Return-risk model
The mean–variance model is the most classic return-risk model. In the mean–variance portfolio model, Markowitz [
1] realizes the quantification of the return and risk of the asset portfolio and systematically investigates the relationship between return and risk for the first time. In this model, it is assumed that investors not only hope that their investment behavior can obtain higher returns but also hope that the risks they face are as small as possible. To meet the needs of investors for risk and return as much as possible, Markowitz uses the mean value of the return on risky assets as the expected return rate of the investment portfolio and uses the variance of the return on each risky asset as a measure of the risk of the investment portfolio. The mean–variance model can be expressed as two goals. One is to achieve the minimum risk under a certain expected rate of return constraint, and the other is to maximize the rate of return given the maximum risk that can be tolerated. This classic theory of Markowitz grasped the quantification of portfolio research for the first time, and this significant contribution laid the foundation for the development of portfolio theory and model research. Brauneis and Mestel [
20] applied Markowitz’ mean–variance framework to evaluate the return risk of cryptocurrency portfolios. Dybvig and Pezzo [
21]studied the mean–variance portfolio rebalancing problem of transaction costs. Pandolfo et al. [
22] used the concept of a weighted Lp depth function to obtain a robust estimate of the mean and covariance matrix of asset returns, which has the advantage of not being affected by parameter assumptions. Compared with traditional technology, it is less sensitive to changes in asset return distribution. Han and Wong [
23] studied the continuous-time mean–variance portfolio selection problem under the Volterra Heston model. Due to the non-Markov and non-semi-martingale properties of the model, the classic stochastic optimal control framework cannot be directly applied to related optimization problems. By constructing an auxiliary stochastic process, they obtained the optimal investment strategy dependent on the solution of the Riccati-Volterra equation. Bauder et al. [
24] solved the problem of optimal portfolio selection when the parameters of asset return distribution, such as the mean vector and covariance matrix, are unknown, and historical data of asset returns need to be used for estimation. Their new method uses a Bayesian posterior prediction distribution, that is, the future realization distribution of asset returns as an observable sample. Shen et al. [
25] studied the issue of asset liability management under the mean–variance criterion with institutional transformation. Among them, the dynamics of assets and liabilities are described by the non-Markov regime transition model.
With the deepening of many scholars' research on investment portfolios, the understanding and definition of investment portfolio risk are also different from each other, which results in different methods of measuring risk in risky assets. Different from using the variance of the return on assets to measure risk, some scholars believe that from the perspective of risk, investors will not pay too much attention to the excess return. Therefore, this part cannot be regarded as a risk, and only the loss faced by investors can be regarded as risk, that is, when the investor's return is lower than the expected return as the main risk factor. Therefore, Mao [
26] and Estrada [
27] established a mean–semivariance model, that is, removing the part whose return is higher than the expected return, and improved the mean–variance model. It is worth noting that this model is more suitable for situations where asset returns are asymmetric. If asset returns are centrally symmetrical, then the semi-variance is exactly half of the variance, which makes the model lose its optimization function. Konno [
28] also proposed the method of optimizing variance to measure risk, namely, to measure portfolio risk with absolute deviation and to replace quadratic programming with linear programming. This model not only retains the advantages of mean–variance but also improves its disadvantages. In addition, Yitzhaki [
29] proposed a new risk measurement method, the Gini mean difference, and on its basis, created the Gini mean portfolio model, carried out numerical analysis, and verified its feasibility. Garcia et al. [
30] extended the mean–semi-variance portfolio selection model to a multiobjective trust model, which considered not only risks and returns but also the price earnings ratio to measure the performance of the portfolio. They used the NSGA-II algorithm to solve the constructed multiobjective model. Chen et al. [
31] discussed the problem of multicycle portfolio selection, whose securities returns experts estimate to be timed. Taking security returns as an uncertain variable, they proposed a multiperiod mean–semi-variance portfolio optimization model considering transaction costs, cardinality and boundary constraints. The equivalent deterministic mean–semi-variance model is given on the premise that the security returns are sawtooth uncertain variables. Salah et al. [
32] used the nonparametric univariate method to obtain income prediction and applied it to the mean–semi-variance model.
