Geometric mean

The average rate that, compounded, turns the starting value into the ending one: multiply one plus each period's return, take the root for the number of periods, and subtract one. It is the true average growth of money over time, and the compound annual growth rate is the same calculation.

A share that rises 50% in one year and falls 50% the next has an arithmetic average return of 0%, yet ₹100 becomes ₹150 and then ₹75. Your average return is zero and you've lost a quarter of your money. Its geometric mean is minus 13.4% a year, and that's the figure that matches what happened to the money.

As volatility rises the gap between the two averages grows, roughly by half the variance, so volatile assets compound at less than their average return suggests. Use the arithmetic mean to estimate next year's expected return and the geometric mean to describe what a holding actually earned.

Money compounds geometrically.