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

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 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.

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].

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.

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 r ˜ i of asset i with the risk attitude parameter k can be defined as follows:

u r ˜ i ( x ) = { 1 - ( a i - x α i ) k , a i - α i ⩽ x < a i , 1 , a i ⩽ x < b i , 1 - ( x - b i β i ) k , b i ⩽ x < b i + β i , 0 , otherwize .(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, r ˜ i is a particular case of LR-power fuzzy numbers when p = q = k. From this perspective, 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 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 r ˜ i = ( a i , b i , α i , β i ) k is

M ¯ ( r ˜ i ) = M * ( r ˜ i ) + M * ( r ˜ i ) 2 = b i + a i 2 + k 2 ( 2 k + 1 ) ( k + 1 ) ( β i - α i ) .(10)

Where k > 0 is the risk attitude parameter.

Proof. According to Definition 2.1, the γ - level set of r ˜ i = ( a i , b i , α i , β i ) k is

[ r ˜ i ] γ = [ r i 1 ( γ ) , r i 2 ( γ ) ] = [ a i - ( 1 - γ ) 1 k α i , b i + ( 1 - γ ) 1 k β i ] , γ ∈ ( 0 , 1 ) .(11)

By Definition 2.4, the lower and upper possibilistic mean values of r ˜ i = ( a i , b i , α i , β i ) k are as follows:

M * ( r ˜ i ) = 2 ∫ 0 1 γ r i 1 ( γ ) d γ = 2 ∫ 0 1 γ [ a i - ( 1 - γ ) 1 k α i ] d γ = a i - 2 k 2 ( 2 k + 1 ) ( k + 1 ) α i .(12)

M * ( r ˜ i ) = 2 ∫ 0 1 γ r i 2 ( γ ) d γ = 2 ∫ 0 1 γ [ b i + ( 1 - γ ) 1 k β i ] d γ = b i + 2 k 2 ( 2 k + 1 ) ( k + 1 ) β i .(13)

Furthmore, the crisp possibilistic mean value of r ˜ i = ( a i , b i , α i , β i ) k is as follows: M ¯ ( r ˜ i ) = M * ( r ˜ i ) + M * ( r ˜ i ) 2 = b i + a i 2 + k 2 ( 2 k + 1 ) ( k + 1 ) ( β i - α i ) .

The proof is complete.

Theorem 3.2. The possibilistic variance of LR-power fuzzy number r ˜ i = ( a i , b i , α i , β i ) k is

Var ( r ˜ i ) = [ b i - a i 2 + k 2 ( α i + β i ) ( 2 k + 1 ) ( k + 1 ) ] 2 + k 2 + 5 k 3 4 ( 2 k + 1 ) 2 ( k + 1 ) 2 ( k + 2 ) ( α i + β i ) 2 .(14)

Proof. According to Definition 2.5, Var ( r ˜ i ) = 1 2 ∫ 0 1 γ [ r i 2 ( γ ) - r i 1 ( γ ) ] 2 d γ = 1 2 ∫ 0 1 γ [ ( b i - a i ) + ( 1 - γ ) 1 k ( β i + α i ) ] 2 d γ = ( b i - a i ) 2 4 + k 2 ( b i - a i ) ( α i + β i ) ( 2 k + 1 ) ( k + 1 ) + k 2 ( α i + β i ) 2 4 ( k + 1 ) ( k + 2 ) = [ b i - a i 2 + k 2 ( α i + β i ) ( 2 k + 1 ) ( k + 1 ) ] 2 + k 2 + 5 k 3 4 ( 2 k + 1 ) 2 ( k + 1 ) 2 ( k + 2 ) ( α i + β i ) 2 .

The proof is complete.

Theorem 3.3. The possibilistic semi-variance of LR-power fuzzy number r ˜ i = ( a i , b i , α i , β i ) k is

Varr - ( r ˜ i ) = [ b i - a i 2 + k 2 ( α i + β i ) ( 2 k + 1 ) ( k + 1 ) ] 2 + k 2 + 5 k 3 ( 2 k + 1 ) 2 ( k + 1 ) 2 ( k + 2 ) α i 2 .(15)

Proof. According to Definition 2.6 and Theorem 2.1,

Var - ( r ˜ i ) = 2 ∫ 0 1 γ [ M ¯ ( r ˜ i ) - r i 1 ( γ ) ] 2 d γ = 1 2 ∫ 0 1 [ b i - a i 2 + 2 k 2 ( β i - α i ) ( 2 k + 1 ) ( k + 1 ) + ( 1 - γ ) 1 k α i ] 2 d γ = [ b i - a i 2 + k 2 ( α i + β i ) ( 2 k + 1 ) ( k + 1 ) ] 2 + k 2 + 5 k 3 ( 2 k + 1 ) 2 ( k + 1 ) 2 ( k + 2 ) α i 2 .

The proof is complete.

