The formula for the present value of an ordinary annuity

An ordinary annuity is a series of equal payments, with all payments being made at the end of each successive period. An example of an ordinary annuity is a series of rent or lease payments. The present value calculation for an ordinary annuity is used to determine the total cost of an annuity if it were to be paid right now.

The formula for calculating the present value of an ordinary annuity is:

P = PMT [(1 - (1 / (1 + r)n)) / r]

Where:

P = The present value of the annuity stream to be paid in the future

PMT = The amount of each annuity payment

r = The interest rate

n = The number of periods over which payments are to be made

For example, ABC International has commited to a legal settlement that requires it to pay $50,000 per year at the end of each of the next ten years. What would it cost ABC if it were to instead settle the claim immediately with a single payment, assuming an interest rate of 5%? The calculation is:

P = $50,000 [(1 - (1/(1+.05)10))/.05]

P = $386,087

As another example, ABC International is contemplating the acquisition of a machinery asset. The supplier offers a financing deal under which ABC can pay $500 per month for 36 months, or the company can pay $15,000 in cash right now. The current market interest rate is 9%. Which is the better offer? The calculation of the present value of the annuity is:

P = $500 [(1 - (1/(1+.0075)36))/.0075]

P = $15,723.40

Since the up-front cash payment is less than the present value of the 36 monthly lease payments, ABC should pay cash for the machinery.

In the calculation, we convert the annual 9% rate to a monthly rate of 3/4%, which is calculated as the 9% annual rate divided by 12 months.

While this formula can be quite useful, it can yield misleading results if actual interest rates vary during the analysis period.