Numerous empirical studies show that portfolio returns are generally asymmetric. In this paper, skewness is considered to measure the asymmetry of portfolio returns and a mean-risk-skewness model for portfolio selection will be proposed in uncertain environment. Here, the returns of the securities are regarded as uncertain variables which are estimated by experienced experts instead of historical data. Furthermore, the corresponding variations and crisp forms of the model are considered. To solve the proposed optimization models, a hybrid intelligent algorithm is designed. Finally, the feasibility and necessity of the hybrid intelligent algorithm and the application of the proposed models are illustrated by two numerical examples.

In the face of complex financial phenomena, describing the high uncertainties in financial markets remains a challenging issue in modelling and decision making. In this paper, chance theory is used to analyse the hybrid uncertainty that combines random returns and uncertain returns. We regard the total return as an uncertain random variable and study the uncertain random portfolio optimization problem. We first define the skewness of an uncertain random variable and derive some important properties and explicit expressions with deterministic distributions. These theoretical results help transform the model into a deterministic form. Then, uncertain random mean–variance–skewness portfolio optimization models and their corresponding equivalents are established to meet the diverse needs of investors. Finally, we provide two numerical experiments to illustrate the applicability of the proposed model, one of which demonstrates that the model can be applied in the area of energy finance. It is shown that when uncertain asset returns are asymmetric, this model can be used to make investment decisions for investors. We also find that a higher return is accompanied by higher risk and higher skewness.

This paper discusses an uncertain portfolio selection problem with consideration of background risk and asset liquidity. In addition, the transaction costs are also considered. The security returns, background asset return, and asset liquidity are estimated by experienced experts instead of historical data. Regarding them as uncertain variables, a mean‐risk model with background risk, liquidity, and transaction costs is proposed for portfolio selection and the crisp forms of the model are provided when security returns obey different uncertainty distributions. Moreover, for better understanding of the impact of background risk and liquidity on portfolio selection, some important theorems are proved. Finally, numerical experiments are presented to illustrate the modeling idea.

The claim process in an insurance risk model with uncertainty is traditionally described by an uncertain renewal reward process. However, the claim process actually includes two processes, which are called the report process and the payment process, respectively. An alternative way is to describe the claim process by an uncertain alternating renewal reward process. Therefore, this paper proposes an insurance risk model under uncertain measure in which the claim process is supposed to be an alternating renewal reward process and the premium process is regarded as a renewal reward process. Then, the paper also gives the inverse uncertainty distribution of the insurance risk process. The expression of ruin index and the uncertainty distribution of the ruin time are derived which both have explicit expressions based on given uncertainty distributions. Finally, several examples are provided to illustrate the modeling ideas.