A Fuzzy Entropy Approach for Portfolio Selection

Abstract

Portfolio management typically aims to achieve better returns per unit of risk by building efficient portfolios. The Markowitz framework is the classic approach used when decision-makers know the expected returns and covariance matrix of assets. However, the theory does not always apply when the time horizon of investments is short; the realized return and covariance of different assets are usually far from the expected values, and considering additional factors, such as diversification and information ambiguity, can lead to better portfolios. This study proposes models for constructing efficient portfolios using fuzzy parameters like entropy, return, variance, and entropy membership functions in multi-criteria optimization models. Our approach leverages aspects related to multi-criteria optimization and Shannon entropy to deal with diversification, and fuzzy and fuzzy entropy variants provide a better representation of the ambiguity of the information according to the investors’ deadline. We compare 418 optimal portfolios for different objectives (return, variance, and entropy), using data from 2003 to 2023 of indexes from the USA, EU, China, and Japan. We use the Sharpe index as a decision variable, in addition to the multi-criteria decision analysis method TOPSIS. Our models provided high-efficiency portfolios, particularly those considering fuzzy entropy membership functions for return and variance.

1. Introduction

The modern portfolio theory (MPT), initiated by Markowitz in 1956 [1], is a set of techniques for constructing portfolios that optimally balance maximum returns and an acceptable level of risk for investors. The mean-variance model serves as the foundation of MPT, which assumes investors display rational behavior and possess complete information. MPT has given rise to several research avenues, including entropy measures and modeling information uncertainty and ambiguity through fuzzy logic. The utilization of entropy measures addresses the issue of the low diversification that can result from the mean-variance model, by providing a more comprehensive outlook on portfolio diversification. Meanwhile, fuzzy logic provides a valuable method for handling uncertainty and ambiguity in information by considering acceptable ranges of values, such as the acceptable range of return levels for the portfolio. This study proposes models that combine entropy, fuzzy logic, and fuzzy entropy modeling within a multi-criteria framework. The models with fuzzy entropy performed well, according to a TOPSIS comparison of different model combinations [2]. To the best of our knowledge, this fuzzy entropy approach has not previously been utilized in portfolio selection. In conclusion, our proposed models incorporating fuzzy entropy showed promising results compared to other models in portfolio selection.

As a general concept, fuzzy entropy measures the vagueness or ambiguity of flexible or imprecise parameters in decision-making systems. This measure can be interpreted as the information gain in fuzzy environments and can be used to compare the ambiguity of different decision-making systems [2]. Mathematically, if we consider a fuzzy set A and a fuzzy membership function ${\mu }_A\left(x\right)\in [0,1]$, defined on a set X, this is given by $H_f\left(A\right)=-{\sum }_{x\in X}{\mu }_A\left(x\right){log}_2\left({\mu }_A\left(x\right)\right)$. Please note that this formula is similar to the Shannon entropy formula, which measures the uncertainty or disorder of a probability distribution. However, we consider the membership degree given by ${\mu }_A\left(x\right)$ in fuzzy entropy.

In our previous work on a fuzzy entropy approach in transportation [3], we maximized information gain and outperformed prior methods when calculating origin–destination matrices; these promising results led us to examine its potential for portfolio selection. Our findings were favorable after conducting model construction, data collection, and portfolio testing. The application of fuzzy entropy in information theory is well-established; we extended this measure by incorporating the return and variance of the portfolio into both the objective function and constraints of our models.

While the developments in post-MPT are numerous, in this paper, we focus on the base mean-variance approach and its variants [4], specifically emphasizing fuzzy entropy. The mean-variance model is well-known and has shown great results in practice, but it has limitations that have led to new models, such as those incorporating entropy measures and fuzzy logic. The mean-variance model can produce highly variable solutions with even small changes to parameters such as the portfolio’s expected return. As noted by [5], it can also assign high values to high-risk assets. This variability is partly due to the model’s assumption of a normal distribution of assets’ expected returns, variances, and covariances, and the expectation that these results will hold in the future.

In the mean-variance model, the expected return level reflects investors’ expectations. It is common practice to perform sensitivity analyses on this parameter, as results have shown that small changes to the expected return can result in significantly different portfolios [6,7]. To address this, we propose adding flexibility to the model by treating the expected return as a range rather than a fixed value. This is achieved using fuzzy intervals and entropy, which we demonstrate in Section 3.1.

The aim of a mean-variance model is typically to achieve the highest possible return level, while minimizing variance. The resulting portfolio may contain high-return assets to meet the desired return, which is also known as a sparse solution [5,8]. Diversification, such as maximizing portfolio entropy [9], can be used to address this issue.

It is usual to assume that the returns follow a normal distribution when estimating means, variances, and covariances in finance. However, asset returns are often non-normally distributed and exhibit skewness and kurtosis, leading to portfolios that appear efficient based on normal assumptions but are sub-optimal when considering the actual distribution of returns [10]. Assuming normal distributions can also result in overly optimistic risk assessments, causing investors to take on more risk than they realize, as normal distributions do not adequately capture rare events such as large market crashes [11]. Additionally, normal assumptions can negatively impact portfolio diversification, as correlations estimated from normal distributions may differ from the actual correlations in the data. Our central hypothesis to address these issues is that using fuzzy entropy on expected returns and variances could help evaluate the performance of portfolios for future unknown values.

There are also modeling approaches using entropy measures to deal with some of the problems with the mean-variance model; the more common use is to produce diversified portfolios [5]. Ref. [9] understood this need for more diversification of the solutions of the mean-variance model as corner solutions and analyzed how different entropy measures could help with this problem.

As indicated by [12], fuzzy approaches deal better with real-world uncertainties in portfolio construction. Nevertheless, as already indicated, entropy measures help diversification by avoiding corner solutions [9]. The evident approach considers multiple objectives; we can mention entropy measures, fuzzy parameters, and return maximization [12,13,14,15].

