On relations between DEA-risk models and stochastic ... - Artax

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On relations between DEA-risk models and stochastic dominance efficiency tests Martin Branda · Miloˇ s Kopa

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Abstract In this paper, several concepts of portfolio efficiency testing are compared, based either on Data Envelopment Analysis (DEA) or the second-order stochastic dominance (SSD) relation: constant return to scale (CRS) DEA models, variable return to scale (VRS) DEA models, diversification-consistent (DC) DEA models, pairwise SSD efficiency tests, convex SSD efficiency tests and full SSD portfolio efficiency tests. Especially, the equivalence between VRS DEA model with binary weights and the SSD pairwise efficiency test is proved. DEA models equivalent to convex SSD efficiency tests and full SSD portfolio efficiency tests are also formulated. In the empirical application, the efficiency testing of 48 US representative industry portfolios using all considered DEA models and SSD tests is presented. The obtained efficiency sets are compared. A special attention is paid to the case of small number of the inputs and outputs. It is empirically shown that DEA models equivalent either to the convex SSD test or to the SSD portfolio efficiency test work well even with quite small number of inputs and outputs. However, the reduced VRS DEA model with binary weights is not able to identify all the pairwise SSD efficient portfolios. Keywords Data Envelopment Analysis · Second-order stochastic dominance · pairwise SSD efficiency · convex SSD efficiency · SSD portfolio efficiency

Martin Branda Charles University in Prague, Faculty of Mathematics and Physics Department of Probability and Mathematical Statistics Sokolovsk´ a 83, Czech Republic Tel.: +420 221 913 404 E-mail: [email protected] Miloˇs Kopa Charles University in Prague, Faculty of Mathematics and Physics Department of Probability and Mathematical Statistics Sokolovsk´ a 83, Czech Republic

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Martin Branda, Miloˇs Kopa

1 Introduction We deal with problem of investors who are looking for “attractive” investment opportunities on financial markets. They can invest into a single asset or combine many assets into a portfolio. The main question is how to select the efficient portfolios. We will discuss several concepts of efficiency used in finance and outline possible relations between these concepts. If a utility functions (von Neumann and Morgenstern (1944)), which models an investor’s risk attitude, is perfectly known, then the optimal decision can be obtained by maximizing expected utility. If that is not the case, one can at least identify the set of efficient portfolios with respect to a chosen class of utility functions. Considering all utility functions, that is, assuming only non-satiation for the investor’s preferences, leads to the first-order stochastic dominance (FSD) relation (see Levy (2006) and references therein). Usually, the investor is risk averse and this reduces the considered utility functions to the concave ones leading to the second-order stochastic dominance rules. A portfolio is classified as pairwise SSD efficient if there is no asset that dominates the portfolio with respect to SSD, see Hanoch and Levy (1969). It means that the portfolio is pairwise SSD inefficient if there exists some asset that is preferred to the portfolio by all risk averse decision makers. This is often a very strong condition and even if it is violated still the portfolio may be the optimal choice for no investor. Therefore, Fishburn (1974) defined the convex stochastic dominance efficiency as follows: a portfolio is convex SSD inefficient (dominated by other assets) if every investor prefers at least another asset. Despite that, it does not cover the full diversification case if investors may combine assets into portfolios. Tests for SSD portfolio efficiency allowing full diversification across the assets were developed in Post (2003), Kuosmanen (2004), Kopa and Chovanec (2008) and Dupaˇcov´ a and Kopa (2012). Performance of these tests was compared by Lizyayev (2012). These tests classify a given portfolio as SSD portfolio efficient if there is no portfolio created from the assets that SSD dominates the portfolio. Alternatively, one can also apply FSD efficiency tests, see Kuosmanen (2004) and Kopa and Post (2009). Several methods have been developed to deal with SSD constrained problems, see, e.g., Dentcheva and Ruszczynski (2006), F´ abi´ an et al. (2011), Gollmer et al. (2011), Luedtke (2008), and Meskarian et al. (2012). The basics of portfolio selection theory were presented in the seminal work of Harry Markowitz (1952). He identified two main components of portfolio performance, mean return and risk represented by variance. He also found the optimal trade-off between these two components which is based on ideas of parametric and multiobjective optimization. A portfolio is considered as mean-variance efficient if there is no portfolio with a higher or equal mean and smaller or equal variance with at least one strict relation. In the last 60 years, the theory of mean-risk models has been enriched by other risk measures and functionals which quantify portfolio risk, for example, semivariance, see Markowitz (1959), Value at Risk (VaR) or Conditional Value at Risk (CVaR), see Pflug (2000), Rockafellar and Uryasev (2000, 2002), deviation measures, see Rockafellar et al. (2006). Various relation between mean-risk and stochastic dominance efficiency can be found in literature, cf. Levy (2006), Ogryczak and Ruszczynski (2001, 2002). Usually, mean-risk efficiency under particular choice of risk measure is only necessary for SD efficiency.

