25
1
1(1 )
1
1(1.12)
$18,500 .12
n
A
A
i
PV A
i
PV
−+
=
−
=
Calculator Solution:
N
I/Y
PV
PMT
FV
25
12
CPT PV
−145,098.07
18,500
0
Answer: $145,098.07
Appendix D
PVA = A × PVIFA (12%, 25 periods)
PVA = $18,500 × 7.843 = $145,096
Yes, the present value of the annuity is worth less than $165,000.
16. Carrie Tune will receive $19,500 for the next 20 years as a payment for a new song she has
written. If a 10 percent rate is applied, should she be willing to sell out her future rights
now for $160,000?
9-16. Solution:
20
1
1(1 )
1
1(1.10)
$19,500 .10
n
A
A
i
PV A
i
PV
−+
=
−
=
Calculator Solution:
N
I/Y
PV
PMT
FV
20
10
CPT PV −166,014.49
19,500
0
Answer: $166,014.49
17. The Clearinghouse Sweepstakes has just informed you that you have won $1 million. The
amount is to be paid out at the rate of $20,000 a year for the next 50 years. With a discount
rate of 10 percent, what is the present value of your winnings?
9-17. Solution:
50
1
1(1 )
1
1(1.10)
$20,000 .10
$198, 296.29
n
A
A
A
i
PV A
i
PV
PV
−+
=
−
=
=
Calculator Solution:
N
I/Y
PV
PMT
FV
50
10
CPT PV −198,296.29
20,000
0
Answer: $198,296.29
Appendix D
PVA = A × PVIFA (10%, 50 periods)
PVA = $20,000 × 9.915 = $198,300
18. Present value (LO9-3) Rita Gonzales won the $41 million lottery. She is to receive
$1.5 million a year for the next 19 years plus an additional lump sum payment of $12.5
million after 19 years. The discount rate is 14 percent. What is the current value of her
winnings?
9-18. Solution:
Annuity Part
19
1
1(1 )
1
1(1.14)
$1,500,000 .14
n
A
A
i
PV A
i
PV
−+
=
−
=
Appendix D
PVA = A × PVIFA (14%, 19 periods)
PVA = $1,500,000 × 6.550 = $9,825,000
Appendix B
PV = FV × PVIF (14%, 19 periods)
PV = $12,500,000 × .083 = $1,037,500
$ 9,825,000
1,037,500
$10,862,500
19. Al Rosen invests $25,000 in a mint condition 1952 Mickey Mantle Topps baseball card. He
expects the card to increase in value 12 percent per year for the next 10 years. How much
will his card be worth after 10 years?
9-19. Solution:
10
(1 )
$25,000 (1.12)
$77,646.21
n
FV PV i
FV
FV
= +
=
=
20. Future value (LO9-2) Christy Reed made a $2,000 deposit in her savings account on her
21st birthday, and she has made another $2,000 deposit on every birthday since then. Her
account earns 7 percent compounded annually. How much will she have in her account
after she makes the deposit on her 32nd birthday?
9-20. Solution:
12
(1 ) 1
(1.07) 1
$2,000 .07
$35,776.90
n
A
A
A
i
FV A
i
FV
FV
+−
=
−
=
=
Calculator Solution:
N
I/Y
PV
PMT
FV
12
7
0
2,000
CPT FV −35,776.90
Answer: $35,776.90
Appendix C
FVA = A × FVIFA (7%, n = 12)
FVA = $2,000 × 17.888 = $35,776.00
21. Future value (LO9-2) At a growth (interest) rate of 10 percent annually, how long will it
take for a sum to double? To triple? Select the year that is closest to the correct answer.
9-21. Solution:
To Double
(1 )
2 1 (1.10)
2 (1.10)
ln(2) ln(1.1)
ln(1.1)
7.27 years
n
n
n
FV PV i
n
n
n
= +
=
=
=
=
=
To Triple
(1 )
3 1 (1.10)
3 (1.10)
ln(3) ln(1.1)
n
n
n
FV PV i
n
= +
=
=
=
Calculator Solution:
To double:
N
I/Y
PV
PMT
FV
CPT N 7.2725
10
1.00
0
−2.00
To triple:
N
I/Y
PV
PMT
FV
CPT N 11.5267
10
1.00
0
−3.00
Answer: 11.53 years
22. Present value (LO9-3) If you owe $35,000 payable at the end of eight years, what amount
should your creditor accept in payment immediately if she could earn 13 percent on her
money?
