# Accumulated value of an Annuity

14th February 2020Annuities are investment contracts sold by financial institutions like insurance companies and banks (generally referred to as the annuity issuer). When you purchase an annuity, you invest your money in a lump sum or gradually during an “accumulation period.” At a specified time the issuer must start making regular cash payments to you for a specified period of time. The future value of an annuity is an analytical tool an annuity issuer uses to estimate the total cost of making the required cash payments to you.

### Identification

When you purchase an annuity, the issuer invests your money to produce income. Annuity issuers make their money by keeping a part of the investment income, which is referred to as the discount rate. However, as each payment is made to you, the income the annuity issuer makes decreases. For the issuer, the total cost of making the annuity payments is the sum of the cash payments made to you plus the total reduction of income the issuer incurs as the payments are made. Issuers calculate the future value of annuities to help them decide how to schedule payments and how large their share (the discount rate) must be to cover expenses and make a profit.

### Types

The formula for the future value of an annuity varies slightly depending on the type of annuity. Ordinary annuities are paid at the end of each time period. Annuities paid at the start of each period are called annuities due. Many annuities are paid yearly. However, some annuities make payments on a semiannual, quarterly or monthly schedule.

### Formula

The basic equation for the future value of an annuity is for an ordinary annuity paid once each year. The formula is F = P * ([1 + I]^N – 1 )/I. P is the payment amount. I is equal to the interest (discount) rate. N is the number of payments (the “^” means N is an exponent). F is the future value of the annuity. For example, if the annuity pays $500 annually for 10 years and the discount rate is 6 percent, you have $500 * ([1 + 0.06]^10 – 1 )/0.06. The future value works out to $6,590.40. This means that, at the end of 10 years, the issuer’s total cost is equal to $6,590.40 ($5,000 in payments plus $1,590.40 in interest not earned).

### Payment Periods

In order to use the formula for the future value of an annuity when the payment interval is less than one year, you must make two adjustments. First, divide the discount rate (I) by the number of payments per year to find the rate of interest paid each month. Use this monthly rate as your value for I. Second, multiply the number of annual payments (N) by the number of payments each year to find the total number of payments and use this value for N.

### Annuity Due

Because payments for an annuity due are made at the beginning of the payment period, the future value of the annuity is increased by the interest earned for one time period. Start by calculating the future value using the equation for an ordinary annuity for the appropriate time period. Then multiply the result by 1 + I where I is equal to the discount rate for the period.