Understanding how to break down an annual rate into smaller periods is crucial for analyzing growth metrics, interest accrual, or subscription models. Whether you are working with an Annual Percentage Rate (APR), an annual growth target, or a yearly depreciation rate, you often need to convert this figure into a daily, weekly, or monthly percentage.
This calculator allows you to convert a Nominal Annual Rate into a Periodic Rate and calculates the Effective Annual Rate (EAR) based on the compounding frequency.
function calculateRates() {
var annualRateInput = document.getElementById('nominalRate').value;
var frequencyInput = document.getElementById('periodFrequency').value;
var resultArea = document.getElementById('result-area');
// Validate inputs
if (annualRateInput === "" || isNaN(annualRateInput)) {
alert("Please enter a valid Nominal Annual Rate.");
return;
}
var annualRate = parseFloat(annualRateInput);
var frequency = parseInt(frequencyInput);
// 1. Calculate Periodic Rate (Nominal / Periods)
var periodicRate = annualRate / frequency;
// 2. Calculate Effective Annual Rate (EAR)
// Formula: (1 + i/n)^n – 1
var decimalRate = annualRate / 100;
var earDecimal = Math.pow((1 + (decimalRate / frequency)), frequency) – 1;
var earPercent = earDecimal * 100;
// 3. Compounding Factor
var compoundingFactor = Math.pow((1 + (decimalRate / frequency)), frequency);
// Get Period Name for display
var periodName = "";
switch(frequency) {
case 12: periodName = "Monthly"; break;
case 52: periodName = "Weekly"; break;
case 26: periodName = "Bi-Weekly"; break;
case 365: periodName = "Daily"; break;
case 4: periodName = "Quarterly"; break;
case 2: periodName = "Semi-Annually"; break;
default: periodName = "Custom Period";
}
// Display Results
document.getElementById('resPeriodName').innerHTML = periodName;
document.getElementById('resPeriodicRate').innerHTML = periodicRate.toFixed(6) + "%";
document.getElementById('resEAR').innerHTML = earPercent.toFixed(4) + "%";
document.getElementById('resFactor').innerHTML = compoundingFactor.toFixed(6);
resultArea.style.display = "block";
}
Understanding the Formulas
1. Periodic Rate Formula
The simplest calculation is finding the percentage rate per period. This assumes simple interest (non-compounding) division for the base rate.
Periodic Rate = Nominal Annual Rate / Number of Periods (n)
Example: If your annual rate is 12% and you want the monthly percentage:
Annual Rate = 12%
Periods (Monthly) = 12
Calculation: 12 / 12 = 1.0% per month
2. Effective Annual Rate (EAR) Formula
When an annual rate is compounded (applied multiple times per year), the actual percentage increase over the year is higher than the nominal rate. This is called the Effective Annual Rate.
EAR = (1 + (r / n))n – 1
Where:
r = Nominal Annual Rate (as a decimal, e.g., 0.12)
n = Number of compounding periods per year
Step-by-Step Calculation Guide
Example 1: Daily Percentage from Annual Rate
Scenario: You have an annual server uptime guarantee or an annual growth rate of 5%, and you want to know the daily equivalent.
Identify the Rate: 5%
Identify Periods: 365 (Days in a year)
Apply Division: 5 / 365 = 0.01369%
Result: The daily rate is approximately 0.0137%.
Example 2: Compounding Weekly Rate
Scenario: An investment offers a 10% nominal annual rate compounded weekly.
Periodic Rate: 10% / 52 weeks = 0.1923% per week.
Effect of Compounding: Using the EAR formula, the effective return is actually higher than 10% because the interest earned each week earns its own interest in subsequent weeks.
Calculation: (1 + 0.10/52)^52 – 1 = 10.506%
Frequently Asked Questions
Why is the Effective Annual Rate higher than the Nominal Rate?
The Effective Annual Rate accounts for compounding. If a rate is applied monthly, the interest gained in January generates its own interest in February, March, and so on. This "interest on interest" effect results in a higher total percentage by the end of the year.
Does this apply to depreciation?
Yes. If an asset depreciates by 20% annually, you can use similar logic to determine how much value it loses on a monthly basis, though depreciation formulas (straight-line vs. declining balance) may vary slightly in application.
What are standard "Period" counts?
Monthly: 12
Bi-weekly: 26 (every two weeks)
Weekly: 52
Daily: 365 (sometimes 360 is used in corporate finance)