## Present Value of Annuity Due

The present value of an annuity due is the total value of a series of future periodic payments, where each payment is received at the beginning of each period, discounted back to the present using a specified rate of return (or discount rate).

An annuity due is different from an ordinary annuity (where payments are made at the end of each period) because of the timing of the payments. In an annuity due, payments are made at the beginning of the period. This timing difference affects the present value calculations because each payment is received sooner, and therefore each payment has less time to be discounted.

The formula to calculate the present value of an annuity due is as follows:

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

Where:

- PV_due is the present value of the annuity due
- Pmt is the amount of each equal payment
- r is the discount rate per period
- n is the number of periods

The (1 + r) at the end of the formula adjusts the calculation for the fact that the payments are received at the beginning of each period.

This formula allows for the comparison of value of cash flows over time and gives insight into the impact of the timing of cash flows on the present value.

## Example of the Present Value of Annuity Due

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

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

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

Substitute the given values into the formula:

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

Perform the calculation:

PV_due = $1,000 * [1 – 0.78353] / 0.05 * 1.05

So:

PV_due = $1,000 * 0.21647 / 0.05 * 1.05

Therefore:

PV_due = $4,545.96

This means that the present value of receiving $1,000 at the beginning of each year for the next 5 years (annuity due), discounted at an annual rate of 5%, is approximately $4,545.96. This is higher than the present value of an ordinary annuity with the same parameters because each payment is received sooner in an annuity due.