Theorem 3.4. The possibilistic semi-absolute deviation of LR-power fuzzy number r ˜ i = ( a i , b i , α i , β i ) k is

Sad ( r ˜ i ) = 1 2 ( b i - a i ) + k 2 ( 2 k + 1 ) ( k + 1 ) ( α i + β i ) .(16)

Proof. Let M ( r ˜ i ) = [ M * ( r ˜ i ) , M * ( r ˜ i ) ] be the interval-valued possibilistic mean of LR-power fuzzy number r ˜ i = ( a i , b i , α i , β i ) k . Indeed, M ( r ˜ i ) can be considered as an LR-power fuzzy number with right and left spread zero, that is M ( r ˜ i ) = ( M * ( r ˜ i ) , M * ( r ˜ i ) , 0 , 0 ) k . According to Definition 2.3 and Definition 2.7, we can obtain

M ( r ˜ i ) - r ˜ i = ( M * ( r ˜ i ) , M * ( r ˜ i ) , 0 , 0 ) k - ( a i , b i , α i , β i ) k = ( a i - 2 k 2 ( 2 k + 1 ) ( k + 1 ) α i , b i + 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 - 2 k 2 α i ( 2 k + 1 ) ( k + 1 ) , b i - a i + 2 k 2 ( 2 k + 1 ) ( k + 1 ) β i , β i , α i ) k ,(17) in addition,

max { 0 , M ( r ˜ i ) - r ˜ i } = ( 0 , b i - a i + 2 k 2 ( 2 k + 1 ) ( k + 1 ) β i , 0 , α i ) k .(18)

Therefore, by Theorem 3.1, the possibilistic semi-absolute deviation of r ˜ i = ( a i , b i , α i , β i ) k is Sad ( r ˜ i ) = M ¯ ( max { 0 , M ( r ˜ i ) - r ˜ i } ) = 1 2 ( b i - a i ) + k 2 ( 2 k + 1 ) ( k + 1 ) ( α i + β i ) .

The proof is complete.

Example 3.1 Let r ˜ i = ( a i , b i , α i , β i ) k be the fuzzy return of asset i, i = 1, 2, ⋯ , n, and let r ˜ p be total return of a portfolio x = (x₁, x₂, ⋯ , x _n )  ^T , (x _i  ⩾ 0, i = 1, 2, ⋯ , n), then we have: M ¯ ( r ˜ p ) = M ¯ ( ∑ i = 1 n r ˜ i x i ) = ∑ i = 1 n [ b i + a i 2 + k 2 ( 2 k + 1 ) ( k + 1 ) ( β i - α i ) ] x i , Var ( r ˜ p ) = ( ∑ i = 1 n [ b i - a i 2 + k 2 ( α i + β i ) ( 2 k + 1 ) ( k + 1 ) ] x i ) 2 + k 2 + 5 k 3 4 ( 2 k + 1 ) 2 ( k + 1 ) 2 ( k + 2 ) ( ∑ i = 1 n ( α i + β i ) x i ) 2 , Var - ( r ˜ p ) = ( ∑ i = 1 n [ b i - a i 2 + k 2 ( α i + β i ) ( 2 k + 1 ) ( k + 1 ) ] x i ) 2 + k 2 + 5 k 3 ( 2 k + 1 ) 2 ( k + 1 ) 2 ( k + 2 ) ( ∑ i = 1 n α i x i ) 2 , Sad ( r ˜ p ) = 1 2 ∑ i = 1 n ( b i - a i ) x i + k 2 ( 2 k + 1 ) ( k + 1 ) ∑ i = 1 n ( α i + β i ) x i .

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:

r ˜ p = ∑ i = 1 n r ˜ i x i = ( ∑ i = 1 n a i x i , ∑ i = 1 n b i x i , ∑ i = 1 n α i x i , ∑ i = 1 n β i x i ) k .(19)

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

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.

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.

OPEN IN VIEWER

Fig. 1

Portfolio efficiencies with different orientations.

P E r = r r 2 , P E v = v 1 v , P E rv = 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.

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),

{ max f ( x ) s . t . g ( x ) ⩽ V 0 , x ∈ Ω .(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 , ⋯ , x n ) T | ∑ i = 1 n x i = 1 , l i ⩽ x i ⩽ u i } , The risk function g (x) is stated as follows:

g ( x ) = ( ∑ i = 1 n [ b i - a i 2 + k 2 ( α i + β i ) ( 2 k + 1 ) ( k + 1 ) ] x i ) 2 + 5 k 3 + k 2 4 ( 2 k + 1 ) 2 ( k + 1 ) 2 ( k + 2 ) ( ∑ i = 1 n ( α i + β i ) x i ) 2 ,(25) and the return function f (x) is as follows:

f ( x ) = ∑ i = 1 n [ b i + a i 2 + 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 ∈ 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 , ⋯ , x n 1 ) T , x 2 = ( x 1 2 , x 2 2 , ⋯ , x n 2 ) T ∈ Ω , λ ∈ R, we have

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 , ⋯ , λ x n 1 + ( 1 - λ ) x n 2 ) T ,(28) and

∑ i = 1 n [ λ x i 1 + ( 1 - λ ) x i 2 ] = λ ∑ i = 1 n x i 1 + ( 1 - λ ) ∑ i = 1 n x i 2 = λ + 1 - λ = 1 ,(29)

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 .(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 ∂ 2 f ( x ) ∂ x 2 = 0 , which can be concluded that f (x) is both a concave function and a convex function. In Equation (27), let

A = ( b 1 - a 1 2 + k 2 ( α 1 + β 1 ) ( 2 k + 1 ) ( k + 1 ) , ⋯ , b i - a i 2 + k 2 ( α i + β i ) ( 2 k + 1 ) ( k + 1 ) ) T , B = ( α 1 + β 1 , ⋯ , α n + β n ) T .(31) Then, the risk function g (x) can be rewritten as

g ( x ) = x T A A T x + 5 k 3 + k 2 4 ( 2 k + 1 ) 2 ( k + 1 ) 2 ( k + 2 ) x T B B T x .(32) And

∂ g ( x ) ∂ x = 2 A A T x + 5 k 3 + k 2 2 ( 2 k + 1 ) 2 ( k + 1 ) 2 ( k + 2 ) B B T x , ∂ 2 g ( x ) ∂ x 2 = 2 A A T + 5 k 3 + k 2 2 ( 2 k + 1 ) 2 ( k + 1 ) 2 ( k + 2 ) B B T > 0 .(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.

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.