Possible Figures Of Merit Related To Return On Investment: The Arithmetic And Geometric Mean

When evaluating the performance of a trading system, I calculate the first moment (an arithmetic mean of the series of returns) as well as the second one (a variance of the series). Originally my “Sharpe-like” ratio, used to adjust the return for the risk, was a ratio of the first moment to the square root of the second. The series of returns would be composed of annualized returns calculated every month.

However there is a subtlety. If {r1,r2,r3,… rn} is a series of monthly returns for some period (ratios of capital at the end of the period to that at the start of the period), then the total return for the same period will be the product r1r2r3… rn.

When each monthly return is already annualized (exponentiated to the 12th power), the opposite operation (applying power 1/12 to the product) is required to get the actual return for the year. This operation is known as taking the geometric mean.

Arithmetic mean is always greater than or equal to the geometric mean. This fact is well known from university math courses.

For example, if the series of returns is 1.0, 1.081, 0.9, 1.02, then the geometric average is 0.998084 and the arithmetic average is 1.00025. In this particular example, the arithmetic average of returns will tell you that you are making money whereas in fact you are losing.

Arithmetic mean equals the geometric mean in the particular case when all elements of the series are equal. The more volatility there is in the series, the more difference between the two means can be expected.

The bottom line is that the results of the research so far should be revisited using the simple return and the Sharpe ratio based on such return instead of the first moment of the series of annualized returns. The first moment gives a biased estimate of the actual return.

Nevertheless, since the main driving logic in the choice of parameters was minimization of the drawdown, the basic conclusions regarding the parameter choices are likely to stay the same.