The effect of underlying distribution of asset returns on efficiency in DEA models

Abstract

According to modern finance theory and increasing need for efficient investments, we evaluate the portfolio performance based on the data envelopment analysis method. By the fact that stock market’s return distributions usually exhibit skewness, kurtosis and heavy-tails, we consider some appropriate underlying distributions that affect the input and output of the model. In this regard, the multivariate skewed t and the multivariate generalized hyperbolic as the heavy-tailed distributions of Normal mean-variance mixture are applied. The models are inspired by the Range Directional Measure (RDM) model to deal with negative values. The value-at-risk (VaR) and conditional VaR (CVaR) as risk measures are used in these optimization problems. We estimate the parameters of such distributions by Expectation Maximization algorithm. Then we present an empirical investigation to measure the relative efficiency of two sets of seven groups of companies from different industries of Iran stock exchange market. By comparing the results of introduced models with previous RDM approach, we show that how well the distribution of assets affect the performance evaluation.

Keywords

Data envelopment analysis normal mean-variance mixture distributions portfolio optimization VaR CVaR

1 Introduction

The mean-variance portfolio selection model is to minimize the variance subject to achieving a prescribed mean target in the investment [34, 36]. Its purpose is to allocate the wealth amongst a basket of financial assets to reach a satisfactory trade-off between the return of the investment and the associated risk. The risk measure and the mean return are two important items in portfolio selection theory. By implementing the investment strategy and measuring the portfolio performance, an investor can construct a profitable portfolio. Data envelopment analysis (DEA) technique has been found useful to measure relative efficiencies and to enhance the portfolio performance [13]. In finance, the DEA models have been frequently applied to measure the performance of the portfolio and operational efficiency. First, a DEA portfolio efficiency index was introduced to measure the relative performance with various transaction fees [27] and its generalization that considers different risk measures as the inputs was proposed [4]. There have been a number of approaches to the development of DEA to nonlinear form by quadratic constraints over a multiple time-horizons [26]. The multi-horizon mean-variance portfolio analysis and diversification was modified [6]. For a better decision making, a mean-variance-skewness model has been proposed into the evaluation of portfolio performance which has an additional constraint for skewness besides the mean and variance. In other words, a non-linear DEA-like framework has been developed by using higher moments in portfolio performance measurement in a three-dimensional space [20]. When the sample size is large enough, it is shown that the DEA frontiers converge to the portfolio efficient frontier [21]. A new DEA-based indices are introduced based on new risk measures concepts into fund performance evaluation [10]. Inspired by multi-objective optimization, an approach to take into account the shape of the distribution of returns using several risk measures at the same time has been proposed [5]. As a most common approach, solving fuzzy programming problem has been used for an uncertain multi-objective mean-variance-skewness-kurtosis portfolio optimization model [9]. Original DEA schemes involve crisp information of inputs and outputs that may not be accessible in real world applications. Also undesirable outputs may be present in the manufacturing system and to obtain a reliable measurement, a neutrosophic DEA model has been proposed [23].

A rate of return in a loss of an investment over a specified period of time may turn into negative. So an approach based on the directional distance function as Range Directional Measure (RDM) model was presented [7, 29]. The RDM model applies positive directions to measure a necessary improvement in the inputs and the outputs to reach the efficient frontier. RDM provided a DEA model which can handle the cases where inputs and outputs take negative values.

Risk is one of the important factor in investment policy and the variance of return is traditionally considered as the risk measure in portfolio management. But it is not appropriate because of its penalizing symmetrically both in profit and loss. Furthermore, it ignores the tail risk and skewness of the distribution. There are alternative measures of risk that have many theoretical and practical advantages. One of them is value-at-risk (VaR) that has been approved by bank regulators as a valid approach for calculating risk charges. However, it is not always sub-additive, and it is generally unable to detect diversification of a portfolio [37]. By employing VaR to measure the risk associated with uncertain random return, some corresponding portfolio optimization models have been studied [31]. Despite its deficiencies, VaR, is still preferred risk measure in portfolio [2 , 37]. Another representative popular risk measure which is always sub-additive is conditional VaR (CVaR) and is more informative than VaR about extreme losses (for more details see [12, 22]). To cope with skewed return distributions, CVaR as a risk measure has been introduced. This measure is called Mean-Excess loss, Mean Shortfall or Tail VaR. This coherent risk measure has better computational characteristics and consequently is well adopted in finance field [1]. CVaR is proved to be stable with respect to the choice of the confidence level [33]. It was demonstrated that linear programming with CVaR constraints can be used for portfolio optimization problems [32]. It should be noted that VaR and CVaR are two tailed-related risk measures. Two fuzzy portfolio selection models integrated with DEA have been proposed by these measures under a credibilistic programming [15].

