Answers and Solutions: 4 -1
Chapter 4
Time Value of Money
ANSWERS TO END-OF-CHAPTER QUESTIONS
4-1 a. PV (present value) is the value today of a future payment, or stream of payments,
discounted at the appropriate rate of interest. PV is also the beginning amount that will
grow to some future value. The parameter i is the periodic interest rate that an account
pays. The parameter INT is the dollars of interest earned each period. FVn (future
value) is the ending amount in an account, where n is the number of periods the money
is left in the account. PVAn is the value today of a future stream of equal payments (an
annuity) and FVAn is the ending value of a stream of equal payments, where n is the
number of payments of the annuity. PMT is equal to the dollar amount of an equal, or
constant cash flow (an annuity). In the EAR equation, m is used to denote the number
of compounding periods per year, while iNom is the nominal, or quoted, interest rate.
b. The opportunity cost rate (i) of an investment is the rate of return available on the best
alternative investment of similar risk.
d. An ordinary annuity has payments occurring at the end of each period. A deferred
annuity is just another name for an ordinary annuity. An annuity due has payments
occurring at the beginning of each period. Most financial calculators will accommodate
either type of annuity. The payment period must be equal to the compounding period.
Answers and Solutions: 4 – 2
g. Compounding is the process of finding the future value of a single payment or series
of payments. Discounting is the process of finding the present value of a single
payment or series of payments; it is the reverse of compounding.
i. The effective annual rate is the rate that, under annual compounding, would have
produced the same future value at the end of 1 year as was produced by more frequent
compounding, say quarterly. The nominal (quoted) interest rate, iNom, is the rate of
interest stated in a contract. If the compounding occurs annually, the effective annual
rate and the nominal rate are the same. If compounding occurs more frequently, the
effective annual rate is greater than the nominal rate. The nominal annual interest rate
is also called the annual percentage rate, or APR. The periodic rate, iPER, is the rate
charged by a lender or paid by a borrower each period. It can be a rate per year, per 6–
month period, per quarter, per month, per day, or per any other time interval (usually
one year or less).
4-2 The opportunity cost rate is the rate of interest one could earn on an alternative investment
with a risk equal to the risk of the investment in question. This is the value of i in the TVM
equations, and it is shown on the top of a time line, between the first and second tick marks.
It is not a single rate—the opportunity cost rate varies depending on the riskiness and
maturity of an investment, and it also varies from year to year depending on inflationary
expectations.
10.
4-4 True, because of compounding effects—growth on growth. The following example
demonstrates the point. The annual growth rate is I in the following equation:
$1(1 + I)10 = $2.
Answers and Solutions: 4 -3
The term (1 + I)10 is the FVIF for I percent, 10 years. We can find I in one of two ways:
1. Using a financial calculator input N = 10, PV = -1, PMT = 0, FV = 2, and I/YR = ?.
Solving for I/YR you obtain 7.18%.
4-5 For the same stated rate, daily compounding is best. You would earn more “interest on
interest.”
Answers and Solutions: 4 – 4
SOLUTIONS TO END–OF-CHAPTER PROBLEMS
4-1 0 1 2 3 4 5
| | | | | |
Cash: PV = 10,000 FV5 = ?
FV5 = $10,000(1.10)5
= $10,000(1.61051) = $16,105.10.
Alternatively, with a financial calculator enter the following: N = 5, I/YR = 10, PV = –
10000, and PMT = 0. Solve for FV = $16,105.10.
7%
4-3 0 18
| |
Cash: PV = $250,000 FV18 = $1,000,000
With a financial calculator enter the following: N = 18, PV = -250000, PMT = 0, and FV
= 1000000. Solve for I/YR = 8.01% ≈ 8%.
4-5 0 1 2 N – 2 N – 1 N
| | | • • • | | |
PV = $42,180.53 $5,000 $5,000 $5,000 $5,000 FV = 250,000
Using your financial calculator, enter the following data: I/YR = 12; PV = -42180.53;
PMT = -5000; FV = 250000; N = ? Solve for N = 11. It will take 11 years to accumulate
$250,000.
12%
I/YR = ?
10%
Answers and Solutions: 4 -5
4-6 Ordinary annuity:
0 1 2 3 4 5
| | | | | |
–300 –300 3–00 –300 -300
FVA5 = ?
4-7 0 1 2 3 4 5 6
| | | | | | |
100 100 100 200 300 500
Cash: PV = ? FV = ?
