9-35. Solution:
15
1
1(1 )
1
1(1 )
$180,000
1
1(1.09)
.09
$22,330.60
n
A
A
n
i
PV A
i
PV
A
i
i
A
A
æ ö
–
ç ÷
+
= ´ ç ÷
ç ÷
ç ÷
è ø
=æ ö
–
ç ÷
+
ç ÷
ç ÷
ç ÷
è ø
=æ ö
–
ç ÷
ç ÷
ç ÷
ç ÷
è ø
=
Calculator Solution:
N I/Y PV PMT FV
A IFA
A PV / PV (9%, 15periods)
$180,000 / 8.061
$22,329.74
=
=
=
36. Morgan Jennings, a geography professor, invests $50,000 in a parcel of land that is
expected to increase in value by 12 percent per year for the next five years. He will take the
proceeds and provide himself with a 10-year annuity. Assuming a 12 percent interest rate,
how much will this annuity be?
9-36. Solution:
Part 1
5
(1 )
$50,000 (1.12)
$88,117.08
n
FV PV i
FV
FV
= ´ +
= ´
=
Part 2
1
1(1 )
1
1(1 )
n
A
A
n
i
PV A
i
PV
A
i
i
æ ö
–
ç ÷
+
= ´ ç ÷
ç ÷
ç ÷
è ø
=æ ö
–
ç ÷
+
ç ÷
ç ÷
ç ÷
è ø
Calculator Solution:
Step one:
N I/Y PV PMT FV
Step two:
N I/Y PV PMT FV
Appendix A
FV = PV × FVIF (12%, 5 periods)
Appendix D
A = PVA/PVIFA (12%, 10 periods)
37. Solving for an annuity (LO9-4) You wish to retire in 14 years, at which time you want to
have accumulated enough money to receive an annual annuity of $17,000 for 19 years after
retirement. During the period before retirement you can earn 8 percent annually, while after
retirement you can earn 10 percent on your money.
What annual contributions to the retirement fund will allow you to receive the $17,000
annuity?
9-37. Solution:
Part 1
19
1
1(1 )
1
1(1.10)
$17,000 .10
$142,203.64
n
A
A
A
i
PV A
i
PV
PV
æ ö
–
ç ÷
+
= ´ ç ÷
ç ÷
ç ÷
è ø
æ ö
–
ç ÷
= ´ ç ÷
ç ÷
ç ÷
è ø
=
Part 2
14
(1 ) 1
(1 ) 1
$142,203.64
(1.08) 1
.08
$5,872.56
n
A
A
n
i
FV A
i
FV
A
i
i
A
A
æ ö
+ –
= ´ ç ÷
è ø
=æ ö
+ –
ç ÷
è ø
=æ ö
–
ç ÷
è ø
=
Calculator Solution:
Determine the present value of a 14-year annuity during retirement:
N I/Y PV PMT FV
To determine the annual deposit into an account earning 8% that is necessary to accumulate
$142,203.64 after 14 years, solve for the annuity:
N I/Y PV PMT FV
Determine the present value of an annuity during retirement:
Appendix D
To determine the annual deposit into an account earning 8 percent
that is necessary to accumulate $142,205 after 14 years, use the
future value of an annuity table. See Appendix C.
A IFA
A FV / FV (8%, 14 years)
$142,205 = $5,872.60 annual contribution
24.215
=
=
38. Del Monty will receive the following payments at the end of the next three years: $2,000,
$3,500, and $4,500. Then, from the end of the 4th through the end of the 10th year, he will
receive an annuity of $5,000 per year. At a discount rate of 9 percent, what is the present
value of all three future benefits?
9-38. Solution:
Payment #1
1
1
(1 )
1
$2,000 (1.09)
$1,834.86
n
PV FV
i
PV
PV
æ ö
= ´ ç ÷
+
è ø
= ´
=
Payment #2
2
1
(1 )
1
$3,500 (1.09)
$2,945.88
n
PV FV
i
PV
PV
æ ö
= ´ ç ÷
+
è ø
= ´
=
Payment #3
3
1
(1 )
1
$4,500 (1.09)
$3,474.83
n
PV FV
i
PV
PV
æ ö
= ´ ç ÷
+
è ø
= ´
=
Annuity Value after three years
7
1
1(1 )
1
1(1.09)
$5,000 .09
$25,164.76
n
A
A
A
i
PV A
i
PV
PV
æ ö
–
ç ÷
+
= ´ ç ÷
ç ÷
ç ÷
è ø
æ ö
–
ç ÷
= ´ ç ÷
ç ÷
ç ÷
è ø
=
3
1
(1 )
1
$25,164.76 (1.09)
$19,431.81
n
PV FV
i
PV
PV
= ´ +
= ´
=
Total Present Value
$ 1,834.86 Payment #1
+ 2,945.88 Payment #2
+ 3,474.83 Payment #3
+19,431.81 Annuity value
$27,687.38 Total present value
Calculator Solution:
First find the present value of the first three payments.
N I/Y PV PMT FV
Answer: $1,834.86
N I/Y PV PMT FV
Answer: $2,945.88
N I/Y PV PMT FV
Then find the present value of the deferred annuity.
N I/Y PV PMT FV
Then, find its PV as of now:
N I/Y PV PMT FV
Finally, find the total present value of all future payments.
First find the present value of the first three payments.
PV = FV × PVIF (Appendix B) i = 9%
Then find the present value of the deferred annuity.
Appendix D will give a factor for a seven period annuity (4th
year through the 10th year) at a discount rate of 9 percent. The
value of the annuity at the beginning of the fourth year is:
A IFA
PV A PV (9%,7 periods)
$5,000 5.033 $25,165
= ´
= ´ =
This value at the beginning of year 4 (end of year 3) must now be
discounted back for three years to get the present value of the
deferred annuity. Use Appendix B.
IF
PV FV PV (9%,3periods)
$25,165 .772 $19.427.38
= ´
= ´ =
Finally, find the total present value of all future payments.
Present value of first three payments $ 8,225.00
39. Bridget Jones has a contract in which she will receive the following payments for the next
five years: $1,000, $2,000, $3,000, $4,000, and $5,000. She will then receive an annuity
of $8,500 a year from the end of the 6th through the end of the 15th year. The appropriate
discount rate is 14 percent. If she is offered $30,000 to cancel the contract, should she do it?