Contemporary financial management 13th edition moyer solutions manual 1

Chapter 5

The Time Value of Money

Contemporary Financial Management

13th Edition by Moyer McGuigan

Rao

ISBN 1285198840 9781285198842

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CHAPTER 5

THE TIME VALUE OF MONEY

ANSWERS TO QUESTIONS:

1. The investment paying five percent compound interest is more attractive because you will receive interest not only on the principal amount each year, but interest will be earned on the previous year's interest as well.

2. The future value interest factor for 10 percent and two years is 1.210, whereas the present value interest factor for 10 percent and two years is 0.826.

3. As the interest rate increases, any annuity amount is being discounted by a higher value, thereby reducing the present value of the annuity. This can be seen in Table IV by looking across any row of successively higher interest rates. In contrast, the future value of an annuity increases as the interest (compounding) rate increases. (See Table III.)

4. Daily compounding is preferred because you will earn interest on the interest earned in the account each day. Table 5-6 illustrates this.

5. Annuity due computations are common for lease contracts and insurance policies, where payments are generally made at the beginning of each period.

6. As can be seen in Table 5-7, the more frequent the compounding period, the lower the present values.

7. a. A marketing manager might use present value concepts to evaluate the success of an advertising or other promotional campaign, the benefits of which are likely to extend beyond one year in time. Also, a firm selling capital goods must be familiar with the type of present value economic analysis that customers will use to evaluate purchases.

b. A personnel manager may need to use present value concepts to evaluate alternative insurance and pension plans.

8. The Rule of 72 can be used to determine the approximate number of years it takes for an amount of money to double, given an interest rate. It also can be used to determine the effective interest rate required for a sum of money to double, given a number of years. To solve for the number of years, the number "72" is divided by the interest rate (in percent). To solve for the percentage interest rate, the number "72" is divided by the number of years.

9. Present value and future value concepts are closely related. For example, PVIF factors are simply the reciprocal of FVIF factors and vice versa. Any problem which can be solved using PVIF factors can also be solved using FVIF factors.

10. An ordinary annuity involves a series of equal, end-of-period payments or receipts. The interest payments on most bonds are ordinary annuities. An annuity due involves a series of equal, beginning-of-period payments or receipts, such as in a lease or some insurance policies.

11. As the required rate of return increases, (a) the present value of an annuity decreases and (b) the future value of an annuity increases.

12. The sinking fund problem tries to find the annuity amount that must be invested each year to produce a future value. If the desired future value is known, it is divided by the FVIFA for the given interest rate and number of years to determine the sinking fund amount.

13. In order to set up a loan amortization schedule, the annual loan payment must first be computed using the appropriate PVIFA from Table IV. The interest portion of each period's payment is equal to the periodic interest rate times the balance outstanding at the beginning of

Chapter 5 The Time Value of Money

each period. The interest is subtracted from the payment to determine the principal portion of the payment. Finally, the principal for the period is subtracted from the beginning of period principal balance to get the new beginning of period balance for the next period.

14. The insurance company is willing to take on the known loss because the settlement of claims of this magnitude often takes five or more years. With the high interest rates prevailing at the time of the disaster, the insurers felt they could earn enough on the new premiums to more than cover their final liability.

15. This means that the basis for interest rate compounding is continuous. However, interest is only credited to your account at the end of each quarter. Thus, in order to earn the effective compounded rate you need to withdraw funds only on the quarterly payment dates.

16. The dividend payments on many preferred stocks are perpetuities. These preferred stocks have no scheduled maturity date and they pay the same dividend each period. A perpetuity is an annuity with no ending date.

17. The present value of an uneven cash flow stream is found by summing the present values of the individual cash flows.

18. This statement is not correct. The powerful microcomputer can be used efficiently to help solve complex problems. The hand calculator, on the other hand, is better suited to solving relatively simple problems because the solution routines are programmed into the calculator and there is a lower start-up cost associated with working the problem on the calculator.

19. The net present value of an investment represents the contribution of that investment to the value of the firm and, accordingly, to the wealth of shareholders. The net present value is a decision criterion that assists managers in achieving the objective of shareholder wealth maximization.

