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 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:

VaR_{t} = (TR_{t} - Z_{c} \times \sigma_{t}) \times V_{t}

where:

TR_{t} is the total return of the index at time t.
Z_{c} is inverse of the normal distribution for c (which is 1-\alpha, where \alpha is the level of significance, here 0.5%)
\sigma_{t} is the volatility of the index at time t
V_{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:

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}


VaR_{t} = (TR_{t} - Z_{cf} \times \sigma_{t}) \times V_{t}

where:

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

K is the excess kurtosis of the return distribution 
\sigma_{t} is the volatility of the index at time t
V_{t} is the value of the index at time 

  • No labels