In my previous article, I spoke about how the mean was a moving target. Almost all popular financial formulations make the assumption of a mean that is normally distributed.

The normal distribution is a probability distribution with assumptions as shown in the chart above. Note that the illustration has been standardized so that the mean is 0. The important thing to know is that 68% of all possible values fall within 1 standard deviation of the mean, 95% of all possible values fall within 2 standard deviations, and over 99% of all possible values fall within 3 standard deviations. If this model was correct, there would be central tendency around the mean.

The long-term average return (the mean) is therefore a weighted average of all possible returns where the weights are the probabilities of the individual returns with the above assumptions.

The concept of an average (or mean) is intimately tied to the concept of probability. So when a probability distribution is presumed normal when in fact it is not, incorrect weights are used in the calculation of the mean.

Recall that returns are either non-normal intrinsically or because they are non-stationary. Even if adjustments were made for non-stationarity, probability is a long-term concept. In the short-run (your investment horizons fall in this category), actual returns will differ from their long-term averages.

Perhaps, this is what prompted Warren Buffet to once say "the correct investment horizon is forever"!

The normal distribution is a probability distribution with assumptions as shown in the chart above. Note that the illustration has been standardized so that the mean is 0. The important thing to know is that 68% of all possible values fall within 1 standard deviation of the mean, 95% of all possible values fall within 2 standard deviations, and over 99% of all possible values fall within 3 standard deviations. If this model was correct, there would be central tendency around the mean.

The long-term average return (the mean) is therefore a weighted average of all possible returns where the weights are the probabilities of the individual returns with the above assumptions.

The concept of an average (or mean) is intimately tied to the concept of probability. So when a probability distribution is presumed normal when in fact it is not, incorrect weights are used in the calculation of the mean.

*Models that use linear regression not only assume normality, they even assume that the standard deviations at different points in the relationship between the dependent and independent variables are the same!*Recall that returns are either non-normal intrinsically or because they are non-stationary. Even if adjustments were made for non-stationarity, probability is a long-term concept. In the short-run (your investment horizons fall in this category), actual returns will differ from their long-term averages.

Perhaps, this is what prompted Warren Buffet to once say "the correct investment horizon is forever"!

**