How to Find the Present Value of an Annuity or Perpetuity
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 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 principal can be applied to a finite series of future annual payments (annuity), or a theoretically infinite series of future payments (perpetuity).
PV of an AnnuityIf 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?
We can weigh the options by comparing either their future values or present values. Since this article is about the latter, we will perform that calculation. The PV of the first option is
(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 PerpetuityIn 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 present value of the perpetuity?
Since we hav P = 500 and r = 0.02, the present value is 500/0.02 = $25000.
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