Ralph Vince - Portfolio Mathematics

ch02 JWBK035-Vince February 12, 2007 6:50 Char Count= 0

50 HANDBOOK OF PORTFOLIO MATHEMATICS

FIGURE 2.3 Skewness alters location

In a symmetrical distribution the mean, median, and mode are all at

the same value. However, when a distribution has a nonzero value for

skewness, this changes as depicted in Figure 2.3. The relationship for a

skewed distribution (any distribution with a nonzero skewness) is:

Mean−Mode= 3 ∗(Mean−Median) (2.08)

As with the first two moments of a distribution, there are numerous

measures for skewness, which most frequently will give different answers.

These measures now follow:

S=(Mean−Mode)/Standard Deviation (2.09)

S=(3∗(Mean−Median))/Standard Deviation (2.10)

These last two equations, (2.09) and (2.10), are often referred to as

Pearson’s first and second coefficients of skewness, respectively. Skew-

ness is also commonly determined as:

S= 1 /N

∑N

i= 1

((Xi−A)/D)^3 (2.11)

where: S=The skewness.

N=The total number of data points.

Xi=The ith data point.

A=The arithmetic average of the data points.

D=The population standard deviation of the data points.