On relations between DEA-risk models and stochastic dominance efficiency tests

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Equivalence can be observed under distributional assumptions (for example the Gaussian distribution) on asset returns. Data Envelopment Analysis (DEA) was introduced by Charnes, Cooper and Rhodes (1978). It serves as a tool to identify efficient units in a set of homogeneous decision making units with the same structure of inputs and outputs. DEA models were applied in finance to analyze mutual fund performance by e.g. Basso and Funari (2001, 2003), Chen and Lin (2006), Murthi, Choi and Desai (1997), Daraio and Simar (2006) or Galagadera and Silvapulle (2002). They considered various risk measures and transaction costs as the inputs and expected return as the output. Recently, Branda and Kopa (2012A) formulated constant return to scale (CRS) DEA-risk models with risk measures as the inputs and the mean gross return as the only output. These DEA-risk models can be seen as a generalization of mean-risk models, because they allow for multiple risk measure application. If a risk measure is considered as the only input, then the DEA-risk efficiency implies the mean-risk efficiency with respect to the same risk measure. This CRS efficiency can be quite stringent for financial applications. In this paper, we consider both basic types of DEA-risk models: constant return to scale (CRS) and variable return to scale (VRS). Various definitions of return to scale were recently proposed by Soleimani-damaneh (2012). We do not consider the transaction costs connected with buying or selling the assets, because we rely only on reward and risk characteristics of the assets. We choose Conditional Value at Risk (CVaR) at several probability levels and lower partial moments as the inputs. The suitable reward measures serves as outputs. Moreover, we formulate input oriented models, where inefficiency is measured with respect to the inputs, as well as input-output oriented models, where the inefficiency is also evaluated with respect to the outputs. We extend diversification-consistent (DC) DEA-risk models introduced and investigated by Lamb and Tee (2012A, 2012B). They considered input oriented models only, whereas we propose input-output oriented tests. Several DEA tests, which are “consistent” with SSD, were proposed by Lozano and Gutierrez (2008). However, these tests state only necessary condition for SSD portfolio efficiency. In Branda and Kopa (2012B), it was shown that SD efficiency can be equivalent to DEA-risk efficiency under suitable choice of the inputs and outputs. In particular, they showed equivalence between the VRS DEA-risk efficiency set and the convex SSD efficiency set as well as between the DC DEA-risk efficient portfolios and the SSD efficient portfolios. In this paper, we enrich this theory by new VRS DEA-risk model with CVaRs on particular levels and binary weights, which is equivalent to pairwise SSD efficiency test. In the empirical study we apply all of the above-mentioned approaches to efficiency analysis. We consider 48 US representative industry portfolios as the basic assets. We compare efficient portfolios selected according to different criteria: CRS DEA-risk efficiency, VRS DEA-risk efficiency, diversification-consistent DEA-risk efficiency, pairwise SSD efficiency, convex SSD efficiency as well as SSD portfolio efficiency. Using 30 years history of monthly returns, we empirically examine the power of the considered efficiency approaches: the smaller an efficiency set is, the more powerful criterion is considered. Contrary to Branda and Kopa (2012C), where preliminary numerical results were presented, we consider more versions of DEA models, namely three new VRS and two new DC DEA-risk tests. Moreover, we take different levels of CVaR. The new structures of the DEA models as well as the new levels of CVaR are chosen in concordance with the theoretical equivalence