Empirical evidence shows that many of financial return series are heavy-tailed and possibly skewed [18]. So, the construction of an optimal portfolio depends on the probability distribution used to model returns. Furthermore, the effectiveness of the underlying distribution is not restricted to the financial markets, but it is applicable to the energy commodity markets and insurance companies. For instance, energy commodity markets’ returns are directly influenced by the volatility of energy prices that are closely related to the economic and financial environment and energy supply-demand situation [34, 39]. Also, the deterministic inputs with stochastic noise as the outputs were introduced according to a family of heavy-tailed stable distribution in DEA framework in insurance companies [28]. The Normal mean-variance mixture are a class of flexible multivariate distributions that can capture heavy tails and skewness [3]. These distributions are well behaved under the linear transformation with nice properties in Monte Carlo simulation and portfolio selection. The multivariate skewed t (mST) and multivariate generalized hyperbolic (mGH) distributions are subfamilies of these distributions. Choosing the proper initial values of the parameters make different skewness and heavy-tailedness in return distributions. Especially mST is the most-heaviest tailed among these distributions [3, 16]. The expectation maximization (EM) algorithm is a preferred numerical method in estimating the parameters of these distributions [18]. This algorithm is a two-step iterative process that obtains the maximum likelihood estimates of parameters. Using current parameter values, and then the function is maximized to produce updated parameter values [11].

In real-world data, due to the non-normality of returns [14 , 30], skewness and leptokurtosis are two important parameters that are taken into account of return distributions. Therefore, the underlying probability distribution effects on each asset performance assessment. The return distributions have impact on the input and output amounts of optimization problem, too. Asset returns are assumed to be normally distributed, while the probabilities of tail events were not considered into the performance assessment process [2 , 27]. So, we use some multivariate distributions that capture heavy tails and skewness and make better fit to real returns. In this paper, we are concerned with the underlying Normal mean-variance mixture distributions. To evaluate the performance of assets by DEA method, we consider the risk and return. We choose risk measure VaR or CVaR as the input and mean return as the only output in our models.

Empirical studies on portfolio performance evaluation with DEA structure showed that risk measure, mean returns and sometimes higher order moments of returns have been computed directly by the sample real data without considering any distribution. While in current study, the return distributions are taken into account to assess the asset performance in DEA framework that makes a proper tool to get accurate results. We focus on the family of Normal mean-variance mixture as the underlying distribution to model the return and use the risk measure of the asset to evaluate the performance. The mean return and risk measure are simulated by the Monte Carlo method with the estimated parameters of the underlying distribution.

To deal with negative values, the RDM model is applied which provides efficiency scores as well as radial efficiencies traditionally used in DEA. This model is one of the common models for evaluating performance of DMUs that we do not focus on the underlying distribution in its inputs and outputs. In addition, two models are proposed based on the mST and mGH distributions inspired by RDM which have impact on input and output. These multivariate continuous distributions are able to cover the characteristics of returns. So, we do not need to add further constraints to the models. In this regard, a two-dimensional mean return-risk space is identified in our models. Comparing the performance of the models indicates that, considering skewness and kurtosis leads to more interpretable efficiency evaluation. Moreover, improper underlying distribution makes underestimated risk measure and mean return. Also, the efficiency scores measured in such models based on mST and mGH distributions are more accurate than the RDM model. In addition, these results help the manager to be aware of company’s performance and can change his policies. Also, if the company is inefficient, the manager finds how much the risk have to be decreased and the mean return be increased. In real data analysis, to apply proposed models, first the parameters of the distributions are estimated by the EM algorithm. Then, the input and output of the models are computed by the simulated log-returns with Monte Carlo technique. All models are applied to Iran stock exchange market that includes two sets of seven groups of companies where each firm named as an asset (financial asset). So, each asset is considered as a DMU that we evaluate the efficiency of it.

We have organized the paper as follows. Section 2 is devoted to the preliminary concepts of CVaR, skewed t and mGH distributions and an introduction to the RDM model. In section 3, the effect of the underlying distribution into the evaluation of the asset performance is provided. The proposed models are based on the RDM model, by considering the mST and mGH as underlying distributions. Empirical illustrations using two sets of seven groups of companies are provided in section 4, and the last section concludes the paper.

2 Preliminaries

In this section, we present some definitions and concepts that are used in the following sections. First, we concentrate on the concept of coherent risk measure, introduced in [1]. Then, definitions of VaR and CVaR are provided. Next, the Normal mean-variance mixture distributions are defined. Finally, the RDM model is represented, briefly.

Definition 1. Let (Ω, F, P) be a probability space and I (Ω, F) a set of random variables of one dimensional on the space. The ρ : I (Ω, F) → R is a coherent risk measure whenever it satisfies following axioms for random variables X, Y ∈ I (Ω, F),

Monotonicity: If X ⩽ Y, then ρ (Y) ⩽ ρ (X)

Subadditivity: ρ (X + Y) ⩽ ρ (X) + ρ (Y)

Translation Invariance: For all α ∈ R, ρ (X + α) = ρ (X) - α

Positive homogeneity: For all λ ⩾ 0, ρ (λX) = λρ (X).

Value at Risk (VaR) as one of the risk measure is a benchmark standard for firm-wide measures of risk.

Definition 2. Suppose w^T = (w₁, w₂, . . . , w_n) is the capital amount which is invested in each asset, and Y^T = (Y₁, Y₂, . . . , Y_n) is the return of each asset. The return of each portfolio is the weighted average of the individual asset return, weighted by capital value. In other words, the return of the portfolio is w^TY. Therefore, the loss of the portfolio over a fixed interval of time is $L (w, Y) = - \sum_{j = 1}^{n} w_{j} Y_{j} = - w^{T} Y$ and F_L is the portfolio’s distribution function.

