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=A * *

r=O

where FVIFAi,, represents the future value interest factor for an n-year annuity compounded at i

percent and can be found in Appendix B.

I50 [CHAR 6

TIME VALUE OF MONEY

EXAMPLE 6.4 Jane Oak wishes to determine the sum of money she will have in her savings account at the end

of 6 years by depositing $1,000 at the end of each year for the next 6 years. The annual interest rate is 8 percent.

Tfie FVIFA846,6ycorsis given in Appendix B as 7.336. Therefore,

S6 = $1,000(FVIFA8,6j $1,000(7.336) = $7,336

=

6.3 PRESENT VALUE-DISCOUNTING

Present value is the present worth of future sums of money. The process of calculating present values,

or discounting, is actually the opposite of finding the compounded future value. In connection with

present value calculations, the iiiterest rate i is called the discomt rate.

Recall that

F,,= P(l + i)"

Therefore,

Where PVIFI,, represents the present value interest factor for $1and is given in Appendix C.

EXAMPLE 6.5 Ron jaffe has been given an opportunity to receive $20,000 6 years from new. If he can earn 10

percent ofikisinvestments, what is the most he should pay for this opportunity? To answer this question, one must

compute the present value of $20,000 to be received 6years from now at a 10 percent rate of discount. F6 is $20,000,

i is 10 percent, which equals 0.1, and n is 6 years. PVIFlo,6from Appendix C is 0.5645.

[(1 1

P = $20,000 o.l)6] = $20,000(PVIFlo,6) $221,000(0.5645) = $11,290

=

+

This means that Ron Jaffe, who can earii 10 percent on his investment, cou!d be indifferent to the choice between

receiving $11,290 now iir $20,000 6 years from now since the amounts are time equivalent. In other words, he could -

invest $11,290 today at 10 percent and have $20,000 in 6 years.

Present Value of Mixed Streams of Cash Flows

The present value of a series of mixed payments (or receipts) is the sum of the present value of each

individua'l payment. We know that the present value of each individual payment is the payment times

the appropriate PVIE

EXAMPLE 6.6 Candy Parker has been offered an opportunity to receive the following mixed stream of revenue

over the next 3 years:˜

Revenue

Year

1 $1,000

2 $2,000

$500

3

If she must earn a minimum of 6 percent on her investment, what is the most she should pay today? The present

value of this series of mixed streams of revenue is as foiiows:

I Year

I

($1 x PVIF = Present Value

Revenue

˜ ˜˜˜˜˜˜˜˜˜˜˜˜˜˜

1,000 0.943

2,000 0.890 1,780

0.840

590 420

I

$3,143

161

TIME VALUE OF MONEY

CHAP. 61

Present Value of an Annuity

Interest received from bonds, pension funds, and insurance obligations all involve annuities. To

compare these financial instruments, we need to know the present value of each. The present value of

an annuity ( P n ) can be found by using the following equation:

.. +A*- 1

.-+. 1

1

P,,=A--+A

(I + i)' (1 + i)"

(1+ i)'

+-_I

1

1

1

= A [ ˜ + ˜ + (1+ I ) " -*

where PVIFAi,, represents the appropriate value for the present value interest factor for a $1annuity

discounted at i percent for n years and is found in Appendix D.

EXAMPLE 6.7 Assume that the revenues in Example 6.6 form an annuity of $ , O

1O Ofor 3 years. Then the present

value is

P, = A PVIFAi,,

*

= $l,OOO(PVIFA6,3)= $1,000(2.6730) = $2,673

P3

Perpetuities

Some annuities go on forever. Such annuities are called perpetuities. An example of a perpetuity

is preferred stock which yields a constant dollar dividend indefinitely. The present value of a perpetuity

is found as follows:

receipt A

--

Present value of a perpetuity =

discount rate i

EXAMPLE 6.8 Assume that a perpetual bond has an $80-per-year interest payment and that the discount rate

is 10 percent. The present value of this perpetuity is:

$800

p = -A - =$80

=

i 0.10

6.4 APPLICATIONS OF FUTURE VALUES AND PRESENT VALUES

Future and present values have numerous applications in financial and investment decisions, which

will be discussed throughout the book. Five of these applications are presented below.

Deposits to Accumulate a Future Sum (or Sinking Fund)

An individual might wish to find the annual deposit (or payment) that is necessary to accumulate

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