Present Value of an Annuity

A stream of equal future payments is worth less than their simple total, because money arriving in ten years buys less than money arriving today. This factor — engineers write it (P/A, i, n) — collapses the whole stream into one number. Worked example: $1,000 a year for ten years at 5% is worth (1 − 1.05⁻¹⁰)/0.05 = (1 − 0.613913)/0.05 = 7.7217 years' worth today, so P = 7,721.73, not 10,000. The missing 2,278 is the discount. Push the same payments out to thirty years and the factor only reaches 15.37: at 5%, no stream of $1 a year is ever worth more than 1/i = $20, however long it runs. That ceiling is the perpetuity, and it is the fastest sanity check on any present-worth answer.

Two conventions decide whether your number matches someone else's. First, this is the ordinary annuity: the first payment lands one full period from today. Rent, leases and insurance are paid at the start of the period instead — an annuity due — and the whole answer is simply multiplied by (1 + i). Second, i and n must describe the same period. A 6% annual rate paid monthly is i = 0.005 with n = 12 per year, and entering 0.06 with n = 120 is the single most common error in the subject. Solved the other way round, for A, this relation becomes the capital recovery factor (A/P) — which is the loan payment formula, which is why a mortgage and a project's annual worth come out of one piece of algebra. There is no closed form for i: finding the rate that makes a stream worth a given price is the internal rate of return, and it is solved by interpolating tabulated factors or by letting a spreadsheet iterate.