The risk measure VaR at confidence level β ∈ (0, 1) is the smallest value l such that the probability that the loss exceeds l is no larger than (1 - β), the other hand $\begin{matrix} {VaR}_{β} & = & inf {l \in R : P (L ⩾ l) ⩽ 1 - β} \\ = & inf {l \in R : F_{L} (l) ⩾ β}, \end{matrix}$

VaR is a coherent measure when the underlying distribution is elliptical [18].

The CVaR which is additionally coherent, at confidence level β ∈ (0, 1) is defined as $C V a R_{β} = E [L | L ⩾ V a R_{β}] .$ By definition, the VaR at confidence level β is never higher than the corresponding CVaR. Consequently, portfolios with a lower CVaR shows a low VaR too.

Definition 3. Normal mean-variance mixture distribution. The d-dimensional random variable X has a multivariate Normal mean-variance mixture distribution if $X \overset{d}{=} μ + T γ + \sqrt{T} Z,$ where $\overset{d}{=}$ denotes equality of finite dimensional distributions and μ, γ ∈ R^d. T is a mixture variable that is nonnegative and scalar-valued. Furthermore, z is a zero mean d-dimensional normal distributed random variable with covariance ∑ denoted by z ∼ N_d (0, ∑). It should be noted that z is independent of T. Conditional on T, X is normal $X | T \sim N_{d} (μ + T γ, T \sum),$ where the random variable T follows generalized inverse Gaussian (GIG) distribution with parameters λ, χ, ψ ∈ R, T ∼ GIG (λ, χ, ψ). So, X has a multivariate generalized hyperbolic (mGH) distribution represented by $X \sim {GH}_{d} (λ, χ, ψ, μ, γ, \sum) .$

In other words, the mGH distributions can be represented as a Normal mean-variance mixture where the mixture variable has GIG distribution. The index parameter λ and concentration parameters χ and ψ are inherited from the mixing distribution that remains the same. Moreover, μ represents the location vector, γ the skewness vector and ∑ is the dispersion matrix. The mGH class offers a natural generalization of the multivariate Gaussian class. Potential distributions from mGH are hyperbolic, normal inverse Gaussian (NIG), variance gamma (VG), student t and skewed t distributions [18]. It should be noted that the mGH is more skewed and has heavier tails than the normal distribution. The parameter λ plays an important role in the mGH distributions.

Both distributions have some parameters in which three of them as λ, χ and ψ are the same. By choosing proper initial value of each parameter at beginning of the procedure of estimation, we can show the behavior of the distributions such as the skewness, kurtosis and heavy-tailedness. So the mGH distributions are appropriately used for cases with lighter tails than the mST. For this purpose,

For this purpose, the effects of parameters on distributions are illustrated in Figs.1 4. Figure s1 and 3 show a comparison among different values that parameters λ and ψ have taken and Figs.2 and 4 have depicted the right tails of them [17].

Fig. 1

The density of generalized hyperbolic and Gaussian.

Fig. 2

The density of Generalized and Gaussian at right tail.

Fig. 3

The density of Gaussian, Skewed t and Hyperbolic.

Fig. 4

The density of Gaussian, Skewed t and Hyperbolic at right tail.

In Fig. 1, it is shown that how the parameter λ influences the tails and kurtosis. Let λ varies from –10 to 10, set μ = 0, γ = 0, ψ = 1, χ = 1 and σ to be a constant so the variance of the generalized hyperbolic distribution is 1 with mean zero.

When |λ| is small, the tails are heavy but when |λ| becomes larger, the tails become thinner and by increasing it, the symmetric mGH distributions tend toward the normal distribution, Fig. 2.

In Fig. 3, for $λ = - \frac{ν}{2}$ where ν is the degree of freedom, χ = ν and ψ goes to 0, we get limiting case which is called the skewed t distribution.

It is shown in Fig. 4 that the skewed t has the heaviest tail among those tested distributions.

The RDM model was proposed by [29] and inspired by the Directional Distance Function model by [7] which can be applied for computing efficiency in the presence of negative data. In the present paper, RDM model is used, since some mean returns are negative.

Definition 4. For DMU_j, j = 1, 2, . . . , n with inputs x_ij, i = 1, 2, . . . , m and outputs y_rj, r = 1, 2, . . . , s in R^m+s and unit o∈ { 1, 2, . . . , n } which is under assessing. The generic directional distance model represents as $\begin{matrix} max {α | \sum_{j = 1}^{n} w_{j} y_{rj} ⩾ y_{ro} + α R_{ro}, r = 1, 2, . . ., s, \\ \sum_{j = 1}^{n} w_{j} x_{ij} ⩽ x_{io} - α R_{io}, i = 1, 2, . . ., m, \\ \sum_{j = 1}^{n} w_{j} = 1, w_{j}, α, R_{ro}, R_{io} ⩾ 0} . \end{matrix}$

The above model is a non-oriented case, where the input contraction and output expansion improve simultaneously. For a given data set, when some of them are negative, an ideal point is defined as $I = (max_{j} y_{j}, r = 1, 2, . . ., s, min_{j} x_{j} = i = 1, 2, . . ., m)$ . The vectors R_ro and R_io which are referred to the range of possible improvement of DMU_o are $\begin{matrix} R_{io} = x_{io} - min_{j} {x_{ij}}, \begin{matrix} i = 1, 2, . . ., m \end{matrix} \\ and R_{ro} = max_{j} {y_{rj}} - y_{ro}, \begin{matrix} r = 1, 2, . . ., s . \end{matrix} \end{matrix}$

At the ideal point I the range of possible improvement can be seen as a surrogate for the maximum improvement that DMU_o could achieve on each input and output. Such an improvement can never be negative [29].

