# Annuity calculation

## Present value of an annuity

The present value of an annuity is easy to calculate: it is simply a DCF.

It can, in fact, be further simplified. To do calculate it this way:

1. Calculate the value of a perpetuity with the same payments, using the appropriate discount rate.
2. Discount the value of this value as though it was a payment made on the termination of the perpetuity.
3. Subtract the discounted value of the perpetuity from the original value.

This assume that the perpetuity is paid at the end of each period. The logic behind it is that the value of a annuity is that of the perpetuity less the value of cash flows after the annuity terminates (which are another perpetuity).

Alternatively you can simply multiply the amount of each payment by the annuity factor:

(1 - (1/(1 + r)n)/r

Where r is the discount rate, and,
n is the number of periods for which the annuity will be paid.

## Annuity payments given present value, rate and dates

Rather than having the payments and the rate and needing the present value, one often needs to calculate the correct payment given the desired present value, rate and the time the annuity will last for. This requires using the formula for the annuity factor and solving for the payment: i.e divide the present value by he annuity factor.