Portfolio Selection Model And Decision Making Under Background Risk Based On Uncertainty Theory

Portfolio selection studies the optimization of asset allocation for the purpose of dispersing risks and improving returns.Markowitz put forward the mean variance model,which laid the foundation of Modern Portfolio Theory.Then,a large number of scholars make a further study on the portfolio theory.Most of the existing researches are conducted when the security parameters are random variables.In fact,historical data on securities returns are often lacking or invalid in financial market,e.g.,IPO,the outbreak of the trade war between China and the United States,the occurrence of COVID-19,etc.At this time,security returns need to be estimated by experts according to experience and regarded as uncertain variables.In reality,investors not only face the financial risk,but also face the background risks.Therefore,this paper studies the portfolio problem under background risk based on uncertainty theory.The main research contents and innovative achievements are as follows:(1)Treating securities returns and background asset returns as uncertain variables,and using risk index as risk measurement,an uncertain mean-risk index portfolio selection model considering background risks is constructed,in which risk index is defined as the average loss lower than the risk-free interest rate.The results show that the expected return of the risk index model considering background risk is less than or equal to that of the risk index model without background risk.The increase of the standard deviation of the return on background assets leads to the decrease or constant of the optimal expected return.In addition,the expected return of mean-risk index model considering background risk is less than or equal to that of mean-chance model.(2)By introducing risk-free asset into the portfolio discussed above,and using the chance as the risk measurement,an uncertain mean-chance portfolio selection model considering risk-free asset under background risk is constructed.Then the solutions of the programming problems with different risk threshold are analyzed.In addition,the effects of the changes of mean and standard deviation of risky assets and background assets on investment decisions are discussed.The results show that when the risk threshold of investors is given,the optimal risky asset proportion increases with the increase of the mean of background assets,and the higher the proportion of background assets to wealth,the faster the optimal proportion of risky assets increases.The optimal risky asset proportion decreases with the increase of the standard deviation of background assets,and the higher the proportion of background assets to wealth,the faster the optimal proportion of risky assets decreases.When the distribution parameters of risky assets change,it has a similar effect on investment decisions.(3)An uncertain mean-variance utility portfolio selection model considering background risk is constructed by using the utility criterion.Then,the effect of the mean and standard deviation of the background assets on the investment decision is discussed when both securities returns and background asset returns follow normal uncertainty distributions.The results show that when the marginal substitution rate is given,the optimal risky investment proportion is the increasing function of the mean of background asset and the decreasing function of the standard deviation of background asset.When the utility function is a specific quadratic function,the optimal investment proportion of risky assets increases with the mean of background assets and is positively correlated with the initial proportion of background assets.The optimal investment proportion of risky assets decreases with the standard deviation of background assets and is negatively correlated with the initial proportion of background assets.(4)An uncertain loss aversion utility portfolio adjustment model considering background risk is constructed.Through the operation criterion of uncertainty theory,the clear equivalent form of securities return and background asset return obeying general uncertainty distribution and normal uncertainty distribution is given respectively.In addition,the sensitivity analysis of the loss aversion coefficient and the mean of background assets are carried out.The numerical results show that with the increase of the loss aversion coefficient,the investment proportion of investors allocated to risk-free assets becomes larger,and the investors' utility value becomes smaller.With the increase of the mean of background assets,the investment proportion of investors allocated to risky assets increases,and the investors'utility value increases.