ESTIMATION OF PORTFOLIO EFFICIENCY VIA STOCHASTIC DEA

. Traditional data envelopment analysis (DEA) and diversification DEA are two common data-driven evaluation approaches, which have been widely used in the estimation of portfolio efficiency. The above two DEA approaches usually use the risk and expected return indicators to build the input-output process of portfolios. However, this input-output process derived from the risk and expected return is inconsistent with the actual investment process, since the real input should be the initial wealth, and the output should be the terminal wealth. To address this problem, we propose a novel input-output process based on the initial and terminal wealth of portfolios. We transform the terminal wealth into the rate of return and construct a stochastic attainable set by using portfolio returns. We provide three deterministic estimation approaches to deal with the stochastic attainable set, and then obtain three deterministic attainable sets. We further propose three stochastic DEA models to estimate the portfolio efficiency by using the above three deterministic attainable sets. Finally, we provide an empirical analysis to assess the portfolio efficiency of 50 open-ended funds in China. The results show that there are some differences in the portfolio efficiency and its ranking between the proposed DEA models and the existing DEA models, which further verify the rationality of the proposed DEA models.


Introduction
Nowadays, the estimation of portfolio efficiency has always been one of the research hotspots in the field of finance.How to estimate the portfolio efficiency has also been a concern of many researchers.DEA approach, as a data-driven evaluation approach, has been considered as an effective tool to solve the efficiency evaluation problem, because it does not need to specify the form of production function.This approach cannot only avoid the setting error of production function, but also provide a benchmark for decision-makers.In recent years, DEA has been widely used in the estimation of portfolio efficiency, among which the traditional DEA and diversification DEA models are two common methods to solve this problem.The above two kinds of evaluation methods need not make assumptions about the effectiveness of the capital market, but only need to use the multidimensional financial indicators (e.g., transaction costs, and the expected return and risk indicators) to make a relative assessment of portfolios.To the best of our knowledge, the existing researches usually use the above financial indicators to build the production possibility set of portfolios.However, the resulting input-output process is also questioned by many researchers, since they believe that these financial indicators cannot fully describe the actual input-output of portfolios (e.g., [25,26,31]).Therefore, how to define the actual input-output of portfolios is an urgent problem to be solved in this paper.
In recent years, the traditional DEA from production theory has been gradually and partially transposed to the analysis of portfolio performance by regarding these financial indicators as the inputs and outputs.To the best of our knowledge, Murthi et al. [23] first applied traditional DEA to estimate the portfolio efficiency and then treated the standard deviation of fund return as an input and the expected return as an output to construct the corresponding DEA model.Basso and Funari [1] took different risk indicators as inputs and the expected return as an output, and also showed the relationship between the proposed DEA model and the traditional evaluation index.Chen and Lin [10] used VaR and CVaR instead of the traditional variance measurement to measure the portfolio's risk and then used them as the input indicators of the proposed DEA model.Basso and Funari [2] proposed a social responsibility measurement index for the social responsibility investment funds, combined the traditional expected return and risk indicators and constructed a DEA model suitable for the evaluation of social responsibility investment funds.Ding et al. [14] studied the applicability of DEA in the performance evaluation of portfolios with margin requirements.Liu et al. [20] proved the convergence of DEA frontier under the generalized mean-risk framework.Basso and Funari [3] studied the relationship between fund size and fund efficiency in combination with DEA and econometric models.Chen et al. [11] proposed three kinds of DEA models to evaluate the performance of portfolios with fuzzy returns.Zhou et al. [30] provided a DEA frontier improvement approach which can effectively improve the convergence speed of DEA frontier.Allevi et al. [4] used a DEA approach to evaluate the performance of green mutual funds, among which the environmental and financial indicators are both considered into the proposed DEA models.
Traditional DEA can be converted into a linear programming problem, in essence, and its calculation is relatively simple, but many researchers have questioned the ability of traditional DEA in diversifying risks.Since Markowitz [22] proposed that the portfolio can diversify risks, there is an ever-growing literature on the different diversification DEA models to estimate the portfolio efficiency.Briec et al. [8] constructed a diversification DEA model under the classical mean-variance criterion, and also distinguished the portfolio efficiency and the allocation efficiency.Joro and Na [16] proposed a diversification DEA model within the mean-variance-skewness framework, and then investigated the influence of skewness on the portfolio efficiency.Inspired by the previous work, Briec et al. [9] further developed a diversification DEA model with consideration of the high-order moment constraints, and also theoretically proved the global optimal solution availability of the proposed model.Lamb and Tee [17] introduced a general class of diversification DEA models based on the multi-return and multi-risk measures, and then systematically examined the relationship between the portfolio diversification, coherent risk measurement and stochastic dominance.Branda [7] constructed several kinds of diversification DEA models under the generalized mean-risk criterion, and also allowed that the negative values of input and output indicators exist in these models.Lin and Li [18] proposed a super-efficiency diversification DEA model based on the directional distance function measure to further distinguish the differences between the effective investment funds.
