A multi-period fuzzy portfolio optimization model with investors’ loss aversion

This paper considers the problem of how to construct the optimal multi-period portfolio for investors with loss aversion in fuzzy environment. Firstly, we regard the return rates of the risky assets as fuzzy numbers and use the value function in prospect theory to transform the return rate of a portfolio into perceived value, which can reflect investors’ loss aversion. Moreover, due to the fact that investors’ perception level toward risk may vary with the loss aversion degree, we propose a new risk measure based on the perceived value. Then, we formulate the objectives of maximizing the cumulative expected perceived value and minimizing the cumulative perceived risk and propose a multi-period portfolio selection model with diversification constraint. Furthermore, to solve the proposed model, we design a multiple particle swarm optimization algorithm with respect to its specific situation. Finally, using the data from real financial market, we construct a real case to illustrate the effectiveness of the model and algorithm. The results show that loss aversion has an important effect on investors’ investment decisions, and the proposed model could provide more reasonable strategies for investors with different loss aversion degrees.


Introduction
How to allocate wealth in multiple assets properly is one of the important problems in financial field.The mean-variance (MV) portfolio selection model proposed by Markowitz (1952) laid the foundation for the modern portfolio theory, where the return of each risky asset is assumed as random variable.Inspired by this pioneering model, many scholars have studied the portfolio problems by using the expected value and variance of portfolio to measure the investment return and risk, respectively, e.g., (Sharpe 1964;Merton 1972;Perold 1984;Best and Hlouskova 2000).Considering the deficiency of variance which views the volatility of high return as risk, Markowitz (1959) proposed the semi-variance, which measures the risk by only taking the volatility of return below the expected value into account.In addition, lower partial moment is also used for measuring the risk, many scholars regarded that this risk measure can characterize the level of investors' perceived risk more properly, such as Harlow and Rao (1989), Chow and Denning (1994) and Jarrow and Zhao (2006).
The above studies assumed the returns of risky assets are random variables and used the historical data to estimate their probability distribution.However, in the real financial market, there exists a lot of non-random factors such as expert opinions, political information and confidence level, which often occur with linguistic descriptions such as high risk, low degree and high level.In this situation, it is impossible for investors to get the precise value of the returns of risky assets which include lots of fuzzy uncertainty.As a powerful tool for describing the fuzzy uncertainty of the real event, the fuzzy set theory was originally proposed by Zadeh (1965).Inspired by the idea of the fuzzy set theory, numerous researchers have applied it to study the fuzzy portfolio selection problem, and they assumed the returns of risky assets are fuzzy numbers and constructed the corresponding optimization model.Assuming the return of each risky asset is trapezoidal fuzzy number, Deng and Li (2012) proposed a fuzzy portfolio selection model with borrowing constraint.Using the turnover rates of risky assets to measure the liquidity of the portfolio, Barak et al. (2013) proposed a fuzzy mean-variance-skewness portfolio selection model with liquidity constraint and cardinality constraint.Chen and Xu (2019) considered transaction costs, liquidity, buy-in thresholds and cardinality constraints, proposed a fuzzy mean-semi-variance-entropy portfolio selection model and designed a hybrid multi-objective bat algorithm to solve the proposed model.By incorporating fuzzy return rates and background risk, He and Lu (2021) proposed a fuzzy portfolio model with cardinality constraint and designed an improved quantum-behaved particle swarm optimization algorithm to solve the model.
In the real investment process, investors often need to adjust the wealth allocated in assets.Therefore, it is reasonable to study the portfolio selection problem in multi-period setting.Sadjadi et al. (2011) proposed a multi-period fuzzy portfolio selection optimization model considering different borrowing rate and lending rate.Using proportion entropy to measure the diversification of portfolio, Zhang et al. (2012) proposed a multi-period fuzzy mean-semi-variance-entropy portfolio selection optimization model with transaction costs.Zhang et al. (2014) proposed the concept of fuzzy semideviation to measure the risk of portfolio and constructed a multi-period fuzzy portfolio selection optimization model with diversification and boundary constraints.Considering the investment horizon may vary with different assets, Guo et al. (2016) proposed a multi-period fuzzy portfolio selection optimization model with different investment horizons and total risk constraint, where the objective is to maximize the final expected return of the portfolio.Mohebbi and Najafi (2018) used scenario tree to characterize the uncertainty of financial market, introduced the cardinality, liquidity and boundary constraints, and then proposed a multi-period fuzzy portfolio selection optimization model with transaction cost.Yang et al. (2021) used expected value and semi-deviation to measure the return and risk of the single asset, respectively, introduced a diversification constraint involving risk-free asset, and proposed a multi-period fuzzy portfolio selection optimization model with the objectives of maximizing the final expected wealth and minimizing the cumulative risk.
