SOLUTIONS TO EXERCISES
EXERCISE 6-1 (5–10 minutes)
(a)
(b)
Rate of Interest
Number of Periods
EXERCISE 6-2 (5–10 minutes)
(a)
Simple interest of $2,400 ($30,000 X 8%) per year X 8 ….
$19,200
Principal …………………………………………………………………..
30,000
Total withdrawn ………………………………………………..
$49,200
Total withdrawn ………………………………………………..
$55,528
X $30,000
Total withdrawn ………………………………………………..
$56,189
EXERCISE 6-3 (10–15 minutes)
(a)
$9,000 X 1.46933 = $13,224.
(b)
$9,000 X .43393 = $3,905.
(c)
$9,000 X 31.77248 = $285,952.
6-22
EXERCISE 6-4 (15–20 minutes)
(a)
$228,809.80
($5,000 X 45.76196)
X 1.08
$ 247,115
(c)
$ 63,544.96
($2,000 X 31.77248)
X 1.10
$ 69,900
(d)
$ 13,457.76
($3,000 X 4.48592)
X 1.09
EXERCISE 6-5 (10–15 minutes)
(a)
$50,000 X 4.96764 = $248,382.
(b)
(c)
($50,000 X 3.03735 X .50663) = $76,941.
(b)
Present value of an ordinary
Factor (1 + .10)
X 1.10
EXERCISE 6-6 (15–20 minutes)
(a)
Future value of $12,000 @ 10% for 10 years
($12,000 X 2.59374) =
$ 31,125
(b)
Future value of an ordinary annuity of $620,000
(c)
$75,000 discounted at 8% for 10 years:
Accept the bonus of $40,000 now.
EXERCISE 6-7 (12–17 minutes)
(a)
$100,000 X .31524
=
$31,524.00
+ $10,000 X 8.55948
=
85,594.80
$ 117,119
(b)
$100,000 X .23939
=
$ 23,939
+ $10,000 X 7.60608
=
76,061
(c)
$100,000 X .18270
=
+ $10,000 X 6.81086
=
EXERCISE 6-8 (10–15 minutes)
(a)
Present value of an ordinary annuity of 1
for 4 periods @ 8%
3.31213
Annual withdrawal
(b)
Fund balance at June 30, 2015
Future value of an ordinary annuity at 8%
EXERCISE 6-9 (5–10 minutes)
The rate of interest is determined by dividing the future value by the present
value and then finding the factor in the FVF table with n = 2 that approxi–
mates that number:
Note: This problem can also be solved using present value tables.
EXERCISE 6-10 (10–15 minutes)
(a) The number of interest periods is calculated by first dividing the future
value of $1,000,000 by $148,644, which is 6.72748—the value $1.00 would
accumulate to at 10% for the unknown number of interest periods. The
EXERCISE 6-11 (10–15 minutes)
(a) Total interest = Total payments—Amount owed today
$155,820 (10 X $15,582) – $100,000 = $55,820.
EXERCISE 6-12 (10–15 minutes)
Building A—PV = $610,000.
Building B—
Rent X (PV of annuity due of 25 periods at 12%) = PV
$70,000 X 8.78432 = PV
$614,902 = PV
Building C—
Rent X (PV of ordinary annuity of 25 periods at 12%) = PV
$6,000 X 7.84314 = PV
$47,059 = PV
Cash purchase price
$650,000
PV of rental income
– 47,059
Net present value
$602,941
6-26
EXERCISE 6-13 (15–20 minutes)
Formula for the interest payments:
PV – OA = R (PVF – OAn, i)
PV – OA = $165,000 (PVF – OA30, 5%)
PV – OA = $165,000 (15.37245)
PV – OA = $2,536,454
Formula for the principal:
The selling price of the bonds = $2,536,454 + $694,140 = $3,230,594.
EXERCISE 6-14 (15–20 minutes)
Time diagram:
i = 8%
Formula: PV – OA = R (PVF – OAn, i)
PV – OA = $800,000 (PVF – OA25–15, 8%)
OR
Time diagram:
i = 8%
6-28
EXERCISE 6-14 (Continued)
(i) Present value of the expected annual pension payments at the end of
the 10th year:
PV – OA = R (PVF – OAn, i)
(ii) Present value of the expected annual pension payments at the begin–
ning of the current year:
PV = FV (PVFn, i)
EXERCISE 6-15 (15–20 minutes)
(a)
i = 6%
PV = $1,000,000 FV = $1,898,000
0 1 2 n = ?
(b) By setting aside $300,000 now, Lee can gradually build the fund to an
amount to establish the foundation.
$? $? $? FV = $1,391,156
0 1 2 8 9
6-30
EXERCISE 6-16 (10–15 minutes)
Amount to be repaid on March 1, 2020.
Time diagram:
Formula: FV = PV (FVFn, i)
FV = $90,000 (FVF20, 6%)
FV = $90,000 (3.20714)
FV = $288,643
Amount of annual contribution to debt retirement fund.
6-31
EXERCISE 6-16 (Continued)
EXERCISE 6-17 (10–15 minutes)
Time diagram:
i = 11%
R R R
PV – OA = $421,087 ? ? ?
0 1 24 25
n = 25
EXERCISE 6-18 (10–15 minutes)
Time diagram:
i = 8%
PV – OA = ? $400,000 $400,000 $400,000 $400,000 $400,000
0 1 2 13 14 15
n = 15
6-33
EXERCISE 6-19 (10–15 minutes)
Time diagram:
i = 8%
PV – AD = ?
