Fuzzy hidden Markov-switching portfolio selection with capital gain tax

•An adjusted fuzzy number is introduced and some properties are studied.•A fuzzy regime switching portfolio selection model is proposed.•The impact of capital gain tax on portfolio selection is studied.•A numerical integral-based particle swarm optimization algorithm is designed. A fuzzy portfolio selection model is considered with a view to incorporating ambiguity about model and data structure. The model features the uncertainty about the exit time of each risky asset within a pre-specified investment horizon and also the presence of transaction costs. However, departing from the traditional paradigm where the transaction costs are often assumed to be unrelated to holding periods, we introduce the capital gain tax of which the realized tax rate is decreasing with respect to the holding periods with a view to encouraging the long-term investment. Meanwhile, the regime switching property of the market state is introduced to fuzzy portfolio selection, where fuzzy random variables are employed to model uncertain returns of risky assets in a Markov-regime switching market. An adjusted L−R fuzzy number is introduced and some of its mathematical properties are studied. In addition, a bi-objective mean-variance model is formulated, and a time varying numerical integral-based particle swarm optimization algorithm (TVNIPSO) is designed to obtain the efficient frontier of the portfolio in the sense of Pareto dominance. Finally, some numerical experiments are provided to validate the effectiveness of the model and the TVNIPSO.