Calculate a loan's APR with fees by the Regulation Z actuarial method, the amount financed, the finance charge and the effective annual rate.
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.
Checked against: 12 CFR 1026 Appendix J, (c)(1) Example (i): annual percentage rate 9.69%; Python decimal 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: Python decimal (prec 60), 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.
Accuracy depends on your inputs and the method's assumptions. Decimal arithmetic uses 50 significant digits, but estimates, numerical methods and source data can be less precise; the displayed rounding does not remove those limits. It is checked against 7 worked examples whose answers come from independent sources; for example, “Regulation Z Appendix J (c)(1) example (i): $5,000, 24 monthly payments of $230” is checked against 12 CFR 1026 Appendix J, (c)(1) Example (i): annual percentage rate 9.69%; Python decimal gives 9.685708 %.
12 CFR 1026, Appendix J (Regulation Z): annual percentage rate computations for closed-end credit; 12 CFR 1026.22: determination of the annual percentage rate and accuracy tolerances; 12 CFR 1026.18(b): amount financed; 12 CFR 1026.4: finance charge.
For planning only. Lenders, tax authorities and markets apply their own rounding, fees and rules; confirm figures with them before you commit.
7 worked examples with independently sourced answers ship with this calculator. They run in the test suite; you can run them here too.
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