In practice, investors are usually bounded rational due to the cognitive factors such as sentiment, confidence level and risk attitude.Kahneman and Tversky (1979) used the value function to characterize decision makers' behavioral preferences such as loss aversion and reference dependence and proposed the prospect theory.Inspired by this pioneering work, numerous researchers introduced the prospect theory to the portfolio selection problem and studied the influences of investors' behavioral preferences in the investment decision.Using the linear value function to characterize investors' loss aversion preference, Fortin and Hlouskova (2011) proposed a portfolio selection optimization model with the objective of maximizing the loss aversion utility.Considering investors' subjective preferences in the mean-variance framework, Fulga (2016) proposed a loss aversion-based downside risk measure and constructed a portfolio selection optimization model with the objectives of maximizing the expected return and minimizing the downside risk.Song et al. (2017) considered investors' loss aversion preferences in the continuous-time investment and consumption process and proposed a continuous portfolio selection optimization model with maximizing the total discounted S-shaped utility from consumption.Wang et al. (2018) considered investors' loss aversion preferences in dynamic scenario, introduced the conditional value at risk (CVaR) constraint, and constructed a portfolio selection model with the objective of maximizing the loss aversion utility.Liu and Zhang (2021) considered the case that investors' loss aversion degrees are relevant to the excess return and proposed a multi-period fuzzy portfolio selection optimization model with the objective of maximizing the time vary loss aversion utility.
In this paper, we study a multi-period fuzzy portfolio selection problem with investors' loss aversion.Investors are often provided information with fuzzy uncertainty when making decisions in multi-period investment process.From this point of view, Zhang et al. (2014) proposed a multi-period fuzzy portfolio selection optimization model with boundary constraints, while Guo et al. (2016) proposed a multi-period fuzzy portfolio selection optimization model with different investment horizons.However, investors are often bounded rational in the uncertain investment process, so loss aversion has an important effect on decision making.Based on this, we introduce the prospect theory to reflect investors' loss aversion and transform the portfolio return rate into perceived value by the value function.Then, to well characterize the bounded rational feature of investors with loss aversion, we take the terminal cumulative expected perceived value as one of the model's objectives.In fuzzy environment, the popular downside risk measures are the semi-variance (see Zhang et al. 2012) and the semi-absolution deviation (see Yang et al. 2021).The limitation of the above risk measures is that they ignored the fact that investors' attitude toward risk varies with the loss aversion.Therefore, we propose an alternative risk measure method, namely perceived risk, which is based on the perceived value and is able to reflect investors' attitude toward risk.Furthermore, we take the terminal cumulative perceived risk as one of the model's objectives.To obtain the optimal portfolio in fuzzy portfolio selection problem, many studies used heuristic algorithm to solve the optimiza-tion model, such as Zhang et al. (2014), Guo et al. (2016), He and Lu (2021).Noting that our proposed model is a nonlinear programming problem with complex constraints, we incorporate the multi-swarm cooperative scheme and the chaos initialization into the heuristic algorithm and design a multiple particle swarm optimization to solve it.
We summarize the main contributions of this study as follows: • A multi-period fuzzy portfolio optimization model considering investors' loss aversion is proposed.• A new method is used to transform the portfolio return rate into perceived value by the value function in prospect theory.
• A risk measure is proposed to characterize investors' loss aversion based on the perceived value.• A multiple particle swarm optimization algorithm is designed to solve the proposed model.
The remainder of this paper is organized as follows.Section 2 introduces some preliminaries.Section 3 establishes a multi-period fuzzy portfolio optimization model considering investors' loss aversion.Section 4 designs a multiple particle swarm optimization algorithm to solve the proposed model.Section 5 conducts a real case study to illustrate the effectiveness of the proposed model and algorithm.Section 6 concludes this paper.

