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Principles of Finance 6e Chapter 9
Besley/Brigham
9-1
CHAPTER 9
ANSWERS
9-1 The opportunity cost rate is the rate of interest one could earn on an alternative investment with a
9-3 True, because of compounding effects—that is, growth on growth. The following example
demonstrates the point. The annual growth rate is r in the following equation:
2. Solve directly for r using the following method:
FVn = PV(1 + r)n
3. Use the “Rate” function on a spreadsheet, which can be set up as follows, the solution is:
Chapter 9 Principles of Finance 6e
Besley/Brigham
9-2
Viewed another way, if earnings had grown at the rate of 10 percent per year for 10 years, then
EPS would have increased from $1.00 to $2.59, found as follows:
9-4 For the same stated rate, more compounding is better. You would earn more “interest on interest.”
Computing the effective annual rate for each alternative shows this to be true:
9-5 To find the present value of an amount to be received in the future, we must take out the interest
that the future amount can earn during the time period in question. The result of “de-interesting” the
future amount is the present value, which represents the amount that must be invested today to
grow to the future value at the given opportunity cost. For example, if you want to invest an amount
today so that you have $500 in three years and your opportunity cost is 7 percent, the following
cash flow time line shows that the present value of the $500 is $408.15:
If the opportunity cost is greater than 7 percent, then the present value will be lower because the
9-7 The concept of a perpetuity implies that payments will be received forever. FV of Perpetuity =
PVP(1 + r)∞ = ∞.
9-8 To compare APRs, you must compute the rEAR for each alternative. APRs are not comparable when
9-9 rEAR = APR is compounding occurs once per year; otherwise, rEAR > APR. This can be seen by
computing rEAR when compounding occurs once per year:
0 3
PV = ? 500
r = 7%
= 408.15
)816298.0(500
)07.1(
500
3==
9-3
9-10 An amortized loan is a loan for which a portion of the periodic payment includes interest that is
____________________________________________________________
SOLUTIONS
(Most solutions are rounded in the final answers, not in the intermediate computations.)
0 1 2
9-1
0 1 2 3 4 5
9-2
PV = ? 1,000
0 1 2 3 4 5 6 7 8 9 10
9-3 (1)
PV = ? 1,552.90
0 1 2 3 4 5 6 7 8 9 10
(2)
PV = ? 1,552.90
6%
12%
6%
6%
Chapter 9 Principles of Finance 6e
Besley/Brigham
9-4
0 1 2 3 4 n = ?
9-4
PV = -200 400
0 1 2 3 4 5 6
9-5
1,000 FV = ?
PV = ? 2,000
0 1 2 3 4 5
9-6
6 12
%87.141487.01
6
12
r
)r1(
5
1
5
==−
=
+
7%
…
r = 14
r = ?
9-5
9-7 The general formula for computing the future value of an ordinary annuity is:
0 1 2 3 4 5 6 7 8 9 10
a.
400 400 400 400 400 400 400 400 400 400
FVA10 = ?
0 1 2 3 4 5
b.
200 200 200 200 200
FVA5 = ?
9-8 The general formula for computing the future value of an annuity due is:
r
n
a. 0 1 2 3 4 5 6 7 8 9 10
400 400 400 400 400 400 400 400 400 400
FVA(DUE)10 = ?
10%
5%
10%
Chapter 9 Principles of Finance 6e
Besley/Brigham
9-6
b. 0 1 2 3 4 5
200 200 200 200 200
FVA(DUE)5 = ?
9-9 The general formula for computing the present value of an ordinary annuity is:
r
)
r + (1 - 1
PMT =
PVA
n-
n
a. 0 1 2 3 4 5 6 7 8 9 10
PVA10 = ? 400 400 400 400 400 400 400 400 400 400
83.457,2)14457.6(400
10.0
)
10(1.
