Definition 4.1 (Skewness)
The skewness of a random variable with mean and variance
is defined as
If the skewness is negative (positive) the distribution is skewed
to the left (right). Normally distributed random variables have a
skewness of zero since the distribution is symmetrical around the
mean. Given a sample of i.i.d. variables
,
Skewness can be estimated by (see Section 3.4)
(4.2)
with
as defined in the previous
section.
Definition 4.2 (Kurtosis)
The kurtosis of a random variable with mean and variance
is defined as
Kurt
Normally distributed random variables have a kurtosis of 3.
Financial data often exhibits higher kurtosis values, indicating
that values close to the mean and extreme positive and negative
outliers appear more frequently than for normally distributed
random variables. Kurtosis can be estimated by
(4.3)
Example 4.1
The empirical standard deviation of monthly DAX data from 1979:1
to 2000:10 is
, which corresponds to a yearly
volatility of
. Later in
Section(6.3.4), we will explain the factor in
detail. The kurtosis of the data is much greater than 3 which
suggests a non-normality of the DAX returns.
SFEsumm.xpl