The aim of our research was to test if the introduction of fuzzy and fuzzy entropy produces better portfolios than those obtained from known approaches, such as the base mean-variance model and models with entropy measures. It is relevant to highlight why our fuzzy and fuzzy entropy approaches are novel. We focus on mean return and variance by applying fuzzy and fuzzy entropy measures in different models. In this context, we represent the information gain as one of the objective functions and control this parameter in the constraints of the optimization problem.

In this paper, we compared the performance of portfolios through the allocation of assets using seven optimization models at different moments. We used various time windows to calculate the resulting portfolio within each model, and within them, we used multiple configurations to analyze the objective functions’ relevance. Then, we calculated the performance of the portfolios for data outside the time window in which we estimated the portfolio. With these results, we ranked them using a TOPSIS approach.

To summarize, our paper showcases the following research highlights:

• We proposed optimization problems with multiple objective functions to construct portfolios and model fuzzy and fuzzy entropy measures on the portfolio’s expected return and variance.
• We analyzed seven models and ran numerical experiments to rank these models using TOPSIS. We considered various time windows and evaluated the portfolios outside the window.
• Among the models tested, the multi-objective model multi-criteria Shannon and fuzzy entropy on target return and variance consistently delivered the highest returns and Sharpe ratios, placing it on the efficient frontier. Based on the TOPSIS methodology applied to return, variance, and portfolio entropy, it delivered consistent results across different configurations. Its key strength was its ability to capture all configurations without imposing significant optimization restrictions, making it the optimal model for constructing the best portfolio. Rather than seeking the theoretical optimal, it tended to converge towards observable centers. Portfolio selection.
• We recommend fuzzy models that contain the entropic membership function in their structure; fuzzy measures provide the flexibility to handle intervals and uncertainties, and fuzzy entropy maximizes the information gain for both return and variance, further reinforcing the models’ flexibility.

The paper is organized into six sections. The following section presents a literature review and positions our paper within the current portfolio-selection literature. We describe the models we considered in Section 3. Section 4 describes the data considered for the numerical experiments and the details of the methodology applied. Section 5 shows the results we obtained, commented on, and discussed in Section 6. Finally, in Section 7, we present our final analysis and conclusions.

2. Literature Review

The field of portfolio optimization is vast and originated with the classic work of [1]. Since then, the number of papers in the area has kept growing, with many surveys in the field, such as the one by [4] focusing on deterministic models or by [16] oriented towards robust optimization methods.

The beginning of the modern portfolio theory (MPT), an investment theory based on the idea that risk-averse investors can construct portfolios to optimize or maximize expected returns based on a given level of market risk, emphasizes that risk is an inherent part of higher rewards [1]. MPT assumes that investors are risk-averse, meaning that given two portfolios that offer the same expected return, investors will prefer the less risky one. Thus, an investor will take on increased risk only if compensated by higher expected returns. Conversely, an investor who wants higher expected returns must accept more risk. The exact trade-off will be the same for all investors, but different investors will evaluate the trade-off differently based on individual risk aversion characteristics. An investor can reduce portfolio risk simply by holding combinations of non-perfectly positively correlated instruments. If all the asset pairs have correlations of 0—they are perfectly uncorrelated—the portfolio’s return variance is the sum over all assets of the square of the fraction held in the asset times the asset’s return variance (and the portfolio standard deviation is the square root of this sum).

The efficient frontier is a cornerstone of modern portfolio theory and is the line that indicates the combination of investments that will provide the highest level of return for the lowest level of risk [17]. When a portfolio falls to the right of the efficient frontier, it possesses greater risk relative to its predicted return. When it falls beneath the slope of the efficient frontier, it offers a lower return relative to risk [1]. It is one of the most important and influential economic theories dealing with finance and investment.

There are five basic assumptions that are fundamentals upon which the MPT is constructed [1]:

• The expected return and the variance are the only parameters that affect an investor’s decision.
• Investors are generally rational and risk-averse. They are completely aware of all the risks in investment and take positions based on the risk determination demanding a higher return for accepting greater volatility.
• There are no transaction costs for buying and selling securities.
• All investors have the same expectations concerning the expected return, variance, and covariance.
• Analysis is based on a single-period investment model.

Despite its theoretical importance, critics of the MPT question whether it is an ideal investment tool, because its model of financial markets does not match the real world in many ways. The risk, return, and correlation measures used by MPT are based on expected values, meaning they are mathematical statements about the future (the expected value of returns is explicit in the above equations and implicit in the definitions of variance and covariance). The mean-variance model is an a priori consideration of the homoscedasticity of the variance of the series, a fact that contradicts one of the stylized facts of the financial series, which is the frequent heteroskedasticity; that is, the variance suffers from systematic changes during the studying time. In practice, investors must substitute predictions based on historical asset return and volatility measurements for these values in the equations. Very often, such expected values fail to take account of new circumstances that did not exist when the historical data were generated, introducing uncertainty in the problem modeled through fuzzy logic [6,7].

In this context, many methods have been used to optimize portfolios, but the attention that fuzzy methods have received has not been as systematically studied as the methodologies in the surveys conducted. An exception is the survey by [13], but their study has not been updated since, and many more developments have appeared since then. In what follows, we will describe other works within portfolio optimization using fuzzy methods.

Since the above survey, approaches such as [18,19] have applied fuzzy methods to model returns, considering its variation as the fuzzy variable, and identifying the variability of returns as a form of uncertainty that can be captured with fuzzy models. These models either made assumptions about the shape of the variance or considered deterministic variance scenarios. Later, researchers focused on keeping the fuzzy view of the uncertainty of returns but modeled the time dynamics of the problem. Papers such as [20,21,22] dealt with time-varying portfolio optimization under different circumstances, ranging from the accumulation of risks from variance, transaction costs, and diversification degrees [20], directly considering the time series of returns as a fuzzy system [23], to applying uncertainty to the returns and the liquidity of the assets within the time series in [22] and later in [24]. In all of these studies, studying the returns across time as a fuzzy system was the focus, as opposed to our paper, in which we consider a two-dimensional fuzzy system over the ball of mean-variance, combining the source of uncertainty.

