Estimating fat tails

One of the important properties of a normal distribution is that we could use mean and standard deviation, the first two moments, to fully define the whole distribution. For n returns of a security, its first four moments are defined in equation (1). The mean or average is defined as follows:

Its (sample) variance is defined by the following equation. The standard deviation, that is, σ, is the squared root of the variance:

The skewness defined by the following formula indicates whether the distribution is skewed to the left or to the right. For a symmetric distribution, its skewness is zero:

The kurtosis reflects the impact of extreme values because of its power of four. There are two types of definitions with and without minus three; refer to the following two equations. The reason behind the deduction of three in equation (4B), is that for a normal distribution, its kurtosis based on equation (4A) is three:

Some books distinguish these two equations by calling equation...