978-1259687969 Chapter 9 Solution Manual Part 4

9-24. Solution:

.a

10

1

1(1 )

1

1(1.10)

$35,000 .10

$215,059.85

n

A

i

PV A

i

PV

–+

= ´

–

= ´

=

.b

Present Value of the Annuity

10

1

1(1 )

1

1(1.10)

$35,000 .10

$215,059.85

n

A

i

PV A

i

PV

–+

= ´

–

= ´

=

Discount two years off

2

1

(1 )

1

$215,059.85 (1.10)

$177,735.41

n

PV FV

i

PV

= ´ +

= ´

=

Calculator Solution:

(a)

N I/Y PV PMT FV

(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

a. PVA = A × PVIFA (10%, 10 periods)

b. Deferred annuity—Appendix D

Now, discount back this value for two periods.

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

= ´ +

= ´

=

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

10

(1 )

$5,500 (1.12)

$17,082.17

n

FV PV i

FV

= ´ +

= ´

=

.b

Semiannually

2

20

(1 )

2

$5,500 (1.06)

$17,639.25

n

i

FV PV

FV

= ´ +

= ´

=

.c

Quarterly

4

40

(1 )

4

$5,500 (1.03)

$17,941.21

n

i

FV PV

FV

= ´ +

= ´

=

Calculator Solution:

(a)

N I/Y PV PMT FV

(b)

N I/Y PV PMT FV

(c)

N I/Y PV PMT FV

Appendix A

c. $5,500 × 3.262 = $17,941 (n = 40; i = 3%)

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

AD

n

i

PV A

i

–

æ ö

–

ç ÷

+

ç ÷

= ´ +

ç ÷

è ø

Likewise, the formula for the future value of an annuity due requires a

modi!cation:

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

$1,800 (41.75328)

$75,155.90

n

AD

i

FV A

i

FV

+

æ ö

+ –

= ´ –

ç ÷

è ø

æ ö

–

= ´ –

ç ÷

è ø

= ´

=

Calculator Solution:

Set calculator beginning

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

AD

n

i

PV A

i

–

æ ö

–

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

i

PV A

i

PV

–

æ ö

–

ç ÷

+

= ´ +

ç ÷

è ø

æ ö

–

ç ÷

= ´ +

ç ÷

è ø

= ´

=

N I/Y PV PMT FV

Answer: $18,984.75

Appendix D

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

you choose? If you could earn 11 percent annually, would you still choose the same

alternative?

9-29. Solution:

10% annually

Option 1

$7,500PV =

Option 2

9

1

1(1 )

1

1(1.10)

$2, 200 .10

$12,669.85

n

A

i

PV A

i

PV

–+

= ´

–

= ´

=

Option 3

9

1

(1 )

1

$31,000 (1.10)

$13,147.03

n

PV FV

i

PV

= ´ +

= ´

=

At a rate of 10 percent annually, the best deal is $31,000 in nine years because it has the

highest present value.

11% annually

Option 1

$7,500PV =

Option 2

9

1

1(1 )

1

1(1.11)

$2, 200 .11

$12,181.50

n

i

PV A

i

PV

–+

= ´

–

= ´

=

Option 3

9

1

(1 )

1

$31, 000 (1.11)

$12,118.67

n

PV FV

i

PV

= ´ +

= ´

=

At a rate of 11 percent annually, the best option is Option 2 because $2,200 a year for

nine years is the best deal.

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

(third alternative) Present value of $31,000 received in nine years:

N I/Y PV PMT FV

(a-2)

Select $31,000 received at end of nine years.

(b-1)

(second alternative) Present value of annuity of $2,200 at 11 percent for nine years:

N I/Y PV PMT FV

(third alternative) Present value of $31,000 received in nine years at 11 percent:

N I/Y PV PMT FV

(b-2)