ANSWERS TO PROBLEMS IN CHAPTER 5
1. a) To generate the tables, enter a column of periods, going from 1 to 30 down a
column and a row of interest rates going from 1% to 20% across a row in a
spreadsheet. Enter the future or present value formula to be calculated in the cell in
the upper left-hand corner of your table. Your formula should refer to two other cells
(not part of your column of period numbers or your row of interest rates) that contain
any specific values for the number of periods and the interest rate. Select the entire
area bounded by this upper left-hand cell, the column of period numbers down the left
side and the row of interest rates across the top. Pull down the Data menu and select
Table. The cell containing the interest rate for your present or future value formula is
your row input cell. The cell containing the period number for your present or future
value formula is your column input cell. Once you have entered these input cells, the
table will fill in. The resulting tables are shown in the spreadsheet pages labeled
“Future value of $1 after T periods,” “Present value of $1 to be received T periods
from now,” “Future value of an annuity of $1 per period for T periods,” and “Present
value of an annuity of $1 per period for T periods.”
b) Take any entry in the “Present value of $1” table, such as the present value of $1
after 20 years at 8%. Add the entries in this 8% column from 1 to 20 periods. The
result is 9.818, which agrees with the entry in the “Present value of an annuity” table
for an 8% interest rate and 20 periods. Similarly, add the entries in “Future value of
$1” table in the 8% column for Periods 1-19. The result is 44.762. Adding 1 give
45.762, which agrees with the entry in the “Future value of an annuity” table for an
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8% interest rate and 20 periods. The reason we add the entries in the future value of
$1 table for one period less than the length of the annuity is that, in the 20-period
case, the first annuity payment is reinvested for 19 periods, the second for 18 periods,
etc. The next to the last annuity payment is reinvested for one period, but the last
payment of $1 is not reinvested at all, so we simply add it to the sum of the previous
19 future value factors.
c) The present and future value tables show only a limited number of entries. The
tables we have generated here, for example, contain no fractional interest rates, even
though these are common in practice. In addition, the values in the table are rounded
off, introducing rounding error into your calculations.
2. We can use Equation (5.2) in the text, with FV/P = 2, to solve for T, the time required
to double our money. Creating a Data Table, as shown in the spreadsheet labeled
“Chap. 5, Prob. 2, we see that the required time ranges from 69.7 years at an interest
rate of 1% to 3.8 years at an interest rate of 20%. The values given by the Rule of 72,
as shown in the next column of the Data Table, are tolerable approximations
throughout this range of interest rates. The Rule of 73 produces the smallest
percentage errors for interest rates in the 7 – 9% range.
3. Using Equation (5.12), or a financial calculator set to beginning-of-period calculation,
the present value of 50/20 = $2.5 million per year for 20 years at 6% is $30.395
million. This is the minimum amount you would be willing to accept, since anything
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less than that would be insufficient to generate an 20-year annuity of $2.5 million per
year. At an interest rate of 10%, the present value, and minimum you would accept is
$23.412 million.
4. a) $100,000
b) P = 150,000/(1.08)5 = $102,087.48
c) P = 8200/.08 = $102,500
d) P 
15000 
1 
1 
  $100,651.22
.08  (1.08)10 
e) P 
14200 
1 
1 
(1.08)  $102,905.81
.08  (1.08)10 
Highest present value of the five.
5. a) The monthly interest rate is 8.4/12 = 0.7%. The mortgage represents a 360-period
annuity. The monthly payment is $1371.31.
b) When the first payment is made, the beginning balance is $180,000. The interest
portion is 0.007(180,000) = $1260 The principal portion is 1371.31 - 1260 = $111.31.
The remaining balance is 180,000 – 111.31 = $179,888.69.
c) In similar fashion:
Payment #
200
360
Beg. Balance
132,178.71
1361.78
Interest
925.25
9.53
Principal
446.06
1361.78
End. Balance
131,732.65
0
6. For the two years in business school, I have outflows of 35,000 in lost salary plus
20,000 in other expenses for a total of 55,000 per year. The future value (at 6%) of
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this two-year annuity is 113,300. My increased salary following the MBA can be
thought of as a 30-year annuity at a 6% interest rate. To break even, the present value
of this annuity must be at least 113,300. Solving for the annuity payment gives a
minimum annual salary increment after the MBA of $8231.
7. Using Equation (5.12) in the text, or with your calculator set to beginning-of-period
calculation, the value, as of Year 25, of 4 years of college costs at $80,000 per year,
with a 5% interest rate, is $297,860. We need to find the annual payment on a 25-year
annuity (end-of-year payments) that has a future value of $297,860 at 5% interest.
