1. Part 2
Valuation
3
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.
2. Chapter 3: The Time Value of Money
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ANSWERS TO QUESTIONS
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1. Simple interest is interest that is paid (earned) on only the
original amount, or principal, borrowed (lent).
2. With compound interest, interest payments are added to the
principal and both then earn interest for subsequent periods.
Hence interest is compounded. The greater the number of periods
and the more times a period interest is paid, the greater the
compounding and future value.
3. The answer here will vary according to the individual. Common
answers include a savings account and a mortgage loan.
4. An annuity is a series of cash receipts of the same amount over a
period of time. It is worth less than a lump sum equal to the sum of
the annuities to be received because of the time value of money.
5. Interest compounded continuously. It will result in the highest
terminal value possible for a given nominal rate of interest.
6. In calculating the future (terminal) value, we need to know the
beginning amount, the interest rate, and the number of periods. In
calculating the present value, we need to know the future value or
cash flow, the interest or discount rate, and the number of peri-
ods. Thus, there is only a switch of two of the four variables.
3. Chapter 3: The Time Value of Money
7. They facilitate calculations by being able to multiply the cash
flow by the appropriate discount factor. Otherwise, it is
necessary to raise 1 plus the discount rate to the nth power and
divide. Prior to electronic calculators, the latter was quite
laborious. With the advent of calculators, it is much easier and
the advantage of present-value tables is lessened.
8. Interest compounded as few times as possible during the five
years. Realistically, it is likely to be at least annually.
Compounding more times will result in a lower present value.
9. For interest rates likely to be encountered in normal business
situations the "Rule of 72" is a pretty accurate money doubling
rule. Since it is easy to remember and involves a calculation that
can be done in your head, it has proven useful.
10. Decreases at a decreasing rate. The present value equation,
1/(1 +i)n, is such that as you divide 1 by increasing (linearly)
amounts of i, present value decreases toward zero, but at a
decreasing rate.
11. Decreases at a decreasing rate. The denominator of the present
value equation increases at an increasing rate with n. Therefore,
present value decreases at a decreasing rate.
4. Chapter 3: The Time Value of Money
12. A lot. Turning to FVIF Table 3-3 in the chapter and tracing down
the 3 percent column to 25 years, we see that he will increase his
weight by a factor of 2.09 on a compound basis. This translates
into a weight of about 418 pounds at age 60.
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SOLUTIONS TO PROBLEMS
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1. a) FVn = P0(1 + i)n
(i) FV3 = $100(2.0)3 = $100(8) = $800
(ii) FV3 = $100(1.10)
3 = $100(1.331) = $133.10
(iii) FV3 = $100(1.0)3 = $100(1) = $100
b) FVn = P0(1 + i)
n; FVAn = R[([1 + i]
n - 1)/i]
(i) FV5 = $500(1.10)5 = $500(1.611) = $ 805.50
FVA5 = $100[([1.10]5 - 1)/(.10)] =
$100(6.105) = 610.50
$1,416.00
(ii) FV5 = $500(1.05)
5 = $500(1.276) = $ 638.00
FVA5 = $100[([1.05]
5 - 1)/(.05)] =
$100(5.526) = 552.60
$1,190.60
(iii) FV5 = $500(1.0)5 = $500(1) = $ 500.00
FVA5 = $100(5)* = 500.00
$1,000.00
*[Note: We had to invoke l'Hospital's rule
in the special case where i = 0; in short,
FVIFAn = n when i = 0.]
