The variables in a future value of an annuity problem include all of the following, except

What Is the Future Value of an Annuity?

The future value of an annuity is the value of a group of recurring payments at a certain date in the future, assuming a particular rate of return, or discount rate. The higher the discount rate, the greater the annuity's future value.

Key Takeaways

• The future value of an annuity is a way of calculating how much money a series of payments will be worth at a certain point in the future.
• By contrast, the present value of an annuity measures how much money will be required to produce a series of future payments.
• In an ordinary annuity, payments are made at the end of each agreed-upon period. In an annuity due, payments are made at the beginning of each period.

Understanding the Future Value of an Annuity

Because of the time value of money, money received or paid out today is worth more than the same amount of money will be in the future. That's because the money can be invested and allowed to grow over time. By the same logic, a lump sum of $5,000 today is worth more than a series of five $1,000 annuity payments spread out over five years.

Ordinary annuities are more common, but an annuity due will result in a higher future value, all else being equal.

Example of the Future Value of an Annuity

The formula for the future value of an ordinary annuity is as follows. (An ordinary annuity pays interest at the end of a particular period, rather than at the beginning, as is the case with an annuity due.)

P = PMT × ( ( 1 + r ) n − 1 ) r where: P = Future value of an annuity stream PMT = Dollar amount of each annuity payment r = Interest rate (also known as discount rate) n = Number of periods in which payments will be made \begin{aligned} &\text{P} = \text{PMT} \times \frac { \big ( (1 + r) ^ n - 1 \big ) }{ r } \\ &\textbf{where:} \\ &\text{P} = \text{Future value of an annuity stream} \\ &\text{PMT} = \text{Dollar amount of each annuity payment} \\ &r = \text{Interest rate (also known as discount rate)} \\ &n = \text{Number of periods in which payments will be made} \\ \end{aligned} ​P=PMT×r((1+r)n−1)​where:P=Future value of an annuity streamPMT=Dollar amount of each annuity paymentr=Interest rate (also known as discount rate)n=Number of periods in which payments will be made​

For example, assume someone decides to invest $125,000 per year for the next five years in an annuity they expect to compound at 8% per year. The expected future value of this payment stream using the above formula is as follows:

Future value = $ 1 2 5 , 0 0 0 × ( ( 1 + 0 . 0 8 ) 5 − 1 ) 0 . 0 8 = $ 7 3 3 , 3 2 5 \begin{aligned} \text{Future value} &= \$125,000 \times \frac { \big ( ( 1 + 0.08 ) ^ 5 - 1 \big ) }{ 0.08 } \\ &= \$733,325 \\ \end{aligned} Future value​=$125,000×0.08((1+0.08)5−1)​=$733,325​

With an annuity due, where payments are made at the beginning of each period, the formula is slightly different. To find the future value of an annuity due, simply multiply the formula above by a factor of (1 + r). So:

P = PMT × ( ( 1 + r ) n − 1 ) r × ( 1 + r ) \begin{aligned} &\text{P} = \text{PMT} \times \frac { \big ( (1 + r) ^ n - 1 \big ) }{ r } \times ( 1 + r ) \\ \end{aligned} ​P=PMT×r((1+r)n−1)​×(1+r)​

If the same example as above were an annuity due, its future value would be calculated as follows:

Future value = $ 1 2 5 , 0 0 0 × ( ( 1 + 0 . 0 8 ) 5 − 1 ) 0 . 0 8 × ( 1 + 0 . 0 8 ) = $ 7 9 1 , 9 9 1 \begin{aligned} \text{Future value} &= \$125,000 \times \frac { \big ( ( 1 + 0.08 ) ^ 5 - 1 \big ) }{ 0.08 } \times ( 1 + 0.08 ) \\ &= \$791,991 \\ \end{aligned} Future value​=$125,000×0.08((1+0.08)5−1)​×(1+0.08)=$791,991​

All else being equal, the future value of an annuity due will be greater than the future value of an ordinary annuity because it has had an extra period to accumulate compounded interest. In this example, the future value of the annuity due is $58,666 more than that of the ordinary annuity.

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