It is not difficult to find that the production possibility sets of the above DEA models are constructed by using the risk and return indicators.In the existing literature, the input-output property of the return and risk mainly includes the following two viewpoints.The first viewpoint treats the portfolio risk as an input indicator and the expected return as an output indicator.The second viewpoint considers that the expected return and risk should be both treated as output indicators, because the expected return and risk are two evaluation indicators derived from the portfolio return.However, the above two input-output assumptions are inconsistent with the actual investment process.In fact, it is more reasonable to construct the input-output process of portfolios by using the assumption that the initial wealth is an input and the terminal wealth is an output.This is also a novel input-output assumption which is different from the existing literature.Since the terminal wealth of a portfolio is a random variable on the portfolio return, thus the resulting production possibility set may also be random.It is also called as the stochastic production possibility set.The stochastic production possibility set can be treated as an extension of the traditional deterministic production possibility set.Obviously, it is unrealistic to estimate the portfolio efficiency directly based on the above stochastic production possibility set.Therefore, how to make a deterministic estimation of this stochastic set is also the fundamental problem to be solved.
To the best of our knowledge, the common method to deal with the stochastic production possibility set is using the classical chance-constrained theory to construct the corresponding chance-constrained DEA models.The related studies include Huang and Li [15], Cooper et al. [13], Tsionas and Papadakis [27], Olesen and Petersen [24] and so on.Obviously, the chance-constrained theory can be used to estimate the stochastic production possibility set of portfolios.In addition to this, the classical mean-risk portfolio optimization theory can also be used to derive the estimation of stochastic production possibility set.In this framework, we can use the return and risk measures (e.g., the expectation and standard deviation measures) to estimate the stochastic constraint conditions in the stochastic set mentioned above.In this paper, we will investigate the deterministic estimation of the stochastic production possibility set of portfolios by using the chance-constrained and mean-standard deviation theories.
Under the proposed input-output process based on the initial and terminal wealth, the difficulty in acquiring the initial wealth of portfolios will make it difficult to estimate the portfolio efficiency.For the above reasons, we standardize the initial and terminal wealth of portfolios, and then the initial wealth can be normalized to one unit wealth and the terminal wealth can be converted into the portfolio return.Generally speaking, the portfolio return (e.g., fund return) is usually observable.Therefore, it is more practical to estimate the portfolio efficiency based on portfolio returns.As the characteristics of data described in Liu et al. [19], the portfolio returns can be treated as the data without explicit inputs.And they proposed a DEA-WEI model to deal with this problem, in which the proposed model needs not know the specific values of inputs, but only needs to use the index data (e.g., the data of fund returns).Yang et al. [28] further proposed a DEA-WEI model with quadratic utility terms, and also compared the difference between the benchmark obtained by the proposed DEA model and that obtained by the traditional DEA model.However, the above studies on the DEA-WEI models all assume that the outputs of decision-making units are the deterministic data, while the data of portfolio returns studied in this paper is still stochastic one.Therefore, how to integrate the DEA-WEI model and stochastic DEA is also the rest of our work.
Motivated by the aforementioned research line, we first define the input-output process of portfolios, and then construct the stochastic attainable set without explicit inputs.Under the framework of the chance-constrained and mean-standard deviation theories, we propose three deterministic estimation approaches to estimate the above stochastic attainable set.Then, we construct three kinds of stochastic DEA models based on the direction distance function measure, i.e., chance-constrained stochastic DEA model, and nonlinear and linear stochastic DEA models under mean-standard deviation criterion.For the proposed chance-constrained DEA models, we assume that the portfolio returns follow a joint normal distribution, and then transform the chance-constrained DEA models into the deterministic quadratic programming problems.Further, we investigate the difference between the proposed stochastic DEA models under mean-standard deviation criterion and the existing DEA models.Finally, we apply the proposed stochastic DEA models to assess the portfolio efficiency of 50 growth open-ended funds in China.The empirical results show that the proposed stochastic DEA models are different from the traditional DEA models in the portfolio efficiency and its ranking.These further validate the rationality of the proposed stochastic DEA models.
The remainder of this paper is organized as follows.In Section 2, we redefine the input-output process of portfolios, and then construct a stochastic attainable set by using portfolio returns.In Section 3, we proposed three deterministic estimation approaches to estimate the stochastic attainable set of portfolios.In Section 4, we provide three stochastic DEA models to estimate the portfolio efficiency.In Section 5, we carry out an empirical analysis to verify the validity of the proposed stochastic DEA models, and then show the difference between the proposed DEA models and the existing DEA models.Finally, some concluding remarks are summarized here.