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results. Finally, we also analyze the DEA models with significantly reduced number of inputs and outputs in order to find whether the equivalences are preserved or not. The remainder of this paper is structured as follows. Section 2 presents the preliminary notation and defines the considered risk measures. In Section 3, the DEA-risk efficiency tests are introduced using risk measures as inputs and reward measures as outputs. Section 4 recalls basic ideas of SSD efficiency approaches and the corresponding linear programming SSD efficiency tests. Moreover, it proposes the equivalent DEA-risk models. Section 5 presents an empirical application comparing all the considered types of efficiency. The paper is concluded in Section 6. 2 Preliminaries and risk measures In all portfolio efficiency tests we consider n assets and we denote by Ri the rate of return of i-th asset with finite mean value. As usual in DEA-risk models or SSD efficiency tests, we assume a discrete probability distribution of rate of returns described by scenarios ri,s , s = 1, . . . , S, that are taken with equal probabilities ps = 1/S. Moreover, the decision maker may combine the assets into portfolios. Each portfolio is represented by its weights x = {x1 , . . . , xn }. We denote by X a considered set of feasible weights. If short sales are not allowed and budget constraint is imposed, the set of feasible weights is

X = {x ∈ R|

n X

xi = 1, xi ≥ 0, i = 1, . . . , n}.

(1)

i=1

More generally, one can consider any bounded polyhedral set of feasible weights. Conditional Value at Risk (CVaR) is often proposed as an alternative measure for Value at Risk (VaR). VaR is widely used in practice, even though it is not an adequate risk measure. CVaR is roughly defined as the conditional mean of losses that are beyond VaR. Below we will provide the formal definition of these measures and we summarize the basic properties under general loss distribution, see Pflug (2000), Rockafellar and Uryasev (2000, 2002). We denote by Z a loss random variable on the probability space (Ω, A, P ). We assume that the expected loss is finite, i.e. E[Z] < ∞. Let F denote the distribution function of the loss random variable, i.e. F (η) = P (Z ≤ η), η ∈ R. Then Value at Risk (VaR) and upper Value at Risk are defined as VaRα (Z) = min{η : F (η) ≥ α}, VaR+ α (Z) = min{η : F (η) > α}, for some probability level α ∈ (0, 1), usually 0.95 or 0.99. CVaR is defined as the mean of losses in the following α-tail distribution F (η) − α , if η ≥ VaRα (Z), 1−α = 0, otherwise.

Fα (η) =


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For application of CVaR in optimization problems, the following minimization formula is of crucial importance, see Rockafellar and Uryasev (2000, 2002): 1 + CVaRα (Z) = min η + E[Z − η] , (2) η∈R 1−α where [·]+ = max{·, 0} denotes the positive part and η is a real auxiliary variable. The optimal solution belongs to the closed interval [VaRα (Z), VaR+ α (Z)]. P In our application, the portfolio loss random variable can be written as Z = − n i=1 Ri xi . Since we consider a discrete probability distribution with equiprobable scenarios, CVaR can be formulated as CVaRS α (−

n X i=1

Ri xi ) = min η + η∈R

S h n i+ X X 1 − xi ri,s − η . (1 − α)S s=1 i=1

3 Data Envelopment Analysis Data Envelopment Analysis (DEA) was introduced by Charnes, Cooper and Rhodes (1978), as a nonparametric tool for efficient units selection among decision making units with the same structure of inputs and outputs, see Cook and Seiford (2009), Cooper, Seiford and Tone (2007), Jablonsky (2012) for recent review of the main results. It should be noted that DEA is primarily a diagnostic tool and does not prescribe any strategies to make inefficient units efficient.

3.1 Constant Return to Scale (CRS) We consider n decision making units which transform K inputs into J outputs. Let Z1i , . . . , ZKi denote the inputs and Y1i , . . . , YJi denote the outputs of the unit i from the considered units. DEA efficiency of unit 0, which belongs to the set of considered units {1, . . . , n}, is then evaluated using the optimal value of the following fractional linear program where the ratio of weighted inputs and weighted outputs of the benchmark is maximized under the conditions that the ratios for all units using the same weights are lower or equal to one, i.e. PJ j=1 yj0 Yj0 max PK yj0 ,˜ yk0 ˜k0 Zk0 k=1 y s.t. (3) PJ j=1 yj0 Yji ≤ 1, i = 1, . . . , n, PK ˜k0 Zki k=1 y y˜k0 ≥ 0, k = 1, . . . , K, yj0 ≥ 0, j = 1, . . . , J, where yj0 , y˜j0 are the weights. The unit 0 is then DEA efficient if the optimal value is equal to 1, otherwise it is DEA inefficient. Infinitesimal non-Archimedean minimal quantity for the weights, which ensures that all inputs and outputs are taken into account, is not considered in this paper. The program can be rewritten

