Based On The Risk Measure Theory Of Portfolio Optimization Study

First, we discuss several theories of risk measures respectively such as theories ofcoherent measures of risk, spectral measures of risk and distortion measures of risk.Within the framework of these theories, we discuss and compare standard deviation,mean absolute deviation, lower partial moment, Gini's difference, VaR, CVaR and soon. We conclude that CVaR is superior to other measures with respect to theoreticalproperties. Definitely, 1) CVaR satisfies the property of sub-additivity, hence acoherent risk measure. 2) CVaR is a risk measure consistent with second orderstochastic dominance (SSD). 3) CVaR is a spectral risk measure as well as distortionrisk measure, thus outperforming other measures despite its imperfect properties asthe case stands.Second, portfolio optimization models are discussed. We conclude that a goodmodel should have perfect theoretical properties, should be easily solved, extendedand implemented. Usually, the properties of risk measures determine the properties ofmodels. Therefore we should consider in the first place the portfolio optimizationmodel based on CVaR measure (CVaR model for short hereinafter). The reason is asfollows. 1) CVaR measure admits diversification effect. 2) The optimal portfolio bysolving CVaR model is also SSD efficient. 3) Usually, CVaR model is convex, thuseffectively avoiding the problem of multiple local extrema. 4) Under finite scenarios,CVaR model reduces to the problem of linear programming and hence being easilysolved and implemented. Also the model is suitable for solving the problem ofmassive portfolio optimization. 5) Since CVaR is a downside risk measure, CVaRmodel is suitable to the optimization of portfolio that contains derivatives such asoptions.Third, CVaR and CVaR model are discussed in more detail. We construct along-term CVaR model with short-term CVaR constraints. We make an empiricalsimulation of the model using scenarios generated by filtered historical simulationapproach and bootstrap approach. The model we construct has practical value to theinstitutional investors such as Fund Management Corporation. Through this model,fund managers can achieve long-term optimal tradeoff of portfolio's mean and CVaRwhile controlling the short-term risk. Consequently, the manager can reduce theshort-term risk such as redemption risk due to the market decline while not alteringthe long-term investment target. The introduction of short-term risk constraints,however, brings down the portfolio's expected long-term return compared to non-constraints. The decrease can be regarded as the cost for investor's controllingportfolio's short-term risk.Finally, we compare CVaR model and Mean-Variance model (MV model). Theformula of Mean-CVaR efficient frontier is developed under the assumption ofelliptical distribution. In this case, CVaR model gives the same optimal portfolio asthe MV model at the appropriate confidence level and expected return level. We thencompare the two models using Monte Carlo simulation method and historicalsimulation. We conclude that, under the assumption of elliptical distribution, thefrontiers produced by two models have minor difference. Yet the difference exists dueto the errors produced by procedure of discretization and linearization of CVaR model.When returns of securities are not elliptically distributed, there exist obviousdistinctions between the optimal portfolios produced by CVaR model and by MVmodel.