CHAPTER 5

Mathematics of Individual Finance

Annuity Due

A regular annuity, similar to that shown in Examples 5-6 and 5-7, is paid at the end of each period. In the world of individual finance, however, many annuities are paid with the first payment beginning immediately. This is called an annuity due. An excellent example is the lottery which pays out equal payments over a large number of years, often 20. In order to generate publicity, the first payment is generally given to the winner immediately, making the payments an annuity due.

Example 5-9

Beth has won the big $5 million lottery and will get paid in 20 annual payments with the first one today. If Beth uses a discount rate of 8 percent, what is the present value of her winnings?

Solution 5-9

First, we must calculate the size of the annual payment which is $5 million divided by 20 or $250,000. She will receive the payments as a single payment today, whose present value is $250,000, and a 19 year annuity of $250,000. The present value of the 19 year annuity of $250,000 at 8% is $2,400,899.80. Adding this to the $250,000 received today, the total present value is $2,650,899.80 or slightly more than half the stated value of the jackpot.

Most financial calculators will automatically calculate annuity due by using the DUE button rather than the CPT button. In this case it is [20 N 8 %i 250000 PMT DUE PV]. Since LOTUS does not have an annuity due function, it must be done in two parts as 250000+@PV(250000,.08,19).

Example 5-10

Frank will retire in 10 years at age 65. How much must he put aside as a lump sum now to insure monthly payments of $1,000 for 15 years using an annual rate of 7.5%?

Solution 5-10

This problem has two parts. First, he must calculate the present value of the annuity at the time of his retirement. Second, he must take the present value of that future amount as of today.

The present value of the annuity at retirement is the value of an annuity with payments of $1,000, a discount rate of 7.5% divided by 12 or 0.625% and 15 years times 12 or 180 periods. The present value is $107,873.43.

Since this amount will be needed in 10 years, we take the present value of the future value of $107,873.43 at 7.5% for 10 years which is $52,339.53. On a financial calculator enter [10 N 7.5 %i 107873.43 FV CPT PV]. On LOTUS you can do it in a single step as @PV(1000,.00625,180)/(1.075)^10.

In financial planning, it is sometimes clear whether one should use a regular annuity or an annuity due to calculate present value. Often, however, the problem is not so simply stated. A common example relates to retirement which will happen in the future. If a person expects to receive a pension of $30,000 per year and has anticipated expenses of $25,000 per year, does one treat these cash flows as regular annuities or annuities due? In other words, should they be treated as happening at the beginning of each year or at the end?

There are a number of possible solutions to this problem. You could treat all cash flows as if they occurred at the end of each period, at the beginning of each period, at the midpoint of each period or more frequently, such as monthly or even weekly during each period. Each solution has its advantages and disadvantages.

For ease of computation, most financial planners will assume cash flows occur at the end of each year when estimating flows in the future. Certainly, monthly or even more frequent periods would make the calculations more accurate since most pension payments and expenses occur throughout the year. In this book, unless otherwise specified, we will use end of period annual payments for calculating present values.

[Future Value Of An Annuity]

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