## Present Value of Annuity Formula

The present value of an annuity formula is a way to calculate the current worth of a series of equal future payments, also known as an annuity. The formula for the present value of an ordinary annuity (where payments are made at the end of each period) is as follows:

PV = Pmt * [1 – (1 + r)^-n] / r

Where:

- PV is the present value of the annuity
- Pmt is the amount of each payment
- r is the discount rate per period (for example, per year if the payments are annual)
- n is the number of periods

This formula is based on the concept of time value of money, which states that a dollar received today is worth more than a dollar received in the future because of the opportunity to earn returns on the money.

If payments are made at the beginning of each period, known as an annuity due, the formula adjusts by multiplying by (1 + r):

PV_due = Pmt * [1 – (1 + r)^-n] / r * (1 + r)

This adjustment accounts for the fact that each payment is received one period earlier, and therefore is discounted one less time, compared to an ordinary annuity.

## Example of the Present Value of Annuity Formula

Let’s calculate the present value of an ordinary annuity.

Suppose you are considering an investment that will pay you $500 per year for the next 10 years. You want to find out what this series of future payments is worth in today’s dollars, given an annual discount rate of 3%.

Here’s how you can calculate it using the formula:

PV = Pmt * [1 – (1 + r)^-n] / r

Substitute the given values into the formula:

PV = $500 * [1 – (1 + 0.03)^-10] / 0.03

Perform the calculation:

PV = $500 * [1 – 0.74409] / 0.03

So:

PV = $500 * 0.25591 / 0.03

Therefore:

PV = $4,265.17

This means that the present value of receiving $500 per year for the next 10 years, discounted at an annual rate of 3%, is approximately $4,265.17. This is the amount you’d need to invest today, at a 3% annual return, to generate a series of $500 payments for the next 10 years.