1
1
400 =
VAP
10
10 ==
−
Using a financial calculator, enter N = 10, I/Y= 10, and PMT = -400; compute PV = 2,457.83
b. 0 1 2 3 4 5
PVA5 = ? 200 200 200 200 200
90.865)32948.4(200
05.0
)
05(1.
1
1
200 =
VAP
5
5==
−
Using a financial calculator, enter N = 5, I/Y= 5, and PMT = -200; compute PV = 865.90
5%
10%
5%
9-7
9-10 The general formula for computing the future value of an annuity due is:
r
)
r + (1
1
1
n
n
−
a. 0 1 2 3 4 5 6 7 8 9 10
400 400 400 400 400 400 400 400 400 400
PVA(DUE)10 = ?
)
10(1.
1
1
10
−
b. 0 1 2 3 4 5
200 200 200 200 200
PVA(DUE)5 = ?
)
05(1.
1
1
5
−
10%
5%
9-8
9-12 Cash Stream A Cash Stream B
a. 0 1 2 3 4 0 1 2 3 4
PV=? 100 400 400 300 PV=? 300 400 400 100
b. PVA = $100 + $400 + $400 + $400 + $300 = $1,200.
9-13 a. 0 1 2 3 4 5 years
-500 FV = ?
b. 1 2 3 4 5 years
0 2 4 6 8 10 six-month periods
-500 FV = ?
c. 1 2 3 4 5 years
0 4 8 12 16 20 quarters
-500 FV = ?
8%
8%
12%
6%
3%
Principles of Finance 6e Chapter 9
Besley/Brigham
9-9
1 2 3 4 5 years
0 12 24 36 38 60 months
d.
-500 FV = ?
9-14 a. 0 1 2 3 4 5 years
PV = ? -500
b. 1 2 3 4 5 years
0 2 4 6 8 10 six-month periods
PV = ? -500
c. 1 2 3 4 5 years
0 4 8 12 16 20 quarters
PV = ? -500
d. 1 2 3 4 5 years
0 12 24 36 38 60 months
1%
…
…
…
…
…
12%
6%
3%
…
…
…
…
…
Chapter 9 Principles of Finance 6e
Besley/Brigham
9-10
PV = ? 500
22.275)55045.0(500
+ 1
1
500 =PV
12
12.0 512 ==
Using a financial calculator, enter N = 60, I/Y= 1, and FV = -500; compute PV = 275.22
9-15 a. 0 1 2 3 9 10 periods
-400 -400 -400 -400 -400
FV = ?
b. 0 1 2 3 4 18 19 20 periods
-200 -200 -200 -200 -200 -200 -200
FV = ?
c. The annuity in part b earns more because some of the money is on deposit for a longer period
9-16 a. 0 1 2 3 9 10 periods
-400 -400 -400 -400 -400
PV = ?
6%
…
3%
…
6%
…
Principles of Finance 6e Chapter 9
Besley/Brigham
9-11
( )
( )
03.944,2)36009.7(400
06.0
)06.1(1
400
m
r
m
r
11
PMTPVA
10
n
n==
−
=
+−
=
−
Using a financial calculator, enter N = 5 x 2 = 10, I/Y= 12/2 = 6, FV = 0, and PMT = -400;
compute PV = 2,944.03.
b. 0 1 2 3 4 18 19 20 periods
-200 -200 -200 -200 -200 -200 -200
PV = ?
c. The annuity in part b requires the first payment to occur in three months, whereas the annuity in
9-17 a. 0 1 2 3 4
30,000 30,000 30,000 30,000
PVA4 = ?
