Value at Risk and Expected Shortfall risk measures using Extreme Value Theory

Abstract

Calculating risk measures as Value at Risk (VaR) and Expected Shortfall (ES) has become popular for institutions and agents in financial markets. A main drawback with these risk measures is that they traditionally assume a specific distribution, as the Normal distribution or the Student’s t distribution. When using Extreme Value Theory (EVT) no assumption of the underlying distribution is necessary as the extreme tails can approximately be described by the Generalized Pareto Distribution. How can EVT be used to calculate VaR and ES for a market index?

The purpose of this study is to calculate VaR and ES risk measures for 10 market indices. The indices are the Stockholm stock exchange index (OMX30S), the Copenhagen stock exchange (OMXC20), the Helsinki stock exchange (OMXH25), the Deutscher Aktienindex (DAX), the Financial Times Stock Exchange (FTSE-100), the Dow Jones Industrial index (DJI), the Standard and Poor's 500 index (SPX), the NASDAQ-100 index (NDX), the Nikkei-225 stock average index (NKY) and the Bombay stock exchange sensitive index (SENSEX). The purpose is also to find which of these indices are exposed to most extreme losses.

Historical data consisting of daily closing prices were collected from Bloomberg for 10 market indices. These data were then processed in Matlab 7.7.0 (R2008b) using Extreme Value Theory to find VaR and ES risk measures. The risk measures were compared to find out which of the indices was exposed to most extreme loss.

This study has examined the VaR and ES risk measures on 10 market indices. The results show that in terms of VaR and ES, NASDAQ is most exposed to extreme losses. VaR and ES equals 5.340% and 7.002% respectively for the left tail and 5.128% and 7.091% respectively for the right tail.