Annuity Calculator | PV, FV, Payment, Term & Perpetuity
Calculate present value, future value, required payment, or term for ordinary annuities and annuities-due. Includes perpetuity mode, step-by-step working, and a year-by-year growth or amortization schedule.
What Is the Annuity Calculator | PV, FV, Payment, Term & Perpetuity?
An annuity is any series of equal payments made at regular intervals, your monthly mortgage payment, a pension you receive in retirement, or a savings plan where you deposit the same amount every month. This calculator covers all six variants of the annuity problem so you can solve whichever unknown you need without rearranging formulas yourself.
What this calculator solves:
- ▸Future Value, the total amount a series of deposits will grow to at a given rate and time horizon.
- ▸Present Value, the lump-sum equivalent today of a future stream of payments, discounted at a given rate.
- ▸Payment from FV, how much you need to save each period to reach a target balance (retirement goal, college fund, etc.).
- ▸Payment from PV (Loan Payment), the periodic payment required to fully repay a loan of a given amount over a given term.
- ▸Solve for Term, how many years and months until you reach a savings goal or pay off a debt.
- ▸Perpetuity, the present value of a payment that lasts indefinitely, with optional growth rate for a growing perpetuity.
Every mode shows a step-by-step breakdown of the calculation and a year-by-year schedule showing how the balance grows (for savings) or shrinks (for loans).
Formula
Every annuity calculation is a variation of two core formulas. All values are computed using the periodic interest rate (r = annual rate ÷ frequency) and total periods (n = years × frequency).
FV = PMT × [(1+r)ⁿ − 1] / r
Annuity-due: × (1+r)
How much your savings will grow to over time.
PV = PMT × [1 − (1+r)^−n] / r
Annuity-due: × (1+r)
What a future stream of payments is worth today.
PMT = FV × r / [(1+r)ⁿ − 1]
or: PMT = PV × r / [1 − (1+r)^−n]
Solves for the required periodic payment.
PV = PMT / r
Growing: PV = PMT / (r − g)
A payment stream that never ends. Used in stock valuation.
How to Use
- 1
Pick the right mode
Choose what you want to find: Future Value, Present Value, Payment, Term, or Perpetuity. Each mode rearranges the annuity formula to solve for a different unknown.
- 2
Enter the known values
Fill in the inputs that appear for your chosen mode, payment amount (or target), annual interest rate, and term in years. Some fields hide automatically when they are the unknown.
- 3
Set the payment frequency
Choose how often payments are made: monthly, quarterly, semi-annually, annually, weekly, or daily. The calculator converts your annual rate to a periodic rate automatically.
- 4
Choose ordinary or annuity-due
Ordinary annuities pay at the end of each period (most loans and investments). Annuities-due pay at the beginning (rent, insurance). Annuity-due values are always slightly higher.
- 5
Review the results
See the primary result, total contributions vs interest, a step-by-step formula walkthrough, and a year-by-year schedule. Click "Show all years" to expand the full schedule.
Example Calculation
Example 1 | Retirement savings goal (FV)
You save $500/month for 30 years at a 7% annual rate (monthly compounding, ordinary annuity).
More than two-thirds of the final balance is interest, a clear demonstration of compound growth over a long horizon.
Example 2 | Mortgage payment (PMT from PV)
You borrow $350,000 at 6.5% for 30 years (monthly payments, ordinary annuity).
Over a 30-year term at 6.5%, the total interest paid exceeds the original loan amount.
Example 3 | Perpetuity (stock valuation)
A preferred stock pays a $4 annual dividend. The required rate of return is 8%.
This is the Gordon Growth Model (with g = 0). If the dividend grows at 2%/year, PV = $4 / (0.08 − 0.02) = $66.67.
Understanding Annuity | PV, FV, Payment, Term & Perpetuity
What Is an Annuity?
