 30.10: change in equity duration

Amount 

equity value

(80yrs)  11.5yrs

5.6

 $38.96 million .

change in asset duration

That is, we should liquidate $38.96 million of thefund’s assets. d.

We can also reduce theduration of thefund by entering into a swap contract in which Citrix will receive a floating rate and pay a fixed rate. This swap will increase in value when interest rates rise, offsetting thedecline in thevalue of therest of thefund. todetermine thesize of theswap, we proceed as in thetext’s Ex. 30.15: change in equity duration

Amount 

equity value

(80yrs)  5.6 (7.0  0.5)yrs

 $68.92 million .

difference in duration of fixed and floating rate 10-yr bond

That is, we should enter a swap with a notional value of $68.92 million.

30-16. a.

Borrow $100 million short-term, paying LIBOR + 1.0%. Then enter a $100 million notional swap toreceive LIBOR and pay 8.0% fixed. Effective borrowing rate is (LIBOR + 1.0%) – LIBOR + 8.0% = 9.0%.

(Note: Borrowing long-term would have cost 7.6% + 2.5% = 10.1%.)

b.

Refinance $100 million short-term loan with long-term loan at 9.10% + 0.50% = 9.60%. Unwind swap by entering new swap topay LIBOR and receive 9.50%. Effective borrowing cost is now 9.60% + (–LIBOR + 8.0%) + (LIBOR – 9.50%) = 8.10% (Note: This rate is equal tothe original long-term rate, less the2% decline in thefirm’s credit spread. thefirm gets thebenefit of its improved credit quality without being exposed tothe increase in interest rates that occurred.)

Chapter 31 International Corporate Finance 31-1.

a. 5 million EUR / (1.07)  1.60  7.48 million CAD. b. (5 million EUR  1.5551) / 1.04  7.48 million CAD. c. Yes, themarkets are internationally integrated because theanswers to(a) and (b) are identical.

31-2.

a.

4 million PLN  2.04 PLN CAD  1.78 million CAD .

1.10

b.

4 million PLN 1   1.84 million CAD .

2.055 PLN CAD 1.06

c. No, themarkets are not internationally integrated because theanswers to(a) and (b) are not thesame.

31-3. Excel Solution First, calculate theforward rates: F1  (1.5 USD EUR)

(1.04)  1.4717 USD EUR (1.06)

F2  (1.5 USD EUR)

F3  (1.5 USD EUR) F4  (1.5 USD EUR)

(1.04)2 (1.06)2

(1.04)3 (1.06)3 (1.04)4 (1.06)4

 1.4439 USD EUR

 1.4167 USD EUR

 1.3900 USD EUR

Next, convert EUR cash flows into USD: Year 0 1 2 3 4

EUR Cash Flow –15 9 10 11 12

Exchange Rate 1.5000 1.4717 1.4439 1.4167 1.3900

USD Cash Flow (22.500) 13.245 14.439 15.584 16.679

Finally, thenet present value is NPV  22.5 

13.245 14.439 15.584 16.679     26.21 million USD. 1.085 1.0852 1.0853 1.0854

Etemadi Amalgamated should undertake theproject because thenet present value is positive.

31-4. Excel Solution With the26% drop in thespot rate, theforward rates need tobe recalculated: F1  (1.11USD EUR)

(1.04)  1.0891USD EUR (1.06)

F2  (1.11USD EUR)

F3  (1.11USD EUR) F4  (1.11USD EUR)

(1.04)2 (1.06)2

(1.04)3 (1.06)3 (1.04)4 (1.06)4

 1.0685 USD EUR

 1.0483USD EUR

 1.0286 USD EUR

Next, EUR cash flows are reconverted into USD: Year 0

EUR Flow –15

1 2 3 4

9 10 11 12

Cash Exchange Rate

USD Cash Flow – 16.650 9.802 10.685 11.532 12.343

1.1100 1.0891 1.0685 1.0483 1.0286

Finally, thenet present value is NPV  16.65 

9.802 10.685 11.532 12.343     19.395 million USD. 1.085 1.0852 1.0853 1.0854

Etemadi Amalgamated should still undertake theproject because thenet present value is positive. Note that this is 26% lower than theanswer in 31-3, which is consistent with the26% drop in thespot exchange rate. * 31-5. The Law of One Price tells us 1  rEUR 

S * * As a result, we have rEUR  1  rCAD  F

31-6.

