Performance evaluation of possibilistic fuzzy portfolios with different investor risk attitudes based on DEA approach

Abstract

By considering the stock market’s fuzzy uncertainty and investors’ psychological factors, this paper studies the portfolio performance evaluation problems with different risk attitudes (optimistic, pessimistic, and neutral) by the Data Envelopment Analysis (DEA) approach. In this work, the return rates of assets are characterized as trapezoidal fuzzy numbers, whose membership functions with risk attitude parameters are described by exponential expression. Firstly, these characteristics with risk attitude are strictly derived including the possibilistic mean, variance, semi-variance, and semi-absolute deviation based on possibility theory. Secondly, three portfolio models (mean-variance, mean-semi-variance, and mean-semi-absolute-deviation) with different risk attitudes are proposed. Thirdly, we prove the real frontiers determined by our models are concave functions through mathematical theoretical derivation. In addition, two novel indicators are defined by difference and ratio formulas to characterize the correlation between DEA efficiency and portfolio efficiency. Finally, numerical examples are given to verify the feasibility and effectiveness of our model. No matter what risk attitude an investor holds, the DEA can generate approximate real frontiers. Correlation analysis indicates that our proposed approach outperforms in evaluating portfolios with risk attitudes. At the same time, our model is an improvement of Tsaur’s work (2013) which did not study the different risk measures, and an extension of Chen et al.’s work (2018) which only considered risk-neutral attitude.

Keywords

Portfolio performance evaluation risk attitude data envelopment analysis (DEA)possibility theory real frontier

1 Introduction

1.1 Literature review

The core of portfolio selection problem involves how to allocate assets among large numbers of alternative securities and construct a satisfying portfolio. The ground-breaking mean-variance (MV) model originally proposed by Markowitz [1] took a major step in the development of modern portfolio theory. After Markowitz’s work, many scholars have improved and extended the M-V models in different ways, wherein one important direction is to explore different alternative measures of risk. Some portfolio optimization models based on different risk measures have been developed, such as mean semi-variance model [2, 3], mean absolute deviation model [4], mean semi-absolute deviation model [5], mean entropy model [6], and CVaR model [7].

In the aforementioned studies, it has been assumed that the financial market is only affected by probabilistic factors and the returns of risky assets are characterized as random variables with probability distributions. However, there exist large numbers of non-probabilistic factors in nature such as vagueness and ambiguity. In this case, it is difficult to capture the fuzzy information through probabilistic methods. Fuzzy set theory proposed by Zadeh [8] has been one of the most popular tools to deal with the fuzzy uncertainty involved in the financial market. Since the possibility theory proposed by Zadeh [9], the fuzzy portfolio problems have received much attention from researchers and investors. For example, Carlsson and Full $\overset{´}{e}$ r [10] proposed the concepts of possibilistic mean and variance of fuzzy variables. Zhang and Nie [11] developed the notions of lower and upper possibilistic variance. Besides, Zhang et al. [12] discussed the possibilistic efficient frontier for the possibilistic mean-variance portfolio selection problem. Huang [13] presented the concept of semi-variance of fuzzy variables and then took it as a risk measure to construct fuzzy portfolio selection models. Vercher et al. [14] studied the fuzzy portfolio optimization under downside risk measures. Chen [15] employed possibilistic semi-absolute deviation as risk measure and some realistic constraints are considered. Yue and Wang [16] attempted to utilize the higher-order moments for the fuzzy multi-objective portfolio selection model. Deng and Chen [17] defined the lower partial moments as the risk measure based on the possibility density function in a novel fuzzy framework. Deng et al. [18] extended the downside risk from stochastic environment to hesitant fuzzy environment and define a new hesitant semi-variance. It can be seen from the above literature that the field of fuzzy portfolio optimization has grown significantly with various risk measures.

Alternatively, the psychological factors of investors play a significant role in the investment decision-making process. The investors will handle the existing information according to their subjective judgment to some degree, and are not completely rational in the fuzzy environment. From this perspective, some researchers attempted to take the psychological factors into account in portfolio management. For instance, Kahneman and Tversky [18] proposed the prospect theory, which can describe the non-rational psychological aspects of investors. Tsaur [20] developed a fuzzy portfolio model that focused on addressing the different investor attitudes. Momen et al. [21] developed a behavioral portfolio selection model that used a robust estimator for expected returns. Zhou et al. [22] studied the fuzzy portfolio selection problem based on varying conservative-neutral-aggressive attitudes. Khayamim [23] presented a two-stage portfolio rebalancing strategy to integrate mean-variance theory with market psychology in a fuzzy environment. Xue et al. [24] applied mental accounts to reflect different attitudes toward risk in the uncertain portfolio selection. Gong et al. [25] proposed a multi-period portfolio selection with investor psychology under the fuzzy environment.

Due to the great development in portfolio optimization, more and more attention has been paid to the field of portfolio performance assessment from both academic and practical viewpoints. In the portfolio performance evaluation, apart from the three well-known performance measures, the Sharp index [26], the Treynor index [27] and the Jenson index [28], another extensively used methodology is the real frontier approach (RFA). However, owing to the complexity of the real financial market, it is difficult to obtain the real portfolio frontier so the method has certain limitations. To tackle this problem, some performance methods have been extended, wherein Data Envelopment Analysis (DEA) technique [29] as an alternative method has created a great deal of interest among researchers. As a non-parametric approach, DEA can integrate multi-performance measures into a key index, which is extremely suitable for portfolio evaluation. Murthi et al. [30] first employed DEA technique to assess portfolio performance and proposed the DEA portfolio efficiency index (DEPI) as a generalization of Sharpe index. Morey and Morey [31] developed quadratic-constrained nonlinear DEA models, in which the expected return was regarded as an output and the variance as an input. Joro and Na [32] discussed portfolio performance evaluation problems in the mean-variance-skewness framework by utilizing DEA. Liu et al. [33] investigated the theoretical foundation of DEA models from the perspective of a sampling portfolio. It further indicated that the classical DEA could be an effective and practical tool for evaluating portfolio efficiency. After this work, Zhou et al [34] innovatively applied a segmented DEA approach to evaluate the performance of the cardinality constrained portfolio selection problem. Branda [35] presented diversification consistent DEA models based on directional distance measures, which could use several risk measures as input and return measures as output. Banihashemi and Navidi [36] concentrated on the portfolio performance evaluation in Mean-CVaR framework. In order to deal with the negative data, one DEA model named Range Directional Measure (RDM) was considered. Zhou et al. [37] proposed a DEA frontier improvement approach under the mean-variance framework, which could provide investors with rebalancing strategies and afford some guidance for future investments. It should be noted that the above-mentioned literature on portfolio performance evaluation is under the assumption that the returns of risky assets are random variables instead of fuzzy variables. Few scholars studied the portfolio efficiency evaluation in the fuzzy environment. Chen et al. [38] proposed the DEA-based possibilistic fuzzy portfolio evaluation models in different risk measures. Gupta et al. [39] employed VaR or CVaR as risk measure under the credibilistic environment to evaluate portfolio efficiency. Xiao [40] proposed an approach under the fuzzy theory framework that can both improve the DEA frontier and suggest a replicable benchmark for investors. Gong [41] proposed a fuzzy multi-objective portfolio selection model involving DEA cross-efficiency and higher moments.

DEA method and its model have been widely used in different industries and departments since it was proposed in 1978. Chen et al. [42] developed a new flexible DEA method for vendor selection and performance evaluation. The case study showed the feasibility and superiority of the proposed approach. Chen et al. [43] applied DEA to analyze the spending efficiency and winning efficiency of the 2019 International Table Tennis Association World Tour in order to develop effective participation strategies for young athletes in future competitions. Le et al. [44] selected 29 river basins in the Dong Nai River Basin as decision-making units and used DEA to study the water resource utilization efficiency in the Dong Nai River Basin in Vietnam during 2010-2017. A [45] applied two-stage data envelopment analysis to the operational efficiency assessment of insurance companies in Thailand.

In order to better sort out the previous articles and compare the differences between our articles and previous work, we made the following table:

From the above table, we can find that the current articles only consider one or two of the three factors: investor psychology, DEA and risk attitude. Before our work, there was no article that considered all three. Compared with previous literature, we can draw the following advantages:

We fully consider the fuzzy uncertainty and psychological factors in portfolio management, and derive some return and risk measures including the possibilistic mean, variance, semi-variance and semi-absolute deviation. Then, three return-return frameworks with risk attitudes are constructed.

We creatively incorporate risk attitude into the portfolio efficiency evaluation model based on DEA. Furthermore, we prove the theoretical basis of DEA model used to estimate fuzzy portfolio with risk attitude.

We propose two new indicators to measure the consistency of portfolio sample evaluation results between the real frontier method and our proposed DEA method.

1.2 Research motivation

Through the review of the above literature, we can find that the previous literature only considered the psychological factors of investors, or only considered the risk attitude of investors, or only considered the application of DEA in portfolio efficiency assessment. In previous work, there is no research on the combination of investor’s psychological factors, risk attitude and DEA. The research motivation of this paper is to make up for the weakness in this aspect. Therefore, this paper comprehensively considers the fuzzy uncertainty of the stock market and the psychological factors of investors, and studies the problems of portfolio optimization and performance evaluation under different risk attitudes in possible environments.

In this work, the return rates of assets are characterized as the trapezoidal fuzzy variables whose membership functions with a risk attitude parameter by exponential expression. Firstly, the return and risk measures with risk attitude including the possibilistic mean, variance, semi-variance and semi-absolute deviation are strictly derived based on possibility theory. Secondly, under three possibilistic return-risk frameworks, several portfolio optimization models with different risk attitudes are proposed. As different risk measures are taken into account, it can be seen as improvements and extensions of the work by Tsaur (2013). Thirdly, the theoretical foundation of DEA models for evaluating the performance of the above portfolio optimization models with risk attitudes is further justified by rigorous mathematical proof. By utilizing DEA approach, the corresponding portfolio efficiency evaluation models are constructed. Indeed, the work by Chen et al. (2018) can be regarded as a special case under the assumption that the investors all hold risk-neutral attitudes. Finally, the comprehensive simulation results show that no matter what risk attitude an investor holds, the DEA frontiers generated by adequate sample size can effectively approximate real frontiers. In addition, two novel indicators are designed as effective supplements characterizing the correlation between DEA scores and real portfolio efficiencies. The results of correlation analysis further indicate the feasibility and effectiveness of our proposed approach in evaluating the performance of possibilistic fuzzy portfolios with risk attitudes.

1.3 Novelty of our method

Although Data Envelopment Analysis (DEA) approach has been employed as an effective tool for portfolio performance evaluation, no researcher has yet considered the important effect of investor risk attitude. In this work, following the work by Chen et al. [38], we are devoted to incorporating risk attitudes of investors into the possibilistic portfolio performance evaluation problem under three return-risk frameworks. The major novel contributions of the proposed method are as follows:

In order to fully consider the fuzzy uncertainty and psychological factors in portfolio management, some return and risk measures including the possibilistic mean, variance, semi-variance and semi-absolute deviation are deduced. Then, three return-return frameworks with risk attitudes are constructed. As different risk measures are taken into account, it can be seen as improvements and extensions of the work by Tsaur [20].

We creatively integrate the risk attitude factor into DEA based portfolio efficiency evaluation model. In the meanwhile, the theoretical foundation of DEA models for estimating the fuzzy portfolio with risk attitudes is justified by rigorous proof. Indeed, the work by Chen et al [38] can be regarded as a special case under the assumption that the investors all hold risk-neutral attitudes.

In the correlation analysis, two novel indexes are proposed to measure the consistency of evaluation results for portfolio samples between the real frontier approach and our proposed DEA methods. Through the comprehensive simulations and correlation analysis, we can conclude that no matter what risk attitude an investor holds, the proposed DEA models can be used as effective tools to estimate the portfolio performance.

1.4 The organization of our paper

The rest of the paper is structured as follows. Section 2 reviews basic knowledge concerning the possibility theory. In Section 3, we first derive several possibilistic return and risk measures with risk attitudes theoretically, and then propose some possibilistic fuzzy portfolio models with investor risk attitudes under three risk-return frameworks. In Section 4, the definition of portfolio efficiency (PE) is introduced. In addition, the theoretical foundation for DEA method to estimate the efficiencies of fuzzy portfolios with risk attitudes is investigated. Then, the corresponding DEA models used to estimate the PE are constructed. In Section 5, a large number of simulations are carried out to illustrate the effectiveness of our proposed models. The conclusion of this paper is summarized in Section 6.