9-22. Solution:
8
1
(1 )
1
$35,000 (1.13)
n
PV FV
i
PV
=
+
=
Calculator Solution:
N
I/Y
PV
PMT
FV
8
13
CPT PV −13,165.60
0
35,000
Answer: $13,165.60
23. Jack Hammer invests in a stock that will pay dividends of $2.00 at the end of the first year;
$2.20 at the end of the second year; and $2.40 at the end of the third year. Also, he believes
that at the end of the third year he will be able to sell the stock for $33. What is the present
value of all future benefits if a discount rate of 11 percent is applied? (Round all values to
two places to the right of the decimal point.)
9-23. Solution:
First Dividend
1
(1 )
1
$2 (1.11)
n
PV FV
i
PV
=
+
=
First dividend:
N
I/Y
PV
PMT
FV
1
11
CPT PV −1.80
0
2.00
Answer: $1.80
Second dividend:
N
I/Y
PV
PMT
FV
2
11
CPT PV −1.79
0
2.20
Answer: $1.79
Third dividend:
N
I/Y
PV
PMT
FV
3
11
CPT PV −1.75
0
2.40
Answer: $1.75
Selling price:
N
I/Y
PV
PMT
FV
3
11
CPT PV −24.13
0
33.00
Answer: $39.15
Total = 1.80 + 1.79 + 1.75 + 24.13 = $29.47
Appendix B
PV = FVIF
Discount rate = 11 percent
$ 2.00 × .901 = $ 1.80
2.20 × .802 = 1.79
2.40 × .731 = 1.75
33.00 × .731 = 24.12
24. Les Moore retired as president of Goodman Snack Foods Company but is currently on a
consulting contract for $35,000 per year for the next 10 years.
a. If Mr. Moore’s opportunity cost (potential return) is 10 percent, what is the present
value of his consulting contract?
b. Assuming Mr. Moore will not retire for two more years and will not start to receive
his 10 payments until the end of the third year, what would be the value of his
deferred annuity?
9-24. Solution:
.a
10
1
1(1 )
1
1(1.10)
$35,000 .10
n
A
A
i
PV A
i
PV
10
1
1(1 )
1
1(1.10)
$35,000 .10
n
A
A
i
PV A
i
PV
−+
=
−
=
Calculator Solution:
(a)
N
I/Y
PV
PMT
FV
10
10
CPT PV
−215,059.85
35,000
0
Answer: $215,059.85
(b)
First part: Find the PV of the 10 payments of annuity:
N
I/Y
PV
PMT
FV
10
10
CPT PV
−215,059.85
35,000
0
Answer: $215,059.85
Second part: Find the PV of the above FV lump sum:
N
I/Y
PV
PMT
FV
2
10
CPT PV
−177,735.41
0
215,059.85
Answer: $177,735.41
Appendix D
a. PVA = A × PVIFA (10%, 10 periods)
PVA = $35,000 × 6.145 = $215,075
b. Deferred annuity—Appendix D
PVA = A × PVIFA (i = 10%, 10 periods)
PVA = $35,000 × 6.145 = $215,075
Now, discount back this value for two periods.
PV = FV × PVIF (i = 10%, 2 periods) Appendix B $215,075 ×
.826
= $177,652
OR
Appendix D
PVA = $35,000 (6.814 – 1.7360, where n = 12, n = 2, and i =
10%)
= $35,000(5.078) = $177,730
The answer is slightly different from the preceding answer due to
rounding in the tables.
25. Juan Garza invested $20,000 10 years ago at 12 percent, compounded quarterly. How much
has he accumulated?
9-25. Solution:
Ten years with quarterly compounding means
10 4 40
.12 .03
4
n
i
= =
==
40
(1 )
$20,000 (1.03)
$65, 240.76
n
FV PV i
FV
FV
= +
=
=
Calculator Solution:
N
I/Y
PV
PMT
FV
40
3
20,000
0
CPT FV −65,240.76
26. Special compounding (LO9-5) Determine the amount of money in a savings account at
the end of 10 years, given an initial deposit of $5,500 and a 12 percent annual interest rate
when interest is compounded (a) annually, (b) semiannually, and (c) quarterly.
