you choose the installment plan. As the time line in Figure 1.9 illustrates, the present

value of the plan™s three cash flows is:

Present Value

Immediate payment $8,000 = $8,000.00

Second payment $4,000/1.08 = 3,703.70

Third payment $4,000/(1.08)2 = 3,429.36

Total present value = $15,133.06

Because the present value of the three payments is less than $15,500, the installment

plan is in fact the cheaper alternative.

The installment plan™s present value equals the amount that you would need to invest

now to cover the three future payments. Let™s check to see that this works. If you start

with the present value of $15,133.06 in the bank, you could make the first $8,000

FIGURE 1.9

Present value of a stream of $8,000

cash flows

$4,000 $4,000

Year

Present value today 0 1 2

(time 0)

$8,000.00

4,000

$3,703.70

1.08

4,000

$3,429.36

(1.08)2

Total $15,133.06

50 SECTION ONE

payment and be left with $7,133.06. After 1 year, your savings would grow with inter-

est to $7,133.06 — 1.08 = $7,703.70. You then would make the second $4,000 payment

and be left with $3,703.70. This sum left in the bank would grow in the last year to

$3,703.70 — 1.08 = $4,000, just enough to make the last payment.

The present value of a stream of future cash flows is the amount you would

have to invest today to generate that stream.

In order to avoid estate taxes, your rich aunt Frederica will pay you $10,000 per year for

Self-Test 5

4 years, starting 1 year from now. What is the present value of your benefactor™s planned

gifts? The interest rate is 7 percent. How much will you have 4 years from now if you

invest each gift at 7 percent?

Level Cash Flows:

Perpetuities and Annuities

Frequently, you may need to value a stream of equal cash flows. For example, a home

Equally

ANNUITY

mortgage might require the homeowner to make equal monthly payments for the life

spaced level stream of cash

of the loan. For a 30-year loan, this would result in 360 equal payments. A 4-year car

flows.

loan might require 48 equal monthly payments. Any such sequence of equally spaced,

level cash flows is called an annuity. If the payment stream lasts forever, it is called a

Stream of

PERPETUITY

perpetuity.

level cash payments that

never ends.

HOW TO VALUE PERPETUITIES

Some time ago the British government borrowed by issuing perpetuities. Instead of re-

paying these loans, the British government pays the investors holding these securities a

fixed annual payment in perpetuity (forever).

The rate of interest on a perpetuity is equal to the promised annual payment C

divided by the present value. For example, if a perpetuity pays $10 per year and you

can buy it for $100, you will earn 10 percent interest each year on your investment. In

general,

cash payment

Interest rate on a perpetuity =

present value

C

r=

PV

We can rearrange this relationship to derive the present value of a perpetuity, given the

interest rate r and the cash payment C:

C cash payment

PV of perpetuity = =

r interest rate

Suppose some worthy person wishes to endow a chair in finance at your university.

If the rate of interest is 10 percent and the aim is to provide $100,000 a year forever, the

amount that must be set aside today is

The Time Value of Money 51

C $100,000

Present value of perpetuity = = = $1,000,000

r .10

Two warnings about the perpetuity formula. First, at a quick glance you can easily

confuse the formula with the present value of a single cash payment. A payment of $1

at the end of 1 year has a present value 1/(1 + r). The perpetuity has a value of 1/r. These

are quite different.

Second, the perpetuity formula tells us the value of a regular stream of payments

starting one period from now. Thus our endowment of $1 million would provide the uni-

versity with its first payment of $100,000 one year hence. If the worthy donor wants to

provide the university with an additional payment of $100,000 up front, he or she would

need to put aside $1,100,000.

Sometimes you may need to calculate the value of a perpetuity that does not start to

make payments for several years. For example, suppose that our philanthropist decides

to provide $100,000 a year with the first payment 4 years from now. We know that in

Year 3, this endowment will be an ordinary perpetuity with payments starting at the end

of 1 year. So our perpetuity formula tells us that in Year 3 the endowment will be worth

$100,000/r. But it is not worth that much now. To find today™s value we need to multi-

ply by the 3-year discount factor. Thus, the “delayed” perpetuity is worth

1 1 1

$100,000 — — = $1,000,000 — = $751,315

r (1 + r)3 (1.10)3

A British government perpetuity pays £4 a year forever and is selling for £48. What is

Self-Test 6

the interest rate?

HOW TO VALUE ANNUITIES

There are two ways to value an annuity, that is, a limited number of cash flows. The

slow way is to value each cash flow separately and add up the present values. The quick

way is to take advantage of the following simplification. Figure 1.10 shows the cash

payments and values of three investments.

