Future Value of an Annuity

An annuity is a series of equal payments made at specified intervals. Interest is compounded on each of these payments. Annuity payments can be made at the beginning or the end of the specified intervals. A payment made at the beginning of the period is called an annuity due; a payment made at the end of the period is called an ordinary annuity. The examples in this book use ordinary annuities, so it always will be assumed that the payment takes place at the end of the period.

Annuities are encountered frequently in business and accounting situations. For example, a lease payment or a mortgage represents an annuity. Life insurance contracts involving a series of equal payments at equal times are another example of an annuity. In some cases, it is appropriate to calculate the future value of the annuity; in other cases, it is appropriate to calculate the present value of the annuity.

Understanding the Future Value of an Annuity

The future value of an annuity is the sum of all the periodic payments plus the interest that has accumulated on them. To demonstrate how to calculate the future value of an annuity, assume that you deposit $1 at the end of each of the next four years in a savings account that pays 10% interest, compounded annually. Figure 11.3 shows how these $1 payments will accumulate to $4.6410 at the end of the fourth period, or year in this case. The future value of each dollar is determined by compounding interest at 10% for the appropriate number of periods. For example, the $1 deposited at the end of the first period earns interest for three periods. It earns interest for only three periods because it was deposited at the end of the first period and earns interest until the end of the fourth. Using the factors from Table III (see Appendix B, the future value of this first $1.00 single payment is $1.3310, determined as follows:

$\text{Future value}=\text{Factor}\times \text{Principal}$

$\mathrm{\$}1.3310=1.3310\times \mathrm{\$}1.00$

The second payment earns interest for two periods and accumulates to $1.2100, and the third payment earns interest for only one period and accumulates to $1.10. The final payment, made at the end of the fourth year, does not earn any interest, because the future value of the annuity is being determined at the end of the fourth period. The total of all payments compounded for the appropriate number of interest periods equals $4.6410—the future value of this ordinary annuity.

Fortunately, you do not have to construct a table like the one in Figure 11.3 in order to determine the future value of an annuity. You can use tables that present the factors necessary to calculate the future value of an annuity of $1, given different periods and interest rates. Table IV in Appendix B is such a table. It is constructed by simply summing the appropriate factors from the compound interest table. For example, the factor for the future value of a $1.00 annuity at the end of four years at 10% compounded annually is $4.6410. This is the same amount determined when the calculation was performed independently by summing the individual factors.

Solving Problems Involving the Future Value of an Annuity

By using the general formula below, you can solve a variety of problems involving the future value of an annuity:

$\text{Future value of an annuity}=\text{Factor}\times \text{Annuity payment}$

As long as you know two of the three variables, you can solve for the third. Thus, you can solve for the future value of the annuity, the annuity payment, the interest rate, or the number of periods.

Determining Future Value

Assume that you deposit in a savings and loan association $4,000 per year at the end of each of the next eight years. How much will you accumulate if you earn 10% compounded annually? The future value of this annuity is $45,743.56, determined as follows:

$\text{Future value of an annuity}=\text{Factor}\times \text{Annuity payment}$

$\mathrm{\$}\mathrm{45,743.56}=11.43589\times \mathrm{\$}\mathrm{4,000}$

The necessary business calculator keystrokes to compute the future value of an annuity are as follows:

Hewlett-Packard Keystrokes:

1. CLEAR ALL.

2. Set P/YR to 1.

3. 4,000 Press PMT

4. 8 Press N

5. 10 Press I/YR

6. Press FV for the answer of $45,743.55.

Determining the Annuity Payment

Assume that by the end of 15 years, you need to have accumulated $100,000 to send your daughter to college. If you can earn 12% at your financial institution, how much must you deposit at the end of each of the next 15 years in order to accumulate this amount? The annual payment is $2,682.42, as determined in the following:

$\text{Future value of an annuity}=\text{Factor}\times \text{Annuity payment}$

$\text{Annuity payment}=\frac{\text{Future value of an annuity}}{\text{Factor}}$

$\mathrm{\$}2,682.42=\frac{\mathrm{\$}100,000}{37.27972}$

The amount of the annuity payment can also be computed as follows:

Hewlett-Packard Keystrokes:

1. CLEAR ALL.

2. Set P/YR to 1.

3. 100,000 Press FV. The calculator considers the annual deposits to be cash outflows.

4. 15 Press N

5. 12 Press I/YR

6. Press PMT for the answer of $2,682.42.

Determining the Interest Rate

In some cases you may want to determine the interest rate that must be earned on an annuity in order to accumulate a predetermined amount. For example, assume that you invest $500 per quarter for ten years and want to have $30,200.99 by the end of the tenth year. What interest rate is required? You need to earn 2% quarterly, or 8% annually, determined as follows:

$\text{Future value of an annuity}=\text{Factor}\times \text{Annuity payment}$

$\text{Factor}=\frac{\text{Future value of an annuity}}{\text{Annuity payment}}$

$60.40198=\frac{\mathrm{\$}30,200.99}{\mathrm{\$}500}$

Because the annuity payments are made quarterly, you must look at Table IV (see Appendix B) across the fortieth-period (10 years × 4) row until you find the factor. In this case it is at the 2% column. Thus the interest rate is 2% quarterly, or 8% annually.

The necessary business calculator keystrokes are as follows:

Hewlett-Packard Keystrokes:

1. CLEAR ALL.

2. Set P/YR to 1.

3. 30,200.99 Press FV. The calculator considers the annual deposits to be cash out-flows.

4. 40 Press N

5. –500 Press PMT

6. Press I/YR for the answer of 2.00% every quarter, or 8.00% compounded quarterly.

In some situations, the interest rate is known, but the number of periods is missing. These problems can be solved by using the same technique you used to determine the interest rate. When the factor is determined, you must be sure to look down the appropriate interest column to find the factor on the annuity table.

Exercises