ReckonBoxWhat APR actually is
APR is the annual rate at which the payments you make are worth exactly the cash you receive, which is what folds a loan's fees back into its interest rate. The CFPB puts it as a broader measure than the interest rate, reflecting the rate plus any points, broker fees and other charges you pay to get the loan. The rate alone prices the principal and nothing else.
On the default: borrow $20,000 at 10% for five years and pay $500 in fees, and the payment is $424.94 a month. You receive $19,500, not $20,000. That gap turns a 10% loan into an 11.099% one.
Why there's no formula to rearrange
APR has no closed form, because it's the rate that solves an equation rather than the output of one. You know the payments and you know the cash received; the APR is the discount rate that makes them equal. That's the same computation as an internal rate of return, and it has to be solved by iteration.
So the formula two ranked pages publish is worth a look. Bankrate and Capital One both print a version of this:
APR = interest plus fees, over the loan amount, divided by the days in the term, times 365
Feed the default loan into it. Interest is $5,496.45, fees are $500, the loan is $20,000 and the term is 1,826 days. Out comes 5.996%.
That's below the note rate. Adding $500 of fees to a 10% loan cannot produce a rate under 10%, so the answer isn't approximate, it's impossible.
The formula isn't nonsense, it's just describing a different loan. It assumes the whole principal stays outstanding for the whole term, which is exactly true of a single-payment advance. Run a $500 payday loan repaid once after 14 days with a $75 fee through it and you get 391.07%, the familiar triple-digit figure and the correct answer. On anything that amortises, the balance falls every month, so treating the full amount as outstanding throughout halves the result.
Neither page says which kind of loan it means. Bankrate's sits on a personal-loan page; Capital One's is about credit cards.
The tolerance nobody mentions
A disclosed APR is allowed to be wrong, within a band written into federal regulation. Under Regulation Z at 12 CFR 1026.22, the CFPB treats a disclosed APR on a closed-end loan as accurate if it falls within one eighth of one percentage point of the true figure. For an irregular transaction, meaning one with multiple advances, irregular payment periods or irregular payment amounts, the band widens to one quarter of a point.
None of the five APR calculators we fetched mention this. It matters most where the number is doing the least work:
| $378,000 mortgage, 6.55%, 30 years, $6,000 fees | |
|---|---|
| Note rate | 6.55% |
| True APR | 6.705% |
| Entire effect of the fees | 0.155 points |
| Legal tolerance either way | 0.125 points |
The whole reason APR exists on that mortgage is to reveal 0.155 points of hidden cost, and the disclosure may legally be off by 0.125 in either direction. The tolerance nearly swallows the signal.
Where the numbers come from
The payment is computed on the amount financed, which is the loan plus any fees you roll in. The cash you receive is the loan minus any fees you pay at closing. The APR is then solved by bisection until the payments discounted at that rate equal the cash received, which is stable in a way Newton's method isn't, and the tool checks itself: with no fees, the APR comes back as the note rate exactly.
Where you pay the fee from changes the answer. Paying $500 at closing gives 11.099%; rolling the same $500 into the balance gives 11.072% and lifts the payment to $435.56. Upfront is dearer because it cuts what you actually receive.
What this does not decide for you
APR compares the cost of credit, and it assumes you hold the loan to term. Sell or refinance early and points paid upfront never earn back, so the loan with the lower APR can be the more expensive one you actually experience. The CFPB also notes that on an adjustable rate mortgage the APR does not reflect the maximum rate the loan can reach. This tool computes a figure, not a recommendation, and it's not a quote or an offer of credit. For a decision on borrowing, speak to a licensed professional.
The APR on your loan estimate is a legal disclosure carrying a tolerance, not a measurement. If it sits a tenth of a point below what you compute here, both numbers can be correct.
Frequently asked questions
What is APR? APR is the annual rate that makes the payments you actually make worth exactly the cash you actually receive, so it folds the fees into the interest rate. Per the CFPB it reflects the interest rate, any points, mortgage broker fees and other charges you pay to get the loan, while the interest rate alone covers none of them. On a $20,000 loan at 10% over 5 years with $500 of fees, the APR is 11.099%.
How is APR calculated? By solving, not by a formula you can rearrange. The APR is the discount rate at which the stream of payments equals the net amount you received, which is the same computation as an internal rate of return and has no closed form. This tool solves it by bisection, so the answer cannot diverge, and with no fees at all it returns the note rate exactly.
Why does the common APR formula give the wrong answer? Because it assumes the whole principal is outstanding for the whole term. Interest plus fees, over loan amount, divided by days, times 365 is correct for a single-payment loan: a $500 advance for 14 days with a $75 fee gives 391.07%, the familiar payday figure. On a loan that amortises the balance falls every month, so the same formula on a $20,000 five-year loan at 10% with a $500 fee returns 5.996%, which is below the note rate and cannot be an APR.
What is the difference between APR and the interest rate? The interest rate prices the principal only, and the APR adds the fees required to get the loan. The gap is the cost of borrowing that the rate hides, which is $500 of fees turning a 10% loan into an 11.099% one here. Two loans at the same rate can carry very different APRs.
How accurate does a disclosed APR have to be by law? Closer than most people assume, and looser than the decimal places suggest. Under Regulation Z at 12 CFR 1026.22 a disclosed APR on a closed-end loan is accurate if it is within one eighth of one percentage point of the true figure, widening to one quarter of a point for an irregular transaction, meaning one with multiple advances or irregular payment periods or amounts. So a quoted 6.500% can sit against a true 6.615% and still comply.
Is APR the same as APY? No. APR is the cost of borrowing and APY is the yield on savings, and APY compounds while a quoted APR does not. A savings account paying 10% compounded monthly yields an APY above 10%, whereas a loan quoted at 10% APR with no fees has an APR of exactly 10%.
Does paying fees upfront or financing them change the APR? Yes, and upfront is the more expensive way round. Paying $500 at closing on a $20,000 loan reduces what you receive to $19,500 and gives an APR of 11.099%; rolling the same $500 into the balance leaves the proceeds at $20,000, raises the payment to $435.56, and gives 11.072%. The difference is small here and grows with the size of the fee.