The Effective Annual Rate (EAR), also known as the annual equivalent rate (AER), is the real return on an investment or the real cost of a loan when the effects of compounding over a specific period are taken into account. While the Nominal Rate (APR) provides a base figure, the EAR reveals the true financial impact because it considers how interest accumulates on interest.
The Mathematical Formula
To find the EAR manually without a financial calculator, use the following formula:
EAR = (1 + i / n)n – 1
i = Nominal annual interest rate (as a decimal)
n = Number of compounding periods per year
How to Find EAR on a Financial Calculator
If you are using a standard financial calculator like the TI BA II Plus or HP 12C, follow these steps:
TI BA II Plus Instructions:
Press [2nd] then [ICONV] (above the number 2).
NOM = Enter the nominal rate (e.g., 10 for 10%) and press [ENTER].
Press the [Down Arrow] to C/Y.
Enter the number of compounding periods (e.g., 12 for monthly) and press [ENTER].
Press the [Down Arrow] to EFF.
Press [CPT] to calculate the Effective Annual Rate.
Comparison Table: APR vs. EAR
Notice how the frequency of compounding increases the effective rate even if the nominal rate remains 10%:
Compounding Frequency
Nominal Rate (APR)
Effective Rate (EAR)
Annual
10.00%
10.00%
Semi-Annual
10.00%
10.25%
Monthly
10.00%
10.47%
Daily
10.00%
10.51%
Real-World Example
Suppose you have a credit card with a nominal interest rate (APR) of 18.99% compounded monthly. To find the EAR:
Nominal Rate (i) = 0.1899
Periods (n) = 12
EAR = (1 + 0.1899 / 12)12 – 1
EAR = (1.015825)12 – 1
EAR = 1.2073 – 1 = 20.73%
In this case, you aren't paying 18.99% interest; you are effectively paying 20.73% due to monthly compounding.
function toggleManualFrequency() {
var selection = document.getElementById("compoundingFrequency").value;
var customContainer = document.getElementById("customFrequencyContainer");
if (selection === "custom") {
customContainer.style.display = "block";
} else {
customContainer.style.display = "none";
}
}
function calculateEffectiveRate() {
var nominalInput = document.getElementById("nominalRate").value;
var frequencySelect = document.getElementById("compoundingFrequency").value;
var customNInput = document.getElementById("customN").value;
var n;
// Validation
if (nominalInput === "" || isNaN(nominalInput)) {
alert("Please enter a valid Nominal interest rate.");
return;
}
if (frequencySelect === "custom") {
if (customNInput === "" || isNaN(customNInput) || customNInput <= 0) {
alert("Please enter a valid number of compounding periods.");
return;
}
n = parseFloat(customNInput);
} else {
n = parseFloat(frequencySelect);
}
var i = parseFloat(nominalInput) / 100;
// EAR Formula: (1 + i/n)^n – 1
var ear = Math.pow((1 + (i / n)), n) – 1;
var earPercentage = (ear * 100).toFixed(4);
// Display results
document.getElementById("resultContainer").style.display = "block";
document.getElementById("earOutput").innerText = earPercentage + "%";
var summaryText = "At " + nominalInput + "% APR compounded " + n + " times per year.";
document.getElementById("calculationSummary").innerText = summaryText;
// Smooth scroll to result
document.getElementById("resultContainer").scrollIntoView({ behavior: 'smooth', block: 'nearest' });
}