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© Pearson Education Limited 2008
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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© Pearson Education Limited 2008
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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© Pearson Education Limited 2008
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
Chapter 3: The Time Value of Money
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© Pearson Education Limited 2008
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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© Pearson Education Limited 2008
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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© Pearson Education Limited 2008
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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© Pearson Education Limited 2008
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.
Chapter 3: The Time Value of Money
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© Pearson Education Limited 2008
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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© Pearson Education Limited 2008
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
Chapter 3: The Time Value of Money
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© Pearson Education Limited 2008
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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© Pearson Education Limited 2008
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
Chapter 3: The Time Value of Money
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© Pearson Education Limited 2008
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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© Pearson Education Limited 2008
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