5. Chapter 3: The Time Value of Money
c) FVn = P0(1 + i)
n; FVADn = R[([1 + i]
n - 1)/i][1 + i]
(i) FV6 = $500(1.10)6 = $500(1.772) = $ 886.00
FVAD5 = $100[([1.10]5 - 1)/(.10)] x [1.10] =
$100(6.105)(1.10) = 671.55
$1,557.55
(ii) FV6 = $500(1.05)6 = $500(1.340) = $ 670.00
FVAD5 = $100[([1.05]5 - 1)/(.05)] x [1.05] =
$100(5.526)(1.05) = 580.23
$1,250.23
(iii) FV6 = $500(1.0)6 = $500(1) = $ 500.00
FVAD5 = $100(5) = 500.00
$1,000.00
d) FVn = PV0(1 + [i/m])
mn
(i) FV3 = $100(1 + [1/4])
12 = $100(14.552) = $1,455.20
(ii) FV3 = $100(1 + [.10/4])12 = $100(1.345) = $ 134.50
e) The more times a year interest is paid, the greater the
future value. It is particularly important when the interest
rate is high, as evidenced by the difference in solutions
between Parts 1.a) (i) and 1.d) (i).
f) FVn = PV0(1 + [i/m])mn; FVn = PV0(e)in
(i) $100(1 + [.10/1])
10 = $100(2.594) = $259.40
(ii) $100(1 + [.10/2])
20 = $100(2.653) = $265.30
(iii) $100(1 + [.10/4])
40 = $100(2.685) = $268.50
(iv) $100(2.71828)1 = $271.83
2. a) P0 = FVn[1/(1 + i)n]
(i) $100[1/(2)
3] = $100(.125) = $12.50
(ii) $100[1/(1.10)3] = $100(.751) = $75.10
(iii) $100[1/(1.0)3] = $100(1) = $100
6. Chapter 3: The Time Value of Money
b) PVAn = R[(1 -[1/(1 + i)n])/i]
(i) $500[(1 -[1/(1 + .04)
3])/.04] = $500(2.775) = $1,387.50
(ii) $500[(1 -[1/(1 + .25)3])/.25] = $500(1.952) = $ 976.00
c) P0 = FVn[1/(1 + i)
n]
(i) $ 100[1/(1.04)1] = $ 100(.962) = $ 96.20
500[1/(1.04)
2] = 500(.925) = 462.50
1,000[1/(1.04)3] = 1,000(.889) = 889.00
$1,447.70
(ii) $ 100[1/(1.25)1] = $ 100(.800) = $ 80.00
500[1/(1.25)2] = 500(.640) = 320.00
1,000[1/(1.25)3] = 1,000(.512) = 512.00
$ 912.00
d) (i) $1,000[1/(1.04)
1] = $1,000(.962) = $ 962.00
500[1/(1.04)
2] = 500(.925) = 462.50
100[1/(1.04)3] = 100(.889) = 88.90
$1,513.40
(ii) $1,000[1/(1.25)
1] = $1,000(.800) = $ 800.00
500[1/(1.25)
2] = 500(.640) = 320.00
100[1/(1.25)
3] = 100(.512) = 51.20
$1,171.20
e) The fact that the cash flows are larger in the first period
for the sequence in Part (d) results in their having a higher
present value. The comparison illustrates the desirability of
early cash flows.
3. $25,000 = R(PVIFA6%,12) = R(8.384)
R = $25,000/8.384 = $2,982
7. Chapter 3: The Time Value of Money
4. $50,000 = R(FVIFA8%,10) = R(14.486)
R = $50,000/14.486 = $3,452
5. $50,000 = R(FVIFA8%,10)(1 + .08) = R(15.645)
R = $50,000/15.645 = $3,196
6. $10,000 = $16,000(PVIFx%,3)
(PVIFx%,3) = $10,000/$16,000 = 0.625
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 table at the back of the book and looking across
the row for n = 4, we find that the discount factor for 6 percent
is 3.465, while for 7 percent it is 3.387. Therefore, the note has
an implied interest rate of almost 7 percent.
