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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 payments 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

PV = 1 1 = 1 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 provides 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 exactly the same cash payments as the investment in row 1. Thus the value of our annuity (row 3) must be equal to the value of the row 1 perpetuity less the value of the delayed row 2 perpetuity:

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

Present value of t-year annuity = C [1 Ј 1 ] r r(1 + r)t

The expression in square brackets shows the present value of a t-year annuity of $1 a year. It is generally known as the t-year annuity factor. Therefore, another way to 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 birthdays. But as long as you bear in mind that an annuity is equivalent to the difference between an immediate and a delayed perpetuity, you shouldn`t have any difficulty.

Back to Kangaroo Autos

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 calculations. The present value of each cash flow is calculated and then the three present values are summed. The annuity formula, however, is much quicker:

Present value = $4,000 [ 1 Ј 1 ] .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 interest rate of 10 percent.

Winning Big at a Slot Machine

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 worthy 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 installments 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:

PV = $465,000 20-year annuity factor = $465,000 [ 1 Ј 1 ] r r(1 + r)20

At an interest rate of 8 percent, the annuity factor is

[ 1 Ј 1 ] = 9.818 .08 .08(1.08)20

(We also could look up the annuity factor in either Table 1.8 or Table A.3.) The present value of the $465,000 annuity is $465,000 9.818 = $4,565,000. That ¬$9.3 million prize ­ has a true value of about $4.6 million.

This present value is the price which investors would be prepared to offer for the series of cash flows. For example, the gambling casino might arrange for an insurance company to actually make the payments to the lucky winner. In this case, the company would charge a bit under $4.6 million to take over the obligation. With this amount in hand today, it could generate enough interest income to make the 20 payments before

running its ¬account ­ down to zero.