Calculate a loan's APR with fees by the Regulation Z actuarial method, the amount financed, the finance charge and the effective annual rate.
By AdminDrafted with AI writing toolsUpdated Checked examples: 7
You receive $294,000.00 and repay 360 payments of $1,896.20, $682,633.47 in all. The rate that makes those payments worth exactly what you received is 0.5579% per period, an APR of 6.695%, against an interest rate of 6.5%. Compounded over a year, that is 6.905%.
| Year | Interest | Principal | Balance |
|---|---|---|---|
| 1 | $19,401.27 | $3,353.18 | $296,646.82 |
| 2 | $19,176.70 | $3,577.74 | $293,069.08 |
| 3 | $18,937.10 | $3,817.35 | $289,251.73 |
| 4 | $18,681.44 | $4,073.01 | $285,178.72 |
| 5 | $18,408.66 | $4,345.79 | $280,832.93 |
| 6 | $18,117.62 | $4,636.83 | $276,196.10 |
| 7 | $17,807.08 | $4,947.37 | $271,248.73 |
| 8 | $17,475.75 | $5,278.70 | $265,970.03 |
| 9 | $17,122.22 | $5,632.23 | $260,337.81 |
| 10 | $16,745.02 | $6,009.43 | $254,328.38 |
Fees paid at closing or deducted from the proceeds are prepaid finance charges and come off the amount financed (12 CFR 1026.18(b)).
No closed form exists; the rate is bracketed by bisection and refined by Newton's method until the two sides agree to 28 decimal places.
12 CFR 1026 Appendix J (b)(1): the rate per unit-period times the unit-periods in a year.
Total of payments minus the amount financed: all interest plus the fees.
The APR is the yearly rate at which the loan payments are worth exactly the money you actually receive, the amount financed. Fees charged for the credit, such as origination fees and points, lower the amount financed when you pay them at closing, or raise the payments when they are added to the loan, so the APR comes out above the interest rate. The calculator solves for that rate numerically, as the actuarial method in Regulation Z, Appendix J, prescribes, and multiplies the rate per payment period by the periods in a year.
With the defaults, $300,000 at 6.5% for 30 years costs $1,896.20 a month. With $6,000 of fees paid upfront, the borrower receives $294,000, and the APR is 6.695%; the effective annual rate is 6.905%.
The APR assumes the loan runs to the end of its term with every payment on time. Paid off early, the fees are spread over fewer years and the true cost is higher.
APR is the yearly rate at which a loan's payments are worth exactly the cash you receive. Borrow 300,000 at 6.5% for 30 years and the payment is 1,896.20 a month. Pay 6,000 of fees at closing, receive 294,000, and the APR is 6.695%. US lenders disclose it under Regulation Z.
Step 3 has no closed formula, so it is solved by trial. In a spreadsheet, =RATE(360, -1896.20, 294000) returns the same 0.55794% a month.
This is the actuarial method of Regulation Z, Appendix J. The finance charge, total payments minus the amount financed, is 388,633.47 here: 382,633.47 of interest plus the 6,000 of fees.
The interest rate, also called the note rate, sets the payment. APR adds the fees charged for the credit, so it is higher whenever there are such fees. With no fees, the two are equal.
| Scenario | Payment per month | Amount financed | APR (%) |
|---|---|---|---|
| Default | 1896.20 | 294000.00 | 6.695 |
| No fees | 1896.20 | 300000.00 | 6.500 |
| Zero note rate | 1000.00 | 11700.00 | 4.700 |
The Default row is the loan above, and No fees is the same loan without the 6,000. Zero note rate is 12,000 repaid over 12 months at 0% with a 300 fee: you receive 11,700 and repay 12,000, an APR of 4.700%.
How a fee is paid changes both the payment and the APR. Take 20,000 at 9% for 3 years with 800 of fees:
Financing the fee gives the lower APR here but costs more cash: 23,811.64 of payments, against 22,895.81 of payments plus 800 at closing, or 23,695.81.
Only charges that are part of the finance charge belong in APR. Points, origination and broker fees, and mortgage insurance that protects the lender count. On a real-estate loan, bona fide title, appraisal and credit-report fees do not (12 CFR 1026.4).
APR is the monthly rate times 12, with no compounding. The effective annual rate compounds it: (1 + 0.0055794)^12 − 1 = 6.905% for the 300,000 loan. Both describe the same cost; the effective rate counts a full year of compounding.
APR assumes you keep the loan to the last payment. Repay early and the fees are spread over fewer months, so the yearly cost rises. For the 300,000 loan with 6,000 of fees, paid off without a prepayment penalty:
A no-fee offer at 6.875% has the higher APR, yet it costs less if you sell or refinance within 7 years (84 payments). From payment 85 on, the 6.5% loan with fees is cheaper. Compare offers over the years you expect to keep the loan.
For the payment and schedule on a loan without fees, use the loan EMI calculator. To weigh a new loan against one you hold, use the refinance calculator, and for a home loan with tax, insurance and PMI, the mortgage calculator.
Checked against: 12 CFR 1026 Appendix J, (c)(1) Example (i): annual percentage rate 9.69%; exact decimal arithmetic gives 9.685708 %
Checked against: calculator.net APR calculator published result: real APR 6.563%, payment $1,110.21, total interest $33,224.60
Checked against: Bisection on the Appendix J equation to 400 halvings: APR 6.695317 %
Checked against: Definition: with no finance-charge fees the amount financed equals the loan; EAR 1.0058333^12 − 1 = 7.2290 %
The interest rate sets the payment; the APR also counts fees charged for the credit and expresses the total cost as a yearly rate. A $100,000 loan at 6% for 10 years costs $1,110.21 a month; with $2,500 of upfront fees the borrower nets $97,500 and the APR is 6.563%. Without fees the APR equals the rate.
Find the rate per payment period at which the present value of all payments equals the amount financed, then multiply by the number of periods in a year. Regulation Z's Appendix J example: $5,000 repaid in 24 monthly payments of $230 gives a monthly rate of 0.80714%, so the APR is 9.69%. There is no closed formula, so the rate is found by iteration.
Charges that are part of the finance charge under Regulation Z: points, loan and finder's fees, and premiums for insurance that protects the lender against default, such as mortgage insurance (12 CFR 1026.4(b)). Charges of a type you would pay in a cash deal are left out, and on real-estate loans so are bona fide title, appraisal, credit-report and document fees (1026.4(c)(7)).
The APR is the rate per period times the periods in a year, with no compounding; the effective annual rate compounds it: (1 + APR ÷ 12)^12 − 1 for monthly payments. A 6.695% APR on a monthly loan is an effective 6.905%. US disclosures use the APR; the effective rate is what the loan costs over a full year of compounding.
Within 1/8 of a percentage point of the true APR for a regular loan, and within 1/4 point for irregular ones with multiple advances or uneven payments, under 12 CFR 1026.22(a). A disclosed APR of 6.75% on the default loan, whose true APR is 6.695%, is inside the tolerance; 6.85% is not.
For planning only. Lenders, tax authorities and markets apply their own rounding, fees and rules; confirm figures with them before you commit.
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