.2

.1

0

2 4 6 8 10 12 14 16 18 20

Number of years

The Time Value of Money 41

TABLE 1.7

Interest Rate per Year

Present value of $1

Number

of Years 5% 6% 7% 8% 9% 10%

1 .952 .943 .935 .926 .917 .909

2 .907 .890 .873 .857 .842 .826

3 .864 .840 .816 .794 .772 .751

4 .823 .792 .763 .735 .708 .683

5 .784 .747 .713 .681 .650 .621

10 .614 .558 .508 .463 .422 .386

20 .377 .312 .258 .215 .178 .149

30 .231 .174 .131 .099 .075 .057

Try using Table 1.7 to check our calculations of how much to put aside for that

$3,000 computer purchase. If the interest rate is 8 percent, the present value of $1 paid

at the end of 1 year is $.926. So the present value of $3,000 is

1

PV = $3,000 — = $3,000 — .926 = $2,778

1.08

which matches the value we obtained in Example 2.

What if the computer purchase is postponed until the end of 2 years? Table 1.7 shows

that the present value of $1 paid at the end of 2 years is .857. So the present value of

$3,000 is

1

PV = $3,000 — = $3,000 — .857 = $2,571

(1.08)2

which differs from the calculation in Example 2 only because of rounding error.

Notice that as you move along the rows in Table 1.7, moving to higher interest rates,

present values decline. As you move down the columns, moving to longer discounting

periods, present values again decline. (Why does this make sense?)

Coca-Cola Enterprises Borrows Some Cash

EXAMPLE 3

In 1995 Coca-Cola Enterprises needed to borrow about a quarter of a billion dollars for

25 years. It did so by selling IOUs, each of which simply promised to pay the holder

$1,000 at the end of 25 years.1 The market interest rate at the time was 8.53 percent.

How much would you have been prepared to pay for one of the company™s IOUs?

To calculate present value we multiply the $1,000 future payment by the 25-year dis-

count factor:

1

PV = $1,000 —

(1.0853)25

= $1,000 — .129 = $129

1 “IOU” means “I owe you.” Coca-Cola™s IOUs are called bonds. Usually, bond investors receive a regular in-

terest or coupon payment. The Coca-Cola Enterprises bond will make only a single payment at the end of Year

25. It was therefore known as a zero-coupon bond. .

42 SECTION ONE

Instead of using a calculator to find the discount factor, we could use Table A.2 at

the end of the material. You can see that the 25-year discount factor is .146 if the inter-

est rate is 8 percent and it is .116 if the rate is 9 percent. For an interest rate of 8.5 per-

cent the discount factor is roughly halfway between at .131, a shade higher than the

exact figure.

Suppose that Coca-Cola had promised to pay $1,000 at the end of 10 years. If the mar-

Self-Test 3

ket interest rate was 8.53 percent, how much would you have been prepared to pay for

a 10-year IOU of $1,000?

Finding the Value of Free Credit

EXAMPLE 4

Kangaroo Autos is offering free credit on a $10,000 car. You pay $4,000 down and then

the balance at the end of 2 years. Turtle Motors next door does not offer free credit but

will give you $500 off the list price. If the interest rate is 10 percent, which company is

offering the better deal?

Notice that you pay more in total by buying through Kangaroo, but, since part of the

payment is postponed, you can keep this money in the bank where it will continue to

earn interest. To compare the two offers, you need to calculate the present value of the

payments to Kangaroo. The time line in Figure 1.6 shows the cash payments to Kanga-

roo. The first payment, $4,000, takes place today. The second payment, $6,000, takes

place at the end of 2 years. To find its present value, we need to multiply by the 2-year

discount factor. The total present value of the payments to Kangaroo is therefore

1

PV = $4,000 + $6,000 —

(1.10)2

= $4,000 + $4,958.68 = $8,958.68

Suppose you start with $8,958.68. You make a down payment of $4,000 to Kanga-

roo Autos and invest the balance of $4,958.68. At an interest rate of 10 percent, this will

grow over 2 years to $4,958.68 — 1.102 = $6,000, just enough to make the final payment

FIGURE 1.6

Present value of the cash $6,000

flows to Kangaroo Autos

$4,000

Year

Present value today 0 1 2

(time 0)

$4,000.00

1

$6,000 $4,958.68

(1.10)2

Total $8,958.68

FINANCE IN ACTION

From Here to Eternity

Under the relentless pressures of compound inter-

Politicians, you may be aware, are fond of urging people

est, the value of future profits is ground to nothing as

to invest in the future. It would appear that some in-

the years go by. Suppose, for example, that you had a

vestors are taking them a bit too literally of late. The lat-

choice between making the following two gifts to a uni-

est fad among emerging-market bond investors, eager to

versity; you could write a cheque for $10,000 today, or

get a piece of the action, is to queue up for bonds with

give $1,000 a year for the next century. The latter dona-

100-year maturities, such as those issued by the Chinese

tion might seem the more generous one, but at a 10%

government and Tenaga Nasional, a Malaysian electrical

interest rate, they are worth the same amount. By the

utility.

