*This is the "advertised" rate required to achieve the input EAR.
function calculateStatedRate() {
// Clear previous errors
document.getElementById('errorMessage').style.display = 'none';
document.getElementById('resultsArea').style.display = 'none';
// Get Inputs
var earInput = document.getElementById('inputEAR').value;
var frequencyInput = document.getElementById('selectCompound').value;
// Validate Inputs
var ear = parseFloat(earInput);
var n = parseInt(frequencyInput);
if (isNaN(ear) || ear < 0 || earInput === "") {
document.getElementById('errorMessage').innerText = "Please enter a valid positive Effective Annual Rate.";
document.getElementById('errorMessage').style.display = 'block';
return;
}
// Logic: Calculate Stated Rate from EAR
// Formula: Stated Rate (r) = n * [ (1 + EAR)^(1/n) – 1 ]
// Note: EAR input is percentage, convert to decimal first
var earDecimal = ear / 100;
// Calculate the rate per period first: (1 + EAR)^(1/n) – 1
var ratePerPeriodDecimal = Math.pow((1 + earDecimal), (1 / n)) – 1;
// Calculate Nominal (Stated) Annual Rate: RatePerPeriod * n
var statedRateDecimal = ratePerPeriodDecimal * n;
// Convert back to percentages
var statedRatePercent = statedRateDecimal * 100;
var ratePerPeriodPercent = ratePerPeriodDecimal * 100;
var difference = ear – statedRatePercent;
// Display Results
document.getElementById('resultNominal').innerText = statedRatePercent.toFixed(4) + "%";
document.getElementById('resultPerPeriod').innerText = ratePerPeriodPercent.toFixed(4) + "%";
document.getElementById('resultDifference').innerText = difference.toFixed(4) + "%";
document.getElementById('resultsArea').style.display = 'block';
}
How to Calculate Stated Rate from Effective Rate
In finance and mathematics, understanding the distinction between the Stated Rate (often called the Nominal Rate) and the Effective Annual Rate (EAR) is crucial for accurate comparison of financial products, investments, or compounding growth scenarios. While the Effective Rate tells you the actual yield realized over a year, the Stated Rate is the raw percentage often quoted in contracts before compounding is taken into account.
This calculator allows you to reverse-engineer the Stated Rate if you already know the Effective Rate and the frequency at which compounding occurs.
The Core Difference
The Stated Rate does not account for intra-year compounding. It is simply the periodic rate multiplied by the number of periods in a year.
The Effective Rate, however, accounts for the effect of interest earning interest (or growth compounding on growth) throughout the year. As the compounding frequency increases (e.g., from annual to monthly to daily), the Stated Rate required to achieve a specific Effective Rate decreases.
Mathematical Formula
To find the Stated Rate ($r$) from a known Effective Annual Rate ($i_{eff}$), we rearrange the standard compounding formula. The mathematical relationship is:
r = n × [ ( 1 + ieff )1/n – 1 ]
Where:
r = The Stated (Nominal) Annual Rate (in decimal form)
ieff = The Effective Annual Rate (in decimal form)
n = The number of compounding periods per year
Calculation Example
Let's say you want an investment to yield an effective return of 10% per year, and the account compounds interest monthly ($n=12$). What Stated Rate must the bank advertise?
Calculate the inner bracket: $1.10^{0.08333} \approx 1.007974$.
Subtract 1: $1.007974 – 1 = 0.007974$ (This is the monthly rate).
Multiply by $n$: $0.007974 \times 12 = 0.095689$.
Convert to percentage: $9.5689\%$.
Therefore, a Stated Rate of roughly 9.57% compounded monthly results in an Effective Rate of 10%.
Why is the Stated Rate Lower?
You will notice that whenever compounding occurs more than once a year ($n > 1$), the Stated Rate is always lower than the Effective Rate. This is because the compounding action "boosts" the stated rate to reach the higher effective yield. The more frequent the compounding, the lower the Stated Rate needs to be to hit the same Effective target.
Applications
This calculation is frequently used in:
Bank Loans: Checking the nominal APR when the effective APY is known.
Bond Yields: Comparing semi-annual coupon bonds with annual yield investments.
Physics & Biology: Calculating base growth rates in population dynamics where growth is continuous or periodic but measured annually.