Preliminaries
In this section, we introduce some preliminaries, which will be used in the following discussion.

Fuzzy number
Definition 1 (Zadeh 1965).A fuzzy number Ã is said to be an L R-power fuzzy number if its membership function has the following form where α, β ≥ 0 represent the left and right spreads, respectively; L, R : [0, 1] → [0, 1] are the left and right functions, Lemma 1 (Dubois and Prade 1980).Let l be a real number, Ã1 = (a 1 , b 1 , α 1 , β 1 ) and Ã2 = (a 2 , b 2 , α 2 , β 2 ) are two trapezoidal fuzzy numbers.Then (3) Definition 2 (Carlsson and Fullér 2001).Let Ã be a fuzzy number with membership function μ Ã(•), γ ∈ (0, 1] be a real number.Define the γ −level set of Ã by Particularly, if Ã is an L R-power fuzzy number, its γ −level set is an interval, denote it by Definition 3 (Carlsson and Fullér 2001).Let Ã be a fuzzy number with γ −level set Define its possibilistic mean value and lower possibilistic semi-variance by and Definition 4 (Zadeh 1965).Let Ã be a fuzzy number with membership function μ Ã(•), φ : R → R be a function.Define a fuzzy number B and its membership function by

Prospect theory
Prospect theory, initially proposed by Kahneman and Tversky (1979), is a descriptive theory for decision makers' bounded rational behaviors under uncertainty.In this pioneering theory, there are three fundamental principles, which describe decision makers' behavioral preferences.The first one is reference dependence, which means that decision makers divide the wealth result into two areas according to a presetting wealth level, i.e., gain area and loss area, which reflect whether the wealth result is better or worse than the presetting wealth level.The second one is loss aversion, which means that decision makers put a higher value on the loss than on the gain when assessing the wealth result.The third one is diminishing sensitivity, which means that decision makers are risk-averse in the area of gain, while risk-seeking in the area of loss.Under the assumption that the decision maker is more focus on the relative wealth amount of gain or loss than the absolute amount of the final wealth, prospect theory can capture decision makers' behavioral preferences by the following value function where r represents the final wealth, r b is the parameter that represents the presetting wealth level, which reflects the decision makers' reference dependence, 0 < δ 1 , δ 2 < 1 are the parameters that represent the decision makers' decreasing sensitivity degree in gain area and loss area, respectively, θ > 1 is the parameter that represents the decision makers' loss aversion degree.A general case of the value function is shown in Fig. 1.
For the influence of loss aversion, different investors may have different risk attitudes toward the same uncertain return of a portfolio.Therefore, an adjustable risk measure should be developed to suit varying loss aversion.In order to incorporate the loss aversion of investors, we define the following perceived risk based on the perceived value.
Definition 5 Let Ã be a fuzzy number.Denote the perceived value of Ã by Ṽ = v( Ã), and its γ −level set

Modeling
In this section, we aim to construct a multi-period fuzzy portfolio selection optimization model with investors' loss aversion.

