978-0273713630 Chapter 3 Solution Manual

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The Time Value of Money

The chief value of money lies in the fact that one lives in

a world in which it is overestimated.

H.L. MENCKEN

From A Mencken Chrestomathy

Chapter 3: The Time Value of Money

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ANSWERS TO QUESTIONS

2. With compound interest, interest payments are added to the principal and both then earn

4. An annuity is a series of cash receipts of the same amount over a period of time. It is worth

6. In calculating the future (terminal) value, we need to know the beginning amount, the

7. They facilitate calculations by being able to multiply the cash-flow by the appropriate

8. Interest compounded as few times as possible during the five years. Realistically, it is likely

9. For interest rates likely to be encountered in normal business situations the ‘‘Rule of 72’’ is

10. Decreases at a decreasing rate. The present value equation, 1/(1 +i)n, is such that as you

12. A lot. Turning to FVIF Table 3.3 in the chapter and tracing down the 3 percent column to

Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition, Instructor’s Manual

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SOLUTIONS TO PROBLEMS

1. a. FVn = P0(1 + i)n

b. FVn = P0(1 + i)n; FVAn = R[([1 + i]n – 1)/i]

(ii) FV5 = $500(1.05)5 = $500(1.276) = $ 638.00

(iii) FV5 = $500(1.0)5 = $500(1) = $ 500.00

*[Note: We had to invoke l’Hospital’s rule in the special case where i = 0; in short,

FVIFAn = n when i = 0.]

c. FVn = P0(1 + i)n; FVADn = R[([1 + i]n – 1)/i][1 + i]

(ii) FV6 = $500(1.05)6 = $500(1.340) = $ 670.00

(iii) FV6 = $500(1.0)6 = $500(1) = $ 500.00

d. FVn = PV0(1 + [i/m])mn

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e. The more times a year interest is paid, the greater the future value. It is particularly

f. FVn = PV0(1 + [i/m])mn; FVn = PV0(e)in

(i) $100(1 + [0.10/1])10 = $100(2.594) = $259.40

2. a. P0 = FVn[1/(1 + i)n]

b. PVAn = R[(1 –[1/(1 + i)n])/i]

c. P0 = FVn[1/(1 + i)n]

(i) $100[1/(1.04)1] = $100(0.962) = $ 96.20

$1,447.70

(ii) $100[1/(1.25)1] = $100(0.800) = $ 80.00

d. (i) $1,000[1/(1.04)1] = $1,000(0.962) = $ 962.00

(ii) $1,000[1/(1.25)1] = $1,000(0.800) = $ 800.00

e. The fact that the cash flows are larger in the first period for the sequence in Part (d)

Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition, Instructor’s Manual

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3. $25,000 = R(PVIFA6%,12) = R(8.384)

4. $50,000 = R(FVIFA8%,10) = R(14.486)

5. $50,000 = R(FVIFA8%,10)(1 + 0.08) = R(15.645)

6. $10,000 = $16,000(PVIFx%,3)

Going to the PVIF table at the back of the book and looking across the row for n = 3, we

find that the discount factor for 17 percent is 0.624 and that is closest to the number above.

7. $10,000 = $3,000(PVIFAx%,4)(PVIFAx%,4) = $10,200/$3,000 = 3.4 Going to the PVIFA

8. Year Sales

1 $ 600,000 = $ 500,000(1.2)

9. Present Value

Year Amount Factor at 14% Present Value

1 $1,200 0.877 $1,052.40

Chapter 3: The Time Value of Money

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10. Amount Present Value Interest Factor Present Value

$1,000 1/(1 + .10)10 = 0.386 $386

11. $1,000,000 = $1,000(1 + x%)100

Going to the FVIF table at the back of the book and looking across the row for n = 50, we

12. a. Annuity of $10,000 per year for 15 years at 5 percent. The discount factor in the PVIFA

b. Discount factor for 10 percent for 15 years is 7.606

c. Annual annuity payment for 5 percent = $30,000/10.380 = $2,890

13. $190,000 = R(PVIFA17%, 20) = R(5.628)

Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition, Instructor’s Manual

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14. a. PV0 = $8,000 = R(PVIFA1%,36)

(1) (2) (3) (4)

End of

Month

Installment

Payment

Monthly Interest

(4)t–1 × 0.01

Principal

Payment

(1) – (2)

Principal Amount

Owing At Month

End (4)t–1 – (3)

0 — — — $8,000.00

1 $ 265.71 $ 80.00 $ 185.71 7,814.29

2 265.71 78.14 187.57 7,626.72

3 265.71 76.27 189.44 7,437.28

4 265.71 74.37 191.34 7,245.94

*The last payment is slightly higher due to rounding throughout.

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b. PV0 = $184,000 = R(PVIFA10%, 25)

(1) (2) (3) (4)

End of Installment Annual Principal Principal Amount

Year Payment Interest Payment Owing At Year End

(4)t–1 × 0.10 (1) – (2) (4)t–1 – (3)

0 — — — $ 184,000.00

1 $ 20,271.01 $ 18,400.00 $ 1,871.01 182,128.99

2 20,271.01 18,212.90 2,058.11 180,070.88

3 20,271.01 18,007.09 2,263.92 177,806.96

4 20,271.01 17,780.70 2,490.31 175,316.65

*The last payment is somewhat lower due to rounding throughout.