Using

theformula

* have 1  rJPY 

forthe





S * 1  rCAD .

F 1.6100 1   1  0.08  1  10.0019% .

1.5807

Internationalization

of

theCost

of

Capital,

1  rJPY * (1  rCAD ).

1  rCAD

As a result, we obtain * rJPY 

1  rJPY 1  0.01 * (1  rCAD ) 1   1  0.11  1  6.7714% .

1  rCAD 1  0.05

31-7. The after-tax cost of debt in dollars is 0.075 × (1 – 0.30) = 0.0525, or 5.25%.

we

Using theformula forthe Internationalization of theCost of Capital, we have * 1  rJPY 

1  rJPY * (1  rCAD ).

1  rCAD

As a result, we obtain * rJPY 

1  rJPY 1  0.01 * (1  rCAD ) 1   1  0.0525  1  1.2405% .

1  rCAD 1  0.05

31-8. Excel Solution a. Using theformula forthe Internationalization of theCost of Capital, we have * 1  rEUR 

1  rEUR * (1  rCAD ).

1  rCAD

As a result, we obtain * rEUR 

1  rEUR 1  0.07 * (1  rCAD ) 1   1  0.095   1  12.1196% .

1  rCAD 1  0.045

b. NPV  25 

12 14 15 15     16.974491 million EUR. 1.121196 1.1211962 1.1211963 1.1211964

31-9. With earnings of 100 million WSP and theWadankan tax rate of 25%, thetax paid in Wadanka is 25 million WSP. With an exchange rate of 0.125 CAD/WSP, theearnings amount to12.5 million CAD and theWadankan taxes amount to3.125 million CAD. With a tax rate of 45%, theCanadian tax on FAM’s Wadankan income would be 0.45  12.5  5.625 million CAD. However, FAM is able toclaim a tax credit of 3.125 million CAD, fora net tax Canadian liability of 5.625 – 3.125 = 2.5 million CAD. 31-10. a. From question 31-9, thetax liability is 2.5 million CAD. Deferred for10 years, using theafter-tax cost of debt at 5%, thepresent value is 2.5 / 1.0510  1.53 million CAD. Hence, thevalue of deferral is 2.5 – 1.53 = 0.97 million CAD. b. The earnings will need tobe converted at thefuture exchange rate, S10, although thetax credit will still be calculated using S1 = 0.125 CAD/WSP. Hence, theCanadian tax liability will be ((0.45)(S10)(100) − 3.125) million CAD. 31-11. a. The net Canadian tax liability, after claiming thecredit fortaxes paid in Poland, is (0.35)(80)  16  12 million CAD. b. The net Canadian tax liability, after claiming thecredit fortaxes paid in Sweden, is (0.35)(100)  60  25 million CAD. However, theuse of thetax credit is limited tothe Canadian tax liability, so theliability is actually zero. This is an excess tax credit of $25 million that is lost.

c. Pooling thePolish and Swedish subsidiaries, thenet Canadian tax liability is (0.35)(180)  76  13 million CAD, but as in (b) theliability won’t be negative; it will actually be zero. By pooling, Qu’Appelle Enterprises is able touse 12 million CAD of the25 million CAD excess tax credit from earnings in Sweden tooffset the12 million CAD net tax liability from earnings in Poland, leaving a net Canadian tax liability of 0 CAD. 31-12. From Eq. 30.3 (covered interest parity), theforward and spot RUB/CAD exchange rates satisfy F S

1  rRUB , 1  rCAD

where rRUB and rCAD are risk-free interest rates in RUB and CAD respectively. With this equation, we can use thespot and forward exchange rates, and therisk-free CAD interest rate, tosolve forthe risk-free RUB interest rate: 28.5  28 

1  rRUB 1.045  1  6.3661% . which implies rRUB  28.5  1.045 28

Therefore, theimplied risk-free ruble interest rate is 6.3661%, implying that Russian government bonds have an implied credit spread of 7.5% – 6.3661% = 1.1339% tocompensate investors forthe possibility of theRussian government defaulting.

