Interest rate calculator

How do I find the interest rate on a loan?

The monthly payment of a fixed-rate loan is P × r ÷ (1 − (1 + r)^−n), with P the amount, n the number of payments and r the monthly rate. There is no formula that gives r back directly, so the calculator solves for it numerically (Newton’s method, with bisection as a fallback) and multiplies it by 12 for the annual rate.

Example

A $200,000 loan paid off over 30 years (360 payments) at $1,199.10 a month works out to 6.00% a year, or 0.5% a month. A $25,000 loan paid with 60 payments of $500 comes to 7.42%.

What is the effective annual rate?

The rate after monthly compounding: (1 + monthly rate)^12 − 1. For a 6% loan it is 6.17% — the same idea as APY on a savings account.

Is this the same as the APR on my loan disclosure?

Only if there are no fees. The APR in a Truth in Lending disclosure also counts prepaid finance charges such as origination fees; use the APR calculator for that. Here the rate is simply the one that makes your payments repay the amount you enter.

What rate turns one amount into another?

Choose “Grow a sum”: rate = n × ((ending ÷ starting)^(1 ÷ (n × years)) − 1), with n the compounding periods a year. $10,000 that grows to $20,000 in 10 years earned 7.18% compounded yearly, or 6.95% compounded monthly.

Why does the calculator say no rate fits?

Because your payments add up to less than the amount borrowed. Payments × number of payments must be at least the loan amount; if it is exactly equal, the rate is 0%. Check that you entered the monthly payment (not the yearly one) and the term in the right unit, years or months.

Can I use this when the dealer only quotes a monthly payment?

Yes, that is its main use. Enter the price after your down payment as the amount, the quoted payment and the number of payments, and the calculator shows the rate hidden in the offer. Fees rolled into the payment are then included in that rate, which is why it can be higher than the advertised one.

What is the difference between the nominal and the effective rate?

The nominal rate is the quoted yearly rate, which is the monthly rate times 12. The effective annual rate adds the effect of compounding within the year, so it is higher: 6% compounded monthly is 6.17% effective. When you compare a savings or growth result with a loan, compare effective rates.

How the rate is found

For a loan with equal payments, the amount borrowed equals the present value of all the payments at the unknown monthly rate r. That equation cannot be rearranged for r, so the calculator starts from a close first guess and refines it with Newton’s method, checking the result by recomputing the payment. The monthly rate times 12 is the annual rate; compounding it over 12 months gives the effective annual rate. In the growth mode the rate comes straight from a closed-form root, since only two amounts and a time are involved.

Worked example: a car loan

You borrow $32,000 and are quoted $560 a month for 72 months. The payments add up to $40,320.00, so $8,320.00 is interest over the life of the loan. Solving gives a monthly rate of 0.6610%, an annual rate of 7.93% and an effective annual rate of 8.23%. Checking: a 7.93% loan of $32,000 over 72 months has a payment of $560.00. Had the same $560 payment been quoted over 84 months for the same $32,000, the rate behind it would be 11.71%: the longer the term for a given payment, the higher the rate. Compare rates, not payments.

Growth mode: $10,000 that becomes $13,000 over 4 years compounded monthly earned 6.58% a year, or 6.78% effective.

Things to watch

• The amount must be what you actually borrow, after any down payment or trade-in. Using the sticker price makes the rate look lower.
• Include only the payment on the loan. Insurance, taxes or escrow added to a mortgage payment would be read as interest.
• The result is the rate that fits the numbers you enter. The CFPB notes that an APR also adds points and other charges to the interest rate, so the APR on a disclosure is usually higher than the plain rate; for fees use the APR calculator.
• The method assumes equal payments at regular intervals. Adjustable rates, balloon payments and skipped payments do not fit it.

Estimates for planning only — not financial advice. Your lender's figures decide. How we check our calculators.

Sources: standard loan-payment (annuity) formula; Consumer Financial Protection Bureau, Ask CFPB, “What is the difference between a mortgage interest rate and an APR?”. Related tools: APR calculator, loan calculator, APY calculator, auto loan calculator.