Revisiting the input-output process of portfolios
Suppose that there are  portfolios to be evaluated in the financial market.Let   denote the random return of the -th portfolio, where  = 1, • • • , .Let r = (  ) and (  ) = √︀ Var(  ) be the expectation and standard deviation of the -th portfolio return, respectively, and Ω , = Cov(  ,   ) represent the covariance of portfolio returns   and   , where ,  = 1, ′ is the vector of expected returns and Ω is the covariance matrix of portfolio return vector r = [ 1 , • • • ,   ] ′ .Based on the above  portfolios to be evaluated, this paper aims to estimate the portfolio efficiency based on production theory, and it is crucial to specify the input-output process of portfolios.Intuitively, for an investment activity in a financial asset, the input should be the initial wealth (i.e., various costs) required to buy a certain amount of the asset and the output should be the gain received when the asset is sold.To this end, we let  0 and  1 be the initial and terminal wealth of a portfolio, respectively.Note that,  0 is a deterministic value, while  1 is a random value on the portfolio return.Therefore, for an actual investment process, we can conclude that the real input is the initial wealth and the output is the terminal wealth.In this case, the details about the input-output process of portfolios are shown in Figure 1a.As shown in Figure 1a, for an investor, the input is the amount of money spent to purchase the portfolio (i.e., initial wealth shown in Figure 1a, which includes transaction costs), and the output is the wealth that the investor will receive when the portfolio is sold (i.e., the terminal wealth shown in Fig. 1a).In practice, the initial wealth of most portfolios (e.g., investment funds) is not always observable, which is also inconvenient for evaluators.However, the initial wealth and the terminal wealth of a portfolio satisfy the following relationship:  1 =  0 ×(1+), where  denotes the portfolio return.Then, the initial wealth of a portfolio can be normalized, that is, its terminal wealth can be replaced by the portfolio return .In this situation, the input of each portfolio can be unified as 1 unit wealth, and the output is the random return .Thus, the input-output process of portfolios can be described by the portfolio return, and the detailed conversion process is shown in Figure 1b.With the input-output setup shown in Figure 1b, the evaluation of different portfolios can be understood as the investor spends 1 unit of wealth on each portfolio and ultimately compares which portfolio has a higher terminal return.Compared with the existing studies (i.e., the DEA portfolio evaluation studies that use the return measure and risk measure of the portfolio return to construct the input-output process of portfolios), it is clear that the input-output process of portfolio defined in Figure 1 is a new understanding.
In this paper, we aims to build the DEA models based on the input-output process of portfolios in Figure 1b.Liu et al. [19] treated the production process shown in Figure 1b as a production process without explicit inputs, and then proposed a DEA-WEI model to deal with this problem.However, Liu et al. [19] assumed that the outputs of decision-making units are all deterministic values, while the portfolio's output  is a random variable.Inspired by the work of Cooper et al. [12], we use the idea of stochastic DEA to construct the stochastic attainable set without explicit inputs.Based on the above  portfolios to be evaluated and their portfolio returns (i.e.,   ,  = 1, • • • , ), we can construct the following stochastic attainable set by using the assumptions of convexity and strong free disposability.
where   denotes the intensity weight, which is used to construct the returns of some virtual portfolios based on the observation portfolios, and  denotes the virtual portfolio return that can be generated based on the observed portfolios.
From the expression of Set (2.1), we can find that Set (2.1) can be treated as the stochastic version of the attainable set provided by Liu et al. [19].Due to the randomness of the portfolio return   , it is difficult for evaluators to directly assess the portfolio efficiency within the framework of the stochastic attainable set (2.1).Obviously, how to reasonably estimate the stochastic constraint in the stochastic attainable set (2.1) is the premise for building an effective DEA model.Under the framework of stochastic DEA, we propose three methods to construct the deterministic estimation of the above stochastic constraint, so as to create the corresponding stochastic DEA model to assess the portfolio efficiency.Further, we analyze the differences and connections between the proposed stochastic DEA models and the existing deterministic DEA models (i.e., traditional DEA and diversification DEA models).