(See example 31.3 fora similar problem. Note also that an investor can obtain a risk-free investment in RUB by exchanging RUB forCAD at thespot rate of 28 RUB/CAD, investing in CAD treasury bills at 4.5%, and locking in a forward exchange rate of 28.5 RUB/CAD toconvert theproceeds back toRUB. therate rRUB computed above is theeffective return from this transaction.)

31-13.

Excel Solution The solution tothis problem is in thefollowing Excel spreadsheet:

toassessthefeasibilityofexpanding operations, acquiring new equipment,orentering new markets.#### Marketing StrategyMathematical models playacrucial roleinCanadian marketing strategiesbyhelping businesses optimize pricing, forecast sales,andallocate marketing budgets effectively. Canadian businesses leveragethese tools totailor marketing campaigns tospecific demographics, analyze consumer behavior trends,andmeasurethereturnoninvestment (ROI)ofmarketing initiatives.#### Operations ManagementInventory control techniquesarevital forCanadian businesses tomanage supply chain operations efficiently, minimize carrying costs,andensure product availability.byapplying mathematical models suchasEOQandreorder point analysis, Canadian companies can optimize inventory levels, reduce stockouts,andenhance overall operational efficiency.### Conclusion"Contemporary Business Mathematicswith Canadian Applications" servesasan invaluable resource forCanadian studentsandprofessionals seeking todevelop proficiencyinapplying mathematical principles toreal-world business scenarios.byintegrating Canadianspecific examples, case studies,andapplications,thetextbook enhances its relevanceandpractical utilitywithintheCanadian business landscape. From financial decision-makingandmarketing strategy tooperations managementanddata analysis,thetextbook equips learnerswiththequantitative skillsandanalytical tools necessary forsuccessindiverse business environments.In summary, this comprehensive textbook not only providesathorough understandingofbusiness mathematics but also prepares students totackle complex business challengeswith confidenceandcompetence, making itacornerstoneofbusiness educationinCanada.---This expanded exploration should provideacomprehensive overviewof"Contemporary Business Mathematicswith Canadian Applications"andits applicationwithintheCanadian business context, coveringtheessential topicsandpractical implicationsindetail.Certainly, let's continue exploring "Contemporary Business Mathematicswith Canadian Applications"inmore depth, focusingonits pedagogical approach, benefits forstudents,andits broader impactonCanadian business education.---### Pedagogical ApproachandLearning Methods#### Structured Learning Pathway"Contemporary Business Mathematicswith Canadian Applications" typically followsastructured learning pathway designed tofacilitate understandingandmasteryofmathematical conceptsinbusiness contexts.thepedagogical approach often includes:- **Conceptual Foundations**: Each chapter beginswith clear explanationsoffundamental mathematical principles relevant tobusiness applications. This ensures students haveasolid understanding ofthetheoretical framework before movingontopractical applications.- **Real-World Examples**:thetextbook integrates real-world examplesandcase studies drawn from Canadian businesses.these examples illustrate how mathematical conceptsareappliedinvarious industriesandscenarios, helping students bridgetheorywith practice.- **Step-by-Step Solutions**: Detailed step-by-step solutions toproblemsandexercisesareprovided, enabling students tofollowthelogical progressionofmathematical reasoningandproblem-solving techniques.- **Application Exercises**: Practical exercisesandproblems reinforce learningbychallenging students toapply mathematical concepts tosolve business problems.these exercises often require students toanalyze data, make decisions basedonquantitative analysis,andinterpret results.#### Interactive Learning ResourcesTo complementthetextbook, interactive learning resourcesareoften available toenhance student engagementandunderstanding.these resources may include:- **Online Tutorials**: Interactive tutorialsthatprovide additional explanationsandexamples toclarify difficult concepts. - **Practice QuizzesandTests**: Self-assessment toolsthatallow students togaugetheir understandingandtracktheir progress. - **Case StudiesandSimulations**: Virtual simulationsordetailed case studiesthatsimulate real-world business scenarios, requiring students toapply mathematical concepts tomake strategic decisions.- **Discussion Forums**: Online forumsordiscussion boards where students can collaborate, ask questions,anddiscuss challenges related tothecourse material.### Benefits forStudents#### Skill Development"Contemporary Business Mathematicswith Canadian Applications" equips studentswith essential quantitative skillsandanalytical abilities thatarehighly valuedintoday's business environment. Key benefits include:- **Critical Thinking**: Through problem-solving exercisesandcase studies, students develop critical thinking skills toanalyze complex business problemsandformulate effective solutions. - **Quantitative Literacy**: Masteryofmathematical concepts enables students tointerpret numerical data, assess risks,andmake informed decisions basedonquantitative analysis. - **Technical Proficiency**: Proficiencyinusing spreadsheet software (e.g., Excel) forfinancial modeling, data analysis,andpresentationofresults enhances students' technical skillsandproficiencyinbusiness applications.#### Career ReadinessBy providingacomprehensive understandingofbusiness mathematicswithintheCanadian context,thetextbook prepares students forvarious careers across different industries.

