Determine the geometric mean growth rate over a set period.
Calculated Growth Rate
function calculateGrowthRate() {
var beginningValue = parseFloat(document.getElementById('beginningValue').value);
var endingValue = parseFloat(document.getElementById('endingValue').value);
var growthPeriods = parseFloat(document.getElementById('growthPeriods').value);
var errorDiv = document.getElementById('growthCalcError');
var resultDiv = document.getElementById('growthCalcResult');
var displayDiv = document.getElementById('growthRateDisplay');
var explanationDiv = document.getElementById('growthExplanation');
errorDiv.style.display = 'none';
resultDiv.style.display = 'none';
if (isNaN(beginningValue) || isNaN(endingValue) || isNaN(growthPeriods)) {
errorDiv.innerText = 'Please enter valid numbers for all fields.';
errorDiv.style.display = 'block';
return;
}
if (beginningValue <= 0) {
errorDiv.innerText = 'Beginning value must be greater than zero.';
errorDiv.style.display = 'block';
return;
}
if (growthPeriods <= 0) {
errorDiv.innerText = 'Number of periods must be greater than zero.';
errorDiv.style.display = 'block';
return;
}
// Formula: ((End / Begin)^(1 / n)) – 1
var growthRate = (Math.pow((endingValue / beginningValue), (1 / growthPeriods)) – 1);
var growthPercentage = growthRate * 100;
displayDiv.innerText = growthPercentage.toFixed(2) + '%';
explanationDiv.innerText = 'This represents a constant growth rate of ' + growthPercentage.toFixed(2) + '% per period required to grow from ' + beginningValue + ' to ' + endingValue + ' over ' + growthPeriods + ' periods.';
resultDiv.style.display = 'block';
}
Understanding the Constant Growth Rate
A Constant Growth Rate, often referred to in finance as the Compound Annual Growth Rate (CAGR) or the Geometric Growth Rate, measures the specific rate at which an investment, population, or business metric grows over a series of time periods, assuming the growth is compounded each period.
Unlike a simple average growth rate (arithmetic mean), the constant growth rate accounts for the effects of compounding. This makes it an essential tool for investors and analysts who need to normalize growth over varying timeframes to compare different assets or performance metrics accurately.
The Constant Growth Rate Formula
To calculate the constant growth rate manually, you use the following mathematical formula:
Ending Value: The final amount at the end of the observation period.
Beginning Value: The initial amount at the start of the period.
n: The number of periods (years, months, quarters, etc.).
Practical Example
Imagine a technology startup that had 1,000 active users at the end of Year 1. By the end of Year 4 (a 3-year growth period), they have 8,000 active users. To find the constant growth rate:
Divide the ending value by the beginning value: 8,000 / 1,000 = 8.
Raise that result to the power of 1 divided by the number of periods (3 years): 8 ^ (1/3) = 2.
Subtract 1: 2 – 1 = 1.
Multiply by 100 to get the percentage: 1 * 100 = 100%.
In this scenario, the startup experienced a constant growth rate of 100% per year.
Why Use Constant Growth Rate Instead of Average Growth?
The arithmetic average can be misleading when dealing with volatility. For example, if an investment grows by 50% in Year 1 and drops by 50% in Year 2, your average growth rate is 0%. However, you actually lost 25% of your total value. The constant growth rate (CAGR) would correctly reflect a negative growth rate (-13.4%), providing a more accurate picture of the realized performance.
Applications in Finance and Business
Dividend Discount Models: The Gordon Growth Model assumes a constant growth rate for dividends to determine the intrinsic value of a stock.
Revenue Forecasting: Businesses use historical constant growth rates to project future sales and budget resources.
Portfolio Comparison: Investors use CAGR to compare the performance of a volatile stock against a stable bond fund over a five-year horizon.