Another focus of many researchers using fuzzy methods has been the multiobjective approach. The recent work by [25] explored how three different objectives can be reconciled when optimizing portfolios using fuzzy logic. Ref. [26] compared the stock return and the general financial situation of a company to build a portfolio following fuzzy logic. Ref. [27] modeled returns as a fuzzy system but included fuzzy targets in the optimization process, including perceptual credibility constraints to the portfolio construction, a similar approach to the work of [28], who considered credibility within the return–liquidity space. Following this credibility approach, the work by [29] considered returns, volatility, skewness, and kurtosis within a fuzzy framework but focused on each as an objective, rather than a source of uncertainty, as we do in our work.

Other approaches in the area have focused, for example, on reducing variability in the decision, such as the semi-entropy approach by [23], which focuses on the downside uncertainty. The impact of rationality in decisions using fuzzy entropy was recently explored by [30], who modeled constraints on portfolios from the rationality expectations of agents. Our work follows the more traditional line of maximizing the portfolio’s risk-adjusted return by explicitly including the uncertainty of the mean-variance surface, as explained in the following sections.

On the other hand, in the context of a decision support system for ranking company stock investments based on value investing, the TOPSIS (technique for order of preference by similarity to ideal solution) approach can be a valuable tool. The approach utilized by this stock ranking decision support system is the TOPSIS procedure. The TOPSIS algorithm’s geometric principle states that the selected alternative must be located at the maximum distance from the negative ideal solution and closest to the positive ideal solution. To ascertain the relative proximity of an alternative to the optimal solution, Euclidean distance is utilized as a method of multi-criteria decision-making that ranks potential solutions according to their similarity to the optimal solution. Several studies have highlighted the applicability of TOPSIS in various domains, including financial performance evaluation [31], group decision making [32], stock portfolio selection [33], and stock portfolio investment [34].

By integrating TOPSIS into the decision-making process, companies can effectively evaluate and rank investment opportunities based on various criteria. This method allows for systematic comparison of stocks, thereby enabling the identification of stocks that outperform those underperforming [35]. Additionally, TOPSIS has been used in the financial sector for applications such as bankruptcy prediction and stock selection for portfolio construction [36]. These applications demonstrate TOPSIS’ flexibility in assisting decision-making processes in portfolio selection. Thus, it is used for making a final decision in order to select the optimal portfolio; the methodology and the involved variables are explained in the following section.

3. Models and Methods

3.1. The Models

First, we define Markowitz’s base model (Model 1), the theoretical foundation for most studies in this field. We then define two variants with entropic measures: Model 2 maximizes Shannon’s entropy with return and variance as constraints, while Model 3 minimizes the variance and maximizes entropy in a single objective function. Model 4 is a multi-criteria model with Shannon’s entropy maximization and fuzzy parameters on the mean and variance of the portfolio. Models 5 and 6 are similar, in that they both maximize entropy, but while Model 5 controls variance with fuzzy intervals, Model 6 only minimizes it. Finally, Model 7 has three objective functions, maximizing entropy, while controlling average return and variance with fuzzy entropy measures.

3.1.1. Model 1: The Markowitz Base Model

Let us consider $i\in \left\{1,\dots ,n\right\}$ assets with returns $r_i$. The decision variables $x_i$ represent the relative weights invested in the portfolio accomplishing Equation (1). The aim is to find the optimal portfolio through the weights, accomplishing the expected return Equation (2) and a variance bound Equation (3) of the portfolio. We can construct the base Markowitz model from this initial setting and the variants and specializations we propose for analysis. In Equation (3), ${\sigma }_i$ are the standard deviations of assets’ returns and ${\rho }_{ij}$ are their correlation coefficients.

$$\sum _{i=1}^n{x}_i=1$$

(1)

$$r_p\left(\mathbf{x}\right)=\sum _{i=1}^n{x}_i\mathbb{E}\left\{r_i\right\}$$

(2)

$$\mathbb{V}\left(\mathbf{x}\right)=\sum _{i=1}^n\sum _{j=1}^n{x}_i{x}_j{\sigma }_i{\sigma }_j{\rho }_{ij}$$

(3)

We define in Equation (4) Markowitz’s base model or Model 1, which minimizes the variance and meets $R_p$ for the portfolios’ expected return.

$$\begin{array}{cc}{\displaystyle \underset{\left\{\mathbf{x}>0\right\}}{min}}& \mathbb{V}\left(\mathbf{x}\right)\\ \mathrm{s}.\mathrm{t}.& r_p\left(\mathbf{x}\right)& =& R_p\\ & {\displaystyle \sum _{i=1}^n{x}_i}& =& 1\end{array}$$

(4)

Let us note that, in Equation (4), the constraint $r_p\left(\mathbf{x}\right)=R_p$ could also be greater than (≥) as in [37].

3.1.2. Model 2: Shannon’s Entropy Maximization

Model 2 includes the maximization of the entropy $H\left(\mathbf{x}\right)$. Equation (5) relates the portfolio to the base model, which helps produce more diversified portfolios.In this optimization problem, the expected returns $R_p$ and variance ${\sigma }_p^2$ are within the constraints. In Equation (6), we define Model 2 that corresponds to this case; this model was also considered by [9,38] among others.

$$H\left(\mathbf{x}\right)=-\sum _{i=1}^n{x}_{i}ln\left(x_i\right)$$

(5)

$$\begin{array}{cc}{\displaystyle \underset{\left\{\mathbf{x}>0\right\}}{max}}& H\left(\mathbf{x}\right)\\ \mathrm{s}.\mathrm{t}.& {\displaystyle r_p\left(\mathbf{x}\right)}& =& R_p\\ & \mathbb{V}\left(\mathbf{x}\right)& \le & {\sigma }_p^2\\ & {\displaystyle \sum _{i=1}^n{x}_i}& =& 1\end{array}$$

(6)

3.1.3. Model 3: Shannon’s Entropy Maximization and Minimum Variance

In Equation (7), we show Model 3, which includes the minimization of the entropy and the minimization of the portfolio’s variance. The model Equation (7) balances within the objective function, with equal emphasis on both the maximization of the entropy and the minimization of the variance of the resulting portfolio, while requiring a specific level $R_p$ for the portfolio return.

