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9-42. Solution:
Part 1
5
(1 )
$10,000 (1.10)
$16,105.10
n
FV PV i
FV
FV
= ´ +
= ´
=
Part 2
12
12
1
1(1 )
1
1(1.11)
.11
$16,105.10
1
1(1.11)
.11
$16,105.10
6.492
$2,480.62
n
A
A
i
PV A
i
PV
A
A
A
A
æ ö
-
ç ÷
+
= ´ ç ÷
ç ÷
ç ÷
è ø
=
-
=
-
=
=
Calculator Solution:
Answer: $16,105.10 is the amount owed after five years.
Step two:
N I/Y PV PMT FV
Answer: $2,480.62 is the annual payment to retire the loan.
Appendix A
IF
FV = PV × FV (10%, 5 periods)
= $10,000 × 1.611
= $16,110 Amount owed after 5 years
Appendix D
A IFA
A = PV /PV (11%, 12periods)
= $16,110/6.492
= $2,482 Annual payments to retire the loan
43. If your uncle borrows $60,000 from the bank at 10 percent interest over the seven-year life
of the loan, what equal annual payments must be made to discharge the loan, plus pay the
bank its required rate of interest (round to the nearest dollar)? How much of his first
payment will be applied to interest? To principal? How much of his second payment will be
applied to each?
9-43. Solution:
Annual Payment
7
1
1(1 )
$60,000
1
1(1.10)
.10
$60,000
6.868
$12,324.33
n
A
i
PV A
i
A
A
A
æ ö
-
ç ÷
+
= ´ ç ÷
ç ÷
ç ÷
è ø
=
-
=
=
Amount of first payment applied to interest and principal
$60,000 (.10) $6,000´ =
First year interest
$12,324.33 $6,000 $6,324.33- =
Applied to principal
Amount of second payment applied to interest and principal
$60,000 $6,324.33 $53,675.67- =
After one year
$53,675.67 (.10) $5,367.57´ =
Second year interest
$12,324.33 $5,367.57 $6,956.76- =
Applied to principal
Calculator Solution:
First payment:
Second payment: First determine remaining principal
Appendix D
A IFA
A = PV /PV (10%,7 periods)
= $60,000/4.868
= $12,325 annual payment
First payment:
Second payment:
First determine remaining principal and then the interest and
principal payment.
44. Larry Davis borrows $80,000 at 14 percent interest toward the purchase of a home. His
mortgage is for 25 years.
a. How much will his annual payments be? (Although home payments are usually on a
monthly basis, we shall do our analysis on an annual basis for ease of computation.
We will get a reasonably accurate answer.)
b. How much interest will he pay over the life of the loan?
c. How much should he be willing to pay to get out of a 14 percent mortgage and into a
10 percent mortgage with 25 years remaining on the mortgage? Assume current
interest rates are 10 percent. Carefully consider the time value of money. Disregard
taxes.
9-44. Solution:
a.
25
1
1(1 )
1
1(1 )
$80,000
1
1(1.14)
.14
$80,000
6.873
$11,639.87
n
A
A
n
i
PV A
i
PV
A
i
i
A
A
A
æ ö
-
ç ÷
+
= ´ ç ÷
ç ÷
ç ÷
è ø
=
-+
=
-
=
=
b.
$11,639.87 25 $290,996.82´ =
Total payments
$290,996.82 $80,000 $210,996.82- =
Total interest paid
c.
25
1
1(1 )
$80,000
1
1(1.10)
.10
$80,000
9.077
A
n
PV
A
i
i
A
A
=
-+
=
-
=
$8,813.45A=
New annual payments
Difference between 14 percent and 10 percent interest
$25,655.53PV =
Amount that could be paid to refinance
Calculator Solution:
(a)
N I/Y PV PMT FV
Answer: Annual payment $11,639.87
(b)
Total payments = 11,639.87 × 25 = $290,996.82
Total interest paid = 290,996.82 – 80,000 = 210,996.82
(c)
N I/Y PV PMT FV
Answer: Annual Payment $8,813.45
Difference between old and new payments = 11,639.87 – 8,813.45 = $2,826.42
P.V. of difference at 10 percent:
N I/Y PV PMT FV
Answer: $25,655.53 is the amount that could be paid to refinance.