2 Preliminaries

In this section, some concepts and main results of possibility theory are reviewed, which will be used in the following sections.

Definition 2.1. [8] Let $\tilde{A}$ be a fuzzy number with membership function $μ_{\tilde{A}} (x)$ . Then the γ - level set $[\tilde{A}]^{γ} = {x \in R | μ_{\tilde{A}} (x) ⩾ γ}$ of $\tilde{A}$ can be written as $[\tilde{A}]^{γ} = [a_{1} (γ), a_{2} (γ)]$ .

Definition 2.2. [8] A fuzzy number $\tilde{A} = {(a, b, α, β)}_{LR}$ is said to be an LR-fuzzy number if its membership function $μ_{\tilde{A}} (x)$ has the following form:

$μ_{\tilde{A}} (x) = {\begin{matrix} L_{\tilde{A}} (\frac{a - x}{α}), a - α ⩽ x < a, \\ 1, a ⩽ x < b, \\ R_{\tilde{A}} (\frac{x - b}{β}), b ⩽ x < b + β, \\ 0, otherwise, \end{matrix}$ (1)

Where a and b are the lower and upper bound of the core of $\tilde{A}$ . $L_{\tilde{A}}, R_{\tilde{A}} : [0, 1] \to [0, 1]$ are reference functions which are continuous non-increasing and satisfy $L_{\tilde{A}} (0) = R_{\tilde{A}} (0) = 1$ , $L_{\tilde{A}} (1) = R_{\tilde{A}} (1) = 0$ .

Example 2.1 (LR-power fuzzy numbers). Let an LR-fuzzy number $\tilde{A} = {(a, b, α, β)}_{LR}$ have the reference functions $L_{\tilde{A}} (t) = max (0, 1 - t^{p})$ , $R_{\tilde{A}} (t) = max (0, 1 - t^{q})$ . Then it can be denoted by $\tilde{A} = {(a, b, α, β)}_{p, q}$ and its membership function $μ_{\tilde{A}} (x)$ has the following form:

$μ_{\tilde{A}} (x) = {\begin{matrix} 1 - {(\frac{a - x}{α})}^{p}, a - α ⩽ x < a, \\ 1, a ⩽ x < b, \\ 1 - {(\frac{a - x}{α})}^{q}, b ⩽ x < b + β, \\ 0, otherwise, \end{matrix}$ (2)

Note a trapezoidal fuzzy number $\tilde{A} = (a, b, α, β)$ is a particular case when p = q = 1. Furthermore, if a = b and p = q = 1 then the number is a triangular number.

Definition 2.3. [14] Let $\tilde{A} = {(a_{1}, b_{1}, α_{1}, β_{1})}_{LR}$ and $\tilde{B} = {(a_{2}, b_{2}, α_{2}, β_{2})}_{LR}$ be two LR-fuzzy numbers and let λ ∈ R be a real number. Then we have $\begin{matrix} \tilde{A} + \tilde{B} = {(a_{1} + a_{2}, b_{1} + b_{2}, α_{1} + α_{2}, β_{1} + β_{2})}_{LR}, \end{matrix}$ $\begin{matrix} λ \tilde{A} = {\begin{matrix} {(λ a_{1}, λ b_{1}, λ α_{1}, λ β_{1})}_{LR} λ ⩾ 0, \\ {(λ b_{1}, λ a_{1}, | λ | β_{1}, | λ | α_{1})}_{LR} λ < 0, \end{matrix} \end{matrix}$ where the above rules for addition and multiplication by a scalar satisfy the sup-min extension principle.

Definition 2.4. [10] Let $\tilde{A}$ be a fuzzy number with γ - level set $[\tilde{A}]^{γ} = [a_{1} (γ), a_{2} (γ)]$ , then the lower and upper possibilistic mean values of $\tilde{A}$ are defined as:

$\begin{matrix} M_{*} (\tilde{A}) = 2 \int_{0}^{1} γ a_{1} (γ) d γ, \\ M^{*} (\tilde{A}) = 2 \int_{0}^{1} γ a_{2} (γ) d γ . \end{matrix}$ (3)

Furthermore, the interval-valued and crisp possibilistic mean values are as follows:

$M (\tilde{A}) = [M_{*} (\tilde{A}), M^{*} (\tilde{A})],$ (4)

$\bar{M} (\tilde{A}) = \frac{1}{2} [M_{*} (\tilde{A}) + M^{*} (\tilde{A})] .$ (5)

Definition 2.5. [10] Let $\tilde{A}$ be a fuzzy number with γ - level set $[\tilde{A}]^{γ} = [a_{1} (γ), a_{2} (γ)]$ , then the possibilistic variance of $\tilde{A}$ is denoted by:

$Var (\tilde{A}) = \frac{1}{2} \int_{0}^{1} γ {(a_{2} (γ) - a_{1} (γ))}^{2} d γ .$ (6)

Definition 2.6. [11] Let $\tilde{A}$ be a fuzzy number with γ - level set $[\tilde{A}]^{γ} = [a_{1} (γ), a_{2} (γ)]$ , then the possibilistic semi-variance of $\tilde{A}$ is denoted by:

${Var}^{-} (\tilde{A}) = 2 \int_{0}^{1} γ {[\bar{M} (\tilde{A}) - a_{1} (γ)]}^{2} d γ .$ (7)

Definition 2.7 [14] Let $\tilde{A}$ be a fuzzy number with γ - level set $[\tilde{A}]^{γ} = [a_{1} (γ), a_{2} (γ)]$ , then the crisp possibilistic semi-absolute deviation of $\tilde{A}$ is represented as follows:

$Sad (\tilde{A}) = \bar{M} (max {0, M (\tilde{A}) - \tilde{A}}) .$ (8)

3 Possibilistic fuzzy portfolio models with different investor risk attitudes

3.1 Possibilistic return and risk measures with different risk attitudes

Based on possibility theory, Tsaur [20] proposed a fuzzy portfolio model with different investor risk attitudes. The fuzzy returns of assets are characterized as triangular fuzzy numbers, where the risk attitude parameter k (k > 0) is introduced into the membership functions. In this paper, the returns of assets are represented as trapezoid fuzzy numbers, which can easily be generalized to the case of possibility distribution of type LR or triangle. The membership function of the fuzzy return ${\tilde{r}}_{i}$ of asset i with the risk attitude parameter k can be defined as follows:

$u_{{\tilde{r}}_{i}} (x) = {\begin{matrix} 1 - {(\frac{a_{i} - x}{α_{i}})}^{k}, & a_{i} - α_{i} ⩽ x < a_{i}, \\ 1, & a_{i} ⩽ x < b_{i}, \\ 1 - {(\frac{x - b_{i}}{β_{i}})}^{k}, & b_{i} ⩽ x < b_{i} + β_{i}, \\ 0, & otherwize . \end{matrix}$ (9)

In the Equation (9), the value of the parameter k varies from the investor risk attitudes. A risk-seeking investor tends to overestimate the positive outcomes and set k > 1, representing the convex utility function. In contrast, a risk-averse investor is eager to overestimate the negative outcomes and set 0 < k < 1, representing the concave utility function. For a risk-neutral investor, he or she has more mild concern about risk and set k = 1, whose utility function is linear. In fact, ${\tilde{r}}_{i}$ is a particular case of LR-power fuzzy numbers when p = q = k. From this perspective, ${\tilde{r}}_{i}$ can be denoted by (a_i, b_i, α_i, β_i) _k more accurately. Next, some relevant possibilistic return and risk measures of ${\tilde{r}}_{i} = {(a_{i}, b_{i}, α_{i}, β_{i})}_{k}$ will be derived theoretically.

Theorem 3.1. The possibilistic mean of LR-power fuzzy number ${\tilde{r}}_{i} = {(a_{i}, b_{i}, α_{i}, β_{i})}_{k}$ is

$\begin{matrix} \bar{M} ({\tilde{r}}_{i}) = & \frac{M_{*} ({\tilde{r}}_{i}) + M^{*} ({\tilde{r}}_{i})}{2} = \frac{b_{i} + a_{i}}{2} \\ + \frac{k^{2}}{(2 k + 1) (k + 1)} (β_{i} - α_{i}) . \end{matrix}$ (10)

Where k > 0 is the risk attitude parameter.

Proof. According to Definition 2.1, the γ - level set of ${\tilde{r}}_{i} = {(a_{i}, b_{i}, α_{i}, β_{i})}_{k}$ is

$\begin{matrix} {[{\tilde{r}}_{i}]}^{γ} = & [r_{i 1} (γ), r_{i 2} (γ)] \\ = [a_{i} - {(1 - γ)}^{\frac{1}{k}} α_{i}, b_{i} + {(1 - γ)}^{\frac{1}{k}} β_{i}], \\ γ \in (0, 1) . \end{matrix}$ (11)

By Definition 2.4, the lower and upper possibilistic mean values of ${\tilde{r}}_{i} = {(a_{i}, b_{i}, α_{i}, β_{i})}_{k}$ are as follows:

$\begin{matrix} M_{*} ({\tilde{r}}_{i}) = 2 \int_{0}^{1} γ r_{i 1} (γ) d γ \\ = 2 \int_{0}^{1} γ [a_{i} - {(1 - γ)}^{\frac{1}{k}} α_{i}] d γ \\ = a_{i} - \frac{2 k^{2}}{(2 k + 1) (k + 1)} α_{i} . \end{matrix}$ (12)

$\begin{matrix} M^{*} ({\tilde{r}}_{i}) & = 2 \int_{0}^{1} γ r_{i 2} (γ) d γ \\ = 2 \int_{0}^{1} γ [b_{i} + {(1 - γ)}^{\frac{1}{k}} β_{i}] d γ \\ = b_{i} + \frac{2 k^{2}}{(2 k + 1) (k + 1)} β_{i} . \end{matrix}$ (13)

Furthmore, the crisp possibilistic mean value of ${\tilde{r}}_{i} = {(a_{i}, b_{i}, α_{i}, β_{i})}_{k}$ is as follows: $\begin{matrix} \bar{M} ({\tilde{r}}_{i}) = & \frac{M_{*} ({\tilde{r}}_{i}) + M^{*} ({\tilde{r}}_{i})}{2} = \frac{b_{i} + a_{i}}{2} \\ + \frac{k^{2}}{(2 k + 1) (k + 1)} (β_{i} - α_{i}) . \end{matrix}$

The proof is complete.

Theorem 3.2. The possibilistic variance of LR-power fuzzy number ${\tilde{r}}_{i} = {(a_{i}, b_{i}, α_{i}, β_{i})}_{k}$ is

$\begin{matrix} Var ({\tilde{r}}_{i}) = & {[\frac{b_{i} - a_{i}}{2} + \frac{k^{2} (α_{i} + β_{i})}{(2 k + 1) (k + 1)}]}^{2} \\ + \frac{k^{2} + 5 k^{3}}{4 {(2 k + 1)}^{2} {(k + 1)}^{2} (k + 2)} {(α_{i} + β_{i})}^{2} . \end{matrix}$ (14)

Proof. According to Definition 2.5, $\begin{matrix} \begin{matrix} Var ({\tilde{r}}_{i}) = \frac{1}{2} \int_{0}^{1} γ {[r_{i 2} (γ) - r_{i 1} (γ)]}^{2} d γ \\ = \frac{1}{2} \int_{0}^{1} γ {[(b_{i} - a_{i}) + {(1 - γ)}^{\frac{1}{k}} (β_{i} + α_{i})]}^{2} d γ \\ = \frac{{(b_{i} - a_{i})}^{2}}{4} + \frac{k^{2} (b_{i} - a_{i}) (α_{i} + β_{i})}{(2 k + 1) (k + 1)} \\ + \frac{k^{2} {(α_{i} + β_{i})}^{2}}{4 (k + 1) (k + 2)} \\ = {[\frac{b_{i} - a_{i}}{2} + \frac{k^{2} (α_{i} + β_{i})}{(2 k + 1) (k + 1)}]}^{2} \\ + \frac{k^{2} + 5 k^{3}}{4 {(2 k + 1)}^{2} {(k + 1)}^{2} (k + 2)} {(α_{i} + β_{i})}^{2} . \end{matrix} \end{matrix}$

The proof is complete.