time compound discounting had finished with it, that

Not to be outdone by these century bonds, Eurotun-

final $1,000 payment would

nel, the beleaguered company that operates the railway

be worth only 7 cents today

beneath the English Channel, is trying to tempt investors Live for today

(see chart).

with a millennium™s worth of profits. Last week, in a bid Present value of $1,000

discounted at 10%

What does this mean for

to sweeten the pot for its shareholders and creditors, received in year 1,000

Eurotunnel™s investors? Ex-

who must agree on an unpalatable financial restructur-

800

tending its franchise by 934

ing, it asked the British and French governments to ex-

years should increase its

tend its operating franchise from a mere 65 years to 999

600

value to today™s investors by

years. By offering investors some windfall profits, the

only 10“15%, after discount-

firm hopes they will be more likely to ratify its plan. Has 400

ing. If they are feeling gener-

the distant future become the latest place to make a fi-

ous, perhaps the British and

nancial killing? 200

French governments should

Alas, the future is not all that it is cracked up to be.

toss in another year and

Although at first glance 999 years of profits would seem 0 10 20 30 40 50 60 70 80 90 100

make the franchise an even

far better than 65 years, those last nine centuries are re-

1,000.

ally nothing to get excited about. The reason is that a

dollar spent today, human nature being what it is, is

worth more to people than a dollar spent tomorrow. So Source: © 1997 The Economist Newspaper Group Inc., Reprinted

when comparing profits in the future with those in the with permission. Further reproduction prohibited. www.economist.

present, the future profits must be “discounted” by a com

suitable interest rate.

on your automobile. The total cost of $8,958.68 is a better deal than the $9,500 charged

by Turtle Motors.

These calculations illustrate how important it is to use present values when compar-

ing alternative patterns of cash payment.

You should never compare cash flows occurring at different times without

first discounting them to a common date. By calculating present values, we

see how much cash must be set aside today to pay future bills.

The importance of discounting is highlighted in the nearby box, which examines the

SEE BOX

value of an extension of Eurotunnel™s operating franchise from 65 to 999 years. While

such an extension sounds as if it would be extremely valuable, the article (and its ac-

companying diagram) points out that profits 65 years or more from now have negligi-

ble present value.

43

FINANCIAL CALCULATOR

An Introduction to Financial Calculators

Financial calculators are designed with present value Future Values

and future value formulas already programmed. There-

Recall Example 3.1, where we calculated the future value

fore, you can readily solve many problems simply by

of Peter Minuit™s $24 investment. Enter 24 into the PV reg-

entering the inputs for the problem and punching a key

ister. (You enter the value by typing 24 and then pushing

for the solution.

the PV key.) We assumed an interest rate of 8 percent, so

The basic financial calculator uses five keys that cor-

enter 8 into the i register. Because the $24 had 374 years

respond to the inputs for common problems involving

to compound, enter 374 into the n register. Enter 0 into

the time value of money.

the PMT register because there is no recurring payment

involved in the calculation. Now ask the calculator to

compute FV. On some calculators you simply press the

n i PV FV PMT

FV key. On others you need to first press the “compute”

key (which may be labeled COMP or CPT), and then

press FV. The exact sequence of keystrokes for three

Each key represents the following input:

popular financial calculators are as follows:1

• n is the number of periods. (We have been using t to denote

the length of time or number of periods. Most calculators Hewlett-Packard Sharpe Texas Instruments

use n for the same concept.) HP-10B EL-733A BA II Plus

• i is the interest rate per period, expressed as a percentage PV PV PV

24 24 24

(not a decimal). For example, if the interest rate is 8 per- n n n

374 374 374

cent, you would enter 8, not .08. On some calculators this

I/YR i I/Y

8 8 8

key is written I/Y or I/YR. (We have been using r to denote

PMT PMT PMT

0 0 0

the interest rate or discount rate.)

FV COMP FV CPT FV

• PV is the present value.

• FV is the future value.

You should find after hitting the FV key that your calcu-

• PMT is the amount of any recurring payment (called an an-

lator shows a value of “75.979 trillion, which, except for

nuity). In single cash-flow problems such as those we have

the minus sign, is the future value of the $24.

considered so far, PMT is zero.

Why does the minus sign appear? Most calculators

Given any four of these inputs, the calculator will solve treat cash flows as either inflows (shown as positive

for the fifth. We will illustrate with several examples. numbers) or outflows (negative numbers). For example,

FINDING THE INTEREST RATE

When we looked at Coca-Cola™s IOUs in the previous section, we used the interest rate

to compute a fair market price for each IOU. Sometimes you are given the price and

have to calculate the interest rate that is being offered.

For example, when Coca-Cola borrowed money, it did not announce an interest rate.

It simply offered to sell each IOU for $129. Thus we know that

1

PV = $1,000 — = $129