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 an