Research On Distributionally Robust Portfolio Optimization Models With Shortfall Risk Measures

The utility-based shortfall risk(abbreviated as SR)measure is more effective than Con-ditional Value at Risk(CVa R)in quantifying the risk of tail losses,and it has the charac-teristics of random invariance,etc.In recent years,it has attracted more and more scholars'attention.In this paper,we use SR to measure the risk and study the SR based distri-butionally robust portfolio optimization model with unknown probability distributions of random variables.The key to distributionally robust optimization is the construction of dis-tribution sets.In this paper,we consider distributionally robust SR portfolios with moment distribution set and Kantorovich ball distribution set.The main elements are as follows:Firstly,since the moment information can be easily obtained from the historical in-formation of random variables,the moment information is used to construct a moment distribution set,and a distributionally robust SR portfolio optimization model is established under this distribution set.In order to solve this problem,methods such as moment theory and duality theory are used to transform the model into an equivalent form that is easy to solve.Finally,select actual stock data and use MATLAB to solve the moment distribution set SR portfolio model.The impact of different parameters,namely the loss risk level,the mean vector control parameter₁and the covariance matrix control parameter₂,on the model is tested.The validity of the model under moment distribution set is verified.Secondly,when the number of samples is not large enough and the only available infor-mation about random variables is empirical data,a Kantorovich spherical distribution set is constructed,and a distributionally robust SR portfolio optimization model is established under this distribution set.In order to solve this problem,the model is transformed into an easy-to-solve approximate model by using the definition of norm and the theory of minimax and so on,and the convergence of the approximation model is proved.Finally,the actual stock data is also selected,and use MATLAB to solve the Kantorovich ball distribution set SR portfolio model.The impact of different parameters,namely the loss risk level,Kan-torovich radius R and the change in sample size M,on the model is tested,which verified the effectiveness of the approximate solution model.