Loan Payment Math: The Fixed-Payment Formula and the Five Bugs Around It

Loan Payment Math: The Fixed-Payment Formula and the Five Bugs Around It The fixed-payment loan formula is one of the most reproduced snippets on the internet, and most copies are subtly wrong in the same three places: the rate period, the final payment, and the difference between the contract rate and the cost the borrower actually pays. The formula Given principal $P$, periodic interest rate $i$, and $N$ total payments: $$M = P \cdot \frac{i(1+i)^N}{(1+i)^N - 1}$$ where $i = r/n$ for annual rate $r$ and $n$ payments per year, and $N = n \times t$ for a term of $t$ years. Where it comes from: the payment $M$ is the level sum whose present value equals the principal. Discount each payment back to today: $$P = \sum_{k=1}^{N} \frac{M}{(1+i)^k} = M \cdot \frac{1 - (1+i)^{-N}}{i}$$ Solving for $M$ gives the formula above. The geometric series is the whole derivation; there is no intuition to substitute for it. function payment(P, annualRate, years, paymentsPerYear = 12) { const i = annualRate / paymentsPerYear; const N = Math.round(years * paymentsPerYear); if (i === 0) return P / N; // 0% promotional loan const disc = Math.pow(1 + i, -N); return P * i / (1 - disc); } Worked example A $25{,}000$ car loan at 7.2% annual, monthly payments, 5 years. $i = 0.072/12 = 0.006$, $N = 60$ $(1.006)^{-60} = 0.69830$ $M = 25000 \times \frac{0.006}{1 - 0.69830} = 25000 \times \frac{0.006}{0.30170} = 25000 \times 0.019887 = \$497.18$ Total paid $= 60 \times 497.18 = \$29{,}830.80$. Total interest $= \$4{,}830.80$. That 19.3% premium over the principal is the number most quotes omit. Edge cases that actually bite Zero interest. At $i = 0$ the formula is $0/0$. A 0% promotional loan is a real product and will NaN your calculator. Return P / N. The final payment. With M computed in floating point, the last payment often leaves a residual of a cent or drives the balance slightly negative. Clamp the final payment to the remaining balance: if (principal > balance) { principal = balance; payment = principal + interest; } Rates near zero but not zero. 1 - (1+i)^-N suffers cancellation for tiny $i$. Use Math.expm1-style log-space evaluation, or the algebraically equivalent i(1+i)^N / ((1+i)^N - 1) form, which behaves better at small $i$. Fractional periods. A 40-month loan has $t = 3.333$ years and $N = 40$. Math.round the period count, not the years — rounding years first introduces a whole payment of error. Balloon and interest-only structures. These are not fixed-payment loans. If the product defers principal, the formula above does not apply and the schedule must be built from the actual payment vector rather than a closed form. Common mistakes Using the annual rate as the periodic rate. 7.2% annual is 0.6% monthly. Forgetting to divide by 12 is the most common source of a 12× overstatement. Confusing the contract rate with the APR. In the US, APR folds in certain fees. A 6.0% contract rate with 1% origination can be a 6.15% APR. Label which one your output is. Reporting the payment without the total cost. $497/month sounds fine; $29,831 over five years is the number the borrower should compare. Ignoring fees entirely. Origination, points, and prepayment penalties all change the effective cost. The payment formula alone cannot show them. Rounding the payment before amortizing. Round only at display time. Rounding $M$ to cents before the schedule drifts the final balance by cents. Assuming a fixed payment for the whole term. Adjustable-rate loans recompute $M$ at each reset. If your calculator only handles fixed rates, say so. Check the numbers yourself If you want to see the payment, the full amortization table, and the total interest for a specific amount and term — with the final-payment correction applied — the loan calculator at https://www.miryori.org/tools/loan-calculator runs the recurrence period by period and shows the cumulative cost. YMYL note: the figures above are illustrative arithmetic, not financial advice. Verify against your loan documents or a qualified adviser before making decisions.