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3 The Time Value of Money
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):
FV5 FOR INDIVIDUAL CASH FLOWS RECEIVED TOTAL
CASH-FLOW AT END OF YEAR FUTURE
STREAM 1 2 3 4 5 VALUE
W $146.40 $266.20 $242.00 $330.00 $ 300.00 $1,284.60
X 878.40 – – – – 878.40
Y – – – – 1,200.00 1,200.00
Z 292.80 – 605.00 – 300.00 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):
PV0 FOR INDIVIDUAL CASH FLOWS RECEIVED TOTAL
CASH-FLOW AT 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
2. a. FV10 Plan 1 = $500(FVIFA 3.5%,20)
= $500{[(1 + 0.035)20 − 1]/[0.035]}
= $14,139.84
b. FV10 Plan 2 = $1,000(FVIFA 7.5%,10)
= $1,000{[(1 + 0.075)10 − 1]/[0.075]}
= $14,147.09
c. Plan 2 would be preferred by a slight margin – $7.25.
d. FV10 Plan 2 = $1,000(FVIFA 7%,10)
= $1,000{[(1 + 0.07)10 − 1]/[0.07]}
= $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%.
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 ) = 21/6 = 20.1667 = 1.1225
i = 0.1225 or X% = 12.25%
4. a. PV0 = $7,000(PVIFA 6%,20) = $7,000(11.470) = $80,290
b. PV0 = $7,000(PVIFA 8%,20) = $7,000(9.818) = $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).
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Part 2 Valuation
b.
(1) (2) (3) (4)
ANNUAL PRINCIPAL PRINCIPAL AMOUNT
END OF INSTALLMENT INTEREST PAYMENT OWING AT YEAR END
YEAR PAYMENT (4)t −1 × 0.14 (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:
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
FVA 2 = $1,000 = R (FVIFA 10%,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.
PVA 20 = $476.19(PVIFA10%,20) = $476.19(8.514) = $4,054.28
7. Effective annual interest rate = (1 + [i/m])m − 1
= (1 + [0.0706/4])4 − 1
= 0.07249 (approximately 7.25%)
Therefore, we have quarterly compounding. And, investing $10,000 at 7.06 percent
compounded quarterly for seven months (Note: Seven months equals 21⁄3 quarter periods),
we get
–
$10,000(1 + [0.0706/4])2.33 = $10,000(1.041669) = $10,416.69
8. FVA 65 = $1,230(FVIFA5%,65)
= $1,230[([1 + 0.05]65 − 1)/(0.05)]
= $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(0.08) = $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 = $24,514.70
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