# APR calculator

> Calculate a loan's APR with fees by the Regulation Z actuarial method, the amount financed, the finance charge and the effective annual rate.

Interactive version: https://www.calcopenly.com/finance/apr-calculator
Subject: Finance calculators

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.

## Inputs

- **I know** (options: The interest rate, The payment)
- **Loan amount**: Before any fees are added or deducted
- **Interest rate (per year)**
- **Payment each period**
- **Term**
- **Term in** (options: Years, Months)
- **Payments every** (options: Week, Two weeks, Half-month, Month, Quarter, Year)
- **Fees in the finance charge**: Origination fees, points, broker fees and similar charges for the credit
- **Fees are** (options: Paid at closing or deducted, Added to the loan)

## Results

- APR — main result
- Effective annual rate
- Interest rate implied by the payment
- Payment each period
- Amount financed
- Finance charge
- Total of payments
- Total interest

## Formula

$$
A_F = P\,\frac{1-(1+i)^{-n}}{i},\qquad \text{APR} = w\,i,\qquad \text{EAR} = (1+i)^{w}-1
$$

## Worked examples

### Regulation Z Appendix J (c)(1) example (i): $5,000, 24 monthly payments of $230

- I know: The payment
- Loan amount: 5000
- Payment each period: 230
- Term: 24
- Term in: Months
- Payments every: Month
- Fees in the finance charge: 0
- Fees are: Paid at closing or deducted
- **APR: 9.686%**
- Checked against: 12 CFR 1026 Appendix J, (c)(1) Example (i): annual percentage rate 9.69%; Python decimal gives 9.685708 %

### $100,000 at 6% for 10 years with $2,500 upfront fees

- I know: The interest rate
- Loan amount: 100,000
- Interest rate (per year): 6%
- Term: 10
- Term in: Years
- Payments every: Month
- Fees in the finance charge: 2500
- Fees are: Paid at closing or deducted
- **APR: 6.563%**
- **Payment each period: 1,110.21**
- **Amount financed: 97,500.00**
- **Total interest: 33,224.60**
- Checked against: calculator.net APR calculator published result: real APR 6.563%, payment $1,110.21, total interest $33,224.60

### Defaults: $300,000 at 6.5% for 30 years, $6,000 upfront

- I know: The interest rate
- Loan amount: 300,000
- Interest rate (per year): 6.5%
- Term: 30
- Term in: Years
- Payments every: Month
- Fees in the finance charge: 6000
- Fees are: Paid at closing or deducted
- **APR: 6.695%**
- **Effective annual rate: 6.905%**
- **Payment each period: 1,896.20**
- **Finance charge: 388,633.47**
- Checked against: Python decimal (prec 60), bisection on the Appendix J equation to 400 halvings: APR 6.695317 %

### No fees: APR equals the rate

- I know: The interest rate
- Loan amount: 20,000
- Interest rate (per year): 7%
- Term: 5
- Term in: Years
- Payments every: Month
- Fees in the finance charge: 0
- Fees are: Paid at closing or deducted
- **APR: 7.000%**
- **Effective annual rate: 7.229%**
- Checked against: Definition: with no finance-charge fees the amount financed equals the loan; EAR 1.0058333^12 − 1 = 7.2290 %

### Fees added to the loan: $20,000 plus $800 at 9% for 3 years

- I know: The interest rate
- Loan amount: 20,000
- Interest rate (per year): 9%
- Term: 3
- Term in: Years
- Payments every: Month
- Fees in the finance charge: 800
- Fees are: Added to the loan
- **APR: 11.701%**
- **Payment each period: 661.43**
- **Total interest: 3,011.64**
- Checked against: Python decimal: payment on 20,800, APR solved against the 20,000 received: 11.701083 %

### Zero-rate loan with a $300 fee

- I know: The interest rate
- Loan amount: 12,000
- Interest rate (per year): 0%
- Term: 12
- Term in: Months
- Payments every: Month
- Fees in the finance charge: 300
- Fees are: Paid at closing or deducted
- **APR: 4.700%**
- **Total interest: 0.00**
- **Finance charge: 300.00**
- Checked against: Python decimal: 11,700 = 1,000 × a(12, i) gives i = 0.391671 %, APR 4.700046 %

## Questions

### What is the difference between APR and interest rate?

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.

### How is APR calculated?

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.

### Which fees are included in the APR?

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)).

### What is the difference between APR and effective annual rate?

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.

### How accurate does a lender's APR have to be?

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.

### How accurate is the APR calculator?

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 %.

### Where does the method come from?

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.

## Sources

- [12 CFR 1026, Appendix J (Regulation Z): annual percentage rate computations for closed-end credit](https://www.ecfr.gov/current/title-12/part-1026/appendix-Appendix%20J%20to%20Part%201026)
- [12 CFR 1026.22: determination of the annual percentage rate and accuracy tolerances](https://www.ecfr.gov/current/title-12/section-1026.22)
- [12 CFR 1026.18(b): amount financed](https://www.ecfr.gov/current/title-12/section-1026.18)
- [12 CFR 1026.4: finance charge](https://www.ecfr.gov/current/title-12/section-1026.4)

_Note: financial information, not professional advice._