Using a financial calculator, enter the following: CF0 = 0; CF1 = 100; Nj = 3; CF4 = 200
(Note calculator will show CF2 on screen.); CF5 = 300 (Note calculator will show CF3 on
screen.); CF6 = 500 (Note calculator will show CF4 on screen.); and I/YR = 8. Solve for
NPV = $923.98.
To solve for the FV of the cash flow stream with a calculator that doesn’t have the
NFV key, do the following: Enter N = 6, I/YR = 8, PV = -923.98, and PMT = 0. Solve
for FV = $1,466.24.
4-8 Using a financial calculator, enter the following: N = 60, I/YR = 1, PV = -20000, and FV
= 0. Solve for PMT = $444.89.
7%
8%
4-9 a. 0 1
| | $500(1.06) = $530.00.
-500 FV = ?
b. 0 1 2
| | | $500(1.06)2 = $561.80.
-500 FV = ?
4-10 a. 0 1 2 3 4 5 6 7 8 9 10 $500(1.06)10 = $895.42.
| | | | | | | | | | |
Cash: -500 FV = ?
b. 0 1 2 3 4 5 6 7 8 9 10 $500(1.12)10 = $1,552.92.
| | | | | | | | | | |
Cash: -500 FV = ?
12%
6%
6%
6%
12%
Answers and Solutions: 4 -7
4-11 a. ?
| |
Cash: $-200 $400
With a financial calculator, enter I/YR = 7, PV = -200, PMT = 0, and FV = 400. Then
press the N key to find N = 10.24 ≈ 10.
c. ?
| |
Cash: $-200 $400 .
With a financial calculator, enter I/YR = 18, PV = -200, PMT = 0, and FV = 400. Then
press the N key to find N = 4.19 ≈ 4.
4-12
a. 0 1 2 3 4 5 6 7 8 9 10
| | | | | | | | | | |
400 400 400 400 400 400 400 400 400 400
FVA10 = ?
7%
18%
10%
Answers and Solutions: 4 – 8
b. 5%
0 1 2 3 4 5
| | | | | |
200 200 200 200 200
FVA5 = ?
With a financial calculator, enter N = 5, I/YR = 5, PV = 0, and PMT =
-200. Then press the FV key to find FV = $1,105.13.
d. To solve Part d using a financial calculator, repeat the procedures discussed in Parts a,
b, and c, but first switch the calculator to “BEG” mode. Make sure you switch the
calculator back to “END” mode after working the problem.
(1) 0 1 2 3 4 5 6 7 8 9 10
| | | | | | | | | | |
400 400 400 400 400 400 400 400 400 400 FVA10 = ?
With a financial calculator set to “BEG” mode, enter N = 10, I/YR = 10, PV = 0, and
PMT = -400. Then press the FV key to find FV = $7,012.46.
(2) 0 1 2 3 4 5
| | | | | |
200 200 200 200 200 FVA5 = ?
10%
5%
Answers and Solutions: 4 -9
4-13
a. 0 1 2 3 4 5 6 7 8 9 10
| | | | | | | | | | |
PV = ? 400 400 400 400 400 400 400 400 400 400
With a financial calculator, enter N = 10, I/YR = 10, PMT = -400, and FV = 0. Then
press the PV key to find PV = $2,457.83.
c. 0 1 2 3 4 5
| | | | | | $400(5) = $2,000.00.
PV = ? 400 400 400 400 400
With a financial calculator, enter N = 5, I/YR = 0, PMT = -400, and FV = 0. Then
press the PV key to find PV = $2,000.
d. (1) 0 1 2 3 4 5 6 7 8 9 10
| | | | | | | | | | |
400 400 400 400 400 400 400 400 400 400
PV = ?
10%
Answers and Solutions: 4 – 10
4-14 a. Cash Stream A Cash Stream B
0 1 2 3 4 5 0 1 2 3 4 5
| | | | | | | | | | | |
PV = ? 100 400 400 400 300 PV = ? 300 400 400 400 100
With a financial calculator, simply enter the cash flows (be sure to enter CF0 = 0), enter
I/YR = 8, and press the NPV key to find NPV = PV = $1,251.25 for the first problem.
Override I = 8 with I = 0 to find the next PV for Cash Stream A. Repeat for Cash Stream
B to get NPV = PV = $1,300.32.
b. PVA = $100 + $400 + $400 + $400 + $300 = $1,600.
