The average return you were quoted is not the return you earned

Up 50, down 50, and down 25 overall

Start with 100. Gain 50% and you have 150. Lose 50% and you have 75. The arithmetic average of the two years is zero, and you are down 25%. The compound annual rate is negative 13.4%, and that is the number that matches the balance.

Nothing about this example is exotic. Any sequence of returns that varies at all produces the same divergence, and the arithmetic average is always the higher of the two. It is not a lie so much as an answer to a question nobody asked: what would a typical single year have looked like, given that you were never invested for a single year.

The gap is about half the variance

The useful approximation is that the compound rate is the arithmetic rate minus half the variance of returns. At a 10% average with 15% annual volatility, that is about 1.1 percentage points of drag. At the same average with 30% volatility, it is about 4.5 points, which is most of the return.

This is why volatility is a cost rather than a personality test. A strategy that is more erratic for the same average year is a strategy that compounds more slowly, and no amount of composure changes the arithmetic. It is also why a leveraged product tracking a volatile index can lose money over a period in which the index finished flat: the daily reset multiplies the variance term without multiplying the average by as much.

The recovery asymmetry is the same fact wearing a hat

Down 20% needs 25% to get back. Down 50% needs 100%. Down 90% needs 900%. People treat this as a motivational poster about risk management, but it is the identical piece of arithmetic: losses and gains of equal size do not cancel because they are multiplicative, and the deeper the hole the more asymmetric the climb becomes.

The practical consequence is that maximum drawdown and compound return are not two independent facts about a track record. A record with a large drawdown has already paid for it in the compound number, whether or not the presentation shows you both.

• Ask whether a quoted return is the average of years or the growth of a dollar.
• If only the average is given, ask for the volatility and subtract half its square.
• Check the largest drawdown, because it is where the divergence was created.
• Compare strategies on compound return, never on average year.

Where the flattering number actually gets used

The best-known example on this site is the 12% claim: an arithmetic average of annual index returns close to 12% presented as what an investor would have earned, against a compound experience nearer 10% before fees. Both numbers are real. Only one of them buys anything.

Our position: quoting an arithmetic average of a volatile series without the compound figure beside it is a presentation choice, and it is always made in the same direction. When you see one number, assume it is the higher one and go find the other.