Loan Payment Formula: M = P × [r(1+r)^n] / [(1+r)^n − 1]

Every fixed-rate installment loan — mortgage, auto loan, personal loan, student loan — uses the same amortization formula to compute the monthly payment. This guide walks you through the math, the variables, and the derivations behind it, with worked examples for the most common loan amounts in 2026.

Quick Answer: Monthly Payments Using the Loan Payment Formula

Precomputed monthly payments for typical 2026 loan amounts and rates, calculated with M = P × [r(1+r)^n] / [(1+r)^n − 1]:

Each M is calculated from the formula above. Plug the same P, r, and n into the calculator above to verify.

The Loan Payment Formula

Worked example: $25,000 auto loan at 7% APR for 5 years

1. P = $25,000
2. Annual APR = 7%, so monthly r = 0.07 ÷ 12 = 0.00583333…
3. 5 years × 12 = 60 monthly payments, so n = 60
4. (1 + r)^n = (1.00583333)^60 = 1.417625
5. Numerator: r × (1+r)^n = 0.00583333 × 1.417625 = 0.008269
6. Denominator: (1+r)^n − 1 = 0.417625
7. Factor = 0.008269 ÷ 0.417625 = 0.019801
8. M = $25,000 × 0.019801 = $495.17 per month
9. Total paid over 60 months = $29,710.20; total interest = $4,710.20

Worked example: $300,000 mortgage at 6.875% for 30 years

1. P = $300,000
2. r = 0.06875 ÷ 12 = 0.00572917
3. n = 360 (30 years × 12)
4. (1+r)^n = (1.00572917)^360 = 7.7957
5. Factor = (0.00572917 × 7.7957) ÷ (7.7957 − 1) = 0.04466 ÷ 6.7957 = 0.006572
6. M = $300,000 × 0.006572 = $1,973.32 / month
7. Total over 360 months = $710,395; total interest = $410,395

Where the loan payment formula comes from

The formula comes from the present value of an ordinary annuity. A loan is just an annuity in reverse: instead of saving monthly and getting a lump sum at the end, you receive a lump sum up front and repay it in equal monthly payments.

The present value of an ordinary annuity is PV = PMT × [1 − (1+r)^(−n)] / r. Solving for PMT (the monthly payment) gives the loan payment formula. Two assumptions sit underneath it:

• The interest rate is fixed for the entire term.
• Payments are made at the end of each month (an "ordinary annuity").

Variable definitions and edge cases

• P — the principal, or amount you borrow. Excludes down payments.
• APR — the annual percentage rate. Convert to monthly by dividing by 12.
• r — the monthly interest rate as a decimal (5% APR → r = 0.05 / 12 = 0.0041667).
• n — total number of monthly payments. A 30-year mortgage has n = 360, not 30.
• Edge case r = 0: if APR is 0%, the formula breaks. Use M = P / n instead.

APR vs APY — which one goes into the formula?

Use the APR, not the APY. APR is the nominal annual rate with monthly compounding, which is exactly the r × 12 you need. APY includes intra-year compounding; if you only know the APY, first convert it back to APR with: APR = 12 × ((1 + APY/12)^12 − 1)... wait, that's wrong — APY is the effective annual rate, so APR (nominal, compounded monthly) is APR = 12 × ((1 + APY)^(1/12) − 1). In practice, lenders quote APR directly on loans and APY on deposits.

Related formulas you should know

• Remaining balance after k payments: B_k = P × [(1+r)^n − (1+r)^k] / [(1+r)^n − 1]
• Interest portion of payment t: I_t = (balance at start of month t) × r
• Principal portion of payment t: principal_t = M − I_t
• Total interest over the loan: total_interest = (M × n) − P