Optimal Portfolio Based On Genera L_∞ Risk Function And Its Empirical Study

Based on the risk dispersion principle of minimizing the maximum individual risk,in this thesis,we propose an axiomatic risk function and establish portfolio optimization models different from the classical mean-variance model.The optimal portfolios are analyzed analytically,structurally and empirically.It is found that the optimal portfolios are consistent with common sense,easy to understand and use,and have good out-of-sample performance.We list the main contents of this thesis as follows.(1)We define the general risk function based on the general deviation measure.There is some equivalence between the two when none of the risky assets is redundant.The general risk function allows the investor to separately measure the downside risk and upside risk.In addition,the general risk function has a linear structure,so that portfolio optimization models constructed with it can be solved analytically.Specifically,for a portfolio,we first measure the risk of random returns associated with each risky asset through a given general deviation measure,and then select the largest one as the risk of the portfolio.We call this method as using the general_∞risk function to measure the risk of a portfolio.This is a risk measurement method that differs from the method used by most portfolio optimization models to directly measure the total random return risk.(2)A bi-criteria portfolio optimization model which maximizes the expected return and minimizes the general_∞risk function is established.We not only provide explicit analytical formulas for all the efficient portfolios,but also explore the structure of the set composed by all effective portfolios such as its dimension and distribution.We generalize the classical Two-fund Theorem by providing some collections of finitely many efficient portfolios to generate the set of all efficient portfolios.We also notice that our efficient portfolios are almost the risk parity ones in the sense that the risks are allocated equally across the investments.In addition,in order to test the performance of some effective portfolios versus inefficient ones,Monte Carlo simulations are carried out under the assumption that the realized return rate of each risk asset falls into an interval based on its point estimate.(3)Under the background that the investor can invest in the risk-free asset,we generalize the classical One-fund Theorem and then the capital allocation line under the general_∞risk function can be obtained.Meanwhile,we prove that the expected excess return of any nonzero portfolio corresponding to the capital allocation line is positive,and lower bounds of the probability that its random excess return rate is positive for some deviation measure are given.Moreover,a conclusion,that can be compared with the mean-variance Capital Asset Pricing Model(CAPM),considered as CAPM-like relationship is derived.We also come up with the concept of master funds of three types,and derive them in an analytical way,in which two thresholds obtained by finite step calculation play a key role.The investor can make certain judgments on the trend of the capital market based on the type of the master fund.It is noted that,no matter what type of master fund the investor makes decisions based on,he/she will buy(short)assets with expected return rates higher(lower)than the risk-free rate,which is consistent with rational behavior.In addition,we also extend the"tangency"relation related to the classical One-fund Theorem through subgradient.(4)In order to test the out-of-sample performance of our master fund and the predictive power of the CAPM-like relationship,numerical experiments are conducted based on three real-world data sets.It can be seen that,in most cases,the investor’s judgment on the capital market trend based on the type of master funds is accurate and the out-of-sample performance of our master fund is superior than that of the global minimum variance portfolio and the equally-weighted portfolio.In particular,on average,the real gain rate of the master fund is slightly better than that of the benchmark index of the data set.Moreover,the predictions of return rate of the equally-weighted portfolio based on our CAPM-like relationship and those based on the classical mean-variance CAPM model are basically consistent,but the sum of squared errors of our model is slightly smaller.