A-1 Solutions
APPENDIX
TIME VALUE OF CASH FLOWS
Questions, Exercises, and Problems: Answers and Solutions
A.2 The value of cash flows differs over time because cash can earn interest.
A.3 In simple interest, only the principal sum earns interest. In compound
A.4 There is no difference; these items refer to the same thing.
A.5 The timing of the first payment for an annuity due is now (at the beginning
(1) Guess a rate.
(2) Compute the net present values of the cash flows using the current
guess.
Solutions A-2
(5) Otherwise, the current guess is the implicit rate of return.
A-3 Solutions
A.9 The formula assumes that the growth [represented by the parameter g in
the formula 1/(r – g)] continues forever. That is a long time. The formula
A.10 a. $5,000 X 3.20714 X 1.06 = $16,998.
A.14 a. ¤90,000 X 14.20679 X 1.05 = ¤90,000 X (15.91713 – 1.0) = ¤1,342,542.
b. ¤90,000 X 18.53117 X 1.10 = ¤90,000 X (21.38428 – 1.0) = ¤1,834,585.
Solutions A-4
A.16 continued.
b. Asking questions about compound interest calculations on
examinations presents a difficult logistical problem to teachers. They
A.17 (Effective interest rates.)
a. 12% per period; 5 periods.
A.18 a. $100 X 1.21665 = $121.67.
A.19 a. $100 X .30832 = $30.83.
A.20 a. $100 X 14.23683 = $1,423.68.
A-5 Solutions
Solutions A-6
A.21 a. $1,000(1.00 + .94340) + $2,000(4.21236 – .94340) + $2,500(6.80169 –
4.21236) = $14,955.
3.79079) = $12,593.
A.22 a. $3,000 + ($3,000/.06) = $53,000.
A.24 a. $60,000 + ($60,000/.1664) = $420,577. (1.08)2 – 1 = .1664.
b. $60,000 + ($60,000/.2544) = $295,850. (1.12)2 – 1 = .2544.
A.27 a. 16% = ($67,280/$50,000)1/2 – 1.
b. Amount
Carrying (Reducing) Carrying
Value Interest Increasing Value
Start for Year Carrying End of Year
A-7 Solutions
A.28 (Berman Company; find implicit interest rate; construct amortization
schedule.)
b. Amount
(Reducing)
Carrying Payment Increasing Carrying
Value Interest End of Carrying Value
Start for Year Year Value End of Year
Year of Year = (2) X .14 (Given) = (3) – (4) = (2) + (5)
(1) (2) (3) (4) (5) (6)
1 $ 86,000 $ 12,040 $ 8,000 $ 4,040 $ 90,040
2 90,040 12,605 8,000 4,605 94,645
A.29 a. Terms of sale of 2/10, net/30 on a $100 gross invoice price, for example,
b. Table 1 can be used. Use the 2% column and the 18-period row to see
that the rate implied by 2/10, net 30 must be at least 42.825% (=
1.42825 – 1).
A.30 (Present value of a perpetuity.)
Solutions A-8
A-9 Solutions
A.32 a. Will: $24,000 + $24,000(3.31213) = $103,488.72 (Preferred).
Dower Option: $300,000/3 = $100,000.
A.34 $1.00(1.00 + .92456 + .85480 + .79031 + .73069) = $1.00 X 4.30036 = $4.30.
A.35 $600/12 = $50 saved per month. $2,000/$50 = 40.0.
A.36 a. $ 3,000,000 X 7.46944 = $ 22,408,320.
Solutions A-10
A.37 (Friendly Loan Company; find implicit interest rate; truth-in lending laws
reduce the type of deception suggested by this problem.)
The effective interest rate is 19.86% and must be found by trial and error.
A.38 (Black & Decker Company; derive net present value/cash flows for decision
to dispose of asset.)
$40,698. The $100,000 is gone and an economic loss of $50,000 was
A-11 Solutions
Solutions A-12
A.39 (Lynch Company/Bages Company; computation of present value of cash
flows; untaxed acquisition, no change in tax basis of assets.)
A.40 (Lynch Company/Bages Company; computation of present value of cash
flows; taxable acquisition, changing tax basis of assets.)
$4,258,199. If the merger is taxable, then the value of the firm V satisfies:
(1) V = 8.51356 X [$700,000 – .40($700,000 – V/20)]
A.41 (Valuation of intangibles with perpetuity formulas.)
a. $50 million = $4 million/.08.
A-13 Solutions
A.42 (Ragazze; analysis of benefits of acquisition of long-term assets.)
a. $270,831.
Cash Cash
Inflows Outflows Total Present
Operating Test (1) + (2) Values at 12%
Dec. 31 Receipts Salvage Maintenance Runs – (3) – (4) Factor Cash Flow
Year (1) (2) (3) (4) (5) (6) (7)
Solutions A-14
A-15 Solutions
A.43 (Gulf Coast Manufacturing; choosing between investment alternatives.)
Basic Data Repeated from Problem Present Value Computations
Mercedes- Mercedes-
Lexus Benz Factor Source [B] Lexus Benz
Initial Cost at the Start of 2008 …………. $60,000 $45,000 1.00000 $ 60,000 $ 45,000
[B]T[i,j,r] means Table i (= Table 2 or Table 4) from the back of the book, row j, interest rate r.
a. Strategy L, buying one Lexus has lower present value of costs, but the difference is so small that we’d encourage the CEO to go
Solutions A-16
A.44 (Wal-Mart Stores; perpetuity growth model derivation of results in Chapter
6.)
a. Reproduce Exhibit 6.22 for Problem A.43.
Growth Rate for Terminal Value: 10.0%
Factor to Present
Discount Value at
End of Cash to End of End of 2008
Year Flow 2008 Origin of Factor 14 = [2] X [3]
[1] [2] [3] [4] [5]
Note A: ($7,884 X 1.10)/(.12 – .10) = $433,620.
Numerator is the amount of the first collection, at the end of
2014.
A-17 Solutions
A.44 continued.
b. Change Growth Rate.
Growth Rate for Terminal Value: 9.0%
Factor to Present
Discount Value at
End of Cash to End of End of 2008
Year Flow 2008 Origin of Factor 14 = [2] X [3]
[1] [2] [3] [4] [5]
c. Change Growth Rate.
Growth Rate for Terminal Value: 5.0%
Factor to Present
Discount Value at
End of Cash to End of End of 2008
Year Flow 2008 Origin of Factor 14 = [2] X [3]
[1] [2] [3] [4] [5]
Solutions A-18
A.44 continued.
d. Comment. In models such as this, the total valuation comes largely
from the terminal value. The terminal value changes more than
A.45 (Fast Growth Start–Up Company; valuation involving perpetuity growth
model assumptions. (Amounts in Millions)
We find the answer with trial and error, starting with 5 years of fast
Free Discount
End of Cash Factors Present Value
Year Flow from Table 2 End of Year 0
0 $ 100 1.00000 $ 100.0
1 125 0.86957 108.7
A-19 Solutions
A.45 continued.
Growth Rate for Early Years of Fast Growth: 25%
Growth Rate for Steady State, Terminal Value: 4%
Discount Rate: 15%
Number of Years of Fast Growth: 6
Free Discount
End of Cash Factors Present Value
Year Flow from Table 2 End of Year 0
0 $ 100 1.00000 $ 100.0
1 125 0.86957 108.7
2 156 0.75614 118.1
Free Discount
End of Cash Factors Present Value
Year Flow from Table 2 End of Year 0
0 $ 100 1.00000 $ 100.0
1 125 0.86957 108.7
Solutions A-20