Loan payment formula (fully explained)

If you’ve ever wondered how lenders turn “loan amount + APR + term” into one clean monthly payment, it’s not magic — it’s amortization. An amortized loan is designed so that each payment includes interest (the cost of borrowing) and principal (paying down what you owe), and the balance reaches $0 at the end of the term.

The standard fixed-rate loan payment formula is:

Payment = P · r · (1+r)ⁿ / ((1+r)ⁿ − 1)

Here’s what each variable means in normal human language:

• P (principal): The amount you borrow. If you take out a $20,000 car loan, P = 20000.
• APR: Your annual percentage rate. APR is an annual number, so we convert it into a periodic rate.
• r (periodic rate): The interest rate per payment period. For monthly payments, r = (APR/100)/12.
• n: The total number of payments. For a 30-year monthly mortgage, n = 30 × 12 = 360.

The reason the formula looks intense is because it’s balancing two goals at once: (1) cover the interest that accrues each period, and (2) pay down principal fast enough to reach zero exactly at payment n.

Interest compounds over time. In an amortized loan, the lender is effectively asking: “What fixed payment makes the present value of all your future payments equal to the amount borrowed?” That’s why you see the exponential term (1+r)ⁿ. It accounts for the time value of money — a dollar today is worth more than a dollar received later.

If APR is 0%, then r = 0 and the formula becomes a division by zero. In real life, a 0% loan is just principal split evenly across n payments. This calculator handles that case by using: Payment = P / n.

Most loan quotes assume monthly payments, but you can choose a different payment frequency. The core idea stays the same — we just change “payments per year” and therefore change r and n:

• Monthly: payments per year = 12, n = years × 12
• Biweekly: payments per year = 26, n = years × 26
• Weekly: payments per year = 52, n = years × 52

A fun “viral” finance fact: paying biweekly instead of monthly can feel like a small change, but it often results in the equivalent of one extra monthly payment per year (because 26 biweekly payments ≈ 13 monthly payments). That’s why biweekly can reduce total interest for many people.

What this calculator does step-by-step

When you press Calculate Payment, the calculator follows the same logic your lender uses. Here’s the exact flow (in plain English):

• 1) Read your inputs: principal P, APR, term (years + optional extra months), payment frequency, and optional extra payment.
• 2) Convert APR into r: r = (APR/100) ÷ paymentsPerYear.
• 3) Convert term into n: totalPayments n = round(termYears × paymentsPerYear + extraMonths × (paymentsPerYear/12)).
• 4) Compute base payment: Use the amortization formula to calculate the minimum payment required to reach zero balance at the end.
• 5) Add extra payment (optional): If you type an extra amount, we add it to each period’s payment and simulate payoff sooner.
• 6) Build a mini amortization snapshot: We show the first 12 periods so you can see how interest vs principal changes.
• 7) Estimate totals: totalPaid = payment × numberOfPeriodsUntilPaidOff, totalInterest = totalPaid − principal.

Early in the loan, your balance is high, so the interest portion is high because interest = balance × r. Over time, your balance falls, so the interest part shrinks, and more of each payment goes toward principal. This is why amortization tables start out feeling unfair and then suddenly “get better” later.

Real lenders round to the nearest cent each period and sometimes apply specific rules for the final payment. This calculator rounds displayed amounts to cents and uses a simple payoff simulation for extra payments. Your statement may differ by a few cents because of lender-specific rounding — that’s normal.

If you pay extra each period, you are directly reducing principal faster. Because interest is computed on the remaining balance, lowering principal earlier reduces interest in every future period. That’s why extra payments can have an outsized impact — you’re not just paying a bit more, you’re shrinking the interest “base” for the rest of the loan.

Loan payment formula examples (with numbers)

Examples make the formula click. Below are realistic scenarios you can copy-paste into the calculator. (All examples assume a fixed rate and fully amortizing payments.)

P = $300,000, APR = 6.5%, term = 30 years, monthly payments. The periodic rate is r = 0.065/12 ≈ 0.0054167 and n = 360. The formula yields a payment around $1,896 (principal+interest only).

• If you add $100 extra per month, payoff can be years earlier, and total interest can drop significantly.
• Why? That extra $100 mostly hits principal after interest is covered, accelerating balance reduction.

P = $25,000, APR = 7.9%, term = 5 years, monthly payments. Here n = 60 and r = 0.079/12. Payments come out around $506 per month (again, principal+interest only).

• Auto loans have shorter terms, so a given APR hurts less than on a mortgage — but it still adds up.
• Try changing APR from 7.9% to 6.9% and screenshot the difference. Rate changes are surprisingly “viral” when you see the dollar impact.

P = $10,000, APR = 12%, term = 3 years, biweekly payments. With 26 payments per year, n = 78 and r = 0.12/26. The biweekly payment is around $155.

• If you switch to monthly for the same term and APR, the monthly payment is higher — but there are fewer payments.
• The total interest can differ slightly because of payment timing and periodic rate conversion.

If your result looks “too low” or “too high,” double-check: (1) APR vs interest rate (APR is annual), (2) term length, and (3) whether you meant monthly vs biweekly. Also remember: mortgages usually show principal + interest separately from taxes and insurance.

Loan payment formula FAQ

• Is this the same formula banks use?

  Yes. For standard fixed-rate, fully amortizing loans (most mortgages, many auto loans, many personal loans), the payment is computed from the amortization formula shown above. Lenders may apply tiny rounding differences.

• Why is my payment mostly interest at first?

  Interest each period is calculated from your remaining balance. Early on the balance is high, so interest is high. Over time, the balance falls, interest shrinks, and more of each payment goes to principal.

• Does paying biweekly always save money?

  Often, yes — because you make 26 half-payments per year (equivalent to 13 full monthly payments), which can accelerate payoff. But the exact savings depends on how your lender applies biweekly payments and on your rate/term.

• What if my loan compounds daily?

  Some loans compute interest using daily accrual. This calculator uses a standard periodic-rate approach tied to payment frequency, which matches most consumer loan disclosures for fixed payments. For daily-accrual specifics, your lender’s statement is the source of truth.

• Do extra payments always reduce interest?

  As long as the extra amount is applied to principal (most loans do this when you specify “principal only”), it reduces the balance earlier and therefore reduces future interest. Confirm with your lender that extra payments are applied correctly.

• Does this include taxes and insurance for mortgages?

  No. Mortgage payments often include escrow (property taxes + homeowners insurance, sometimes PMI). This calculator outputs principal + interest only.

• Can I use this for student loans?

  You can for simple fixed-rate repayment. But income-driven repayment plans, deferment/forbearance, or changing rates aren’t captured here.