Principles of Finance 6e Chapter 9
Numerical solution:
r
1)r1(
PMTFVA
n
n
−+
=
Chapter 9 Principles of Finance 6e
Besley/Brigham
Future Value of the Annuity
Numerical solution:
)r1(
r
1)r1(
PMT)DUE(FVA
n
n
+
−+
=
Principles of Finance 6e Chapter 9
Besley/Brigham
9-37
0 1 2 3 4
90.91 100 300 300 –50
225.39
Numerical solution:
g. 0 1 2 3
Numerical solution:
%0.8080.00.1
00.100$
97.125$
r
)r1(100$97.125$
)r1(PVFV
3
1
3
n
==−
=
+=
+=
h. (1) Investments that pay interest more frequently than once per year, for example—
10%
r = ?
(2) The quoted, or simple, rate is merely the quoted percentage rate of return that is used to
compute the periodic rate of return—it is the same as the APR; the periodic rate is the rate
(3) The effective annual rate for 10 percent semiannual compounding, is 10.25 percent:
1.0 –
m
r
+ 1 = r
SIMPLE
m
EAR
360
(4) With semiannual compounding, the $100 is compounded over six semiannual periods at a
5.0 percent periodic rate:
1 2 3 Years
0 1 2 3 4 5 6 Six–month periods
–100 FV=?
Numerical Solution:
m
r
1PVFV
nm
SIMPLE
n
+=
Another approach here would be to use the effective annual rate and compound over
annual periods:
5%
Principles of Finance 6e Chapter 9
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9-39
i. If annual compounding is used, then the simple rate will be equal to the effective annual rate. If
more frequent compounding is used, the effective annual rate will be greater than the simple
rate. That is, rSIMPLE = rPER = rEAR when interest is compounded annually, whereas rSIMPLE < rEAR
when interest is compounded more than once per year.
0 1 2 3
100 100 100.00
or (2) treat the cash flows as an ordinary annuity, but use the effective annual rate:
10.25%. = 1 –
2
0.10
+ 1 = 1 –
m
r
+ 1 = r
2
SIMPLE
m
EAR
Now we have this three-period annuity:
5%
Chapter 9 Principles of Finance 6e
Besley/Brigham
9-40
0 1 2 3
100 100 100 100.00 = 100(1.1025)0
Numerical solution:
r
1)r1(
PMTFVA
n
n
−+
=
(2) 0 1 2 3
90.70 100 100 100
Numerical solution:
r
1
PMTPVA
n
)r1(
1
n
−
=+
5%
5%
(3) The payment stream is an annuity in the sense of constant amounts at regular intervals,
k. (1) To begin, note that the face amount of the loan, $1,000, is the present value of a three-year
annuity at a 10 percent rate:
0 1 2 3
–1,000 PMT PMT PMT
Numerical solution:
r
1
PMTPVA
n
)r1(
1
n
−
=+
10%
Chapter 9 Principles of Finance 6e
Besley/Brigham
9-42
(ii) The repayment of principal is the difference between the $402.11 annual payment
and the interest payment:
(vii) The payment might have to be increased by a few cents in the final year to take care
of rounding errors and make the final payment produce a zero ending balance.
The amortization schedule would be:
Beginning Interest Principal Ending
Year Balance Payment @ 10% Repayment Balance
1 $1,000.00 $402.11 $100.00 $302.11 $697.89
l. First, determine the effective annual rate of interest, with daily compounding:
12.0%. = 0.12 = 1
365
0.1133463
+ 1 = EAR
365
−
Thus, if you left your money on deposit for an entire year, you would earn $12 of interest, and
you would end up with $112. The question, however, is: How much will be in your account on
October 1, 2005?
–100 FV=? 112
12%
Principles of Finance 6e Chapter 9
Besley/Brigham
9-43
Numerical solution:
0 1 1.75 2 Years
–100 112 FV=? 125.44
Numerical solution:
n. You can solve this problem in three ways: (1) by compounding the $850 now in the bank for 15
months and comparing that FV with the $1,000 the note will pay; (2) by finding the PV of the
note and then comparing it with the $850 cost; and (3) by finding the effective annual rate of
return on the note and comparing that rate with the 7 percent you are now earning, which is
your opportunity cost of capital. All three procedures lead to the same conclusion. Here is the
cash flow time line:
0 1 1.25 2 Years
(1) Future Value
Numerical solution:
(2) Present Value
12%
7%
Chapter 9 Principles of Finance 6e
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9-44
Numerical solution:
(3) Effective Annual Rate of the Investment
o. Here is the cash flow time line:
¼ ½ ¾ 1 1¼ Years
0 1 2 3 4 5 Quarters
(1) Future Value
Numerical solution:
1)0170585.1(
5
−
(2) Present Value
1.706%
Principles of Finance 6e Chapter 9
Besley/Brigham
9-45
Numerical solution:
25.903)75397.4(190$
0170585.0
1
190$PVA )0170585.1(
1
==
−
=
Financial calculator solution: Input N = 5, I/Y = 1.70585, PMT = 190, and FV = 0; compute
PV = -903.25
Spreadsheet solution: Use the PV financial function that is available on the spreadsheet.