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as a linear program which is based on fractional programming reformulation, cf. Charnes and Cooper (1962):

J X

max

yj0 ,˜ yk0

yj0 Yj0

j=1

s.t. K X

y˜k0 Zk0 = 1,

k=1

−

K X

y˜k0 Zki +

J X

yj0 Yji ≤ 0, i = 1, . . . , n,

j=1

k=1

y˜k0 ≥ 0, k = 1, . . . , K, yj0 ≥ 0, j = 1, . . . , J.

This model is usually solved in its dual form where nonnegative linear combinations of inputs are compared with the benchmark inputs under the condition that the nonnegative linear combinations of outputs are higher or equal to the benchmark outputs, i.e.

min θ θ,xi

s.t. n X i=1 n X

(4)

xi Yji ≥ Yj0 , j = 1, . . . , J,

xi Zki ≤ θ · Zk0 , k = 1, . . . , K,

i=1

xi ≥ 0, i = 1, . . . , n,

where xi are the (dual) weights and θ represents a possible improvement in all inputs simultaneously. If such improvement is possible, i.e. θ < 1, than the unit 0 is DEA inefficient. Note that the program is always feasible because the benchmark is included between other units.


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3.2 Variable Return to Scale (VRS) The main difference between CRS and VRS DEA model can be seen from the dual representation which is given below, see Banker, Charnes and Cooper (1984): min θ θ,xi

s.t. n X i=1 n X

(5)

xi Yji ≥ Yj0 , j = 1, . . . , J,

xi Zki ≤ θ · Zk0 , k = 1, . . . , K,

i=1 n X

xi = 1, xi ≥ 0, i = 1, . . . , n.

i=1

P The condition n i=1 xi = 1 is added to restrict the possible linear combinations of the inputs and outputs, i.e. only convex combinations are allowed. The primal program can be rewritten as: J X

max

yj0 ,˜ yk0 ,y0

yj0 Yj0 − y0

j=1

s.t. K X

y˜k0 Zk0 = 1,

k=1

−

K X

y˜k0 Zki +

J X

yj0 Yji − y0 ≤ 0, i = 1, . . . , n,

j=1

k=1

y˜k0 ≥ 0, k = 1, . . . , K, yj0 ≥ 0, j = 1, . . . , J. Finally, the fractional programming problem is PJ max

yj0 ,˜ yk0 ,y0

j=1

PK

yj0 Yj0 − y0

k=1

y˜k0 Zk0

s.t. y Y − y 0 j=1 j0 ji ≤ 1, i = 1, . . . , n, PK y ˜ Z k=1 k0 ki y˜k0 ≥ 0, k = 1, . . . , K,

PJ

yj0 ≥ 0, j = 1, . . . , J, where the free variable y0 is added in comparison with the model (3). The model can be extended to measure inefficiency with respect to any of the considered inputs. The resulting DEA test is then stronger than the previous

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model. We can employ the dual formulation and modify it to the following program

min

θk ,xi

K 1 X θk K k=1

s.t. n X i=1 n X

(6)

xi Yji ≥ Yj0 , j = 1, . . . , J,

xi Zki ≤ θk · Zk0 , k = 1, . . . , K,

i=1

0 ≤ θk ≤ 1, k = 1, . . . , K n X

xi = 1, xi ≥ 0, i = 1, . . . , n.

i=1

Moreover, inefficiency can be measured also with respect to any of the considered outputs leading to nonlinear input-output oriented VRS model:

min

θk ,ϕj ,xi

1 K +L

X K

θk +

J X 1 ϕj

j=1

k=1

s.t. n X i=1 n X

(7)

xi Yji ≥ ϕj · Yj0 , j = 1, . . . , J,

xi Zki ≤ θk · Zk0 , k = 1, . . . , K,

i=1

0 ≤ θk ≤ 1, ϕj ≥ 1, n X

xi = 1, xi ≥ 0, i = 1, . . . , n,

i=1

where ϕj represents a possible improvement in the corresponding output. The unit 0 is inefficient if there is a convex combination of inputs or outputs which outperforms the benchmark in at least one input or output and at the same time the other constraints are fulfilled.

