A Mean-Variance Hybrid-Entropy Model for Portfolio Selection with Fuzzy Returns

s: In this paper, we define the portfolio return as fuzzy average yield and risk as hybrid-entropy and variance to deal with the portfolio selection problem with both random uncertainty and fuzzy uncertainty, and propose a mean-variance hybrid-entropy model (MVHEM). A multi-objective genetic algorithm named Non-dominated Sorting Genetic Algorithm II (NSGA-II) is introduced to solve the model. We make empirical comparisons by using the data from the Shanghai and Shenzhen stock exchanges in China. The results show that the MVHEM generally performs better than the traditional portfolio selection models.


Introduction
The financial market is a complex system, in which investors need to make a tradeoff between return and risk in an uncertain environment.Modern portfolio theory was first proposed in 1952 by Markowitz [1], who put forward the classic mean-variance model (MVM) as also created a precedent for the use of quantitative methods in investment portfolio selection.MVM has been proved effective and useful, but its disadvantages such as "corner solutions", low diversity and poor out-of-sample

The Expected Value and Variance of the Triangular Fuzzy Returns
Suppose that ξ = (a, b, c) is a triangular fuzzy variable with the following membership function: According to the Credibility Measure Theory (see Liu [25]), the expected value and variance of the triangular fuzzy returns of stocks are given by: Based on the Zedeh Extension Principle, when ̃ = ( , , ) ( = 1,2, … , ) are all triangular fuzzy variables, ∑ ̃ = (∑ , ∑ , ∑ ) are triangular fuzzy variables as well.
Therefore, we get:

Prediction of Stock Returns
Taking randomness and fuzziness of stock price volatility into consideration, stock yields are set as triangular fuzzy random variables.Then let ̃ be the return rate of stock i, ̃ = ( , , ), = 1,2, … , .Let denote the mathematical expectation of the random variable of ̃ ; and are the ceiling return and the floor return, respectively.Hence, the membership function of ̃ is defined by the following functions: Next, we use the Markov method to predict the fuzzy returns of stock following steps as below: Step 1. Collect the historical trading data in a sample period (in this paper, it is one year or three years), including the opening price , closing price , ceiling price and floor price , = 1,2, … , , where N is the number of sub-intervals.Then we calculate the possible average rates of returns = , the highest possible rates of returns = , and the lowest possible rates of returns = .
Step 2. Use the classic K-Means cluster analysis method to get the step transition matrix.We divide the range of rate of return into M intervals called state spaces and get mid-points ( = 1,2, … , ) and probability ( , = 1,2, … , ) that the return is in space j if it was in space i the last state.Then form one step transition matrix by these probabilities: Step 3. Develop the state transition equation.The probability of stock return in state space i can be calculated by: = where = ( , , … , ) , , , … , ≥ 0, ∑ = 1 .The unique solution of the equation is = ( , , … , ) .Therefore the probabilities of the stock return in state space i after a long enough time are , , … , .
Step 4. Compute the prediction of the stock return by function = ∑ .The highest possible rate of return and lowest possible rate of return can be calculated in the same way.That is how we get the value of the ceiling return and the floor return .Hence, the prediction of our triangular fuzzy returns turns out to be ̃ = ( , , ).

Hybrid Entropy
Hybrid entropy originated from Shannon entropy.Discrete Shannon entropy is referred to as the uncertainty of discrete random variable P in a probability space.That is: Similarly, the uncertainty of discrete fuzzy random variable M in a fuzzy space can be expressed by De Luca-Termini [26] hybrid entropy as follows: Hybrid entropy is an effective tool to measure financial risk caused by both randomness and fuzziness simultaneously (Xu et al. [21]).The hybrid entropy is defined as: This should meet the following requirements: (1) will reach its biggest value if and only if μ = 0.5 and = 1/ ( = 1,2, … , ); (2) will reach its smallest value 0 if and only if μ = 0 or 1 ( = 1,2, … , ), = 1, and = 0( ≠ , , = 1,2, … , ); (3) When randomness (ambiguity) disappears, hybrid entropy should be reduced to a normal probability entropy (fuzzy entropy).
Taking both the randomness and fuzziness of stock returns into consideration, we choose the discrete hybrid entropy to measure the risk of stock return.

Portfolio Optimization Model
In this section, we present the well-known traditional portfolio selection models and also develop a mean-variance hybrid-entropy model with fuzzy returns.

MVM and MEM
Markowitz's MVM was the first risk measurement model to apply the mathematical methods into portfolio selection.In this model, mean measures the expected return and variance measures the risk.The two objectives of the investors are getting higher returns and lower risk relatively.The model is given as follows: where the yield vector of N securities R = ( , , … , ) ; the wealth fraction invested in the securities X = ( , , … , ) ; C is the covariance matrix and c represents the given expected return.
But only taking advantage of variance to measure risk is not sufficient.That is why many scholars put forward the MEM, which uses the entropy instead of variance to measure the portfolio's risk.The MEM is given as follows: where the yield vector of N securities = ( , , … , ) , the wealth fraction invested in the securities = ( , , … , ) , and the risk vector of N securities = (ℎ , ℎ , … , ℎ ) .

MVHEM
We can see that the formula of hybrid entropy is improperly complex, and so is the hybrid entropy formula expressed by triangular fuzzy variables.The applicability will be greatly reduced if the model is too complex.Therefore, we use the approximate formula instead.The formula is: Kalyagin et al. [27] once discussed the specifics of financial modeling and emphasized the multidimensional aspects of financial decisions, so here we choose multi-objective programming.This paper intends to measure the returns by the possibilistic mean of the fuzzy rate of return, and measure the risk by both variance and entropy, and thus form the mean-variance hybrid-entropy model for portfolio selection with fuzzy returns: MVHEM is a multi-objective optimization model.We use the improved algorithm NSGA-II proposed by Deb [28] to solve this optimization problem, which can effectively reduce the computational complexity. Let which can be solved directly by MATLAB.