Using a financial calculator or Equation (5.13), the answer is $6240.90.
8. You expect to live 360 monthly periods. The monthly interest rate is .06/12 = .005.
The present value of your annuity is $2 million. Using Equation (5.11) or a financial
calculator, the monthly annuity payment is $11.991.01.
9. a) Modifying the perpetuity formula, Equation (5.8) to account for the immediate
payment, we can write:
 1
8000  12001  
 r
Solving, we find r = .1765. This rate is the APR, but since it is calculated using an
annual compounding period, it is also the EAR.
b) With monthly payments,
 1
8000  1101  
 r
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and r = .01394. The APR is 12(.01394) = .1673. However, the EAR is found from
text Equation (5.6) as EAR = .1807. Note that, because the compounding period is
monthly in this case, the EAR is not equal to the APR. Since we can only compare
rates that have been calculated using the same compounding period, the monthly
payment plan’s higher EAR means that the yearly payment plan is better for Faust.
10. a) At the end of Year 10, the present value of the payments to be made to you is
1000/.1 = $10,000. From text Equation (5.13), the value at Year 10 of the payments
you are to make to the annuity company is


1000
(1.1)10  1  $15,937.42
.1
Thus the offer is not a good deal.
b) For this to be a good deal, the future value of the outflows, as of Year 10, must be
no greater than the present value of the inflows at the same time. The breakeven
interest rate is thus given by:


1000
1000
(1  r )10  1 
r
r
This implies r = .0718. For any interest rate below 7.18%, the proposition is
acceptable.
11. a) In 30 years, you expect your salary to be 100,000(1.04)29 = $311,865.15 (i.e.,
salary for Year 1 is $100K, and there are 29 years of growth after that), so your
annual pension payment is 25% of that, or $77,966.29.
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b) In Year 30, the value of 20 of these annual pension payments at 8% is
$765,484.49. If the pension is eliminated, you will need an annual salary bonus with a
future value in 30 years of $765,484.49 to compensate you. The annual payment on a
30-year annuity with a future value of $765,484.49 is $6,757.26, which is the amount
of the breakeven bonus.
12. a) Using a financial calculator, IRRA = 50% and IRRB = 24.7%.
b) At a discount rate of 10%, we can use a financial calculator to find NPVA = 85.95
and NPVB = 112.84. Because of its higher NPV, Project B is preferred at a discount
rate of 10%. Project A promises a higher yearly return, but it does so for only two
years. Because 10% is investors’ opportunity cost, it is safest to assume that any
money returned to shareholders could be invested to earn only 10% per year.
However, Project B offers a yearly return substantially in excess of 10% and does so
over a period of 20 years. If the company could keep on repeating Project A every
two years, then it would be preferred. However, if Project A is a one-shot deal, we
would prefer Project B at a 10% cost of capital.
13. a) Using a financial calculator, IRRA = 10.73% and IRRB = 46.17%.
b) The net present value profiles are shown on the graph labeled “Chap. 5, Problem
14, NPV Profiles.” For discount rates less than the rate at which the two profiles
intersect, Project A is preferred to Project B, because it has the higher NPV. This
intersection point appears from the graph to occur around 5%. We can solve for this
point explicitly if we define a dummy project, (A-B), whose yearly cash flows equal
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the cash flow from A minus the cash flow from B. The IRR of this dummy project is
5.22%, which is the intersection, or crossover, point between the two NPV profiles.
For discount rates between 5.22% and 46.17%, Project B is preferred to A, since it
has the larger NPV. For discount rates greater than 46.17%, neither project is
acceptable, since both have a negative NPV.
c) Project A’s cash flows are more heavily loaded toward the later years than are
Project B’s. At low discount rates, these later cash flows are worth more in present
value terms, and Project A has the higher NPV. Project B’s cash flows are front-end
loaded, on the other hand, so it is preferred at higher discount rates, although at rates
in excess of 46.17%, neither project is acceptable.
14. a) A financial calculator gives the IRR as 25.10%.
b) Using the IRR function in Excel and a guess of 10%, we find an IRR of 25.10%.
However, using a guess of 150%, we find an IRR of 149.34%.
c) The NPV profile (see “Chap. 5, Prob. 15: NPV Profile”) shows that the project has
two IRR values. The project is acceptable (i.e., NPV > 0) for all discount rates less
than 25.1% and for all rates in excess of 149.34%. However, the project is
unacceptable for discount rates between 25.1% and 149.34%.