Problem description and notations
We consider a portfolio optimization problem with several risky assets.Assume that an investor with initial wealth W 0 gets into the financial market at the beginning of the first period horizon, he/she intends to invest the wealth among n risky assets and get the final wealth after T periods.He/she can adjust the proportion of wealth invested in each risky asset at the beginning of each period.We formulate the portfolio optimization model under fuzzy uncertainty, in which the return rate of each risky asset is regarded as trapezoidal fuzzy number.We consider investors' loss aversion, which means the investor usually presets the reference level of return rate according to his/her subjective judgment, and puts a higher value on the loss than on the gain.In fact, we only care whether portfolio selection is affected by his/her loss aversion under the fuzzy uncertainty environment or not.Specifically, we characterize investors' loss aversion through two aspects: one is transforming the portfolio return rate into the perceived value by the value function in prospect theory; the other is using the perceived risk measure to capture investors' attitude toward risk under different loss aversion degrees.
We introduce the following notations for convenience.rt,i : the fuzzy return rate of asset i during period t, which is represented by a trapezoidal fuzzy number, i.e., rt,i = (a t,i , b t,i , α t,i , β t,i ); x t,i : the investment proportion of asset i during period t; x t : the portfolio of period t, which is represented by x: the investment strategy for T periods, where c + : the transaction cost rate of purchasing risky asset; c − : the transaction cost rate of selling risky asset.Transaction costs, occurring as adjusting the portfolio, have a significant impact on the investment decision making.Investors need to pay transaction costs when purchasing or selling risky assets.Denote the adjustment of the purchasing and selling proportion of risky asset i at period t by , respectively.Then the amount of transaction cost paid at period t is Then the return rate of the portfolio at period t is Then the return rate of the portfolio at period t is represented by The expected wealth of the portfolio at the end of period t is and the terminal cumulative wealth of the investment strategy is The semi-variance of the portfolio at period t is and the terminal cumulative semi-variance of the investment strategy is

Objective function
Most fuzzy portfolio optimization models are proposed with the objectives of maximizing the terminal cumulative wealth CW T and minimizing the terminal cumulative semi-variance of the investment strategy C R T .The objectives of maximizing CW T and minimizing C R T can't reflect investors' loss aversion during the investment process.However, investors' portfolio decision making may be significantly affected by the loss aversion.Specifically, investors' psychological perceived value toward the portfolio may vary with different loss aversion degree, and then he/she may make different portfolio selection due to the varying satisfaction degree.Therefore, we consider the objectives of maximizing the terminal cumulative expected perceived value CEPV T and minimizing the terminal cumulative perceived risk CPR T to reflect the loss aversion of different investors, instead of taking the maximization of CW T and the minimization of C R T .Given a return rate Rt , we generate its expected perceived value by the following two steps.Firstly, let Ṽt = v( Rt ), which is called the perceived value of Rt .Since Rt is a fuzzy number, from Definition 4, the perceived value Ṽt is also a fuzzy number, its membership function can be represented by Then, from Definition 3 and the above membership function, we can calculate the expected value of the perceived value Ṽt and denote it by EPV t , which represents the expected perceived value of the portfolio at period t.We only discuss the most common cases while where the expected perceived value EPV t can be calculated as follows.
(1) If a t − α t ≤ r b < a t , we have 123 According to the above discussion, the terminal cumulative expected perceived value of the investment strategy is From Definition 5, we can calculate the perceived risk of the portfolio at period t as follows.
(1) If r b ∈ (a t − α t , a t ], we have Then, the terminal cumulative perceived risk of the investment strategy is We use an example to explain the relation between the loss aversion degree and investors' decision preferences.Consider two portfolios with trapezoidal fuzzy return rates R1 and R2 , where R1 = (0.2, 0.3, 0.3, 0.6) and R2 = (0.3, 0.4, 0.5, 0.1), as shown in Fig. 2.Then, set δ 1 = δ 2 = 0.88, r b = 0.20 and plot the expected perceived value and perceived risk of the two portfolios with different θ , respectively, as shown in Fig. 3.We can see from Fig. 3(a) that the expected perceived value of Portfolio 1 is higher than that of Portfolio 2 when θ < 1.75, while it is lower than that of Portfolio 2 when θ > 1.75, and from Fig. 3(b) that the perceived risk of Portfolio 1 is always lower than that of Portfolio 2.Then, we find that the investor prefers Portfolio 1 to Portfolio 2 when θ < 1.75, while he/she may have different selections when θ > 1.75.This indicates that investors' attitude toward portfolios varies with the loss aversion degree θ .
To sum up, we can characterize investors' decision preferences by the loss aversion degree θ .The investor with higher loss aversion degree gets lower perceived value and higher perceived risk from the fuzzy return rates of the portfolio.Therefore, by considering the objectives of maximizing the terminal cumulative expected perceived value CEPV T and minimizing the terminal cumulative perceived risk CPR T , investors can choose the appropriate parameter θ according Fig. 2 The membership functions of the two portfolios' fuzzy return rates to their loss aversion degree and generate the optimal investment strategy.

