## Geometric vs. Arithmetic Mean

The geometric mean is a type of mean that indicates the central tendency of a set of numbers by using their product as opposed to their sum which refers to the arithmetic mean.

For n numbers the geometric mean is equal to (x1 * x2 * … * xn) ^ 1/n whereas the arithmetic mean is equal to (x1 + x2 + … + xn) / n.

Many times investment managers when presenting performance metrics they utilize arithmetic means which is a practice that has been debated over time. The reason is that the arithmetic mean downplays the impact of a bad year in the total investment and does not accurately reflect the growth of the investment from initiation and throughout the measurement period.

For example, if someone invested \$1 in a fund at its initiation (let’s assume 5-years ago) and the fund had the following annual returns:

The arithmetic mean implies a positive return whereas the initial investment has actually been wiped out.

Of course other measures such as Sharpe ratio would have given an indication for bad performance and the volatility of the returns would practically have been very high, but still this extreme example highlights the fact that the arithmetic mean can sometimes be fairly deceiving.