An annuity is a financial contract that involves a series of equal payments made at regular intervals over a fixed period. The word comes from the Latin annuus (yearly), though modern annuities can be paid monthly, quarterly, or at any other frequency. They appear everywhere in personal finance, your mortgage, your retirement plan contributions, a car loan, a pension you receive in old age, even a streaming subscription.
The mathematics behind annuities answers a fundamental question about money: is it better to have money now or in the future? The annuity formula captures the time value of money, the principle that a dollar received today is worth more than a dollar received a year from now, because today's dollar can be invested and earn a return.
The Six Types of Annuity Problems
| Problem | Unknown | Real-World Example |
|---|---|---|
| Future Value | How much will I have? | Retirement savings projection |
| Present Value | What is this worth today? | Bond pricing, pension valuation |
| Payment from FV | How much do I need to save? | Monthly savings to reach $1M |
| Payment from PV | What is my loan payment? | Mortgage, car loan, student loan |
| Term (Periods) | How long will this take? | Years to pay off a debt at a given rate |
| Perpetuity | Value of an infinite stream | Preferred stock, endowment, consol bond |
Ordinary Annuity vs. Annuity-Due, The Timing Difference
The only difference between an ordinary annuity and an annuity-due is when each payment is made. That single timing shift, end of period vs. start of period, changes every formula result by a factor of (1 + r).
- ▸Ordinary annuity (end of period): Most loans, investment contributions, and retirement income streams. Payments earn one period less of interest.
- ▸Annuity-due (start of period): Rent, insurance premiums, leases, and some pension systems. Each payment sits in the account one period longer, so the FV is always slightly higher and the PV is also slightly higher.
Payment Frequency and Compounding
How often payments are made affects the result, more frequent payments mean more compounding events. Here is how a $500/month equivalent deposit at 6% annual rate over 20 years compares across frequencies (same total annual deposit of $6,000):
| Frequency | Payment | FV (20 years, 6%) |
|---|---|---|
| Annually | $6,000/yr | $220,714 |
| Semi-annually | $3,000/6mo | $232,175 |
| Quarterly | $1,500/qtr | $232,175 |
| Monthly | $500/mo | $232,175 |
Figures are illustrative. For exact results use the calculator above with your specific inputs.
Perpetuities, Annuities That Pay Forever
A perpetuity is a theoretical annuity with an infinite number of payments. Despite sounding exotic, perpetuities are widely used in finance:
- ▸Preferred stocks: Pay a fixed dividend indefinitely. Valued using PV = dividend ÷ required return.
- ▸UK consol bonds: Issued for centuries with no maturity date. Valued as a perpetuity.
- ▸University endowments: A perpetuity structure where only interest is spent and the principal is preserved forever.
- ▸Gordon Growth Model: Values a stock as a growing perpetuity, PV = D₁ ÷ (r − g), where D₁ is next year's dividend and g is the dividend growth rate.
The growing perpetuity formula only works when the growth rate g is strictly less than the discount rate r. If g ≥ r, the present value would be infinite (payments grow faster than they are discounted), which has no financial meaning.
How Annuities Power Everyday Financial Decisions
- ▸Mortgages: A home loan is a PV annuity problem. The bank gives you a lump sum (PV) today in exchange for equal monthly payments over 15 or 30 years.
- ▸Car loans: Same structure as a mortgage, PV known, solving for PMT.
- ▸401(k) / pension planning: Modelled as an FV annuity, how much will regular contributions grow to by retirement age?
- ▸Structured settlements: Courts award compensation as annuities rather than lump sums to prevent rapid spending. The value of the settlement is calculated as a PV annuity.
- ▸Savings plans (SIPs): Systematic Investment Plans in mutual funds work exactly like an ordinary annuity, equal monthly amounts invested at a projected average return.
- ▸Lease payments: Commercial leases and equipment leases are annuity-due structures, where the first payment is made at signing.
