Fitting extreme gains and losses of the Prague Stock Exchange Index

Abstract

In this paper we focused on the daily log returns of investment in the Prague stock exchange index, PX-Index. Considering an investment trust that takes a „passive“ investment strategy and invests its assets in a specified stock-market index - the PX Index. We analysed data from January 1st, 1995 to February 20th, 2014. A popular model for stock market returns is that the log investment returns are independent and identically distributed (i.i.d.) normal random variables. We focused on the daily log returns and analysed the distribution of these returns. By means of the well-known Jarque-Bera test we reject the i.i.d. normal hypothesis of daily log returns. We emphasize this by looking at the data using graphical techniques, such as histogram and Q-Q plot. We can see that the data has fatter left and right-hand tails than the normal distribution. Conclusions of our basic analysis are that the daily log returns are leptokurtic and heavy tailed. They are not i.i.d. and volatility varies over time. Also we can say that extreme daily log returns appear in clusters. Further we investigated a simple model which incorporates stochastic volatility. We analysed volatility-standardised residuals using a GARCH approach. We can see that standardised residuals do not show any clusters of high and low volatility. Plotted standardised residuals also show that there are more exceedances of the lower threshold than the upper and that they are larger. International banking regulations require banks to pay specific attention to the probability of large losses over short periods of time. We were focusing on the tails of the standardised residual. We fitted tail data separately using a Pareto distribution. Estimated parameters of the Pareto distributions show us that the Pareto distribution gives a generally better fit over the tails than t and non-central t distribution.