APR Calculator
Result
Annualised rate (APR)
- Monthly payment
- 1,199.10
- Rate increase from the fees
- 0.142%
- Total interest
- 231,677.04
An APR calculator answers the question a quoted rate leaves open: what does this money cost once the fees paid up front are counted? Enter the amount borrowed, the nominal annual rate, the term in months and the upfront fees charged at closing, and the page returns the annual percentage rate together with the monthly payment, the total interest and the number of percentage points those fees added. The arithmetic is the one Regulation Z sets out in Appendix J: the annual percentage rate is not a formula applied to the amount borrowed, it is the rate that solves an equation in which the amount actually advanced to the borrower — the principal less the prepaid finance charge — equals the present value of the payments. A 200,000 loan at 6 percent over 30 years carries a monthly payment of 1,199.10; if 3,000 of fees come out of the proceeds at closing the borrower receives 197,000 and still repays that same stream of payments, which is a cost of 6.142 percent a year rather than 6. The extra 0.142 of a percentage point is the whole reason the page exists: the rate printed on the contract describes the interest, and the annual percentage rate describes the deal. Three things move it. The size of the fee, which the reference table walks from nothing up to 5,000 in steps of 1,000. The length of the term, because the same fee spread over 30 years is diluted far more than over one — 2,500 on a one-year 100,000 loan lifts a 5 percent rate to 9.756 percent, where a 30-year term would barely move it. And the rate itself, because fees are a fixed dollar amount while interest scales with the balance, so at a high rate the fees are a smaller share of the total cost. Two consequences are worth knowing before you read the panel. A loan with no fees has an annual percentage rate equal to its nominal rate. And a loan at zero percent with fees still has a positive annual percentage rate, because the fees are a cost of borrowing and the equation does not care what the interest rate is — only what the money cost.
3,000 in fees on a 200,000 30-year loan at 6%, moved in 1,000 steps
| Upfront fees | Annual percentage rate | Rate increase from the fees | Monthly payment | Total interest |
|---|---|---|---|---|
| 0 | 6 | 0 | 1199.1 | 231677.04 |
| 1000 | 6.047 | 0.047 | 1199.1 | 231677.04 |
| 2000 | 6.094 | 0.094 | 1199.1 | 231677.04 |
| 3000 | 6.142 | 0.142 | 1199.1 | 231677.04 |
| 4000 | 6.189 | 0.189 | 1199.1 | 231677.04 |
| 5000 | 6.238 | 0.238 | 1199.1 | 231677.04 |
Read the fourth column first, because it is the one that does not move: at 1,199.10 across all six rows, the fees change nothing about what is paid each month. They are withheld from the proceeds, so the borrower's obligation is identical in every row and the only thing that changed is how much money was actually handed over — 200,000 in the first row, 195,000 in the last. That is why the first three columns and the fifth move in lockstep: five thousand dollars of fees on a 200,000 loan is 2.5 percent of the money advanced, and annualised over 30 years it adds 0.238 of a percentage point to the rate, taking 6 percent to 6.238 percent. The column headings say what each figure is: the annual percentage rate is the cost including the fee, the rate increase is the difference between that and the 6 percent the loan nominally charges, and the total interest is what the 360 payments cost above the principal. Notice how small those increments are — each 1,000 of fees is worth 0.047 to 0.048 of a point, and it is worth knowing that this is a property of the term rather than of the loan size, since a longer term divides the fee across more payments and a shorter one concentrates it.