(3) Effective Annual Rate of the Investment
−
=+
r
1
190$850$ )r1(
1
9-45 Computer-Related Problem
a. INPUT DATA: KEY OUTPUT:
Loan amount 30,000 Payment 3,523.79
Interest rate 10.00%
Number of years 20
MODEL-GENERATED DATA:
Amortization schedule:
Principal Remaining PV of
Year Payment Interest Repayment Balance Payments
1 3,523.79 3,000.00 523.79 29,476.21 3,203.44
2 3,523.79 2,947.62 576.17 28,900.04 2,912.22
3 3,523.79 2,890.00 633.78 28,266.26 2,647.47
Chapter 9 Principles of Finance 6e
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9-46
14 3,523.79 1,715.53 1,808.26 15,347.02 927.92
b. INPUT DATA: KEY OUTPUT:
Loan amount 60,000 Payment 7,047.58
Interest rate 10.00%
Number of years 20
MODEL-GENERATED DATA:
Amortization schedule:
Principal Remaining PV of
Year Payment Interest Repayment Balance Payments
1 7,047.58 6,000.00 1,047.58 58,952.42 6,406.89
2 7,047.58 5,895.24 1,152.34 57,800.09 5,824.44
11 7,047.58 4,330.43 2,717.15 40,587.17 2,470.13
12 7,047.58 4,058.72 2,988.86 37,598.31 2,245.58
13 7,047.58 3,759.83 3,287.75 34,310.56 2,041.43
14 7,047.58 3,431.06 3,616.52 30,694.04 1,855.85
15 7,047.58 3,069.40 3,978.17 26,715.86 1,687.13
c. INPUT DATA: KEY OUTPUT:
Loan amount 60,000 Payment 13,321.29
Interest rate 20.00%
Number of years 20
MODEL-GENERATED DATA:
Principles of Finance 6e Chapter 9
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9-47
Amortization schedule:
Principal Remaining PV of
Year Payment Interest Repayment Balance Payments
1 12,321.39 12,000.00 321.39 59,678.61 10,267.83
2 12,321.39 11,935.72 385.67 59,292.94 8,556.52
11 12,321.39 10,331.42 1,989.97 49,667.12 1,658.31
12 12,321.39 9,933.42 2,387.97 47,279.15 1,381.93
13 12,321.39 9,455.83 2,865.56 44,413.59 1,151.61
14 12,321.39 8,882.72 3,438.67 40,974.91 959.67
15 12,321.39 8,194.98 4,126.41 36,848.50 799.73
ETHICAL DILEMMA
It’s All Chinese to Me!
Ethical dilemma:
Terry Zupita must decide whether she should invest in Universal Autos, which is an American firm that is
partnering with the Chinese government to manufacture U.S. automobiles in China. If she makes the
investment, it appears that Ms. Zupita’s money will grow substantially during the next few years. However,
friends and relatives have told her that UA might use child labor in its Chinese plants, and that employees
often are abused in such plants. Should Terry purchase UA? If she does, she could be supporting
exploitation of children and mistreatment of employees.
Discussion questions:
• Is there an ethical problem? If so, what is it?
Chapter 9 Principles of Finance 6e
Besley/Brigham
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with a certain product or service or otherwise on a password-protected website for classroom use.
9-48
comments she has heard about the treatment of Chinese employees in the UA plant in Shanghai.
• What are the implications if Terry invests in UA?
If the information about child labor and employee abuse are correct and Terry invests in UA, she would
• Should Terry invest in UA?
You might get some interesting responses to this question. Some students might say that it would be
immoral to invest in UA, given the rumors about its manufacturing plant in China. Others might say that
References:
The following articles might be assigned for background material:
Yochi J. Dreazen, “U.S. Investigates Firm Building Embassy in Iraq,” The Wall Street Journal, June 7, 2007,
A1+.
Betsy Atkins, “Is Corporate Social Responsibility Responsible?,” November 28, 2006, Forbes.com.
Ruth David, “Indian Law Does Little for Littlest Laborers,” October 10, 2006, Forbes.com.
Yochi J. Dreazen, “Probe Targets Ex–Navy Official for Link to Disgraced Contractor,” The Wall Street
Journal, May 5, 2006, A1+.
Deborah Orr, “Slave Chocolate?,” April 24, 2006, Forbes.com.