3.3 Diversification-consistent tests If values of asset risk measures are used as the inputs in the previous DEA models, the linear combinations overestimate the true risk of asset linear combinations. Therefore, general DEA tests with diversification effect were recently introduced by Lamb and Tee (2012). They have formulated the input oriented diversification-


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consistent model for a benchmark with return R0 as θDC (R0 ) = min θ θ,xi

s.t. n X E( Ri xi ) ≥ E(R0 ), CVaR+ αk (−

i=1 n X

(8)

Ri xi ) ≤ θ · CVaR+ αk (−R0 ), k = 1, . . . , K,

i=1 n X

xi = 1, xi ≥ 0, i = 1, . . . , n,

i=1

+ where CVaR+ α denotes the positive part of CVaRα , i.e. CVaRα = max{CVaRα , 0}, and αk are different levels. Since the positive parts of CVaRs serve as the inputs and the expected return as the output, we may see this model as a diversificationconsistent modification of the DEA-risk models analysed in Branda and Kopa (2012A).

3.4 General diversification-consistent tests with Russell measure Let CVaRα (−R0 ) > 0 for α ∈ {α1 , . . . , αK } and CVaRε (−R0 ) < 0 for ε ∈ {ε1 , . . . , εJ }. The positive CVaRs are used as the inputs and the negative as the outputs leading to the following input-output oriented DC model

θ

DC I−O I

1 1 θ+ (R0 ) = min θ,ϕ,xi K + J ϕ s.t. (9) n X Ri xi ) ≥ ϕ · (−CVaRεj (−R0 )), j = 1, . . . , J, −CVaRεj (− i=1

CVaRαk (−

n X

Ri xi ) ≤ θ · CVaRαk (−R0 ), k = 1, . . . , K,

i=1

0 ≤ θ ≤ 1, ϕ ≥ 1, n X

xi = 1, xi ≥ 0, i = 1, . . . , n.

i=1

The model leads to convex programming problem. Note that CVaR0 (−R0 ) = E[−R0 ], i.e. the expected loss (return) can be also included into this model without any changes in its formulation. The benchmark is inefficient if there is a portfolio which has at least as good inputs and outputs and all inputs or all outputs are strictly better.

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We can extend the last model to take into account possible inefficiency with respect to any of the inputs or outputs. X K J X 1 1 DC I−O II θ (R0 ) = min θk + θk ,ϕj ,xi K + J ϕj j=1

k=1

s.t. −CVaRεj (− CVaRαk (−

n X i=1 n X

(10)

Ri xi ) ≥ ϕj · (−CVaRεj (−R0 )), j = 1, . . . , J, Ri xi ) ≤ θk · CVaRαk (−R0 ), k = 1, . . . , K,

i=1

0 ≤ θk ≤ 1, ϕj ≥ 1, n X

xi = 1, xi ≥ 0, i = 1, . . . , n.

i=1

These DEA-risk models can be seen as modifications of model (7) and Russel measure DEA model (see Cook and Seiford (2009)). Moreover, they extend the diversification consistent models introduced by Lamb and Tee (2012) and the DEA-risk models preconsidered in Branda and Kopa (2012A). 4 Second-order stochastic dominance relation Stochastic dominance approach allows for portfolio comparisons where each portfolio is represented by its random return. Assuming only non-satiated and risk averse decision makers we consider the second-order stochastic dominance crite¯ dominates rion. Following Levy (2006) and references therein, portfolio x Pn portfolio ¯i ) ≥ x with respect to second-order stochastic dominance (SSD) if Eu( i=1 Ri x P R x ) for all non-decreasing and concave utility functions with strict inEu( n i i i=1 equality for at least one such utility function. Let Fx (η) denote the cumulative probability distribution function of returns of portfolio x, i.e. Fx (t) = P (−

n X

Ri xi ≤ t).

i=1

The twice cumulative probability distribution function of returns of portfolio x is defined as: Z t Fx(2) (t) = Fx (η) dη. −∞

¯ dominates portfolio Then the SSD relation can be verified as follows: portfolio x x with respect to second-order stochastic dominance if and only if (2)

Fx¯ (t) ≤ Fx(2) (t)