8. Year Sales
1 $ 600,000 = $ 500,000(1.2)
2 720,000 = 600,000(1.2)
3 864,000 = 720,000(1.2)
4 1,036,800 = 864,000(1.2)
5 1,244,160 = 1,036,800(1.2)
6 1,492,992 = 1,244,160(1.2)
8. Chapter 3: The Time Value of Money
9. Present Value
Year Amount Factor at 14% Present Value
1 $1,200 .877 $1,052.40
2 2,000 .769 1,538.00
3 2,400 .675 1,620.00
4 1,900 .592 1,124.80
5 1,600 .519 830.40
Subtotal (a) ........................... $6,165.60
1-10 (annuity) 1,400 5.216 $7,302.40
1-5 (annuity) 1,400 3.433 -4,806.20
Subtotal (b) ........................... $2,496.20
Total Present Value (a + b) ............ $8,661.80
10. Amount Present Value Interest Factor Present Value
$1,000 1/(1 + .10)
10
= .386 $386
1,000 1/(1 + .025)40
= .372 372
1,000 1/e(.10)(10) = .368 368
11. $1,000,000 = $1,000(1 + x%)100
(1 + x%)100 = $1,000,000/$1,000 = 1,000
Taking the square root of both sides of the above equation
gives (1 + x%)50 = (FVIFAx%,50) = 31.623
Going to the FVIF table at the back of the book and looking across
the row for n = 50, we find that the interest factor for 7 percent
is 29.457, while for 8 percent it is 46.901. Therefore, the
implicit interest rate is slightly more than 7 percent.
9. Chapter 3: The Time Value of Money
12. a) Annuity of $10,000 per year for 15 years at 5 percent. The
discount factor in the PVIFA table at the end of the book is
10.380.
Purchase price = $10,000 X 10.380 = $103,800
b) Discount factor for 10 percent for 15 years is 7.606
Purchase price = $10,000 X 7.606 = $76,060
As the insurance company is able to earn more on the amount
put up, it requires a lower purchase price.
c) Annual annuity payment for 5 percent = $30,000/10.380
= $2,890
Annual annuity payment for 10 percent = $30,000/7.606
= $3,944
The higher the interest rate embodied in the yield calcula-
tions, the higher the annual payments.
13. $190,000 = R(PVIFA17%,20) = R(5.628)
R = $190,000/5.628 = $33,760
12. Chapter 3: The Time Value of Money
15. $14,300 = $3,000(PVIFA15%,n)
(PVIFA15%,n) = $14,300/$3,000 = 4.767
Going to the PVIFA table at the back of the book and looking down
the column for i = 15%, we find that the discount factor for 8
years is 4.487, while the discount factor for 9 years is 4.772.
Thus, it will take approximately 9 years of payments before the
loan is retired.
16. a) $5,000,000 = R[1 + (.20/1)]5 = R(2.488)
R = $5,000,000/2.488 = $2,009,646
b) $5,000,000 = R[1 + (.20/2)]
10 = R(2.594)
R = $5,000,000/2.594 = $1,927,525
c) $5,000,000 = R[1 + (.20/4)]
20 = R(2.653)
R = $5,000,000/2.653 = $1,884,659
d) $5,000,000 = R(e)
(.20)(5) = R(2.71828)
(1)
R = $5,000,000/2.71828 = $1,839,398
17. FV of Earl's plan = ($2,000) x (FVIFA7%,10) x (FVIF7%,35)
= ($2,000) x (13.816) x (10.677)
= $295,027
FV of Ivana's plan = ($2,000) x (FVIFA7%,35)
= ($2,000) x (138.237)
= $276,474
Earl's investment program is worth ($295,027 - $276,474) = $18,553
more at retirement than Ivana's program.
13. Chapter 3: The Time Value of Money
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 assumed $1,000 deposits never occurred.
So, we need to then subtract three future values from our "trial" ending
balance: 1) the future value of $1,000 compounded for 25 - 5 = 20 years; 2)
the future value of $1,000 compounded for 25 - 7 = 18 years; and 3) the
future value of $1,000 compounded for 25 - 11 = 14 years. After collecting
terms, we get the following:
FV25 = $1,000[(FVIFA5%,25) - (FVIF5%,20) - (FVIF5%,18) - (FVIF5%,14)]
= $1,000[ (47.727) - (2.653) - (2.407) - (1.980) ]
$1,000[40.687] = $40,687
8. There are many ways to solve this problem correctly. Here are two:
Cashwithdrawals at the END of year ...
0 1 2 3 4 5 6 7 8 9
R R R R R R
Alt. #1 This above pattern is equivalent to ...