Appendix D
A IFA
a. A = PV /PV (14%, 25periods)
= $80,000/6.873
= $11,639.75
b. $11,639.75 Annual payments
× 25 Years
$290,993.75 Total payment
80,000.00 Repayment of principal
$210,993.75 Total interest paid
-
Appendix D
c. New payments at 10 percent
A IFA
A = PV /PV (10%, 25periods)
= $80,000/9.077
= $8,813.48
9-44. (Connued)
Difference between old and new payments
$11,639.75 Old
8,813.48 New
$ 2,826.27 Difference
PV of difference – Appendix D
A IFA
PV A PV (assumes 10% discount rate, 25 periods)
$ 2,826.27 9.077
$25,654.05 Amount that could be paid to refinance
= ´
= ´
=
45. Annuity with changing interest rates (LO9-4) You are chairperson of the investment
fund for the Continental Soccer League. You are asked to set up a fund of semiannual
payments to be compounded semiannually to accumulate a sum of $250,000 after nine
years at a 10 percent annual rate (18 payments). The first payment into the fund is to take
place six months from today, and the last payment is to take place at the end of the ninth
year.
a. Determine how much the semiannual payment should be. (Round to whole numbers.)
On the day after the sixth payment is made (the beginning of the fourth year), the interest
rate goes up to a 12 percent annual rate, and you can earn a 12 percent annual rate on funds
that have been accumulated as well as all future payments into the fund. Interest is to be
compounded semiannually on all funds.
b. Determine how much the revised semiannual payments should be after this rate
change (there are 12 payments and compounding dates). The next payment will be in
the middle of the fourth year. (Round all values to whole numbers.)
9-45. Solution:
a.
18
(1 ) 1
(1 ) 1
$250,000
(1.05) 1
.05
$250,000
28.132
$8,886.68
n
A
A
n
i
FV A
i
FV
A
i
i
A
A
A
+ -
= ´
=+ -
=-
=
=
b. Part 1: Value of first six payments at the beginning of year 4
6
(1 ) 1
(1.05) 1
$8,886.68 .05
$8,886.68 (6.802)
$60,446.42
n
A
A
A
A
i
FV A
i
FV
FV
FV
+ -
= ´
-
= ´
= ´
=
Part 2 : FV of first six payments at the end of year 9
12
(1 )
$60,446.42 (1.06)
$60,446.42 (2.012)
n
FV PV i
FV
FV
= ´ +
= ´
= ´
Part 3: Additional amount required
Part 4 : New payment level
12
(1 ) 1
$128,369.90
(1.06) 1
.06
$128,369.90
16.870
$7,609.39
A
n
FV
A
i
i
A
A
A
=+ -
=-
=
=
Calculator Solution:
(a)
N I/Y PV PMT FV
Answer: $8,886.68
(b)
First determine how much the old payments are equal to after 6 periods at 5%.
N I/Y PV PMT FV
Answer: $60,446.40
Then, determine how much this value will grow to after 12 periods at 6 percent (semiannual
Then, determine how much this value will grow to after 12 periods at 6 percent.
N I/Y PV PMT FV
Answer: $121,629.23
Subtract this value ($121,629.00) from $250,000 to determine how much you need to
accumulate on the next 12 payments.
250,000 – 121,629 = $128,371
Determine the revised semiannual payment necessary to accumulate this sum after 12 periods at
6 percent.
N I/Y PV PMT FV
Answer: $7,609.39 is the revised semiannual payment.
Appendix C
A IFA
a. A FV / FV
$250,000 / 28.132 (5%, 18 periods)
$8,887
=
=
=
b. First determine how much the old payments are equal to after
six periods at 5 percent. Use Appendix C.
A IFA
FV = A × FV (5%, 6 periods)
= $8,887 × 6.802
= $60, 449
Then, determine how much this value will grow to after 12
periods at 6 percent (semiannual rate).
Appendix A
IF
FV = PV × FV (6%, 12 periods)
= $60, 449 × 2.012
= $121,623
Subtract this value from $250,000 to determine how much you
need to accumulate on the next 12 payments.
$250, 000
121, 623
$128,377
-
Determine the revised semiannual payment necessary to
accumulate this sum after 12 periods at 6 percent.
Appendix C
A = FVA/FVIFA
46. Your younger sister, Linda, will start college in five years. She has just informed your
parents that she wants to go to Hampton University, which will cost $17,000 per year for four
years (cost assumed to come at the end of each year). Anticipating Linda’s ambitions, your
parents started investing $2,000 per year five years ago and will continue to do so for five more
years. How much more will your parents have to invest each year (A?) for the next five years to
have the necessary funds for Linda’s education? Use 10 percent as the appropriate interest rate
throughout this problem (for discounting or compounding). This timeline depicts the cash flows
described (in thousands of dollars.)