The deterministic estimation of stochastic attainable set
In this section, we first use the chance-constrained theory to deal with the above stochastic attainable set (2.1).In this case, the stochastic constraint shown in Set (2.1)only needs to be true at a given probability level 1 − (0 <  ≤ 1).Then, the corresponding deterministic estimation of the stochastic attainable set (2.1) can be expressed as It is easy to find that PPS (1) denotes the set of the virtual portfolio returns (i.e., ) that satisfy the above chance-constrained condition.Inspired by the classical mean-standard deviation portfolio optimization theory, we also provide a second method to deal with Set (2.1).Under the mean-standard deviation framework, we quantify the stochastic constraint in Set (2.1) by using the mean and standard deviation measures.Under the risk averse preference assumption, investors want the mean of portfolio return to be larger and the standard deviation of portfolio return to be smaller.Similar to the assumption using in Set (2.1), we consider that the mean and standard deviation of portfolio return satisfy the strong free disposability1 .Then, the corresponding deterministic estimation can be described as Difference from PPS (1) , PPS (2) denotes the set of the virtual portfolio returns (i.e., ) that meet the above mean-standard deviation framework.Referring to the traditional DEA models, the third method is to deal with the stochastic constraint in Set (2.1) by linearizing the nonlinear constraints shown in PPS (2) .In this situation, the corresponding deterministic estimation can be expressed as The relationship between PPS (2) and PPS (3) is that the latter can be consider as a linear estimation of the former, since the conclusions of  (3) ignores the risk diversification, the resulting linear model is more convenient to be solved, which is consistent with the characteristic of traditional DEA.Moreover, it follows from the conclusion of Liu et al. [20] that as the portfolio sample size increases, PPS (3) converges to PPS (2) with probability one.Therefore, it is also worth considering PPS (3) as a feasible estimation of Set (3.1).
In the following, this paper will construct the corresponding stochastic DEA model based on the above deterministic estimations, i.e., PPS (1) , PPS (2) and PPS (3) .

Estimation of portfolio efficiency via stochastic DEA
Because of the negative value of portfolio returns, we will construct the corresponding stochastic DEA model based on the directional distance function.For any portfolio return   , we assume that,   +   ∈ PPS (1) ,   +   ∈ PPS (2) and   +   ∈ PPS (3) , where   > 0, and  denotes the proportion of the portfolio return that can be expanded within different attainable sets.Under the three scenarios described above, the investor wants to find the maximum expansion proportion under a given attainable set and treats it as a measure of the scope for improvement in the portfolio to be evaluated.Then, we have the following evaluation models: }︁ .The detailed forms of the above stochastic DEA models are shown in Sections 4.1-4.3.