∀ t ∈ R,

with strict inequality for at least one t ∈ R. ¯ with correIn the following sections, we Pwill test efficiency of a portfolio x sponding random return R0 = n ¯i , which will be used as the benchmark i=1 Ri x in DEA discrete distribution we denote the realizations of R0 as Pmodels. Under s ¯ r0,s = n r x ¯ = r x , where rs = (r1,s , ..., rn,s ). i,s i i=1


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4.1 Pairwise second-order stochastic dominance efficiency In this section, we firstly formulate a computationally attractive necessary and sufficient condition of SSD relation and then we apply it for pairwise SSD efficiency testing. The idea is based on dual SSD expressions in terms of cumulative returns. Following Levy (2006), let vxs , s = 1, 2, .., S, denote the ordered returns of portfolio x in ascending order, that is, vx1 ≤ vx2 ≤ ... ≤ vxS . As a special case, let vi1 ≤ vi2 ≤ ... ≤ viS denote the ordered returns of i-th asset. Then portfolio x dominates ¯ with respect to second-order stochastic dominance if and only if portfolio x

s X

vxt ≥

t=1

s X

vxt¯ , s = 1, 2, ..., S,

t=1

¯ as pairwise with at least one strict inequality. Moreover, we classify the portfolio x ¯ . Otherwise, portfolio SSD inefficient if there exists an asset that SSD dominates x ¯ is pairwise SSD efficient. Therefore, the algorithm for testing pairwise SSD effix ¯ consists of two steps. Firstly, we compute cumulative returns ciency of portfolio x P of assets: st=1 vit for all i = 1, 2, ..., N , s = 1, 2, .., S, and cumulative returns of the portfolio: vx1¯ ≤ vx2¯ ≤ ... ≤ vxS ¯ . Secondly, we try to find some i such that

s X

vit ≥

t=1

s X

vxt¯ , ∀s = 1, 2, ..., S.

(11)

t=1

¯ is pairwise SSD inefficient, if not then portfolio x ¯ If such i exists then portfolio x is pairwise SSD efficient.

4.1.1 Equivalent DEA-risk test We propose new DEA-risk test, which is equivalent to the pairwise SSD efficiency test and is based on a relation between CVaR and SSD. More details are left to ¯ as the proof of theorem. We consider the random return of the tested portfolio x the benchmark. Let αs = s/S, s ∈ Γ = {0, 1, . . . , S − 1}, and s˜ = arg max{s ∈ Γ : CVaRαs (−R0 ) < 0}. We assume that no CVaRαs of the benchmark is equal to zero1 . To ensure feasibility of the problem we include the tested portfolio x as the additional asset. Then the input-output oriented VRS DEA-risk test can be 1 The existence of zero value of CVaR αs (−R0 ) can be detected prior to testing and in the positive case, the data matrix can be shifted by an arbitrary constant. This data transformation would be harmless because SSD efficiency is shift equivariant.

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constructed as θV RS

I−O II

(R0 ) =

min

θk ,ϕk ,xi

1 S

S−1 X

θk +

k=˜ s+1

s ˜ X 1 ϕk

k=0

s.t. n X

(12)

xi CVaRαk (Ri ) ≥ ϕk · CVaRαk (R0 ), k = 0, . . . , s˜,

i=0 n X

xi CVaRαk (−Ri ) ≤ θk · CVaRαk (−R0 ), k = s˜ + 1, . . . , S − 1,

i=0

0 ≤ θk ≤ 1, ϕk ≥ 1, n X

xi = 1, xi ∈ {0, 1}, i = 0, . . . , n.

i=0

This model corresponds to the DEA-risk model (7) with particular choices of the probability levels and binary weights. Proposition 1 The input-output oriented VRS DEA-risk model (12) with binary weights is equivalent to pairwise SSD efficiency test, that is, a benchmark R0 = P n ¯ ¯i is DEA-risk efficient (θV RS I−O II (R0 ) = 1) if and only if portfolio x i=1 Ri x is pairwise SSD efficient. Proof We can prove the proposition using the Theorem 4 proposed by Kopa and Chovanec (2008), which states that two discretely distributed random variables can be compared with respect to strict SSD relation using CVaRs on finite number of levels αs , s ∈ Γ , in particular X1 SSD X2 if and only if CVaRαs (−X1 ) ≤ CVaRαs (−X2 ), s ∈ Γ with at least one strict inequality. Using DEA model, we are looking for an asset which has at least one strictly lower CVaR for a particular level and lower or equal CVaRs for the other levels. If such an asset exists, then the portfolio is pairwise SSD inefficient, otherwise it is pairwise SSD efficient. We can search for this asset by solving the DEA model (12) where the binary weights ensure that exactly one asset is selected. The CVaRs of the benchmark portfolio are included in order to insure the feasibility of the program.

