Based on Extreme Value Theory asset allocation study
|Keywords||assets allocation extreme value theory extreme dependence copula function hybrid genetic algorithm|
In recent years, financial crisis occurred frequently, which has resulted in a great loss to numerous investors. It is a great task for financial enterprises and theory community to take precautions against and alleviate extreme financial risk caused by extremal market. Investment diversification is one of the most common strategies in financial risk management, and extreme value theory is an efficient tool for analyzing extreme events. So this dissertation commits itself to the research of the extreme value theory and its practice in assets allocation, and aim to construct an assets allocation that can alleviate extreme financial risk. In this dissertation, the extreme value theory and extreme dependence are studied intensively. As a result of those researches, an assets allocation model based extreme value theory are proposed. Finally, according to the data from representative share indexes of ripe and newly rising stock market, the article does empirical research for the portfolio selection model, and copare this model with M-V model. The result demonstrates that this model can alleviate extreme financial risk. The key points of this dissertation are listed as follows:1、The literature related to the assets allocation theory and extreme value theory is reviewed and special emphasis was put on the extreme value theory and its application. There are two risk measure models based on extreme value theory: static model and dynamic model. Static model presumes the data of financial return are independent and identically distributed; dynamic model presumes the data are autoregressive and heteroscedastic.2、Peak over threshold model(POT model) is an important branch of extreme value thoery and has became the most widely used model, because it effectively used the very little extremal data. Threshold selection is the key in application of the POT model. The usual methods of threshold selection are high subjective, which fix a threshold by examining the figure. An objective method based on Monte Carlo simulation is designed in this dissertation and empirical analysis is made to test its validity. The result demonstrates that the method based on Monte Carlo simulation can effectively divide the sample data and fix the right threshold.3、Value-at-Risk(VaR) is a commonly used tool to measure risk and usual estimated by historical simulation、Variance-Covariance method、Monte Carlo simulation、GARCH model and so on. In this paper risk measure models based on extreme value theory is compared with the the above-mentioned methods and an empirical analysis is made by using The Shanghai composite index and S & P 500. The empirical results show the static model and dynamic model do better than other methods, expecially at high confidence level.4、It is a premise for the risk measure of portfolio to map the correlation between financial asset returns rightly. The methods of correlation analysis is introduced such as linear correlation coefficient、rank correlation coefficient and tail correlation coefficient. Because copula function can map correlation between financial asset returns, this paper focus on the concept of copula、the method of estimation and so on, and analyze the relation between tail dependence and copula function. An empirical analysis is made to test tail dependence between representative stock indexes. The result show the stock indexes measured in this paper is asymptotically independent.5、Extreme risk caused by extreme market(for example plummeted stock market) is descibed by means of VaR and ES at high confidence level. In extreme market cases, investors commonly concern the amount of loss, so only at the condition of extreme risk, the portfolio that has the minimal risk is optimal in this paper. The article constructed a portfolio selection model based on extreme value theory, which measures the risk by VaR and RS, reflects the tail distributions by EVT and kernel estimation, reflects the dependence of financial assets returns by copula function. An empirical analysis is made using the representative stock indexes of ripe and newly Rising Stock Market.6、According to the fact VaR’s no convexity, which make the computation of the portfolio selection optimization based on VaR very complex and inaccurate, the article designed a hybrid genetic quantitative algorithm based on traditional genetic algorithms and pattern search method. Traditional genetic algorithms has capability of searching the optimal solution within global space, and pattern search method has capability within local space. Empirical result demonstrates the hybrid genetic algorithm has the reliability in the process of portfolio selection.7、The assets allocation model based on extreme value theory is compared with M-V model in terms of accumulated income、sharpe ratio and daily maximum loss. The empirical result demonstrates that M-V model is superior in accumulated income, but the assets allocation model based on extreme value theory is superior in aspect of risk premium per unit of risk and withstanding extreme risk,which shows the validity of the model in aspect of withstanding extreme risk.The main innovation of this work are listed as follows:1、The method to fix threshold is designed based on Monte Carlo.Peak over threshold model(POT model) is an important branch of extreme value thoery, which use all data that are extreme in the sense that they exceed a particular designated high level. Because it effectively used the very little extremal data, the POT model has became the most widely used model. It is crucial to select an appropriate threshold about the POT model. A threshold being too low is likely to violate the asymptotic basis of the model that leads to bias. A threshold being too high will generate few excesses with which the model can be estimated that leads to large variation. There are some drawbacks in the present methods of selecting threshold. So this paper designed a quantitative method by means of Monte Carlo and two samples k-s test. Empirical result shows this method can divide the data efficiently, fix an appropriate threshold, and Parameter estimation is relative stable in this method.2、A semi-parametric model is designed to estimate the distribution of assets loss.In financial risk management, it is a premise for risk measurement to assume the distribution of assets loss, especial tail distribution, rightly. The loss of assets often has leptokurtic distribution that is peaked near the center and has fat tails. In this paper, a semi-parametric model is designed, which estimate two tails using extreme value theory, and estimate the center using kernel estimation method. This method combines advantages of parametric and noparametric method, which has more applicability.3、An assets allocation model is constructed based on extreme value theory.In extreme market cases, investors commonly concern the amount of loss, so only at the condition of extreme risk, the portfolio that has the minimal risk is optimal in this paper. An assets allocation model is constructed based on extreme value theory and copula function, and an empirical study is made using major stock indexes in the world. This model is compared with M-V model, and the empirical result shows this model can minimize the risk of portfolio.4、A hybrid genetic algorithm is designed to solve the assets allocation model constructed in the paper.Because the distribution of assets loss often do not follow the normal distribution, VaR is discete、discontinuity and no convexity, and do not satisfy subadditivity. There are maybe some local optimum solution in the assets allocation using VaR as the risk measurement index, and traditional optimization algorithms maybe can not find the global optimum solution, which can minimize the risk based on VaR. So the article designed a hybrid genetic quantitative algorithm based on traditional genetic algorithms and pattern search method. The hybrid genetic algorithm combines advantages of traditional genetic algorithms and pattern search method. Empirical result demonstrates the hybrid genetic algorithm has the reliability in the process of portfolio selection.