The mean-CVaR model is another classic return-risk model. Researchers have made corresponding improvements to the narrow hypothesis of mean variance based on the actual situation of the financial market, added more realistic factors when constructing the portfolio model, such as transaction costs, background risks, liquidity, personal preferences, and transactions, and conducted more applicable research. Xia et al. [
33] studied the impact of transaction costs on the investment portfolio, studied the effective frontier of the portfolio model when the transaction costs were constant, linear and V-type functions and compared and analyzed the impact degree of different transaction cost functions on the investment portfolio model. However, it is worth noting that the introduction of transaction costs in the model will lead to a nonconvex minimization of the problem. Strub et al. [
34] investigated a discrete-time mean-CVaR portfolio selection problem. Wang and Chen [
35] investigated a mean-CVaR combinatorial optimization problem considering realistic constraints to maximize the mean return and minimize CVaR and proved that this was an NP-hard problem. To solve this complex problem, it is divided into asset selection and proportional distribution. Kang et al. [
36] studied the data-driven robust mean-CVaR portfolio selection problem under fuzzy distribution. Liu et al. [
37] studied closed-form optimal portfolios with unknown mean and variance of the closed-form optimal portfolios of the cubic mean-CVaR problem. Forsyth [
38] proposed the multiperiod, time-consistent mean-CVaR (conditional risk value) asset allocation problem in the form of easy numerical calculation. Setiawan [
39] studied the risk-averse mean-CVaR optimization problem and several biology-based heuristic algorithms. Kobayashi et al. [
40] studied the mean-risk portfolio optimization model by taking CVaR as a risk measurement, and this model used cardinality constraints to limit the amount of invested assets. Cui et al. [
41] deduced the time-consistent strategy of the multiphase mean conditional value-at-risk model and proposed the self-coordination strategy of the multiphase mean conditional value-at-risk model.
Expected utility maximization model
Based on the expected utility maximization model, some scholars have applied prospect theory to the problem of portfolio selection [
42]. Kahneman and Tversky [
43] showed that investment decisions are guided and influenced by psychological, emotional and behavioral factors. The behavioral framework links investment objectives with transaction behaviors. Behavioral portfolio theory emphasizes the role of behavioral preferences in portfolio selection and investor investment [
44]. Grishina et al. [
45] established the model of prospect theory–based portfolio optimization and designed the corresponding differential evolution algorithm and genetic algorithm. Gong et al. [
46] studied the portfolio selection problem under cumulative prospect theory and gave a portfolio optimization model. Scene generation technology is combined with a genetic algorithm to solve the model. In addition, an adaptive real-coded genetic algorithm (ARCGA) is proposed to find the optimal solution of the model. The calculation results show that this method solves the portfolio selection model, and the ARCGA algorithm is effective and stable. Liu et al. [
47] proposed a PBES (photovoltaic/battery energy storage/electric vehicle charging stations) portfolio optimization model based on the sustainability perspective. Kwak and Pirvu [
48] studied the problem of portfolio optimization over a period when investors with multiple risky assets and one risk-free asset had accumulated prospect theory (CPT). The returns of multiple risk assets follow the multiple generalized hyperbolic t element distribution, and the results of three-fund separation of two risk portfolios and risk-free assets were obtained. Gong et al. [
49], based on the coupling of a genetic algorithm and a bootstrap method, proposed a single-cycle portfolio optimization model solution method based on cumulative prospect theory. Calculation experiments show that by using this method, investors in CPT (cumulative prospect theory) display behavioral characteristics when facing a portfolio composed of risky assets. Deng et al. [
50] studied the multiperiod optimal investment portfolio considering transaction costs under cumulative prospect theory and analyzed the impact of transaction costs on the choice of optimal investment portfolio. Chang and Young [
51] used the additional information provided by behavioral stocks to propose a portfolio selection model to produce a profitable portfolio superior to the traditional investment benchmarks (market indexes, mutual funds and exchange-traded funds). Using the embedded holding period information of behavioral stocks, a strategy of buying and holding behavioral stocks is studied with the goal of maximizing the expected effect.
In summary, the theoretical research on stochastic uncertain portfolios has been thorough, and the research on this type of model is becoming increasingly mature. There are also some studies on portfolio optimization based on prospect theory, but research on the behavioral portfolio problem and its high-performance algorithm based on uncertain information is rare. For this reason, this paper studies the optimization problem of an uncertain return portfolio and its high-performance optimization algorithm based on prospect theory.