How the math works
Our calculator methodology, step by step
This calculator methodology explains the formulas and conventions behind every result on the site: how payments are amortized, how annual rates become periodic rates, how compounding and day counts are handled, and how money is rounded. It also lists what the calculators leave out, how we test them against independent computations, and how changes and corrections are recorded.
What formula do the loan calculators use?
Fixed-payment loans use the standard amortization formula: M = P × r(1 + r)^n ÷ ((1 + r)^n − 1), where P is the amount borrowed, r is the periodic rate and n is the number of payments. Each payment covers that period's interest first, and the rest reduces the balance.
How is a fixed loan payment worked out?
A fixed payment is the amount that brings the balance to exactly zero after the last scheduled payment. The formula is M = P × r(1 + r)^n ÷ ((1 + r)^n − 1), with r equal to the annual rate divided by the number of payments per year.
Example: $20,000 at 7% for 60 monthly payments. The monthly rate is 0.07 ÷ 12 = 0.0058333. The formula gives $396.0240, which rounds to a $396.02 payment. In each month, interest is the balance times 0.0058333, rounded to the cent, and the rest of the payment reduces principal.
Because the regular payment is rounded down by a fraction of a cent, a tiny balance builds up. The last payment absorbs it: in this example the 60th payment is $396.30, and total interest over the loan is $3,761.48. Lenders handle this the same way, which is why a final payment often differs slightly.
How are annual rates turned into periodic rates?
Unless a calculator says otherwise, it uses a nominal annual rate divided evenly across the periods in a year.
| Payment or compounding frequency | Periods per year | Periodic rate at 6% |
|---|---|---|
| Monthly | 12 | 0.5% |
| Biweekly | 26 | 0.2308% |
| Weekly | 52 | 0.1154% |
| Quarterly | 4 | 1.5% |
| Daily (365-day year) | 365 | 0.01644% |
| Annually | 1 | 6% |
Which conventions change the answer?
Three conventions explain most of the small differences between our results and a lender's or bank's figures.
| Convention | What it means | How the calculators handle it |
|---|---|---|
| APR vs. APY | APR is the yearly borrowing rate before compounding; APY is the yearly savings yield after compounding. | Loans take an APR or interest rate. Savings tools show APY = (1 + r ÷ m)^m − 1; 6% compounded monthly gives 6.168%. |
| Day count: 365/360 | Daily interest uses the annual rate ÷ 360, but every actual day is charged. | Used only by the 365/360 calculator. On $10,000 at 6%, 31 days cost $51.67. |
| Day count: 365/365 | Daily interest uses the annual rate ÷ 365 for each actual day. | Used by the 365/365 calculator. The same 31 days cost $50.96. |
| Payment timing | Payments or deposits made at the start or end of each period. | End of period unless the calculator offers a choice. |
| Rounding | Amounts rounded to cents each period, or only at the end. | Schedules round each period to the cent; summary totals add up the rounded rows. |
What do the calculators leave out?
Unless a field asks for it, the calculators do not include:
- Origination, documentation or dealer fees
- Sales tax, title and registration costs
- Income tax on interest earned
- Insurance, including credit, gap or property insurance
- Late fees and penalty rates
- Variable-rate changes after the start date
- Early withdrawal penalties on CDs
- Credit approval, which depends on the lender
How are the calculators tested?
We check results against independent computations before a page goes live and after any change.
Independent re-computation
Every worked example is recomputed in a separate script, written apart from the calculator, and the outputs are compared to the cent. A mismatch blocks publishing until we find the cause.
Spot-checks against regulators
Where a government tool exists, we compare. For example, savings results are spot-checked against the SEC's compound interest calculator on Investor.gov using the same inputs and compounding.
Edge cases
We test zero rates, one-period terms, very long terms and payments that are too small to cover interest, and make sure the calculator explains rather than returns a misleading number.
What are the known limits, and how are changes recorded?
The calculators model a clean version of each product. Real loans can have irregular first periods, payment holidays, daily accrual tied to the exact date a payment arrives, or rate changes. Credit card calculators assume a fixed APR and no new purchases unless you add them. Results are estimates in US dollars.
When a formula or default changes, the page's updated date changes and the change is noted in the site changelog. If you report a calculation error and we confirm it, we fix it within 5 business days and note the correction on the page. Our editorial policy explains the corrections process in full.
Frequently asked questions
Why does my last loan payment differ from the others?
Because the regular payment is rounded to the cent, the balance after the second-to-last payment is rarely an exact match for one more regular payment. The final payment absorbs the difference. On a $20,000 loan at 7% for 60 months, the regular payment is $396.02 and the last one is $396.30.
Does a 365/360 loan really cost more than a 365/365 loan?
Yes, at the same stated rate. The 365/360 method divides the annual rate by 360 to get a daily rate, then charges it for all 365 days, so the effective yearly cost is about 1.39% higher in relative terms. On $10,000 at 6%, a full year of interest is $608.33 instead of $600.00.
How do I convert an APR into an APY?
Divide the rate by the number of compounding periods in a year, add 1, raise it to the power of that number of periods, then subtract 1. For 6% compounded monthly: (1 + 0.06 ÷ 12)^12 − 1 = 0.06168, or 6.168%. The savings calculators show this figure for you.
Do the calculators include daily interest accrual?
Most installment loan calculators use equal monthly periods, which matches how most fixed-rate loans are quoted. The 365/360 and 365/365 calculators model daily accrual directly. Credit card calculators use the APR divided by 12 as a monthly approximation, which is close to, but not identical to, a card issuer's daily balance method.
Why don't the results include fees or taxes?
Fees and taxes vary by lender, state and product, so a default value would often be wrong. Where a cost reliably affects the answer, such as sales tax on a car loan, the calculator gives you a field for it. Otherwise, add the costs from your own disclosure to the result.
How are compounding periods handled for savings and CDs?
Savings tools let you pick how often interest compounds, such as daily, monthly or quarterly. The calculator divides the annual rate by that number of periods and applies it each period. If a bank quotes an APY instead of a rate, enter the APY in a field that asks for one, or convert it first.
How do you test the calculators against independent tools?
Each worked example is recomputed in a separate script and compared to the cent with the calculator's output. Where a regulator publishes a calculator, such as the SEC's compound interest calculator on Investor.gov, we run the same inputs through both and compare. Any mismatch is investigated before publishing.
What happens when a formula or default changes?
The calculator page's updated date changes, the change is listed in the site changelog, and if it corrects an error, the page says so. We do not silently change results. If a change affects a worked example, the example is recomputed and rechecked.
Are the results precise enough to plan a payoff date?
They are close enough to plan around but not to replace your lender's payoff quote. A real payoff amount includes interest up to the exact day the payment arrives and any fees. Ask your lender for a payoff letter before you send a final payment.
Check our math against your own
Run a worked example by hand, then tell us if anything does not match.