$$\begin{array}{cc}{\displaystyle \underset{\left\{\mathbf{x}>0\right\}}{min}}& \mathbb{V}\left(\mathbf{x}\right)-H\left(\mathbf{x}\right)\\ \mathrm{s}.\mathrm{t}.& {\displaystyle r_p\left(\mathbf{x}\right)}& =& R_p\\ & {\displaystyle \sum _{i=1}^n{x}_i}& =& 1\end{array}$$

(7)

3.1.4. Model 4: Multi-Criteria with Shannon’s Entropy Maximization and Fuzziness on Target Return and Variance

In Model 4, we introduce a multi-criteria optimization scheme, by adding fuzzy parameters for the portfolio’s target expected return and variance. We define this case as the bi-objective model or Model 4 in Equation (8), where we maximize the entropy and control the expected return and variance with trapezoidal intervals.

In Equation (8), the parameters $\left(\underline{r},r_1,r_2,\overline{r}\right)$ and $\left({\underline{\sigma }}^2,{\sigma }_1^2,{\sigma }_2^2,{\overline{\sigma }}^2\right)$ correspond to the representation of trapezoidal fuzzy membership functions for the expected return $R_p$ and variance ${\sigma }_p^2$.

$${\displaystyle \begin{array}{ll}{\displaystyle \underset{\left\{\mathbf{x}>0,\lambda \in \left[0,1\right]\right\}}{max}}& {\displaystyle \left\{H\left(\mathbf{x}\right),\lambda \right\}}\\ \mathrm{s}.\mathrm{t}.& {\displaystyle r_p\left(\mathbf{x}\right) \ge \lambda r_1+\left(1-\lambda \right)\underline{r}}\\ & {\displaystyle r_p\left(\mathbf{x}\right) \le \lambda r_2+\left(1-\lambda \right)\overline{r}}\\ & {\displaystyle \mathbb{V}\left(\mathbf{x}\right) \ge \lambda {\sigma }_1^2+\left(1-\lambda \right){\underline{\sigma }}^2}\\ & {\displaystyle \mathbb{V}\left(\mathbf{x}\right) \le \lambda {\sigma }_2^2+\left(1-\lambda \right){\overline{\sigma }}^2}\\ & {\displaystyle \sum _{i=1}^n{x}_i = 1}\end{array}}$$

(8)

In (8), ($\underline{r},r_1,r_2,\overline{r}$) and (${\underline{\sigma }}^2,{\sigma }_1^2,{\sigma }_2^2,{\overline{\sigma }}^2$) are the parameters of trapezoidal fuzzy sets associated with the expected return and variance of the portfolio. We use the trapezoidal form of a fuzzy parameter; in Figure 1, we show how the return $R_p$ is represented by a trapezoid and a membership function $\mu \left(R_p\right)$, which in (8) is controlled by the decision variable $\mu$. Please note that the case for the trapezoid membership function for the variance is analogous, and when $r_1=r_2$, the trapezoid shape turns into a triangular membership function.

For the return trapezoid, at every point in time, we calculated the maximum possible return and defined every parameter of the trapezoid based on this number. 0 was always the lower bound for the return trapezoid, because nobody wanted to invest in a negative expected return portfolio. We calculated the minimum and maximum volatility values for the volatility case at each point in time. We set the trapezoid parameter to 1.25 times the minimum sigma and the other parameter to 0.75 times the maximum sigma.

3.1.5. Model 5: Multi-Criteria Shannon, Fuzzy Variance and Fuzzy Entropy on Target Return

In Equation (10), we define a model with a trapezoidal fuzzy bound. This is a tri-objective approach; its first objective function is maximizing the entropy $H\left(\mathbf{x}\right)$. The second objective is a fuzzy entropy approach for maximizing the entropy of the fuzzy portfolio’s return $g_1\left(\alpha \right)$ (see Equation (9)). The third is maximizing the membership for the variance using the decision variable $\lambda$.

$$g_1\left(\alpha \right)=-\alpha ln\left(\alpha \right)-\left(1-\alpha \right)ln\left(1-\alpha \right)$$

(9)

$${\displaystyle \begin{array}{ll}{\displaystyle \underset{\left\{\mathbf{x}>0,\alpha \in \left[0,1\right],\lambda \in \left[0,1\right]\right\}}{max}}& {\displaystyle \left\{H\left(\mathbf{x}\right),g_1\left(\alpha \right),\lambda \right\}}\\ \mathrm{s}.\mathrm{t}.& r_p\left(\mathbf{x}\right) = \alpha \underline{r}+\left(1-\alpha \right)\overline{r}\\ & \mathbb{V}\left(\mathbf{x}\right) \ge \lambda {\sigma }_1^2+\left(1-\lambda \right){\underline{\sigma }}^2\\ & \mathbb{V}\left(\mathbf{x}\right) \le \lambda {\sigma }_2^2+\left(1-\lambda \right){\overline{\sigma }}^2\\ & {\displaystyle \sum _{i=1}^n{x}_i = 1}\end{array}}$$

(10)

The model in expression Equation (11) is a particular case of Equation (10) in which ${\sigma }_p^2={\sigma }_1^2={\sigma }_2^2$ and $\lambda =1$, and in which we represented the constraint associated with the variance as an upper-bound, considering that if there is any solution with a variance lower than ${\sigma }_p^2$, then we consider this a good solution.

$${\displaystyle \begin{array}{ll}{\displaystyle \underset{\left\{\mathbf{x}>0,\alpha \in \left[0,1\right]\right\}}{max}}& {\displaystyle \left\{H\left(\mathbf{x}\right),g_1\left(\alpha \right)\right\}}\\ \mathrm{s}.\mathrm{t}.& r_p\left(\mathbf{x}\right) = \alpha \underline{r}+\left(1-\alpha \right)\overline{r}\\ & \mathbb{V}\left(\mathbf{x}\right) \le {\sigma }_p^2\\ & {\displaystyle \sum _{i=1}^n{x}_i = 1}\end{array}}$$

(11)

We show the fuzzy entropy function $g_1\left(\alpha \right)$ in Figure 2. In this case, $\alpha$ controls the value of $R_p\in \left[\underline{r},\overline{r}\right]$.

