Annuities Defined

To understand how to calculate an annuity, it's useful to understand the variables that impact the calculation. An annuity is essentially a loan, a multi-period investment that is paid back over a fixed (or perpetual, in the case of a perpetuity)  period of time. The amount paid back over time is relative to the amount of time it takes to pay it back, the interest rate being applied, and the principal (when creating the annuity, this is the present value). 

Generally speaking, annuities and perpetuities will have consistent payments over time. However, it is also an option to scale payments up or down, for various reasons. 

Variables

This gives us six simple variables to use in our calculations:

1. Present Value (PV) - This is the value of the annuity at time 0 (when the annuity is first created)
2. Future Value (FV) - This is the value of the annuity at time n (i.e. at the conclusion of the life of the annuity).
3. Payments (A) - Each period will require individual payments that will be represented by this amount. 
4. Number of Payments (n) - The number of payments (A) will equate to the number of expected periods of payment over the life of the annuity.
5. Interest (i) - Annuities occur over time, and thus a given rate of return (interest) is applied to capture the time value of money.
6. Growth (g) - For annuities that have changes in payments, there is a growth rate applied to these payments over time.

Calculating Annuities

With all of the inputs above at hand, it's fairly simply to value various types of annuities. Generally investors, lenders, and borrowers are interested in the present and future value of annuities.

Present Value

The present value of an annuity can be calculated as follows:

${\displaystyle PV(A)\,=\,{\frac {A}{i}}\cdot \left[{1-{\frac {1}{\left(1+i\right)^{n}}}}\right]}$

For a growth annuity (where the payment amount changes at a predetermined rate over the life of the annuity), the present value can be calculated as follows:

${\displaystyle PV\,=\,{A \over (i-g)}\left[1-\left({1+g \over 1+i}\right)^{n}\right]}$

Future Value

The future value of an annuity can be determined using this equation:

${\displaystyle FV(A)\,=\,A\cdot {\frac {\left(1+i\right)^{n}-1}{i}}}$

In a situation where payments grow over time, the future value can be determined using this equation:

${\displaystyle FV(A)\,=\,A\cdot {\frac {\left(1+i\right)^{n}-\left(1+g\right)^{n}}{i-g}}}$

Various Formula Arrangements

It is also possible to use existing information to solve for missing information. Which is to say, if you know interest and time, you can solve for the following (given the following):

Annuities Equations

This table is a useful way to view the calculation of annuities variables from a number of directions. Understanding how to manipulate the formula will underline the relationship between the variables, and provide some conceptual clarity as to what annuities are.

This table is a useful way to view the calculation of annuities variables from a number of directions. Understanding how to manipulate the formula will underline the relationship between the variables, and provide some conceptual clarity as to what annuities are.