PVB = $300 + $400 + $400 + $400 + $100 = $1,600
8%
8%
Answers and Solutions: 4 -11
d. 0 1 2 3 4 5
| | | | | |
+9,000 -2,684.80 -2,684.80 -2,684.80 -2,684.80 -2,684.80
With a financial calculator, enter N = 5, PV = 9,000, PMT = -2,684.8, and FV = 0.
Then press the I/YR key to find I/YR = 15%.
4-16 a. 0 12% 1 2 3 4 5
| | | | | |
-500 FV = ?
With a financial calculator, enter N = 5, I/YR = 12, PV = -500, and PMT = 0, and then
press FV to obtain FV = $881.17.
c. 0 4 8 12 16 20
| | | | | |
-500 FV = ?
With a financial calculator, enter N = 20, I/YR = 3, PV = -500, and PMT = 0, and then
press FV to obtain FV = $903.06.
I = ?
3%
Answers and Solutions: 4 – 12
4-17 a. 0 2 4 6 8 10
| | | | | |
PV = ? 500
With a financial calculator, enter N = 10, I/YR = 6, PMT = 0, and FV = –500. Then
press the PV key to find PV = $279.20.
06.1
b. 0 4 8 12 16 20
| | | | | |
PV = ? 500
With a financial calculator, enter N = 20, I/YR = 3, PMT = 0, and FV = -500. Then
press the PV key to find PV = $276.84, or
PV = $500
)5(4
4
12.0
1
1
+
= $500
20
03.1
1
= $276.84.
c. 0 1 2 12
| | | • • • |
PV = ? 500
6%
3%
1%
Answers and Solutions: 4 -13
c. The annuity payments in Part b occur more frequently than those in Part a, which means
that interest is earned on interest more frequently. In addition, the first annuity payment
for Part b occurs earlier in the year than the first payment for the annuity in Part a, so
interest on Part b’s Month-3 payment begins compounding before interest begins
compounding on Part a’s Month–6 semiannual payment. The same is true for Part b’s
Month–9 payment relative to Part a’s Month-12 end-of-year payment
4-19 a. Universal Bank: Effective rate = 7%.
b. If funds must be left on deposit until the end of the compounding period (1 year for
Universal and 1 quarter for Regional), and you think there is a high probability that you
will make a withdrawal during the year, the Regional account might be preferable. For
example, if the withdrawal is made after 10 months, you would earn nothing on the
Universal account but (1.015)3 – 1.0 = 4.57% on the Regional account.
Ten or more years ago, most banks and S&Ls were set up as described above, but
now virtually all are computerized and pay interest from the day of deposit to the day
of withdrawal, provided at least $1 is in the account at the end of the period.
Answers and Solutions: 4 – 14
4-20 a. With a financial calculator, enter N = 5, I/YR = 10, PV = -25000, and FV = 0, and then
press the PMT key to get PMT = $6,594.94. Then go through the amortization
procedure as described in your calculator manual to get the entries for the amortization
table.
Repayment Remaining
Year Payment Interest of Principal Balance
1 $ 6,594.94 $2,500.00 $ 4,094.94 $20,905.06
2 6,594.94 2,090.51 4,504.43 16,400.63
*The last payment must be smaller to force the ending balance to zero.
b. Here the loan size is doubled, so the payments also double in size to $13,189.87: enter
N = 5, I/YR = 10, PV = –50000, and FV = 0, and then press the PMT key to get PMT
= $13,189.87.
4-21 a. 0 I=? 1 2 3 4 5
| | | | | |
-6 12 (in millions)
With a calculator, enter N = 5, PV = -6, PMT = 0, FV = 12, and then solve for I/YR =
14.87% ≈ 15%.
b. The calculation described in the quotation fails to take account of the compounding
effect. It can be demonstrated to be incorrect as follows:
Answers and Solutions: 4 -15
4-22 0 1 2 3 4 5 6 7 8 9 10
| | | | | | | | | | |
-4 8 (in millions)
With a financial calculator, enter N = 10, PV = –4, PMT = 0, FV = 8, and then solve for
I/YR = 7.18%.