3.1.6. Model 6: Multi-Criteria Shannon, Fuzzy Entropy on Target Return and Min Variance

In Equation (12), we define Model 6 with a fuzzy entropy representation for the expected return and with three objective functions. The first is the maximization of the portfolio’s entropy; the second is the maximization of the fuzzy entropy of the expected return; and the third is the minimization of the portfolio’s variance.

$${\displaystyle \begin{array}{ll}{\displaystyle \underset{\left\{\mathbf{x}>0,\alpha \in \left[0,1\right]\right\}}{max}}& {\displaystyle \left\{H\left(\mathbf{x}\right),g_1\left(\alpha \right),-\mathbb{V}\left(\mathbf{x}\right)\right\}}\\ \mathrm{s}.\mathrm{t}.& r_p\left(\mathbf{x}\right) = \alpha \underline{r}+\left(1-\alpha \right)\overline{r}\\ & {\displaystyle \sum _{i=1}^n{x}_i = 1}\end{array}}$$

(12)

3.1.7. Model 7: Multi-Criteria Shannon and Fuzzy Entropy on Target Return and Variance

In Equation (14), we propose Model 7. It has three objective functions. The first is the maximization of the portfolio’s entropy $H\left(\mathbf{x}\right)$, the second is the maximization of the fuzzy entropy function of the portfolio’s return $g_1\left(\alpha \right)$, and the third is the minimization of the portfolio’s variance $g_2\left(\beta \right)$, defined in Equation (13). Note that the shape of the fuzzy entropy function $g_2\left(\mathbf{x}\right)$ is the same as $g_1\left(\mathbf{x}\right)$.

$$g_2\left(\beta \right)=-\beta ln\left(\beta \right)-\left(1-\beta \right)ln\left(1-\beta \right)$$

(13)

$${\displaystyle \begin{array}{ll}{\displaystyle \underset{\left\{\mathbf{x}>0,\alpha \in \left[0,1\right],\beta \in \left[0,1\right]\right\}}{max}}& {\displaystyle \left\{H\left(\mathbf{x}\right),g_1\left(\alpha \right),g_2\left(\beta \right)\right\}}\\ \mathrm{s}.\mathrm{t}.& r_p\left(\mathbf{x}\right) = \alpha \underline{r}+\left(1-\alpha \right)\overline{r}\\ & \mathbb{V}\left(\mathbf{x}\right) = \beta {\underline{\sigma }}^2+\left(1-\beta \right){\overline{\sigma }}^2\\ & {\displaystyle \sum _{i=1}^n{x}_i = 1}\end{array}}$$

(14)

The bases of the fuzzy models (4, 5, 6, and 7) can be seen in the Appendix A, with the mathematical deduction of the models with fuzzy parameters, and where some definitions of fuzzy sets and fuzzy optimization are described.

4. Methodology

In this section, we describe the data we considered for our numerical experimentation and explain how we ordered the numerical instances; we also give details on how we solved the optimization problems and, finally, describe how we applied TOPSIS.

4.1. Data Description

The database we considered has monthly registers of stock indices of the USA Standard & Poor’s 500, Europe Euro Stoxx 50, Japan Nikkei 225, and China HSCEI from 2003 to 2023. The relevance of using indices in the portfolio resides in the fact that this strategy is scalable to different investment amounts because the volumes traded with the instruments are very high. The values are monthly from the year 2003 to 2023.

The USA Standard & Poor’s 500 index, also known as S&P 500, is one of the most important stock indexes in the United States; it is considered the most representative index of the real market situation. The index is based on the market capitalization of 500 large companies that own shares listed on the NYSE or NASDAQ exchanges and captures approximately 80% of all market capitalization in the United States. The Europe Euro Stoxx 50 index represents the performance of the 50 largest companies among the 19 super sectors regarding market capitalization in 11 eurozone countries. These countries are Germany, Austria, Belgium, Spain, Finland, France, Ireland, Italy, Luxembourg, the Netherlands, and Portugal. The Japan Nikkei 225 index, also called the Nikkei index, is the most popular stock index in the Japanese market, comprising the 225 most liquid securities listed on the Tokyo Stock Exchange. The China HSCEI index is the main Chinese stock index of Hong Kong. It records and monitors the daily changes in the largest Hong Kong companies on the stock market. It consists of 33 companies, representing 65% of the Hong Kong Stock Exchange.

4.2. Description of Numerical Instances

From the entire dataset, with monthly values from the year 2003 to 2023, corresponding to 235 samples of each stock index, we divided the dataset into five non-overlapping subsets, performing a temporal partition with the same length (47 samples). The objective was to test the model under different market conditions, denoted as independent analysis sets $\left(S1, S2, S3, S4, S5\right)$. In

Table 1. Subset.

Dates/Scenario	1	2	3	4	5
Start date	31 May 2003	31 May 2007	30 April 2011	31 March 2016	29 February 2020
End date	31 March 2007	31 March 2011	28 February 2015	31 January 2020	31 December 2023

, we show the time interval for each scenario.

To evaluate the performance of each model in each subset, we used a rolling window approach, where we calculated the expected returns and covariance matrices for each window. Please see the rolling time window methodology in Figure 3.

We consider it relevant to explain that, as we considered four indexes, the variances and covariances do not constitute high-dimensional matrices. However, when modelers consider several stock prices, indexes, or exchange-traded funds, they will face high-dimensional variances and covariances matrices. Although this is not relevant in our case, we can recommend literature for performing analysis under these circumstances [39,40,41].

There were 47 samples (monthly returns) in the five scenarios analyzed. The methodology used to estimate the returns and the covariance matrix was based on a moving window of 20 samples of the returns of the stock indices (from $t-19$ to t) used in this study; the last 20 samples were considered. Therefore, in one scenario, there were 27 portfolios obtained, since new data (returns) were included for calculating the estimation of returns and the covariance matrix. Based on these estimates, we proceeded to optimize the portfolios. The advantage of using this method lies in capturing possible changes over time in the statistics of the indices and thus making a dynamic estimation that helps to improve the optimization of the model [42]. Finally, we compounded each model’s returns (portfolio) over all the subsets, to evaluate each model and obtain its total returns.