4.2 Convex second-order stochastic dominance efficiency The pairwise efficiency test described in previous section classifies a portfolio as SSD inefficient if and only if at least one asset SSD dominates the portfolio. It means that there exists an asset such that all risk averse investors prefer the asset to the portfolio. If this is violated a portfolio is classified as SSD pairwise efficient. This however does not necessarily mean that at least one investor prefers the portfolio to all assets or is indifferent between the portfolio and assets. Therefore, Fishburn (1974) defines a concept of convex stochastic dominance. We say that ¯ is convex SSD inefficient if every investor prefers some of the assets portfolio x ¯ . Hence, contrary to the pairwise efficiency, the investors may prefer to portfolio x ¯ is convex different portfolios from each others. Alternatively, we say that portfolio x SSD efficient if at least one risk averse investor prefers the portfolio to all assets


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or is indifferent between them. Let D = {d1 , d2 , ..., d(n+1)S } be the set of all scenario returns of the assets and portfolio x, that is, for every i ∈ {0, 1, 2, ..., n} and s ∈ {1, 2, ..., S} exists k ∈ {1, 2, ..., (n + 1)S} such that ri,s = dk and vice (2) versa. We denote by Fi (t) the twice cumulative probability distribution function of returns of i-th asset. Following Bawa et al. (1985) we test convex SSD efficiency ¯ using the following linear programming problem: of portfolio x

(n+1)S

δ ∗ (¯ x) = max δk ,xi

(2)

s.t. Fx¯ (dk ) −

n X

X

δk

(13)

k=1 (2)

xi Fi (dk ) ≥ δk , k = 1, 2, ..., (n + 1)S,

i=1

δk ≥ 0,

k = 1, 2, ..., (n + 1)S,

x ∈ X,

where δk are slack variables, which measure possible improvement in convex combination of asset twice cumulative distribution functions compared with portfolio twice cumulative distribution function. If such improvement is possible, i.e. if ¯ is convex SSD inefficient. Otherwise, portfolio x ¯ is δ ∗ (¯ x) > 0, a given portfolio x convex SSD efficient. The test can be rewritten using lower partial moments

Li (d) = 1/S

S X

[ri,s − d]+ ,

s=1

which can be used in the places of integrated distribution functions

(n+1)S ∗

X

δ (¯ x) = max δk ,xi

s.t. Lx¯ (dk ) −

n X

δk

(14)

k=1

xi Li (dk ) ≥ δk , k = 1, 2, ..., (n + 1)S,

i=1

δk ≥ 0,

k = 1, 2, ..., (n + 1)S,

x ∈ X.

4.2.1 Equivalent DEA-risk test Find the index of the lowest return scenario of benchmark for which the corre˜ = arg min{k : Lx¯ (dk ) > 0}. Then sponding lower partial moment is positive, i.e. k

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the DEA-risk model with variable return to scale can be formulated as δ CDEA (R0 ) = min

θk ,xi

1 ˜+1 (n + 1)S − k

(n+1)S X

θk

˜ k=k

s.t. n X

(15)

xi Li (dk ) ≤ θk · Lx¯ (dk ), k = 1, . . . , (n + 1)S,

i=1

0 ≤ θk ≤ 1, n X

xi = 1, xi ≥ 0, i = 1, . . . , n,

i=1

where θk measure possible improvements in positive lower partial moments. It was shown by Branda and Kopa (2012B) that the DEA-risk model P (15) is equivalent to convex SSD efficiency test (13), that is, a benchmark R0 = n ¯i is DEA-risk i=1 Ri x ¯ is convex SSD efficient (δ ∗ (¯ efficient (θCDEA (R0 ) = 1) if and only if x x) = 0).

