Chapter 28
Time Value of Money
ANSWERS TO END-OF-CHAPTER QUESTIONS
28-1 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
b. The opportunity cost rate (i) of an investment is the rate of return available on the best
alternative investment of similar risk.
c. An annuity is a series of payments of a fixed amount for a specified number of periods.
A single sum, or lump sum payment, as opposed to an annuity, consists of one payment
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: 28 -1
representation which is used to show the timing of cash flows. The terminal value is
the future value of an uneven cash flow stream.
g. Compounding is the process of finding the future value of a single payment or series
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
j. An amortization schedule is a table that breaks down the periodic fixed payment of an
28-2 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
28-3 True. The second series is an uneven payment stream, but it contains an annuity of $400
Answers and Solutions: 28 – 2
28-4 True, because of compounding effects—growth on growth. The following example
28-5 For the same stated rate, daily compounding is best. You would earn more “interest on
interest.“
Answers and Solutions: 28 -3
SOLUTIONS TO END-OF-CHAPTER PROBLEMS
28-1 0 1 2 3 4 5
| | | | | |
PV = 10,000 FV5 = ?
28-2 0 5 10 15 20
| | | | |
PV = ? FV20 = 5,000
28-4 0 N = ?
| |
PV = 1
FVN = 2
$2 = $1(1.065)N.
7%
6.5%
10%
Answers and Solutions: 28 – 4
28-6 Ordinary annuity:
0 1 2 3 4 5
| | | | | |
300 300 300 300 300
FVA5 = ?
28-7 0 1 2 3 4 5 6
| | | | | | |
100 100 100 200 300 500
PV = ? FV = ?
28-8 Using a financial calculator, enter the following: N = 60, I/YR = 1, PV = -20000, and FV
= 0. Solve for PMT = $444.89.
7%
8%
Answers and Solutions: 28 -5
28-9 a. 0 1
| | $500(1.06) = $530.00.
-500 FV = ?
28-10 a. 0 1 2 3 4 5 6 7 8 9 10 $500(1.06)10 = $895.42.
| | | | | | | | | | |
-500 FV = ?
6%
6%
28-11 a. ?
| |
-200 400
c. ?
| |
-200 400 .
28-12
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: 28 -7
b. 5%
0 1 2 3 4 5
c. 0 1 2 3 4 5
| | | | | |
400 400 400 400 400
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.
(3) 0 0% 1 2 3 4 5
| | | | | |
400 400 400 400 400 FVA5 = ?
0%
Answers and Solutions: 28 – 8
28-13
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.
10%
Answers and Solutions: 28 -9
28-14 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
28-15 These problems can all be solved using a financial calculator by entering the known values
shown on the time lines and then pressing the I/YR button.
a. 0 1
| |
+700 -749
8%
8%
I = ?
Answers and Solutions: 28 – 10
c. 0 10
| |
+85,000 -201,229
With a financial calculator, enter N = 10, PV = 85,000, PMT = 0, and FV = -201,229.
Then press the I/YR key to find I/YR = 9%.
28-16 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.
d. 0 12 24 36 48 60
| | | | | |
-500 ?
With a financial calculator, enter N = 60, I/YR = 1, PV = –500, and PMT = 0, and then
press FV to obtain FV = $908.35.
I = ?
1%
Answers and Solutions: 28 -11
28-17 a. 0 2 4 6 8 10
| | | | | |
PV = ? 500
b. 0 4 8 12 16 20
| | | | | |
PV = ? 500
c. 0 1 2 12
| | | • • • |
PV = ? 500
With a financial calculator, enter N = 12, I/YR = 1, PMT = 0, and FV = –500. Then
press the PV key to find PV = $443.72, or
6%
3%
1%
Answers and Solutions: 28 – 12
28-18 a. 0 1 2 3 9 10
| | | | • • • | |
400 400 400 400 400
FVA10 = ?
28-19 a. Universal Bank: Effective rate = 7%.
Regional Bank:
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
6%
Answers and Solutions: 28 -13
28-20 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
b. Here the loan size is doubled, so the payments also double in size to $13,189.87: enter
c. The annual payment on a $50,000, 10-year loan at 10 percent interest would be
28-21 a. 0 I=? 1 2 3 4 5
| | | | | |
-6 12 (in millions)
Answers and Solutions: 28 – 14
28-23 0 1 2 3 4 30
| | | | | • • • |
85,000 -8,273.59 -8,273.59 -8,273.59 -8,273.59 -8,273.59
With a calculator, enter N = 30, PV = 85000, PMT = -8273.59, FV = 0, and then solve for
I/YR = 9%.
28-24 a. 0 1 2 3 4
| | | | |
PV = ? -10,000 -10,000 -10,000 -10,000
28-25 0 1 2 ?
| | | • • • |
12,000 -1,500 -1,500 -1,500
I = ?
7%
9%
Answers and Solutions: 28 -15
I = ?
28-26 0 1 2 3 4 5 6
| | | | | | |
1,250 1,250 1,250 1,250 1,250 ?
FV = 10,000
28-27 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.
28-28 0 1 2 3 4
| | | | |
PV = ? 50 50 50 1,050
8.24%
12%
Answers and Solutions: 28 – 16
28-29 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,
28-30 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
28-31 a. Begin with a time line:
Since the first payment is made today, we have a 5-period annuity due. The applicable
interest rate is I = 12/2 = 6 per period, N = 5, PV = 0, and PMT = –100. Setting the
calculator on “BEG,” we find FVA (Annuity due) = $597.53. That will be the value at
Answers and Solutions: 28 -17
b. 1 10 years
0 1 2 3 4 5 40 quarters
| | | | | | • • • |
PMT PMT PMT PMT PMT FV = 1,432.02
The time line depicting the problem is shown above. Because the payments only
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: 28 – 18
28-32 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.
First, we must find the EAR = EFF% on the bank loan. Enter NOM% = 15, N = P/YR
= 12, and press EFF% to get EAR = 16.08%.
Now recognize that giving 3 months of credit is equivalent to quarterly compounding-
Nominal rate that should be quoted to customers:
Answers and Solutions: 28 -19
28-33 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
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
6. The $535,272.85 is the FV of a 10-year ordinary annuity. The payments will be
deposited in the bank and earn 8 percent interest. Therefore, set the calculator to