1
4
(1.07)
1
PVA 30,000 30,000(3.3872113) 101,616.34
0.07
−
= = =
Using a calculator, enter N = 4, I/Y= 7, PMT = 30,000, and FV = 0; compute PV = -101,616.34.
b. (1) At this point, we have a three-year $30,000 annuity at 7 percent. Input N = 3 to override the
number of years from part a in your calculator’s TVM register, and you will find PV =
r = 7%
3%
…
Chapter 9 Principles of Finance 6e
Besley/Brigham
9-12
07.0
1
3
)07.1(
1
−
9-18 0 1 2 10 11 12 Quarters
12,000 PMT PMT PMT PMT PMT
1
r
1
PMTPVA
12
n
)03.1(
1
)r1(
1
−
−
=+
9-19 0 1 2 n-1 n Years
12,000 -1,500 -1,500 -1,500 -1,500
−
=+
r
1
PMTPVA
n
)r1(
1
−
=09.0
1
500,1000,12
n
)09.1(
1
Using a calculator, enter I/Y= 9, PV = 12,000, FV = 0, and PMT = -1,500; compute N = 14.77 ≈ 15
years
9-20 0 1 2 n-1 n
-1,750 -1,750 -1,750 PMT = ?
FVA = 10,000
r=12%/4=3%
…
r = 9%
…
r = 6%
…
Principles of Finance 6e Chapter 9
Besley/Brigham
9-13
−
=
−+
=
06.0
1)06.1(
750,1000,10
r
1)r1(
PMTFVA
n
n
Financial calculator: I/Y= 6, PV = 0, FV = 10,000, and PMT = -1,750; compute N = 5.06. This
answer assumes that a payment of $1,750 will be made 6/100 of the way through Year 6.
Now find the FV of $1,750 for 5 years at 6 percent; it is $9,864.91.
91.864,9)63709.5(750,1
06.0
1)06.1(
750,1FVA
5
==
−
=
Using a calculator, enter N = 5, I/Y= 6, PV = 0, and PMT = -1,750; compute FV = 9,864.91
So the payment at the end of Year 5 will include an additional $135.09 = $10,000 - $9,864.91,
which means the last investment will total $1,885.09 = $1,750 + $135.09. It will take 5 years to
accumulate the $10,000 if, beginning one year from today, $1,750 is invested each year for the next
four years at 6 percent, and a $1,885.09 investment is made at the end of Year 5.
9-21 The $2.9 million 30-year payment represents an annuity due. Therefore, compute the present value
of the annuity due.
( )
30
1
1(1.05)
PVA(DUE) ($2.9 million) 1.05 ($2.9 million)(16.1410736) $46,809,113
0.05
−
= = =
9-22 a. The $3.5 million 30-year payment represents an annuity due. Therefore, compute the present
value of the annuity due.
( )
30
1
1(1.06)
PVA(DUE) ($3.5 million) 1.06 ($3.5 million)(14.590721) $51,067,524
0.06
−
= = =
Financial calculator: Switch to the BGN mode, n= 30, I/Y= 6, PMT = 3,500,000, and FV = 0;
compute PV = -51,067,524.
Because PVA(DUE) = $51,067,524, which is less than the lump-sum payment of $54 million,
the lump-sum option should be chosen.
Chapter 9 Principles of Finance 6e
Besley/Brigham
9-14
9-23 a. 0 1 2 3 4
-1,000 FV = ?
b. 0 4 8 12 16
-1,000 FV = ?
nm
r
FV PV 1 m
=+
c. 0 1 2 3 4
-250 -250 -250 -250
d. 0 1 2 3 4
PMT PMT PMT PMT
FV = 1,360.49
N = 4; I = 8%; PV = 0; FV = $1,360.49; PMT = ? = $301.92
8%
8%
8%
8%
9-15
9-24 a. Set up a time line like the one in the preceding problem:
0 1 2 3 4
PV = ? 1,000
( )
1.08
b. 0 1 2 3 4
PMT PMT PMT PMT
FV = 1,000
$221.92
4.50611
$1,000
PMT
==
c. This problem can be approached in several ways. Perhaps the simplest is to ask this
question: “If I received $750 in one year and deposited it to earn 8 percent, would I have
$1,000 in four years?” The answer is no:
0 1 2 3 4
-750 FV = ?
