CAGR Explained: Formula, Example and Limits

CAGR, short for compound annual growth rate, is the single steady yearly rate that would take a value from where it started to where it ended. It is a way to compare growth over different periods using one comparable number. It tells you nothing about the bumps along the way, which is the thing most worth remembering about it.

The formula

Wikipedia gives the formula as the ending value divided by the starting value, raised to the power of one over the number of years, minus one.

CAGR = (ending value / starting value) ^ (1 / years) - 1

Multiply the result by 100 for a percentage. The same formula works for an investment, a company's sales, a population or a country's output. For quarterly or monthly data, Wikipedia says the exponent's top number changes to 4 or 12 and the bottom counts the number of those periods.

A worked example

Suppose a holding grows from ₹1,00,000 to ₹1,80,000 over 4 years. This example is made up for illustration.

• Divide the end by the start: 1,80,000 / 1,00,000 = 1.8.
• Raise it to 1/4 (the fourth root): 1.8 ^ 0.25 is about 1.1583.
• Subtract 1: 0.1583, or about 15.8% a year.

You can check it by compounding: 1,00,000 grows to about 1,15,830 after year one, about 1,34,170 after year two, about 1,55,410 after year three and 1,80,000 after year four. The Motley Fool's explainer shows the same check with a $50 stock that grows to $150 in five years, which gives about 24.6%.

CAGR is not the average of yearly returns

The simple average adds up each year's growth and divides by the number of years. The Motley Fool illustrates the gap: a $50 stock stays flat for four years and then gains 200% in year five. The average annual growth rate is 40%, but the CAGR is 24.6%, and only the second number describes what an investor in the stock ended up with.

A second, made-up example shows how the average misleads. A holding rises 50% in year one, then falls 50% in year two. The simple average is 0%. But 100 becomes 150, then 75, which is a loss. The CAGR is the square root of 0.75, minus 1, or about −13.4% a year.

What CAGR leaves out

• Volatility. The Motley Fool says a path of steep falls and rises can give the same CAGR as steady growth. The number reflects only the start and the end.
• Cash going in or out. The formula assumes one starting amount. The Motley Fool notes that added contributions would overstate growth, and withdrawals would understate it. A regular monthly investment needs a different measure, such as the internal rate of return. Excel's IRR function and its RRI function are related tools.
• Risk. CAGR measures growth only, not how risky the path was.
• The future. Wikipedia says forecasting by compounding a past CAGR forward carries error. It is a description of what happened, not a promise.

Choosing the dates

The start and end dates change the answer a lot. A period that begins just after a fall and ends just after a rise will look strong; the reverse will look weak. When someone quotes a CAGR, check the start date, the end date and whether the figure includes dividends or other income.

Where you see it

Wikipedia lists uses including mean returns of investment funds, comparing stock, bond and savings returns, business measures such as sales or market share, and economic growth like GDP. It is also common in market reports that say a market will grow at some CAGR to a future year; those figures are forecasts by whoever wrote the report.

Sources

• Compound annual growth rate, Wikipedia
• Compound Annual Growth Rate (CAGR), The Motley Fool