9-24. Solution:
.a
10
1
1(1 )
1
1(1.10)
$35,000 .10
$215,059.85
n
A
A
A
i
PV A
i
PV
PV
–+
= ´
–
= ´
=
.b
Present Value of the Annuity
10
1
1(1 )
1
1(1.10)
$35,000 .10
$215,059.85
n
A
A
A
i
PV A
i
PV
PV
–+
= ´
–
= ´
=
Discount two years off
2
1
(1 )
1
$215,059.85 (1.10)
$177,735.41
n
PV FV
i
PV
PV
= ´ +
= ´
=
Calculator Solution:
N I/Y PV PMT FV
Answer: $215,059.85
(b)
First part: Find the PV of the 10 payments of annuity:
N I/Y PV PMT FV
Answer: $215,059.85
Second part: Find the PV of the above FV lump sum:
N I/Y PV PMT FV
Appendix D
b. Deferred annuity—Appendix D
OR
Appendix D
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
Answer: $65,240.76
Appendix A
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
.b
Semiannually
.c
Quarterly
Calculator Solution:
(a)
N I/Y PV PMT FV
Answer: $17,082.17
(b)
N I/Y PV PMT FV
Answer: $17,639.25
(c)
N I/Y PV PMT FV
Answer: $17,941.21
Appendix A
FV = PV × FVIF
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 ) 1
AD
n
i
PV A
i
–
æ ö
–
ç ÷
+
ç ÷
= ´ +
ç ÷
ç ÷
è ø
Likewise, the formula for the future value of an annuity due requires a modicaon:
1
(1 ) 1 1
n
AD
i
FV A
i
+
æ ö
+ –
= ´ –
ç ÷
è ø
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
$75,155.90
n
AD
AD
AD
i
FV A
i
FV
FV
+
æ ö
+ –
= ´ –
ç ÷
è ø
æ ö
–
= ´ –
ç ÷
è ø
=
Calculator Solution:
Set calculator beginning
N I/Y PV PMT FV
Answer: $75,155.90
Appendix C
FVA = A × FVIFA
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
–
æ ö
–
ç ÷
+
ç ÷
= ´ +
ç ÷
ç ÷
è ø
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
–
æ ö
–
ç ÷
+
= ´ +
ç ÷
ç ÷
ç ÷
è ø
æ ö
–
ç ÷
= ´ +
ç ÷
ç ÷
ç ÷
è ø
= ´
=
Calculator Solution:
Set calculator to beginning
N I/Y PV PMT FV
Answer: $18,984.75
Appendix D
PVA = A × PVIFA
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
9-29. Solution:
Option 3
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
Answer: $12,669.85
(third alternative) Present value of $31,000 received in nine years:
N I/Y PV PMT FV
(a-2)
(b-1)
(second alternative) Present value of annuity of $2,200 at 11 percent for nine years:
N I/Y PV PMT FV
Answer: $12,181.50
(third alternative) Present value of $31,000 received in nine years at 11 percent:
N I/Y PV PMT FV
Answer: $12,118.67
(b-2)
Select $2,200 received for nine years.
(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
$12,670
=
= ´
= ´
=
(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
A IFA
IFA
PV A PV
$2, 200 PV (11%, 9 years)
$2, 200 5.537
$12,181
= ´
= ´
= ´
=
(third alternative) Present value of $31,000 received in nine years
at 11 percent: Appendix B
IF
IF
PV = FV×PV
= $31,000×PV (11%, 9 years)
= $31,000×.391
= $12,121
Select $2,200 a year for nine years. As the interest rate (discount
rate) increases, the present value declines.
30. You need $28,974 at the end of 10 years, and your only investment outlet is an 8 percent
long-term certificate of deposit (compounded annually). With the certificate of deposit, you
make an initial investment at the beginning of the first year.
a. What single payment could be made at the beginning of the first year to achieve this
objective?
b. What amount could you pay at the end of each year annually for 10 years to achieve
this same objective?