Chance-constrained stochastic DEA model
For the first evaluation model }︁ , the details can be expressed as follows.
Here  is a predetermined parameter, and it satisfies that 0 <  ≤ 1. Parameter  can be treated as the modeler's risk level, indicating the probability measure of the extent to which chance constraint violation is admitted.According to Model (4.1), we can find that, the lower the value of , the higher the modeler's confidence about the portfolio being evaluated and the lower the modeler's risk, and vice versa.Referring to Zhou et al. [5], in most cases, the modeler's confidence is relatively high.In the following, we further assume that the predetermined parameter  satisfies  ≤ 0.5.
Due to the existence of chance-constrained condition in Model (4.1), it is difficult for decision-makers to directly solve it.The traditional method of solving this kind of problem is to assume that the random output follows a given distribution function, and then transforms the chance constraint into the deterministic equivalent constraint.In this case, the above chance-constrained DEA model can be transformed into the corresponding equivalent model.Since the equivalent model only depends on the distribution of portfolio returns, we can estimate the distribution parameters of portfolio returns by using the historical data.Then, the portfolio efficiency can be assessed by using the equivalent model.
In the following, we assume that the portfolio returns follow a joint normal distribution, that is, r ∼  (r, Ω), where r ′ .Without loss of generality, this paper assumes that the direction   can be respectively valued in deterministic direction and exogenous random direction, and then Model (4.1) can be transformed into the corresponding deterministic equivalent model2 .The relevant conclusions are shown in Theorems 4.1 and 4.2.
Theorem 4.1.Suppose that the portfolio returns satisfy r ∼  (r, Ω) and the direction   is a deterministic value, and then Model (4.1) can be equivalent to the following deterministic programming problem. where Proof.
when the portfolio returns meet that r ∼  (r, Ω), we can conclude that the random variable follows the standard normal distribution.Then, the chance constraint condition (4.3) can be equivalent to the following inequality condition.
To sum up, we can conclude that Theorem 4.1 is true here.
Theorem 4.2.Assume that the portfolio return vector r ∼  (r, Ω) and the direction   is an exogenous random variable, and then Model (4.1) can also be equivalent to the following deterministic programming problem. ) (4.1) only needs to assume that the portfolio return vector follows a continuous distribution (see, [5]), but for comparison with the traditional DEA and diversification DEA models in the mean-standard deviation framework, we suppose that the portfolio return vector obeys a joint normal distribution.
Proof.Since the direction   is an exogenous variable, then we can derive the following conclusion: Cov(  ,   ) = 0,  = 1, Then, we can conclude that Theorem 4.2 holds here.
In this section, we define In addition, we let ̃︀ Ω denote the covariance matrix of the random vector ̃︀ r, where ̃︀ Ω is a semi-positive definite matrix.Due the fact that Model (4.2) is an extreme case of Model (4.6) (i.e., Var(  ) = 0), then () and (, ) can be uniformly represented as where ‖•‖ denotes the standard Euclidean norm, and ̃︀ Ω 1/2 is the matrix that satisfies the condition ̃︀ Therefore, Models (4.2) and (4.6) are respectively rewritten as the following second-order cone programming (SOCP). (4.10) Based on the findings in Lobo et al. [21] and Boyd and Vandenberghe [6] on SOCP and convex optimization, it is clear that Models (4.10) and (4.11) are both the convex programming when  ≤ 0.5.
Further, from the definition of stochastic effective shown in Cooper et al. [13] and Zhou et al. [29], for a given risk level , when the optimal value of Model (4.2)/(4.6) is equal to 0, then the portfolio being evaluated is called as stochastic effective.Further, for any portfolio under evaluation, the portfolio efficiency can be expressed as

Nonlinear stochastic DEA model under mean-standard deviation criterion
Section 4.1 mainly uses the chance-constrained theory to deal with Set (3.1).In fact, in addition to this, the classical mean-standard deviation portfolio optimization theory can also be used to derive the deterministic estimation of Set (3.1).Under this framework, we have that  * 2 = max  {︁ |  +   ∈ PPS (2)  }︁ .The detailed model can be described as Similar to Models (4.2) and (4.6), we assume that the direction   can be both evaluated in the deterministic direction and exogenous random direction shown above.First, when   is a deterministic value, then we obtain the conclusion that (  +   ) = (  ) +   and (  +   ) = (  ).Further, Model (4.12) can be simplified to the following DEA model.
Under the mean-standard deviation criterion, when the optimal value of Model (4.13)/(4.14) is equal to 0, the portfolio being evaluated is said to be stochastic effective.In addition to this, the portfolio efficiency can be defined as

