Two Different Averages
The arithmetic mean simply adds returns and divides by the number of periods. The geometric mean accounts for compounding and shows what you actually earned. With returns of +50% and -50%, the arithmetic mean is 0%, but $100 becomes $150 then $75, a geometric mean of approximately -13.4%. The arithmetic mean always equals or exceeds the geometric mean, with the gap widening as volatility increases.
Why Arithmetic Means Mislead
When a fund advertises '12% average annual return,' it may be using the arithmetic mean. The geometric mean (CAGR) could be significantly lower. Returns of +40%, -20%, +30%, -10%, +20% produce a 12% arithmetic mean but only 10.4% geometric mean. Over 30 years, this 1.6% gap compounds into a 30%+ difference in final wealth. Always verify which average is being reported.
When to Use Each
Use the geometric mean (CAGR) for evaluating past performance. Use the arithmetic mean for estimating expected future returns over a single period. To project long-term growth from arithmetic mean expectations, convert using the approximation: geometric mean ≈ arithmetic mean - (variance / 2). This adjustment accounts for the drag that volatility imposes on compound growth.