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10, press = and discover that the answer is 1.791. (Try this!)

If you donâ€™t have a calculator, you can use a table of future values such as Table 1.6.

Check that you can use it to work out the future value of a 10-year investment at 6 per-

cent. First find the row corresponding to 10 years. Now work along that row until you

reach the column for a 6 percent interest rate. The entry shows that $1 invested for 10

years at 6 percent grows to $1.791.

Now try one more example. If you invest $1 for 20 years at 10 percent and do not

withdraw any money, what will you have at the end? Your answer should be $6.727.

Table 1.6 gives futures values for only a small selection of years and interest rates.

Table A.1 at the end of the material is a bigger version of Table 1.6. It presents the fu-

ture value of a $1 investment for a wide range of time periods and interest rates.

Future value tables are tedious, and as Table 1.6 demonstrates, they show future val-

ues only for a limited set of interest rates and time periods. For example, suppose that

you want to calculate future values using an interest rate of 7.835 percent. The power

FIGURE 1.3

Compound interest 140

120

Value in your account, dollars

100

This yearâ€™s

interest

80

Interest from

previous years

60

Original

investment

40

20

0

1 2 3 4 5

Year

The Time Value of Money 37

FIGURE 1.4

Future values with compound 18

interest

16

r=0

Future value of $1, dollars

14 r = 5%

r = 10%

12

r = 15%

10

8

6

4

2

0

2 4 6 8 10 12 14 16 18 20

Number of years

TABLE 1.6

Interest Rate per Year

Future value of $1

Number

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

1 1.050 1.060 1.070 1.080 1.090 1.100

2 1.103 1.124 1.145 1.166 1.188 1.210

3 1.158 1.191 1.225 1.260 1.295 1.331

4 1.216 1.262 1.311 1.360 1.412 1.464

5 1.276 1.338 1.403 1.469 1.539 1.611

10 1.629 1.791 1.967 2.159 2.367 2.594

20 2.653 3.207 3.870 4.661 5.604 6.727

30 4.322 5.743 7.612 10.063 13.268 17.449

key on your calculator will be faster and easier than future value tables. A third alter-

native is to use a financial calculator. These are discussed in two boxes later.

Manhattan Island

EXAMPLE 1

Almost everyoneâ€™s favorite example of the power of compound interest is the sale of

Manhattan Island for $24 in 1626 to Peter Minuit. Based on New York real estate prices

today, it seems that Minuit got a great deal. But consider the future value of that $24 if

it had been invested for 374 years (2000 minus 1626) at an interest rate of 8 percent per

year:

$24 Ã— (1.08)374 = $75,979,000,000,000

= $75.979 trillion

Perhaps the deal wasnâ€™t as good as it appeared. The total value of land on Manhattan

today is only a fraction of $75 trillion.

38 SECTION ONE

Though entertaining, this analysis is actually somewhat misleading. First, the 8 per-

cent interest rate weâ€™ve used to compute future values is quite high by historical stan-

dards. At a 3.5 percent interest rate, more consistent with historical experience, the fu-

ture value of the $24 would be dramatically lower, only $24 Ã— (1.035)374 = $9,287,569!

Second, we have understated the returns to Mr. Minuit and his successors: we have ig-

nored all the rental income that the islandâ€™s land has generated over the last three or four

centuries.

All things considered, if we had been around in 1626, we would have gladly paid $24

for the island.

The power of compounding is not restricted to money. Foresters try to forecast the

compound growth rate of trees, demographers the compound growth rate of population.

A social commentator once observed that the number of lawyers in the United States is

increasing at a higher compound rate than the population as a whole (3.6 vs. .9 percent

in the 1980s) and calculated that in about two centuries there will be more lawyers than

people. In all these cases, the principle is the same:

Compound growth means that value increases each period by the factor (1 +

growth rate). The value after t periods will equal the initial value times (1 +

growth rate)t. When money is invested at compound interest, the growth rate

is the interest rate.

Suppose that Peter Minuit did not become the first New York real estate tycoon, but in-

Self-Test 1

stead had invested his $24 at a 5 percent interest rate in New Amsterdam Savings Bank.

What would have been the balance in his account after 5 years? 50 years?

Start-up Enterprises had sales last year of only $.5 million. However, a stock market an-

Self-Test 2

alyst is bullish on the company and predicts that sales will double each year for 4 years.