Row 1. The investment shown in the first row provides a perpetual stream of $1 pay-

ments starting in Year 1. We have already seen that this perpetuity has a present value

of 1/r.

Row 2. Now look at the investment shown in the second row of Figure 1.10. It also

provides a perpetual stream of $1 payments, but these payments don™t start until Year 4.

This stream of payments is identical to the delayed perpetuity that we just valued. In

Year 3, the investment will be an ordinary perpetuity with payments starting in 1 year

and will therefore be worth 1/r in Year 3. To find the value today, we simply multiply

this figure by the 3-year discount factor. Thus

1 1 1

—

PV = =

r (1 + r)3 r(1 + r)3

Row 3. Finally, look at the investment shown in the third row of Figure 1.10. This pro-

vides a level payment of $1 a year for each of three years. In other words, it is a 3-year

annuity. You can also see that, taken together, the investments in rows 2 and 3 provide

52 SECTION ONE

FIGURE 1.10

Cash Flow

Valuing an annuity

Year: 1 2 3 4 5 6... Present Value

1

1. Perpetuity A $1 $1 $1 $1 $1 $1 . . .

r

1

2. Perpetuity B $1 $1 $1 . . .

r(1 + r)3

1 1

3. Three-year annuity $1 $1 $1

“

r r(1 + r)3

exactly the same cash payments as the investment in row 1. Thus the value of our an-

nuity (row 3) must be equal to the value of the row 1 perpetuity less the value of the de-

layed row 2 perpetuity:

1 1

Present value of a 3-year $1 annuity = “

r r(1 + r)3

The general formula for the value of an annuity that pays C dollars a year for each

of t years is

[ ]

1 1

Present value of t-year annuity = C “

r r(1 + r)t

The expression in square brackets shows the present value of a t-year annuity of $1

ANNUITY FACTOR

a year. It is generally known as the t-year annuity factor. Therefore, another way to

Present value of a $1 annuity.

write the value of an annuity is

Present value of t-year annuity = payment annuity factor

Remembering formulas is about as difficult as remembering other people™s birth-

days. But as long as you bear in mind that an annuity is equivalent to the difference be-

tween an immediate and a delayed perpetuity, you shouldn™t have any difficulty.

Back to Kangaroo Autos

EXAMPLE 8

Let us return to Kangaroo Autos for (almost) the last time. Most installment plans call

for level streams of payments. So let us suppose that this time Kangaroo offers an “easy

payment” scheme of $4,000 a year at the end of each of the next 3 years. First let™s do

the calculations the slow way, to show that if the interest rate is 10%, the present value

of the three payments is $9,947.41. The time line in Figure 1.11 shows these calcula-

tions. The present value of each cash flow is calculated and then the three present val-

ues are summed. The annuity formula, however, is much quicker:

[ ]

1 1

Present value = $4,000 — “

.10 .10(1.10)3

= $4,000 — 2.48685 = $9,947.41

You can use a calculator to work out annuity factors or you can use a set of annuity

tables. Table 1.8 is an abridged annuity table (an extended version is shown in Table A.3

at the end of the material). Check that you can find the 3-year annuity factor for an in-

terest rate of 10 percent.

The Time Value of Money 53

FIGURE 1.11

Time line for Kangaroo Autos $4,000 $4,000 $4,000

Year

0 1 2 3

Present value

4,000

$3,636.36

1.10

4,000

$3,305.79

(1.10)2

4,000

$3,005.26

(1.10)3

Total $9,947.41

If the interest rate is 8 percent, what is the 4-year discount factor? What is the 4-year

Self-Test 7

annuity factor? What is the relationship between these two numbers? Explain.

Winning Big at a Slot Machine

EXAMPLE 9

In May 1992, a 60-year-old nurse plunked down $12 in a Reno casino and walked away

with the biggest jackpot to that date”$9.3 million. We suspect she received unsolicited

congratulations, good wishes, and requests for money from dozens of more or less wor-

thy charities, relatives, and newly devoted friends. In response she could fairly point out

that her prize wasn™t really worth $9.3 million. That sum was to be paid in 20 annual in-

stallments of $465,000 each. What is the present value of the jackpot? The interest rate

at the time was about 8 percent.

The present value of these payments is simply the sum of the present values of each

payment. But rather than valuing each payment separately, it is much easier to treat the

cash payments as a 20-year annuity. To value this annuity we simply multiply $465,000

by the 20-year annuity factor:

TABLE 1.8

Interest Rate per Year