Realistic constraints
To control the non-systematic risk borrowed from the wealth over-concentration, investors are usually more willing to take the diversify portfolio than the centralize portfolio.In this study, we use the proportion entropy to measure the diversification of portfolios.Then, the diversification constraint at period t can be represented by We consider the case that short selling is not allowed.Then the boundary constraint can be represented by In addition, the budget constraint at period t can be represented by

Portfolio optimization model
Assume that the investor aims to maximize the terminal cumulative expected perceived value and to minimize the terminal cumulative perceived risk.Assume that short selling is not allowed in the whole investment process and the investor is self-financing.Meanwhile, assume the investor requires the portfolio to satisfy the diversification degree at Fig. 3 The expected perceived value and perceived risk of the two portfolios with different θ each period.Then, the multi-period portfolio optimization model can be established as Considering that P 1 is a bi-objective model, we transform it into the following single objective model by using the weighted max-min fuzzy programing method proposed by Lin (2004).
where CEPV + and CEPV − are the ideal solution and the antiideal solution of the cumulative expected perceived value, respectively; CPR + and CPR − are the ideal solution and the anti-ideal solution of the cumulative perceived risk, respectively; ω denotes the degree of importance of the objective CEPV T , while 1 − ω represents the degree of importance of the objective CPR T , where ω ∈ [0, 1].When ω = 1, the investor only concerns the objective CEPV T and ignores the objective CPR T ; when ω = 0, the investor only concerns the objective CPR T and ignores the objective CEPV T .We can obtain a Pareto-optimal solution of model P 1 by solving the above model P 2 under a certain value of ω.Hence, the Pareto-frontier of model P 1 can be obtained by solving model P 2 with ω ranging over [0, 1].

Multiple particle swarm optimization
This section aims to design a multiple particle swarm optimization (MPSO) to solve the following general form of model P 2 Particle swarm optimization (PSO), based on the idea of the collaborative process of the flock of birds searching for food, is a heuristic algorithm proposed by Kennedy and Eberhart (1995).In PSO, a swarm is formed by a group of particles, each particle searches for the optimal solution and exchanges the information with other particles in the search space.The global optimal position of the current swarm is obtained by comparing all of the particles, and the personal optimal position of each particle is represented by its historical optimal position.The above two positions provide the social learning information and self-learning information for each particle in the swarm to adjust its position in the iterative process.The social learning ability and self-learning ability can improve the swarm's global exploration ability and the particle's local exploitation ability, respectively.Since it is difficult to combine the two kinds of abilities, PSO may suffer some deficiencies, such as parameter sensitivity and premature convergence.Then, some researchers, such as Niu et al. (2007) and Ma et al. (2018), introduced multi-swarm cooperative scheme to improve the performance of PSO.
Next, we propose a multiple particle swarm optimization, in which the population is divided into a leader swarm and several subsidiary swarms.The self-learning rates and social learning rates vary with different swarms.When the particle is in the subsidiary swarm, it adjusts its position based on the optimal particle information of the subsidiary swarm, while each particle of the leader swarm adjusts its position based on the optimal particle information of the leader swarm and all subsidiary swarms.Through the above collaborative approach, the leader swarm can obtain additional reference information from the subsidiary swarms and thus reduce the possibility of converging to a local optimal solution.In addition, we introduce chaos initialization, which can help to expand search space and improve the population stability.

Encoding and decoding
) that represents a particle, where p t,i ∈ [0, 1].Define the search space by (28) For a position vector P ∈ S P, the actual investment proportion of asset i at period t is decoded by It can be seen that the corresponding solution x obtained by any particle in the search space satisfies constraints n i=1 x t,i = 1 and x t,i ≥ 0.