Linear stochastic DEA model under mean-standard deviation criterion
The above DEA evaluation models are all nonlinear in nature.Although they all consider the diversification role of portfolios, they may not have advantages in solving these models.Referring to the traditional linear DEA model, the stochastic DEA model with linear constraints can be constructed under the meanstandard deviation criterion.For any portfolio being evaluated, the evaluation model can be expressed as }︁ .The details are shown as follows.
Similarly, when   is a deterministic value, Model (4.15) can be simplified as follows.
when   is an exogenous random variable, Model (4.15) can be rewritten as In this case, for any portfolio being evaluated, its efficiency is defined as  4.4.The proposed stochastic DEA models vs. the existing DEA models In order to further investigate the difference between the proposed stochastic DEA models and the existing DEA models, this paper will revisit the existing DEA models and their production possibility set.To the best of our knowledge, the diversification DEA and traditional DEA are two common approaches used in estimating the portfolio efficiency.However, different from the proposed models, the input-output process shown in the existing studies is constructed by using the return and risk indicators of portfolio returns.Under the meanstandard deviation criterion, the production possibility set of the diversification DEA and traditional DEA can be respectively expressed as follows.
Based on Sets PPS (4) and PPS (5) , we assume that Point (  ,   ) is made up of the standard deviation and expected return of the portfolio under evaluation.From the existing literature, we find that the corresponding DEA models are usually constructed by using Sets PPS (4) and PPS (5) , and the input and output indicators are also characterized by the portfolio's returns and risks.At present, there are two different viewpoints in the selection of input and output indicators.The first viewpoint is that, the portfolio risk is regarded as an input indicator, while the expected return of a portfolio is regarded as an output indicator.The second viewpoint is that, the expected return and risk should be both treated as output indicators, because the expected return and risk are two evaluation indicators derived from the portfolio return.In particular, the expected return is treated as a desirable output and the risk is treated as an undesirable output.To further show the difference between the proposed DEA models and the existing DEA models, we will compare the proposed models and the existing DEA models under the different input-output assumptions.Since the proposed DEA models in this paper are all output-oriented ones, thus we mainly focus on the comparative analysis under the output-oriented measure.
First, when the portfolio risk is regarded as an input indicator, and the expected return is regarded as an output indicator, using Sets PPS (4) and PPS (5) , the following DEA models can be derived under the outputoriented measure.
In order to further investigate the relationship between the proposed DEA models and the existing DEA models, we assume that   is an exogenous random direction and also let    = (  ) and    = (  ).In this case, we can find that Models (4.22) and (4.23) are consistent with the stochastic DEA models (4.14) and (4.17) proposed in this paper.This further verifies that the proposed DEA models have good uniformity.
In the existing literature, diversification DEA and traditional DEA models (4.22) and (4.23) are generally assumed that    < 0 and    > 0, because decision-makers are willing to find a benchmark with higher expected return and less risk as the goal for improvement.However, in the stochastic DEA models (4.13) and (4.16), the exogenous random direction   satisfies the following conditions: (  ) > 0 and (  ) > 0. At this time, the benchmark provided by the proposed DEA models has the characteristics of higher return and higher risk compared with the portfolio under evaluation.To analyze the difference between the proposed DEA models and the existing DEA models more intuitively, the direction of Models (4.22) and (4.23) might as well be ⃗  = (−(  ), (  )) ′ , and the improvement direction of risk and return derived from the exogenous random ′ .In this case, the detailed relationship between ⃗  and ⃗  1 is shown in Figure 2. As can be seen from Figure 2, we can find that, the existing diversification DEA and traditional DEA models usually adopt a more conservative improvement direction because the benchmark of an ineffective portfolio has the characteristics of higher return and lower risk.However, the benchmark sought by the proposed models (4.13) and (4.16) is characterized by higher return and higher risk.Therefore, compared with the existing DEA models, the stochastic DEA models (4.13) and (4.16) are more optimistic evaluation methods.

Empirical analysis
In order to verify the effectiveness of the above DEA models and discuss the difference between the proposed DEA models and the existing DEA models, this paper selects 50 growing open-ended funds from China's fund market, whose fund codes are shown in Table 1.The range of weekly historical returns is from January 1, 2017 to December 31, 2019, and the data is downloaded from RESSET database (http://db.resset.com/).Before beginning the following empirical analysis, this paper still needs to check the normality of the fund returns selected here.As far as we know, Jarque-Bera test is a common test method to check the normality.Here, Jarque-Bera test (JB test for short) is conducted at the significance level of 5%.If the JB statistic is greater than 5.99, the test result rejects the null hypothesis of normal distribution.The specific test results are shown in Table 1.
According to the results in Table 1, we find that the historical weekly returns of the above funds are all subject to normal distribution.This also indicates that the 50 funds selected here can be used to check the effectiveness of the proposed DEA models.