What are projected sales at the end of this period?

Present Values

Money can be invested to earn interest. If you are offered the choice between $100,000

now and $100,000 at the end of the year, you naturally take the money now to get a yearâ€™s

interest. Financial managers make the same point when they say that money in hand

today has a time value or when they quote perhaps the most basic financial principle:

A dollar today is worth more than a dollar tomorrow.

We have seen that $100 invested for 1 year at 6 percent will grow to a future value

of 100 Ã— 1.06 = $106. Letâ€™s turn this around: How much do we need to invest now in

order to produce $106 at the end of the year? Financial managers refer to this as the

present value (PV) of the $106 payoff.

Future value is calculated by multiplying the present investment by 1 plus the inter-

The Time Value of Money 39

est rate, .06, or 1.06. To calculate present value, we simply reverse the process and di-

vide the future value by 1.06:

future value $106

Present value = PV = = = $100

1.06 1.06

What is the present value of, say, $112.36 to be received 2 years from now? Again

we ask, â€œHow much would we need to invest now to produce $112.36 after 2 years?â€

The answer is obviously $100; weâ€™ve already calculated that at 6 percent $100 grows to

$112.36:

$100 Ã— (1.06)2 = $112.36

However, if we donâ€™t know, or forgot the answer, we just divide future value by (1.06)2:

$112.36

Present value = PV = = $100

(1.06)2

In general, for a future value or payment t periods away, present value is

future value after t periods

Present value =

(1 + r)t

DISCOUNT RATE Interest In this context the interest rate r is known as the discount rate and the present value is

rate used to compute present often called the discounted value of the future payment. To calculate present value, we

values of future cash flows. discounted the future value at the interest r.

Saving to Buy a New Computer

EXAMPLE 2

Suppose you need $3,000 next year to buy a new computer. The interest rate is 8 per-

cent per year. How much money should you set aside now in order to pay for the pur-

chase? Just calculate the present value at an 8 percent interest rate of a $3,000 payment

at the end of one year. This value is

$3,000

PV = = $2,778

1.08

Notice that $2,778 invested for 1 year at 8 percent will prove just enough to buy your

computer:

Future value = $2,778 Ã— 1.08 = $3,000

The longer the time before you must make a payment, the less you need to invest

today. For example, suppose that you can postpone buying that computer until the end

of 2 years. In this case we calculate the present value of the future payment by dividing

$3,000 by (1.08)2:

$3,000

PV = = $2,572

(1.08)2

Thus you need to invest $2,778 today to provide $3,000 in 1 year but only $2,572 to

provide the same $3,000 in 2 years.

40 SECTION ONE

We repeat the basic procedure:

To work out how much you will have in the future if you invest for t years at

an interest rate r, multiply the initial investment by (1 + r)t. To find the present

value of a future payment, run the process in reverse and divide by (1 + r)t.

Present values are always calculated using compound interest. Whereas the as-

cending lines in Figure 1.4 showed the future value of $100 invested with compound in-

terest, when we calculate present values we move back along the lines from future to

present.

Thus present values decline, other things equal, when future cash payments are de-

layed. The longer you have to wait for money, the less itâ€™s worth today, as we see in Fig-

ure 1.5. Notice how very small variations in the interest rate can have a powerful effect

on the value of distant cash flows. At an interest rate of 10 percent, a payment of $1 in

Year 20 is worth $.15 today. If the interest rate increases to 15 percent, the value of the

future payment falls by about 60 percent to $.06.

The present value formula is sometimes written differently. Instead of dividing the

future payment by (1 + r)t, we could equally well multiply it by 1/(1 + r)t:

future payment

PV =

(1 + r)t

1

= future payment Ã—

(1 + r)t

DISCOUNT FACTOR The expression 1/(1 + r)t is called the discount factor. It measures the present value of

Present value of a $1 future $1 received in year t.

payment. The simplest way to find the discount factor is to use a calculator, but financial man-

agers sometimes find it convenient to use tables of discount factors. For example, Table

1.7 shows discount factors for a small range of years and interest rates. Table A.2 at the

end of the material provides a set of discount factors for a wide range of years and in-

terest rates.

FIGURE 1.5

Present value of a future

cash flow of $1 1

.9

.8 r = 5%

Present value of $1, dollars

r = 10%

.7

r = 15%

.6

.5

.4

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