Interest Rate Formula
To find the interest rate on a loan or investment, you rearrange the core interest equation to solve for the rate. Whether you have the total future value and principal, the interest amount paid, or regular payments, the same principles apply. The most common interest rate formulas cover simple interest, compound interest, effective annual rate (EAR), and the rate embedded in amortized loans.
What is the formula for simple interest rate?
Simple interest grows linearly. The accumulated amount \(A\) after \(t\) years is:
\[ A = P(1 + r \cdot t) \]where \(P\) is the principal, \(r\) is the annual interest rate as a decimal, and \(t\) is time in years. To solve for the rate:
\[ r = \frac{A - P}{P \cdot t} = \frac{I}{P \cdot t} \]where \(I = A - P\) is the total interest earned or paid.
Example: You borrow $5,000 and repay $6,500 after 3 years. The interest rate is:
\[ r = \frac{6500 - 5000}{5000 \times 3} = \frac{1500}{15000} = 0.10 \quad \text{or 10\% per year} \]The same logic works for finding the rate when you know only the interest earned: deposit $10,000, receive $1,200 in interest after 2 years – the annual rate is \(1{,}200 / (10{,}000 \times 2) = 0.06\) (6%).
Formulas Used
Simple Interest Rate: r = (A − P) / (P × t) = I / (P × t)
Compound Interest Rate: r = n × [(A/P)1/(n×t) − 1]
Effective Annual Rate: EAR = (1 + r/n)n − 1
Continuous EAR: EAR = er − 1
Loan Payment: M = P × r(1+r)N / [(1+r)N − 1] (solved numerically for r)
Rate Conversion: EAR = (1 + rfrom/nfrom)nfrom − 1; rto = nto × [(1+EAR)1/nto − 1]
Enter any three values – principal, total amount, time, or interest – and the calculator solves for the missing rate using the formulas on this page.
How do you calculate compound interest rate?
Compound interest adds earned interest back to the balance. The future value is:
\[ A = P \left(1 + \frac{r}{n}\right)^{n \cdot t} \]\(n\) is the number of compounding periods per year. Solve for the nominal annual rate \(r\):
\[ r = n \left[ \left(\frac{A}{P}\right)^{\frac{1}{n \cdot t}} - 1 \right] \]Example: $2,000 grows to $2,430.58 in 2 years with quarterly compounding (\(n = 4\)). What is the annual nominal rate?
\[ \frac{A}{P} = \frac{2430.58}{2000} = 1.21529 \]\[ \left(1.21529\right)^{\frac{1}{4 \times 2}} = 1.21529^{0.125} \approx 1.024695 \]\[ r = 4 \times (1.024695 - 1) = 0.09878 \quad \text{or roughly 9.88\%} \]The same formula works monthly, daily, or any compounding frequency. The calculator above automatically handles this when you select the compounding option.
Effective annual rate (EAR) formula
The nominal rate doesn’t reflect the true cost or yield when compounding occurs more than once a year. The effective annual rate does:
\[ \text{EAR} = \left(1 + \frac{r\_{\text{nom}}}{n}\right)^n - 1 \]If a credit card charges 18% compounded monthly, the EAR is:
\[ \text{EAR} = \left(1 + \frac{0.18}{12}\right)^{12} - 1 \approx 0.1956 \quad \text{or 19.56\%} \]For continuous compounding, replace the discrete formula with \( \text{EAR} = e^{r} - 1 \), where \(e \approx 2.71828\).
This adjustment is critical when comparing financial products. Lenders often quote APR (nominal), while the effective rate shows the actual annual cost.
How to find the interest rate on an amortized loan
Installment loans (mortgages, car loans) use a fixed payment formula:
\[ M = P \cdot \frac{r(1 + r)^N}{(1 + r)^N - 1} \]where \(M\) is the regular payment, \(r\) is the periodic interest rate (monthly rate if payments are monthly), and \(N\) is the total number of payments. This equation cannot be rearranged algebraically to isolate \(r\). In practice, the rate is found through numerical methods (trial and error, Newton’s method) or a dedicated calculator.
The widget above solves for the periodic rate directly from your loan inputs: enter the principal, monthly payment, and loan term, and it will compute the nominal annual rate.
Converting between different interest rate periods
- Annual to monthly nominal rate: \( r*{\text{monthly}} = \frac{r*{\text{annual}}}{12} \)
- Annual to daily nominal rate: \( r*{\text{daily}} = \frac{r*{\text{annual}}}{365} \)
- Monthly to annual effective: \( r*{\text{annual, eff}} = (1 + r*{\text{monthly}})^{12} - 1 \)
- Daily to annual effective: \( r*{\text{annual, eff}} = \left(1 + \frac{r*{\text{annual, nom}}}{365}\right)^{365} - 1 \)
Example: A payday loan charges 15% per month. The effective annual rate is \( (1.15)^{12} - 1 \approx 4.35 \) or 435%, not 180% from simple multiplication.
When entering values into the formula, always match the time unit of the rate with the period used for \(t\) and \(n\).
The formulas and examples are for educational purposes. Actual loan agreements may include fees, grace periods, or variable rates. Consult a financial professional for personal decisions.