Why the rate and the APR are different numbers
Your note rate does one job: it produces your payment. Multiply the loan by the periodic rate, run it through the amortisation formula, and out comes the amount due each month. Fees have no part in that calculation at all.
The annual percentage rate answers a different question. It asks: given the payment you are committed to making, and given that some of the money you borrowed went straight back to the lender in fees, what interest rate did you actually pay on the money you got to keep? A $300,000 loan with $4,200 in origination and underwriting charges puts $295,800 at your disposal but bills you as though you borrowed $300,000. The APR is the rate that reconciles those two facts.
Regulation Z, which implements the Truth in Lending Act, requires lenders to disclose the APR precisely so that offers with different fee structures can be compared on one number. The mechanics are set out in 12 CFR Part 1026 and its Appendix J: identify the finance charge, subtract the prepaid portion from the amount borrowed to get the amount financed, and then solve for the rate that equates the payment stream to that amount. The regulation also sets an accuracy tolerance — for a regular closed-end transaction, a disclosed APR is considered accurate if it is within one eighth of one percentage point of the correct figure.
The practical consequence is simple. A lender offering 6.25% with two points and a lender offering 6.625% with none may be selling identical value. You cannot tell from the rates. You can tell from the APRs.
The three quantities and the equation that links them
The payment comes from the note amount at the note rate, using the ordinary amortised loan formula M = L·r/(1 − (1+r)−n). Fees never enter here. If your lender says the payment changes when you add points, they mean the note rate changed, not the fee treatment.
The amount financed is the note amount minus the prepaid finance charges. What counts as a prepaid finance charge is the part people get wrong. It is a charge imposed by the creditor as an incident to the extension of credit and paid before or at closing: origination fees, discount points, underwriting and processing fees, prepaid interest to the first payment date, and mortgage insurance premiums. It is not charges you would pay in a cash transaction or can shop for freely — appraisals, title insurance you select, recording fees, property taxes. Getting this boundary right matters, because every dollar you misclassify moves the APR.
The APR is then the root of
amount financed = M × (1 − (1 + r)−n) / r
There is no algebraic solution for r. This calculator brackets the root between essentially zero and 500% per period and bisects until the two sides agree, which is robust regardless of how extreme the inputs are. The annual figure is the periodic root multiplied by the number of payments a year — the nominal convention Regulation Z uses, not an effective annual rate that compounds.
That last distinction is worth holding onto. A 12% APR on a monthly loan means 1% a month; compounded twelve times that is 12.68% effective. The APR is deliberately not the effective rate, so that it can be compared with the note rate on the same footing.
Worked example: $20,000 personal loan at 12% with a 5% origination fee
A lender offers $20,000 at 12% over 60 monthly payments, with a 5% origination fee deducted from the proceeds.
- Periodic rate. 12% ÷ 12 = 0.01 a month, and n = 60.
- Payment. (1.01)60 = 1.816697, so (1.01)−60 = 0.550450 and 1 − 0.550450 = 0.449550. Then M = 20,000 × 0.01 ÷ 0.449550 = $444.89. That matches the published factor of 22.2444 per $1,000: 22.2444 × 20 = 444.89.
- Prepaid finance charge. 5% × $20,000 = $1,000, taken out of the proceeds.
- Amount financed. 20,000 − 1,000 = $19,000. That is the cash that reaches you.
- Required annuity factor. 19,000 ÷ 444.89 = 42.70705. You need the rate at which 60 payments are worth 42.70705 times one payment.
- Solve. At 1.19% a month the factor is 42.70176 — slightly too small. At 1.18% it is larger. The root is 1.18949% a month.
- APR. 1.18949 × 12 = 14.274%, against a note rate of 12%.
- Finance charge. Total of payments is 444.89 × 60 = $26,693.40. Less the $19,000 financed, the finance charge is $7,693.40.
The fee is 5% of the loan, and it raised the cost by 2.274 percentage points a year for five years. Spread over a 30-year mortgage the same 5% would raise the rate by roughly half a point. Short loans punish fees; that asymmetry is the single most useful thing this calculation teaches.
How to use the spread between rate and APR
Read the spread as a fee gauge. A spread under an eighth of a point means the loan carries almost no prepaid finance charges — that is what a genuine no-fee offer looks like, and it happens to be the same eighth-of-a-point tolerance Regulation Z allows for disclosure accuracy on regular transactions. A spread of a quarter to half a point on a 30-year mortgage is typical of a loan with one point plus routine lender charges. A spread above half a point means the fees deserve an explicit question.