Chaos initialization
Denote the number of the swarms by L, the number of particles in each swarm by N P, where the swarms 1, 2, • • • , L −1 are subsidiary swarms, and the swarm L is the leader swarm.
Denote the maximum number of iterations by G max .Denote the upper limit for the velocity of the particles by v max .Denote the maximum and minimum of the inertia weight are w max and w min , respectively.
In this subsection, we use the following chaos mapping method to initialize each population Specifically, in the search space, we randomly generate a particle ), then the above chaos mapping method is used to generate a new particle, and the same operation is performed on the newly generated particle to continue generating the next new particle, repeating this process N P − 1 times to generate N P particles.

Fitness
We adapt the following penalty function to handle the constraint g k (x) ≤ 0. For the k-th constraint, the penalty value is The total penalty value of the solution x is If x satisfies the constraint g k (x) ≤ 0, p(x) = 0; otherwise, p(x) > 0.
We design the following fitness function to evaluate each particle in the search space.
where f (x) is the objective function value of the solution x, M is a positive number that sufficiently large.It can be seen that in a series of particles that satisfy the constraint g k (x) ≤ 0, the higher the objective function value of a particle, the higher its fitness; while if a particle is not satisfying the constraint g k (x) ≤ 0, its fitness is lower than any particle that satisfies the constraint g k (x) ≤ 0.

Update process
Denote the position and velocity of particle j of swarm l in the current generation g by P l j (g) and V l j (g), respectively.Denote the best previous position of particle j, which represents the personal optimal position, by Pbest l j (g), i.e., For the global optimal position of subsidiary swarm For the global optimal position of leader swarm L, denote it by Gbest L (g), i.e., (36) Then, the following equations are used to update the velocity and position of each particle at each iteration where w = w max − (w max − w min ) • g/G max is the inertia weight proposed by Tripathi et al. (2007); c l 1 and c l 2 are the self-learning rate and social learning rate of swarm l, respectively; r 1 and r 2 are two numbers that randomly generated from interval [0, 1].

Main procedure
The main procedure of the MPSO is summarized as follows.
Step 1 Set parameters used in the algorithm including L, N P, v max , w max , w min , c l 1 , c l 2 , G max .
Step 2 Let g = 1, and generate a series of particles for each swarm, where the initial positions of the particles in each swarm are generated by the chaos mapping method, and the velocity of each particle is randomly generated on interval [−v max , v max ].
Step 3 Calculate the fitness of the position of each particle, and update the personal optimal position and the global optimal position.
Step 4 Update the velocity of each particle.If an element of the velocity matrices is excluded from the interval [−v max , v max ], replace it by the closer interval endpoint.
Step 5 Update the position of each particle.If an element of the position vector is excluded from interval [0, 1], replace it by the closer interval endpoint.
Step 6 If g = G max , quit the iteration and report corresponding solution decoded by Gbest L (g); otherwise, let g ← g + 1, and return to Step 3.

Real case study
In this section, we give a real case study to illustrate the practicability and effectiveness of the proposed model and algorithm.We assume the investor plans to invest his/her wealth in five stocks from Shanghai Stock Exchange, i.e., A 1 (600004), A 2 (600008), A 3 (600011), A 4 (600017) and A 5 (600023).We collect the weekly historical data of the above stocks over six years, i.e., January 1, 2016, to December 31, 2021, and divide it into three periods evenly.We adopt the method proposed by Mehlawat (2016) to estimate the weekly fuzzy return rates of the stocks at each period, as shown in Table 1.
Assume the initial wealth is W 0 = 10000, and the transaction cost rates of purchasing and selling the risky assets are c + = 0.003 and c − = 0.004, respectively.Assume the reference level of the return rate is r b = 0.02, the decreasing sensitivity degree is δ 1 = δ 2 = 0.5, the loss aversion degree is θ = 1.1, and the diversification level is e = 1.13 except sensitivity analysis.
In the proposed algorithm MPSO, the parameters are set as follows: the total number of swarms is L = 6; the number of particles in each swarm is N P = 300; the upper velocity of each particle is v max = 3; the maximum and minimum inertia weights are w max = 0.7 and w min = 0.3, respectively; the self-learning rate and social learning rate are c l Fig. 4 The performances of the six algorithms in the four tests