R R R R R R R R R
PVA9
-- minus --
R R R
PVA3
PVA9 - PVA3 = $100,000
R(PVIFA.05,9) - R(PVIFA.05,3) = $100,000
R(7.108) - R(2.723) = $100,000
R(4.385) = $100,000
R = $100,000/(4.385) = $22,805.02
14. Chapter 3: The Time Value of Money
Cash withdrawals at the END of year ...
0 1 2 3 4 5 6 7 8 9
R R R R R R
Alt. #2 This above pattern is equivalent to ...
R R R R R R
$100,000
PVA6
PVA6 x (PVIF.05,3) = $100,000
R(PVIFA.05,6) x (PVIF.05,3) = $100,000
R(5.076) x (.864) = $100,000
R(4.386) = $100,000
R = $100,000/(4.386) = $22,799.82
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 + [.096/1])
1
- 1 = .0960
b. (semiannually) = (1 + [.096/2])2
- 1 = .0983
c. (quarterly) = (1 + [.096/4])
4
- 1 = .0995
d. (monthly) = (1 + [.096/12])
12
- 1 = .1003
e. (daily) = (1 + [.096/365])
365
- 1 = .1007
Effective annual interest rate
with continuous compounding = (e)i – 1
f. (continuous) = (2.71828).096 - 1 = .1008
15. Chapter 3: The Time Value of Money
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 + .08)($25,000)(PVIFA8%,40)
= (1.08)($25,000)(11.925) = $321,975
Alt. 2: PVAD40 = ($25,000)(PVIFA8%,39) + $25,000
= ($25,000)[(1 - [1/(1 + .08)
39])/.08] + $25,000
= ($25,000)(11.879) + $25,000 = $321,950
NOTE: Answers to Alt. 1 and Alt. 2 differ slightly due to rounding.
22. For approximate answers, we can make use of the "Rule of 72"
as follows:
i) 72/14 = 5.14 or 5 percent (to the nearest whole percent)
ii) 72/8 = 9 percent
iii) 72/2 = 36 percent
For greater accuracy, we proceed as follows:
i) (1 + i)
14 = 2
(1 + i) = 21/14 = 2.07143 = 1.0508
i = 5 percent (to the nearest whole percent)
ii) (1 + i)8 = 2
(1 + i) = 21/8 = 2.125 = 1.0905
i = 9 percent (to the nearest whole percent)
iii) (1 + i)
2 = 2
(1 + i) = 21/2 = 2.5 = 1.4142
i = 41 percent (to the nearest whole percent)
Notice how the "Rule of 72" does not work quite so well for high rates
of growth such as that seen in situation (iii).
16. Chapter 3: The Time Value of Money
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SOLUTIONS TO SELF-CORRECTION PROBLEMS
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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):
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FV5 FOR INDIVIDUAL CASH FLOWS RECEIVED AT TOTAL
CASH-FLOW END OF YEAR FUTURE
STREAM 1 2 3 4 5 VALUE
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W $146.40 $266.20 $242 $330 $ 300 $1,284.60
X 878.40 -- -- -- -- 878.40
Y -- -- -- -- 1,200 1,200.00
Z 292.80 -- 605 -- 300 1,197.80
__________
______________________________________________________________
b. Present value of each cash flow and total present value of
each stream (using Table II in the end-of-book Appendix):
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PV0 FOR INDIVIDUAL CASH FLOWS RECEIVED AT TOTAL
CASH-FLOW END OF YEAR PRESENT
STREAM 1 2 3 4 5 VALUE
________________________________________________________________________
W $ 87.70 $153.80 $135.00 $177.60 $155.70 $709.80
X 526.20 -- -- -- -- 526.20
Y -- -- -- -- 622.80 622.80
Z 175.40 -- 337.50 -- 155.70 668.60
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2. a. FV10 Plan 1 = $500(FVIFA3.5%,20)
= $500([(1 + .035)20 - 1]/[.035]) = $14,139.84
b. FV10 Plan 2 = $1,000(FVIFA7.5%,10)
= $1,000([(1 + .075)
10 - 1]/[.075]) = $14,147.09
17. Chapter 3: The Time Value of Money
c. Plan 2 would be preferred by a slight margin -- $7.25.
d. FV10 Plan 2 = $1,000(FVIFA7%,10)
= $1,000([1 + .07)10 - 1]/[.07]) = $13,816.45
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 provide an equivalent $50,000 cash flow in
year 12. In short, $25,000 would double in 6 years. Using the
"Rule of 72," 72/6 = 12 percent.
