APR Calculator | True Cost of Any Loan
Calculate the Annual Percentage Rate (APR) and Effective Annual Rate (EAR) for any loan. Accounts for upfront fees, shows step-by-step Newton-Raphson solution, and compares two loan offers side by side.
What Is the APR Calculator | True Cost of Any Loan?
The Annual Percentage Rate (APR) is the true annualised cost of borrowing, it captures not just the interest rate but also the effect of any upfront fees, making it a reliable measure for comparing loan products. Lenders are legally required to disclose APR in most jurisdictions precisely because it is harder to manipulate than a headline interest rate.
- ▸Single loan mode: Enter principal, monthly payment, term, and optional fees to find APR, EAR, and total cost.
- ▸Compare two loans: Enter details for two different offers side by side to see which costs less in total, useful for comparing banks or refinancing options.
- ▸Step-by-step calculation: See exactly how Newton-Raphson solves for the monthly rate and how it converts to APR and EAR.
- ▸Interest breakdown bar: Visual split showing what percentage of your total payments is principal vs interest.
Formula
APR is solved by finding the periodic interest rate r that makes the present value of all payments equal to the effective loan amount (principal minus upfront fees).
P = PMT × [1 − (1+r)^−n] / r
P = effective principal, r = monthly rate, n = term in months, PMT = monthly payment.
APR = r × 12
Multiply the solved monthly rate by 12. This is the nominal APR (not compounded).
EAR = (1 + APR/12)¹² − 1
EAR accounts for monthly compounding. Always slightly higher than APR for monthly loans.
P_eff = Principal − Fees Solve r using P_eff (not full principal)
Fees reduce the effective amount received, so you pay the same PMT on less money, raising the true rate.
How to Use
- 1
Choose single or comparison mode
Single mode calculates one loan. Compare mode lets you enter two loans and see which has the lower APR and total cost.
- 2
Enter loan amount
This is the principal, the total amount borrowed before any fees. Do not include fees here; enter them separately.
- 3
Enter the monthly payment
The fixed amount you will pay each month. If you know the interest rate and need to find the payment, use the Annuity Calculator instead.
- 4
Set the loan term in months
For a 5-year loan enter 60, for 30-year mortgage enter 360.
- 5
Add upfront fees (optional)
Enter origination fees, mortgage points, or closing costs. These reduce the effective principal you receive, which raises the true APR above the quoted rate.
Example Calculation
Example 1 | Personal loan, no fees
Example 2 | Same loan with $300 origination fee
Upfront fees can meaningfully raise your true borrowing cost, especially on shorter-term loans.
Example 3 | APR vs nominal rate on a mortgage
On a 30-year mortgage, $8,000 in upfront costs raises the APR by 0.41 percentage points, this is why APR matters more than the headline rate for mortgages.
Understanding APR | True Cost of Any Loan
What Is APR and Why Does It Matter?
The Annual Percentage Rate (APR) is the standardised measure of a loan's true yearly cost. Unlike the nominal interest rate, APR folds in fees and other charges, giving borrowers a single number they can use to compare products from different lenders on equal terms. Consumer protection laws in most countries, including the Truth in Lending Act (TILA) in the US and the Consumer Credit Directive in the EU, require lenders to disclose APR precisely because headline rates can obscure the real cost of borrowing.
APR vs Nominal Rate vs EAR
| Rate type | What it includes | When to use it |
|---|---|---|
| Nominal rate | Interest only, no fees | Never rely on this alone for comparisons |
| APR | Interest + fees, annualised | Comparing loan products |
| EAR (effective annual rate) | APR adjusted for compounding | Comparing across compounding frequencies |
How Fees Affect APR
Upfront fees reduce the actual money you receive while keeping your payment obligation the same. This makes the true cost of borrowing higher than the face rate. The shorter the loan term, the larger the APR impact of a given fee amount, because the fee is effectively spread over fewer months.
