Annually (1x per year)
Semi-Annually (2x per year)
Quarterly (4x per year)
Monthly (12x per year)
Bi-Weekly (26x per year)
Weekly (52x per year)
Daily (365x per year)
Continuously
Effective Annual Rate (EAR)
0.00%
Nominal Rate Input:0.00%
Period Rate:0.00%
function calculateEAR() {
// Get input values
var nominalInput = document.getElementById('nominalRate').value;
var frequency = document.getElementById('compoundingFreq').value;
var resultBox = document.getElementById('resultBox');
// Validation
if (nominalInput === "" || isNaN(nominalInput)) {
alert("Please enter a valid numeric Nominal Interest Rate.");
return;
}
var r = parseFloat(nominalInput); // Nominal rate in percent
var r_decimal = r / 100; // Nominal rate in decimal
var ear_decimal = 0;
var period_rate = 0;
var n = 0;
// Calculation Logic
if (frequency === 'continuous') {
// Formula for continuous compounding: e^r – 1
ear_decimal = Math.exp(r_decimal) – 1;
period_rate = 0; // Not applicable really, effectively infinitesimal
} else {
// Standard formula: (1 + r/n)^n – 1
n = parseFloat(frequency);
var base = 1 + (r_decimal / n);
ear_decimal = Math.pow(base, n) – 1;
period_rate = (r / n);
}
// Convert back to percentage
var ear_percent = ear_decimal * 100;
// Display Results
document.getElementById('earResult').innerText = ear_percent.toFixed(4) + "%";
document.getElementById('nominalDisplay').innerText = r.toFixed(2) + "%";
if (frequency === 'continuous') {
document.getElementById('periodRateDisplay').innerText = "N/A (Continuous)";
} else {
document.getElementById('periodRateDisplay').innerText = period_rate.toFixed(4) + "%";
}
resultBox.style.display = "block";
}
Understanding the Effective Annual Rate (EAR)
When evaluating investment opportunities or loan costs, the "nominal" interest rate often tells only half the story. The Effective Annual Rate (EAR), also known as the Annual Percentage Yield (APY) in savings contexts, provides a more accurate measure of the true interest accumulated over one year because it accounts for the effects of compounding.
What is the Effective Annual Rate?
The EAR represents the actual interest rate earned or paid after compounding is taken into consideration. While a nominal rate is simply the periodic rate multiplied by the number of periods in a year, the effective rate grows exponentially as the frequency of compounding increases.
For example, a loan with a 10% nominal annual rate compounded monthly will actually cost you more than 10% per year because you are paying interest on the interest added every month.
The EAR Formula
The mathematical formula used to convert a nominal rate to an effective annual rate depends on the compounding frequency. For standard discrete compounding:
EAR = (1 + i/n)n – 1
Where:
i = The nominal annual interest rate (expressed as a decimal).
n = The number of compounding periods per year (e.g., 12 for monthly, 4 for quarterly).
Continuous Compounding
In theoretical finance, if compounding happens infinitely often, we use the continuous compounding formula:
EAR = ei – 1
Where e is Euler's number (approximately 2.71828).
Why Compounding Frequency Matters
The frequency at which interest is added to the principal balance significantly impacts the final effective rate. The more frequently interest is compounded, the higher the EAR will be compared to the nominal rate.
Consider a nominal rate of 12%:
Annually (n=1): EAR = 12.00%
Semi-Annually (n=2): EAR = 12.36%
Quarterly (n=4): EAR = 12.55%
Monthly (n=12): EAR = 12.68%
Daily (n=365): EAR = 12.75%
How to Use This Calculator
Nominal Annual Interest Rate: Enter the advertised or stated interest rate (percentage). Do not adjust for periods; enter the full annual figure.
Compounding Frequency: Select how often interest is calculated and added to the balance (e.g., Monthly for most credit cards, Semi-Annually for many bonds).
Calculate: Click the button to see the true annual cost or yield expressed as a percentage.
Real-World Applications
For Savers: When comparing high-yield savings accounts, banks often advertise the APY (which is the EAR). However, if you are looking at a Certificate of Deposit (CD) that quotes a nominal rate compounded daily, you can use this calculator to verify the true annual yield.
For Borrowers: Credit card issuers state an APR (Annual Percentage Rate). If your card compounds interest daily (which most do), the effective rate you pay is higher than the stated APR. This calculator helps you understand the true cost of carrying a balance.