Theorem 3.3. The possibilistic semi-variance of LR-power fuzzy number ${\tilde{r}}_{i} = {(a_{i}, b_{i}, α_{i}, β_{i})}_{k}$ is

$\begin{matrix} {Varr}^{-} ({\tilde{r}}_{i}) = & {[\frac{b_{i} - a_{i}}{2} + \frac{k^{2} (α_{i} + β_{i})}{(2 k + 1) (k + 1)}]}^{2} \\ + \frac{k^{2} + 5 k^{3}}{{(2 k + 1)}^{2} {(k + 1)}^{2} (k + 2)} α_{i}^{2} . \end{matrix}$ (15)

Proof. According to Definition 2.6 and Theorem 2.1,

$\begin{matrix} \begin{matrix} {Var}^{-} ({\tilde{r}}_{i}) = 2 \int_{0}^{1} γ {[\bar{M} ({\tilde{r}}_{i}) - r_{i 1} (γ)]}^{2} d γ \\ = \frac{1}{2} \int_{0}^{1} {[\frac{b_{i} - a_{i}}{2} + \frac{2 k^{2} (β_{i} - α_{i})}{(2 k + 1) (k + 1)} + {(1 - γ)}^{\frac{1}{k}} α_{i}]}^{2} d γ \\ = {[\frac{b_{i} - a_{i}}{2} + \frac{k^{2} (α_{i} + β_{i})}{(2 k + 1) (k + 1)}]}^{2} \\ + \frac{k^{2} + 5 k^{3}}{{(2 k + 1)}^{2} {(k + 1)}^{2} (k + 2)} α_{i}^{2} . \end{matrix} \end{matrix}$

The proof is complete.

Theorem 3.4. The possibilistic semi-absolute deviation of LR-power fuzzy number ${\tilde{r}}_{i} = {(a_{i}, b_{i}, α_{i}, β_{i})}_{k}$ is

$Sad ({\tilde{r}}_{i}) = \frac{1}{2} (b_{i} - a_{i}) + \frac{k^{2}}{(2 k + 1) (k + 1)} (α_{i} + β_{i}) .$ (16)

Proof. Let $M ({\tilde{r}}_{i}) = [M_{*} ({\tilde{r}}_{i}), M^{*} ({\tilde{r}}_{i})]$ be the interval-valued possibilistic mean of LR-power fuzzy number ${\tilde{r}}_{i} = {(a_{i}, b_{i}, α_{i}, β_{i})}_{k}$ . Indeed, $M ({\tilde{r}}_{i})$ can be considered as an LR-power fuzzy number with right and left spread zero, that is $M ({\tilde{r}}_{i}) = {(M_{*} ({\tilde{r}}_{i}), M^{*} ({\tilde{r}}_{i}), 0, 0)}_{k}$ . According to Definition 2.3 and Definition 2.7, we can obtain

$\begin{matrix} M ({\tilde{r}}_{i}) - {\tilde{r}}_{i} \\ = {(M_{*} ({\tilde{r}}_{i}), M^{*} ({\tilde{r}}_{i}), 0, 0)}_{k} - {(a_{i}, b_{i}, α_{i}, β_{i})}_{k} \\ = (a_{i} - \frac{2 k^{2}}{(2 k + 1) (k + 1)} α_{i}, b_{i} \\ {+ \frac{2 k^{2}}{(2 k + 1) (k + 1)} β_{i}, 0, 0)}_{k} + {(- b_{i}, - a_{i}, β_{i}, α_{i})}_{k} \\ = (a_{i} - b_{i} - \frac{2 k^{2} α_{i}}{(2 k + 1) (k + 1)}, b_{i} - a_{i} \\ {+ \frac{2 k^{2}}{(2 k + 1) (k + 1)} β_{i}, β_{i}, α_{i})}_{k}, \end{matrix}$ (17) in addition,

$\begin{matrix} max {0, M ({\tilde{r}}_{i}) - {\tilde{r}}_{i}} \\ = {(0, b_{i} - a_{i} + \frac{2 k^{2}}{(2 k + 1) (k + 1)} β_{i}, 0, α_{i})}_{k} . \end{matrix}$ (18)

Therefore, by Theorem 3.1, the possibilistic semi-absolute deviation of ${\tilde{r}}_{i} = {(a_{i}, b_{i}, α_{i}, β_{i})}_{k}$ is $\begin{matrix} Sad ({\tilde{r}}_{i}) = & \bar{M} (max {0, M ({\tilde{r}}_{i}) - {\tilde{r}}_{i}}) = \frac{1}{2} (b_{i} - a_{i}) \\ + \frac{k^{2}}{(2 k + 1) (k + 1)} (α_{i} + β_{i}) . \end{matrix}$

The proof is complete.

Example 3.1 Let ${\tilde{r}}_{i} = {(a_{i}, b_{i}, α_{i}, β_{i})}_{k}$ be the fuzzy return of asset i, i = 1, 2, ⋯ , n, and let ${\tilde{r}}_{p}$ be total return of a portfolio x = (x₁, x₂, ⋯ , x_n) ^T, (x_i ⩾ 0, i = 1, 2, ⋯ , n), then we have: $\begin{matrix} \bar{M} ({\tilde{r}}_{p}) = \bar{M} (\sum_{i = 1}^{n} {\tilde{r}}_{i} x_{i}) \\ = \sum_{i = 1}^{n} [\frac{b_{i} + a_{i}}{2} + \frac{k^{2}}{(2 k + 1) (k + 1)} (β_{i} - α_{i})] x_{i}, \end{matrix}$ $\begin{matrix} Var ({\tilde{r}}_{p}) = {(\sum_{i = 1}^{n} [\frac{b_{i} - a_{i}}{2} + \frac{k^{2} (α_{i} + β_{i})}{(2 k + 1) (k + 1)}] x_{i})}^{2} \\ + \frac{k^{2} + 5 k^{3}}{4 {(2 k + 1)}^{2} {(k + 1)}^{2} (k + 2)} {(\sum_{i = 1}^{n} (α_{i} + β_{i}) x_{i})}^{2}, \end{matrix}$ $\begin{matrix} {Var}^{-} ({\tilde{r}}_{p}) \\ = {(\sum_{i = 1}^{n} [\frac{b_{i} - a_{i}}{2} + \frac{k^{2} (α_{i} + β_{i})}{(2 k + 1) (k + 1)}] x_{i})}^{2} \\ + \frac{k^{2} + 5 k^{3}}{{(2 k + 1)}^{2} {(k + 1)}^{2} (k + 2)} {(\sum_{i = 1}^{n} α_{i} x_{i})}^{2}, \end{matrix}$ $\begin{matrix} Sad ({\tilde{r}}_{p}) = & \frac{1}{2} \sum_{i = 1}^{n} (b_{i} - a_{i}) x_{i} + \frac{k^{2}}{(2 k + 1) (k + 1)} \\ \sum_{i = 1}^{n} (α_{i} + β_{i}) x_{i} . \end{matrix}$

Proof. By Definition 2.3, the total return of a portfolio x = (x₁, x₂, ⋯ , x_n) ^T is still an LR-power fuzzy number in the form as follows:

$\begin{matrix} {\tilde{r}}_{p} = \sum_{i = 1}^{n} {\tilde{r}}_{i} x_{i} \\ = {(\sum_{i = 1}^{n} a_{i} x_{i}, \sum_{i = 1}^{n} b_{i} x_{i}, \sum_{i = 1}^{n} α_{i} x_{i}, \sum_{i = 1}^{n} β_{i} x_{i})}_{k} . \end{matrix}$ (19)

Based on Theorems 3.1–3.4, it is easy to obtain the above conclusions (a)–(d). Therefore, the proof is complete.

3.2 Possibilistic portfolio optimization models with risk attitudes

Based on the possibilistic return and risk measures with risk attitude parameter derived from the previous section, some possibilistic fuzzy portfolio optimization models with different investor risk attitudes under three risk-return frameworks would be constructed.

3.2.1 Possibilistic mean variance portfolio model with risk attitudes

Assume that there are n risky assets to be invested, where the fuzzy return of asset i is characterized as \hfilneg a fuzzy variable ${\tilde{r}}_{i} = {(a_{i}, b_{i}, α_{i}, β_{i})}_{k}$ , and x_i is the proportion of asset i. It is worth noting that there are various market frictions and restrictions in the actual investment environment. In this paper, linear transaction cost and investment ratio restrictions are mainly considered. Suppose that c_i is the unit transaction cost for asset i. Then the total transaction cost for the portfolio x = (x₁, x₂, ⋯ , x_n) ^T can be stated as $\sum_{i = 1}^{n} c_{i} x_{i}$ . Hence, the possibilistic mean of the net return of portfolio is formulated as follows:

$\bar{M} (\sum_{i = 1}^{n} {\tilde{r}}_{i} x_{i} - \sum_{i = 1}^{n} c_{i} x_{i}) = \sum_{i = 1}^{n} [\frac{b_{i} + a_{i}}{2} + \frac{k^{2} (β_{i} - α_{i})}{(2 k + 1) (k + 1)} - c_{i}] x_{i}$ (20)ased on the traditional Markowitz’s mean variance model, let the possibilistic mean and variance be considered as the main objectives when optimizing the portfolio selection respectively. The possibilistic mean-variance portfolio models with risk attitudes is constructed as follows:

{\begin{cases} \max \sum_{i = 1}^{n} [\frac{b_{i} + a_{i}}{2} + \frac{k^{2} (β_{i} - α_{i})}{(2 k + 1) (k + 1)} - c_{i}] x_{i} \\ s . t . {(\sum_{i = 1}^{n} [\frac{b_{i} - a_{i}}{2} + \frac{k^{2} (α_{i} + β_{i})}{(2 k + 1) (k + 1)}] x_{i})}^{2} + \frac{(5 k^{3} + k^{2}) {(\sum_{i = 1}^{n} (α_{i} + β_{i}) x_{i})}^{2}}{4 {(2 k + 1)}^{2} {(k + 1)}^{2} (k + 2)} \leq σ_{0}, \\ \sum_{i = 1}^{n} x_{i} = 1, \\ 0 \leq l_{i} \leq x_{i} \leq u_{i}, i = 1, 2, \dots, n . \end{cases}

(21)

where k is the risk attitude parameter, σ₀ is the preset maximum risk by the investor, and l_i, u_i are the upper and lower limits of the investment ratio, respectively.

3.2.2 Possibilistic mean semi-variance portfolio model with risk attitudes

The main drawback of possibilistic variance as a risk measure is that it makes no distinction between the deviations higher or lower than expected returns. It is inconsistent with the actual risk preference of investors. As a downside risk, the probabilistic semi-variance penalizes the deviations below the expected return, which can better describe the realistic risk perception. From this perspective, taking transaction costs and investment ratio restrictions into account, the possibilistic mean semi-variance portfolio models with risk attitudes are formulated as follows: ${\begin{matrix} max \sum_{i = 1}^{n} [\frac{b_{i} + a_{i}}{2} + \frac{k^{2}}{(2 k + 1) (k + 1)} (β_{i} - α_{i}) - c_{i}] x_{i} \\ s . t . {(\sum_{i = 1}^{n} [\frac{b_{i} - a_{i}}{2} + \frac{k^{2} (α_{i} + β_{i})}{(2 k + 1) (k + 1)}] x_{i})}^{2} + \frac{(k^{2} + 5 k^{3})}{{(2 k + 1)}^{2} {(k + 1)}^{2} (k + 2)} {(\sum_{i = 1}^{n} α_{i} x_{i})}^{2} ⩽ σ_{0}, \\ \sum_{i = 1}^{n} x_{i} = 1, \\ 0 ⩽ l_{i} ⩽ x_{i} ⩽ u_{i}, i = 1, 2, \dots, n . \end{matrix}$ (22)

3.2.3 Possibilistic mean semi-absolute deviation portfolio model with risk attitudes

Compared with the measure of possibilistic semi-variance, the risk function of possibilistic semi-absolute deviation as another measure of downside risk is a linear form and can be more easily evaluated. Using possibilistic semi-absolute deviation as the risk of portfolio can not only be closer to the actual risk perception of investors, but also reduce the complexity of the task of solving the portfolio optimization model. Considering transaction costs and investment ratio restrictions, the possibilistic mean semi-absolute deviation portfolio models with risk attitudes are established as follows:

{\begin{matrix} max \sum_{i = 1}^{n} [\frac{b_{i} + a_{i}}{2} + \frac{k^{2}}{(2 k + 1) (k + 1)} (β_{i} - α_{i}) - c_{i}] x_{i} \\ s . t . \sum_{i = 1}^{n} [\frac{b_{i} - a_{i}}{2} + \frac{k^{2}}{(2 k + 1) (k + 1)} (β_{i} + α_{i})] x_{i} ⩽ σ_{0}, \\ \sum_{i = 1}^{n} x_{i} = 1, \\ 0 ⩽ l_{i} ⩽ x_{i} ⩽ u_{i}, i = 1, 2, \dots, n . (23) \end{matrix}

4 Portfolio efficiency evaluation models with risk attitudes via DEA method

4.1 Portfolio efficiency (PE) definition

According to the notion of technical efficiency in classical economics, Morey and Morey (1999) put forward the definition of portfolio efficiency (PE) based on the real frontier approach. As shown in Figure 1, the abscissa represents risk, while the ordinate indicates return. The curve B₁CB₂ represents the real efficient frontier without risk-free assets under the possibilistic mean variance framework, where B₁ (r₁, v₁), B₂ (r₂, v₂) and C (r₃, v₃) are optimal portfolios on the efficient frontier. Let A (r, v) denote a portfolio under evaluation, whose portfolio efficiency can be defined by the relative distances from the reference points on the efficient frontier. According to three different projection ways, we can obtain the following portfolio efficiencies with different orientations.

