The Financial Formula for Annual Annuity

by Edriaan Koening ; Updated April 19, 2017

An annual annuity refers to a series of payments that occur every year. Depending on the particular annuity, you may pay into the annuity or receive payments from the annuity. For example, a mortgage loan is an annuity you pay into, while a retirement annuity provides you with regular income from a bank or insurance company. There are three types of annual annuity: ordinary annuity, annuity due and growing annuity. When conducting calculations, you need to first determine the type of annuity and the appropriate formula to use.

Present Value Formula

With an ordinary annual annuity, you receive a fixed amount of money at the end of every year during the annuity term. The present value of an annuity refers to the value of the annuity at the beginning of the annuity term, before any interest accumulates. The formula for calculating the present value of an ordinary annuity is as follows: PV = C X {[1 - (1+r)^(-n)] / r}. In the formula, PV stands for present value, C for the amount of each annual payment, r for the annual interest rate and n for the number of payments. In other words, start by adding 1 and the interest rate, then raise the resulting figure to the power of the negative form of the number of payments. Deduct this from 1 and divide everything by the interest rate. Multiply the result by the amount of each payment.

Future Value Formula

An annuity's future value shows the value of the annuity at the end of the annuity term, after taking interest into account. The formula for the future value of an ordinary annuity is as follows: FV = C X {[(1 + r)^n - 1] / r}. FV in this formula represents future value, while all other letters in the formula mean the same as they do in the present value formula. Start by adding 1 and the interest rate together, then raise the result to the power of the number of payments. Deduct 1 from the result, then divide everything by the interest rate. Multiply the result by the number of payments.

Annuity Due Formula

An annuity due differs from an ordinary annuity because the payments take place at the beginning of each year instead of at the end. As such, calculations of an annuity due requires different formulas. To calculate the present value of an annuity due, use the following formula: PV = C X {[1 - (1+r)^(-n)] / r} X (1+r). Add 1 and the interest rate together, then raise it to the power of the negative form of the number of payments. Deduct the result from 1 and divide everything by the interest rate. Multiply this by the amount of each payment, then multiply the result again by the sum of 1 and the interest rate. The formula for future value of an annuity due is as follows: FV = C X {[(1+r)^n - 1] / r} X (1+r). Add 1 and the interest rate together, then raise it to the power of the number of payments. Deduct 1 from the result and divide it by the interest rate. Multiply the result by the amount of each payment, then multiply it again by the sum of 1 and the interest rate.

Growing Annuity Formula

With both the ordinary annuity and annuity due, the payments remain the same every year. A growing annuity involves payments that become larger every year, growing at a certain rate. In the following formulas, g stands for the constant growth rate of the annuity's annual payments. For this type of annuity, the present value formula is: PV = [C / (r - g)] X {1 - [(1 + g) / (1 + r)]^n}. Divide the amount of each payment by the difference between the interest rate and the growth rate. Set the resulting figure aside. Deduct the sum of 1 and the growth rate from 1, then divide the result by the sum of 1 and the interest rate, raised to the power of the number of payments. Multiply the result by the figure you set aside earlier. To calculate the future value of a growing annuity, use the following formula: FV = [(1 + r)^n - (1+g)^n] / (r - g). Add 1 and the interest rate together, then raise it to the power of the number of payments. From this, deduct the sum of 1 and the growth rate, raised to the power of the number of payments. Divide everything by the difference between the interest rate and the growth rate.

References

About the Author

Edriaan Koening began writing professionally in 2005, while studying toward her Bachelor of Arts in media and communications at the University of Melbourne. She has since written for several magazines and websites. Koening also holds a Master of Commerce in funds management and accounting from the University of New South Wales.

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