Annuity Calculator

An annuity, in the mathematical sense, is simply a stream of equal payments made at regular intervals. Four quantities describe it completely: the payment amount, the number of periods, the periodic interest rate, and the present value of the whole stream. Fix any three and the fourth is determined. This tool solves for whichever one you leave out. The core relationship is the present-value formula for an ordinary annuity: PV equals the payment multiplied by one minus (1 plus i) to the power of negative n, all divided by i. Here i is the rate per period and n is the number of periods. The bracketed term is the annuity factor — the present value of one dollar per period — and it is the whole calculation in one number. When the rate is exactly zero the formula divides by zero, and the correct answer is trivially payment times periods, which is handled as an explicit branch. Solving for payment inverts the same expression algebraically. Solving for the number of periods requires logarithms: n equals the natural log of PMT divided by (PMT minus PV times i), over the natural log of (1 plus i). Solving for the rate has no closed form at all — the equation cannot be rearranged for i — so a bisection routine narrows the interval until the annuity factor matches. That is not a shortcut; it is the standard approach, and the same one financial calculators use internally. The rate you enter is a nominal annual rate. It is divided by the periods per year to get the periodic rate — 6% with monthly periods becomes 0.5% per month — following the same convention used for loan quotes. This is not the effective annual rate; twelve months at 0.5% compounds to 6.17%, not 6%. The timing assumption is the thing that quietly changes the answer. This calculator models an ORDINARY annuity, where each payment lands at the END of its period. An annuity DUE pays at the beginning instead, and every payment therefore sits one full period closer to the present. The conversion is exact and refreshingly simple: multiply the ordinary present value by (1 plus i). At 6% annual with monthly payments that is a 0.5% difference; at 6% annual with yearly payments it is a full 6%. Rent and insurance premiums are typically annuities due, while bond coupons and most loan payments are ordinary. If a quoted figure differs from yours by a fraction of a percent, timing convention is almost always the reason. Notice what solving for n reveals. If your payment is less than or equal to PV times i, no finite n exists — the payment never covers the interest, and the tool refuses rather than returning nonsense. That threshold is the dividing line between a stream that terminates and a perpetuity that does not. Two more caveats: level payments and a constant rate are assumed throughout, and no inflation adjustment is applied, so a payment stream of fixed nominal dollars loses purchasing power over a long term. For an income stream that rises with prices across a retirement, use the retirement calculator, and for pure lump-sum growth without a payment stream, the compound interest calculator.