Compare like with like on term. Because the fee is recovered over the life of the loan, the same dollar amount produces a much larger APR spread on a short loan than a long one. The reference table below quantifies that: 1% of the loan in fees adds about a tenth of a point on a 30-year mortgage and about 0.44 of a point on a five-year personal loan. Never compare the APR spread on a car loan against the spread on a mortgage and conclude the car lender is greedier.
Watch the horizon assumption. The APR assumes you keep the loan for its full term. If you sell or refinance early, the fees are recovered over fewer payments and your realised cost is higher than the disclosed APR. This is the same effect that drives the discount points break-even calculation, and it is why a low-rate high-fee loan is a bad choice for someone who expects to move.
Finally, do not treat APR as the whole answer on a mortgage. It ignores the mortgage insurance dropping off later, the possibility of recasting, and any rate adjustment on an adjustable loan. It is a comparison tool that assumes the loan runs to term exactly as written.
What prepaid fees do to the APR, by loan length
| Prepaid finance charge | 30-year loan at 6.5% | 5-year loan at 12% |
|---|---|---|
| 0.5% of loan | +0.05 pp | +0.22 pp |
| 1.0% of loan | +0.10 pp | +0.44 pp |
| 2.0% of loan | +0.20 pp | +0.89 pp |
| 3.0% of loan | +0.30 pp | +1.35 pp |
Each figure is the root of M·(1 − (1+r)⁻ⁿ)/r = (1 − fee) with M held at the note-rate payment factor, converted to an annual rate and less the note rate. Roughly four times as much APR damage per dollar of fee on the five-year loan, because the fee is spread over 60 payments rather than 360.
What counts as a prepaid finance charge, and what does not
- Counts: origination fee, discount points, underwriting fee, processing fee, document preparation charged by the lender, prepaid interest to the first payment date, mortgage insurance premiums, and lender-required broker compensation.
- Does not count: appraisal and credit report fees in most cases, title insurance and settlement services you are free to shop for, recording fees and transfer taxes, property taxes and homeowners insurance escrowed at closing, and any charge you would incur in a comparable cash transaction.
- Depends: some charges are finance charges only when the creditor requires a particular provider. The classification rules live in 12 CFR 1026.4, and lenders sometimes disagree at the margins.
- Never counts: fees payable only on default — late charges, prepayment penalties that may never be triggered, and returned-payment fees.
- Lender credits work in reverse. A credit toward closing costs raises the amount financed above the note amount and pulls the APR below the note rate. This calculator does not model negative fees; treat a credit as a reason the disclosed APR is lower than what you compute here.
Regulation Z tolerances
A disclosed APR is treated as accurate under 12 CFR 1026.22 if it is within one eighth of one percentage point of the rate determined under the regulation for a regular transaction, or one quarter of a point for an irregular transaction — one with multiple advance amounts, irregular payment periods, or unequal payments. Tighter tolerances apply to certain mortgage disclosures. If your own calculation and the lender's disclosure differ by more than these amounts, the likely cause is a disagreement about which fees are finance charges, not an arithmetic error.
Where APR helps and where it misleads
APR is at its best comparing two fixed-rate closed-end loans of the same amount and term that you intend to keep to maturity. That is exactly the case it was designed for, and there it is close to a complete answer.
It gets weaker fast outside that case. On an adjustable-rate mortgage, the disclosed APR uses the fully indexed rate at consummation and cannot know where the index will go. On a loan you will refinance in four years, the APR understates your cost because the fees never get their full term to amortise over. On a credit card, APR is a plain periodic rate annualised and there are no fees folded in at all — the finance charge arithmetic there works completely differently, as the credit card interest charge calculator shows.
For those cases, use a cash-flow comparison instead of a single rate. The refinance break-even calculator prices the fees against your actual expected horizon, and the debt consolidation calculator compares a fee-bearing new loan against the debts it would replace rather than against an abstract rate. If what you want is the payment rather than the cost of the money, the personal loan payment calculator and the auto loan payment calculator work from the note rate directly.
One last practical note. When a lender quotes you an APR, ask for the Loan Estimate rather than a verbal figure. The form separates the loan amount, the interest rate, the payment, the finance charge, the amount financed and the APR into labelled boxes, and it is the only version of those numbers you can hold the lender to.