## Present Value of an Annuity

The present value of an annuity is the total value of a series of future periodic payments (an annuity), discounted back to the present using a specific rate of return, also known as the discount rate.

Annuities are common in finance and they include regular payments for mortgages, rents, and savings for retirement. An annuity is a series of equal payments made at regular intervals (for example, monthly or annually).

The formula to calculate the present value of an ordinary annuity (payments 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 equal payment
- r is the discount rate per period
- n is the number of periods

This formula calculates the value in today’s dollars of a series of future payments. It allows us to answer questions like “how much money would I need in the bank today, in order to generate a series of $500 payments for the next 10 years, given a yearly interest rate of 5%?”

The present value of an annuity is useful for comparing the value of payments or cash flows over time and allows for the comparison of monetary amounts from different time periods on a similar footing.

## Example of the Present Value of an Annuity

Let’s say you are considering an investment that will pay you $1,000 per year for the next 5 years (an annuity). You want to find out what this series of future payments is worth in today’s dollars, given an annual discount rate of 5%.

The formula to calculate the present value of this annuity is:

PV = Pmt * [1 – (1 + r)^-n] / r

Substituting the given values into the formula:

PV = $1,000 * [1 – (1 + 0.05)^-5] / 0.05

Perform the calculation:

PV = $1,000 * [1 – 0.78353] / 0.05

So:

PV = $1,000 * 0.21647 / 0.05

Therefore:

PV = $4,329.48

This means that the present value of receiving $1,000 per year for the next 5 years, discounted at an annual rate of 5%, is approximately $4,329.48. This means that, assuming a 5% return per year, you should be indifferent between receiving $4,329.48 today and receiving $1,000 per year for the next 5 years.