Sample Data
In order to avoid drastic fluctuations in portfolio returns which may result from industrial risk, we select 10 listed stocks from 10 different industries in Shanghai Stock Exchange (SHSE) and Shenzhen Stock Exchange (SZSE) in China, respectively.The original data obtained from the Straight Flush Software are one-year and three-year weekly data covering from 1 January in 2011 to 1 January in 2014, from which weekly yields, the highest possible yields and the lowest possible yields can be calculated.According to the Markov Method mentioned above, we obtain the prediction of stock returns as shown in Tables 1 and 2.
From Tables 1 and 2, we can observe that compared to the one-year fuzzy returns, most of the three-year results display a trend of convergence, which means less volatility with shorter distance between the left endpoint and right endpoint, and the three-year fuzzy returns' center point is much closer to zero.
Based on the fuzzy returns calculated and the Formulae ( 6) and ( 8), we have the possibilistic mean and the value of hybrid entropy of the sample stocks.The computational results are shown in Tables 3 and 4.    It can be seen from Tables 3 and 4 that the three-year means are all smaller than the one-year means both in the SHSE and SZSE.Not all of the three-year sample's hybrid entropy are smaller than the one-year sample's, and we attribute this to the lack of fuzziness of the three-year sample, which affects the magnitude of the hybrid entropy to some extent.

The Empirical Comparisons among MVM, MEM and MVHEM
On the basis of the real historical data in Tables 1 and 2, we develop a portfolio model composed by ten stocks.The essence of this model is multi-objective.By using the NSGA-II, we get several Pareto optimal solutions for the stock portfolio.These optimal solutions can be divided into two categories: one category is income-oriented and focused on the pursuit of higher yields, the other is risk-oriented and focused on pursuing low risks.We single out one group from each type (MVHEM-I is income-oriented, and MVHEM-II is risk-oriented) as representations, and we also list results of two other classic models (MVM and MEM) for comparison.The results are shown in Tables 5-8.
The above results from Tables 5 to 8 show that we have constructed a portfolio with better dispersion compared to the other two portfolio selection models, that is, the results will not be skewed by one or two stocks.We also draw intuitive figures to display this characteristic (See Figures 1-4).The price data from 1 January to 1 July in 2014 of each stock mentioned above are used to predict the relative cumulative returns of each model.Figures 1 and 2 show the relative cumulative returns of sample stocks in the Shanghai and Shenzhen stock exchange markets generated from the one-year sample period.Although there are a few flaws such as the returns of MVHEM are not always the highest, it can be seen from the figures above that when the sample period is short (such as one year), the MVHEM portfolio optimization model can help investors earn higher revenue or reduce investment risk most of the time.Besides, the model we proposed may balance the returns and risk more comprehensively, and thus allocate the proportion more properly.
The following Figures 3 and 4 show the relative cumulative returns of sample stocks in Shanghai and Shenzhen stock exchange market generated from the three-year sample period.Figure 4 shows the same features as Figure 1 (even better than Figure 1), but compared with Figure 4, Figure 3 seems a little bit unusual, because the relative cumulative returns generated from the MVHEM are lower than those of the other two models except for the two points in the end.We attribute this to the market volatility, which may affect the results significantly.In all, the MVHEM we propose generates higher cumulated returns and smaller volatilities than the MVM and MEM.Besides, when we use the one-year sample, the MVHEM obtains an even more favorable effect, which is consistent with the Markov method's feature of being more suitable for short-term data.

Conclusions
This paper considers stock yields as triangular fuzzy random variables, and uses the Markov method to predict stock returns.Then we measure the portfolio income by fuzzy average yield and the portfolio risk by hybrid entropy and variance, on the basis of which we build a mean-variance hybrid-entropy portfolio optimization model.This model measures the risk of local deviation from the mean as well as the risk of overall deviation from the uniform distribution, and turns out to be more comprehensive and effective than other classic models.Besides, taking the randomness and fuzziness of financial systems into consideration, we fuzzify the investment target and use linear membership functions to measure degree of satisfaction.To solve the proposed model, a multi-objective genetic algorithm is employed.Future work could consider transaction costs as well as investors' trade-off between low-risk and high return objectives in the research.
Then we could form a new multi-objective function as follows:

Figure 1 .
Figure 1.The relative cumulative returns of one-year period sample stocks in SHSE.

Figure 2 .
Figure 2. The relative cumulative returns of one-year period sample stocks in SZSE.

Figure 3 .
Figure 3.The relative cumulative returns of three-year period sample stocks in SHSE.

Figure 4 .
Figure 4.The relative cumulative returns of three-year period sample stocks in SZSE.

Table 1 .
The fuzzy returns of different sample period stocks in SHSE.

Table 2 .
The fuzzy returns of different sample period stocks in SZSE.

Table 3 .
Expected values and hybrid entropy calculated from different sample period stocks in SHSE.

Table 4 .
Expected values and hybrid entropy calculated from different sample period stocks in SZSE.

Table 5 .
The proportion of one-year period sample stocks in different portfolio selection models in SHSE.

Table 6 .
The proportion of three-year period sample stocks in different portfolio selection models in SHSE.

Table 7 .
The proportion of one-year period sample stocks in different portfolio selection models in SZSE.

Table 8 .
The proportion of three-year period sample stocks in different portfolio selection models in SZSE.