## Formula for Future Value of an Ordinary Annuity

The future value (FV) of an ordinary annuity (which is an annuity where payments are made at the end of each period) can be calculated using the following formula:

\(FV = P \times \frac{(1 + r)^n – 1}{r} \)

Where:

- FV is the future value of the annuity
- P is the payment amount per period
- r is the interest rate per period
- n is the number of periods

This formula calculates the future value of a series of equal payments (P) made at regular intervals for n periods, assuming interest is compounded each period at a rate of r. This future value represents the amount of money that will be in the account after the last payment is made (that is why it’s an “ordinary” or “deferred” annuity, as the payment is made at the end of the period).

Please note that the interest rate (r) and the number of periods (n) must correspond with each other. For example, if the payments are made annually and you are given an annual interest rate, then r is the annual interest rate and n is the number of years. If payments are made monthly, you’ll need to adjust the interest rate to a monthly rate (usually by dividing the annual rate by 12) and count the number of periods in months.

Also, the formula assumes that the interest rate is constant over the entire period. If the interest rate varies, the formula would have to be adjusted accordingly or the calculation would have to be done on a period by period basis.

## Example of the Formula for Future Value of an Ordinary Annuity

Let’s consider a scenario where you decide to save $1,000 per year in an account that offers a 5% annual interest rate. You plan to make these payments at the end of each year for the next 10 years. What will be the future value of this ordinary annuity?

To calculate this, we can use the future value formula for an ordinary annuity:

\(FV = P \times \frac{(1 + r)^n – 1}{r} \)

Substituting the values into the formula, we have:

P = $1,000 (the payment per period)

r = 0.05 (the annual interest rate expressed as a decimal)

n = 10 (the number of periods, in years)

So, the calculation becomes:

\(FV = \$1,000 \times \frac{(1 + 0.05)^10 – 1}{0.05} \)

Performing the calculations within the brackets first:

\(FV = \$1,000 \times \frac{1.62889 – 1}{0.05} \)

\(FV = \$1,000 \times \frac{0..62889}{0.05} = \$12,577.8 \)

So, the future value of this ordinary annuity, after 10 years, will be approximately $12,577.8. This means that if you deposit $1,000 at the end of each year for 10 years into an account with an annual interest rate of 5%, you would have $12,577.8 in the account after the last payment.