The annuity formula is one of the most widely used tools in finance. Whether you are calculating mortgage payments, projecting retirement savings, or valuing a stream of bond coupon payments, annuity and perpetuity formulas provide elegant shortcuts for determining the present or future value of regular cash flows — without discounting each payment individually. This guide covers everything you need to know about the annuity formula, perpetuities, growing cash flows, and how to apply them to real financial decisions.
What is an Annuity?
An annuity is a series of equal cash flows occurring at regular intervals for a finite number of periods. Annuities are everywhere in finance — from mortgage payments and car loans to bond coupon streams and retirement withdrawals.
An annuity consists of a fixed payment C made at regular intervals (monthly, quarterly, annually) for a specified number of periods n. By convention, the first payment arrives one period from now (called payment in arrears). This timing assumption is built into every standard annuity formula — getting it wrong shifts all cash flows by one period.
A perpetuity is simply an annuity that continues forever — the number of payments is infinite. While true perpetuities are rare, the perpetuity formula serves as the foundation from which annuity formulas are derived and is used to value instruments like preferred stock dividends and endowment payouts.
The Annuity Formula
The annuity formula allows you to calculate the present value of a stream of equal payments, the future value of regular contributions, or the payment amount required to repay a loan. All three versions are rearrangements of the same core relationship.
Where:
- C (or PMT) — the fixed payment per period (first payment is one period from now)
- r — the discount rate per payment period (not the annual rate unless payments are annual)
- n — the total number of payment periods
- PV — the present value (lump sum today)
- FV — the future value (accumulated amount at period n)
The PMT formula is simply the PV formula solved for C. Many searchers looking for the “annuity formula” actually want this loan payment version — it tells you exactly how much you need to pay each month on a mortgage, car loan, or student loan. Try our Annuity Calculator to compute payments instantly.
The intuition behind the PV formula comes from a clever insight: an n-period annuity is equivalent to a perpetuity whose first payment arrives in period 1 minus a perpetuity whose first payment arrives in period n + 1. Since the second perpetuity’s value at time n is C/r, its present value today is (C/r) / (1 + r)n. Subtracting gives the annuity formula: PV = C/r − (C/r) / (1 + r)n = C × [1 − (1 + r)−n] / r.
Future Value of an Annuity
While the present value formula answers “what is this stream of payments worth today?”, the future value formula answers “how much will I have after making regular contributions?” This is the essential question for retirement planning, college savings, and any goal-based investing strategy.
Ellen is 35 and plans to save $10,000 at the end of each year until she retires at 65. Her investments earn 10% per year. How much will she have at retirement?
Using the FV formula with C = $10,000, r = 0.10, n = 30:
FV = $10,000 × [(1.10)30 − 1] / 0.10 = $10,000 × 164.49 = $1,644,940
Ellen invests a total of $300,000 over 30 years, but compounding adds $1,344,940 in investment gains — more than four times her total contributions. This dramatic effect illustrates the power of starting early and contributing consistently.
Annuity Due vs Ordinary Annuity
The timing of the first payment creates two distinct types of annuities. Confusing them is one of the most common mistakes in financial calculations.
Ordinary Annuity
- Payments at the end of each period
- First payment occurs one period from now
- Examples: mortgage payments, bond coupons, loan repayments
- The standard assumption in all annuity formulas
Annuity Due
- Payments at the beginning of each period
- First payment occurs immediately (today)
- Examples: rent, lease payments, insurance premiums
- Each payment earns one extra period of interest
Converting between the two is straightforward — since every payment in an annuity due arrives one period earlier, each earns one additional period of interest:
What is a Perpetuity?
A perpetuity is an annuity with no end date — it pays the same amount forever. While no financial instrument truly lasts forever, the perpetuity formula is remarkably useful as both a valuation shortcut and the mathematical foundation of annuity pricing.
A perpetuity pays a fixed amount C at the end of every period, indefinitely. Its present value is simply the payment divided by the discount rate: PV = C / r. This elegantly simple result arises because, at rate r, a principal of C/r generates exactly C in interest each period — forever.
Historical examples bring perpetuities to life. British consol bonds, issued by the UK government, promised fixed coupon payments indefinitely — some dating to the eighteenth century. The oldest known perpetuities still making payments were issued in 1624 by a Dutch water board responsible for maintaining local dikes. Preferred stock dividends and university endowment payouts are modern examples that approximate perpetuities in practice.
Growing Annuity and Growing Perpetuity
When cash flows grow at a constant rate each period — rising salary contributions, dividends that increase annually, or pension payments indexed to inflation — the standard formulas need adjustment.
For growing perpetuities, the formula requires g < r. When the growth rate equals or exceeds the discount rate, the present value is infinite — no finite price can purchase an endlessly growing stream that outpaces discounting. For finite growing annuities, however, g > r can still be valued because the payments eventually stop. The special case where g = r simplifies to: PV = C × n / (1 + r).
The growing perpetuity formula is the foundation of the Gordon Growth Model used in stock valuation, where C represents the next expected dividend, r is the required return, and g is the expected dividend growth rate. For rate conversion details when growth and discount rates use different compounding, see our interest rates guide.
Annuity Example
A homebuyer takes a $300,000 mortgage at 6% annual interest (0.5% monthly) with 360 monthly payments. What is the monthly payment?
Using the PMT formula with PV = $300,000, r = 0.005, n = 360:
PMT = $300,000 × 0.005 / [1 − (1.005)−360] = $1,500 / 0.8337 = $1,798.65
Over 30 years, the borrower pays a total of approximately $647,515 — meaning roughly $347,515 is interest on the original $300,000 loan. The amortization schedule below shows how early payments are dominated by interest:
| Month | Payment | Interest | Principal | Remaining Balance |
|---|---|---|---|---|
| 1 | $1,799 | $1,500 | $299 | $299,701 |
| 2 | $1,799 | $1,499 | $300 | $299,401 |
| 3 | $1,799 | $1,497 | $302 | $299,100 |
| 4 | $1,799 | $1,495 | $303 | $298,796 |
| 5 | $1,799 | $1,494 | $305 | $298,492 |
In the first month, 83% of the payment covers interest. This ratio gradually shifts as the balance decreases.
Suppose Ellen from the earlier example expects her salary to rise, allowing her to increase savings by 5% per year. Her first deposit is still $10,000, investments still earn 10%, and she saves for 30 years. How much more does she accumulate?
Using the growing annuity PV formula, then compounding forward:
PV = $10,000 × [1 − (1.05/1.10)30] / (0.10 − 0.05) = $150,463
FV = $150,463 × (1.10)30 = $2,625,000
By increasing contributions 5% annually, Ellen accumulates $2.625 million — nearly $1 million more than the $1.645 million from fixed contributions.
How to Calculate Annuities
Follow these steps to solve any annuity problem:
- Identify the annuity type: Is it ordinary (end-of-period) or annuity due (beginning)? Is it growing or constant? Finite or perpetual?
- Convert the rate to the effective periodic rate: If payments are monthly but you have an annual rate, convert to the monthly rate. For an APR with monthly compounding, divide by 12. For an EAR, use (1 + EAR)1/12 − 1. See our interest rates guide for detailed conversion methods.
- Match n to the number of payment periods: A 30-year monthly mortgage has n = 360, not n = 30.
- Apply the correct formula: PV for valuation, FV for accumulation, PMT for loan payments.
- Adjust for annuity due if needed: Multiply PV or FV by (1 + r). For PMT, divide the ordinary-annuity payment by (1 + r) instead.
Common Mistakes
These errors appear frequently in finance courses and professional practice. Avoiding them requires careful attention to timing and rate conventions.
1. Using the annual rate with periodic payments. A 6% annual rate does not mean r = 0.06 for monthly payments. You must convert to the effective rate per payment period. For a 6% APR with monthly compounding, the monthly rate is 0.5%. For a 6% EAR, the equivalent monthly rate is (1.06)1/12 − 1 = 0.487%. These are different numbers — using the wrong one compounds errors across hundreds of payments. See Interest Rates: EAR, APR & the Yield Curve for conversion details.
2. Confusing annuity due and ordinary annuity. Standard mortgage payments occur at the end of each month (ordinary annuity). Rent is paid at the beginning (annuity due). Applying the wrong convention shifts every cash flow by one period, causing the PV to be off by a factor of (1 + r).
3. Applying the perpetuity formula to a finite stream. If a bond pays coupons for 20 years, using PV = C/r instead of the annuity formula overstates the present value by ignoring the fact that payments stop. The error grows larger for shorter-duration streams and higher discount rates.
4. Using the growing perpetuity formula when g ≥ r. The formula PV = C/(r − g) produces a negative or infinite result when the growth rate meets or exceeds the discount rate. This is mathematically invalid for perpetuities. Always verify that g < r before applying it.
5. Using the most recent payment instead of the next payment in growing formulas. In both the growing perpetuity and growing annuity formulas, C represents the next payment — the one arriving one period from now. If a company just paid a $2 dividend growing at 5%, C = $2 × 1.05 = $2.10, not $2.00. Using the wrong value understates the present value by exactly (1 + g).
Limitations of Annuities and Perpetuities
Annuity and perpetuity formulas assume perfectly regular, constant-sized payments at a fixed discount rate. Real-world cash flows rarely match these assumptions exactly.
Constant payments and timing. The standard formulas assume every payment is identical and arrives at a perfectly regular interval. In reality, borrowers may miss payments, variable-rate loans change the payment amount, and some instruments have irregular schedules.
Taxes, fees, and transaction costs. Annuity formulas compute gross present values. They do not account for income taxes on interest, origination fees on loans, or management fees on retirement accounts — all of which reduce the actual value received.
The perpetuity assumption is an approximation. No entity truly pays forever. Companies go bankrupt, governments default, and endowments can be depleted. The perpetuity formula works well when the payment horizon is long enough that distant cash flows contribute negligibly to the present value.
Bond coupon streams are annuities, but bonds are not. A bond’s coupon payments form an annuity, but full bond pricing also requires discounting the lump-sum principal repayment at maturity — the bond is an annuity plus a single future cash flow.
Frequently Asked Questions
Disclaimer
This article is for educational and informational purposes only and does not constitute financial or investment advice. All formulas, examples, and calculations are based on standard academic finance as presented in Berk, DeMarzo & Harford’s “Fundamentals of Corporate Finance.” Actual loan terms, investment returns, and financial products may differ from the simplified examples shown. Always consult a qualified financial advisor before making financial decisions.