4.3. Solving the Optimization Problems

We solved these models using the Optimization Toolbox of Matlab v9.6 (Release R2019a; MathWorks Inc., Natick, MA, USA).

Models 1, 2, and 3 are nonlinear and have a single objective function. It is worth noting that, even though Model 3 has both variance minimization and entropy maximization objectives, we used the approach proposed by [43] to solve it as a single-objective problem. For Model 1, we performed optimization by minimizing the constrained variance to eleven different return values, from $r_{min}$ to $r_{max}$, where $r_{min}$ is the lowest possible return and $r_{max}$ is the highest possible return given the window under analysis. This return interval was moved with a step of 0.1 times the length of the return interval, generating 11 optimal solutions, one for each restricted return. In Model 2, the maximization of the portfolio entropy was constrained to return and variance values. The return and variance interval was moved with a step of 0.1 times the length of the interval (return and variance), generating 121 solutions (11 returns × 11 variances).

In the case of Model 3, we minimized the variance and maximized the entropy in the target function, for which we constrained the return; we implemented this constraint as an equality and an inequality. The path of the return interval was carried out at a step of 0.1 times the length of the interval, generating 22 solutions.

Models 4 to 7 are nonlinear and have multiple objective functions. We used the $ϵ$-constraint methodology [44] to find solutions. This methodology solves a multi-objective problem by minimizing a single function, while leaving the others as constraints. The $ϵ$-constraint approach converts the multiple objective functions into constraints controlled by a right-hand-side parameter $ϵ$. We could estimate the efficient frontier of the solutions by performing a sensitivity analysis on the $ϵ$ parameters. The efficient frontier concept, in this case, refers to the optimization problem with multiple objective functions, where we aimed to find the set of optimal solutions that maximized one objective function, while satisfying the constraints imposed by the other objective functions.

In mathematical terms, let us consider a general multiobjective optimization problem as defined in (15), with ${\left\{f_i\left(\mathbf{x}\right)\right\}}_{i=0}^n$ objective functions and $j\in \left\{1,\dots ,m\right\}$ constraints, defined by the functions $g_j\left(\mathbf{x}\right)$ and its right-hand-side parameters $b_j$. The $ϵ$-constraint approach is shown in (16) when keeping $f_0\left(\mathbf{x}\right)$ as a single objective function, and considering ${\left\{f_i\left(\mathbf{x}\right)\right\}}_{i=1}^n$ as constraints with right-hand-side parameters ${\left\{{ϵ}_i\right\}}_{i=1}^n$.

$$\begin{array}{ll}{\displaystyle \underset{\left\{\mathbf{x}\ge 0\right\}}{min}}& \left\{f_0\left(\mathbf{x}\right),\dots ,f_n\left(\mathbf{x}\right)\right\}\\ \mathrm{s}.\mathrm{t}.& g_j\left(\mathbf{x}\right) \ge b_j \forall j\in \left\{1,\dots ,m\right\}\end{array}$$

(15)

$$\begin{array}{ll}{\displaystyle \underset{\left\{\mathbf{x}\ge 0\right\}}{min}}& f_0\left(\mathbf{x}\right)\\ \mathrm{s}.\mathrm{t}.& g_j\left(\mathbf{x}\right) \ge b_j \forall j\in \left\{1,\dots ,m\right\}\\ & f_i\left(\mathbf{x}\right) \ge {ϵ}_i \forall i\in \left\{1,\dots ,n\right\}\end{array}$$

(16)

We applied the $ϵ$-constraint approach for each model, as indicated below:

• Model 4—Maximum entropy, fuzzy in return and variance: This model has two objective functions for the entropy maximization, and the control parameter $\lambda$ for the membership function, and for variance and the expected return of the portfolio. We kept the entropy as the single objective function in this case, turning the $\lambda$ into an $ϵ$-constraint. We performed numerical experiments by carrying out sensitivity analysis on the corresponding $ϵ$ bounding $\lambda$. Then, these intervals were controlled by $\lambda$; this variable takes values between 0 and 1, moving at a step of 0.1, generating 11 solutions.
• Model 5—Entropy maximization, fuzzy entropy in return, fuzzy in variance: This model has three objective functions: entropy maximization, the fuzzy entropic function for the expected return, and the control parameter $\lambda$ of the membership function for portfolio’s variance. In this case, we kept the entropy as the single objective function, turning the other objective functions into two $ϵ$-constraints. We performed the numerical experiments through sensitivity analysis on the corresponding $ϵ$ parameters bounding $\lambda$ and $g_1\left(\mathbf{x}\right)$.
  Using the same approach as for Model 4, $\alpha$ and $\lambda$ took values between 0 and 1; with $\alpha$ acting on the return constraint, and $\lambda$ modulating the variance interval (trapezium); these intervals were constraints in the optimization process. The path of $\alpha$ and $\lambda$ had a step of 0.1, which generated 121 solutions.
• Model 6—Entropy maximization, fuzzy entropy in return and variance: This model had three objective functions: entropy maximization, the fuzzy entropic for the expected return, and the portfolio’s variance minimization functions. In this case, we kept the entropy as a single objective function, turning the other objective functions into two $ϵ$-constraints. We performed the numerical experiments through a sensitivity analysis on the corresponding $ϵ$ parameters bounding $g_1\left(\mathbf{x}\right)$ and $\mathbb{V}\left(\mathbf{x}\right)$. Continuing with the same methodology used in the other models, $\alpha$ was a variable that took values between 0 and 1, running this interval at a step of 0.1, generating 11 solutions.
• Model 7—Entropy maximization and fuzzy entropy in return and variance: This model had three objective functions: the entropy maximization and the fuzzy entropic functions for the expected return and variance of the portfolio. In this case, we kept the entropy as a single objective function, turning the other objective functions into two $ϵ$-constraints. We performed the numerical experiments by carrying ou sensitivity analysis on the corresponding $ϵ$ parameters bounding $g_1\left(\mathbf{x}\right)$ and $g_2\left(\mathbf{x}\right)$, moving through these parameters at a step of 0.1, and 121 portfolios were generated.
  The optimization process of each model ended when we selected the optimal portfolio for each (Models 1 to 7) and computed their returns. Finally, we used TOPSIS to select the best model regarding their three main features: expected return, variance, and entropy. We applied sensitivity analysis of the relevance of each variable to represent different scenarios; of course, the ranking of the models changed in each case.