Chapter 5 The Time Value of Money

SOLUTIONS TO PROBLEMS:

1. a. FV3 = $1,000(FVIF.06,3) = $1,000(1.191) = $1,191

b. FV5 = $1,000(FVIF.06,5) = $1,000(1.338) = $1,338

c. FV10 = $1,000(FVIF.06,10) = $1,000(1.791) = $1,791

2.

a. Present value of $5,000 today = $5,000

b. Present value of $15,000 received in 5 years at 9%:

PV0 = $15,000(PVIF.09,5) = $15,000 (0.650) = $9,750 (tables)

$9749 (calculator)

c. Present value of a 15 year, $1,000 annuity at 9%:

PVAN0 = $1,000 (PVIFA.09,15) = $1,000(8.061) = $8,061

Therefore, you prefer $15,000 in five years because it has the highest present value.

3. FVAND8 = $20,000(FVIFA.09,8)(1 + 0.09) = $20,000(11.028)(1.09)

= $240,410.40 ($240,420.73 with a calculator)

4. Alternative a: PVAND0 = $1,200(PVIFA.08,12)(1 + 0.08) = $1,200(7.536)(1.08)

= $9,766.66 (tables); $9,766.76 (calculator)

Alternative b:

Present value cost equals $10,000 (given). Therefore, choose Alternative (a) because it has a lower present value cost.

5. a. PV0 = $50,000 /[1 + (0.06/2)]2x5 = $37,204.70 (calculator)

b. PV0 = $50,000 /[1 + (0.06/4)]4x5 = $37,123.52 (calculator)

6. $1,000 = $333.33(FVIFi,9)

FVIFi,9 = 3.000

i  13% from Table I. (12.98% by calculator)

7. a. PV0 = $800(PVIF.04,8) = $800 (0.731) = $584.80 (tables)

$584.55 (calculator)

b. PV0 = $800(0.540) = $432

c. PV = $800(PVIF.05,32) = $167.89 (by calculator)

d. PV = $800(1.000) = $800

8. PVAN0 = $60,000 - $10,000 = $50,000

$50,000 = PMT(PVIFA.10,25) = PMT(9.077)

PMT = $5,508.43 (tables)

$5,508.40 (calculator)

Interest (first year) = .10($50,000) = $5,000

Principal reduction = $5,508.43 - $5,000 = $508.43

9. $200,000 = $41,067(PVIFAi,20)

PVIFAi,20 = 4.870; From Table IV, i = 20%

10. $600,000 = PMT(FVIFA.09,25) = PMT(84.701)

PMT = $7,083.74 (tables); $7,083.75 (calculator)

11. a. PV0 = $70(PVIFA.05,25) + $1000(PVIF.05,25) = $70(14.094) + $1000(0.295) = $1,281.58 (tables); $1,281.88 (calculator)

b. PV0 = $70(11.654) + 1000(0.184) = $1,000 ($999.78 using tables; difference from $1,000 due to rounding)

c. PV0 = $70(7.843) + $1000(0.059) = $608.01 (tables)

$607.84 (calculator)

12. ieff = [ 1 + (inom/m)]m - 1 = [ 1 + (.08/4)]4 -1 = 0.0824 or 8.24%

13. NPV1 = -$10,000 + $5,000(0.909) + $6,000(0.826) + $7,000(0.751) +

$8,000(0.683) = $10,222

NPV2 = -$10,000 + $8,000(0.909) + $7,000(0.826) + $6,000(0.751) +

$5,000(0.683) = $10,975 This is the preferred alternative.

14. PVAN0 = $80,000 = PMT(PVIFA.10,10) = PMT(6.145)

PMT = $13,018.71 (Calculator solution = $13,019.63)

15. PVAN0 = $30,000 - $5,000(down) - $750 (loan origination fee) = $24,250

Origination fee = 0.03 x $25,000 = $750

$24,250 = $3,188(PVIFAi,15)

PVIFAi,15 = 7.607

Therefore, i  10% from Table IV and calculator

16. a. PV0 = $6,000(PVIFA.12,5) + $4,000(PVIFA.12,5)(PVIF.12,5) = $6,000(3.605) + $4,000(3.605)(0.567) = $29,806 (tables)

$29,810 (calculator)

