annuity at the beginning of the year it starts (2011) is:
A IFA
PV A PV (13%, 10periods)
$8,000 5.426
$43,408
=
=
=
The present value at the beginning of 2014 is found using
Appendix B (two years at 13 percent). The factor is .783. Note
we are discounting from the beginning of 2016 to the beginning
of 2014.
IF
PV = FV × PV (13%, 2periods)
= $43,408 × .783
= $33,988
The maximum that should be paid for the annuity is $33,988.
41. Yield (LO9-4) If you borrow $9,441 and are required to pay back the loan in five equal
annual installments of $2,750, what is the interest rate associated with the loan?
9-41. Solution:
Calculator Solution:
N
I/Y
PV
PMT
FV
5
CPT I/Y 14.00
−9,441.00
2,750.00
0
Answer: 14.00%
Appendix D
IFA A
PV PV / A (5 periods)
$9,441/$2,750
3.433
=
=
=
Interest rate = 14%
Go across period 5 until you find 3.433. Go up to the percentage
at the top of the column and find 14 percent.
42. Cal Lury owes $10,000 now. A lender will carry the debt for five more years at 10 percent
interest. That is, in this particular case, the amount owed will go up by 10 percent per year
for five years. The lender then will require that Cal pay off the loan over the next 12 years
at 11 percent interest. What will his annual payment be?
9-42. Solution:
Part 1
5
(1 )
n
FV PV i
= +
12
1
1(1 )
1
1(1.11)
.11
$16,105.10
n
A
A
i
PV A
i
PV
A
−
+
=
=
−
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
1
1(1 )
$60,000
n
A
i
PV A
i
A
−
+
=
=
Calculator Solution:
N
I/Y
PV
PMT
FV
7
10
60,000.00
CPT PMT −12,324.33
0
First payment:
$60,000 × .10 = $6,000 first year interest
$12,324.33 – $6,000 = $6,324.33 applied to principal
Second payment: First determine remaining principal
$60,000 – $6,324.33 = $53,675.67
$53,675.67× .10 = $5,367.57 second year interest
$12,324.33 – $5,367.57 = $6,956.76 applied to principal
Appendix D
A IFA
A = PV /PV (10%,7periods)
= $12,325 annual payment
First payment:
$60,000 × .10 = $6,000 interest
$12,325 – $6,000 = $6,325 applied to principal
Second payment:
First determine remaining principal and then the interest and
$60,000 – $6,325 = $53,675 remaining principal
$53,675 × .10 = $ 5,368 interest
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.
1
1(1 )
1
1(1 )
$80,000
n
A
A
n
i
PV A
i
PV
A
i
i
−
+
=
=
−+
25
1
1(1 )
$80,000
1
1(1.10)
.10
$80,000
9.077
A
n
PV
A
i
i
A
A
=
−+
=
−
=
Difference between 14 percent and 10 percent interest
$11,639.87
$2,826.42 (9.077)
PV
=
$25,655.53PV =
Amount that could be paid to refinance
Calculator Solution:
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
25
10
80,000
CPT PMT −8,813.45
0
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
25
10
CPT PV −25,655.53
2,826.42
0
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)
= $8,813.48
9-44. (Continued)
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
n
A
i
FV A
i
+−
=
−
12
(1 ) 1
$128,369.90
A
n
FV
A
i
i
A
=+−
=−
−7,609.45
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
Appendix A
IF
FV = PV × FV (6%, 12 periods)
need to accumulate on the next 12 payments.
$250,000
121,623
$128,377
−
Determine the revised semiannual payment necessary to
Appendix C
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.)
9-46. Solution:
PV of college costs five years from today (Part 1)
1
1(1 )
1
n
A
i
PV A
i
−+
=