Annually (Once per year)
Semi-Annually (Twice per year)
Quarterly (4 times per year)
Monthly (12 times per year)
Bi-Weekly (26 times per year)
Weekly (52 times per year)
Daily (365 times per year)
Continuous Compounding
Effective Annual Rate (EAR)
0.00%
function calculateEffectiveRate() {
// Get input values
var nominalRateInput = document.getElementById("nominalRate").value;
var frequencyValue = document.getElementById("compoundingFreq").value;
// Basic Validation
if (nominalRateInput === "" || nominalRateInput === null) {
alert("Please enter a nominal interest rate.");
return;
}
var r = parseFloat(nominalRateInput);
if (isNaN(r) || r < 0) {
alert("Please enter a valid positive number for the interest rate.");
return;
}
// Convert percentage to decimal
var decimalRate = r / 100;
var ear = 0;
// Calculate based on frequency
if (frequencyValue === "CONTINUOUS") {
// Formula: e^r – 1
ear = Math.exp(decimalRate) – 1;
} else {
// Formula: (1 + r/n)^n – 1
var n = parseInt(frequencyValue);
ear = Math.pow((1 + (decimalRate / n)), n) – 1;
}
// Convert back to percentage
var earPercentage = ear * 100;
// Calculate difference for display
var difference = earPercentage – r;
// Display results
var resultContainer = document.getElementById("result-container");
var finalEARDisplay = document.getElementById("finalEAR");
var diffDisplay = document.getElementById("diffDisplay");
resultContainer.style.display = "block";
finalEARDisplay.innerHTML = earPercentage.toFixed(4) + "%";
diffDisplay.innerHTML = "This is " + difference.toFixed(4) + "% higher than the nominal rate of " + r + "%.";
}
Understanding the Effective Rate Formula
The Effective Annual Rate (EAR), also known as the annual equivalent rate (AER) or effective interest rate, represents the true interest rate earned or paid on an investment or loan after accounting for the effects of compounding periods within a year. While the "Nominal Rate" is the stated headline percentage, the Effective Rate reveals the actual financial outcome.
Standard Formula:
EAR = (1 + r/n)n – 1
Continuous Compounding Formula:
EAR = er – 1
Where:
r = The nominal annual interest rate (in decimal form).
n = The number of compounding periods per year (e.g., 12 for monthly, 4 for quarterly).
e = The mathematical constant approximately equal to 2.71828 (Euler's number).
Why is the Effective Rate Higher?
The discrepancy between the nominal and effective rate occurs because interest is earned on top of previously earned interest. The more frequently interest is compounded, the higher the effective rate will be. This concept is crucial for comparing financial products with different compounding schedules.
Step 3: Raise to power of n: (1.008166…)12 ≈ 1.1025
Step 4: Subtract 1: 0.1025 or 10.25%
Even though Account B has a lower nominal rate (9.8%), its effective return is 10.25%, which is higher than Account A's 10%. This demonstrates why calculating EAR is vital for accurate financial comparisons.
Compounding Frequency Reference Table
When using the effective rate formula, use the following values for n:
Frequency
Periods per Year (n)
Annually
1
Semi-Annually
2
Quarterly
4
Monthly
12
Weekly
52
Daily
365
Nominal vs. Effective Rate
In lending, the Nominal rate is often referred to as the simple interest rate. However, if you are a borrower, the Effective Rate (often reflected in the APR, though APR may also include fees) tells you the true cost of the loan. If you are an investor or saver, the Effective Rate tells you your true yield (APY).