Two-step Kernel Estimation Of Expected Shortfall For α-Mixing Time Series

Risk management play a very important role in the financial,economical and insurance area.We must measure the risk before we do the risk management,so choicing the fight tool for risk measure is very important.The tools which have been used or proposed mainly are Standard Deviation,Absolute Deviation,Value at Risk(VaR),Conditional Value at Risk(CVaR),Worst Conditional Expectation(WCE),Expected Shortfall(ES).VaR has been widely accepted as a tool for financial risk management.Financial regulators all over the world have adopted VaR for risk management in investment and securities companies,also in banks and financial institutions.VaR is simply a quantile of the loss distribution over a prescribed holding period.But researches show that VaR has some defects. Artzner et al(1997)showed that VaR is not sub-additive,which is part of the necessary requirements to be a coherent measure of risk.A coherent risk measure satisfy a set of four desirable properties:monotonicity,sub-additivity,positively homogeneity and translation invariance.VaR is not sub-additive expresses the idea that the total risk on a portfolio is greater than the sum of the individual risks,so it damage the risk diversification principle of the portfolio.Moreover,VaR tells us nothing about the potential size of the loss that exceeds it,so the information from VaR may mislead investors.ES was proposed and discussed by Acerbi et al.(2001)after the concept of VaR was proposed.ES is the conditional expectation knowing that the loss is above VaR.Acerbi et al.(2002a,2002b)showed that ES was a coherent risk measure.ES is superior than VaR.First,ES is sub-additive,whereas VaR is not.Second,VaR tells us nothing about the potential size of the loss that exceeds it,but ES does.Moreover,ES is easy to compute and estimate.The use of ES rather than VaR in risk measurement has been recently advocated by scholars.Nowadays more and more scholars do the research of ES because of its better properties.However,there is very little research on the estimation of ES,particularly in the area of nonparametric estimation.In 2004,Scaillet presented nonparametric estimation and sensitivity analysis of ES.In 2005,Scaillet presented a two-step estimator of ES. Moreover,there is much work about ES in Chen(2006).Chen and Tang(2005)establish the Bahadur representation of VaR estimation in senses of convergence in probability and provide the strong consistence in the context ofα-mixing coefficient which is geometric decay.Using the results of VaR in Chen and Tang(2005),Chen(2006)establish the Bahadur representation of ES estimation in senses of convergence in probability,also provide the uniformly asymptotic normality but don't give the convergence rates.In this paper,we do the research in the context of a stationary process satisfying poly-nomial decay coefficients of strong mixing conditions.This mixing coefficients is weaker than the geometric rate of convergence which is required in most of theoretical investigations such as Chen(2006).We discuss the two-step kernel estimation of ES.The first step is the kernel estimation of VaR.We establish the Bahadur representation of VaR estimation in senses of almost sure convergence and prove the strong consistence of the kernel VaR estimator and its convergence rate,which improve the results of Chen and Tang(2005).The second step is the kernel estimation of ES.We establish the Bahadur representation of ES estimation in senses of almost sure convergence.Furthermore,we give the mean square error and prove the uniformly asymptotic normality and convergence rates of ES estimation, which improve the results of Chen(2006).These results can be used to construct asymptotic confidence intervals of ES and to carry out the hypothesis test regarding ES.Then we report results from a simulation study designed to evaluate the performance of the nonparametric ES estimation for three commonly used financial time-series models.We conclude that the kernel estimation of ES is more robust than the sample estimation of ES.Finally,we apply the proposed kernel estimation of ES to estimate the risk of two financial time series,we conclude that Shen zheng stock index is more risky than Shanghai stock index.Stationary is basic requirement in time series.A lot of research illustrates that a huge number of financial and economic time series are strong mixing series.Hence,the stationary and strong mixing conditions are very common in financial time series.The paper is organized as follows.In chapter 1,some new progress in financial risk measurement are introduced,theoretical models and nonparametric estimation of VaR and ES are given,Meanwhile,the main work about this paper are showed.In chapter 2,some basic assumptions are given,meanwhile,the main results in this paper and some remarks about the results are showed.In chapter 3,the Bahadur representation of the kernel estimation of ES is derived.The lemma about moment inequality and iterated Logarithm for strong mixing sequences are the work of predecessors.We prove the strong consistence of the kernel VaR estimation and its convergence rate.After that,we derive the Bahadur representation of the kernel estimation of ES using these lemmas.In chapter 4,the mean,the variance and the mean square error of the kernel estimation are derived.We prove our own results from lemma 4.1 to lemma 4.7 under the assumptions of this paper based on the work of predecessors.After that,we prove theorem 2.2 using these lemmas. In chapter 5,we prove the uniformly asymptotic normality and convergence rates of ES estimation by using the method of block.In chapter 6,numerical simulation.In this section we report results from a simulation study designed to evaluate the performance of the nonparametric ES estimation for three commonly used financial time-series models.We conclude that the kernel estimation of ES is more robust than the sample estimation of ES.In chapter 7,empirical studies on real financial return series.We conclude that Shen zheng stock index is more risky than Shanghai stock index.In chapter 8,we summarized the main results and this paper can be extended to other dependence conditions.After now,we may do some risk management applications.