3 Models under different distributions

In this section, we apply RDM model to indicate that the input and the output of a model are affected by the return distributions. As previously mentioned, the return distributions exhibit skewness, kurtosis and are heavy-tailed and they impact on the performance evaluation that leads to more precise efficiency scores. The DEA is used as an efficiency assessment tool, but the traditional DEA models are restricted to non-negative data. So because the variables in financial field such as returns take positive and negative values, we employ one of the DEA-based model that deals with negative values as RDM. In the empirical example, we show that how the data underlying distribution influences the asset performance assessment while in RDM model, it is ignored.

First, we apply RDM model as one of the existing models for evaluating efficiency on data from the stock market. It evaluates the asset performance without considering the type of return distributions. Also, in this model, risk and mean return are the only input and output, respectively. Let’s assume Y₁, Y₂, . . . , Y_n be the log-returns of the n assets’ prices in stock market. For a specific asset return Y_o where o∈ { 1, 2, . . . , n } and regarding to the negative returns value, the vector $g^{T} = (R_{{CVaR}_{β}^{o}}, R_{E (Y_{o})})$ , where $R_{{CVaR}_{β}^{o}} = ({CVaR}_{β}^{o} - \min ({CVaR}_{β}^{j} : j = 1, . . ., n))$ and R_{E(Y_o)} = (max(E (Y_j) : j = 1, . . . , n) - E (Y_o)). This vector is a range of possible improvement in the input and the output. The ${CVaR}_{β}^{o}$ where o∈ { 1, 2, . . . , n } is the value of risk and E (Y_o) is the mean return of asset under evaluation. Then, we solve the following linear model

$\begin{matrix} max \begin{matrix} α \end{matrix} \\ s . t . \begin{matrix} \end{matrix} E (Y (w)) ⩾ E (Y_{o}) + α R_{E (Y_{o})} \\ \begin{matrix} {CVaR}_{β} (Y (w)) \end{matrix} ⩽ {CVaR}_{β}^{o} - α R_{{CVaR}_{β}^{o}} \\ \begin{matrix} \begin{matrix} \end{matrix} & e^{T} w = 1 \end{matrix} \\ \begin{matrix} \begin{matrix} \end{matrix} & w ⩾ 0, α ⩾ 0, \end{matrix} \end{matrix}$ (1)

Where $Y (w) = \sum_{j = 1}^{n} w_{j} Y_{j}$ is the return of a portfolio and $E (Y (w)) = \sum_{j = 1}^{n} w_{j} E (Y_{j})$ the weighted average of each mean return. The optimal value of α which is shown by α^* indicates the distance between the asset under evaluation and the efficient frontier. In other words, α^* represents the inefficiency score of the asset under evaluation and 1 - α^* is the amount of efficiency. The vector w^T = (w₁, w₂, . . . , w_n) is the proportions of initial capital of n assets in a portfolio and e denotes the n-dimensional vector of ones. Apparently, if α^* equals zero the asset is located on the efficient frontier and it is called efficient asset. Otherwise, the amount of inefficiency means that its mean return and CVaR should be changed in order to result in an efficient point on the efficient frontier.

In this framework, CVaR and the mean return are the only input and output of the model, respectively. But significant value of skewness and kurtosis, indicating that the data are not normally distributed. The results reveal that VaR and CVaR tend to underestimate the risk. Therefore, this problem is carried over into the asset performance assessment.

In contrast, the mST and mGH distributions are flexible in their tails behavior. Since the return distributions exhibit skewness and leptokurtosis, the mST and mGH are appropriate candidates for return distributions. In the following two models, the mST and mGH distributions are considered as the underlying distribution, in models (2) and (3) respectively with n financial assets.

