Calculating the Present and Future Value of Annuities

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April 10, 2025
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First determine what type of annuity you have

Fact checked by Suzanne Kvilhaug
Reviewed by David Kindness

Recurring or ongoing payments are technically annuities. Whether making a series of fixed payments over a period, such as rent or car loan, or receiving periodic income from a bond or certificate of deposit (CD), you can calculate the present value (PV) or future value (FV) of an annuity.

Key Takeaways

Recurring payments like rent on an apartment or interest on a bond can be considered annuities.
Ordinary annuities and annuities due differ in the timing of those recurring payments.
The future value of an annuity is the total value of payments at a future point in time.
The present value is the money required now to produce those future payments.

Types of Annuities

Annuities as ongoing payments can be defined as ordinary annuities or annuities due.

Ordinary Annuities: An ordinary annuity makes (or requires) payments at the end of a particular period. For example, bonds generally pay interest at the end of every six months.
Annuities Due: An annuity due, by contrast, involves payments made at the beginning of each period. Rent or loan payments required at the beginning of each month are examples.

With ordinary annuities, payments are made at the end of a specific period. Annuities due are made at the beginning of the period.

Future Value of an Ordinary Annuity

FV measures how much a series of regular payments will be worth at some point in the future, given a specified interest rate. If you plan to invest a certain amount each month or year, FV will tell you how much you will accumulate. If you are making regular payments on a loan, the FV helps determine the total cost of the loan.

Consider 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.

Suppose you invest $1,000 annually for five years at 5% interest. Below is how much you would have at the end of the five years.

*Image by Julie Bang © Investopedia 2019*

Or use the Future Value formula:

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}

$FV_{Ordinary Annuity} = C \times [\frac{( 1 + i ) ^{n} - 1}{i}] where: C = cash flow per period i = interest rate n = number of payments$

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}

$FV_{Ordinary Annuity} = $1, 000 \times [\frac{( 1 + 0.05 ) ^{5} - 1}{0.05}] = $1, 000 \times 5.53 = $5, 525.63$

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

Present Value of an Ordinary Annuity

In contrast to the FV calculation, the PV calculation tells you how much money is 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 five years, here is how a PV 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}

$PV_{Ordinary Annuity} = C \times [\frac{1 - ( 1 + i ) ^{- n}}{i}]$

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}

$PV_{Ordinary Annuity} = $1, 000 \times [\frac{1 - ( 1 + 0.05 ) ^{- 5}}{0.05}] = $1, 000 \times 4.33 = $4, 329.48$

Future Value of an Annuity Due

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, the ordinary annuity FV formula above requires a slight modification. It then results in the higher values 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 FV of an annuity due is:

begin{aligned} text{FV}_{text{Annuity Due}} &= text{C} times left [ frac{ (1 + i) ^ n – 1}{ i } right ] times (1 + i) \ end{aligned}

$FV_{Annuity Due} = C \times [\frac{( 1 + i ) ^{n} - 1}{i}] \times (1 + i)$

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}

$FV_{Annuity Due} = $1, 000 \times [\frac{( 1 + 0.05 ) ^{5} - 1}{0.05}] \times (1 + 0.05) = $1, 000 \times 5.53 \times 1.05 = $5, 801.91$

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

Present Value of an Annuity Due

Similarly, the formula for calculating the PV of an annuity due considers that payments are made at the beginning rather than the end of each period.

For example, you could use this formula to calculate the PV 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 cost at present value, assuming you kept your money in an account earning 5% interest.

This is the formula for calculating the PV 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}

$PV_{Annuity Due} = C \times [\frac{1 - ( 1 + i ) ^{- n}}{i}] \times (1 + i)$

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}

$PV_{Annuity Due} = $1, 000 \times [\frac{( 1 - ( 1 + 0.05 ) ^{- 5}}{0.05}] \times (1 + 0.05) = $1, 000 \times 4.33 \times 1.05 = $4, 545.95$

What Is an Example of an Ordinary Annuity Payment?

An ordinary annuity is a series of recurring payments made at the end of a period, such as payments for quarterly stock dividends.

What Is the Difference Between Amortization and Annuity Due?

Amortization schedules are given to borrowers by a lender, like a mortgage company. They outline the payments needed to pay off a loan and how the portion allocated to principal versus interest changes over time. An annuity due is the total payment required at the beginning of the payment schedule, such as the 1st of the month.

What Is a Deferred Annuity?

A deferred annuity is a contract with an insurance company that promises to pay the owner a regular income or lump sum at a future date. Deferred annuities differ from immediate annuities, which begin making payments right away.

The Bottom Line

Present value and future value formulas help individuals determine what an ordinary annuity or an annuity due is worth now or later. Such calculations and their results help with financial planning and investment decision-making.