Fig. 1

Portfolio efficiencies with different orientations.

$P E_{r} = \frac{r}{r_{2}}, P E_{v} = \frac{v_{1}}{v}, P E_{rv} = \frac{1 - (v - v_{3}) / - v}{1 + (r_{3} - r) / - r},$ (23) where PE_r represents the return-oriented efficiency, PE_v represents the risk-oriented efficiency, and PE_rv is the non-oriented efficiency.

As described in the literature (Joro and Na 2006; Liu et al. 2015; Zhou et al. 2018; Branda 2015), due to the complexity of securities market environment, it is difficult to obtain the analytical solution to the efficient frontier of portfolio. That is to say, there are some limitations in obtaining PE with the real frontier approach. Some scholars have been devoted to developing alternative methods about the performance evaluation of portfolios. When it comes to the performance evaluation problems, suitable DEA models would always be good choices to estimate the efficiencies of fuzzy portfolios with various restrictions. In the following, we will discuss the theoretical foundation of DEA method for the portfolio efficiency evaluation problem.

4.2 Theoretical foundation of DEA method: convergence property

According to the convergence theory proposed by Banker et al [46], Liu et al. [33] firstly investigated the theoretical foundation of using DEA approach to estimate the efficiency of portfolio under the random environment. In addition, they pointed out that the frontiers of suitable DEA models with adequate samples can effectively approximate real frontiers for portfolios if the real efficient frontiers are concave. Inspired by the above work, here we further construct the theoretical foundation for DEA method to evaluate the efficiencies of fuzzy portfolios with different investor risk attitudes under the possibilistic fuzzy environment.

Lemma 4.1 [33] Consider the following general portfolio model (25),

${\begin{matrix} max f (x) \\ s . t . g (x) ⩽ V_{0}, \\ x \in Ω . \end{matrix}$ (24)

If the risk function g (x) is convex, and the return function f (x) is concave with a convex set Ω. Then the real frontier R = h (V) determined by Model (25) is concave.

Lemma 4.2 [33] Suppose that R = h (V) is the real frontier of model (25) and $h_{N}^{*} (V)$ is the DEA-BCC frontier with N portfolio samples. Then $h_{N}^{*} (V)$ converges to R = h (V) in probability when N→ ∞.

Theorem 4.1 The real frontier determined by Model (21) is a concave function.

Proof: In Model (21), let a domain $Ω = {x = {(x_{1}, x_{2}, \dots, x_{n})}^{T} | \sum_{i = 1}^{n} x_{i} = 1, l_{i} ⩽ x_{i} ⩽ u_{i}}$ , The risk function g (x) is stated as follows:

$\begin{matrix} g (x) = & {(\sum_{i = 1}^{n} [\frac{b_{i} - a_{i}}{2} + \frac{k^{2} (α_{i} + β_{i})}{(2 k + 1) (k + 1)}] x_{i})}^{2} \\ + \frac{5 k^{3} + k^{2}}{4 {(2 k + 1)}^{2} {(k + 1)}^{2} (k + 2)} \\ {(\sum_{i = 1}^{n} (α_{i} + β_{i}) x_{i})}^{2}, \end{matrix}$ (25) and the return function f (x) is as follows:

$f (x) = \sum_{i = 1}^{n} [\frac{b_{i} + a_{i}}{2} + \frac{k^{2} (β_{i} - α_{i})}{(2 k + 1) (k + 1)} - c_{i}] x_{i} .$ (26)

Then, the fuzzy portfolio possibility set of Model (21) can be written as

$P = {(V, R) | x \in Ω, g (x) ⩽ V, f (x) ⩾ R} .$ (27)

Thus, the real frontier determined by Model (21) can be expressed as h (V) = sup {R| (V, R) ∈ P}. Next, we prove that h (V) is a concave function by three steps.

Step 1: For any $x^{1} = {(x_{1}^{1}, x_{2}^{1}, \dots, x_{n}^{1})}^{T}, x^{2} = {(x_{1}^{2}, x_{2}^{2}, \dots, x_{n}^{2})}^{T} \in Ω$ , λ ∈ R, we have

$\begin{matrix} λ x^{1} + (1 - λ) x^{2} = (λ x_{1}^{1} + (1 - λ) x_{1}^{2}, λ x_{2}^{1} \\ {+ (1 - λ) x_{2}^{2}, \dots, λ x_{n}^{1} + (1 - λ) x_{n}^{2})}^{T}, \end{matrix}$ (28) and

$\begin{matrix} \sum_{i = 1}^{n} [λ x_{i}^{1} + (1 - λ) x_{i}^{2}] \\ = λ \sum_{i = 1}^{n} x_{i}^{1} + (1 - λ) \sum_{i = 1}^{n} x_{i}^{2} = λ + 1 - λ = 1, \end{matrix}$ (29)

$\begin{matrix} l_{i} = λ l_{i} + (1 - λ) l_{i} ⩽ λ x_{i}^{1} + (1 - λ) x_{i}^{2} \\ ⩽ λ u_{i} + (1 - λ) u_{i} = u_{i} . \end{matrix}$ (30)

It is evident that λx¹ + (1 - λ) x² ∈ Ω. Hence Ω is a convex set.

Step 2: It is necessary to prove that the return function f (x) is concave function, while the risk function g (x) is a convex function. Note that the return function f (x) is a linear function. That is, we have $\frac{\partial^{2} f (x)}{\partial x^{2}} = 0$ , which can be concluded that f (x) is both a concave function and a convex function. In Equation (27), let

$\begin{matrix} A = & (\frac{b_{1} - a_{1}}{2} + \frac{k^{2} (α_{1} + β_{1})}{(2 k + 1) (k + 1)}, \dots, \\ {\frac{b_{i} - a_{i}}{2} + \frac{k^{2} (α_{i} + β_{i})}{(2 k + 1) (k + 1)})}^{T}, \\ B = {(α_{1} + β_{1}, \dots, α_{n} + β_{n})}^{T} . \end{matrix}$ (31) Then, the risk function g (x) can be rewritten as

$\begin{matrix} g (x) = & x^{T} A A^{T} x + \frac{5 k^{3} + k^{2}}{4 {(2 k + 1)}^{2} {(k + 1)}^{2} (k + 2)} \\ x^{T} B B^{T} x . \end{matrix}$ (32) And

$\begin{matrix} \frac{\partial g (x)}{\partial x} = 2 A A^{T} x + \frac{5 k^{3} + k^{2}}{2 {(2 k + 1)}^{2} {(k + 1)}^{2} (k + 2)} \\ B B^{T} x, \frac{\partial^{2} g (x)}{\partial x^{2}} \\ = 2 A A^{T} + \frac{5 k^{3} + k^{2}}{2 {(2 k + 1)}^{2} {(k + 1)}^{2} (k + 2)} B B^{T} > 0 . \end{matrix}$ (33)

Therefore, g (x) is a convex function. Based on Lemma 3.1, the real frontier h (V) = sup {R| (V, R) ∈ P} determined by Model (21) is a concave function. The proof is complete.

Theorem 4.2 The real frontiers determined by Models (22) and (23) are concave.

Proof: Analogous to the idea of the proof of Theorem 3.3, we can easily obtain that the risk functions in Models (22) and (23) is convex functions, and the return function are concave with a concave set Ω. Then the real frontiers obtained by Models (22) and (23) are concave.

The above theorems essentially indicate that the real frontiers determined by Models (21)-(23) are concave functions. It lays the theoretical foundation of the DEA models for evaluating the efficiencies of the possibilistic fuzzy models with risk attitudes.

4.3 Model development

In this section, DEA models are used to evaluate the performance of possibilistic fuzzy portfolios with risk attitudes under three return-risk frameworks. There are many kinds of DEA models, such as CCR model, BCC model, FG model, and so forth. Indeed, due to the characteristics of the real frontier shown in Fig. 1, BCC models with variable returns to scale can be adopted to construct the corresponding portfolio efficiency evaluation models.

4.3.1 DEA-BCC models for portfolios with risk attitudes under possibilistic mean variance framework

Suppose there are totally N portfolios to be evaluated, which are considered as DMUs. For the j-th portfolio, the investment weight vector of risky assets is $x^{j} = {(x_{1}^{j}, x_{2}^{j}, \dots, x_{n}^{j})}^{T}$ . In DEA-BCC models, let the expected return be considered as the output factor, and the possibilistic variance with risk attitudes as the input factor, respectively. For DMU_{j
₀} under evaluation, the risk-oriented DEA-BCC model under the possibilistic variance framework can be formulated as follows:

${\begin{matrix} min θ_{j_{0}} \\ s . t \sum_{j = 1}^{N} λ_{j} {{(\sum_{i = 1}^{n} [\frac{b_{i} - a_{i}}{2} + \frac{k^{2} (α_{i} + β_{i})}{(2 k + 1) (k + 1)}] x_{i}^{j})}^{2} + \frac{5 k^{3} + k^{2}}{4 {(2 k + 1)}^{2} {(k + 1)}^{2} (k + 2)} {(\sum_{i = 1}^{n} (α_{i} + β_{i}) x_{i}^{j})}^{2}} \\ ⩽ θ_{j_{0}} {(\sum_{i = 1}^{n} [\frac{b_{i} - a_{i}}{2} + \frac{k^{2} (α_{i} + β_{i})}{(2 k + 1) (k + 1)}] x_{i}^{j_{0}})}^{2} + \frac{5 k^{3} + k^{2}}{4 {(2 k + 1)}^{2} {(k + 1)}^{2} (k + 2)} {(\sum_{i = 1}^{n} (α_{i} + β_{i}) x_{i}^{j_{0}})}^{2}, \\ \sum_{j = 1}^{N} λ_{j} {\sum_{i = 1}^{n} [\frac{b_{i} + a_{i}}{2} + \frac{k^{2} (β_{i} - α_{i})}{(2 k + 1) (k + 1)} - c_{i}] x_{i}^{j}} ⩾ \sum_{i = 1}^{n} [\frac{b_{i} + a_{i}}{2} + \frac{k^{2} (β_{i} - α_{i})}{(2 k + 1) (k + 1)} - c_{i}] x_{i}^{j_{0}}, \\ \sum_{j = 1}^{N} λ_{j} = 1, λ_{j} ⩾ 0, j = 1, 2, \dots, N . \end{matrix}$ (34)

In Model (35), the optimal value $θ_{j_{0}}^{*}$ represents the risk-oriented efficiency score of DMU_{j
₀}. For convenience, denote the risk-oriented efficiency value $D E_{v} = θ_{j_{0}}^{*} \in (0, 1]$ . DMU_{j
₀} is said to be BCC-efficient and on the frontier if $θ_{j_{0}}^{*} = 1$ .