In model (2) called RDM-mST model, Y_j is the j-th asset where Y_j ∼ mST (ν, μ_j, γ_j, ∑_jj) , j = 1, . . . , n and ν, μ_j, γ_j and ∑_jj are parameters of mST distribution. According to the directional vector, we solve the following model $\begin{matrix} max \begin{matrix} α \end{matrix} \\ s . t . \begin{matrix} \end{matrix} E (Y (w)) ⩾ E (Y_{o}) + α R_{E (Y_{o})} \\ \begin{matrix} CVaR (Y (w)) \end{matrix} ⩽ {CVaR}_{β}^{o} - α R_{{CVaR}_{β}^{o}} \\ \begin{matrix} \begin{matrix} \end{matrix} & e^{T} w = 1 \end{matrix} \\ \begin{matrix} \begin{matrix} \end{matrix} & w ⩾ 0, α ⩾ 0 \end{matrix}, \\ \begin{matrix} \begin{matrix} \end{matrix} \end{matrix} where \\ \begin{matrix} \begin{matrix} \begin{matrix} \end{matrix} \end{matrix} & Y_{j} \sim mST (ν, μ_{j}, γ_{j}, \sum_{jj}), \begin{matrix} j = 1, . . ., n . \end{matrix} \end{matrix} \end{matrix}$ (2) Now, we introduce the following model whereas the underlying distribution is mGH and we call it RDM-mGH model where Y_j ∼ mGH (λ, χ, ψ, μ_j, γ_j, ∑_jj) , j = 1, 2, . . . , n and λ, χ, ψ, μ_j, γ_j and ∑_jj are parameters of mGH distribution. $\begin{matrix} max \begin{matrix} α \end{matrix} \\ s . t . E (Y (w)) ⩾ E (Y_{o}) + α R_{E (Y_{o})} \\ \begin{matrix} CVaR (Y (w)) \end{matrix} ⩽ {CVaR}_{β}^{o} - α R_{{CVaR}_{β}^{o}} \\ \begin{matrix} \begin{matrix} \end{matrix} & e^{T} w = 1 \end{matrix} \\ \begin{matrix} \begin{matrix} \end{matrix} & w ⩾ 0, α ⩾ 0, \end{matrix} \\ \begin{matrix} \begin{matrix} \begin{matrix} \end{matrix} \end{matrix} & where \end{matrix} \\ \begin{matrix} \end{matrix} \begin{matrix} \begin{matrix} \end{matrix} & Y_{j} \sim m GH (λ, χ, ψ, μ_{j}, γ_{j}, \sum_{jj}), \begin{matrix} j = 1, . . ., n . \end{matrix} \end{matrix} \end{matrix}$ (3)

It should be noted that the proper choice of the initial values of parameters λ, χ and ψ in mGH distributions, we can control their tails behavior and show the skewness and kurtosis values of the distributions. So the mGH distributions are appropriately used for cases which have less heavy tailed than the mST. When ψ goes to zero, a subclass or limiting distribution of mGH is asymmetric or skewed t distribution [17]. As it is cited, the RDM model underestimates the probability of the skewness and tail events, while the heavy tail properties of the mST and mGH distributions describe them well. Therefore, if we do not employ appropriate distributions including tails behavior, risk measures will be underestimated and efficiency scores will be inaccurate.

The models (2) and (3) are in two-dimensional mean return-risk space and cover the higher moments of returns, so it is not needed to consider additional constraints for them.

We remind that the optimal solution α^* in models (2) and (3) indicates the inefficiency score of asset under evaluation and the asset is efficient when the inefficiency score is zero.

In order to solve the models (2) and (3), EM algorithm and Monte Carlo simulation are applied according to the following steps

Step 1. By each financial asset return data, the mST and mGH distributions parameters are estimated by EM algorithm.

Step 2. With the estimated parameters, for each asset sufficiently large number of the scenarios returns, Y_j, j = 1, 2, . . . , n, are simulated by Monte Carlo technique.

Step 3. The asset’s VaR, CVaR and mean return are computed by Y_j, j = 1, 2, . . . , n, then the models (2) and (3) are solved.

The above models, CVaR can be substituted by VaR as a risk measure.

In investment policy, it is concerned to have the highest return with lowest risk. So, these items have the main role on the assets performance assessment as the input and the output in our models. In addition to the type of input and output which are effective in asset portfolio evaluation, the type of distribution also affects. So in Table 1. we highlight the novelties of the proposed models in compare with commonly used RDM model.

Table 1

Comparing proposed models by RDM model

Attributes	Model (1) RDM model	Model (2) RDM-mST	Model (3) RDM-mGH
Mean-Return (Output)	✓	✓	✓
VaR or CVaR (Input)	✓	✓	✓
Environment	deterministic	stochastic	stochastic
Sample data (input/output)	✓	×	×
Simulated data (input/output)-Monte Carlo	×	✓	✓
Underlying asset	×	Multivariate skewed t	Multivariate generalized hyperbolic
Portfolio performance evaluation	DEA	DEA	DEA
Efficiency score	Not precise	Relatively accurate	Relatively accurate

4 Empirical analysis

We compare the introduced models for some stock companies of Iranian financial market. Each company is considered as a financial asset. We have daily logarithmic returns of two sets of seven groups of different kinds of industries. The public information of the companies is given from Tehran Stock Exchange (TSE) market. The daily price is considered as the closed price of each asset. The first seven groups data being recorded from 18/07/2016 to 19/07/2017, and the second seven groups is 25/03/2018 to 29/04/2019 which are illustrated in Figs.5 and 6.

Fig. 5

Stock price of first seven groups.

Fig. 6

Stock price of second seven groups.

As in practice, real data for stock prices returns are often characterized by skewness and kurtosis and have heavy tails, first we find these numerical measures of the shape of these two data sets. As it is shown in Table 2, the skewness and kurtosis of each asset are meaningfully different from the normal distribution. So we employ other probability distributions that can be efficiently captured heavy tails and skewness in the return distributions.

