# How to Find the Future Value of an Annuity

 Payment Amount \$ Number of Payments Starting... now (annuity due)    one year from now Annual Rate (Decimal)

The future value of a series of payments is the effective monetary value of those funds when the payments stop. For instance, suppose you are given three equal payments: \$1300 in one year, \$1300 in two years, and \$1300 in three years. If you could invest your money in an account that earns 4% interest per year, then in three years you would have

\$1300(1.042) + \$1300(1.04) + \$1300
= \$4058.08.

Thus, the future value of this annuity is \$4058.08. Knowing how to compute FV will help you compare different annuity and lump sum options.

## FV of an Annuity

If you are set to receive yearly payments of \$P for n years (starting 1 year from now and finishing at the end of Year n ), and the annual interest rate is r (expressed as a decimal), then the future value of those payments is

FV = P(1+r)n-1 + P(1+r)n-2 + ... + P(1+r) + P
= P[1+ (1+r) + (1+r)2 + ... + (1+r)n-2 + ... + (1+r)n-1]
= (P/r)[(1+r)n - 1]

## FV of an Annuity Due

An annuity due is when the payments start immediately. To compute the future value of an annuity due at the end of n years, just multiply the formula above by a factor of (1+r). This factor accounts for the extra year of interest.

FVDue = (P/r)[(1+r)n+1 - (1+r)]

## Example

You are presented with three options for payment. Option A: \$7000 now. Option B: \$15000 five years from now. Option C: five equal annual payments of \$2500, receiving with the first payment now, and the last payment at the beginning of the fifth year. (Option C is an annuity due.) Assume that you can invest your money in a scheme that earns 6.5% annually.

We can compare these three choices by computing future values at the end of 5 years. We use n = 5 for the number of payments and r = 0.065

FVA = \$7000(1.065)5 = \$9590.61

FVB = \$15000 (because it is already five years in the future!)

FVC = (\$2500/0.065)[1.0656 - 1.065] = \$15159.32

Option C has the highest future value under these conditions.