Annuity Calculator

Computes annuity payments, present value, or future value given the other variables: payment amount, interest rate, number of periods, and timing (ordinary annuity vs annuity-due). Standard formulas apply: PV = PMT × [(1 − (1+r)^−n) / r] for ordinary annuities.

Distinguishes ordinary annuities (payment at period end) from annuities-due (payment at period start). The latter has slightly higher present value because each payment compounds for one extra period.

The annuity formulas derive from the geometric series. The present value of n equal payments PMT at periodic rate r is the sum of PMT/(1+r)^k for k from 1 to n, which collapses to the closed-form expression. Future value follows from PV × (1+r)^n. Annuities-due multiply both by (1+r) because each payment shifts forward one period. The arithmetic is mechanical; the substantive question is which rate to use.

The discount rate captures opportunity cost and risk. For a guaranteed payment stream (Treasury-backed annuity), the risk-free rate is appropriate. For a private pension or insurance company annuity, the rate should reflect default risk premium. For an investment evaluation, the rate is the return available on alternative investments of similar risk. Wrong rate selection dominates calculation accuracy: a 1% change in discount rate over 30 years can change present value by 25%.

A worked example: a 30-year mortgage of $400,000 at 6% annual rate, 360 monthly payments. Using the annuity payment formula: PMT = PV × [r(1+r)^n / ((1+r)^n − 1)] where r is the monthly rate (0.005) and n is 360. PMT = $2,398. Total payments over 30 years: $863,353. Total interest: $463,353. The mortgage is an annuity from the lender's perspective, with the borrower making the level payments. Front-loaded interest (the standard amortization) means early payments are mostly interest while late payments are mostly principal.

Annuities-due versus ordinary annuities matter most for retirement income planning. A retiree receiving payments at the start of each month (annuity-due, common for pension distributions) gets each payment compounding for an extra month versus an ordinary annuity. Over a 25-year retirement at 4% real return, the annuity-due is roughly 4% more valuable in present-value terms.

Limitations: the formulas assume constant payments, constant rate, and a fixed period. Real annuities frequently have variable rates (TIPS-linked annuities), graduated payments (Social Security cost-of-living adjustments), and contingent durations (lifetime annuities tied to mortality). Lifetime annuity valuation requires actuarial tables and a separate framework. The calculator handles fixed-term, fixed-payment cases cleanly and approximates more complex cases with explicit assumptions.