## Annuities

Ordinary Annuity. An annuity is a series of equal papnents or receipts occurring over a specified number of periods. In an ordinary annuity, payments or receipts occur at the end of each period. Figure 3.3 shows the cash-flow sequence for an ordinary annuity on a time line. Assume that Figure 3.3 represents your receiving \$1,000 a year for three years. Now iet's further assume that you deposit each annual receipt in a savings aciount earning B percent compound annual interest. How much money will you have at the end of three years?

Expressed algebraically, with FVA, defined as the future (compound) value of an annuity, R the periodic receipt (or payment), and n the length of the annuity, the formula for FVA.

As you can see, FVA,, is simply equal to the periodic receipt (R) times the "sum of the future vaiue interest factors at I percent for time periods 0 to r - 1." Luckily, we have two shorthand ways of stating this mathematically.

Bill Veeck once bought the Chicago White Sox baseball team franchise for \$10 million and then sold it 5 years Iater for \$20 million. In short, he doubled his money in 5 years. What compound rate of return did Veeck earn on his investment?

A quick way to handle compound interest problems involving doubling your money makes use of the "Rule of 72." This rule states that if the number of years, n, for ivhich an investment will be held is divided into the value 72, we will get the approximate interest rate, /, required for the investment to double in value. In Veeck's case, the rule gives.

Alternatively, if Veeck had taken his initial investment and placed it in a savings account earniag 6 percent compound interest, he would have had to wait approximately 12 years for his money to have doubled.

Indeed, for most interest rates we encounter, the "Rule of 72" gives a good approximation of the interest rate - or the number ofyears - required to double your money. But the answer is not exact. For example, money doubling in 5 years would have to earn at a 14.87 percent compound annual rate [(1 + 0.1487)5 = 2]; the "Rule of 72" says 14.4 percent. Also, money invested at 6 percent interest would actually require only 11.9 years to double [(1 + 0.06)'1'o = 2]; the "Rule of 72" suggests 12.

However, for bailpark-close money-doubling approximations that can be done in your head, the "Rule of 72" comes in pretty handy.