Payment amount
\$
Interest rate
%
Annuity term
yrs
mos
Compounding frequency
Yearly
Payment frequency
Yearly
Type of annuity
Ordinary Annuity
Future value of annuity
\$

# Future Value of Annuity Calculator

By Tibor Pal, PhD candidate

The future value of annuity calculator is a compact tool that helps you to compute the value of a series of equal cash flows at a future date. In other words, with this annuity calculator, you can estimate the future value of a series of periodic payments. You can also use it to find out what is an annuity payment, periods, or interest rate if other values are given. Besides, you can read about different types of annuities, and get some insight into the the analytical background.

## What is an annuity?

Annuity refers to a specific type of financial construction that involves a series of payments over a certain period of time, regardless of the direction of the flow of the money (i.e., the money being paid to you or you paying the money to someone else). Annuities must also satisfy two conditions: that the payments are equal and are made at fixed intervals. For example, 200 dollars paid at the end of each of the next ten years is a 10-year annuity.

If you happen to deal with an annuity, there are two aspects to be considered: the present and the future value of the annuity. This calculator will estimate the future value of annuities for you, but if you are interested in finding out the present value of an annuity, please visit our present value of annuity calculator.

## Types of annuities

There are multiple ways to classify annuities. You may hear about a life annuity where payments are handed out for the rest of the purchaser's (annuitant) life. Since this kind of annuity is only paid under particular circumstances, it is called a contingent annuity (i.e., it is contingent on how long the annuitant lives for). If the contract specifies the period in advance, we call it a certain or guaranteed annuity.

Annuities are also distinguished according to the variability of payments. There are fixed annuities, where the payments are constant, but there are also variable annuities that allow you to accumulate the payments and then invest them on a tax-deferred basis. There are also equity-indexed annuities were payments are linked to an index.

The most important way to differentiate annuities from the view of the present calculator is the timing of the payments.

In this context, there are two types of annuities:

• Ordinary annuity (or deferred annuity): payments are made at the ends of the periods - mortgages, car loans, and student loans are conventionally ordinary annuities.

• Annuity due: Payments are made at the beginning of each period - rental lease payments, life insurance premiums, and lottery payoffs (if you have the fortune to win one!)

The easiest way to understand the difference between these types of annuities is to consider a simple example. Let's assume that you deposit 100 dollars annually for three years, and the interest rate is 5 percent; thus, you have a \$100, 3-year, 5% annuity.

`Payment Amount = 100 dollars`

`Interest Rate = 5%`

`Annuity Term = 3 years`

The graph below shows the timelines of the two types of the annuity with their future values. As you can see, in the case of annuity due, each payment occurs a year before the payment at the ordinary annuity. The advanced payments have an immediate effect on the future value of the annuity as the money stays in your bank for longer, and therefore earns interest for one additional period. Therefore with the annuity due, the future value of the annuity is higher than with the ordinary annuity. The graph also serves to visually explain how the future value of an annuity is calculated: it is merely the sum of compounded cash flows estimated in each year. How to compute these individual payments? Look at our example for the ordinary annuity. The first payment earns interest for two periods, the second for one period, and the third earns no interest because it is made at the end of the annuity's life. This is an example of compound interest, a common feature in finance where interest is calculated on the interest.

This approach may sound straightforward, but if the annuity covers an extended interval, the computation may become burdensome. Besides, other factors that need to be take into consideration may appear and complicate the estimation even further. In the following section, you can learn how to apply our future value annuity calculator to any scenario, no matter how complex.

## How to use our annuity calculator?

In the previous section, we hope we provided some insight into how a simple annuity works. However, you can apply our future value of annuity calculator to help solve some more sophisticated financial problems. In this section, you can learn how to use this calculator and the mathematical background that governs it.

To start, let's have a quick look at the parameters and terms you may encounter in our calculator:

• Payment amount (PMT) is the amount paid in or out (cash flow) for each period.

• Interest rate (r) is the annual nominal interest rate expressed as a percentage.

• Annuity term constitutes the lifespan of the annuity.

• Compounding frequency (m) refers to the number of times the interest is compounded. For example, when compounding is applied annually, m=1, when quarterly, m=4, monthly, m=12, etc. You can choose the frequency as continuous as well, which is an extreme form and the theoretical limit of compounding frequency. In such a case, m=infinity.

• Payment frequency (q) indicates how often the payments will materialize.

• Type of annuity (T) signifies the timing of the payment in each payment period (ordinary annuity: end of each payment period; annuity due: the beginning of each payment period).

• Future value of annuity (FVA) the future value of any present value cash flows (payments).

In advanced mode, you can also see the following fields:

• Growth rate of annuity (g) is the percentage increase of an annuity in the case of a growing annuity.

• Number of periods (t) shows the annuity term in years.

• Equivalent interest rate and Periodic equivalent interest rate are the interest rates computed when the payments and compounding occur at a different frequency (cannot be set manually).

Now that you are (hopefully) familiar with the financial jargon applied in this calculator, we will provide an overview of the equations involved in the computation.

The two basic annuity formulas are as follows:

• Ordinary Annuity:

`FVA = PMT / i * ((1 + i) ^ n - 1)`

• Annuity Due:

`FVA = PMT / i * ((1 + i) ^ n - 1) * (1 + i)`

`n = m * t` where `n` is the total number of compounding intervals

`i = r / m` where `i` is the periodic interest rate (rate over the compounding intervals)

For the matter of simplicity, in the following specifications, we refer to the ordinary annuity.

• Future Value of a Growing Annuity (g ≠ i):

`FVA = PMT / (i - g) * ((1 + i) ^ n - (1 + g) ^ n)`

• Future Value of a Growing Annuity (g = i):

`FVA = PMT * n * (1 + i) ^ (n - 1)`

• Future Value of an Annuity with Continuous Compounding (m → ∞)

`FVA = PMT / (eʳ - 1) * (eʳᵗ - 1)`

where `e` stands for the exponential constant, which is approximately 2.718.

• Cipra Tomas: Financial and Insurance Formulas - 2010, Springer Heidelberg Dordrecht London New York
• Eugene F. Brigham and Michael C. Ehrhardt: Financial Management - Theory & Practice (15e) - 2017, Cengage Learning
• Zwillinger Dan: CRC standard mathematical tables and formulas - 33rd edition - 2018, Taylor & Francis Group