Formula
Annual percentage rate = 12 × i, where i is the monthly rate that solves: (principal − upfront fees) = monthly payment × [1 − (1 + i)^(−termMonths)] ÷ i
- Principal
- The amount borrowed, before anything is deducted from it — the number on the note, not the number the borrower actually receives
- Upfront fees
- Costs paid at closing and withheld from the proceeds rather than added to the balance — an origination or arrangement fee, points, a processing charge. Regulation Z calls these prepaid finance charges, and they are the entire difference between this page and a plain rate calculation
- Net proceeds
- The principal minus the upfront fees, which is what the borrower actually walks away with and the figure the annual percentage rate is solved against — it is the left-hand side of the equation, and moving it is what moves the answer
- Monthly payment
- The level payment on the loan, set by the principal, the nominal rate and the term alone. The fees do not appear in it, which is why a loan with a higher annual percentage rate can carry exactly the same monthly payment as the same loan without the fees
- i
- The monthly rate that makes the two sides equal, found by search rather than by a closed-form formula. Multiply it by twelve and you have the annual percentage rate — the same annualisation Regulation Z applies to a periodic rate, applied here to the rate implied by the money actually advanced
Use it whenever money is paid at closing, because that is the only situation in which the annual percentage rate and the nominal rate differ. A loan with no upfront fees has an annual percentage rate identical to its rate, and the page will show you that in the third decimal, which is a useful thing to confirm once. The comparison it is built for is between two offers on the same loan: a lower rate with a large fee against a higher rate with none. Quoting them side by side by nominal rate is misleading, because the rate ignores the fee entirely, and the fee is money the borrower parts with on day one. The annual percentage rate folds both into one number, so the cheaper offer is the one with the lower annual percentage rate — with one caveat that matters in practice: it assumes the loan runs to the end of its term. Fees are paid once and their effect is spread over every month of the loan, so a borrower who refinances or sells after three years pays the whole fee for a fraction of the dilution. That is why a fee-heavy loan looks much worse on a five-year comparison than a 30-year one, and the term field is how you test that: shorten the term and watch the fee lift column grow. The other thing to keep straight is that this page includes only the fees you enter. Discount points, a broker fee and a processing charge are all prepaid finance charges, while an appraisal, title insurance and recording fees generally are not part of the finance charge, so entering every closing cost would overstate the annual percentage rate. Enter what the disclosure would call a finance charge. The equation itself has no algebraic solution for i — it is the ordinary annuity equation every loan calculator solves, with the principal replaced by the net proceeds, and the page finds i by search rather than by a formula.
Worked examples
200,000 at 6% for 30 years with 3,000 in fees
- Monthly rate: 6% ÷ 12 = 0.5%
- Monthly payment: 200,000 × 0.005 ÷ (1 − 1.005⁻³⁶⁰) = 1,199.10
- Net proceeds: 200,000 − 3,000 = 197,000
- Solve 197,000 = 1,199.10 × [1 − (1 + i)⁻³⁶⁰] ÷ i → i = 0.5118% a month
- Annual percentage rate: 0.5118% × 12 = 6.142%, so the fees added 0.142 of a percentage point
The default case, and the clearest view of what the fees did. The monthly payment is unchanged at 1,199.10 because the fees came out of the proceeds rather than being added to the balance — the borrower pays the same 360 payments on 3,000 less money, and the annual percentage rate is the only figure on the panel that says so. Spread over 30 years, 3,000 works out at a little over 8 a month in extra cost, which is why the lift is 0.142 of a point rather than something dramatic.
The same fees on a one-year loan — 2,500 on 100,000 at 5%
- Monthly rate: 5% ÷ 12 = 0.41667%
- Monthly payment: 100,000 × 0.0041667 ÷ (1 − 1.0041667⁻¹²) = 8,560.75
- Net proceeds: 100,000 − 2,500 = 97,500
- Solve 97,500 = 8,560.75 × [1 − (1 + i)⁻¹²] ÷ i → i = 0.8130% a month
- Annual percentage rate: 0.8130% × 12 = 9.756%, so the fees added 4.756 of a percentage point
The same 2,500 would add about a tenth of a point on a 30-year loan; over twelve months it adds nearly five. The fee is a fixed cost and the term is what divides it, so a short term concentrates it. This is the arithmetic behind the rule of thumb that paying points only makes sense if you keep the loan long enough — the lift is not a property of the fee, it is a property of the fee divided by the time you have to earn it back.
No fees at all — the two rates meet
- Monthly rate: 6% ÷ 12 = 0.5%
- Monthly payment: 200,000 × 0.005 ÷ (1 − 1.005⁻³⁶⁰) = 1,199.10
- Net proceeds: 200,000 − 0 = 200,000
- Solve 200,000 = 1,199.10 × [1 − (1 + i)⁻³⁶⁰] ÷ i → i = 0.5% a month
- Annual percentage rate: 0.5% × 12 = 6%, identical to the nominal rate
Worth running once, because it is the check that the page is doing what it claims: with nothing withheld, the amount advanced equals the amount borrowed and the equation collapses to the ordinary loan equation, whose solution is exactly the nominal rate divided by twelve. Every other row on the panel can then be read as a deviation from this one.