You could also compare the $750 with the PV of the payments:
0 1 2 3 4
PV = ? -221.92 -221.92 -221.92 -221.92
8%
8%
8%
8%
Chapter 9 Principles of Finance 6e
Besley/Brigham
9-16
d. 0 1 2 3 4
-750 1,000
e. 0 1 2 3 4
-186.29 -186.29 -186.29 -186.29
f. 0 1 2 3 4
400 PMT PMT PMT PMT PMT PMT
FV = 1,000
Alternative solution: N = 6; I = 4%; PV = -400; FV = 1,000; solve for PMT = 74.46
r = ?
r = ?
r = 8%/2
Principles of Finance 6e Chapter 9
Besley/Brigham
9-17
m2
SIMPLE
r0.08
m2
9-25 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/Y button.
a. 0 1
+700 -749
b. 0 1
-700 +749
c. 0 1 2 3 4 5 6 7 8 9 10
+85,000 -201,229
r = ?
r = ?
r = ?
Chapter 9 Principles of Finance 6e
Besley/Brigham
9-18
d. 0 1 2 3 4 5
+9,000 -2,684.80 -2,684.80 -2,684.80 -2,684.80 -2,684.80
9-26 a. First City Bank: Effective rate = 7%.
Second City Bank:
0.065
4
b. If funds must be left on deposit until the end of the compounding period (one year for First City
and one quarter for Second City), and you think there is a high probability that you will make a
9-27 0 1 2 3 4 17 18 months
-150,000 168,925
r = ?
r = ?
…
Principles of Finance 6e Chapter 9
Besley/Brigham
9-19
period per %0.202.01)12617.1(1
000,150
925,168
r
)r1(
1
000,150925,168
)r1(
1
PVFV
16667.0
6
1
6
n
==−=−
=
+
=
+
=
Using a calculator, enter N = 6 quarters = [18 months/(3 months per quarter)], PV = -150,000, PMT
= 0, and FV = 168,925; compute I/Y= 2% per quarter.
So the APR for this investment equals 2% x 4 = 8%. The effective annual rate of interest is:
rEAR = (1.02)4 - 1 = 8.24%
9-28 0 1 2 8 9 10
-13,250 2,345.05 2,345.05 2,345.05 2,345.05 2,345.05
9-29 0 1 2 28 29 30
85,000 8,273.59 8,273.59 8,273.59 8,273.59 8,273.59
9-30 Bank A’s effective annual rate is 8.24 percent:
4
4
0.08
Effective annual rate 1 1.0 (1.02) 1 1.0824-1 0.0824 8.24%
4
= + − = − = = =
Now Bank B must have the same effective annual rate:
r = ?
…
r = ?
…
Chapter 9 Principles of Finance 6e
Besley/Brigham
Thus, the two banks have different quoted rates—Bank A’s quoted rate is 8 percent, while Bank
B’s quoted rate is 7.94 percent; however, both banks have the same effective annual rate of
8.24 percent. The difference in their quoted rates is due to the difference in compounding
frequency.
9-31 a. Cost of points = ($250,000 - $40,000)(0.035) = $7,350
b. Bank of Middle Texas:
n = 30 x 12 = 360, r = 6.9%/12 = 0.575%, mortgage = $250,000 - $40,000 + $7,350 = $217,350
−
=
−
=+
00575.0
1
PMT350,217
r
1
PMTPVA
360
n
)00575.1(
1
)r1(
1
PMT = $217,350/151.8372 = $1,431.47
Calculator solution: N = 360, I/Y= 0.575, PV = 217,350, and FV = 0; PMT = ? = -1,431.47
Bank of South Alaska:
n = 30 x 12 = 360, r = 7.2%/12 = 0.6%, mortgage = $250,000 - $40,000 = $210,000
c. First, determine the present value of the payment of the mortgage from Bank of South Alaska—
that is, $1,425.46—using the interest rate of the Bank of Middle Texas—that is, 6.9 percent.
Because the amount needed to purchase the house is $210,000, the cost of the points must be
$6,437.86. As a percent, the points would be:
