The annuity formula can solve for the equal periodic payment needed to repay a known loan or reach a future savings target. Use the present-value payment formula for a balance owed today, and the future-value deposit formula for an amount wanted later.
Start with the date of the known amount. A loan balance exists today, before the scheduled payments, so solve for a payment from present value. A savings goal exists at a future date, after the deposits accumulate, so solve for a deposit from future value.
Both formulas assume equal payments at equal intervals and a constant rate per interval. The standard versions below assume each payment occurs at the end of the period, which is an ordinary annuity. If the first payment occurs immediately, adjust for beginning-of-period timing after solving the ordinary amount.
PMT = PV × r ÷ [1 − (1 + r)^(−n)]
— Equal payment made at the end of each period
— Loan balance or fund amount one period before the first payment
— Interest rate per payment period, written as a decimal
— Total number of equal payments
A level-payment loan charges interest on the unpaid balance while each payment covers that interest and reduces principal. The formula finds the single payment amount that brings the balance to zero after exactly n payments, assuming the rate and timing remain unchanged.
Before substituting values, match the rate to the payment interval. Monthly payments need a monthly rate and a number of months. For a nominal annual rate compounded monthly, divide the annual decimal rate by 12. Multiply the loan term in years by 12 to find the payment count.
A student finances $28,750 of graduate-school living costs with a four-year loan at a nominal annual rate of 7.2%, compounded monthly. Payments are due at each month-end. The monthly rate is 0.6%, or 0.006, and four years gives 48 payments.
| Step | Calculation | Result |
|---|---|---|
| Periodic rate | 0.072 ÷ 12 | 0.006 per month |
| Number of payments | 4 × 12 | 48 payments |
| Payment denominator | 1 − (1.006)^−48 | 0.2495929 |
| Monthly payment | $28,750 × 0.006 ÷ 0.2495929 | $691.13 |
| Total of 48 payments | $691.125385… × 48 | $33,174.02 |
| Total interest | $33,174.02 − $28,750.00 | $4,424.02 |
The required month-end payment is approximately $691.13. Early payments contain more interest because the unpaid balance is larger. Later payments contain more principal. The total interest of $4,424.02 is not found by multiplying $28,750 by 7.2% by four; the loan balance declines every month.
A useful check is direction. At a positive rate, the payment must exceed PV ÷ n, which would be $598.96 before interest. A calculated payment below that amount could not repay both principal and interest within 48 months.
The formula does not include origination fees, late charges, insurance, or a final balloon payment. If those cash flows are part of the problem, place them separately on the timeline rather than hiding them inside the level payment.
PMT = FV × r ÷ [(1 + r)^n − 1]
— Equal deposit made at the end of each period
— Target account balance at the date of the final deposit
— Return per deposit period as a decimal
— Total number of equal deposits
The future-value payment formula works backward from a target balance. It spreads the required funding across equal deposits while recognizing that early deposits earn returns for longer. The final ordinary-annuity deposit arrives on the target date and therefore earns no return before measurement.
Suppose a family wants $52,500 in five years for tuition and expects an account to earn a nominal 4.8% annual rate compounded monthly. Deposits occur at each month-end. The periodic rate is 0.4%, or 0.004, and there are 60 deposits.
| Step | Calculation | Result |
|---|---|---|
| Periodic rate | 0.048 ÷ 12 | 0.004 per month |
| Number of deposits | 5 × 12 | 60 deposits |
| Future-value annuity factor | [(1.004)^60 − 1] ÷ 0.004 | 67.6601797 |
| Required monthly deposit | $52,500 ÷ 67.6601797 | $775.94 |
| Total deposits | $775.936456… × 60 | $46,556.19 |
| Account growth | $52,500.00 − $46,556.19 | $5,943.81 |
The family needs to deposit approximately $775.94 per month. The deposits total about $46,556.19, while the remaining $5,943.81 comes from the assumed investment growth. That growth is a model result, not a guarantee; a variable-return investment may finish above or below the target.
The answer should be lower than $52,500 ÷ 60, or $875, when the assumed return is positive because earlier deposits earn growth. If the calculated payment exceeds $875 under these assumptions, recheck the sign of the exponent, rate conversion, or formula choice.
A payment stream beginning today is an annuity due. Every payment occurs one period earlier than in an ordinary annuity, so each payment receives one additional period of interest effect. To reach the same loan payoff or savings target, divide the ordinary payment by 1 + r.
Beginning-of-period PMT = End-of-period PMT ÷ (1 + r)
— Equal payment when the first payment occurs immediately
— Payment produced by the ordinary-annuity formula
— Rate per payment period
For the savings example, beginning-of-month deposits would be approximately $772.85 rather than $775.94. The lower amount works because each deposit earns one extra month of return. This timing adjustment is about solving for the payment; it does not require revaluing the full payment stream in this article.
Using an annual rate with monthly payments. Convert both the rate and period count. Using 0.048 as a monthly rate would overstate growth dramatically.
Entering a percentage as a whole number. A 7.2% annual rate is 0.072, not 7.2.
Choosing the wrong known amount. A balance today points to the PV payment formula. A goal at the end points to the FV deposit formula.
Ignoring payment timing. “First payment today” means beginning-of-period timing. “First payment one month from now” means end-of-period timing.
Rounding the periodic rate or payment too early. Keep several decimal places through the calculation. Real payment schedules may round each payment to cents and adjust the final one.
Treating a forecast return as certain. The savings formula states the deposit needed under an assumed constant return. It does not promise that a market investment will earn that return.
Write four lines before calculating: known balance and its date, payment interval, rate per interval, and total payment count. Then draw the first and last payments. This makes the choice between the two payment formulas visible and catches most timing errors.
In Excel or Google Sheets, PMT(rate, nper, pv, fv, type) handles both cases. For a loan, enter the current balance as pv and normally leave fv at zero. For a savings target, enter zero for pv and the target as fv. The returned payment often has the opposite sign from the known balance because spreadsheets distinguish cash received from cash paid. Use type = 0 for end-of-period payments and type = 1 for beginning-of-period payments. Always label the interval; $691.13 is incomplete unless the reader knows it is monthly.
Use the PV-based formula when today’s balance is known and the FV-based formula when a future target is known. Match rate and payment intervals, then adjust only if payments begin immediately.
For equal end-of-period payments, PMT equals PV times the periodic rate divided by one minus one plus the rate raised to negative n.
Use the future-value payment formula with the target balance, monthly rate, and number of monthly deposits. The result assumes deposits occur at each month-end.
Use the PV formula when the known balance is before the payments, such as a loan today. Use the FV formula when the known target is after the deposits.
For a nominal annual rate compounded monthly, divide the decimal annual rate by 12 and multiply years by 12. Effective annual rates require an effective monthly conversion instead.
That is beginning-of-period timing. Divide the ordinary end-of-period payment by one plus the periodic rate to find the equal payment for the same balance or target.
Financial functions use signs to show cash-flow direction. A positive amount received today commonly produces negative payments because repayment cash flows move in the opposite direction.
By the FinanceBrain Team · Last verified July 12, 2026 · How we produce and verify articles