Limitations
It includes the fees you enter and nothing else. Whether a given closing cost belongs on this page is a legal question the page cannot answer for you: under Regulation Z the finance charge covers interest plus the costs of obtaining credit, which is why an origination fee or discount points count while an appraisal, title insurance or a recording fee generally do not, and entering every line of a closing disclosure would overstate the annual percentage rate. It assumes the loan runs to the end of its term. A borrower who repays early, refinances or sells pays the whole upfront fee while getting only part of the dilution, so the annual percentage rate understates the cost of a loan that will not be held to maturity — over a short holding period the same fee is far more expensive than the figure on this panel. It assumes fixed payments at a fixed rate for the whole term; adjustable rates, interest-only periods, balloon payments and skipped first payments are all outside what the equation can express, and each of them changes the answer in a way that a single rate cannot describe. It does not model the time value of the fees in any way other than through the equation itself, so it has nothing to say about what the money not paid as fees could have earned if it had stayed invested. And it is a calculator, not a disclosure: the annual percentage rate a lender must give you is produced by the creditor's own software under the regulation as written, and small differences between this page and a disclosure can be legitimate — Regulation Z lets a disclosed rate be considered accurate within an eighth of a percentage point, or a quarter point for an irregular transaction.
Frequently asked questions
- What is the difference between the interest rate and the annual percentage rate?
- The interest rate prices the money you borrow; the annual percentage rate prices the money you borrow plus the money you pay to borrow it. If nothing is withheld at closing the two are the same number. If 3,000 of upfront fees come out of the proceeds of a 200,000 loan at 6 percent, the borrower receives 197,000 and makes the same payments as if they had received the full 200,000, which costs 6.142 percent a year. The gap between the two figures is the fee, annualised over the life of the loan.
- Why does the monthly payment not change when I add fees?
- Because the fees are withheld from the proceeds rather than added to the balance. The monthly payment is set by the principal, the rate and the term, and none of those three changed — the borrower simply receives less money for the same 360 payments. That is exactly why the annual percentage rate is worth computing: the payment looks identical in both cases and the cost does not.
- How do upfront fees affect the annual percentage rate?
- They raise it, and how much depends on the size of the fee and the length of the term. A fixed fee spread over a long loan lifts the rate only slightly, because it is divided across many payments: 3,000 over 30 years on a 200,000 loan adds 0.142 of a percentage point. The same kind of fee over a short term is concentrated: 2,500 on a one-year 100,000 loan at 5 percent adds 4.756 points, taking the annual percentage rate to 9.756 percent. The reference table holds everything else fixed and walks the fee up in steps so the first column's effect is visible on its own.
- Can the annual percentage rate be higher than the interest rate on a zero percent loan?
- Yes, and it usually is. A zero percent loan with an origination fee has a positive annual percentage rate, because the fee is a cost of borrowing and the equation measures cost against the money actually advanced. A 50,000 loan at zero percent for 24 months with 500 in fees has an annual percentage rate of 0.967 percent — there is no interest at all, and the cost of credit is real.
- Which fees should I include?
- The ones the regulation counts as a finance charge: an origination or arrangement fee, discount points, a broker fee, a processing charge, and anything else withheld from the proceeds for the creditor's benefit. Appraisal fees, title insurance, credit report fees, recording fees and transfer taxes generally are not finance charges, so including them would overstate the annual percentage rate. If you are comparing two offers, use the same basket of fees for both — the comparison is only meaningful if both sides are measured the same way.
- Is the annual percentage rate the same as the effective annual rate?
- No, and they usually differ. The annual percentage rate is an annualised periodic rate: a monthly rate multiplied by twelve, which is the convention Regulation Z uses so the disclosed number can be compared across loans with different payment schedules. The effective annual rate takes compounding into account and is the figure that describes what a balance actually earns or costs over a year. On a loan whose interest is computed monthly on a declining balance, the annual percentage rate is the lower of the two, exactly as a nominal rate is lower than an effective one. This page reports the first, since that is what a loan disclosure reports.
References
- 12 CFR Part 1026, Appendix J (Annual Percentage Rate Computations for Closed-End Credit Transactions) — the general rule that the annual percentage rate is the unit-period rate multiplied by the number of unit periods in a year — Consumer Financial Protection Bureau regulation, via the Electronic Code of Federal Regulations (United States)
- 12 CFR 1026.22(a) — accuracy of the annual percentage rate: it is determined by the actuarial method of Appendix J, and is considered accurate if it is within 1/8 of 1 percentage point of that figure (1/4 of a point in an irregular transaction) — Consumer Financial Protection Bureau regulation, via the Electronic Code of Federal Regulations (United States)
- 12 CFR 1026.2(a)(23) — prepaid finance charge: any finance charge paid separately in cash or by check before or at consummation, or withheld from the proceeds of the credit at any time — Consumer Financial Protection Bureau regulation, via the Electronic Code of Federal Regulations (United States)