APR calculator
A $200,000 loan over 30 years at 6.5% with $4,000 of fees has an APR of 6.70%. The rate prices the money and the APR prices the deal, which is why two offers at the same rate are not the same offer. It is solved for here rather than approximated, because the closed form does not exist and the usual shortcut drifts on short terms.
What a loan really costs once its fees are counted — expressed as the one number two offers can be compared on, and solved for rather than approximated.
Solver checked against the discounted cash flow it solves · How we check
Annual percentage rate
6.695%
The note rate is 6.5%. The fees add 0.195 percentage points — that is what they are worth expressed as a rate, and it is the number two offers can actually be compared on.
What the APR is measuring
| Monthly payment | $1,896.20 |
| Amount actually advanced | $294,000.00 |
| Paid over the term | $682,633.47 |
| Finance charge | $388,633.47 |
| Payments | 360 |
The payment is set by the note rate on the full loan. The APR is the rate at which those payments discount back to what you actually received — the loan less the fees — which is why it is higher and why it cannot be worked out with a formula. It is solved for.
Why this needs a solver
Every other figure on this site comes out of a formula. This one cannot: the rate sits inside an exponent in every term of the payment stream, and no rearrangement isolates it. The only way to find it is to guess, check whether the discounted payments come to more or less than the amount advanced, and narrow the bracket — fifty times over, which takes a fraction of a millisecond and no algebra at all.
That is also why the APR on a disclosure is worth trusting: it is not an estimate a lender chose, it is the solution to an equation the regulation defines. Where two lenders disagree on an APR for the same loan, they disagree about which charges are finance charges, not about the arithmetic.
Questions people actually ask
- What is the difference between the interest rate and the APR?
- The interest rate sets your payment. The APR expresses the rate plus the fees as a single figure, by asking what rate would produce those payments if you had received the loan minus the fees. A 6.5% loan with $6,000 of costs on $300,000 is a 6.695% APR — the payment is unchanged, and what you got for it was less.
- Why can’t APR be worked out with a formula?
- Because the equation cannot be rearranged for the rate. The rate appears inside an exponent in every term of the payment stream, and there is no algebraic solution — you have to try values until the discounted stream matches the amount advanced. This page bisects toward it, which is what a lender’s software does too.
- Does a lower APR always mean a cheaper loan?
- Only if you keep it to term. APR spreads the fees over the whole life of the loan, so a low rate with high fees looks good over thirty years and is expensive if you sell or refinance in five. That is the single most common way APR misleads, and it is why the break-even against a no-fee option matters more than the APR itself.
- What counts as a fee in the APR?
- Under Regulation Z, the finance charges: origination fees, points, mortgage insurance premiums, and lender-required services. Not the ones you would pay anyway — appraisal in some cases, title insurance you shop for yourself, recording fees, prepaid taxes. Which items are included is a rule about the loan rather than about the arithmetic, and lenders do occasionally differ.
- Is APR the same on a credit card?
- The word is, the arithmetic is not. A card APR is a simple annual rate divided into daily periodic rates, with no fees amortised into it, and compounding makes the effective annual rate higher than the stated APR. A 24.99% card APR compounds daily to about 28.4% effective — which the card does not have to advertise.