Reading the Year-by-Year Schedule
The growth/amortization schedule this calculator produces shows how money moves every year. Understanding each column helps you make better decisions:
| Column | Meaning | What to watch |
|---|---|---|
| Annual Contribution | Your deposits that year | Stays constant for regular saving plans |
| Interest Earned | What compounding adds | Should grow each year as the balance rises |
| Cumulative Interest | Total interest earned so far | Compare to total contributions, shows growth power |
| Balance | Total value at year end | Should grow faster over time (compound effect) |
| Interest Paid (loan) | Cost of borrowing that year | Should decrease each year as balance falls |
| Principal Paid | Debt actually eliminated | Should increase each year as interest shrinks |
Common Mistakes When Using Annuity Formulas
- ▸Using the annual rate instead of the periodic rate, divide by payment frequency first.
- ▸Forgetting to convert years to periods, multiply years by payments per year to get n.
- ▸Mixing up ordinary annuity and annuity-due, a one-period timing difference affects every result.
- ▸Ignoring compounding frequency when comparing loan offers with the same stated annual rate.
- ▸Assuming perpetuity formulas work when the growth rate exceeds the discount rate, they do not.
Frequently Asked Questions
What is the difference between an ordinary annuity and an annuity-due?
An ordinary annuity (also called annuity-immediate) makes payments at the end of each period, this is how most mortgages, car loans, and investment contributions work. An annuity-due makes payments at the start of each period, like rent or insurance premiums. Because annuity-due payments sit in the account one full period longer, its future value is always higher by a factor of (1 + r) compared to an otherwise identical ordinary annuity.
How is an annuity different from a lump sum investment?
A lump sum is deposited once and grows through compounding alone. An annuity is a stream of equal payments, it benefits from both compounding and the accumulation of regular deposits. For long time horizons, regular annuity contributions often outperform a single equivalent lump sum because money invested early has the longest time to grow.
What is a perpetuity and who uses it?
A perpetuity is an annuity with no end date, it pays forever. Its present value is simply PMT ÷ r. Perpetuities are used to value preferred stocks (fixed dividends with no maturity), government consol bonds, and endowments. A growing perpetuity, where each payment increases by a fixed percentage, has PV = PMT ÷ (r − g), and is the basis of the Gordon Growth Model for equity valuation.
What happens when I change the payment frequency?
Higher payment frequency increases the future value slightly because each payment enters the account sooner and earns more compound interest. Monthly compounding produces a slightly higher FV than quarterly, which beats semi-annual, which beats annual, for the same stated annual rate and total payments. The difference is real but modest unless the rate is very high or the term is very long.
What does the 'Solve for Term' mode actually calculate?
It uses the inverse of the standard FV or PV formula to find the number of periods (n) needed to reach a goal or pay off a loan, given a fixed payment. The formula is n = ln(FV × r / PMT + 1) / ln(1 + r) for savings, and n = −ln(1 − PV × r / PMT) / ln(1 + r) for loans. The result is converted into years and months for readability.
Why does the year-by-year schedule sometimes stop before the full term?
For loan amortization schedules, the balance reaches zero once the loan is fully paid off, which can happen slightly before the stated term due to rounding of the periodic payment. The schedule stops at that point. For growth schedules, the table is capped at 50 years for performance reasons.
How is the interest rate converted between annual and periodic?
The periodic rate is r = annual rate ÷ number of payments per year. For a 6% annual rate with monthly payments, r = 0.06 ÷ 12 = 0.005 (0.5%) per month. This is the nominal rate approach. Note that if the stated rate is an effective annual rate (EAR), you would convert differently: r = (1 + EAR)^(1/freq) − 1. This calculator uses the nominal approach (dividing directly), which matches how most consumer loans and investment products are quoted.
What is the annuity formula used for in real life?
The annuity formula underpins nearly every area of personal finance. Mortgages and auto loans are PV annuity calculations. Retirement savings projections use the FV formula. Pension valuations, lottery structured settlements, bond pricing, and insurance premium calculations all rely on some form of the annuity equation. Understanding the formula helps you compare products, negotiate better terms, and plan with confidence.