4.4. TOPSIS in Detail

The TOPSIS methodology is a multi-criteria decision methodology that allows choosing a solution to a problem based on multiple attributes. Typically, the attributes are opposing characteristics, such as return and volatility. The methodology determines a score based on the Euclidean distance to a positive ideal element and a negative ideal element. The closer the element is to the positive ideal element, and the further it is from the negative ideal element, the better the score. These ideal elements are constructed by taking the highest and lowest values of the available elements.

In this case, we considered the attributes for $k\in \left\{1,\dots ,K\right\}$ portfolios, including their expected returns, variances, and entropies. The ideal positive element was the one with the highest return, lowest variances, and highest entropy. The ideal negative element was the one with the lowest return, highest volatility, and lowest entropy. Let us consider the attributes ${\mathbf{a}}_k$ as indicated in Equation (17), where we calculated the values of each attribute by considering the portfolio ${\mathbf{x}}_k$ and its performance based on data outside the time window.

$${\mathbf{a}}_k=\left(r_p\left({\mathbf{x}}_k\right), \mathbb{V}\left({\mathbf{x}}_k\right), H\left({\mathbf{x}}_k\right)\right)$$

(17)

We performed the following steps:

• Define the performance matrix: The performance matrix corresponds to the collection of attributes of the portfolios. It has three columns (attributes): return, volatility, and entropy.
• Normalization: This step corresponds to turning the values of the performance matrix $\forall k\in \left\{1,\dots ,K\right\}$ into values between 0 and 1:

  $$\begin{array}{cccccc}\overline{{\mathbf{a}}_k}& =& {\displaystyle \frac{{\mathbf{a}}_k}{\parallel {\mathbf{a}}_k\parallel },}& \parallel {\mathbf{a}}_k\parallel & =& {\displaystyle \sqrt{\sum _{l=1}^3{a}_{k,l}^2}}\end{array}$$

  (18)
• Weighted normalized matrix: In this step, we multiplied the columns of the normalized performance matrix by a scalar value, where the sum of these scalar values equaled 1.

  $$\begin{array}{ccc}{\displaystyle \sum _{l=1}^3{w}_l}& =& 1,\end{array}$$

  (19)

  where $w_l$ are the scalars that multiply the columns; these values represent the relative relevance of the attributes. We considered multiple configurations.
• Positive and negative ideals: In this step, we defined the positive and negative ideal portfolio; we used the highest and lowest values of each portfolio’s attributes.

  $$\begin{array}{ccc}P& =& \left[{\displaystyle \underset{k\in \left\{1,\dots ,K\right\}}{max}{r}_p\left({\mathbf{x}}_k\right);\underset{k\in \left\{1,\dots ,K\right\}}{min}\mathbb{V}\left({\mathbf{x}}_k\right);\underset{k\in \left\{1,\dots ,K\right\}}{max}H\left({\mathbf{x}}_k\right)}\right]\end{array}$$

  (20)

  $$\begin{array}{cccc}& N& =& \left[{\displaystyle \underset{k\in \left\{1,\dots ,K\right\}}{min}{r}_p\left({\mathbf{x}}_k\right);\underset{k\in \left\{1,\dots ,K\right\}}{max}\mathbb{V}\left({\mathbf{x}}_k\right);\underset{k\in \left\{1,\dots ,K\right\}}{min}H\left({\mathbf{x}}_k\right)}\right]\end{array}$$

  (21)
• Distance calculation: $\forall k\in \left\{1,\dots ,K\right\}$ we calculated the Euclidean distance to the positive and negative ideal portfolios.

  $$\begin{array}{ccc}{d}_k^P& =& {\displaystyle \sqrt{\sum _{l=1}^3{(a_{k,l}-p_l)}^2}}\\ d_k^N& =& {\displaystyle \sqrt{\sum _{l=1}^3{(a_{k,l}-n_l)}^2}}\end{array}$$

  (22)

  where $p_l$ and $n_l$ are the components (return, variance, entropy) of the positive ideal and negative ideal elements, respectively.
• Relative closeness: we calculated the relative closeness to the ideal solution, corresponding to the distance ratio to the negative ideal portfolio divided by the sum of the distances to the negative and positive ideal.

  $$\begin{array}{ccc}{C}_k& =& {\displaystyle \frac{d_k^N}{d_k^N+d_k^P}}\end{array}$$

  (23)
• Ranking results: we ranked the solution from the highest relative closeness to the lowest, as calculated in the previous step.

According to this proposed approach, we considered nine different configurations for the weights, to analyze rankings and discriminate which models were better; these configurations considered the return more relevant. The main reason we had to do this was to represent that the return is dominant over other metrics in the eyes of investors when they select their portfolio. In

Table 2. TOPSIS weight configurations analyzed.

Attribute Weight/Scenario	1	2	3	4	5	6	7	8	9
Return [%]	33	50	50	50	70	60	60	70	60
Variance [%]	33	50	0	25	15	40	0	30	30
Entropy [%]	33	0	50	25	15	0	40	0	10

, we show the nine sets of weights $e_i$ for the portfolios’ attributes.