Similarly, the return-oriented DEA-BCC model for evaluating DMU_{j
₀} can be constructed as follows:

$\begin{matrix} {\begin{matrix} max φ_{j_{0}} \\ s . t \sum_{j = 1}^{N} λ_{j} {{(\sum_{i = 1}^{n} [\frac{b_{i} - a_{i}}{2} + \frac{k^{2} (α_{i} + β_{i})}{(2 k + 1) (k + 1)}] x_{i}^{j})}^{2} + \frac{5 k^{3} + k^{2}}{4 {(2 k + 1)}^{2} {(k + 1)}^{2} (k + 2)} {(\sum_{i = 1}^{n} (α_{i} + β_{i}) x_{i}^{j})}^{2}} \\ ⩽ {(\sum_{i = 1}^{n} [\frac{b_{i} - a_{i}}{2} + \frac{k^{2} (α_{i} + β_{i})}{(2 k + 1) (k + 1)}] x_{i}^{j_{0}})}^{2} + \frac{5 k^{3} + k^{2}}{4 {(2 k + 1)}^{2} {(k + 1)}^{2} (k + 2)} {(\sum_{i = 1}^{n} (α_{i} + β_{i}) x_{i}^{j_{0}})}^{2}, \\ \sum_{j = 1}^{N} λ_{j} {\sum_{i = 1}^{n} [\frac{b_{i} + a_{i}}{2} + \frac{k^{2} \cdot (β_{i} - α_{i})}{(2 k + 1) (k + 1)} - c_{i}] x_{i}^{j}} ⩾ φ_{j_{0}} \sum_{i = 1}^{n} [\frac{b_{i} + a_{i}}{2} + \frac{k^{2} \cdot (β_{i} - α_{i})}{(2 k + 1) (k + 1)} - c_{i}] x_{i}^{j_{0}}, \\ \sum_{j = 1}^{N} λ_{j} = 1, λ_{j} ⩾ 0, j = 1, 2, \dots, N . \end{matrix} \end{matrix}$ (35)

In Model (36), the optimal value $φ_{j_{0}}^{*} ⩾ 1$ , and $φ_{j_{0}}^{*} = 1$ indicates that DMU_{j
₀} is BCC-efficient. Let the return-oriented efficiency score denote $D E_{r} = 1 / φ_{j_{0}}^{*} \in (0, 1]$ .

As discussed in section 4.2, the DEA-BCC frontiers in Models (35) and (36) would approximate the real frontier of possibilistic mean variance portfolio model with risk attitudes. With adequate portfolio samples, the efficiency scores of DEA-BCC models can be good estimates for the real portfolio efficiencies.

4.3.2 DEA-BCC models for portfolios with risk attitudes under possibilistic mean semi-variance framework

Analogously, let the expected return be considered as the output factor, and the possibilistic semi-variance with risk attitudes as the input factor in DEA-BCC models. For DMU_{j
₀} under evaluation, the risk-oriented DEA-BCC model under the possibilistic mean semi-variance framework can be expressed as follows:

${\begin{matrix} min θ_{j_{0}} \\ s . t \sum_{j = 1}^{N} λ_{j} {{(\sum_{i = 1}^{n} [\frac{b_{i} - a_{i}}{2} + \frac{k^{2} (α_{i} + β_{i})}{(2 k + 1) (k + 1)}] \cdot x_{i}^{j})}^{2} + \frac{5 k^{3} + k^{2}}{{(2 k + 1)}^{2} {(k + 1)}^{2} (k + 2)} {(\sum_{i = 1}^{n} α_{i} x_{i}^{j})}^{2}} \\ ⩽ θ_{j_{0}} {(\sum_{i = 1}^{n} [\frac{b_{i} - a_{i}}{2} + \frac{k^{2} (α_{i} + β_{i})}{(2 k + 1) (k + 1)}] x_{i}^{j_{0}})}^{2} + \frac{5 k^{3} + k^{2}}{{(2 k + 1)}^{2} {(k + 1)}^{2} (k + 2)} {(\sum_{i = 1}^{n} α_{i} x_{i}^{j_{0}})}^{2}, \\ \sum_{j = 1}^{N} λ_{j} {\sum_{i = 1}^{n} [\frac{b_{i} + a_{i}}{2} + \frac{k^{2} (β_{i} - α_{i})}{(2 k + 1) (k + 1)} - c_{i}] x_{i}^{j}} ⩾ \sum_{i = 1}^{n} [\frac{b_{i} + a_{i}}{2} + \frac{k^{2} (β_{i} - α_{i})}{(2 k + 1) (k + 1)} - c_{i}] x_{i}^{j_{0}}, \\ \sum_{j = 1}^{N} λ_{j} = 1, λ_{j} ⩾ 0, j = 1, 2, \dots, N . \end{matrix}$ (36)

And the corresponding return-oriented DEA-BCC model under the possibilistic mean semi-variance framework can be stated as follows:

$\begin{matrix} {\begin{matrix} max φ_{j_{0}} \\ s . t \sum_{j = 1}^{N} λ_{j} {{(\sum_{i = 1}^{n} [\frac{b_{i} - a_{i}}{2} + \frac{k^{2} (α_{i} + β_{i})}{(2 k + 1) (k + 1)}] x_{i}^{j})}^{2} + \frac{5 k^{3} + k^{2}}{{(2 k + 1)}^{2} {(k + 1)}^{2} (k + 2)} {(\sum_{i = 1}^{n} α_{i} x_{i}^{j})}^{2}} \\ ⩽ {(\sum_{i = 1}^{n} [\frac{b_{i} - a_{i}}{2} + \frac{k^{2} (α_{i} + β_{i})}{(2 k + 1) (k + 1)}] x_{i}^{j_{0}})}^{2} + \frac{5 k^{3} + k^{2}}{{(2 k + 1)}^{2} {(k + 1)}^{2} (k + 2)} {(\sum_{i = 1}^{n} α_{i} x_{i}^{j_{0}})}^{2}, \\ \sum_{j = 1}^{N} λ_{j} {\sum_{i = 1}^{n} [\frac{b_{i} + a_{i}}{2} + \frac{k^{2} (β_{i} - α_{i})}{(2 k + 1) (k + 1)} - c_{i}] x_{i}^{j}} ⩾ φ_{j_{0}} \sum_{i = 1}^{n} [\frac{b_{i} + a_{i}}{2} + \frac{k^{2} (β_{i} - α_{i})}{(2 k + 1) (k + 1)} - c_{i}] x_{i}^{j_{0}}, \\ \sum_{j = 1}^{N} λ_{j} = 1, λ_{j} ⩾ 0, j = 1, 2, \dots, N . \end{matrix} \end{matrix}$ (37)

4.3.3 DEA-BCC models for portfolios with risk attitudes under possibilistic mean semi-absolute deviation framework

Similarly, let the expected return be considered as the output factor, and the possibilistic semi-absolute with risk attitudes as the input factor in DEA-BCC models. For DMU_{j
₀} under evaluation, the risk-oriented DEA-BCC model under possibilistic semi-absolute deviation framework can be shown as below:

${\begin{matrix} min θ_{j_{0}} \\ s . t \sum_{j = 1}^{N} λ_{j} {\sum_{i = 1}^{n} [\frac{b_{i} - a_{i}}{2} + \frac{k^{2} (β_{i} + α_{i})}{(2 k + 1) (k + 1)}] x_{i}^{j}} ⩽ θ_{j_{0}} \sum_{i = 1}^{n} [\frac{b_{i} - a_{i}}{2} + \frac{k^{2} (β_{i} + α_{i})}{(2 k + 1) (k + 1)}] x_{i}^{j_{0}}, \\ \sum_{j = 1}^{N} λ_{j} {\sum_{i = 1}^{n} [\frac{b_{i} + a_{i}}{2} + \frac{k^{2} (β_{i} - α_{i})}{(2 k + 1) (k + 1)} - c_{i}] x_{i}^{j}} ⩾ \sum_{i = 1}^{n} [\frac{b_{i} + a_{i}}{2} + \frac{k^{2} (β_{i} - α_{i})}{(2 k + 1) (k + 1)} - c_{i}] x_{i}^{j_{0}}, \\ \sum_{j = 1}^{N} λ_{j} = 1, λ_{j} ⩾ 0, j = 1, 2, \dots, N . \end{matrix}$ (38)

And the corresponding return-oriented DEA-BCC model can be built as below:

${\begin{matrix} max φ_{j_{0}} \\ s . t \sum_{j = 1}^{N} λ_{j} {\sum_{i = 1}^{n} [\frac{b_{i} - a_{i}}{2} + \frac{k^{2} (β_{i} + α_{i})}{(2 k + 1) (k + 1)}] x_{i}^{j}} ⩽ \sum_{i = 1}^{n} [\frac{b_{i} - a_{i}}{2} + \frac{k^{2} (β_{i} + α_{i})}{(2 k + 1) (k + 1)}] x_{i}^{j_{0}}, \\ \sum_{j = 1}^{N} λ_{j} {\sum_{i = 1}^{n} [\frac{b_{i} + a_{i}}{2} + \frac{k^{2} (β_{i} - α_{i})}{(2 k + 1) (k + 1)} - c_{i}] x_{i}^{j}} ⩾ φ_{j_{0}} \sum_{i = 1}^{n} [\frac{b_{i} + a_{i}}{2} + \frac{k^{2} (β_{i} - α_{i})}{(2 k + 1) (k + 1)} - c_{i}] x_{i}^{j_{0}}, \\ \sum_{j = 1}^{N} λ_{j} = 1, λ_{j} ⩾ 0, j = 1, 2, \dots, N . \end{matrix}$ (39)

In particular, if we consider the risk attitude parameter k = 1, the above DEA-BCC models would degenerate into the models proposed by Chen et al. (2018). In their work, the fuzzy portfolio efficiency evaluation problems are discussed under the assumption that investors are all risk-neutral. Indeed, the assumption is not always true because of systematic biases that exist in human psychology. In this paper, risk attitude factor is incorporated into the fuzzy portfolio efficiency evaluation problem. From this perspective, the proposed evaluation models are the extensions of Chen et al. (2018).

4.4 Two new definitions of correlation coefficient between PE and DEA efficiency (DE)

The classical Pearson correlation coefficient [47] r_p is introduced to describe the correlation between PE and DEA efficiency (DE). Also, Spearman coefficient [48] r_s is employed to measure the correlation of rankings between PE and DE. It is worth noting that r_p is an index to measure the degree of linear correlation. When r_p = 0, it only indicates there are no linear correlation. Except for the above classical correlation coefficients, we would attempt to construct two novel indicators to describe the closeness of the DEA frontier and the real portfolio frontier from the geometric viewpoint, which can fully characterize the consistency between PE and DE.

Definition 4.1 Suppose there are N portfolio samples to be evaluated. For the i-th sample, the portfolio efficiency and DEA efficiency can be denoted by PE_i and DE_i, respectively. For any i∈ { 1, 2, ⋯ , N }, 0 < PE_i ⩽ DE_i ⩽ 1, then two indicators can be defined as follows:

$r_{1} = 1 - \frac{1}{N} \sum_{i = 1}^{N} (D E_{i} - P E_{i}), r_{2} = \frac{1}{N} \sum_{i = 1}^{N} \frac{P E_{i}}{D E_{i}} .$ (40)

Among them, r₁ and r₂ are defined by difference and ratio formula, respectively. It is easy to obtain r₁, r₂ ∈ [0, 1]. If the values of r₁, r₂ are close to unit 1, it reveals that DEA frontier generated by N portfolio samples can be extremely close to the real portfolio frontier. In this sense, there is a small difference between DEA scores and portfolio efficiency values. Additionally, with the increase of two indictors, the efficiency evaluation results between the two approaches are higher consistent. In particular, if considering the extreme situations r₁ = 1 or r₂ = 1, we can easily conclude that the classical correlation coefficients r_s and r_p must achieve unit 1. From the above perspective, r₁, r₂ can be used as good indicators to illustrate the effectiveness and practicality of DEA-BCC approach.

5 Numerical illustration

In this section, we carry out numerous simulations to verify the validity of the DEA-BCC efficiency evaluation models with risk attitudes under three possibilistic return-risk frameworks. Assume investors select five risky assets to construct portfolios. Consider an example introduced in the literature (Chen et al. 2018), the possibilistic distributions of returns are shown in Table 2.

