Estimation And Application Of Value At Risk Based On Mixture Distributions

With the derivatives are flourishing and development of information technology, the transactions are conducted more quickly and more widely in financial markets and the volume is booming. Finacial markets in different districtions are connected more and more closely. If financial risk incident occurs, all the connected financial markets will be involved even cause financial crisis. Financial risk management attracts the attentions of financial institutions. After VaR(Value at risk) is introduced in Basel Capital Accord amendment to measure the risk of commercial banks’ capital. VaR is applied widely and become a common risk measure instrument.There are three kinds of method for VaR estimation. The first kind is parametric method such as GARCH type models. The second kind is semiparametric method such as extreme value theory. The last kind is nonparametric method such as historical simulation and Monte Carlo method. Almost all the exsiting models assume the distribution of return as a single distrubiton such as normal distribution or student-t distribution. The distribution of return is usually asymmetric and heavy tailed. A mixture of distributions can generate an asymmetric and heavy tailed distribution easily. This paper assumes the return distribution as a mixture of distributions, presents a nonparametric VaR estimation method based on mixture distribution and parametric NM-GARCH-POT model. In the former method, returns in different market states were assumed as generating from different sub distrbutions. The samples were classified by ADF test on 1-minute closing price series. Probability density function of sub distributions were estimated by kenel density estimation based on Bootstrap. At last the overall VaR were calculated by Monte Carlo method. Empirical results show that u-MD model and c-MD model can estimate VaR validly and VaR of short position better than GARCH type model. They are more robust when using different data. Parametric NM-GARCH-POT model assumed the innovation as a normal mixture with two sub distributions. Return series were modeled using NM-GARCH model to estimate the conditional mean and conditional variance. Then the standardized innovations were calculated and modeled by POT model and its quantiles were estimated. At last the VaR of return is calculated. Empirical results show that the NM-GARCH-POT model can model the tail of return distribution accurately and can estimate the VaR of both long and short positions validly.