(Note: $4,000(PVIFA.12,5) gives the present value of that annuity at the end of five years. Hence, it must be discounted back to time 0 at a 12% rate.)

b. Both terms in the Part (a) solution need to be multiplied by (1 + 0.12). Hence, the annuity due solution to this problem is equal to $29,806 (1.12)

Chapter 5 The Time Value of Money

= $33,383 (tables); $33,387 (calculator)

17. $919 = $87.5(1 - 0.28)(PVIFAi,20) + [$919 + ($1000 - $919) x (1 - 0.28)](PVIFi,20)

$919 = $63(PVIFAi,20) + $977.32(PVIFi,20)

Try i = 7%

$919 = $63(10.594) + $977.32(0.258) = $919.57

Therefore, i = 7%. (7.01% by calculator)

18. FV25 = $1000(FVIF.05,25) = $1000(3.386) = $3,386

19. PVAN0 = $6,000(PVIFA.08,15) = $6,000(8.559) = $51,354(tables)

$51,357(calculator)

Because the lifetime annuity has a higher expected present value than the $50,000 lump sum payment, she should take the annuity.

20. FVAN10 = $10,000,000 = PMT(FVIFA.08,10) = PMT(14.487)

PMT = $690,274 (tables); $690,295 (calculator)

21. $30,000 = PMT(PVIFA.11,3) = PMT(2.444)

PMT = $12,275 (tables); $12,276 (calculator)

Chapter 5 The Time Value of Money

* difference from zero due to rounding in tables

22. a. PV0 = $6,000(PVIFA.12,5) + $3,000(PVIFA.12,5)(PVIF.12,5) + $2,000(PVIFA.12,10)(PVIF.12,10)

PV0 = $6,000(3.605) + $3,000(3.605)(0.567) + $2,000(5.650)(0.322)

PV0 = $31,401

b. PV of beginning of year receipts = $31,401(1.12) = $35,169

23. PVAND30 = $250,000(PVIFA.10,5)(1 + .10) = $250,000(3.791)(1.1) = $1,042,525

FVAN30 = $1,042,525 = PMT(FVIFA.10,30) = PMT(164.494)

PMT = $6,338 (tables); $6,337 (calculator)

24. FVAN25 = $4,500(FVIFA.10,25) = $4,500(98.347)

FVAN25 = $442,561.50 (amount in account at the end of 25 years)

PVAN0 = $442,561.50 = PMT(PVIFA.10,20) = PMT(8.514)

PMT = $51,980.44 (Calculator solution = $51,983)

25. FVAN4 = $10,000(FVIFA.12,4) = $10,000(4.779) = $47,790 (balance in the account at the end of four years)

FV6 = $47,790(FVIF.12,6) = $47,790(1.974) = $94,337 (balance in the account at the end of ten years) (tables); $94,335 (calculator)

26. a. FVn = PV0 [ 1 + (inom /m)]mn

FV3 = $10,000 [ 1 + (0.08/2)]6 = $12,653.19

b. FV3 = $10,000 [ 1 + (0.08/4)]12 = $12,682.42

c. FV3 = $10,000 [ 1 + (0.08/12)]36 = $12,702.37 (by calculator)

27. NPV = $40,000(PVIFA.20,5)(PVIF.20,3) - $100,000

NPV = $40,000(2.991)(0.579) - $100,000

NPV = $-30,728 (tables) (The project should not be undertaken.)

$-30,773(calculator)

28. $100,000 = $60,000(PVIFi,1) + $79,350(PVIFi,2)

Try i = 24%

$100,000  $60,000(0.806) + $79,350(0.650)

$100,000  $99,937.5, hence i  24%

29. PV0 = $20,000(PVIF.15,1) + $30,000(PVIF.15,2) + $15,000(PVIF.15,3)

PV0 = $20,000(0.870) + $30,000(0.756) + $15,000(0.658)

PV0 = $49,950

$49,950 = PMT(PVIFA.15,3) = PMT(2.283)

PMT = $21,879 (tables); $21,872 (calculator)

30. Amount needed by 18th birthday: PV0 = $18,000(PVIF.10,0) + $19,000(PVIF.10,1)

Chapter 5 The Time Value of Money

+ $20,000(PVIF.10,2) + $21,000(PVIF.10,3)