Table 2

Skewness and Kurtosis of the first and second groups

Asset	The first seven groups	The second seven groups
	Skewness	Kurtosis	Skewness	Kurtosis
1	–0.7964	11.9567	–2.0873	27.6953
2	–0.6158	7.9799	–0.0088	2.46360
3	1.6351	16.7113	–8.8993	110.8949
4	–0.0827	4.5408	1.0106	9.45560
5	–0.5895	3.9204	–4.6107	52.8753
6	–0.2610	2.8998	–6.7550	83.3473
7	1.3113	13.3225	–0.20910	4.06960

The input and output of models (2) and (3) come from the simulated log-returns which they are obtained by estimated parameters of mST and mGH distributions and then simulated by Monte Carlo technique. Tables 3, 4, 5 and 6 represent the mST and mGH estimated parameters of two sets of seven groups assets by EM algorithm.

Table 3

Estimated parameters of mST of the first seven groups for λ = -2.5, χ = 5 and ψ = 10^-6

Asset	μ	γ	Σ
$\begin{matrix} 1 \\ 2 \\ 3 \\ 4 \\ 5 \\ 6 \\ 7 \end{matrix}$	$(\begin{matrix} - 0.00402 \\ - 0.00263 \\ 0.000097 \\ 0.001229 \\ 0.000463 \\ 0.002552 \\ 0.003590 \end{matrix})$	$(\begin{matrix} 0.004159 \\ 0.000999 \\ 0.001311 \\ - 0.00091 \\ - 0.00152 \\ - 0.00175 \\ - 0.00087 \end{matrix})$	$(\begin{matrix} 0.000214 & 0.000021 & - 0.000014 & 0.000011 & - 0.000005 & - 0.000006 & 0.000009 \\ 0.000021 & 0.000292 & 0.000003 & 0.000005 & 0.000015 & - 0.000006 & - 0.000003 \\ - 0.000014 & 0.000003 & 0.000178 & - 0.000019 & - 0.000007 & 0.0000009 & 0.000022 \\ 0.000011 & 0.000005 & - 0.000019 & 0.000144 & 0.000005 & 0.000012 & 0.000020 \\ - 0.000005 & 0.000015 & - 0.000007 & 0.000005 & 0.000206 & 0.000030 & - 0.000019 \\ - 0.000006 & - 0.000006 & 0.0000009 & 0.000012 & 0.000030 & 0.000324 & - 0.000001 \\ 0.000009 & - 0.000003 & 0.000023 & 0.000020 & - 0.000019 & - 0.000001 & 0.000332 \end{matrix})$

Table 4

Estimated parameters of mGH of the first seven groups for λ = 8, χ = 9.714 and ψ = 13.766

Asset	μ	γ	Σ
$\begin{matrix} 1 \\ 2 \\ 3 \\ 4 \\ 5 \\ 6 \\ 7 \end{matrix}$	$(\begin{matrix} - 0.0060 \\ - 0.0013 \\ - 0.0037 \\ 0.0016 \\ 0.0020 \\ 0.0049 \\ 0.0004 \end{matrix})$	$(\begin{matrix} 0.0051 \\ 0.0000 \\ 0.0037 \\ - 0.0011 \\ - 0.0024 \\ - 0.0032 \\ 0.0013 \end{matrix})$	$(\begin{matrix} 0.000207 & 0.000022 & - 0.000015 & 0.000011 & - 0.000007 & - 0.000005 & 0.000011 \\ 0.000022 & 0.000272 & 0.000004 & 0.0000003 & 0.000014 & - 0.000007 & - 0.000002 \\ - 0.000015 & 0.000004 & 0.000194 & - 0.000022 & - 0.000006 & 0.000007 & 0.000026 \\ 0.000011 & 0.000003 & - 0.000022 & 0.000128 & 0.000003 & 0.000008 & 0.000017 \\ - 0.000007 & 0.000014 & - 0.000006 & 0.000003 & 0.000178 & 0.000025 & - 0.000014 \\ - 0.000005 & - 0.000007 & 0.000007 & 0.000008 & 0.000025 & 0.00028 & - 0.000005 \\ 0.000011 & - 0.000002 & 0.000026 & 0.000017 & - 0.000014 & - 0.000005 & 0.000308 \end{matrix})$

Table 5

Estimated parameters of mST of the second seven groups for λ = -1.5, χ = 3 andψ = 10^-2

Asset	μ	γ	Σ
$\begin{matrix} 1 \\ 2 \\ 3 \\ 4 \\ 5 \\ 6 \\ 7 \end{matrix}$	$(\begin{matrix} 0.0045 \\ 0.0021 \\ 0.0128 \\ - 0.0015 \\ 0.0012 \\ 0.0081 \\ - 0.0002 \end{matrix})$	$(\begin{matrix} - 0.0015 \\ - 0.0002 \\ - 0.0073 \\ 0.0026 \\ - 0.0014 \\ - 0.003 \\ 0.0018 \end{matrix})$	$(\begin{matrix} 0.000379 & 0.000049 & 0.000090 & 0.000115 & 0.000039 & 0.000236 & 0.000183 \\ 0.000049 & 0.000357 & 0.000148 & 0.000053 & 0.000086 & 0.000228 & 0.000090 \\ 0.000090 & 0.000149 & 0.000546 & 0.000087 & - 0.000018 & 0.000120 & 0.000171 \\ 0.000115 & 0.000053 & 0.000087 & 0.000176 & 0.000037 & 0.000140 & 0.000110 \\ 0.000039 & 0.000086 & - 0.000018 & 0.000037 & 0.000258 & 0.000074 & 0.000035 \\ 0.000236 & 0.000228 & 0.000120 & 0.000140 & 0.000074 & 0.000884 & 0.000160 \\ 0.000183 & 0.000090 & 0.000170 & 0.000110 & 0.000035 & 0.000160 & 0.000557 \end{matrix})$