Table 2
The possibilistic distributions of returns of 5 risky assets

Asset a _i b _i α_i β_i

1 0.0220 0.0260 0.0200 0.0300

2 0.0530 0.0600 0.0500 0.0540

3 0.0780 0.0850 0.0720 0.0940

4 0.1200 0.1640 0.0080 0.1760

5 0.0060 0.0860 0.0720 0.0900

Asset	a _i	b _i	α_i	β_i
1	0.0220	0.0260	0.0200	0.0300
2	0.0530	0.0600	0.0500	0.0540
3	0.0780	0.0850	0.0720	0.0940
4	0.1200	0.1640	0.0080	0.1760
5	0.0060	0.0860	0.0720	0.0900

Table 1

Literature comparison

Literature	Risk measure	Environment	Psychological	DEA	Risk
factor	attitude
Markowitz (1952)	Mean-variance	Random	×	×	×
Speranza (1993)	Mean semi-absolute deviation	Random	×	×	×
Grootveld and Hallerbach (1999)	Mean semi-variance	Random	×	×	×
Morey and Morey (1999)	Mean-variance	Random	×	✓	×
Joro and Na (2006)	Mean-variance-skewness	Random	×	✓	×
Vercher et al. (2007)	Mean semi-variance	Fuzzy	×	×	×
Zhang et al. (2007)	Mean variance	Fuzzy	×	×	×
Huang (2008)	Semi-variance	Fuzzy	×	×	×
Tsaur (2013)	Mean-standard deviation	Fuzzy	✓	×	✓
Yu et al. (2014)	Mean entropy	Random	×	×	×
Chen (2015)	Semi-absolute deviation	Fuzzy	×	×	×
Yue and Wang (2017)	Higher order moments	Fuzzy	×	×	×
Banihashemi and Navidi (2017)	Mean-CVaR	Random	×	✓	×
Zhou et al. (2018)	Mean-VaR	Fuzzy	✓	×	✓
Khayamim (2018)	Mean-variance	Fuzzy	✓	×	✓
Zhou et al. (2018)	Mean variance	Random	×	✓	×
Chen et al. (2018)	Variance, semi-variance and semi-absolute deviation	Fuzzy	×	✓	×
Xue et al. (2019)	Mean-variance	Uncertain	✓	×	✓
Gupta et al. (2020)	VaR and CVaR	Fuzzy	×	✓	×
Gong (2021)	Mean-variance-skewness	Fuzzy	✓	✓	×
Deng and Chen (2022)	Lower partial moments	Fuzzy	×	×	✓
Deng et al. (2022)	Hesitant semi-variance	Hesitant fuzzy	✓	×	✓
Gong et al. (2022)	Mean-variance	Fuzzy	✓	×	✓
Our paper	Variance, semi-variance and semi-absolute deviation	Fuzzy	✓	✓	✓

In the experiments, the upper and lower bounds of investment proportion are given by l_i = 0, u_i = 0.6 and i = 1, 2, ⋯ , n. The linear unit transaction cost vector is c = (0 . 003, 0 . 003, 0 . 003, 0 . 003, 0 . 003) ^T. The values of risk attitude parameter k are taken by 0.5, 1, 2 respectively, representing risk-averse, risk-neutral and risk-seeking psychological states. First, the exact portfolio frontiers with risk attitudes in section 3.1 would be calculated. Second, we randomly generate investment weights to construct portfolio samples with known expected returns and risks. In the following, the portfolio efficiencies of these samples are calculated by the real frontier approach and the proposed DEA-BCC models, respectively. Finally, correlation analysis is performed to illustrate the effectiveness of the DEA-BCC models for portfolios with risk attitudes.

5.1 The evaluation results under possibilistic mean variance framework

Randomly generate N = 20, 100, 500, 2000 investment weight vectors $x^{j} = {(x_{1}^{j}, x_{2}^{j}, \dots, x_{n}^{j})}^{T} \in Ω$ , j = 1, 2, ⋯ , N, and $Ω = {x = {(x_{1}, x_{2}, \dots, x_{5})}^{T} | \sum_{i = 1}^{5} x_{i} = 1, 0 ⩽ x_{i} ⩽ 0.6}$ . Then we can obtain the possibilistic mean and variance of these samples. Take N = 20 as an example, Table 3 presents the possibilistic mean and variance with different risk attitudes of 20 DMUs. For each DMU, it can be found that risk-seeking investors have the highest values of possibilistic mean and variance, while risk-averse investors hold the lowest values. It indicates that risk-seeking investors have the characteristics of high return and risk.

Table 3
Possibilistic mean and variance with different risk attitudes of 20 DMUs

DMU Risk-averse (k = 0.5) Risk-neutral (k = 1.0) Risk-seeking (k = 2.0)

Return Risk Return Risk Return Risk

1 0.0721 0.0008 0.0765 0.0015 0.0818 0.0025

2 0.0880 0.0015 0.0937 0.0027 0.1004 0.0045

3 0.0719 0.0009 0.0750 0.0018 0.0786 0.0031

4 0.0815 0.0012 0.0860 0.0023 0.0913 0.0039

5 0.0817 0.0010 0.0868 0.0020 0.0930 0.0033

6 0.0721 0.0005 0.0761 0.0011 0.0810 0.0019

7 0.0438 0.0003 0.0455 0.0006 0.0474 0.0011

8 0.0808 0.0005 0.0860 0.0011 0.0924 0.0020

9 0.0863 0.0007 0.0915 0.0015 0.0977 0.0027

10 0.0751 0.0010 0.0793 0.0019 0.0843 0.0032

11 0.0663 0.0005 0.0684 0.0010 0.0711 0.0019

12 0.0726 0.0006 0.0764 0.0012 0.0810 0.0021

13 0.0701 0.0010 0.0742 0.0019 0.0791 0.0031

14 0.0467 0.0004 0.0485 0.0007 0.0507 0.0013

15 0.0571 0.0009 0.0594 0.0016 0.0622 0.0027

16 0.0815 0.0011 0.0868 0.0021 0.0933 0.0034

17 0.0694 0.0010 0.0727 0.0019 0.0766 0.0032

18 0.0696 0.0008 0.0725 0.0016 0.0760 0.0028

19 0.0876 0.0017 0.0932 0.0030 0.1000 0.0050

20 0.0580 0.0007 0.0592 0.0014 0.0607 0.0024

DMU	Risk-averse (k = 0.5)	Risk-neutral (k = 1.0)	Risk-seeking (k = 2.0)
1	0.0721	0.0008	0.0765	0.0015	0.0818	0.0025
2	0.0880	0.0015	0.0937	0.0027	0.1004	0.0045
3	0.0719	0.0009	0.0750	0.0018	0.0786	0.0031
4	0.0815	0.0012	0.0860	0.0023	0.0913	0.0039
5	0.0817	0.0010	0.0868	0.0020	0.0930	0.0033
6	0.0721	0.0005	0.0761	0.0011	0.0810	0.0019
7	0.0438	0.0003	0.0455	0.0006	0.0474	0.0011
8	0.0808	0.0005	0.0860	0.0011	0.0924	0.0020
9	0.0863	0.0007	0.0915	0.0015	0.0977	0.0027
10	0.0751	0.0010	0.0793	0.0019	0.0843	0.0032
11	0.0663	0.0005	0.0684	0.0010	0.0711	0.0019
12	0.0726	0.0006	0.0764	0.0012	0.0810	0.0021
13	0.0701	0.0010	0.0742	0.0019	0.0791	0.0031
14	0.0467	0.0004	0.0485	0.0007	0.0507	0.0013
15	0.0571	0.0009	0.0594	0.0016	0.0622	0.0027
16	0.0815	0.0011	0.0868	0.0021	0.0933	0.0034
17	0.0694	0.0010	0.0727	0.0019	0.0766	0.0032
18	0.0696	0.0008	0.0725	0.0016	0.0760	0.0028
19	0.0876	0.0017	0.0932	0.0030	0.1000	0.0050
20	0.0580	0.0007	0.0592	0.0014	0.0607	0.0024

In the following, the data of Table 3 are used as the input and output factors for Models (35) and (36). Then the risk-oriented DEA efficiency DE_v and return-oriented DEA efficiency DE_r of each DMU are obtained. In the meanwhile, based on the optimization model (21), the exact portfolio efficiencies PE_r and PE_v of each DMU can be calculated by comparing the relative distance to the optimal point on the real frontier. Table 4 presents the efficiency scores and rankings of 20 DMUs with three risk attitudes under the possibilistic mean variance framework. In Table 3, it can be seen that DEA efficiency scores of each DMU are always higher than the efficiency values based on the real efficient frontier. This is because the real frontiers of portfolios are always higher than DEA-BCC frontiers. That is, under the same return level, the value of risk on the real frontiers smaller than that of DEA-BCC frontier, while under the same risk level, the value of return on the real frontier is greater that of DEA-BCC frontier. Therefore, it is natural to hold DE_σ > PE_σ and DE_r > PE_r for each DMU.

Table 4(a)

Evaluation results with risk-averse under possibilistic mean variance framework

DMU	Risk-averse (k = 0.5)
	DE_v	rankings	PE_v	rankings	DE_r	rankings	PE_r	rankings
1	0.6078	10	0.5286	8	0.8343	14	0.7043	9
2	1.0000	1	0.4006	16	1.0000	1	0.7104	7
3	0.5261	14	0.4568	12	0.8297	15	0.6518	16
4	0.4581	19	0.4213	15	0.9318	9	0.6575	14
5	0.5636	13	0.5140	9	0.9395	6	0.7001	10
6	0.9012	6	0.7835	2	0.9045	10	0.8716	3
7	1.0000	1	0.6272	6	1.0000	1	0.7545	6
8	1.0000	1	0.9482	1	1.0000	1	0.9694	1
9	1.0000	1	0.7796	3	1.0000	1	0.8729	2
10	0.5057	16	0.4536	13	0.8644	12	0.6518	15
11	0.9306	5	0.7608	4	0.9331	8	0.8562	4
12	0.8324	8	0.7283	5	0.8868	11	0.8373	5
13	0.4594	18	0.3914	18	0.8061	16	0.6011	19
14	0.8334	7	0.5423	7	0.8413	13	0.6969	11
15	0.4443	20	0.3278	20	0.6601	20	0.5376	20
16	0.5109	15	0.4703	11	0.9348	7	0.6703	13
17	0.4671	17	0.3951	17	0.7984	18	0.6030	18
18	0.5766	11	0.4884	10	0.8051	17	0.6741	12
19	0.7860	9	0.3555	19	0.9952	5	0.7070	8
20	0.5673	12	0.4228	14	0.6865	19	0.6165	17

Table 4(b)

Evaluation results with risk-neutral under possibilistic mean variance framework

DMU	Risk-neutral (k = 1.0)
	DE_v	rankings	PE_v	rankings	DE_r	rankings	PE_r	rankings
1	0.6652	10	0.5484	8	0.8392	14	0.7020	9
2	1.0000	1	0.4221	15	1.0000	1	0.7039	8
3	0.5468	16	0.4455	13	0.8150	15	0.6336	16
4	0.4781	20	0.4259	14	0.9247	8	0.6493	14
5	0.6030	11	0.5188	9	0.9411	6	0.7060	7
6	0.9053	5	0.7440	2	0.9024	9	0.8364	2
7	1.0000	1	0.6427	6	1.0000	1	0.7274	6
8	1.0000	1	0.8914	1	1.0000	1	0.9349	1
9	1.0000	1	0.7267	3	1.0000	1	0.8330	3
10	0.5480	15	0.4614	12	0.8607	12	0.6542	13
11	0.8643	7	0.6694	5	0.8600	13	0.7773	5
12	0.8379	8	0.6904	4	0.8785	10	0.8011	4
13	0.5150	17	0.4171	16	0.8053	16	0.6128	17
14	0.8748	6	0.5766	7	0.8691	11	0.6813	12
15	0.4874	19	0.3517	20	0.6486	20	0.5282	20
16	0.5722	13	0.4919	10	0.9393	7	0.6909	11
17	0.5085	18	0.4070	18	0.7890	18	0.6023	18
18	0.5942	12	0.4750	11	0.7917	17	0.6466	15
19	0.8292	9	0.3799	19	0.9955	5	0.7007	10
20	0.5697	14	0.4103	17	0.6623	19	0.5725	19

Table 4(c)

Evaluation results with risk-seeking under possibilistic mean variance framework

DMU	Risk-seeking (k = 2.0)
	DE_v	rankings	PE_v	rankings	DE_r	rankings	PE_r	rankings
1	0.7012	10	0.5528	8	0.8478	13	0.6905	11
2	1.0000	1	0.4259	15	1.0000	1	0.6973	8
3	0.5554	16	0.4301	13	0.7996	16	0.6121	17
4	0.4992	20	0.4182	14	0.9173	8	0.6487	13
5	0.6243	11	0.5113	9	0.9428	7	0.7037	6
6	0.9061	5	0.7111	2	0.8982	9	0.8014	3
7	1.0000	1	0.6502	6	1.0000	1	0.6981	7
8	1.0000	1	0.8442	1	1.0000	1	0.9017	1
9	1.0000	1	0.6815	3	1.0000	1	0.8020	2
10	0.5729	14	0.4581	12	0.8568	12	0.6482	14
11	0.8210	9	0.6090	5	0.8056	14	0.7118	5
12	0.8385	8	0.6580	4	0.8688	11	0.7647	4
13	0.5504	17	0.4275	16	0.8045	15	0.6126	16
14	0.8988	6	0.5935	7	0.8854	10	0.6587	12
15	0.5124	19	0.3612	20	0.6364	19	0.5098	20
16	0.6144	12	0.4961	10	0.9442	6	0.6969	9
17	0.5322	18	0.4075	18	0.7791	17	0.5923	18
18	0.6008	13	0.4586	11	0.7775	18	0.6181	15
19	0.8564	7	0.3868	19	0.9957	5	0.6943	10
20	0.5647	15	0.3945	17	0.6352	20	0.5284	19

Figures 2–4 present the real portfolio frontiers and risk-oriented DEA frontiers in four sample sizes under different risk attitudes. As we can see, Fig. 2 displays the efficient frontiers of portfolio and DEA with risk-averse attitude in different sample sizes. With the increase of the sample size, the DEA frontier would be significantly approaching the real frontier of possibilistic mean variance model with risk-averse attitude. In the same way, from Figs. 3 and 4, we can draw same conclusions that the DEA frontiers with adequate sample size can be well approximate real frontiers with risk attitudes.