= $18,000(1.0) + $19,000(0.909) + $20,000(0.826) + $21,000(0.751)

= $67,562

FV8 (at age 18) = PMT(FVIFA.10,8)

$67,562 = PMT(11.436)

PMT = $5,907.83 (tables) amount to be deposited in account on 11th through 18th birthdays; $5,909.40 (calculator)

31. FVANn = PMT(FVIFAi,n); n = 5 years x 4 quarters/year = 20 periods

i = 0.20/4 = 0.05 per period

FVAN20 = $10,000 = PMT(FVIFA.05,20)

$10,000 = PMT(33.066)

PMT = $302.43 (amount to be deposited each quarter)

32. Amount needed in account after final deposit on your 60th birthday:

PV0 = $120,000(PVIFA.12,15) + $250,000(PVIF.12,15)

PV0 = $120,000(6.811) + $250,000(0.183)

PV0 = $863,070

$863,070 = PMT(FVIFA.12,30) = PMT(241.333)

PMT = $3,576

33. Present value of payments to first child for college:

$10,000 (PVIF.13,10) = $10,000 (0.295) = $2,950

Chapter 5 The Time Value of Money

$11,000(0.261) = $2,871

$12,000(0.231) = $2,772

$13,000(0.204) = $2,652

Total $11,245

Present value of payments to second child for college:

$15,000(PVIF.13,15) = $15,000(0.160) = $2,400

$16,000(0.141) = $2,256

$17,000(0.125) = $2,125

$18,000(0.111) = $1,998

Total $8,779

Present value of retirement annuity: PV0 = $50,000(PVIFA.13,20)(PVIF.13,30)

PV0 = $50,000(7.025)(0.026) = $9,133

Present value of funds needed = $11,245 + $8,779 + $9,133 = $29,157

Payment needed for 30 years: $29,157 = PMT(PVIFA.13,30) = PMT(7.496)

PMT = $3,890 (tables); $3,869 (calculator)

34.

$100,000(8.514)(1 + .10) = $936,540 needed

a. PVAND0 = $100,000(PVIFA.10,20)(1 + .10)

at age 60

Chapter 5 The Time Value of Money

PV (at age 45) = $936,540 (PVIF.10,15)

= $936,540(0.239) = $223,833 (tables); $224,189 (calculator)

b. PV (at age 30) = $223,833(PVIF.12,15)

= $223,833(0.183) = $40,961

With $10,000 available, you must save an annuity amount at the end of each of the next 15 years that has a present value equal to $30,961, or:

$30,961 = PMT (PVIFA.12,15) = PMT (6.811)

PMT = $4,546 (tables); $4,545 (calculator)

35. PVAN0 = PMT(PVIFA0.10,t)

$400,000 = $40,000(PVIFA0.10,t)

PVIFA0.10,t = 10

Therefore, at 10% per year his $400,000 savings will last forever, i. e., $400,000 x 0.10 = $40,000

36. FV7/1/2024 = $2,000 (FVIF0.07,10) + $1,000 (FVIFA0.07,6) x (FVIF0.07,4) - $3,000 (FVIF0.07,2)

Chapter 5 The Time Value of Money

= $2,000 (1.967) + $1,000 (7.153)(1.311)

- $3,000 (1.145) = $9,877 (tables); $9,880 (calculator)

37. 10% pretax x (1 - T) = 7% after tax

PVo = $15,000 (PVIF0.07,3) + $16,000 (PVIF0.07,4)

+ $17,000 (PVIF0.07,5) + $18,000 (PVIF0.07,6) + $25,000 (PVIF0.07,7)

= $15,000 (0.816) + $16,000 (0.763) + $17,000 (0.713) + $18,000 (0.666) + $25,000 (0.623) = $64,132

Amount needed = $64,132 - $8,000 = $56,132

PVAN0 = PMT (PVIFA0.07,6)

$56,132 = PMT(4.766)

PMT = $11,778

38. 10% pretax x (1 - T) = 7% after tax

PV0 = $25,000 (PVIFA0.07,4) (PVIF0.07,9)

+ $55,000 (PVIF0.07,14)

