Arithmetic and Geometric Averages in Trading and Investing: Position Sizing and the Kelly Criterion

The arithmetic vs geometric averages can be difficult to grasp. Albert Einstein is famous for saying that compounding is the eighth wonder. But what if he is wrong? Perhaps multiplicative compounding is the most destructive force in the universe? The sequence of returns and the different alternative paths ultimately determine your geometrical average which might be far away from the arithmetic average. Don’t be fooled by the arithmetic average.

The arithmetic and geometric averages/means and returns differ in trading and investing because the arithmetic average is mainly a theoretical average, while the geometric average takes into account the sequence of returns (or paths) of an investment.

The arithmetic average might be positive, but you can still end up with losses – even ruin. The reason is the volatility tax. The multiplicative effects of compounding might leave you with losses you never recover from. Your sequence of returns is dependent on your position sizing. Thus, we end the article by explaining the Kelly Criterion which is all about finding the optimal position/betting sizing.

If your strategy has a positive expected average gain per trade, the end result still might be catastrophic. The reason is due to path/sequence and compound average growth rate (CAGR). CAGR (geometric return), mean, or average, is the correct measurement and is different from the arithmetic return.

What is the arithmetic average?

Let’s assume you have a strategy that returns the following sequence of ten trades measured in percentage: 11, 33, 6, -5, 7, 21, -19, -9, 29, and -24. If you add all the numbers and divide by the number of observations (10), you get 5%. This means that the average gain per trade was 5%. This is the arithmetic average.

But the problem with the arithmetic average is that it doesn’t indicate your compounded return on your trades and your end result.

The difference between your starting capital and how much you end up with is not the arithmetic return, but the CAGR or geometric average/geometric mean:

What is the geometric average?

Now, if you start with 100 000 and add or deduct the above ten trades, you end up with 139 092. This equals a geometric return (CAGR) of 3.35%, much lower than the arithmetic average of 5%.

The compound annual growth rate (CAGR) is the return on an investment over a certain period of time. This is why it differs from the arithmetic average. It takes into account the compounding from the start to the finish. This means that the arithmetic and geometric average are two completely different things.

Arithmetic averages vs geometric averages: Why do they differ?

Arithmetic averages and geometric averages differ because the arithmetic average is mainly a theoretical value while the geometrical average is what you get in real life based on the sequence of returns. Mark Spitznagel writes in his Safe Haven book: you get what you get, not what you expect.

If you roll the dice you might not get the theoretical average because of the sequence and order of the rolling of the dice. You might get 1, 4, and 3 when you roll, while the next sequence might be 2, 6, and 1. Depending on the expected gain on each number of the dice, the end result fluctuates wildly. We show this by an example further down.

In the example above we got a lower CAGR and geometric average than the arithmetic average. Why do we get worse results when using the geometric return? It’s because of the volatility tax:

What is the volatility tax?

Over my 25yrs in the game of quantitative investing, and drenched in equations and models, I’ve honed in on the only math that really matters: Minimize drawdowns during unfavorable times. The multiplicative nature of positive returns during good times does all the rest.

Wayne Himelsein on Twitter

The name volatility tax is taken from Mark Spitznagel’s book called Safe Haven – Investing For Financial Storms. Spitznagel defines the volatility tax like this:

It’s a tax extracted by the multiplicative dynamics of compounding, what I have dubbed the volatility tax.

The volatility tax is a tax well hidden in the arithmetic average. The geometric average is lower because you suffer drawdowns that are hard to recover from. If you have a 33% loss in one year, you need a 49% return to get back to even. If you lose 50%, you need to get 100% to recoup.