APR Across Different Loan Types
- ▸Mortgages: APR includes points, origination fees, and mandatory mortgage insurance. Compare APRs, not just rates, when shopping lenders.
- ▸Auto loans: APR is usually close to the stated rate because fees are lower. Still check for documentation fees and dealer markups.
- ▸Personal loans: Origination fees of 1–8% are common and can significantly raise APR on shorter terms.
- ▸Credit cards: APR is quoted as an annual rate but interest accrues daily. Paying in full each month avoids all interest charges.
- ▸Payday loans: Two-week fees translate to APRs of 300–500%+ when annualised, illustrating why APR disclosure is so important for consumer protection.
How Newton-Raphson Solves for APR
Because the loan equation P = PMT × [1−(1+r)^−n] / r has no algebraic solution for r, numerical methods are needed. Newton-Raphson starts with an initial estimate (derived from a linear approximation), then iteratively refines it:
- ▸Start: r₀ ≈ (PMT × n − P) / (P × n)
- ▸Compute: f(r) = PMT × [1−(1+r)^−n] / r − P
- ▸Compute: f′(r) = derivative of f with respect to r
- ▸Update: r_new = r − f(r) / f′(r)
- ▸Repeat until |r_new − r| < 10⁻¹⁰ (typically 5–15 iterations)
Practical Tips for Comparing Loan APRs
- ▸Always compare APRs, not just interest rates, especially for mortgages where fees vary widely between lenders.
- ▸Check whether the APR includes all fees. Some lenders exclude certain costs that others include.
- ▸For short-term loans, fee impact on APR is amplified. A $200 fee on a 6-month $5,000 loan adds about 8% to your APR.
- ▸Refinancing only makes sense if the APR savings over the remaining loan life exceed the closing costs of refinancing.
- ▸Use the comparison mode in this calculator to paste two competing loan offers and see the total cost difference in dollars.
Frequently Asked Questions
What is the difference between APR and the interest rate?
The nominal interest rate only reflects the cost of borrowing the principal. APR also includes fees such as origination charges, mortgage points, and certain closing costs, making it the more complete cost figure. Two loans can have identical nominal rates but different APRs because one has higher fees.
What is the difference between APR and EAR?
APR is a nominal annualised rate (monthly rate × 12). It does not account for the effect of compounding within the year. EAR (Effective Annual Rate) does: EAR = (1 + APR/12)¹² − 1. For a 12% APR compounded monthly, EAR ≈ 12.68%. Use EAR when comparing investments or loans with different compounding frequencies.
Why does the calculator use Newton-Raphson instead of a simple formula?
The standard present value equation P = PMT × [1−(1+r)^−n] / r cannot be rearranged algebraically to solve for r. Newton-Raphson is an iterative root-finding method that starts with an estimate and refines it until the error is below 10⁻¹⁰. It typically converges in fewer than 20 iterations.
My calculated APR is higher than my lender quotes. Why?
Lenders often advertise the nominal interest rate without fees. Your calculated APR includes all upfront costs distributed over the loan term. This is the correct 'true cost' figure and will always be higher than the naked interest rate when fees are present.
Does a higher monthly payment always mean a lower APR?
Not necessarily. A higher payment could simply mean a shorter loan term. APR depends on the relationship between the amount received, the payment size, and the number of periods. You can have a high payment and a high APR (short term, expensive loan) or a high payment and a low APR (short term, cheap loan).
How does APR work for credit cards?
For credit cards, APR is the daily periodic rate multiplied by 365. Most credit cards are quoted as annual APR. The effective rate on carried balances is compounded daily, so the EAR on a 20% APR credit card is about 22.1%. To avoid interest entirely, pay the statement balance in full each month.
What fees are typically included in a mortgage APR?
In the US (under TILA/Regulation Z), mortgage APR includes origination fees, discount points, mortgage broker fees, prepaid mortgage insurance, and certain closing costs. It excludes title insurance, appraisal, credit report fees, and transfer taxes. This varies by jurisdiction.