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Basic procedure

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 ascending lines in Figure 1.4 showed the future value of $100 invested with compound interest, 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 delayed. The longer you have to wait for money, the less it`s worth today, as we see in Figure 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:

PV = future payment (1 + r)t = future payment 1 (1 + r)t

The expression 1/(1 + r)t is called the discount factor. It measures the present value of $1 received in year t.

The simplest way to find the discount factor is to use a calculator, but financial managers 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 interest rates.

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

PV = $3,000 1 = $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

PV = $3,000 1 = $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

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 discount factor:

PV = $1,000 1 (1.0853)25 = $1,000 .129 = $129 TABLE 1.7

Present value of $1 Interest Rate per Year

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

1 ¬IOU ­ means ¬I owe you. ­ Coca-Cola`s IOUs are called bonds. Usually, bond investors receive a regular interest 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. .

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 interest rate is 8 percent and it is .116 if the rate is 9 percent. For an interest rate of 8.5 percent the discount factor is roughly halfway between at .131, a shade higher than the exact figure.

Finding the Value of Free Credit

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 Kangaroo. 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

PV = $4,000 + $6,000 1 (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 Kangaroo 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 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 comparing 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 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 accompanying diagram) points out that profits 65 years or more from now have negligible

present value.

DISCOUNT FACTOR

Present value of a $1 future payment.