Fund efficiency analysis under the deterministic direction
When the improvement direction   is a deterministic value, this paper will discuss the difference between the three proposed stochastic DEA models in the portfolio efficiency and its ranking.Since the chance-constrained stochastic DEA model (4.2) has taken into account the risk preference of decision-makers, that is, the portfolio efficiency obtained from Model (4.2) is dependent on the risk preference .In this paper, it is assumed that  can be respectively evaluated at 0.3, 0.35, 0.4, 0.45 and 0.5.In addition this, we might as well assume that the direction satisfies   = max  ((  )) − min  ((  )).Based on Models (4.2), (4.13) and (4.16), the evaluation results can be obtained as shown in Table 2.
It can be seen from Table 2 that when the risk preference of decision-makers  is lower (e.g., 0.3 and 0.35), most of the funds being evaluated are stochastic effective.It also suggests that at a lower level of risk preference, it is harder for decision-makers to judge the efficiency differences between the 50 funds selected above.This is because the lower the level of risk preference, the more cautious decision-makers are, so that decision-makers cannot find an absolute dominant investment fund among the 50 funds selected here.When the risk preference of decision-makers is higher (e.g., 0.4, 0.45 and 0.5), the ability of chance-constrained DEA model (4.2) to identify funds also increases.In addition, different from Model (4.2), the stochastic DEA models (4.13) and (4.16) do not consider the risk preference of decision-makers, but only alone measure the expected return and the risk of a portfolio under the mean-standard deviation framework.That is to say, Models (4.13) and (4.16) do not consider the return and risk of a portfolio as a whole, and the evaluation results are more relaxed, therefore they have a higher degree of differentiation of funds.
Next, we further quantify the differences between the efficiencies (or rankings) obtained from different DEA models.Referring to the existing literature (e.g., [7,20]), we use the Pearson correlation test to analyze the fund efficiencies and rankings in Table 2. Based on the efficiency series and ranking series in Table 2, the correlation coefficients between the efficiency (or ranking) series obtained from different DEA models can be obtained, and the results are shown in Tables 3 and 4. As shown in Table 3, under the framework of chance-constrained DEA model (4.2), the correlation coefficients of fund efficiencies obtained at different risk levels are still quite different, especially when the risk preferences of decision-makers are quite different (e.g.,  = 0.3 and  = 0.5).This indicates that the decision-makers' risk preference will have a greater impact on the fund efficiency.On the other hand, under the mean-standard deviation criterion, we find that the fund efficiencies obtained by using the nonlinear and linear stochastic DEA models (i.e., Models (4.13) and (4.16)) have a higher correlation (i.e., the correlation coefficient is 0.9992).This also shows that, although Model (4.16) does not consider the diversification effect between portfolios, it can be still better approximate the fund efficiency obtained from Model (4.13).Most importantly, as the risk preference of decision-makers increases, the correlation coefficient between the fund efficiency from Model (4.2) and that from Model (4.13) or Model (4.16) also increases.Table 4 shows that, for Model (4.2), the risk preference of decision-makers will also have a great impact on the fund ranking.In addition, Models (4.13) and (4.16) have a good consistency in the fund ranking.Further, the correlation coefficient between the fund ranking of Model (4.2) and that of Model (4.13) or Model (4.16) is increasing with the risk preference of decision-makers.Obviously, the conclusion in Table 4 is also in line with that in Table 3.

