# How to Find the Present Value of an Annuity or Perpetuity

 Annuity               Perpetuity        n = Payment Amount \$ Annual Rate (Decimal) Present Value = \$

The present value of a future payment is the current monetary value of funds you will receive in the future. Suppose you are offered the choice between receiving \$1000 one year from now, and an equivalent (not necessarily equal) sum now. If you could earn a 5% annual return on money you hold now, then the present value of that \$1000 would equal \$1000/1.05 = \$952.38.

This means that if you accepted \$952.38 right now and invested it in a scheme with a 5% return, you would have (\$952.38)(1.05) = \$1000 in a year. In other words, \$952.38 now is equivalent to \$1000 in a year.

This general principle can be applied to a finite series of future annual payments (annuity), or a theoretically infinite series of future payments (perpetuity).

## PV of an Annuity

If you are set to receive payments of \$P each year for n years (starting 1 year from now), and the annual interest rate is r (expressed as a decimal), then the present value of those payments is

P/(1+r) + P/(1+r)2 + P/(1+r)3 + ... + P/(1+r)n
= P[(1+r)-1 + (1+r)-2 + (1+r)-3 + ... + (1+r)-n]
= (P/r)(1 - (1+r)-n)

Example: You are given the choice between receiving \$1500 every year for 10 years (starting 1 year from now), or \$12000 upfront. Assume that you can invest any money received into an account that earns 3.5% annually. Which option is better?

(1500/0.035)(1 - 1.035-10) = (42857.1429)(1 - 0.7089) = \$12474.91

And the present value of the second option is simply \$12000. Therefore the first option is a better deal.

## PV of a Perpetuity

In a perpetuity, the value of n is infinity. If you take the limit of (P/r)(1 - (1+r)-n) as n goes to infinity, then the factor on the right becomes (1-0), so the PV of a perpetuity is simply P/r.

Example: You will be given \$500 per year until your passing, at which point the payments will continue onto your heirs. If the money could be invested in an account that earns 2% annually, what is the present value of the perpetuity?

Since we have P = 500 and r = 0.02, the present value is 500/0.02 = \$25000.