The 3 annuity formulas (PMT, PV, FV)

This calculator uses the classic “time value of money” annuity formulas. The logic is simple: every payment earns interest for some number of periods, and you add them up. Because the payments are equal and the interest rate per period is assumed constant, the math collapses into a clean closed-form equation.

Most people think in annual interest rates (like 6% per year), but annuity payments typically happen monthly or quarterly. So we convert the annual rate (APR) into a periodic rate:

• r = (APR / 100) / paymentsPerYear
• n = years × paymentsPerYear

Example: 6% APR, monthly payments → r = 0.06 / 12 = 0.005 (0.5% per month), and 20 years means n = 20 × 12 = 240 payments.

The only difference is timing. If payments happen at the end of each period, it’s an ordinary annuity. If payments happen at the beginning, it’s an annuity due. “Due” payments earn one extra period of interest, so you multiply by (1 + r):

• Ordinary: use the base formulas below.
• Due: multiply PV or FV by (1 + r), or divide PMT by (1 + r) when solving for PMT.

Future value is “how much money will I have at the end?” For an ordinary annuity:

• FV = PMT × [((1 + r)^n − 1) / r]

Intuition: the first payment sits and earns interest for many periods; the last payment earns interest for almost none. The bracket term is basically a shortcut for “add up all those compounded payments.” For an annuity due, each payment happens one period earlier, so:

• FV_due = FV_ordinary × (1 + r)

Present value is “what is this stream of payments worth today as a lump sum?” For an ordinary annuity:

• PV = PMT × [1 − (1 + r)^(-n)] / r

Intuition: you’re discounting each future payment back to today. Because the payments are equal and regular, the discounts create a geometric series that collapses into the bracket term. For an annuity due:

• PV_due = PV_ordinary × (1 + r)

Sometimes your goal is the payment itself. That usually happens in two scenarios: (1) “How much do I need to deposit every month to hit a future goal?” or (2) “If I have a lump sum today, how much can I withdraw each month?”.

• PMT from FV (ordinary): PMT = FV × r / ((1 + r)^n − 1)
• PMT from PV (ordinary): PMT = PV × r / (1 − (1 + r)^(-n))
• Annuity due adjustment: PMT_due = PMT_ordinary / (1 + r)

If r = 0, the formulas above would divide by zero. But the logic becomes simple: without interest, the future value is just payments added up. So:

• FV = PMT × n
• PV = PMT × n
• PMT = FV / n or PMT = PV / n

The calculator automatically switches to these versions when the periodic rate is zero (or extremely close).

Real-world examples you can copy

Goal: reach $250,000 in 20 years. Rate: 6% APR. Payments: monthly. Solve for PMT (ordinary annuity).

• r = 0.06 / 12 = 0.005, n = 20×12 = 240
• PMT = FV × r / ((1+r)^n − 1)

Plugging in values gives a monthly payment around the mid-hundreds (your exact result depends on rounding). This is the classic “how much should I invest per month?” calculation.

Pension pays $2,000 per month for 25 years. Discount rate: 5% APR. Payments monthly. Solve for PV (ordinary annuity).

• r = 0.05 / 12, n = 25×12
• PV = PMT × [1 − (1+r)^(-n)] / r

The result is the lump-sum “equivalent value” of that pension stream today. This is how people compare taking a pension vs taking a lump sum.

You invest $500 per month for 10 years at 7% APR (monthly). Solve for FV (ordinary).

• FV = 500 × [((1+r)^n − 1)/r]

The interesting part is the split between contributions vs growth. Even if you deposit $60,000 total, compound growth often adds a meaningful extra chunk by the end.

Same as Example 3, but payments happen at the beginning of the month (annuity due). Because each deposit happens one period earlier, FV increases by a factor of (1+r). Over many payments, this can noticeably boost the end value.

Viral challenge idea: run the calculator with “due” vs “ordinary” and screenshot the difference — it’s a quick “timing matters” lesson people actually share.

What the calculator is doing (plain English)

When you press Calculate, the page does four things:

• Validates inputs: makes sure required fields are filled and numbers are reasonable.
• Converts units: turns APR into a periodic rate and years into total number of payments.
• Applies the correct annuity formula: based on whether you’re solving for PMT, PV, or FV — and whether it’s ordinary or due.
• Builds a step-by-step explanation: so you can see the exact formula path and copy it if needed.

Think of annuity math as a “stack of repeated mini-investments.” Each payment is like its own tiny deposit. The first payment has the longest time to grow. The last payment has the shortest time. The future value formula is simply the sum of all those deposits after interest.

• Mixing annual vs monthly rates: 6% per year is not 6% per month. The tool converts for you.
• Ignoring payment timing: beginning-of-month vs end-of-month changes the result.
• Forgetting the zero-rate case: the formula changes when r = 0.
• Using years as n: n is the number of payments, not years.

If you want to go deeper: annuity formulas come from a geometric series. The multiplier ((1+r)^n − 1)/r is the closed-form sum of compounded payments, and [1 − (1+r)^(-n)]/r is the closed-form sum of discounted payments.

Frequently Asked Questions

• What’s the difference between PV and FV in an annuity?

  PV (present value) is the lump sum today that is “equivalent” to the payment stream. FV (future value) is the amount the payment stream grows to at the end of the term. PV discounts payments backward; FV compounds them forward.

• Ordinary annuity vs annuity due — which one should I pick?

  Choose ordinary if payments happen at the end of the period (common for many investments and loans). Choose due if payments happen at the beginning (common for rent, some pensions, and some retirement withdrawals). Due typically produces a slightly higher PV/FV for the same PMT.

• Is APR the same as APY?

  Not exactly. APR is a nominal annual rate; APY reflects compounding. This calculator uses APR and converts it to a periodic rate using the payment frequency. If you have an APY and want precision, convert it to an effective periodic rate first.

• What if my interest rate changes over time?

  These formulas assume a constant rate. If rates change, you need a schedule or a year-by-year model. For quick planning, you can run multiple scenarios (e.g., 5%, 6%, 7%) and compare outcomes.

• Does this include taxes and fees?

  No. Taxes, annuity fees, and investment expenses can significantly change outcomes. Use this as a clean math baseline, then adjust for real-world costs.

• Can I use this for retirement withdrawals?

  Yes — PV and PMT relationships are the same. Just interpret PMT as a withdrawal rather than a deposit. If withdrawals happen at the start of each month, select “annuity due.”

• What does “payments per year” do?

  It sets how many payments happen (and how often interest compounds in this model). Monthly means 12 payments per year. The calculator then uses r = APR/12 and n = years×12.

• Why is my result slightly different from another site?

  Differences usually come from (1) rounding the periodic rate, (2) using APR vs APY, (3) assumptions about compounding frequency, or (4) whether it’s ordinary or due. Make sure all inputs match exactly.

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