Fund efficiency analysis under the exogenous random direction
In this section, we assume that the improvement direction   is an exogenous random variable.Similar to Section 5.1, we will discuss the difference between the fund efficiencies/rankings of the three proposed DEA models.In addition, we assume that  can be respectively evaluated at 0.3, 0.35, 0.4, 0.45 and 0.5.Further, we also suppose that the random direction   satisfies the conditions that (  ) = max  ((  )) − min  ((  )) and (  ) = max  ((  )) − min  ((  )) 3 .Using Models (4.6), (4.14) and (4.17), we can obtain the evaluation results as shown in Table 5.
Similar to Section 5.1, this paper will make a correlation test of the fund efficiency and its ranking, and then analyze the differences among different stochastic DEA models.The detailed results are shown in Tables 6 and 7.
For Model (4.6), Tables 6 and 7 show that decision-makers' risk preference will also have a great impact on both the fund efficiency and the fund ranking.In addition, from the results shown in Tables 6 and 7, we find that Models (4.14) and (4.17) also have good consistency in the fund efficiency and its ranking.Further, under the framework of Models (4.6), (4.14) and (4.17), the corresponding correlation coefficient of between fund efficiencies/rankings increases with the increase of decision-makers' risk preference.It is not difficult to find that the conclusions in Tables 6 and 7 are consistent with those in Tables 3 and 4.This also indicates that the different improvement directions will not change the properties of the stochastic DEA models.

Comparative analysis of different DEA models
Sections 5.1 and 5.2 mainly focus on the analysis of the differences between the proposed stochastic DEA models.In the following, we will investigate the difference between the stochastic DEA models and the existing DEA models.To keep consistent with the directions selected in Sections 5.1 and 5.  Table 10 shows the difference between the efficiency rankings of stochastic DEA models and those of the existing DEA models.It can be seen from Table 10 that there are also certain differences between the above stochastic DEA models and the existing DEA models in the efficiency ranking.This further validates the conclusions in Table 9.Since the proposed stochastic DEA models are all derived from the actual input-output process of portfolios, they can better reflect the real portfolio efficiency.Therefore, the results in Tables 9 and 10 further indicate the rationality of the stochastic DEA models proposed in this paper.

Conclusion
Portfolio efficiency evaluation has always been one of the research hotspots in the financial field.To the best of our knowledge, traditional DEA and diversification DEA models are two common nonparametric evaluation methods, which have been widely used in the portfolio efficiency evaluation.The existing researches usually use the expected return and risk indicators to build the input-output process of portfolios.At present, there are mainly two viewpoints on this issue.The first viewpoint considers the portfolio's risk as an input and the expected return as an output.The second viewpoint is that the expected return and risk of a portfolio should be both regarded as output indicators, because they are two evaluation indicators derived from the portfolio return.It is not difficult to find that these existing input-output assumptions are inconsistent with the actual investment process, because the real input should be the initial wealth and the output be the terminal wealth.In this input-output framework, this paper standardizes the initial wealth of a portfolio and converts the terminal wealth value into the form of return rate, which is usually observable.
In this paper, we first clarify the actual input-output process of portfolios, and then construct the corresponding stochastic production possibility set.In the following, we propose three estimation methods to deal with the stochastic production possibility sets constructed above, and then propose three deterministic sets as the estimations of the above stochastic production possibility set.Using the above three kinds of deterministic estimation, three kinds of stochastic DEA models are directly constructed, including the chance-constrained DEA model, and the nonlinear and linear stochastic DEA models under the mean-standard deviation criterion.
Finally, the proposed stochastic DEA models are used to assess the efficiency of 50 open growth models in China's fund market.By comparing with the existing DEA models, the effectiveness of the proposed stochastic DEA models is further verified.

Figure 1 .
Figure 1.The input-output process of portfolios.

Figure 2 .
Figure 2. The difference between the projection directions of different DEA models.

Table 1 .
The normality test of the return rates of funds being evaluated.

Table 2 .
Stochastic DEA efficiencies and rankings of funds being evaluated.

Table 3 .
The correlation of fund efficiencies from different stochastic DEA models.

Table 4 .
The correlation of efficiency rankings from different stochastic DEA models.

Table 5 .
Stochastic DEA efficiencies and rankings of funds being evaluated.

Table 8 .Table 6 .
2, we assume that the directions of traditional DEA models (4.20)-(4.23)satisfy that   =   The correlation of fund efficiencies from different stochastic DEA models.

Table 8 .
The fund efficiencies and rankings derived from traditional DEA models.

Table 9 .
The correlation of fund efficiencies from different DEA models.

Table 10 .
The correlation of efficiency rankings from different DEA models.