Index VaR

One-year VaR is calculated at a 99.5% and a 95% confidence interval at each point in time from the mean of total index returns and Historical Volatility. Rolling five- and 10-year windows are used to compute the mean return and volatility, and the following two parametric approaches of computation are applied:

Gaussian VaR

This approach assumes a normal distribution of returns and computes VaR as follows:

$$//<![CDATA[ \begin{array}{l}\displaystyle VaR_{t} = (TR_{t} - Z_{c} \times \sigma_{t}) \times V_{t}\end{array} //]]>$$

where:

$//<![CDATA[ \begin{array}{l}TR_{t}\end{array} //]]>$ is the total return of the index at time t.
$//<![CDATA[ \begin{array}{l}Z_{c}\end{array} //]]>$ is inverse of the normal distribution for c (which is 1- $//<![CDATA[ \begin{array}{l}\alpha\end{array} //]]>$, where  $//<![CDATA[ \begin{array}{l}\alpha\end{array} //]]>$ is the level of significance, here 0.5%)
$//<![CDATA[ \begin{array}{l}\sigma_{t}\end{array} //]]>$ is the volatility of the index at time t
$//<![CDATA[ \begin{array}{l}V_{t}\end{array} //]]>$ is the value of the index at time 

Cornish-Fisher VaR

This approach is a modification of the Gaussian VaR and accounts for the skewness and excess kurtosis in the returns distribution:

$$//<![CDATA[ \begin{array}{l}\displaystyle Z_{cf} = Z_{c} + \frac {(Z_{c}^2-1)S} {6} + \frac {(Z_{c}^3 - 3Z_{c})K} {24} - \frac {(2Z_{c}^3 - 5Z_{c})S^2} {36}\end{array} //]]>$$

$$//<![CDATA[ \begin{array}{l}\displaystyle VaR_{t} = (TR_{t} - Z_{cf} \times \sigma_{t}) \times V_{t}\end{array} //]]>$$

where:

$//<![CDATA[ \begin{array}{l}TR_{t}\end{array} //]]>$ is the total return of the index at time t.
$//<![CDATA[ \begin{array}{l}Z_{c}\end{array} //]]>$ is the inverse of the normal distribution for c (which is 1- $//<![CDATA[ \begin{array}{l}\alpha\end{array} //]]>$, where  $//<![CDATA[ \begin{array}{l}\alpha\end{array} //]]>$ is the level of significance, here 0.5%)
$//<![CDATA[ \begin{array}{l}Z_{cf}\end{array} //]]>$ is the modified z-score accounting for the non-normality in the returns distribution
$//<![CDATA[ \begin{array}{l}S\end{array} //]]>$ is the skewness of the return distribution

$//<![CDATA[ \begin{array}{l}K\end{array} //]]>$ is the excess kurtosis of the return distribution 
$//<![CDATA[ \begin{array}{l}\sigma_{t}\end{array} //]]>$ is the volatility of the index at time t
$//<![CDATA[ \begin{array}{l}V_{t}\end{array} //]]>$ is the value of the index at time