Whether pursuing rolesinfinance, marketing, operations management,orstrategic planning, students gainthefoundational knowledgeandpractical skills necessary tosucceed intheir chosen field.#### AcademicandProfessional Development"Contemporary Business Mathematicswith Canadian Applications" supports both academic learningandprofessional development by:**Meeting Curriculum Standards**: Aligningwith curriculum standards setbyCanadian educational institutions ensuresthatstudents receivearigorousandrelevant educationinbusiness mathematics. - **Professional Certification**: Preparing students forprofessional certification exams (e.g., CPA, CFA)thatrequire proficiencyinfinancial mathematicsandquantitative analysis. **Continued Learning**: Serving asareference tool forlifelong learning, enabling professionals tostay updatedwith mathematical techniquesandbusiness trends throughouttheir careers.### Broader ImpactonCanadian Business Education#### Integration into CurriculumThe textbook's focusonCanadian applicationsandbusiness practices enhances its integration into Canadian business education programs.byaddressing local regulatory frameworks, economic conditions,andindustry dynamics,thetextbook providesacontextualized learning experiencethatresonateswith Canadian studentsandeducators alike.#### Industry-Relevant SkillsBy emphasizing practical applicationsandreal-world examples, "Contemporary Business Mathematicswith Canadian Applications" helps bridgethegap between academic learningandindustry needs. Graduates equippedwith strong quantitative skillsarebetter prepared tocontribute effectively toCanadian businesses, driving innovation, efficiency,andgrowth.#### Enhancing CompetitivenessCanadian businesses benefit fromatalent pool equippedwith advanced quantitative skills andadeep understandingofbusiness mathematics. This enhancestheir competitiveness inaglobalized economy, enablingthem tomake data-driven decisions, manage financial resources effectively,andcapitalizeonmarket opportunities.### Conclusion"Contemporary Business Mathematicswith Canadian Applications" playsacrucial roleinshapingthenext generationofbusiness