= $25,000 (3.387) (0.544) + $55,000 (0.388) = $67,403

Amount needed = $67,403 - $10,000 - $25,000 (PVIF0.07,9) = $43,803

Chapter 5

The Time Value of Money

PVAN = PMT (PVIFA0.07,10)

$43,803 = PMT (7.024)

PMT = $6,236 (tables); $6,235 (calculator)

39. Amount needed by 60th birthday = $100,000(PVIFA.07,20)(1+0.07) = $1,133,560

Future value of $35,000 at the end of year 10: = $35,000(FVIF.05,10) = $57,011

Future value of $57,011 at the end of year 30: = $57,011(FVIF.07,20) = $220,615

Future value of $5,000 annuity at the end of year 10: = $5,000(FVIFA.05,10) = $62,889

Future value of $62,889 at the end of year 30: = $62,889(FVIF.07,20) = $243,361

Net amount needed on 60th birthday: = $1,133,560 - $220,615 - $243,361 = $669,584

Payment needed years 11-20: = $669,584 = PMT(FVIFA.07,20)

PMT = $16,333

40. FV = $200,000 = $10,000(FVIF.07,5)(FVIF.09,5) + PMT(FVIFA.07,5)(FVIF.09,5) + PMT(FVIFA.09,5)

$200,000 = $10,000(1.403)(1.539) + PMT(5.751)(1.539) + PMT(5.985)

PMT = $12,025

41. $1,000,000 = PMT (PVIFA0.1125, 5)

Chapter 5 The Time Value of Money

PMT = $272,274 (by calculator)

*Differs from $0 due to rounding.

42. Amount needed at 60th birthday = $500,000(PVIF0.07, 20)

PMT = $15,859 (tables); $15,864 (calculator)

43. Amount needed by year 35 = $200,000 (PVIFA.08,25) = $2,134,955

Chapter 5

The Time Value of Money

Value of $5,000 annuity at year 35 = $5,000(FVIFA.12,15)(FVIF.12, 20) = $1,798,049

Future Value of amount needed from 20 year annuity payments:

$2,134,955 - $1,798,049 = $336,905

$336,905 = PMT (FVIFA.12, 20)

PMT = $4,676 (calculator accuracy)

44. FVAND30 = $5,000 (FVIFA0.10,30)(1.10)

= $5,000 (164.494)(1.10) = $904,717

PVAN0 = PMT (PVIFA0.12,20)(1.12)

$904,717 = PMT (7.469)(1.12)

PMT = $108,151

Integrative Case Problem:

Chapter 5 The Time Value of Money

1. At age 65: n = 35; i = 0.03; PV0 = $60,000

2. At age 65: n = 15; PMT = $168,831.75 (annuity due); FV15 = $1,000,000

a. i = 0.11

c. i = 0.085

3. i = 0.11; FV needed = $1,556,596.62

a. At age 30: n = 35

$1,556,596.62 = PMT(FVIFA.11,35)

PMT = $4,556.92

b. At age 40: n = 25

$1,556,596.62 = PMT(FVIFA.11,25)

PMT = $13,605.03

c. At age 50: n = 15

$1,556,596.62 = PMT(FVIFA.11,15) PMT = $45,242.85

4. i = 0.06; FV needed = $2,155,385.21

a. At age 30: n = 35

$2,155,385.21 = PMT(FVIFA.06, 35)

PMT = $19,342.12

b. At age 40: n = 25

$2,155,385.21 = PMT(FVIFA.06, 25)

PMT = $39,285.60

c. At age 50: n = 15

$2,155,385.21 = PMT(FVIFA.06, 15)

PMT = $92,601.31

5. i = 0.085; FV needed = $1,815,330.28

a. At age 30: n = 35 $1,815,330.28 = PMT(FVIFA

PMT = $9,420.42

b. At age 40: n = 25

$1,815,330.28 = PMT(FVIFA.085, 25) PMT = $23,075.90

c. At age 50: n = 15

$1,815,330.28 = PMT(FVIFA.085, 15)

Chapter 5

The Time Value of Money

PMT = $64,299.84

6. The earlier one begins investing, the lower the annual payments required. Also, the greater the returns earned on investments, the lower the annual payments required.