Fig. 2

Efficient frontiers of portfolio and DEA with risk-averse attitude k = 0.5 in different sample sizes.

Fig. 3

Efficient frontiers of portfolio and DEA with risk-neutral attitude k = 1 in different sample sizes.

Fig. 4

Efficient frontiers of portfolio and DEA with risk-seeking attitude k = 2.0 in different sample sizes.

Furthermore, Fig. 5 displays the frontiers of portfolio with risk attitudes and the corresponding DEA frontiers of 2000 samples. The following conclusions can be drawn. (1) When the sample size is N = 2000, the frontiers of DEA-BCC with different risk attitudes will be close to the corresponding real frontiers and always be below the real frontiers. (2) The investors with different risk attitudes have quite different efficient frontiers of portfolio and DEA. It can be seen that the DEA frontiers and exact efficient frontiers for risk-neutral investors are between those for risk- averse and risk-seeking investors. At a small risk level, the risk-averse investors can obtain greater values of expected return than risk- neutral and risk-seeking investors. In addition, it can be seen that the feasible minimal values of risk on the DEA and exact efficient frontiers for risk-seeking investors are higher than those of risk-neutral and risk-averse investors. This occurs when risk-seeking investors are more interested in higher risk and return.

Fig. 5

Efficient frontiers with different risk attitudes under the possibilistic mean variance framework.

Finally, to further illustrate the effectiveness of DEA approach for evaluating the possibilistic mean variance models with different risk attitudes, the quantitative correlation analysis is performed.

Under the possibilistic mean variance framework, Table 5 presents the results of correlation analysis considering risk attitudes and sample sizes. It is obvious that with the increase of sample size, the four indexes r_p, r_s, r₁ and r₂ are also increasing. When the sample size increases to 2000, the classical correlation coefficients r_p and r_s are over 0.99, and the other indicators r₁ and r₂ also achieve 0.9. Therefore, the results of correlation analysis further indicate that the effectiveness and practicality of DEA approach for evaluating the efficiency of portfolios with risk attitudes.

It is worth noting that the values of r_p and r_s are extremely high and obtained over 0.99 when the sample size increases to 100. However, from Figs. 2–4, there are some differences between the real frontiers and DEA frontier in 100 sample size. It reveals that the classical correlation coefficients r_p and r_s may not work well in some cases. The validity of DEA method cannot be determined solely by relying on r_p and r_s. Therefore, the two indicators r₁ and r₂ can be used as effective supplements to characterize the effectiveness and practicality of DEA method.

Table 5

Correlation of efficiencies with risk attitudes under possibilistic mean variance framework

Index	Sample	Risk-oriented			Return-oriented
	size N	Risk-averse (k = 0.5)	Risk-neutral (k = 1.0)	Risk-seeking (k = 2.0)	Risk-averse (k = 0.5)	Risk-neutral (k = 1.0)	Risk-seeking (k = 2.0)
r _p	20	0.7173	0.7191	0.7264	0.6716	0.7123	0.7695
	100	0.9900	0.9937	0.9945	0.9647	0.9932	0.9878
	500	0.9912	0.9946	0.9949	0.9838	0.9958	0.9927
	2000	0.9967	0.9989	0.9973	0.9976	0.9971	0.9979
r _s	20	0.7045	0.7113	0.6925	0.7752	0.7707	0.7707
	100	0.9886	0.9955	0.9936	0.9538	0.9940	0.9874
	500	0.9936	0.9961	0.9958	0.9829	0.9960	0.9927
	2000	0.9971	0.9988	0.9978	0.9972	0.9976	0.9989
r ₁	20	0.8413	0.8148	0.7914	0.8346	0.8236	0.8098
	100	0.9231	0.8669	0.8384	0.9277	0.9040	0.8900
	500	0.9780	0.9325	0.8985	0.9627	0.9412	0.9259
	2000	0.9792	0.9462	0.9297	0.9830	0.9688	0.9636
r ₂	20	0.7918	0.7582	0.7281	0.8137	0.8006	0.7837
	100	0.8784	0.8042	0.7675	0.9062	0.8788	0.8607
	500	0.9618	0.8874	0.8372	0.9481	0.9206	0.9001
	2000	0.9698	0.9272	0.9005	0.9763	0.9575	0.9498

5.2 The evaluation results under the possibilistic mean semi-variance framework

In this section, the simulation would be conducted to illustrate the efficiency evaluation models with risk attitudes under the possibilistic mean semi-variance framework. Analogous to the simulation performed in the previous section, we first randomly generate N = 20, 100, 500, 2000 portfolio samples. Then the efficiency sores of these samples can be obtained by the real frontier approach and DEA-BCC models, respectively.

Figures 6–8 present the frontier comparisons under three different risk attitudes. For example, Fig. 6 compares the real frontier and DEA frontiers with risk-averse attitude in different sample sizes. Obviously, as the sample size increases, the frontiers of DEA-BCC models would be gradually approaching the real portfolio frontier with risk-averse attitude. Also, from Figs. 7 and 8, similar conclusions can be drawn. Additionally, the real efficient frontier with three different risk attitudes and the corresponding DEA frontiers of 2000 samples are whole presented in Fig. 9. It can be found that no matter what risk attitude investors have taken, the frontiers of DEA-BCC models with adequate samples are significantly approaching the real frontiers under the possibilistic mean semi-variance framework.

Fig.6

Frontier comparison with risk-averse attitude k = 0.5.

Fig.7

Frontier comparison with risk-neutral attitude k = 1.0.

Fig. 8

Frontier comparison with risk-seeking attitude k = 2.0.

Fig. 9

Efficient frontiers with different risk attitudes under the possibilistic mean semi-variance framework.

Under the possibilistic mean semi-variance framework, Table 6 presents the correlations of PE and DE with different risk attitudes and sample sizes. It is clear that the four correlation indexes r_p, r_s, r₁ and r₂ increase with the sample size. In addition, the four correlation indexes have reached more than 0.9 with a sample size of 2000. Therefore, the results further verify that the proposed DEA-BCC approach can well characterize the real portfolio frontiers with different risk attitudes mathematically.

Table 6

Correlation of efficiencies with risk attitudes under possibilistic mean semi-variance framework

Index	Sample	Risk-oriented			Return-oriented
	size N	Risk-averse (k = 0.5)	Risk-neutral (k = 1.0)	Risk-seeking (k = 2.0)	Risk-averse (k = 0.5)	Risk-neutral (k = 1.0)	Risk-seeking (k = 2.0)
r _p	20	0.7312	0.7325	0.7361	0.6699	0.7073	0.7616
	100	0.9878	0.9943	0.9956	0.9594	0.9896	0.9867
	500	0.9913	0.9975	0.9971	0.9821	0.9970	0.9920
	2000	0.9979	0.9985	0.9986	0.9965	0.9973	0.9925
r _s	20	0.7105	0.7053	0.7135	0.7857	0.7684	0.7000
	100	0.9896	0.9963	0.9965	0.9531	0.9868	0.9868
	500	0.9911	0.9981	0.9976	0.9815	0.9978	0.9924
	2000	0.9978	0.9989	0.9991	0.9961	0.9973	0.9925
r ₁	20	0.8371	0.8074	0.7883	0.8134	0.8095	0.8025
	100	0.9164	0.8590	0.8342	0.9133	0.8958	0.8872
	500	0.9715	0.9241	0.8951	0.9541	0.9368	0.9251
	2000	0.9834	0.9557	0.9421	0.9837	0.9740	0.9695
r ₂	20	0.7744	0.7388	0.7187	0.7863	0.7822	0.7741
	100	0.8580	0.7834	0.7567	0.8846	0.8653	0.8556
	500	0.9424	0.8668	0.8275	0.9341	0.9125	0.8976
	2000	0.9643	0.9169	0.9064	0.9756	0.9631	0.9569

5.3 The evaluation results under possibilistic mean semi-absolute deviation framework

Under possibilistic mean semi-absolute deviation framework, the simulation is performed to illustrate the effectiveness and feasibility of DEA-BCC evaluation models with risk attitudes. Analogously, first randomly generate N = 20, 100, 500, 2000 portfolio samples. Then the efficiency scores of these samples can be obtained by the real frontier approach and DEA-BCC models, respectively. Figures 10–12 display the frontier comparisons under three different risk attitudes. As we can see, the frontiers of DEA-BCC models are very reasonable approximations of the real portfolio frontiers with a sample size of 2000 under three risk attitudes. From Fig. 13, it is clear that no matter what risk attitude investors have taken, the frontiers of DEA-BCC models with adequate samples are significantly approaching the real frontiers under the possibilistic mean semi-absolute deviation framework.

Fig. 10

Frontier comparison with risk-averse attitude k = 0.5.

Fig. 11

Frontier comparison with risk-neutral attitude k = 1.0.

Fig. 12

Frontier comparison with risk-seeking attitude k = 2.0.

Fig. 13

Efficient frontiers with different risk attitudes under the possibilistic mean semi-absolute deviation framework.

Under the possibilistic mean semi-absolute deviation framework, Table 7 reports the results of correlations between PE and DE with different risk attitudes and sample sizes. It can be seen that for a sample size of 2000, the four correlation indexes s r_p, r_s, r₁ and r₂ are all above 0.95, indicating the effectiveness of DEA-BCC models with risk attitudes.

Table 7

Correlation analysis considering risk attitudes under possibilistic mean semi-absolute deviation framework

Index	Sample	Risk-oriented			Return-oriented
	size N	Risk-averse (k = 0.5)	Risk-neutral (k = 1.0)	Risk-seeking (k = 2.0)	Risk-averse (k = 0.5)	Risk-neutral (k = 1.0)	Risk-seeking (k = 2.0)
r _p	20	0.7597	0.7144	0.7243	0.7454	0.6915	0.7605
	100	0.9809	0.9878	0.9984	0.9549	0.9834	0.9932
	500	0.9854	0.9927	0.9953	0.9795	0.9932	0.9953
	2000	0.9977	0.9988	0.9992	0.9979	0.9987	0.9948
r _s	20	0.7173	0.7203	0.7143	0.8451	0.7820	0.7669
	100	0.9801	0.9867	0.9979	0.9490	0.9797	0.9936
	500	0.9879	0.9952	0.9960	0.9792	0.9936	0.9957
	2000	0.9980	0.9990	0.9993	0.9980	0.9986	0.9949
r ₁	20	0.8441	0.8936	0.8753	0.8036	0.8279	0.8125
	100	0.9309	0.9370	0.9055	0.9117	0.9174	0.8931
500	0.9614	0.9697	0.9395	0.9513	0.9512	0.9300
	2000	0.9780	0.9812	0.9665	0.9774	0.9813	0.9748
r ₂	20	0.8119	0.8772	0.8564	0.7777	0.8048	0.7862
	100	0.9050	0.9206	0.8846	0.8839	0.8931	0.8640
	500	0.9420	0.9593	0.9220	0.9312	0.9328	0.9050
	2000	0.9670	0.9742	0.9550	0.9671	0.9739	0.9648

In summary, under the above three possibilistic return-risk frameworks, the numerous simulations and correlation analysis have illustrated that the proposed DEA-BCC approach can effectively estimate the efficiencies of portfolios with risk attitudes.