R =
$400,000 $400,000 $400,000 $400,000 $400,000
0 1 2 13 14 15
n = 15
Formula:
EXERCISE 6-20 (15–20 minutes)
Expected
Cash Flow Probability Cash
Estimate X Assessment = Flow
(a) $ 4,800 20% $ 960
(b) $ 5,400 30% $ 1,620
7,200 50% 3,600
8,400 20% 1,680
EXERCISE 6-21 (10–15 minutes)
Estimated
Cash Probability Expected
Outflow X Assessment = Cash Flow
$200 10% $ 20
6-35
EXERCISE 6-22 (15–20 minutes)
(a) This exercise determines the present value of an ordinary annuity or
expected cash flows as a fair value estimate.
Cash flow Probability Expected
Estimate X Assessment = Cash Flow
$ 380,000 20% $ 76,000
The fair value estimate of the trade name exceeds the carrying value;
thus, no impairment is recorded.
TIME AND PURPOSE OF PROBLEMS
Problem 6-1 (Time 15–20 minutes)
Purpose—to present an opportunity for the student to determine how to use the present value tables in
various situations. Each of the situations presented emphasizes either a present value of 1 or a present
value of an ordinary annuity situation. Two of the situations will be more difficult for the student because
a noninterest-bearing note and bonds are involved.
Problem 6-2 (Time 15–20 minutes)
Purpose—to present an opportunity for the student to determine solutions to four present and future
value situations. The student is required to determine the number of years over which certain amounts
will accumulate, the rate of interest required to accumulate a given amount, and the unknown amount
of periodic payments. The problem develops the student’s ability to set up present and future value
equations and solve for unknown quantities.
Problem 6-3 (Time 20–30 minutes)
Purpose—to present the student with an opportunity to determine the present value of the costs of
competing contracts. The student is required to decide which contract to accept.
Problem 6-4 (Time 20–30 minutes)
Purpose—to present the student with an opportunity to determine the present value of two lottery
payout alternatives. The student is required to decide which payout option to choose.
Problem 6-5 (Time 20–25 minutes)
Purpose—to provide the student with an opportunity to determine which of four insurance options results
in the largest present value. The student is required to determine the present value of options which
include the immediate receipt of cash, an ordinary annuity, an annuity due, and an annuity of changing
amounts. The student must also deal with interest compounded quarterly. This problem is a good
summary of the application of present value techniques.
Problem 6-6 (Time 25–30 minutes)
Purpose—to present an opportunity for the student to determine the present value of a series of
deferred annuities. The student must deal with both cash inflows and outflows to arrive at a present
value of net cash inflows. A good problem to develop the student’s ability to manipulate the present
value table factors to efficiently solve the problem.
Problem 6-7 (Time 30–35 minutes)
Purpose—to present the student an opportunity to use time value concepts in business situations.
Some of the situations are fairly complex and will require the student to think a great deal before
answering the question. For example, in one situation a student must discount a note and in another
must find the proper interest rate to use in a purchase transaction.
Problem 6-8 (Time 20–30 minutes)
Purpose—to present the student with an opportunity to determine the present value of an ordinary
annuity and annuity due for three different cash payment situations. The student must then decide
which cash payment plan should be undertaken.
6-37
Time and Purpose of Problems (Continued)
Problem 6-9 (Time 30–35 minutes)
Purpose—to present the student with the opportunity to work three different problems related to time
value concepts: purchase versus lease, determination of fair value of a note, and appropriateness of
taking a cash discount.
Problem 6-10 (Time 30–35 minutes)
Purpose—to present the student with the opportunity to assess whether a company should purchase or
lease. The computations for this problem are relatively complicated.
Problem 6-11 (Time 25–30 minutes)
Purpose—to present the student an opportunity to apply present value to retirement funding problems,
including deferred annuities.
Problem 6-12 (Time 20–25 minutes)
Purpose—to provide the student an opportunity to explore the ethical issues inherent in applying time
value of money concepts to retirement plan decisions.
Problem 6-13 (Time 20–25 minutes)
Purpose—to present the student an opportunity to compute expected cash flows and then apply
present value techniques to determine a warranty liability.
Problem 6-14 (Time 20–25 minutes)
Purpose—to present the student an opportunity to compute expected cash flows and then apply
present value techniques to determine the fair value of an asset.
Problems 6-15 (Time 20–25 minutes)
Purpose—to present the student an opportunity to estimate fair value by computing expected cash
flows and then applying present value techniques to value an asset retirement obligation.
SOLUTIONS TO PROBLEMS
PROBLEM 6-1
(a) Given no established value for the building, the fair market value of
the note would be estimated to value the building.
Time diagram:
Formula: PV = FV (PVFn, i)
PV = $240,000 (PVF3, 9%)
6-39
PROBLEM 6-1 (Continued)
(b) Time diagram:
i = 11%
Principal
$300,000
Interest
PV – OA = ? $27,000 $27,000 $27,000 $27,000
1/1/12 1/1/13 1/1/14 1/1/21 1/1/22
n = 10
(c) Time diagram:
i = 8%
PV – OA = ? $4,000 $4,000 $4,000 $4,000 $4,000
0 1 2 8 9 10
n = 10
PROBLEM 6-1 (Continued)
(d) Time diagram:
i = 12%
PV – OA = ?
$20,000 $5,000 $5,000 $5,000 $5,000 $5,000 $5,000 $5,000 $5,000
0 1 2 3 4 5 6 7 8
n = 8
Cost of tractor = $20,000 + $24,838.20 = $44,838
(e) Time diagram:
i = 11%
PV – OA = ? $120,000 $120,000 $120,000 $120,000