Ralph Vince - Portfolio Mathematics

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

Probability Distributions 51

FIGURE 2.4 Kurtosis

Finally, thefourth momentof a distribution,kurtosis(see Figure 2.4),

measures the peakedness or flatness of a distribution (relative to the Nor-

mal Distribution). Like skewness, it is a nondimensional quantity. A curve

less peaked than the Normal is said to beplatykurtic(kurtosis will be neg-

ative), and a curve more peaked than the Normal is calledleptokurtic(kur-

tosis will be positive). When the peak of the curve resembles the Normal

Distribution curve, kurtosis equals zero, and we call this type of peak on a

distributionmesokurtic.

Like the preceding moments, kurtosis has more than one measure. The

two most common are:

K=Q/P (2.12)

where: K=The kurtosis.

Q=The semi-interquartile range.

P=The 10–90 percentile range.

K=

(

1 /N

∑N

i= 1

((Xi−A)/D)^4

)

− 3 (2.13)

where: K=The kurtosis.

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.