A novel probabilistic hesitant fuzzy portfolio selection model with value-at-risk and safety level of score

Purpose This paper aims to propose two portfolio selection models with hesitant value-at-risk (HVaR) – HVaR fuzzy portfolio selection model (HVaR-FPSM) and HVaR-score fuzzy portfolio selection model (HVaR-S-FPSM) – to help investors solve the problem that how bad a portfolio can be under probabilistic hesitant fuzzy environment. Design/methodology/approach It is strictly proved that the higher the probability threshold, the higher the HVaR in HVaR-S-FPSM. Numerical examples and a case study are used to illustrate the steps of building the proposed models and the importance of the HVaR and score constraint. In case study, the authors conduct a sensitivity analysis and compare the proposed models with decision-making models and hesitant fuzzy portfolio models. Findings The score constraint can make sure that the portfolio selected is profitable, but will not cause the HVaR to decrease dramatically. The investment proportions of stocks are mainly affected by their HVaRs, which is consistent with the fact that the stock having good performance is usually desirable in portfolio selection. The HVaR-S-FPSM can find portfolios with higher HVaR than each single stock and has little sacrifice of extreme returns. Originality/value This paper fulfills a need to construct portfolio selection models with HVaR under probabilistic hesitant fuzzy environment. As a downside risk, the HVaR is more consistent with investors’ intuitions about risks. Moreover, the score constraint makes sure that undesirable portfolios will not be selected.