Bayesian Estimation Of Value At Risk And Related Risk Measurement

In risk management, risk measurement is very important. Value at risk (VaR) which is taken as the mainstream of the fmancial market risk measurement tool, is the expected losses of investors during certain period of time and under a certain confidence level. In normal market conditions, most VaR risk measurement method is very effective. But these traditional methods are only using the sample information, and ignore prior information of the risk and risk parameters. So, in this paper, based on bayesian framework, we discuss Bayesian estimation and linear bayesian estimation of VaR. as well as its statistical properties. However, when in extreme market conditions. VaR as a risk measurement tools does not meet the additivity, can not be regarded as a good estimate for the existing risk. Through improving VaR measure, the TVaR is derived which meet additivity In addition, the Bayesian estimation of TVaR are discussed. By doing simulation, the results show that the estimator of TVaR is better than that of VaR..This article focuses on the framework of Bayesian estimation of VaR and TVaR. On the one hand.under the appropriate loss function, we also discuss the statistical properties of Bayesian estimator of VaR and TVaR. On the other hand, the linear bayes estimation and statistical prop-erties of the VaR are considered. Finally, the corresponding numerical simulation are done to show the convergence speed of those estimator.After introduced relevant knowledge of preparation, the third chapter get the Bayesian esti-mate of VaR under Bayesian framework. However, there is no display expression in general case. hence we discuss estimator under exponential-gamma model and derive the Bayesian estimation of VaR. Through the numerical simulation, the results show that there is good efficiency for the estimator of VaR. Especially the sample size is very small, the mean square error is still very small. Furthermore, we derive the Bayesian estimator for VaR and TVaR under Pareto-exponential model. We also find that the effects of the estimate under this model are good enough to meet the needs of actual applications.The fourth chapter get the linear bayesian estimation of VaR by combining with the a prior information and sample information. The linear Bayesian estimator (Credibility estimator) of VaR is derived. In addition, the statistical properties of estimator are proved. Numerical simulation results show that the linear bayes estimation is better than the nonparametric estimator of VaR. And even in the case of small sample size, the estimation can be used effectively.In the fifth chapter of this paper.the paper are summarized, and the future research directions are suggested based this research.