352259.Rozga.Arneric_-_Full_paper.doc - DEPENDENCE BETWEEN...

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DEPENDENCE BETWEEN VOLATILITY PERSISTENCE, KURTOSIS AND DEGREES OF FREEDOM Ante Rozga, e-mail: [email protected] Faculty of Economics, University of Split, Croatia Josip Arnerić, e-mail: [email protected] Faculty of Economics, University of Split, Croatia ABSTRACT In this paper the dependence between volatility persistence, kurtosis and degrees of freedom from Student’s t-distribution will be presented in estimation alternative risk measures on simulated returns. As the most used measure of market risk is standard deviation of returns, i.e. volatility. However, based on volatility alternative risk measures can be estimated, for example Value-at-Risk (VaR). There are many methodologies for calculating VaR, but for simplicity they can be classified into parametric and nonparametric models. In category of parametric models the GARCH(p,q) model is used for modeling time-varying variance of returns. 1. INTRODUCTION It isn’t easy to estimate VaR when stochastic process which generates distribution of returns is not known. Unfortunately the assumption that the returns are independently and identically normally distributed is often unrealistic. Furthermore, empirical research about financial markets reveals following facts about financial time series: financial return distributions are leptokurtic, i.e. they have heavy and fat tails, equity returns are typically negatively skewed and squared return series shows significant autocorrelation, i.e. volatilities tend to cluster According to first two facts it is important to examine which probability density function capture heavy tails and asymmetry the best. According to the third fact it is important to correctly specify conditional mean and conditional variance equations from GARCH family models. Therefore, high kurtosis exists within financial time series of high frequencies (observed on daily or weekly basis). This confirms the fact that distribution of returns generated by GARCH(p,q) model is always leptokurtic, even when normality assumption is introduced. It is important to note that kurtosis is both a measure of peakdness and fat tails of the distribution. Hence, in this paper distributional properties of returns generated using GARCHH(1,1) model with high volatility persistence will be compared to distributional properties of returns generated by the same model but with low volatility persistence. The effect between differences in distributional properties on alternative risk measures will be also examined. 2. KURTOSIS OF GARCH(1,1) PROCESS If it is assumed GARCH(1,1) process: 1 2 1 t 1 2 1 t 1 0 2 t t 2 t t t t t 1 , 0 N . d . i . i u ; u r , (1) the second moment of innovation process t equals: 1 1 0 t 2 t 1 Var E , (2) while the fourth moment is given as: 1 Engle assumes multiplicative structure of innovation process.
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2 1 1 1 2 1 1 1 1 1 2 0 4 t 2 3 1 1 1 3 E
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