Finance & Economics · FIRE & Retirement
Annuity Calculator
Calculates the present value, future value, and periodic payment of an ordinary annuity or annuity-due given a fixed interest rate and number of periods.
Calculator
Formula
PMT is the periodic payment amount. r is the interest rate per period (annual rate divided by periods per year). n is the total number of payment periods. t = 0 for an ordinary annuity (payments at end of period) and t = 1 for an annuity-due (payments at beginning of period). PV is the present value — the lump sum today equivalent to all future payments. FV is the future value — the accumulated value of all payments at the end of the annuity term.
Source: Brealey, Myers & Allen, Principles of Corporate Finance, 13th Ed., McGraw-Hill; CFA Institute, Fixed Income Analysis.
How it works
An annuity is a contract that delivers a stream of equal cash flows — called payments or coupons — at regular intervals over a defined period. Annuities are the backbone of retirement income planning, fixed-income securities, mortgage amortization, and structured settlements. Two primary variants exist: the ordinary annuity, in which payments occur at the end of each period (the default for most loans and bonds), and the annuity-due, in which payments occur at the beginning of each period (common in lease agreements and insurance premiums). The annuity-due is always worth slightly more than an equivalent ordinary annuity because each payment has one additional period of compounding.
The present value (PV) of an annuity answers the question: what single lump sum today is economically equivalent to receiving all future payments, discounted at a given interest rate? The formula is PV = PMT × [1 − (1 + r)^(−n)] / r, multiplied by (1 + r) for an annuity-due. The future value (FV) answers the inverse question: if you invest each payment at the given rate, how much will accumulate by the end of the term? FV = PMT × [(1 + r)^n − 1] / r, again multiplied by (1 + r) for annuity-due. Here, r is the periodic interest rate (annual rate ÷ periods per year) and n is the total number of periods (years × periods per year). Total interest earned is simply FV minus total nominal contributions, revealing the pure effect of compounding.
Practical applications span every corner of personal and institutional finance. A retiree can determine the lump-sum cost of purchasing a lifetime annuity paying $2,000 per month. A corporate treasurer can price a bond's coupon stream. A mortgage borrower can verify how much interest accumulates over a 30-year loan. Investment advisers use the future value formula to project the terminal portfolio balance of a client making regular 401(k) contributions. Because the formula assumes a constant interest rate and equal payments, it provides a clean, transparent baseline for comparing alternatives — variable products, lump-sum investments, or inflation-adjusted income streams require additional adjustments beyond this model.
Worked example
Scenario: A 40-year-old investor wants to save for retirement by contributing $500 per month into a tax-deferred account earning 7% annually, compounded monthly, for 25 years. Payments are made at the end of each month (ordinary annuity).
Step 1 — Identify inputs: PMT = $500, Annual Rate = 7%, Periods per Year = 12, Years = 25, t = 0 (ordinary annuity).
Step 2 — Compute periodic rate: r = 7% ÷ 12 = 0.5833% = 0.005833 per month.
Step 3 — Compute total periods: n = 25 × 12 = 300 months.
Step 4 — Future Value: FV = 500 × [(1.005833)^300 − 1] / 0.005833 = 500 × [5.8916 − 1] / 0.005833 = 500 × 4.8916 / 0.005833 = 500 × 838.85 ≈ $419,427.
Step 5 — Present Value: PV = 500 × [1 − (1.005833)^(−300)] / 0.005833 = 500 × [1 − 0.1697] / 0.005833 = 500 × 143.02 ≈ $71,513. This is the lump sum needed today to fund the same 25 years of $500/month withdrawals at 7%.
Step 6 — Total contributions: 500 × 300 = $150,000.
Step 7 — Total interest earned: $419,427 − $150,000 = $269,427. Compounding nearly triples the nominal contribution, illustrating the power of consistent long-term investing.
Limitations & notes
The annuity formula assumes a constant, fixed interest rate for the entire term — real-world returns fluctuate, and variable annuities or investment accounts will deviate from this projection. The model also assumes payments are perfectly equal and on schedule; any missed, early, or irregular payment breaks the formula's symmetry and requires more complex cash-flow discounting. Inflation is not accounted for in the basic model: a nominal future value of $400,000 in 25 years has substantially less purchasing power than today. Taxes on earnings — relevant in non-qualified accounts — reduce effective accumulation and are not modeled here. For perpetuities (infinite payment streams), the formula simplifies to PV = PMT / r, and the future value is undefined (infinite). Finally, this calculator does not account for fees, commissions, or surrender charges common in commercial annuity products, which can meaningfully reduce net returns.
Frequently asked questions
What is the difference between an ordinary annuity and an annuity-due?
An ordinary annuity makes payments at the end of each period (e.g., most mortgages and bonds), while an annuity-due makes payments at the beginning of each period (e.g., most leases and insurance premiums). Because annuity-due payments are received one period earlier, they compound for one extra period, making the present value and future value each exactly (1 + r) times higher than the equivalent ordinary annuity.
How do I use this calculator to figure out my retirement savings goal?
Enter your planned monthly contribution as PMT, your expected annual return as the interest rate, your savings timeline in years, and select monthly compounding. The Future Value output shows how much your contributions will accumulate to at retirement — your target nest egg. The Total Interest Earned field shows how much of that balance comes purely from compounding rather than your own deposits.
Can I use this calculator for loan payments?
Yes — for a loan, the present value is the loan principal you receive today, and PMT is the fixed periodic repayment. If you know the loan amount, rate, and term, you can rearrange: PMT = PV × r / [1 − (1 + r)^(−n)]. Our dedicated mortgage and loan calculators solve directly for payment, but the annuity present value formula is mathematically identical to standard loan amortization.
What does the present value of an annuity actually mean?
The present value represents the lump-sum amount today that is economically equivalent to receiving all future annuity payments. It answers the question: 'How much money would I need right now, invested at the given rate, to exactly fund every scheduled payment?' It is widely used to price bonds, value pension obligations, and compare lump-sum buyouts against structured payment streams.
Why does compounding frequency matter for the annuity calculation?
Higher compounding frequency (e.g., monthly vs. annually) slightly increases both present and future values because interest is applied more often, generating more sub-annual compounding. For a given annual rate, monthly compounding produces a marginally higher FV than annual compounding because each month's interest earns interest in subsequent months. The difference grows more pronounced over longer terms and higher rates.
Last updated: 2025-01-15 · Formula verified against primary sources.