## Definition

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.

## Formula

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:

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.

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

## Examples

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.

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.

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