Algorithm testing
To test the performance of the proposed MPSO, we adapt it to solve the programming model of form P 4 .Specifically, we take model P 2 as an example by setting the objective weight ω = 0.5, and run the MPSO four times.The convergence process of the MPSO is presented in Fig. 4. The convergence processes of the PSOs corresponding to the subsidiary swarms are also presented for convenient comparison.
From Fig. 4, we can see that the proposed algorithm MPSO gets better performance than that of the PSOs in the four dependent tests.This means that gathering the information from other PSOs can effectively help the MPSO to reduce the possibility of premature convergence.Therefore, we will adopt the MPSO to solve the portfolio optimization problems.

Solving process
In this subsection, we introduce how to get the efficient frontier of model P 1 by using the designed algorithm MPSO to solve model P 2 .As mentioned in the above section, we solve model P 2 with ω ranging over [0, 1].Then, we obtain a series of efficient solutions, which form the efficient frontier of model P 1 , as shown in Fig. 5. From Fig. 5, we can see that there is a positive correlation between cumulative expected perceived value and cumulative perceived risk.Specifically, as the given cumulative perceived risk level increases, the cumulative expected perceived value of the optimal strategy also increases.This indicates that the greater the given cumulative perceived risk level, the investor has a higher tolerate level toward risk and pays more attention to the perceived value.Therefore, our proposed model can provide the suitable investment strategies for investors with respect to their personal preferences.

Comparison results
In order to illustrate the effectiveness of the proposed model P 1 in reflecting investors' loss aversion, we compare it with a classical form of multi-period fuzzy portfolio model, which is as follows To evaluate the excess return per unit downside risk, Keating and Shadwick (2002) proposed the omega ratio.Here, we propose a similar ratio, named perceived ratio, to measure the perceived value per unit perceived risk, which is defined as follows where x represents the investment strategy and CEPV(x) and CPR(x) represent the terminal cumulative expected perceived value and terminal cumulative perceived risk corresponding to the investment strategy x, respectively.To measure the differences between two efficient frontiers, Fulga (2016) proposed a dissimilarity index by comparing each pair of efficient portfolios.Inspired by this idea, we aim to assess the performance of the efficient frontier based on the above perceived ratio (PR) and define the perceived ratio of frontier (PRF) as follows where x * j represents one of the investment strategies in the efficient frontier; D represents the set of all of the investment strategies in the efficient frontier; J represents the number of investment strategies selected from the efficient frontier.
Next, we use the PRF to compare the efficient portfolios that created by model P 1 and MSV.Specifically, given the selected number J ranged from [10, 90] with the interval of 10, we use MPSO to solve the proposed model P 1 and model MSV and compare the PRF values of the above two models with respect to different J , as shown in Fig. 6.
From Fig. 6, we can see that the PRF value of our proposed model P 1 is higher than that of the model MSV, which indicates that it performs better than the model MSV.The main reason is that our proposed model P 1 incorporates investors' loss aversion into portfolio decisions.In summary, the proposed model has a good performance in reflecting investors' subject preferences and providing more reasonable investment strategies.

