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Answers and Solutions: 4 -1
or in part.
Chapter 4
Time Value of Money
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.
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 occurring now or at some future time. A cash flow can be an inflow (a
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
f. An outflow is a deposit, a cost, or an amount paid, while an inflow is a receipt. A
time line is an important tool used in time value of money analysis; it is a graphical
Answers and Solutions: 4 - 2
© 2017 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole
or in part.
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
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.
h. Annual compounding means that interest is paid once a year. In semiannual,
quarterly, monthly, and daily compounding, interest is paid 2, 4, 12, and 365 times
per year respectively. When compounding occurs more frequently than once a year,
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
is one that is repaid in equal periodic amounts (or "killed off" over time).
4-2 The opportunity cost rate is the rate of interest one could earn on an alternative
the riskiness and maturity of an investment, and it also varies from year to year
4-3 True. The second series is an uneven payment stream, but it contains an annuity of $400
or in part.
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:
2. Using a financial calculator, input N = 10, I/YR = 10, PV = -1, PMT = 0, and FV = ?.
Solving for FV you obtain $2.59. This formulation recognizes the "interest on
4-5 For the same stated rate, daily compounding is best. You would earn more "interest on
interest."
Answers and Solutions: 4 - 4
or in part.
SOLUTIONS TO END-OF-CHAPTER PROBLEMS
4-1 0 1 2 3 4 5
= $10,000(1.61051) = $16,105.10.
4-2 0 5 10 15 20
5000. Solve for PV = $1,292.10.
4-3 0 18
= 1000000. Solve for I/YR = 8.01% ≈ 8%.
4-4 0 N = ?
$2 = $1(1.065)N.
4-5 0 1 2 N – 2 N – 1 N
| | | | | |
7%
6.5%
12%
I/YR = ?
10%
Answers and Solutions: 4 -5
4-6 Ordinary annuity:
0 1 2 3 4 5
Solve for FV = $1,725.22.
FVA5 = ?
4-7 0 1 2 3 4 5 6
| | | | | | |
screen.); CF6 = 500 (Note calculator will show CF4 on screen.); and I/YR = 8. Solve for
NPV = $923.98.
4-8 Using a financial calculator, enter the following: N = 60, I/YR = 1, PV = -20000, and FV
= 0. Solve for PMT = $444.89.
EAR =
M
NOM
M
I
1
– 1.0
= (1.01)12 – 1.0
= 12.68%.
Alternatively, using a financial calculator, enter the following: NOM% = 12 and P/YR =
12. Solve for EFF% = 12.6825%. Remember to change back to P/YR = 1 on
your calculator.
7%
8%
4-9 a. 0 1
| | $500(1.06) = $530.00.
-500 FV = ?
c. 0 1
PV = ? 500
4-10 a. 0 1 2 3 4 5 6 7 8 9 10 $500(1.06)10 = $895.42.
| | | | | | | | | | |
-500 FV = ?
d. 0 1 2 3 4 5 6 7 8 9 10
| | | | | | | | | | | $500(1/1.12)10 = $160.99
6%
6%
6%
12%
Answers and Solutions: 4 -7
4-11 a. ?
| |
b. ?
| |
c. ?
| |
d. 100% ?
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 = ?
With a financial calculator, enter N = 10, I/YR = 10, PV = 0, and PMT = -400. Then
press the FV key to find FV = $6,374.97.
7%
10%
18%
10%
Answers and Solutions: 4 - 8
or in part.
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.
c. 0 1 2 3 4 5
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.
(2) 0 1 2 3 4 5
| | | | | |
200 200 200 200 200 FVA5 = ?
With a financial calculator set to “BEG” mode, enter N = 5, I/YR = 0, PV = 0, and
PMT =
-400. Then press the FV key to find FV = $2,000.
0%
10%
5%
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 = 5, I/YR = 5, PMT = -200, and FV = 0. Then
press the PV key to find PV = $865.90.
d. (1) 0 1 2 3 4 5 6 7 8 9 10
| | | | | | | | | | |
400 400 400 400 400 400 400 400 400 400
| | | | | |
200 200 200 200 200
PV = ?
With a financial calculator set to “BEG” mode, enter N = 5, I/YR = 5, PMT = -
10%
(3) 0 1 2 3 4 5
4-14 a. Cash Stream A Cash Stream B
0 1 2 3 4 5 0 1 2 3 4 5
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.
4-15 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.
With a financial calculator, enter N = 1, PV = 700, PMT = 0, and FV = -749. Then
press the I/YR key to find I/YR = 7%.
With a financial calculator, enter N = 1, PV = -700, PMT = 0, and FV = 749. Then
0%
8%
8%