Alternatively, note that $50,000 = $25,000(FVIFX%,6). Therefore,
(FVIFX%,6) = $50,000/$25,000 = 2. In Table I in the Appendix at
the end of the book, the interest factor for 6 years at 12 percent
is 1.974 and that for 13 percent is 2.082. Interpolating, we have
2.000 - 1.974
X% = 12% + ___________________________ = 12.24%
2.082 - 1.974
as the interest rate implied in the contract.
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:
(1 + i)6 = 2
(1 + i) = 2
1/6 = 2
.1667 = 1.1225
i = .1225 or X% = 12.25%
4. a. PV0 = $7,000(PVIFA6%,20) = $7,000(11.470) = $80,290
b. PV0 = $7,000(PVIFA8%,20) = $7,000(19.818) = $68,726
5. a. PV0 = $10,000 = R(PVIFA14%,4) = R(2.914)
Therefore, R = $10,000/2.914 = $3,432 (to the nearest dollar).
18. Chapter 3: The Time Value of Money
b.
(1) (2) (3) (4)
ANNUAL PRINCIPAL PRINCIPAL AMOUNT
END OF INSTALLMENT INTEREST PAYMENT OWING AT YEAR END
YEAR PAYMENT (4)t-1 x .14 (1) - (2) (4)t-1 - (3)
0 -- -- -- $10,000
1 $ 3,432 $1,400 $ 2,032 7,968
2 3,432 1,116 2,316 5,652
3 3,432 791 2,641 3,011
4 3,432 421 3,011 0
$13,728 $3,728 $10,000
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:
0 1 2 3 4 19 20
|_________|_________|_________|_________|_____ _____|_________|
//
$1,000 $1,000 $1,000
TIP: Convert $1,000 every 2 years into an equivalent annual annuity
(i.e., an annuity that would provide an equivalent present or
future value to the actual cash flows) pattern. Solving for a 2-
year annuity that is equivalent to a future $1,000 to be received
at the end of year 2, we get
FVA2 = $1,000 = R(FVIFA10%,2) = R(2.100)
Therefore, R = $1,000/2.100 = $476.19. Replacing every $1,000 with
an equivalent two-year annuity gives us $476.19 for 20 years.
0 1 2 3 4 19 20
|_________|_________|_________|_________|_____ _____|_________|
//
$476.19 $476.19 $476.19 $476.19 $476.19 $476.19
PVA20 = $476.19(PVIFA10%,20) = $476.19(8.514) = $4,054.28
19. Chapter 3: The Time Value of Money
7. Effective annual
interest rate = (1 + [i/m])m -1
= (1 + [.0706/4])4 - 1 = .07249 (approx. 7.25%)
Therefore, we have quarterly compounding. And, investing $10,000
at 7.06% compounded quarterly for 7 months (Note: 7 months equals 2
and 1/3 quarter periods), we get
_
$10,000(1 + [.0706/4])2.33 = $10,000(1.041669) = $10,416.69
8. FVA65 = $1,230(FVIFA5%,65) = $1,230[([1 + .05]65 - 1)/(.05)]
= $1,230(456.798) = $561,861.54
Our "penny saver" would have been better off by ($561,861.54 -
$80,000) = $481,861.54 -- or 48,186,154 pennies -- by depositing
the pennies saved each year into a savings account earning 5
percent compound annual interest.
9. a. $50,000(.08) = $4,000 interest payment
$7,451.47 - $4,000 = $3,451.47 principal payment
b. Total installment payments - total principal payments
= total interest payments
$74,514.70 - $50,000 = $24,514.