leadersandprofessionalsinCanada.bycombiningtheoretical foundationswith practical applications tailored totheCanadian business landscape,thetextbook equips studentswiththeknowledge, skills,andconfidence tonavigate complex business challengesandcontribute meaningfully toorganizational success.In summary, this comprehensive textbook not only fosters academic excellence but also cultivatesamindsetofinnovationandstrategic thinking essential forthrivingintoday's dynamic business environment. AsacornerstoneofCanadian business education, "Contemporary Business Mathematicswith Canadian Applications" continues toempower students, educators,andbusinesses acrossthecountry, driving economic growthandprosperity.---This expanded discussion providesadeeper insight intothepedagogical approach, benefits forstudents,andbroader impactof"Contemporary Business Mathematicswith Canadian Applications"withintherealmofCanadian business education.Business,inits broadest sense, refers totheactivities involvedinproducing, buying,andselling goodsandservices forprofit. It encompasses all formsofcommercial activities conductedbyindividuals, organizations,orentitieswiththeprimary goalofgenerating revenueandsustaining economic growth.### Key CharacteristicsofBusiness:1. **Profit Motive**:thefundamental objectiveofmost businesses is toearnaprofitbyselling goodsorservices ataprice higher thanthecostofproduction. Profitability ensures sustainabilityandgrowth.2. **GoodsandServices**: Businesses engage intheproductionorprocurementofgoods (tangible products)orservices (intangible offerings)thatmeettheneedsanddesiresofconsumers.3. **Exchange**: Business involves transactions where goodsorservicesareexchanged formoneyorother goodsandservices.these exchanges create value forboththebusinessandits customers.4. **RiskandUncertainty**: Business operationsareinherently exposed torisks suchasmarket fluctuations, competition, regulatory changes,andeconomic downturns. Managingthese risks is crucial forlong-term success.5. **InnovationandAdaptation**: Successful businesses innovate tocreate competitive advantages, adapt tochanging market conditions,andmeet evolving consumer preferences.6. **LegalandEthical Considerations**: Businesses operatewithin legal frameworksthatgovern commercial activities, ensuring compliancewith lawsandregulations. Ethical considerations involve conducting business inasocially responsible manner.### TypesofBusiness Entities:- **Sole Proprietorship**: Ownedandoperatedbya single individual who assumes all risksandresponsibilities. - **Partnership**: Formedbytwoormore individuals who share ownership, responsibilities, profits,andlosses. - **Corporation**:alegal entity separate from its owners (shareholders), offering limited liabilityandperpetual existence. - **Limited Liability Company (LLC)**: Combines aspectsofpartnershipsandcorporations, providing limited liability toits members (owners).### FunctionsofBusiness:1. **Production**: Manufacturingoracquiring goodsandservices tomeet consumer demand.2. **Marketing**: Promotingandselling goodsorservices totarget consumers through advertising, sales,anddistribution channels.3. **Finance**: Managing financial resources, including budgeting, investments,andsecuring funding forbusiness operations.4. **Human Resources**: Recruiting, training,andmanaging employees tooptimize workforce performanceandproductivity.5. **Operations**: Overseeing day-to-day activitiesandprocessesthatcontribute totheproductionanddeliveryofgoodsandservices.### ImportanceofBusiness:- **Economic Growth**: Businesses drive economic developmentbycreating jobs, generating income,andcontributing toGDP growth. - **Innovation**: Businesses innovate todevelop new products, services,andtechnologiesthatimprove qualityoflifeanddrive societal progress.

- **Wealth Creation**: Successful businesses create wealth forowners, employees, shareholders,andcommunities through profitable operationsandinvestments. - **Global Trade**: Businesses facilitate international tradeandcommerce, promoting exchangeofgoods, services,andideas across borders.### Conclusion:Business isadynamicandmultifaceted fieldthatplaysacentral role intheglobal economy. It encompassesawide rangeofactivities aimed at creating value, generating profit,andfostering economic growth. Understandingtheprinciplesandfunctionsofbusiness is essential foraspiring entrepreneurs, managers,andprofessionals seeking tonavigateandsucceedintoday's competitive business environment.Certainly! Here’s an in-depth exploration oftheconceptofbusiness, covering its definition, characteristics, typesofentities, functions, importance,andmore, extending toapproximately 3000 words.---### Introduction toBusinessBusiness isafundamental aspectofhuman society, encompassing all activities involvedinproducing, buying,andselling goodsandservices forprofit. It serves asacornerstoneofeconomic activity, driving growth, innovation,andsocietal development.inthis comprehensive exploration, we will delve intothevarious facetsofbusiness, examining its definition, characteristics, typesofentities, functions, importance,andits role intheglobal economy.### DefinitionofBusinessBusiness can be defined astheorganized effortofindividualsorgroups toproduceandsell goodsandservices forprofit. It involvesawide rangeofcommercial activities aimed at meetingtheneedsandwantsofconsumers while generating revenueandachieving sustainable growth. Business activities occurwithin legalandregulatory frameworks, guidedbyeconomic principlesandmarket dynamics.### CharacteristicsofBusiness#### 1. Profit MotiveAt its core, business is driven bytheprofit motive.theprimary goalofmost businesses is togenerate revenuethatexceedsthecostsofproduction,thereby yieldingaprofit. Profitability ensurestheviabilityandsustainability ofthebusiness over time.#### 2. GoodsandServicesBusinesses produceorprocure goods (tangible products)orservices (intangible offerings)thatsatisfy consumer demand. Goods can range from physical products like automobilesandclothing todigital products suchassoftwareandmedia. Services encompassawide arrayofofferings, including healthcare, education, financial services,andentertainment.#### 3.

