In finance and economics, comparing interest rates can be misleading if the compounding frequencies differ. A 5% rate compounded annually is not the same as a 5% rate compounded monthly. To compare "apples to apples," investors and borrowers must convert these rates into an Equivalent Rate or an Effective Annual Rate (EAR).
What is an Equivalent Rate?
An equivalent rate is a nominal interest rate that, given a specific compounding frequency, yields the same amount of interest at the end of a year as another nominal rate with a different compounding frequency. The most common conversion is finding the Effective Annual Rate (EAR), which represents the true return on an investment or the true cost of a loan when compounding is taken into account.
The Conversion Formulas
To convert a Nominal Rate with compounding frequency $n$ to an Effective Annual Rate (EAR):
EAR = (1 + r / n)n – 1
Where: r = Nominal Annual Interest Rate (as a decimal) n = Number of compounding periods per year
To convert a rate from one compounding frequency ($n$) to another target frequency ($m$):
rtarget = m × [ (1 + r / n)(n/m) – 1 ]
Where: rtarget = The new nominal annual rate compounded $m$ times per year m = Target compounding frequency
Example Calculation
Suppose you have a loan with a nominal rate of 6% compounded monthly ($n=12$). You want to know the equivalent rate if it were compounded semi-annually ($m=2$).
Convert to the semi-annual basis: $2 \times [(1.061677)^{(1/2)} – 1]$.
Result: $2 \times [1.0304 – 1] = 0.0609$ or 6.09%.
This means 6% compounded monthly generates the same amount of interest as 6.09% compounded semi-annually.
Why This Matters
Understanding equivalent rates is crucial for financial decision-making. Banks often advertise nominal rates to make loans look cheaper or savings accounts look less lucrative than they actually are. By converting everything to an Effective Annual Rate or a common compounding period, you ensure you are making the most mathematically sound choice.
function calculateEquivalentRate() {
// 1. Get Input Values
var nominalRateInput = document.getElementById("nominalRate").value;
var currentFreqInput = document.getElementById("currentFreq").value;
var targetFreqInput = document.getElementById("targetFreq").value;
// 2. Validate Inputs
if (nominalRateInput === "" || isNaN(nominalRateInput)) {
alert("Please enter a valid nominal interest rate.");
return;
}
var r = parseFloat(nominalRateInput); // Percentage form, e.g., 5.5
var n = parseInt(currentFreqInput);
var m = parseInt(targetFreqInput);
if (r < 0) {
alert("Interest rate cannot be negative.");
return;
}
// 3. Perform Calculations
// Convert percentage to decimal
var rDecimal = r / 100;
// Calculate Effective Annual Rate (EAR)
// Formula: (1 + r/n)^n – 1
var base = 1 + (rDecimal / n);
var earDecimal = Math.pow(base, n) – 1;
// Calculate Equivalent Nominal Rate for Target Frequency (m)
// Formula: m * [ (1 + EAR)^(1/m) – 1 ]
// Note: (1+EAR) is effectively Math.pow(base, n)
var targetBase = 1 + earDecimal;
var equivalentRateDecimal = m * (Math.pow(targetBase, (1 / m)) – 1);
// Calculate Periodic Rate for Target Frequency
var periodicRateDecimal = equivalentRateDecimal / m;
// 4. Update UI
// Convert back to percentage strings
var earPercent = (earDecimal * 100).toFixed(4);
var equivalentPercent = (equivalentRateDecimal * 100).toFixed(4);
var periodicPercent = (periodicRateDecimal * 100).toFixed(4);
document.getElementById("resEAR").innerHTML = earPercent + "%";
document.getElementById("resTargetRate").innerHTML = equivalentPercent + "%";
document.getElementById("resPeriodicRate").innerHTML = periodicPercent + "%";
document.getElementById("result-area").style.display = "block";
}