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# How to Find the Future Value of an Annuity

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(1.04^{2}) + $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.

FV

_{Due}= (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

FV

_{A}= $7000(1.065)

^{5}= $9590.61

FV

_{B}= $15000 (because it is already five years in the future!)

FV

_{C}= ($2500/0.065)[1.065

^{6}- 1.065] = $15159.32

Option C has the highest future value under these conditions.

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