Annuity Calculator

Background.

An annuity, in the general finance-math sense this calculator uses, is simply a series of equal payments made at regular intervals — a $500 monthly savings deposit, a $2,000 annual bond coupon, a fixed pension payout, a level loan payment. That is a different, broader meaning than the insurance-industry "annuity product" sold by insurers as a retirement-income contract, though the underlying arithmetic is exactly the mathematics that prices those insurance products too. This calculator computes both the present value and the future value of a level payment stream from one set of inputs, for either of the two standard payment-timing conventions: ordinary annuities, where payments land at the end of each period, and annuity-due, where payments land at the beginning.

The distinction between the two timing conventions is not academic hair-splitting — it changes the answer by a predictable, calculable amount. Because an annuity-due payment arrives one period earlier than the equivalent ordinary-annuity payment, it has one extra period to either compound (for future value) or avoid being discounted (for present value), which makes every annuity-due figure exactly (1 plus the per-period rate) times larger than its ordinary-annuity counterpart. Ordinary annuities are the default convention for bonds, mortgages, and most retirement-plan contribution formulas; annuity-due is the standard convention for leases, rent, and insurance premiums, all of which are paid in advance of the period they cover.

This calculator deliberately reports present value and future value together rather than asking you to pick a "solve for" mode, because both numbers answer genuinely different, commonly asked questions from the same payment stream: present value tells you what a pension, bond, or structured settlement is worth if you had to buy it today in one lump sum; future value tells you what a savings or contribution habit will accumulate to by a target date. A 0% interest rate is handled as an explicit, fully valid case rather than a division-by-zero error — with no interest at all, ordinary and annuity-due timing make no difference whatsoever, because timing advantages only exist when money can actually earn something during the head start.

The same annuity math that prices a savings plan also prices a bond's coupon stream, a fixed-rate mortgage's payment schedule, a structured legal settlement, and a defined-benefit pension's promised payout, discounted or compounded at whatever rate is appropriate to the instrument in question. Enter the periodic payment amount, the nominal annual rate, the number of years, how often payments occur, and the timing convention, and the calculator returns both the present and future value of the stream, the total nominal payments made with no interest applied, the total interest the stream earns by the end of the horizon, and the effective annual rate implied by your chosen compounding frequency.

What is annuity calculator?

An annuity, in financial mathematics, is a finite series of equal payments made at fixed, regular intervals. The present value of an annuity is what that entire future payment stream is worth today, once every individual payment is discounted back to the present at a chosen rate; the future value is what the stream will have accumulated to by its final payment date, once every earlier payment has had time to compound. Both quantities use the same underlying geometric-series mathematics, differing only in whether payments are discounted backward or compounded forward. An ordinary annuity assumes payments occur at the end of each period — the convention used by nearly every standard bond, mortgage, and textbook annuity formula. An annuity-due assumes payments occur at the beginning of each period — the convention used by leases, rent, and insurance premiums, which are paid in advance of the coverage they provide. Because each annuity-due payment is effectively one period closer to today than its ordinary-annuity equivalent, both its present value and its future value are always exactly (1 plus the per-period interest rate) times larger than the ordinary-annuity figure for identical payment amounts, rates, and horizons.

How to use this calculator.

1. 01Enter the periodic payment amount — the level sum paid or received every period.
2. 02Enter the nominal annual interest rate. The calculator divides by the payment frequency to get the per-period rate used internally.
3. 03Enter the number of years the payment stream runs for.
4. 04Choose how often payments occur — this doubles as the compounding frequency used throughout the calculation.
5. 05Choose the annuity type. Ordinary annuity assumes payments at the end of each period (bonds, mortgages, most textbook formulas). Annuity-due assumes payments at the beginning of each period (leases, rent, insurance premiums).
6. 06Read both the present value and the future value of the stream, along with total payments made, total interest earned by the end of the horizon, and the effective annual rate implied by your chosen compounding frequency.

The formula.

The calculator computes both quantities from the same inputs. Let i be the per-period rate (the nominal annual rate divided by the number of payments per year) and n be the total number of periods (years multiplied by payments per year). The future value of an ordinary annuity is FV = PMT × [(1+i)^n − 1] / i — each payment compounds forward for however many periods remain until the end of the stream. The present value of an ordinary annuity is PV = PMT × [1 − (1+i)^−n] / i — each payment is discounted back to today by however many periods it is away. For an annuity-due, where every payment lands one period earlier than the ordinary-annuity convention, both formulas are simply multiplied by an additional factor of (1+i), since every payment gets one extra period of either compounding or discounting benefit. When the per-period rate i is exactly zero, both formulas would otherwise divide by zero; the calculator instead uses the straight-line limit of each formula, PMT × n, which is mathematically what both expressions converge to as i approaches zero. Critically, the due-multiplier (1+i) also collapses to exactly 1 when i = 0, so ordinary and annuity-due annuities produce identical present and future values whenever the interest rate is zero — timing has no effect on value when money earns nothing in the meantime, which is exactly the correct real-world behavior. Total payments is simply PMT × n with no interest applied, and total interest earned is future value minus total payments — the portion of the final balance attributable purely to compounding. The effective annual rate is (1+i) raised to the payments-per-year power, minus one, reported as a percentage, and is unaffected by the ordinary-versus-due choice since it describes only the compounding of the rate itself.

A worked example.

Frequently asked questions.

References& sources.

1. [1]Brealey, R.A., Myers, S.C., and Allen, F. — Principles of Corporate Finance, 13th edition. Chapters 2-3 derive the present-value and future-value annuity formulas and the ordinary-versus-due distinction.
2. [2]Bodie, Z., Kane, A., and Marcus, A.J. — Investments, 12th edition. §5 covers annuity valuation as applied to bond pricing and retirement-plan projections.
3. [3]CFA Institute — CFA Program Curriculum, Quantitative Methods topic area: annuity present value and future value formulas.
4. [4]U.S. Securities and Exchange Commission, Investor.gov — Compound Interest Calculator and time-value-of-money investor education materials.