4-24 a. 0 1 2 3 4
| | | | |
PV = ? -10,000 -10,000 -10,000 -10,000
With a calculator, enter N = 4, I/YR = 7, PMT = -10000, and FV = 0. Then press PV
to get PV = $33,872.11.
b. (1) At this point, we have a 3-year, 7% annuity of $10,000 whose present value is
$26,243.16: N = 3, I/YR = 7, PMT = -10000, and FV = 0. Then press PV to get
PV = $26,243.16. You can also think of the problem as follows:
(Beginning balance)(1+I) – PMT = Ending balance
$33,872.11 (1.07) ─ $10,000 = $26,243.16.
(2) Zero after the last withdrawal.
I = ?
7%
Answers and Solutions: 4 – 16
4-26 0 1 2 3 4 5 6
| | | | | | |
1,250 1,250 1,250 1,250 1,250 ?
FV = 10,000
With a financial calculator, get a “ballpark” estimate of the years by entering I/YR = 12,
PV = 0, PMT = -1250, and FV = 10000, and then pressing the N key to find N = 5.94 years.
This answer assumes that a payment of $1,250 will be made 94/100th of the way through
4-27 PV = $100/0.07 = $1,428.57. PV = $100/0.14 = $714.29.
When the interest rate is doubled, the PV of the perpetuity is halved.
4-28 0 1 2 3 4
| | | | |
PV = ? 50 50 50 1,050
Discount rate: Effective rate on bank deposit:
8.24%
12%
Answers and Solutions: 4 -17
4-29 This can be done with a calculator by specifying an interest rate of 5% per period for 20
periods with 1 payment per period to get the payment each 6 months: N = 10 2 = 20,
I/YR = 10%/2 = 5, PV = -10000. FV = 0. Solve for PMT = $802.43. Set up amortization
table as below:
Pmt of Pmt of
Period Beg Bal Payment Interest Principal End Bal
1 $10,000.00 $802.43 $500.00 $302.43 $9,697.57
You can also work the problem with a calculator having an amortization function. Find
the interest in each 6-month period, sum them, and you have the answer. Even simpler,
4-30 First, find PMT by using a financial calculator: N = 5, I/YR = 15, PV = -1000000, and FV
= 0. Solve for PMT = $298,315.55. Then set up the amortization table:
Beginning Ending
Year Balance Payment Interest Principal Balance
1 $1,000,000.00 $298,315.55 $150,000.00 $148,315.55 $851,684.45
4-31 a. Begin with a time line:
6-mos. 0 1 2 3 4 5 6 8 10 12 14 16 18 20
Years 0 6% 1 2 3 4 5 6 7 8 9 10
| | | | | | | | | | | | | | | | | | | | |
100 100 100 100 100 FVA
Answers and Solutions: 4 – 18
b. 1 10 years
0 1 2 3 4 5 40 quarters
| | | | | | • • • |
PMT PMT PMT PMT PMT FV = 1,432.02
(1) Discount the $1,432.02 back to the end of Quarter 5 to obtain the PV of that future
amount at Quarter 5.
(2) Then solve for PMT using the value solved in Step 1 as the FV of the five-period
annuity due.
Step 1: Input the following into your calculator: N = 35, I/YR = 3, PMT = 0, FV =
1432.02, and solve for PV at Quarter 5. PV = $508.92.
3%
Answers and Solutions: 4 -19
4-32 Here we want to have the same effective annual rate on the credit extended as on the bank
loan that will be used to finance the credit extension.
the cost of the bank loan will be covered.
Alternative solution: We need to find the effective annual rate (EAR) the bank is
charging first. Then, we can use this EAR to calculate the nominal rate that should be
quoted to the customers.
Bank EAR: EAR = (1 + INOM/M)M – 1 = (1 + 0.15/12)12 – 1 = 16.08%.
Nominal rate that should be quoted to customers:
Answers and Solutions: 4 – 20
4-33 Information given:
1. Will save for 10 years, then receive payments for 25 years.
3. He now has $100,000 in an account which pays 8 percent, annual compounding. We
need to find the FV of the $100,000 after 10 years. Enter N = 10, I/YR = 8, PV = –
100000, PMT = 0, and press FV to get FV = $215,892.50.
4. He wants to withdraw, or have payments of, $65,155.79 per year for 25 years, with the
first payment made at the beginning of the first retirement year. So, we have a 25-year
annuity due with PMT = 65,155.79, at an interest rate of 8 percent. (The interest rate
is 8 percent annually, so no adjustment is required.) Set the calculator to “BEG” mode,
then enter N = 25, I/YR = 8, PMT = 65155.79, FV = 0, and press PV to get PV =
$751,165.35. This amount must be on hand to make the 25 payments.