Table 6

Estimated parameters of mGH of the second seven groups for λ = 0.2, χ = 1.518 and ψ = 6.432

Asset	μ	γ	Σ
$\begin{matrix} 1 \\ 2 \\ 3 \\ 4 \\ 5 \\ 6 \\ 7 \end{matrix}$	$(\begin{matrix} 0.0049 \\ 0.0021 \\ 0.0143 \\ - 0.002 \\ 0.0016 \\ 0.0091 \\ - 0.0006 \end{matrix})$	$(\begin{matrix} - 0.0035 \\ - 0.0004 \\ - 0.0163 \\ 0.0056 \\ - 0.0032 \\ - 0.0071 \\ 0.0040 \end{matrix})$	$(\begin{matrix} 0.000754 & 0.000095 & 0.000172 & 0.000224 & 0.000076 & 0.000454 & 0.000355 \\ 0.000095 & 0.000686 & 0.000287 & 0.000103 & 0.000168 & 0.000440 & 0.000174 \\ 0.000172 & 0.000287 & 0.001109 & 0.000173 & - 0.000041 & 0.000227 & 0.000330 \\ 0.000224 & 0.000103 & 0.000173 & 0.000347 & 0.000073 & 0.000273 & 0.000215 \\ 0.000076 & 0.000168 & - 0.000041 & 0.000073 & 0.000511 & 0.000139 & 0.000069 \\ 0.000454 & 0.000440 & 0.000227 & 0.000273 & 0.000139 & 0.001743 & 0.000309 \\ 0.000355 & 0.000174 & 0.000330 & 0.000215 & 0.000069 & 0.000309 & 0.001078 \end{matrix})$

As it is shown in Table 2, the skewness and kurtosis values of each group are different from normal distribution. We set the initial values of the parameters of mST and mGH distributions.

Tables 7 and 8 represent the values of VaR, CVaR and mean return of both seven groups of assets. In model (1), the calculated risk measures and the mean returns depend on the use of log-returns of stock prices whereas in models (2) and (3), they are obtained by simulated returns on estimated parameters. According to the VaR and CVaR definition, we arbitrarily choose the confidence level β = 0.90. These data are used to compute the inefficiency scores in models (1) to (3).

Table 7

VaR, CVaR and mean-Return of models (1), (2) and (3) of the first seven groups

	The first seven groups
	Model (1)			Model (2)			Model (3)
Asset	VaR	CVaR	mean-Return	VaR	CVaR	mean-Return	VaR	CVaR	mean-Return
1	0.0133	0.0307	0.0018	0.0182	0.0253	0.0013	0.0189	0.0274	0.0025
2	0.0228	0.0406	–0.0012	0.0266	0.0371	–0.0015	0.0290	0.0382	–0.0012
3	0.0075	0.0332	0.0019	0.0180	0.0268	0.0012	0.0209	0.0306	0.0015
4	0.0176	0.0276	–0.0001	0.0186	0.0271	–0.0002	0.0170	0.0240	0.0000
5	0.0218	0.0366	–0.0017	0.0220	0.0303	–0.0009	0.0226	0.0311	–0.0015
6	0.0273	0.0403	0.0001	0.0270	0.0401	0.0001	0.0251	0.0364	0.0010
7	0.0253	0.0376	0.0024	0.0242	0.0371	0.0022	0.0264	0.0363	0.0009

Table 8

VaR, CVaR and mean-Return of models (1), (2) and (3) of the second seven groups

	The second seven groups
	Model (1)			Model (2)			Model (3)
Asset	VaR	CVaR	mean-Return	VaR	CVaR	mean-Return	VaR	CVaR	mean-Return
1	0.0226	0.0494	0.0017	0.0311	0.0480	0.0008	0.0305	0.0453	0.0019
2	0.0282	0.0374	0.0017	0.0292	0.0458	0.0020	0.0271	0.0415	0.0005
3	0.0283	0.0795	–0.0004	0.0387	0.0567	0.0012	0.0430	0.0636	–0.0019
4	0.0134	0.0266	0.0031	0.0187	0.0275	0.0025	0.0185	0.0275	0.0028
5	0.0257	0.0450	–0.0013	0.0285	0.0394	–0.0015	0.0280	0.0411	–0.0015
6	0.0425	0.0728	0.0027	0.0454	0.0646	0.0034	0.0466	0.0656	0.0036
7	0.0391	0.0522	0.0030	0.0343	0.0514	0.0029	0.0347	0.0488	0.0016

In order to show the effect of the underlying distribution on the assets performance, we find the inefficiency scores of each asset by models (1), (2) and (3), recorded in Table 9.