Analysis of investors' loss aversion
This subsection discusses the effect of investors' loss aversion on investment decision through the following aspects.First, we consider three cases of the loss aversion degree, i.e., θ = 1.1, 1.2 and 1.3.By using the MPSO to solve the Fig. 6 The performances of the two models under different J optimization problems, we obtain the efficient frontiers of model P 1 under different loss aversions in the CEPV-CPR space, as shown in Fig. 7. From Fig. 7, we can see that the loss aversion has a significant effect on the efficient frontier in the CEPV-CPR space.Specifically, as the loss aversion degree increases, the efficient frontier obtained by model P 1 moves right down.The main reason is that the investor with higher loss aversion degree is more pessimistic about the future returns of the risky assets and thus gets a lower cumulative expected perceived value and a higher cumulative perceived risk from the investment strategy.This indicates that an investor with high loss aversion degree may be more conservative than the one with low loss aversion degree, and thus he/she may not invest wealth in risky assets when their risk level is excessing the maximum risk level that he/she can tolerate.
Next, we analyze the effect of the loss aversion on the optimal investment strategy with respect to the cumulative Fig. 7 The efficient frontiers of model P 1 under different loss aversions in the CEPV-CPR space Fig. 8 The optimal strategies obtained by model P 2 under different loss aversion degree θ wealth (CW) and cumulative risk (CR).Specifically, we set the objective weight ω = 0.5, and the loss aversion degree θ ranged from [1.1, 5.1] with the interval of 0.5, respectively, then obtain the optimal investment strategies of model P 1 by solving model P 2 under the above different parameter settings, and the cumulative wealth and cumulative risk of these investment strategies, as shown in Fig. 8. From Fig. 8, we find that the loss aversion has an effect on the investment strategy.As the loss aversion degree θ increases, the investor intends to select the investment strategy of lower cumulative wealth and lower cumulative risk, which indicates that as the higher the given loss aversion degree, the investor tends to be more conservative.From the above analysis, it is meaningful to incorporate investors' loss aversion into the portfolio selection model.Therefore, investors can determine the parameter θ according to their loss aversion degree in the practical investment management.

Analysis of diversification
In this subsection, we analyze the effect of the diversification level on investment decision.We consider three cases of the diversification level, i.e., e = 0.93, 1.13 and 1.33.By using the MPSO to solve the optimization problems, we obtain the efficient frontiers of model P 1 under different diversification levels, as shown in Fig. 9. From Fig. 9, we can see that the diversification level has a significant effect on the efficient frontier.Specifically, in the CEPV-CPR space, as the diversification level increases, the efficient frontier obtained by model P 1 gets shorter and moves right down.The main reason is that an efficient solution to the model with high diversification level may be dominated by the one with low diversification level, and thus it cannot be efficient solution to the latter.This indicates that as the higher the diversification level, the investor is more willing to reduce the non-systematic risk rather than increasing the Fig. 9 The efficient frontiers of model P 1 under different diversification levels excess return.Therefore, investors can determine e according to their diversification requirement in the practical investment management.

Algorithms' comparison
To illustrate the performance of our proposed MPSO, we compare the efficient frontier obtained by the MPSO with that obtained by other methods.We consider two related methods as follows: (1) The multiple population genetic algorithm (MPGA), which incorporates the similar multi-swarm cooperative scheme into the genetic algorithm.
(2) The multiple population sine cosine algorithm (MPSCA), which incorporates the similar multi-swarm cooperative scheme into the sine cosine algorithm.
We compare the efficient frontiers obtained by the three methods, as shown in Fig. 10.From Fig. 10, we can see that Fig. 10 The efficient frontiers obtained by the three methods the efficient frontier obtained by our proposed MPSO is on the upper-left side of that obtained by the other two methods.For given cumulative perceived risk level, it can provide investment strategies with higher cumulative expected perceived value for investors; for given cumulative expected perceived value level, it can provide investment strategies with lower cumulative perceived risk, which indicates that it performs better than the compared methods.

Conclusion
This paper addressed the multi-period portfolio selection problem considering investors' loss aversion in fuzzy environment.Firstly, the return rate of each risky asset is described by a trapezoidal fuzzy number, which captures the uncertainty in financial market.Next, by employing the value function in prospect theory, we transformed the portfolio return rate into the perceived value and used the perceived risk to measure the portfolio's risk.Then, we proposed a multiperiod fuzzy portfolio selection model with investors' loss aversion.Moreover, in order to solve the proposed model, we designed a multiple particle swarm optimization with respect to its specific situation.Finally, we illustrated the effectiveness of the designed algorithm and proposed model through a real case.The results show that the proposed model could provide decision support for investors with different loss aversion degrees.

Fig. 5
Fig.5The efficient frontier of the proposed model

Table 1
The fuzzy return rates of the five risky assets on the three periods