Research On Modeling And Backtesting Value-at-Risk And Expected Shortfall

In financial market,whether for the pricing of financial derivatives,portfolio se-lection,or for the purpose of risk management,the quantification of market risk has always been the interest of researchers and financial institutions.Researchers have been committed to building more optimized risk models and more effective backtesting of risk models.Consideration of the above two aspects of the actual situation,based on the related work during my doctoral period,we obtain the following results:(i)When the risk variables are modeled by the time series ARMA-GARCH model,we propose a two-step inference method by estimating the parameters via a weighted quasi-maximum likelihood and then fitting a parametric family to residuals from the first step.The strong assumption that when the model fit to financial data requires fourth moment to have a normal limit will be relaxed.By giving a weight to the quasi information matrix,we achieve moment reduction.Then,the optimization of the model is realized.(ii)In this paper,based on the improved ARMA-GARCH model,we use two-step model inference procedure to develop novel empirical likelihood backtesting the conditional Value-at-Risk(VaR)and conditional Expected Shortfall(ES),which has the advantages of simple calculation,less estimated parameters in the process,and more simplified weight equation.Through simulation studies and three groups of empirical analysis,the effectiveness of the backtesting is proved in the face of extreme thick tail of risk mutation.There are two important risk measurement indexes in financial market,which are defined as Value-at-Risk VaRx(?)=inf{x:P(X?x)??},and Expected Shortfall(?) where X is a risk variable or return.VaR includes all possible maximum losses of portfolio or financial derivatives in the future within a fixed confidence level and a certain target time range.ES is the average loss degree when the loss of financial instruments exceeds the VaR.It can be seen that ES defines the conditional expected loss when the loss exceeds VaR.Compared with VaR,ES can calculate the tail risk of extreme cases.Due to various limitations of VaR,the development of ES gets promoted.According to the Third Basel document on May 3,2012,the Basel Committee clearly put forward the idea of phasing out VaR and replacing it with ES.Because the risk is an observation series which changes with time,in order to detect the dynamic risk,we select ARMA-GARCH model to model the risk.When the actual financial risk variables are fitted by the time series model,the real-time information given by the prediction risk becomes natural and informative.Under this condition,the conditional risk measure is often used.For GARCH model,because the variance lag term is added to its variance influencing factors,it can well describe the aggregation effect of risk,so it is often used as a standard model to describe risk(or volatility),this paper considers the following ARM A(p,q)-GARCH(r,S)process,for a sequence of observed risks or returns {Xt}tn=1:(?) Here ?1?(?0,?1,…,?p,?1,…,?q)T and ?2?(?,a1,?,ar,b1,…,bs)T are param-eters in ?1?Rp+q+1and?2?[0,?)1+r+s,respectively,{?t}t=1nis a sequence of in-dependent and identically distributed random variables with zero mean,variance one,and distribution function F?(·;?3)with parameters ?3={?1,…,?qj)T in ?3(?)Rq1,and AT denotes the transpose of the vector or matrix A.Let Ft denote the ?-field generated by {?:s:?t}.Given the observations {X1,…Xn} and the initial values{X0,X-1,…},the parametric form of ARMA-GARCH model is(?) for t=1,…,n,where ?12=(?1T,?2T)T.Therefore,the one-step ahead conditional Value-at-Risk and conditional Expected Shortfall under ARMA-GARCH model are,respectively,defined as(?)However,when the risks change abruptly,conditional risk measurement may pro-vide inaccurate information or lose specificity.Therefore,to measure the effectiveness of risk measurement in system risk monitoring,backtesting is a very important method to detect the profitability and risk of a trading strategy.Standard model selection often makes ?t in ARMA-GARCH model adapt to a parameter distribution.Although the parameter estimation of the model plays an important role in deriving the asymptotic distribution of backtest,the existing methods generally assume that the parameter estimation is normal.But this assumption of normality is proved to be a big prob-lem,because when ?t has a the normal distribution,conditional maximum likelihood estimation is equivalent to quasi maximum likelihood estimation,which has a normal limit if and only if E?t4<? and E?t4<?.However,when ARMA-GARCH model is suitable for specific financial data,(?)often tends to 1,so the assumption of E?t4<? becomes unreasonable.One of our research results of is to solve the above problems,we relax the strong assumption of fourth moment requirement.We propose a two-step model inference procedure to ensure a normal limit without assuming E?t4<?.The two-step infer-ence method is different from the commonly used method of separately estimating the parameters in the model.By estimating fewer parameters,the calculation is convenient and the robustness of risk prediction is enhanced.It also lays a theoretical foundation for the hypothesis that the empirical likelihood backtesting requires fewer moment as-sumptions.Because the risk prediction depends on the distribution of ?t,we further propose adding information for the goodness-of-fit test into the model inference to add more information,which makes the backtesting more effective and persuasive.In addition to the theoretical optimization of the model,researchers and finan-cial institutions also focus on whether the established risk measurement model can accurately predict the risk.As a common statistical method of model validation,back-testing is used to check whether the actual loss is consistent with the predicted value.Backtesting is often based on the combination of different estimates.For example,recursion,scrolling,or fixed window is used to calculate the size of the violation-that is,the part of the loss that exceeds the predicted value of the risk.A failed backtesting means that either the model can not successfully describe the abnormal proportion of violations in the outer regions of the sample,such as when the out-of-sample period occurs in a recession,the financial data during this period tends to be extremely thick tailed.A backtesting of a failed result can be a good warning that our risk model needs to be revised,or the risk prediction model can not respond to worse scenarios.The other main result of our research is that in order to avoid the above men-tioned bias,we propose a more effective two sample empirical likelihood backtesting for conditional Value-at-Risk and conditional Expected Shortfall.We select the in-sample period observations to capture the normal economic time,and establish ARM A(p,q)-GARCH(1,1)model,We use empirical likelihood to replace the commonly used maximum likelihood estimation,because empirical likelihood has been proved to be better in both interval estimation and hypothesis test.Finally,by comparing with the results of Du and escanciano,we consider the external effective evaluation of the U.S.economic recession in history and the capitalization rate of intermediaries in the 2007-2009 financial crisis,we make an empirical analysis of our empirical likeli-hood backtesting based on ARMA-GARCH model.Our work analyzes and explains the universality,advantages and disadvantages of the new two-step empirical likeli-hood backtesting method.To sum up,this paper proposes a more effective empirical likelihood backtesting for conditional VaR and conditional ES by two-step method,and proves the importance of backtesting for the establishment of risk model and its forecasting effect on financial market.