What is the formula for the present value of an ordinary annuity?

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 made

For example, ABC International has commited to make a legal settlement in the amount of $50,000 per year for each of the next ten years. What would it cost ABC if it were to settle the claim immediately, 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.

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