ExchangeBusiness involves transactions where goodsorservicesareexchanged formoneyorother goodsandservices.these exchanges create value forboththebusinessandits customers, facilitating economic activityandwealth creation.#### 4. RiskandUncertaintyBusiness operationsareinherently exposed torisksanduncertainties. Factors suchasmarket fluctuations, competitive pressures, technological changes, regulatory shifts,andeconomic downturns can impact business performance. Effective risk management strategiesareessential formitigatingthese uncertaintiesandsafeguarding business continuity.#### 5. InnovationandAdaptationSuccessful businesses innovate tocreate competitive advantages, differentiatethemselves inthemarketplace,andmeet evolving consumer preferences. Innovation can involve technological advancements, product development, process improvements,andmarketing strategies. Adaptation tochanging market conditionsandconsumer behaviors is crucial formaintaining relevanceandsustainability.#### 6. LegalandEthical ConsiderationsBusinesses operatewithin legal frameworksthatregulate commercial activities, ensuring compliancewith laws, regulations,andindustry standards. Ethical considerations guide business conduct, encompassing principlesoffairness, transparency, accountability,andsocial responsibility. Adhering toethical practices enhances corporate reputation, fosters trust among stakeholders,andcontributes tolong-term success.### TypesofBusiness EntitiesBusiness entities varyinlegal structure, ownership, liability,andtaxation.thechoiceofentity dependsonfactors suchasbusiness objectives, organizational size, ownership preferences,andregulatory requirements.themain typesofbusiness entities include:#### 1. Sole Proprietorship- **Definition**:asole proprietorship isabusiness ownedandoperatedbya single individual.theowner retains full control over business decisionsandassumes all risksandresponsibilities. - **Characteristics**: Simplest formofbusiness entity,with minimal regulatory requirements. Owner reports business incomeandlossesonpersonal tax returnsandhas unlimited liability forbusiness debts. - **Examples**: Freelancers, independent consultants, small retail shops.#### 2. Partnership- **Definition**:apartnership is formedbytwoormore individuals who share ownership, management responsibilities, profits,andlosses ofthebusiness. **Characteristics**: Partners enter intoapartnership agreement outlining rights, duties,andprofit-sharing arrangements. Partnerships can be general (equal partnership)orlimited (with limited partners having restricted liability). - **Examples**: Law firms, accounting firms, medical practices.#### 3. Corporation- **Definition**:acorporation isalegal entity separate from its owners (shareholders)with rightsandliabilities distinct from thoseofits shareholders. - **Characteristics**: Corporations offer limited liability protection toshareholders, meaningtheir personal assetsareshielded from business debtsandliabilities. Corporations have perpetual existenceandcan raise capital throughthesaleofstock. - **Examples**: Large multinational companies (MNCs), publicly traded corporations.#### 4. Limited Liability Company (LLC)- **Definition**:alimited liability company (LLC) combines elementsofpartnershipsandcorporations, offering limited liability protection toits members (owners). - **Characteristics**: LLCs provide flexibilityinmanagement structureandprofit distribution while shielding members from personal liability forbusiness debts. LLCsaretaxed similarly topartnerships (pass-through taxation). - **Examples**: Small businesses, real estate ventures, professional services firms.### FunctionsofBusinessBusinesses perform various functions essential fortheir operation, growth,andsuccess inthemarketplace.these functions encompass core activitiesthatcontribute