2.6.5 Index VaR

Definition

Value at Risk (VaR) is a statistical technique used to measure and quantify the level of financial risk within a firm, portfolio or index over a specific time frame. VaR is calculated by assessing the amount of potential loss, the probability of occurrence for the amount of loss and the time frame. A 3% one-month VaR of 2% means that there is a 3% chance of the asset/portfolio/index declining in value by 2% during the on-month time frame.

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 Debt - 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:

where:

is the total return of the index at time t.
is inverse of the normal distribution for c (which is 1-, where  is the level of significance, here 0.5%)
is the volatility of the index at time t
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:

where:

is the total return of the index at time t.
is the inverse of the normal distribution for c (which is 1-, where  is the level of significance, here 0.5%)
is the modified z-score accounting for the non-normality in the returns distribution
is the skewness of the return distribution

is the excess kurtosis of the return distribution
is the volatility of the index at time t
is the value of the index at time

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