Lesson 9 of 20 · 7 min

Why the average is a bad summary of a leveraged outcome

Averages add, but wealth multiplies, and leverage widens the gap between the two.

This lesson is arithmetic with a large consequence. There are two ways to average returns, they give different answers, and only one of them describes what happened to your money.

The arithmetic mean adds the returns and divides. The geometric mean multiplies the growth factors and takes the root. Returns of plus fifty percent and minus forty percent have an arithmetic mean of plus five percent, which sounds fine. Multiply them instead: 1.5 times 0.6 is 0.9, so the portfolio is down ten percent. The geometric mean is about minus 5.1 percent per period.

The gap between the two is called volatility drag, and it grows with the square of volatility. Wider swings make the gap larger, always in the same direction — the geometric mean is never above the arithmetic mean, and they are equal only when every period's return is identical.

Leverage widens the gap because it magnifies each period's return, and the drag scales with the square of that magnification. Double the swings and the drag roughly quadruples. This is the mechanism behind the decay of daily-reset leveraged products, and it operates on any leveraged portfolio, not only on packaged ones.

There is a further effect that is specific to borrowing against collateral: an average is computed over paths, and some paths end early. A path that crosses a maintenance threshold produces a forced sale, after which the position is permanently smaller and cannot participate fully in a recovery. Averaging outcomes that include truncated paths alongside intact ones produces a number that describes neither.

Key terms

Arithmetic mean: The sum of returns divided by their count. Describes a typical single period, not the outcome of a sequence.
Geometric mean: The constant rate that would produce the same final wealth. This is the one that describes what happened to the money.
Volatility drag: The gap between arithmetic and geometric means. Grows with the square of volatility, and therefore with leverage.

Worked example

Two averages, one portfolio

Two periods of returns: +50% then −40%, applied to $100,000.

Arithmetic mean: +5.0%
After period one: $150,000
After period two: $90,000
Actual total return: −10.0%
Geometric mean: −5.1% per period
Gap between the two averages: 10.1 points per period

The arithmetic mean said plus five percent while the account lost ten. The wider the swings, the larger this gap becomes.

Check yourself

A portfolio returns +50% then −40%. The arithmetic mean is +5%. What actually happened, and why does leverage make the gap worse?

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Educational only. Nothing in this lesson is investment, legal or tax advice, or a recommendation to enter into any transaction. Figures are illustrative and computed from the inputs stated in the lesson.