5. Results

The results obtained from the different models in the risk return space showed three groups, as can be seen in Figure 4. This figure shows the portfolios’ standard deviation versus its return in this 20-year evaluation. The chart below shows the compound return over a period (cumulative or compound). We identified three groups according to the level of return. The first group had the highest returns between 1200% and 1400% or between 1.11% to 1.04% in monthly terms, corresponding to 6.2% (red) of the total portfolios; a second group had returns oscillating around 800% (equivalent to 0.87% monthly return), with a 6.2% (grey) of the total portfolios; and finally, a third group, with the largest number of elements and oscillating returns around 400%(equivalent to 0.58% monthly return) corresponding to 87% (yellow), which is considered more representative of the total.

The first group showed low entropy, around 0.13, dominated by Model 2 and 5 portfolios. The second group had an entropy of around 0.59, where Model 2 and 5 portfolios dominated the group in quantity. Having a similar behavior, the third group had an entropy of 0.83, which was high compared to the other models. The dominant portfolios in this group belonged to Model 7.

The colors of the points represent the relation between standard deviation and compound return of the portfolios for each model, a total of 418, three separated groups were naturally obtained according to the level of return, risk, and entropy, as indicated before. The third group (yellow) is selected to analyze, with the largest number of elements, corresponding to 87%, oscillating returns around 0.58% monthly return, and the highest entropy of 0.83; it’s more representative of the total, and a cluster with better portfolios according to the parameters like mean Sharpe index and entropy like a measure of diversification.

In Figure 5, we present the portfolio performance of the third group selected (blue points), the resulting portfolio of Model 7 ($\alpha =0.3, \beta =0.9)$, in terms of the compound return and Sharpe index, which is situated in the upper right quadrant and demonstrates the above-average performance in both return and efficiency. Furthermore, it stands out for its position relative to other portfolios in the same quadrant.

Figure 6 illustrates that Model 7 lies on the efficient frontier when considering the models in terms of minimizing risk and maximizing returns. It is noteworthy that Model 7, with $\alpha =0.3$ and $\beta =0.9$, exhibited one of the highest returns and was one of the most efficient among the portfolios derived from the different models.

In Figure 7, we present a performance comparison of the models across the nine scenarios proposed, considering the differences in the values of their attributes (return, variance, and entropy). The purpose of the TOPSIS analysis predicted in the chart was to show the ranking position of each portfolio against the rest of the portfolios in each configuration. The length of the bar represents the number of models analyzed, and the position of the dot in the bar is how good the result was. The lower the position in the bar, the better the result.

Models 5 and 7 exhibited superior performance compared to the other models across the different configurations of TOPSIS.

Model 5 performed well and was comparable to Model 7; however, it did not outperform Model 7.

Models 2 and 4 fell within the top half of the TOPSIS chart. In particular, Model 4 consistently fell around the middle of the ranking in all scenarios. In configurations where the return was relevant, Model 2 was the worst. This situation was the opposite of Model 4 when there is a low importance of return. This model did not perform well, but Model 1 was the worst.

It is relevant to note that Model 4 had a stable behavior, meaning its rankings were similar among scenarios.

While Model 6 was better than 3, its behavior was similar to 3, which performed better than the models in the middle of the ranking.

Lastly, in general terms, you can see that Model 7 consistently exhibited superior performance to the other models (obtaining the first position of the ranking) across the different configurations of the TOPSIS decision variables. Therefore, it can be considered the best model under the different scenarios.

6. Discussion

The results of our analysis suggest that, for the data and methodology considered, models that incorporated fuzzy measures and fuzzy entropy measures in multiobjective settings—such as Models 5, 6, and 7—demonstrated superior performance compared to Models 1, 2, and 3. Is important to note the Model 4 did not have a good performance in comparison with the other fuzzy models, and it id is worth noting that this model does not have an entropic membership function. In our opinion, fuzzy measures provide the flexibility to handle intervals and uncertainties, and fuzzy entropy maximizes the information gain for both return and variance, further reinforcing the models’ flexibility.

Across the full range of results, there was a notable dispersion in portfolio performance, as measured by their actual returns and variances. In our view, this dispersion can be attributed to two primary factors: (i) the use of various variance bounds, not only in the Markowitz base model but also in all of the other models that incorporated entropy maximization in their $ϵ$-constraint versions, and (ii) the fact that the results of the fuzzy entropy models matched the Markowitz results for low variances, due to the inherent design of the model, which seeks to minimize variance while maximizing return.

In the case of the multiobjective models 4 to 7, the actual variances concentrated in the mid-range, because we used objective functions to maximize the parameters’ fuzzy membership functions. For both considered cases, $\lambda$ was for the fuzzy instances and $\alpha$ and $\beta$ for the fuzzy entropy cases.

Model 5 exhibited the best ranking results when $\alpha =0.3$ and $\lambda$ values covered the interval 0–1, indicating that the variance is modulated by $\lambda$. Models 5, 6, and 7 showed that the actual variance tended to concentrate in the mid-range, which resulted from optimizing the membership functions of the parameters $\alpha$ and $\beta$.

Model 7, with $\alpha =0.3$ and $\beta =0.9$, achieved a higher ranking across the various configurations with different attribute weights, indicating the effectiveness of fuzzy entropy approaches for return and variance optimization in our data.

7. Conclusions

Among the models tested, the multi-objective Model 7 with $\alpha =0.3$ and $\beta =0.9$ consistently delivered the highest returns and Sharpe ratios, placing it on the efficient frontier. Based on the TOPSIS methodology applied to return, variance, and portfolio entropy, it delivered consistent results across different configurations. Its key strength was its ability to capture all configurations, without imposing significant optimization restrictions, making it the optimal model for constructing the best portfolio. Rather than seeking a theoretical optimal, it tended to converge towards observable centers.

Based on that fact, Model 4 did not have a good performance in comparison with the other fuzzy models; noting that this model does not have an entropic membership function, our opinion is that the fuzzy measures provide the flexibility to handle intervals and uncertainties, and fuzzy entropy maximizes the information gain for both return and variance, further reinforcing the models’ flexibility.

For portfolio selection, we recommend fuzzy models that contain an entropic membership function in their structure. In conclusion, fuzzy measures provide the flexibility to handle intervals and uncertainties, and fuzzy entropy maximizes the information gain for both return and variance, further reinforcing the models’ flexibility.