4.3 Skewness and Kurtosis

Definition 4.1 (Skewness)  
The skewness of a random variable $ X$ with mean $ \mu$ and variance $ \sigma^2$ is defined as

$\displaystyle S(X) = \frac{\mathop{\text{\rm\sf E}}[(X-\mu)^3]}{\sigma^3}.
$

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 $ X_1,\ldots,X_n$, Skewness can be estimated by (see Section 3.4)

$\displaystyle \hat{S}(X) = \frac{\frac{1}{n} \sum_{t=1}^n(X_t-\hat{\mu})^3}{\hat{\sigma}^3},$ (4.2)

with $ \hat{\mu}, \hat{\sigma}^2$ as defined in the previous section.

Definition 4.2 (Kurtosis)  
The kurtosis of a random variable $ X$ with mean $ \mu$ and variance $ \sigma^2$ is defined as

   Kurt$\displaystyle (X) = \frac{\mathop{\text{\rm\sf E}}[(X-\mu)^4]}{\sigma^4}.
$

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

$\displaystyle \widehat{\mathop{\text{\rm Kurt}}}(X) = \frac{\frac{1}{n} \sum_{t=1}^n(X_t-\hat{\mu})^4}{\hat{\sigma}^4}.$ (4.3)

Example 4.1   The empirical standard deviation of monthly DAX data from 1979:1 to 2000:10 is $ \hat{\sigma}=0.056$, which corresponds to a yearly volatility of $ \hat{\sigma}\cdot \sqrt{12}=0.195$. Later in Section(6.3.4), we will explain the factor $ \sqrt{12}$ in detail. The kurtosis of the data is much greater than 3 which suggests a non-normality of the DAX returns.
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