This study proposes a new uncertain risk index model with background risk and presents its deterministic equivalents. The security returns and background asset returns are assumed as uncertain variables and estimated by experts. To discuss the influence of background risk on investment decisions, we compare the proposed model with a variant without background risk and find that the portfolio with background risk produces an equal or lower return than the one without background risk. The effects of changes in the standard deviation of background asset and the risk-free interest rate on optimal expected value are discussed. Two different risk measures for portfolio optimization model with background risk are compared, viz., the risk index model with background risk is further compared with the mean chance model with background risk. The nonlinear risk index model is solved by using a genetic algorithm. The efficiency of the genetic algorithm and the applications of the proposed models are illustrated through numerical experiments.

这项研究提出了一个新的具有背景风险的不确定风险指数模型，并提出了其确定性等价物。证券收益和背景资产收益被假定为不确定变量，并由专家估算。为了讨论背景风险对投资决策的影响，我们将提出的模型与无背景风险的变体进行了比较，发现具有背景风险的投资组合的收益等于或低于无背景风险的投资收益。讨论了背景资产标准差和无风险利率的变化对最优期望值的影响。比较了具有背景风险的投资组合优化模型的两种不同风险度量，即具有背景风险的风险指数模型与具有背景风险的平均机会模型的比较。非线性风险指数模型通过遗传算法求解。通过数值实验说明了遗传算法的有效性和所提出模型的应用。