# Future Value of Annuities

Annuities are level streams of payments. Each payment is the same amount, and occurs at a regular interval. Sometimes, one may be curious to learn how much a recurring stream of payments will grow to after a number of periods.

# Future Value of an Annuity Due

An annuity due (also known as an annuity in advance) involves a level stream of payments, with the payments being made at the beginning of each time period. For instance, perhaps you plan on saving for retirement by investing \$5,000 at the beginning of each year for the next 5 years. If the annual interest rate is 10% per year, how much will you accumulate by the end of the 5-year period? The following graphic shows how each of the five individual payments would grow, and the accumulated total would reach \$33,578: Although the graphic provides a useful explanatory tool, it is a bit cumbersome to implement. The same conclusion can be reached by reference to a FUTURE VALUE OF AN ANNUITY DUE TABLE. Examine the table linked at the website to find the value of 6.71561 (10% column/5-period row). Multiplying the \$5,000 annual payment by this factor yields \$33,578 (\$5,000 X 6.71561). This means that the \$25,000 paid in will have grown to \$33,578; perhaps Albert Einstein was right!

# Future Value of an Ordinary Annuity

Sometimes an annuity will be based on "end of period" payments. These annuities are called ordinary annuities (also known as annuities in arrears). The next graphic portrays a 5-year, 10%, ordinary annuity involving level payments of \$5,000 each. Notice the similarity to the preceding graphic - except that each year's payment is shifted to the end of the year. This means each payment will accumulate interest for one less year, and the final payment will accumulate no interest! Be sure to note the striking difference between the accumulated total under an annuity due versus and ordinary annuity (\$33,578 vs. \$30,526). The moral is to save early and save often (and live long!) to take advantage of the power of compound interest. As you might have guessed, there are also tables that reflect the FUTURE VALUE OF AN ORDINARY ANNUITY. Review the table found in the appendix to satisfy yourself about the \$30,526 amount (\$5,000 X 6.10510).

# Present Value

Future value calculations provide useful tools for financial planning. But, many decisions and accounting measurements will be based on a reciprocal concept known as present value. Present value (also known as discounting) determines the current worth of cash to be received in the future. For instance, how much would you be willing to take today, in lieu of \$1 in one year. If the interest rate is 10%, presumably you would accept the sum that would grow to \$1 in one year if it were invested at 10%. This happens to be \$0.90909. In other words, invest 90.90 for a year at 10%, and it will grow to \$1 (\$0.90909 X 1.1 = \$1). Thus, present value calculations are simply the reciprocal of future value calculations: Where "i" is the interest rate per period and "n" is the number of periods

The PRESENT VALUE OF \$1 TABLE (found in the appendix) reveals predetermined values for calculating the present value of \$1, based on alternative assumptions about interest rates and time periods. To illustrate, a \$25,000 lump sum amount to be received at the end of 10 years, at 8% annual interest, with semiannual compounding, would have a present value of \$11,410 (recall the earlier discussion, and use the 4% column/20-period row - \$25,000 X 0.45639).

# Present Value of an Annuity Due

Present value calculations are also applicable to annuities. Perhaps you are considering buying an investment that returns \$5,000 per year for five years, with the first payment to be received immediately. What should you pay for this investment in you have a target rate of return of 10%? The graphic shows that the annuity has a present value of \$20,849. Of course, there is a PRESENT VALUE OF AN ANNUITY DUE TABLE (see the link at the companion website) to ease the burden of this calculation (\$5,000 X 4.16897 = \$20,849).