Nominal against effective
The two rates answer different questions. The nominal rate is what a contract quotes; the effective rate is what a year of monthly compounding actually costs. Comparing a nominal rate on one offer with an effective rate on another will always flatter the second.
| Rate | What it is | Use it for |
|---|---|---|
| Monthly rate | The rate applied to the balance each month | Reconstructing the amortization |
| Nominal annual rate | The monthly rate multiplied by 12 | Comparing with a lender's quoted rate |
| Effective annual rate | The true annual cost with monthly compounding | Comparing products with different compounding |
- This is the interest rate implied by the cash flows, not an APR. An APR also includes fees and charges, so where a loan carries arrangement fees the APR will be higher than the rate found here.
- If the total of the payments does not exceed the amount borrowed, no non-negative rate exists and the calculator says so rather than returning a negative rate as though it were a discount.
What the implied interest rate is
Four quantities describe a fixed-rate loan: the amount, the payment, the term and the rate. Fix any three and the fourth follows. This calculator fixes the amount, the payment and the term, and finds the rate that makes them consistent.
Unlike the payment or the term, the rate cannot be isolated algebraically — it appears both as a factor and inside an exponent. It is therefore found by numerical search, narrowing an interval that is known to contain the answer until it is pinned to well beyond display precision.
The nominal annual rate is simply the monthly rate multiplied by twelve, which is how lenders usually quote. The effective annual rate accounts for interest compounding monthly, and is always the larger of the two.
How to use the interest rate calculator
- Enter the amount actually borrowed, not the purchase price.
- Enter the contractual monthly payment.
- Enter the term in months.
- Read the nominal annual rate, which is the figure comparable with a lender's quoted rate, and the effective annual rate beside it.
How the rate is found
The rate i solves P = payment x (1 - (1 + i)^-n) / i. Because i cannot be isolated, the calculator evaluates the difference between the two sides across a bracket known to contain a sign change and halves that interval repeatedly — the bisection method. Bisection is used rather than a derivative-based method because it cannot diverge once a bracket exists.
The search runs to a tolerance far finer than the displayed precision, so the reported rate is limited by rounding for display, not by the method.
Common mistakes
- Treating the result as an APR. It is the interest rate implied by the payments, before fees.
- Entering the price rather than the amount financed.
- Comparing this nominal rate against another product's effective rate.
- Entering the total repayable as the monthly payment.
- Expecting a rate when the payments never repay the principal — in that case none exists.
Pertanyaan yang sering diajukan
How do you calculate the interest rate on a loan?
From the amount, the payment and the term, using P = payment x (1 - (1 + i)^-n) / i. The rate i cannot be isolated algebraically, so it is found numerically. Borrowing 20,000 and paying 387 a month for 60 months implies a monthly rate of 0.5031%, a nominal annual rate of 6.037% and an effective annual rate of 6.207%.
Why can't the interest rate be solved directly?
Because the rate appears both as a multiplier and inside an exponent in the annuity equation, which has no algebraic rearrangement for i. Financial calculators and spreadsheets all use a numerical search; this one uses bisection, which converges reliably without needing a derivative.
What is the difference between nominal and effective annual rates?
The nominal rate is the monthly rate multiplied by twelve and ignores compounding within the year. The effective rate is (1 + i)^12 - 1 and includes it. At a monthly rate of 0.5031% the nominal rate is 6.037% while the effective rate is 6.207%.
Is this the same as the APR?
No. The APR is a regulated disclosure that includes fees and certain charges alongside interest, so it is generally higher. Use this figure to check the interest rate implied by the payments, and the lender's disclosed APR to compare total cost.
What if the payments total less than I borrowed?
Then no non-negative interest rate can explain the agreement, and the calculator reports the payment as too small rather than returning a negative rate. In practice this usually means one of the three inputs has been entered incorrectly.
Referensi
- Brealey RA, Myers SC, Allen F. Principles of Corporate Finance, 13th edition. McGraw-Hill Education, 2019 — internal rate of return and annuity valuation.
- Press WH, Teukolsky SA, Vetterling WT, Flannery BP. Numerical Recipes: The Art of Scientific Computing, 3rd edition. Cambridge University Press, 2007 — bisection and bracketing of roots.
- US Consumer Financial Protection Bureau. Regulation Z (12 CFR Part 1026), Appendix J — annual percentage rate computations for closed-end credit.