15. $14,300 = $3,000(PVIFA15% ,n)

16. a. $5,000,000 = R[1 + (0.20/1)]5 = R(2.488)

Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition, Instructor’s Manual

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b. $5,000,000 = R[1 + (0.20/2)]10 = R(2.594)

17. FV of Earl’s plan = ($2,000) × (FVIFA7%,10) × (FVIF7%,35)

18. Tip: First find the future value of a $1,000-a-year ordinary annuity that runs for 25 years.

Unfortunately, this future value overstates our “true” ending balance because three of the

FV25 = $1,000[(FVIFA5%, 25) – (FVIF5%, 20) – (FVIF5%, 18) – (FVIF5%,14)]

19. There are many ways to solve this problem correctly. Here are two:

Cash withdrawals at the END of year …

Alt. %1 This above pattern is equivalent to …

PVA9

— minus —

PVA3

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PVA9 – PVA3 = $100,000

PVA6 × (PVIF.05, 3) = $100,000

NOTE: Answers to Alt. #1 and Alt. #2 differ slightly due to rounding in the tables.

20. Effective annual interest rate = (1 + [i/m])m – 1

a. (annually) = (1 + [0.096/1])1 – 1 = 0.0960

b. (semiannually) = (1 + [0.096/2])2 – 1 = 0.0983

21. (Note: You are faced with determining the present value of an annuity due. And, (PVIFA8%, 40)

can be found in Table IV at the end of the textbook, while (PVIFA8%, 39) is not listed in the

table.)

Alt. 1: PVAD40 = (1 + 0.08)($25,000)(PVIFA8%, 40)

Alt. 2: PVAD40 = ($25,000)(PVIFA8%, 39) + $25,000

NOTE: Answers to Alt. 1 and Alt. 2 differ slightly due to rounding.

Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition, Instructor’s Manual

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22. For approximate answers, we can make use of the ‘‘Rule of 72’’ as follows:

(i) 72/14 = 5.14 or 5% (to the nearest whole percent)

For greater accuracy, we proceed as follows:

(i) (1 + i)14 = 2

(ii) (1 + i)8 = 2

(iii) (1 + i)2 = 2

Notice how the “Rule of 72” does not work quite so well for high rates of growth such as that

seen in situation (iii).

SOLUTIONS TO SELF-CORRECTION PROBLEMS

1. a. Future (terminal) value of each cash-flow and total future value of each stream are as

follows (using Table I in the end-of-book Appendix):

CASH-

FLOW

STREAM

PV0 FOR INDIVIDUAL CASH FLOWS RECEIVED AT

END OF YEAR

1 2 3 4 5

TOTAL

FUTURE

VALUE

W $146.40 $266.20 $242 $330 $ 300 $1,284.60

b. Present value of each cash-flow and total present value of each stream (using Table II in

the end-of-book Appendix):

CASH-

FLOW

STREAM

PV0 FOR INDIVIDUAL CASH FLOWS RECEIVED AT

END OF YEAR

1 2 3 4 5

TOTAL

FUTURE

VALUE

W $ 87.70 $153.80 $135.80 $177.60 $155.70 $709.80

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2. a. FV10 Plan 1 = $500(FVIFA3.5%,20)

b. FV10 Plan 2 = $1,000(FVIFA7.5%, 10)

Now, Plan 1 would be preferred by a nontrivial $323.37 margin.

3. Indifference implies that you could reinvest the $25,000 receipt for 6 years at X% to

Alternatively, note that $50,000 = $25,000(FVIFX%,6). Therefore, (FVIFX%,6) = $50,000/

2.000 1.974

−

For an even more accurate answer, recognize that FVIFX%, 6 can also be written as (1 + i)6.

Then we can solve directly for i (and X% = i(100)) as follows:

4. a. PV0 = $7,000(PVIFA6%, 20) = $7,000(11.470) = $80,290

b.

End of

Year

(1)

Installment

Payment

(2)

Annual Interest

(4)t–1 × 0.14

(3)

Principal

Payment

(4)

Principal Amount

Owing At Year End

(4)t–1 – (3)

0 — — — $10,000

1 $ 3,432 $1,400 $ 2,032 7,968

Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition, Instructor’s Manual

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6. When we draw a picture of the problem, we get $1,000 at the end of every even-numbered

year for years 1 through 20:

TIP: Convert $1,000 every 2 years into an equivalent annual annuity (i.e., an annuity that

7. Effective annual interest rate = (1 + [i/m])m –1

Therefore, we have quarterly compounding. And, investing $10,000 at 7.06% compounded

8. FVA65 = $1,230(FVIFA5%, 65) = $1,230[([1 + 0.05]65 – 1)/(0.05)]

9. a. $50,000(0.08) = $4,000 interest payment

b. Total installment payments – total principal payments