A Theoretical And Empirical Study On The CVaR Portfolio Optimization Models

As a core theory of the modern investment field, portfolio selection is dedicated to solving the under-mentioned problem:how should investors allocate their money to different assets so that they can gain the most or suffer the least? There, which risk measure we should use to determine the risk has always been controversial among scholars.As we all know, economic and financial globalization has reached to such a degree that any crisis that happens in a certain area will promptly spread to other parts of the world. A case in point is the subprime crisis. In addition, not only is the Chinese financial market still developing, but it is also gradually open to foreigners, which, absolutely, will intensify fluctuation. Therefore, finding a suitable measure is also urgent in practice.In this thesis, the newly born measure-CVaR-is introduced to determine portfolio risk and then construct portfolio optimization models, for it can correctly determine the loss investors face and can fully reflect the information contained in extreme financial events, and for it is a coherent and convex risk measure and can be solved and expanded easily, which makes CVaR superior to measures such as variance, semi-variance, MAD, absolute semi-deviation, LPM and VaR. This dissertation employs a combination of theoretical, empirical and comparative analysis method to elaborate the following three aspects:1) compare VaR and CVaR in the framework of coherent risk measure;2) systematically construct single-and multiple-period, stationary and dynamic CVaR models in order to satisfy the needs of different investors;3) compare CVaR model with MV model so that readers can clearly know the differences between the two optimization models. Detailed outline and conclusions are presented as follows.Firstly, because of the correlation between VaR and CVaR, following Chapter Ⅰ (Introduction) and Chapter Ⅱ (Review of the literature), Chapter Ⅲ is introduced to elaborate the definition and calculation method of VaR and CVaR and compare the two risk measures in the framework of coherent risk measure. We then conclude that CVaR excels VaR with regard to statistical properties and calculation methods. Exactly speaking,1) CVaR is a coherent and convex risk measure, but VaR is not sub-additive, thus not a coherent or convex measure;1) After constructing a continuous, differentiable and convex auxiliary function, we can get portfolio’s VaR and CVaR value at the same time. In this sense, not only do CVaR models own excellent theoretical properties, but they can be easily implemented and expanded. Reasons are as follows.1) CVaR satisfies the diversification effect;2) CVaR models can be viewed as convex optimization, hence avoiding multiple extremum problems;3) Under finite scenarios, CVaR models can be transferred into linear optimization through discretization and linearization.Secondly, we introduce two chapters to fully elaborate single-and multiple-period, stationary and dynamic CVaR models and construct three relative models, all of which are made an empirical analysis. Chapter Ⅳ discusses the single-period CVaR model both without and with the transaction cost constraint and arrives at the under-mentioned conclusions.1) With the increase of the confidence level, CVaR value increases and the efficient frontier of the mean-CVaR model moves right;2) With the increase of the portfolio’s expected return, CVaR value increases too, reflecting the relationship between the return and risk;3) When portfolio’s expected return grows, investors will mainly allocate their money to those stocks with high expected return, proving the existence of the diversification effect;4) When transaction cost increases, CVaR value will grow, but the lower and upper limits of the efficient frontier of the mean-CVaR model will decrease, meaning that the efficient frontier will move bottom right. In Chapter Ⅴ, we study the CVaR model with short-run CVaR constraint and reach the following conclusions.1) The efficient frontier of the CVaR model with constraints locates to the right of the CVaR model without constraints;2) With the increase of the short-term CVaR value, the efficient frontier of the CVaR model with constraints will approach the model without constraints. And then, Chapter V also discusses the multiple-stage CVaR model and concludes that this model can effectively offset part systematical risk.Thirdly, Chapter Ⅵ compares mean-CVaR model with the basic MV model theoretically and empirically and concludes that under normal distribution assumptions, the two models are inter-changeable.Finally, Chapter VII summarizes the whole dissertation and presents some aspects for future study.Following is the innovative points of the thesis.1) This dissertation systematically constructs single-and multiple-period, stationary and dynamic CVaR models so that investors can choose the model satisfying their own needs;2) When comparing the mean-CVaR model and the MV model, we not only study the non-normal case but also employ the Monte Carlo simulation to generate scenarios and prove that they are equivalent under normal distribution assumptions;3) We make an empirical analysis on each of the CVaR models constructed and analyze the influence of confidence level, transaction cost, short-run CVaR constraints and investment stages on relative models.