Table 9

The inefficiency scores under mean return-VaR and mean return-CVaR framework in models (1), (2) and (3) for each seven group at β = 0.90

	The First Seven Groups’ Inefficiency Scores						The Second Seven Groups’ Inefficiency Scores
	Model (1)		Model (2)		Model (3)		Model (1)		Model (2)		Model (3)
Asset	Return-VaR	Return-CVaR	Return-VaR	Return-CVaR	Return-VaR	Return-CVaR	Return-VaR	Return-CVaR	Return-VaR	Return-CVaR	Return-VaR	Return-CVaR
1	0.34	0.00	0.00	0.00	0.00	0.00	1.00	1.00	0.71	0.72	0.60	0.60
2	0.88	0.78	0.81	0.80	0.87	0.82	1.00	1.00	0.50	0.52	0.75	0.76
3	0.00	0.12	0.00	0.20	0.59	0.57	1.00	1.00	0.70	0.70	0.87	0.87
4	0.82	0.00	0.63	0.65	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
5	0.89	0.74	0.75	0.74	0.78	0.73	1.00	1.00	0.83	0.83	0.85	0.85
6	0.82	0.74	0.73	0.72	0.79	0.76	1.00	1.00	0.00	0.00	0.00	0.00
7	0.00	0.00	0.00	0.00	0.82	0.76	1.00	1.00	0.14	0.19	0.67	0.66

From the results of Table 2, the skewness and kurtosis values of the first seven groups are less heavy tailed than the second groups. So, the normal distribution is not well-fitted for this group and the inefficiency scores are inaccurate. Furthermore, the mST distribution is not an appropriate underlying distribution, because it is one of the most-heaviest tailed distribution among the mGH distributions. Therefore, by proper choice of the initial values, the mGH distribution is well-fitted for the first seven groups. For instance, asset 4 is efficient based on model (3) but it has the high inefficiency score under mean return-VaR framework in model (1) and also it is inefficient in model (2). The underlying distributions are not normal and mST, therefore the VaR, CVaR and mean returns values are underestimated with these distributions. Moreover, the small skewness and kurtosis makes mGH to be an appropriate underlying distribution in this group. As a result, the inefficiency scores gained by model (3) are reliable to models (1) and (2).

The skewness and kurtosis of each asset in the second seven groups are significantly different from zero and 3 as the skewness and kurtosis of the normal distribution. As it is shown in Table 9, the inefficiency scores provided by model (1) are only zero and 1. For more details, the forth asset’s mean return is the highest one among these seven assets and its risk measures are the lowest among others. For the reasons indicated, asset 4 is an efficient asset based on model (1). Therefore, the mST and mGH distributions are well-fitted to the second seven groups and the inefficiency scores are precise and interpretable. Moreover, asset 6 is completely inefficient by model (1), while it is efficient in models (2) and (3). Since mST is a heavy-tailed distribution among the mGH distributions family, from the inefficiency scores, we conclude that mST is a well-fitted distribution to the second seven groups. In models (2) and (3), asset 7 is inefficient under the mST distribution, has the highest mean return and under mGH distribution has the fourth highest mean return among others. However, at the same time it has a high risk measures. So, it leads to higher inefficiency score in model (3) rather than model (2).

Comparing the models (2) and (3) with the results of model (1) we find that the inefficiency scores of these two seven groups of assets in models (2) and (3) are more accurate than model (1). Therefore, considering the different distributions into the asset performance evaluation makes different results of inefficiency scores and may effect on risk measures and mean returns values of assets. It should be noted that the initial values of λ, χ and ψ in mGH distributions depend on the skewness and kurtosis of the distributions.

5 Conclusion

Empirical studies show that the asset return distributions are leptokurtic with nonzero skewness and are not taken into account in performance evaluation. In order to evaluate the financial assets performance, we have applied the DEA method. In portfolio performance assessment in DEA framework, there is no study that focus on return distributions. Therefore, the main idea is to demonstrate how the types of return distributions affect the portfolio evaluation. For this purpose, we have introduced some models where the underlying distributions such as the mST and mGH are able to cover the skewness and heavy-tailedness of data sets. The VaR and CVaR risk measures are applied as the input of the models and the mean return is as the only output. The optimal objective value of each model indicates different maximum proportionally changes in the risk measure and the mean return of the asset under evaluation. We have given an example of two groups of assets to compare the models. For return distributions which they exhibit skewness and kurtosis, the mST and mGH distributions describe the performance of assets much better than the normal distribution and more accurately than the RDM model. In other words, the inefficiency scores measured by DEA models based on mST and mGH distributions are more accurate than the RDM model. We have observed from the first seven groups that because of the small skewness and kurtosis values, the mGH distribution is well-fitted to those data sets. It is shown that how the evaluation of the asset performance based on the return distributions in DEA framework depends on the underlying distribution. The multivariate continuous distributions mST and mGH capture more characteristics of financial data and it is not needed further constraints for them in the models. To describe the assets’ return distributions, we can apply the other distributions of Normal mean-variance mixture class and also the other classes of risk measures instead of VaR and CVaR to evaluate the asset performance with DEA method for future work. Moreover, by taking only a single input and a single output in the proposed model we are able to find the efficiency score but it can be extended to some inputs and outputs.

In addition, we can employ the traditional portfolio performance measures Treynor [38], Sharpe [35] and Jensen’s alpha [19] indices as the outputs of the model for more accuracy in asset performance.

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