6 Conclusion

Due to the significant effect of investors’ psychological factors on portfolio management, the aim of this study is to incorporate risk attitude into the fuzzy portfolio optimization and performance evaluation analysis. Firstly, in order to consider the risk attitudes of investors in the possibilistic fuzzy environment, the parameter k is introduced into the membership function of trapezoidal fuzzy number. Then the return and risk measures including the possibilistic mean, variance, semi-variance and semi-absolute deviation with risk attitude are derived. Secondly, some possibilistic fuzzy portfolio optimization models with different investor risk attitudes under three return-risk frameworks are established. To make the above models more practical, some realistic constraints such as transaction costs and investment bounds are also considered. Additionally, the theoretical foundation of DEA-BCC approach for evaluating the efficiencies of fuzzy portfolios with risk attitudes is developed. Afterwards, the corresponding portfolio efficiency evaluation models with risk attitudes based on DEA approach are constructed. Finally, several simulations and correlation analysis are conducted to illustrate the feasibility and effectiveness of our proposed efficiency evaluation models with risk attitudes. In particular, except for the classical correlation coefficients, two novel indicators are designed as effective supplements characterizing the correlation between DEA efficiency scores and real portfolio efficiencies. The results have shown that no matter what risk attitude an investor holds, the corresponding DEA frontiers can effectively approximate the real frontiers with adequate samples under the three possibilistic return-risk frameworks. In other words, the proposed approach can be used as a reasonable way to estimate the efficiencies of portfolios with risk attitudes.

For further research, the proposed portfolio efficiency evaluation models with risk attitudes would be extended to other fuzzy environments. For example, credibilistic environment, hesitant fuzzy and random fuzzy environment. In addition, more psychological characteristics of investors and real constraints can be incorporated into the portfolio selection and efficiency analysis.

Footnotes

Acknowledgments

This research was supported by “the National Social Science Foundation Projects of China, No. 21BTJ069”, “the Fundamental Research Funds for the Central Universities, No. ZDPY202213” and “the Double First-Class Construction Funds of South China University of Technology, No. x2 lx/D6223910”. The authors are highly grateful to the referees and editor in-chief for their very helpful advice and comments.

Compliance with ethical standards

Funding

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

References

Markowitz

H.M.

, Portfolio selection, The Journal of Finance 7(1) (1952), 77–91.

Markowitz

H.M.

Portfolio selection: Efficient diversification of investments. New York: John Wiley & Sons (1959).

Grootveld

and Hallerbach

, Variance vs downside risk: is therereally that much difference? European Journal of OperationalResearch 114 (1999), 304–319.

Konno

and Yamazaki

, Mean-absolute deviation portfoliooptimization model and its applications to Tokyo stock market, Management Science 37(5) (1991), 519–531.

Speranza

M.G.

, Linear programming models for portfolio optimization, Finance 14 (1993), 107–123.

J.R.

, Lee

W.Y.

and Chiou

W.J.P.

, Diversified portfolios withdifferent entropy measures, Applied Mathematics andComputation 241 (2014), 47–63.

Tong

X.J.

, Qi

L.Q.

, Wu

and Zhou

, A smoothing method forsolving portfolio optimization with CVaR and applications inallocation of generation asset, Applied Mathematics &Computation 216(6) (2010), 1723–1740.

Zadeh

L.A.

, Fuzzy sets, Information and Control 8(3) (1965), 338–353.

Zadeh

L.A.

, Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets and Systems 1 (1978), 3–28.

10.

Carlsson

and Fulér

, On possibilistic mean valueand variance of fuzzy numbers, Fuzzy Sets and Systems 122 (2001), 315–326.

11.

Zhang

W.G.

and Nie

Z.K.

, On possibilistic variance of fuzzy number, Lecture Notes in Computer Science 2639 (2003), 398–402.

12.

Zhang

W.G.

, Wang

Y.L.

, Chen

Z.P.

and Nie

Z.K.

, Possibilisticmean–variance models and efficient frontiers for portfolioselection problem, Information Sciences 177 (2007), 2787–2801.

13.

Huang

, Mean-semivariance models for fuzzy portfolio selection, Journal of Computational and Applied Mathematics 217(2008), 1–8.

14.

Vercher

, Bermúdez

J.D.

and Segura

J.V.

, Fuzzy portfoliooptimization under downside risk measures, Fuzzy Sets andSystems 158(7) (2007), 769–782.

15.

Chen

, Artificial bee colony algorithm for constrainedpossibilistic portfolio optimization problem, Physica A 429 (2015), 125–139.

16.

Yue

and Wang

Y.P.

, A new fuzzy multi-objective higher ordermoment portfolio selection model for diversified portfolios, Physica A 465 (2017), 124–140.

17.

Deng

and Chen

J.X.

, The lower partial moments risk measure in anovel fuzzy framework based on possibility density function, Computers & Industrial Engineering (2022), 108309.

18.

Deng

, Li

W.M.

and Liu

Y.Y.

, Hesitant fuzzy portfolio selectionmodel with score and novel hesitant semi-variance, Computers &Industrial Engineering 164 (2022), 107879.

19.

Kahneman

and Tversky

, Prospect theory: An analysis of decisionunder risk, Econometrica 47(2) (1979), 263–292.

20.

Tsaur

R.C.

, Fuzzy portfolio model with different investor riskattitudes, European Journal of Operational Research 227(2) (2013), 385–390.

21.

Momen

, Esfahanipour

and Seifi

, A robust behavioralportfolio selection: model with investor attitudes and biases, Operational Research: An International Journal 2 (2017), 1–20.

22.

Zhou

X.Y.

, Wang

, Yang

X.P.

, Lev

, Tu

and WangPortfolio

S.Y.

, selection under different attitudes in fuzzy environment, Information Sciences 462 (2018), 278–289.

23.

Khayamin

, Mirzazadeh

and Naderi

, Portfolio rebalancing withrespect to market psychology in a fuzzy environment: A case study inTehran Stock Exchange, Applied Soft Computing 64 (2018), 244–259.

24.

Xue

, Di

, Zhao

X.J.

and Zhang

Z.L.

, uncertain portfolioselection with mental accounts with realistic constraints, Journal of Computational and Applies Mathematics 346 (2019), 42–52.

25.

Gong

X.M.

, Yu

L.Y.

and Min

C.R.

, Multi-period portfolio selectionunder the coherent fuzzy environment with dynamic risk-tolerance andexpected-return levels, Applied Soft Computing 114(2022) 108104 doi: 10.1016/j.asoc.2021.108104.

26.

Sharpe

, Mutual fund performance, The Journal of Business 39 (1966), 119–138.

27.

Treynor

, How to rate management investment funds, HarvardBusiness Review 44 (1965), 131–136.

28.

Jensen

M.C.

, The performance of mutual funds in the period, The Journal of Finance 23 (1968), 389–416.

29.

Charnes

, Cooper

W.W.

and Rhodes

, Measuring the efficiency ofdecision-making units, European Journal of Operational Research 2 (1978), 429–444.

30.

Murthi

B.P.S.

, Choi

Y.K.

and Desai

, Efficiency of mutual fundsand portfolio performance measurement: A non-parametric approach, European Journal of Operational Research 98(2) (1997), 408–418.

31.

Morey

M.R.

and Morey

R.C.

, Mutual fund performance appraisals: amulti-horizon perspective with endogenous benchmarking, Omega 27(2) (1999), 241–58.

32.

Joro

and Na

, Portfolio performance evaluation inmean-variance-skewness framework, European Journal ofOperational Research 175(1) (2016), 446–461.

33.

Liu

W.B.

, Zhou

Z.B.

, Liu

D.B.

and Xiao

H.L.

, Estimation of portfolioefficiency via DEA, Omega 52 (2015), 107–118.

34.

Zhou

Z.B.

, Jin

Q.Y.

, Xiao

H.L.

, Wu

and Liu

W.B.

, Estimation ofcardinality constrained portfolio efficiency via segmented DEA, Omega 76 (2018), 28–37.

35.

Branda

, Diversification-consistent data envelopment analysisbased on directional-distance measures, Omega 52 (2015), 65–76.

36.

Banihashemi

and Navidi

, Portfolio performance evaluation inMean-CVaR framework: A comparison with non-parametric methods valueat risk in Mean-VaR analysis, Operations Research Perspectives 4 (2017), 21–28.

37.

Zhou

, Xiao

, Jin

and Liu

, DEA frontier improvement andportfolio rebalancing: An application of China mutual funds onconsidering sustainability information disclosure, EuropeanJournal of Operational Research 269 (2018), 111–131.

38.

Chen

, Gai

and Gupta

, Efficiency evaluation of fuzzyportfolio in different risk measures via DEA, Annals ofOperations Research 269 (2018), 103–127.

39.

Gupta

, Mehlawat

M.K.

, Kumar

, Yadav

, AggarwalCredibilistic

Credibilistic fuzzy DEA approach for portfo-lio efficiency evaluation and rebalancing toward benchmark portfolios using positive and negative returns, International Journal of Fuzzy Systems (2020), 10.1007/s40815-020-00801-4.

40.

Xiao

H.L.

, Ren

T.T.

and Ren

, Estimation of fuzzy portfolioefficiency via an improved DEA approach, INFOR 58(3) (2020), 478–510.

41.

Gong

X.M.

, Yu

C.R.

, Min

L.Y.

and Ge

Z.P.

, Regret theory-based fuzzymulti-objective portfolio selection model involving DEAcross-efficiency and higher moments, Applied Soft Computing 100 (2021) 106958.doi: 10.1016/j.asoc.2020.106958.

42.

Chen

H.D.

, Wang

Z.P.

, Feng

Z.F.

and Li

Y.J.

, A flexible dataenvelopment analysis model for vendor selection and performanceevaluation. ICIC express letters. Part B, Applications: AnInternational Journal of Research and Surveys 4(2) (2013), 301–306.

43.

Chen

Y.C.

, Hu

L.H.

, Lu

W.C.

, Wu

J.Z.

and Yang

J.J.

, Multiplecriteria decision-making for developing an international gameparticipation strategy: a novel application of the data envelopmentanalysis (DEA) two-stage efficiency process, Mathematics 9(14) (2021), 1700.

44.

N.T.

, Thinh

N.A.

and Ha

N.T.V.

, Measuring water resource useefficiency of the Dong Nai River Basin (Vietnam): an application ofthe two-stage data envelopment analysis (DEA), Environment,Development and Sustainability 24(10) (2022), 12427–12445.

45.

W.C.A. Operational efficiency assessment of thai insurance companies: an application of two-stage data envelopment analysis, International Conference on Power Electronics and Energy Engineering (2015), 252–256.

46.

Banker

R.D.

, Charnes

and Cooper

W.W.

, Some models for estimatingtechnical and scale inefficiencies in data envelopment analysis, Management Science 30(9) (1984), 1078–1092.

47.

Pearson

, Notes on the history of correlation, Biometrika 13(1) (1920), 25–45.

48.

Spearman

, The proof and measurement of association between twothings, International Journal of Epidemiology 39(5) (2010), 1137–50.

Performance evaluation of possibilistic fuzzy portfolios with different investor risk attitudes based on DEA approach

Abstract

Keywords

1 Introduction

1.1 Literature review

1.2 Research motivation

1.3 Novelty of our method

1.4 The organization of our paper

2 Preliminaries

3.1 Possibilistic return and risk measures with different risk attitudes

3.2.1 Possibilistic mean variance portfolio model with risk attitudes

4 Portfolio efficiency evaluation models with risk attitudes via DEA method

4.1 Portfolio efficiency (PE) definition

4.3.1 DEA-BCC models for portfolios with risk attitudes under possibilistic mean variance framework

Table 2 The possibilistic distributions of returns of 5 risky assets Asset a i b i α i β i 1 0.0220 0.0260 0.0200 0.0300 2 0.0530 0.0600 0.0500 0.0540 3 0.0780 0.0850 0.0720 0.0940 4 0.1200 0.1640 0.0080 0.1760 5 0.0060 0.0860 0.0720 0.0900

Footnotes

Acknowledgments

Compliance with ethical standards

Funding

Ethical approval

References

Table 2
The possibilistic distributions of returns of 5 risky assets

Asset a _i b _i α_i β_i

1 0.0220 0.0260 0.0200 0.0300

2 0.0530 0.0600 0.0500 0.0540

3 0.0780 0.0850 0.0720 0.0940

4 0.1200 0.1640 0.0080 0.1760

5 0.0060 0.0860 0.0720 0.0900