The Annuity Formula for the Present and Future Value of Annuities

Most of us have had the experience of making a series of fixed payments over a period of time—such as rent or car payments—or receiving a series of payments for a period of time, such as interest from a bond or certificate of deposit (CD). These recurring or ongoing payments are technically referred to as annuities (not to be confused with the financial product called an annuity, though the two are related).

There are several ways to measure the cost of making such payments or what they're ultimately worth. Here's what you need to know about calculating the present value (PV) or future value (FV) of an annuity.

Two Types of Annuities

Annuities, in this sense of the word, break down into two basic types: ordinary annuities and annuities due.

• Ordinary Annuities: An ordinary annuity makes (or requires) payments at the end of each period. For example, bonds generally pay interest at the end of every six months.
• Annuities Due: With an annuity due, by contrast, payments come at the beginning of each period. Rent, which landlords typically require at the beginning of each month, is a common example.
  

You can calculate the present or future value for an ordinary annuity or an annuity due using the following formulas.

Calculating the Future Value of an Ordinary Annuity

Future value (FV) is a measure of how much a series of regular payments will be worth at some point in the future, given a specified interest rate. So, for example, if you plan to invest a certain amount each month or year, it will tell you how much you'll have accumulated as of a future date. If you are making regular payments on a loan, the future value is useful in determining the total cost of the loan.

Consider, for example, a series of five $1,000 payments made at regular intervals.

Because of the time value of money—the concept that any given sum is worth more now than it will be in the future because it can be invested in the meantime—the first $1,000 payment is worth more than the second, and so on. So, let's assume that you invest $1,000 every year for the next five years, at 5% interest. Below is how much you would have at the end of the five-year period.

Rather than calculating each payment individually and then adding them all up, however, you can use the following formula, which will tell you how much money you'd have in the end:

 $\begin{aligned} &\text{FV}_{\text{Ordinary~Annuity}} = \text{C} \times \left [\frac { (1 + i) ^ n - 1 }{ i } \right] \\ &\textbf{where:} \\ &\text{C} = \text{cash flow per period} \\ &i = \text{interest rate} \\ &n = \text{number of payments} \\ \end{aligned}$​FVOrdinary Annuity​=C×[i(1+i)n−1​]where:C=cash flow per periodi=interest raten=number of payments​

Using the example above, here's how it would work:

 $\begin{aligned} \text{FV}_{\text{Ordinary~Annuity}} &= \$1,000 \times \left [\frac { (1 + 0.05) ^ 5 -1 }{ 0.05 } \right ] \\ &= \$1,000 \times 5.53 \\ &= \$5,525.63 \\ \end{aligned}$FVOrdinary Annuity​​=$1,000×[0.05(1+0.05)5−1​]=$1,000×5.53=$5,525.63​

Note that the one-cent difference in these results, $5,525.64 vs. $5,525.63, is due to rounding in the first calculation.

Calculating the Present Value of an Ordinary Annuity

In contrast to the future value calculation, a present value (PV) calculation tells you how much money would be required now to produce a series of payments in the future, again assuming a set interest rate.

Using the same example of five $1,000 payments made over a period of five years, here is how a present value calculation would look. It shows that $4,329.48, invested at 5% interest, would be sufficient to produce those five $1,000 payments.

This is the applicable formula:

 $\begin{aligned} &\text{PV}_{\text{Ordinary~Annuity}} = \text{C} \times \left [ \frac { 1 - (1 + i) ^ { -n }}{ i } \right ] \\ \end{aligned}$​PVOrdinary Annuity​=C×[i1−(1+i)−n​]​

If we plug the same numbers as above into the equation, here is the result:

 $\begin{aligned} \text{PV}_{\text{Ordinary~Annuity}} &= \$1,000 \times \left [ \frac {1 - (1 + 0.05) ^ { -5 } }{ 0.05 } \right ] \\ &=\$1,000 \times 4.33 \\ &=\$4,329.48 \\ \end{aligned}$PVOrdinary Annuity​​=$1,000×[0.051−(1+0.05)−5​]=$1,000×4.33=$4,329.48​

Calculating the Future Value of an Annuity Due

​An annuity due differs from an ordinary annuity in that the annuity due's payments are made at the beginning, rather than the end, of each period.

To account for payments occurring at the beginning of each period, it requires a slight modification to the formula used to calculate the future value of an ordinary annuity and results in higher values, as shown below.

The reason the values are higher is that payments made at the beginning of the period have more time to earn interest. For example, if the $1,000 was invested on January 1 rather than January 31, it would have an additional month to grow.

The formula for the future value of an annuity due is as follows:

$\begin{aligned} \text{FV}_{\text{Annuity Due}} &= \text{C} \times \left [ \frac{ (1 + i) ^ n - 1}{ i } \right ] \times (1 + i) \\ \end{aligned}$FVAnnuity Due​​=C×[i(1+i)n−1​]×(1+i)​

Here, we use the same numbers as in our previous examples:

$\begin{aligned} \text{FV}_{\text{Annuity Due}} &= \$1,000 \times \left [ \frac{ (1 + 0.05)^5 - 1}{ 0.05 } \right ] \times (1 + 0.05) \\ &= \$1,000 \times 5.53 \times 1.05 \\ &= \$5,801.91 \\ \end{aligned}$FVAnnuity Due​​=$1,000×[0.05(1+0.05)5−1​]×(1+0.05)=$1,000×5.53×1.05=$5,801.91​

Again, please note that the one-cent difference in these results, $5,801.92 vs. $5,801.91, is due to rounding in the first calculation.

Calculating the Present Value of an Annuity Due

Similarly, the formula for calculating the present value of an annuity due takes into account the fact that payments are made at the beginning rather than the end of each period.

For example, you could use this formula to calculate the present value of your future rent payments as specified in your lease. Let's say you pay $1,000 a month in rent. Below, we can see what the next five months would cost you, in terms of present value, assuming you kept your money in an account earning 5% interest.

This is the formula for calculating the present value of an annuity due:

 $\begin{aligned} \text{PV}_{\text{Annuity Due}} = \text{C} \times \left [ \frac{1 - (1 + i) ^ { -n } }{ i } \right ] \times (1 + i) \\ \end{aligned}$PVAnnuity Due​=C×[i1−(1+i)−n​]×(1+i)​

So, in this example:

 $\begin{aligned} \text{PV}_{\text{Annuity Due}} &= \$1,000 \times \left [ \tfrac{ (1 - (1 + 0.05) ^{ -5 } }{ 0.05 } \right] \times (1 + 0.05) \\ &= \$1,000 \times 4.33 \times1.05 \\ &= \$4,545.95 \\ \end{aligned}$PVAnnuity Due​​=$1,000×[0.05(1−(1+0.05)−5​]×(1+0.05)=$1,000×4.33×1.05=$4,545.95​

The Bottom Line

The formulas described above make it possible—and relatively easy, if you don't mind the math—to determine the present or future value of either an ordinary annuity or an annuity due. Excel can help with calculating the PV of fixed annuities. Financial calculators also have the ability to calculate these for you with the correct inputs.