By Yuriy Smirnov Ph.D.

In general terms, an annuity is a series of equal cash inflows or outflows made at fixed intervals. A series of coupon payments of a fixed-rate bond is an example of an annuity. So, the future value of an annuity (FVA) is a value at a specific date in the future based on a regular cash flow amount and interest rate.

Depending on the moment the regular payment is made, annuities can be classified as two types: an ordinary annuity is when cash flow comes in at the end of a relevant period. Its future value can be calculated as follows:

FVA_{Ordinary Annuity} = |
P × | (1 + r)^{N} - 1 |

r |

Here P is payment or cash flow per period, r represents the interest rate per period, and N is the number of periods.

If a regular payment is made at the beginning of the relevant period, we have an example of an annuity due. The formula to find its future value is shown below.

FVA_{Annuity Due} = |
P × | (1 + r)^{N} - 1 |
× (1 + r) |

r |

In turn, the equation describing the relationship between the future value of an ordinary annuity and annuity due is as follows:

FVA _{Annuity Due} = FVA _{Ordinary Annuity} × (1 + r)

Let’s assume that someone invests $1,000 each year for 5 years at an annual interest rate of 7.5%. If $1,000 is invested at the end of each year, we have an ordinary annuity. The future values of each cash flow are schematically presented in the chart below.

The first $1,000 invested at the end of the first year will earn interest during 4 years, and its future value will be $1,335.47. The next $1,000 invested at the end of the second year will earn interest during 3 years and will have a future value of $1,242.30. The same calculations are made for the other cash flows.

FV_{1} = $1,000 × (1 + 0,075)^{4} = $1,335.47

FV_{2} = $1,000 × (1 + 0,075)^{3} = $1,242.30

FV_{3} = $1,000 × (1 + 0,075)^{2} = $1,155.63

FV_{4} = $1,000 × (1 + 0,075)^{1} = $1,075.00

FV_{5} = $1,000 × (1 + 0,075)^{0} = $1,000.00

Thus, the total future value of all cash flows will be $5,808.39. We can also get the same result using the formula of future value of an ordinary annuity mentioned above.

FVA _{Ordinary Annuity} = |
$1,000 × | (1 + 0.075)^{5} - 1 |
= $5,808.39 |

0.075 |

Let’s assume that $1,000 is invested at the beginning of each year. Under such a scenario, we have the annuity due, and its cash flows will be as follows:

The first payment made at the beginning of the first year will earn interest during 5 years, and its future value will reach $1,435.63. The second $1,000 invested at the beginning of the second year will earn interest during 4 years, etc.

FV_{0} = $1,000 × (1 + 0,075)^{5} = $1,435.63

FV_{1} = $1,000 × (1 + 0,075)^{4} = $1,335.47

FV_{2} = $1,000 × (1 + 0,075)^{3} = $1,242.30

FV_{3} = $1,000 × (1 + 0,075)^{2} = $1,155.63

FV_{4} = $1,000 × (1+0,075)^{1} = $1,075.00

The total future value of cash flows will reach $6,244.02 at the end of the fifth year. The same amount can be calculated using the formula of future value of an annuity due.

FVA _{Annuity Due} = |
$1,000 × | (1 + 0.075)^{5} - 1 |
× (1 + 0.075) | = $6,244.02 |

0.075 |

or

FVA _{Annuity Due} = FVA _{Ordinary Annuity} × (1 + r) = $5,808.39 × (1 + 0.075) = $6,244.02

You can also calculate the future value of an annuity using our online calculator or discount table.

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