Formulas and assumptions

This page lists the formulas and assumptions behind every calculator on calculate-loan.com, so you know exactly what a number means and when it will differ from your lender’s figure. Every result is an estimate for education, not financial advice.

1. Level payment (amortizing loan, EMI)

M = P × r ÷ (1 − (1 + r)⁻ⁿ). P is the amount borrowed, r is the periodic rate (for monthly payments, APR ÷ 12 ÷ 100) and n is the number of payments. At a 0% rate the payment is simply P ÷ n. Total paid = M × n and total interest = M × n − P. “EMI” (equated monthly instalment) is the same formula.

2. Balance after k payments

B_k = P(1 + r)^k − M((1 + r)^k − 1) ÷ r. Principal repaid so far = P − B_k; interest paid so far = M × k − (P − B_k). A balloon payment is the balance left when the loan ends before it is fully amortized.

3. Months to pay off with a given payment

n = −ln(1 − r × P ÷ M) ÷ ln(1 + r). If the payment is not larger than the first month’s interest (M ≤ r × P), the balance never falls and the calculator shows “Cannot compute”. Extra payments use the same formula with M + extra; the saving is the interest without the extra minus the interest with it.

4. APR when there are fees

When a fee is paid out of the loan or up front, you receive less than you repay on. The APR is the yearly rate i that makes the payments worth exactly the cash received: (P − fees) = M × (1 − (1 + i ÷ 12)⁻ⁿ) ÷ (i ÷ 12). There is no closed form, so the calculator solves it numerically (repeated halving of the interval until the rate is fixed to well under 0.0001%). The effective annual rate of an APR compounded monthly is (1 + APR ÷ 12)¹² − 1.

5. Simple, daily and interest-only interest

Simple interest I = P × rate × time. Daily interest = balance × APR ÷ 365 (some lenders use 360; the pages say which). An interest-only payment is P × APR ÷ 12; it repays no principal.

6. Ratios and budgets

Debt-to-income (DTI) = monthly debt payments ÷ gross monthly income × 100. Loan-to-value (LTV) = loan ÷ property value × 100. The 50/30/20 split is a common rule of thumb for dividing take-home pay, not a rule any lender applies. Lenders set their own limits; the pages never say what a lender will accept.

7. Assumptions on every page

• Fixed rate for the whole term, payments of equal size at the end of each period, interest compounded once per payment period.
• Fees, taxes, insurance, escrow, prepayment penalties and late charges are left out unless the calculator has a field for them.
• No government program, credit-score or tax rules are modelled.
• Inputs outside a sensible range are refused with a message (for example an APR above 100%, or a term longer than 50 years).

8. Rounding

Money is shown to the cent; months are whole numbers rounded up where a part-month payment is still a payment. The calculators do not round each month’s interest to the cent as a lender’s system does, so a real schedule can differ by a few cents.

9. How the calculators are checked

Each calculator is run against an independent calculation (for example the closed-form balance against a month-by-month schedule, or a Newton solve against the bisection used for APR) on test cases with known answers, including a 0% rate, very short and very long terms, and payments too small to repay the loan. The same formula runs in your browser and on the server, and both are tested. Report an error.