9-26. Solution:
.a
Annually
10
(1 )
$5,500 (1.12)
$17,082.17
n
FV PV i
FV
FV
= +
=
=
.b
Semiannually
2
20
(1 )
2
$5,500 (1.06)
$17,639.25
n
i
FV PV
FV
FV
= +
=
=
.c
Quarterly
4
40
(1 )
4
$5,500 (1.03)
$17,941.21
n
i
FV PV
FV
FV
= +
=
=
Calculator Solution:
(a)
N
I/Y
PV
PMT
FV
10
12
5,500
0
CPT FV −17,082.17
Answer: $17,082.17
(b)
N
I/Y
PV
PMT
FV
20
6
5,500
0
CPT FV −17,639.25
Answer: $17,639.25
(c)
N
I/Y
PV
PMT
FV
40
3
5,500
0
CPT FV −17,941.21
Answer: $17,941.21
Appendix A
27. Annuity due (LO9-4) As stated in the chapter, annuity payments are assumed to come at
the end of each payment period (termed an ordinary annuity). However, an exception
occurs when the annuity payments come at the beginning of each period (termed an annuity
due). To find the present value of an annuity due, subtract 1 from n and add 1 to the tabular
value. To find the present value of an annuity due, the annuity formula must be adjusted as
to the following:
1
1
1(1 ) 1AD
n
i
PV A
i
−
−
+
= +
Likewise, the formula for the future value of an annuity due requires a modification:
1
(1 ) 1 1
n
AD
i
FV A
i
+
+−
= −
What is the future value of a 15-year annuity of $1,800 per period where payments come at
the beginning of each period? The interest rate is 12 percent.
9-27. Solution:
1
16
(1 ) 1 1
(1.12) 1
$1,800 1
.12
$1,800 (41.75328)
$75,155.90
n
AD
AD
AD
AD
i
FV A
i
FV
FV
FV
+
+−
= −
−
= −
=
=
Calculator Solution:
Set calculator beginning
N
I/Y
PV
PMT
FV
15
12
0
1,800
CPT FV 75,155.90
Answer: $75,155.90
Appendix C
FVA = A × FVIFA
n = 16, i = 12% 42.753 – 1 = 41.753
FVA = $1,800 × 41.753 = $75,155
28. Annuity due (LO9-4) What is the present value of a 10-year annuity of $3,000 per period
in which payments come at the beginning of each period? The interest rate is 12 percent.
1
1
1(1 ) 1
AD
n
i
PV A
i
−
−
+
= +
9-28. Solution:
1
9
1
1(1 ) 1
1
1(1.12)
$3,000 1
.12
$3,000 (6.32825)
$18,984.75
n
AD
AD
AD
AD
i
PV A
i
PV
PV
PV
−
−
+
= +
−
= +
=
=
Calculator Solution:
Set calculator to beginning
N
I/Y
PV
PMT
FV
10
12
CPT PV
−18,984.75
3,000
0
Answer: $18,984.75
Appendix D
PVA = A × PVIFA
n = 9, i = 12% 5.328 + 1 = 6.328
PVA = $3,000 × 6.328 = $18,984
29. Present value alternative (LO9-3) Your grandfather has offered you a choice of one of
the three following alternatives: $7,500 now; $2,200 a year for nine years; or $31,000 at the
end of nine years. Assuming you could earn 10 percent annually, which alternative should
Option 2
9
1
1(1 )
1
1(1.10)
$2, 200 .10
n
A
A
i
PV A
i
PV
−+
=
−
=
9
1
(1 )
1
$31,000 (1.11)
n
PV FV
i
PV
=
+
=
Calculator Solution:
(a-1)
(first alternative) Present value of $7,500 received now: $7,500
(second alternative) Present value of annuity of $2,200 for nine years:
N
I/Y
PV
PMT
FV
9
10
CPT PV –12,669.85
2,200
0
Answer: $12,669.85
(third alternative) Present value of $31,000 received in nine years:
N
I/Y
PV
PMT
FV
9
10
CPT PV –13,147.03
0
31,000
Answer: $13,147.03
(a-2)
Select $31,000 received at end of nine years.
(b-1)
(first alternative) Present value of $7,500 received today: $7,500
(second alternative) Present value of annuity of $2,200 at 11 percent for nine years:
N
I/Y
PV
PMT
FV
9
11
CPT PV –12,181.50
2,200
0
Answer: $12,181.50
(third alternative) Present value of $31,000 received in nine years at 11 percent:
N
I/Y
PV
PMT
FV
9
11
CPT PV –12,118.67
0
31,000
Answer: $12,118.67
(b-2)
Select $2,200 received for nine years.
(first alternative) Present value of $7,500 received now:
$7,500
(second alternative) Present value of annuity of $2,200 for nine
years: Appendix D
A IFA
IFA
PV A×PV
$2, 200 PV (10%, 9 years)
$2, 200 5.759
=
=
=
(third alternative) Present value of $31,000 received in nine
years: Appendix B
IF
IF
PV FV×PV
$31,000×PV (10%, 9 years)
$31,000×.424
$13,144
=
=
=
=
Select $31,000 to be received in nine years.
9-29. (Continued)
Revised answers based on 11 percent.
(first alternative) Present value of $7,500 received today: $7,500
(second alternative) Present value of annuity of $2,200 at 11
percent for nine years: Appendix D