# Future value of an annuity

## What is the Future Value of an Annuity?

The amount of a series of payments or receipts taken to a future date at a specified interest rate is called the **future value of an annuity**.

## Explanation

An **annuity** is a series of equal payments made at specified intervals. Interest is compounded on each of these payments. Annuities are often called rents because they are like the payment of monthly rentals. Annuity payments can be made at the beginning or the end of the specified intervals. If they are made at the beginning of the period, the annuity is called an annuity due, if the payment is made at the end of the period, it is called an ordinary annuity. The examples in this article use ordinary annuities, and so we will always assume that the payment takes place at the end of the period.

### Examples of annuities

Annuities are commonly encountered in business and accounting situations. For example, 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.

## How to determine the future value of an annuity

In some cases, it is appropriate to calculate the future value of the annuity, and in other cases, it is appropriate to calculate the present value of the annuity. We will first explain how to determine 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 4 years in a savings account that pays 10% interest compounded annually. The following table 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 3 periods. It earns interest for only 3 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, the future value of this first $1.00 single payment is $1.3310, determined as follows:

Future Value = Factor x Principal

= $1.3310 x $1.00

= $1.3310

The second payment earns interest for 2 periods and accumulates to $1.2100, and the third payment earns interest for only 1 period and accumulates to $1.10. The final payment, made at the end of the fourth year, does not earn any interest because we are determining the future value of the annuity at the end of the fourth period. The total of all payments compounded for the appropriate number of interest periods equals $4.6410 and represents the future value of this ordinary annuity.

Fortunately, we do not have to construct a table like this one in order to determine the future value of an annuity. We 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 1 is such a table. This table 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 4 years at 10% compounded annually is $4.6410, which is the amount we determined when we performed the calculation independently by summing the individual factors.

### Problems Involving the Future Value of an Annuity

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

### Formula

**Future Value of an annuity = Factor x Annuity Payment**

As long as we know two of the three variables, we can solve for the third. Thus, we 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 deposited in a savings and loan association $4,000 per year at the end of each of the next 8 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:

Future Value of an annuity = Factor x Annuity Payment

= 11.43589 x $4.000

= $45,743.56

### Determining the Annuity Payment

Assume that at the end of 15 years, you need to accumulate $100,000 to send your daughter to college. If you can earn 12% at your local savings and loan association, how much must you deposit at the end of each of the next 15 years in order to accumulate the $100,000 at the end of the fifteenth year? The annual payment is $2,682.42, as determined in the following:

Future Value of an annuity = Factor x Annuity Payment

Annuity payment = Future value of an annuity / Factor

= $100,000 / 37.27972

= $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 10 years and want to accumulate $30,200.99 at the end of the tenth year. What interest rate is required? You need to earn 2% quarterly, or 8% annually, determined as follows:

Future Value of an annuity = Factor x Annuity Payment

Factor = Future value of an annuity / Annuity payment

= $30,200.99 / $500

= 60.40198

Because the annuity payments are made quarterly, we must look across the fortieth period (10 years x 4) row until we find the factor. In this case, it is at the 2% column. Thus the interest rate is 2% quarterly or 8% annually.

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 we 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.