4.3 Second-order stochastic dominance portfolio efficiency Contrary to the previous cases, we may compare a tested portfolio with all other feasible portfolios. In this case, we define (full) SSD portfolio efficiency as the efficiency with respect to all possible portfolios that can be created from the con¯ is SSD portfolio inefficient if there exists another sidered assets. A given portfolio x ¯ by SSD. Otherwise, portfolio x ¯ is SSD portfolio portfolio x ∈ X that dominates x efficient. Using minimax theorem, we can see that SSD efficient portfolio is the expected utility maximizer for at least one concave utility function, that is, the optimal choice for at least one risk averse decision maker. 4.3.1 Post (2003) test for SSD portfolio efficiency ¯ . Let ¯ < ... < rS x ¯ < r2 x Let the return scenarios be ordered such that: r1 x θP (¯ x) = min θ θ,βs

s.t.

S X

¯ − ri,s ) + θ ≥ 0, βs (rs x

i = 1, 2, ..., n,

s=1

βs − βs+1 ≥ 0,

s = 1, 2, ..., S − 1,

βs ≥ 0,

s = 1, 2, ..., S − 1,

βS = 1. ¯ is SSD portfolio efficient then θP (¯ If portfolio x x) = 0. This test generally provides only a necessary condition for SSD portfolio efficiency, however, in financial empirical applications it is a very powerful tool and efficiency results are empirically indistinguishable from the following two tests. Moreover, it is the most computationally attractive test.


15

4.3.2 Kuosmanen (2004) SSD portfolio efficiency test The Kuosmanen test is based on double stochastic matrix W = {wst }S s,t=1 application and majorization principle, see Hardy, Littlewood and P´ olya (1934). To simplify the notation, let all scenarios be collected in matrix M , where rs is the s-th row of matrix M . Let ¯) θK1 (¯ x) = min(1T M x − 1T W M x W,x

¯, Mx ≥ W Mx

s.t. S X

wst = 1,

t = 1, 2, ..., S,

wst = 1,

s = 1, 2, ..., S,

s=1 S X t=1

x ∈ X, where 1T = (1, .., 1) and x) = θK2 (¯

min

W,x,C + ,C −

S X S X

− (c+ st + cst )

s=1 t=1

¯, Mx = W Mx 1 − − cst = wst − , s, t = 1, 2, ..., S, 2 − s, t = 1, 2, ..., S, c+ st , cst , wst ≥ 0,

s.t.

c+ st

S X

wst = 1,

t = 1, 2, ..., S,

wst = 1,

s = 1, 2, ..., S,

s=1 S X t=1

x ∈ X, S − S S where C + = {c+ = {c− st }s,t=1 , C st }s,t=1 and W = {wst }s,t=1 . ¯ . Then portfolio x ¯ is SSD portfolio Let k denote the number of k-way ties in M x efficient if and only if S

x) = x) = 0 ∧ θK2 (¯ θK1 (¯

S2 X − kk . 2 k=1

¯ is SSD portfolio inefficient then every risk averse Moreover if the tested portfolio x ¯. investor prefers the solution portfolio x∗ to portfolio x 4.3.3 Kopa and Chovanec (2008) SSD portfolio efficiency test The Kopa and Chovanec test is based on CVaRs comparisons at particular levels, see Ogryczak and Ruszczynski (2002) for more details. Let αs = s/S, s ∈ Γ =

16


{0, 1, . . . , S − 1}. Let δ K−C (¯ x) = max δs ,λi

s.t. CVaRαs (−

n X

Ri x ¯i ) − CVaRαs (−

i=1

S−1 X

Ds

(16)

s=0

n X

Ri λi ) ≥ δs , s ∈ Γ,

i=1

δs ≥ 0,

s ∈ Γ,

x ∈ X. Using linear programming reformulation of CVaR (2) we can compute the measure of inefficiency δ K−C (¯ x) as follows:

δ K−C (¯ x) =

s.t. CVaR s−1 (− S

n X

Ri x ¯ i ) − bs −

i=1

maxt

δs ,λi ,as ,bs

1 (1 −

S X

δs

s=1 T X

ats s−1 )S S t=1 t a s + rt x ats

≥ δs ,

s = 1, ..., S,

≥ −bs , s, t = 1, ..., S, ≥ 0,

s, t = 1, ..., S,

δs ≥ 0,

s = 1, ..., S,

x ∈ X. ¯ is SSD portfolio inefficient and the solution portfolio x∗ If δ K−C (¯ x) > 0 then x ¯ . Otherwise, δ K−C (¯ ¯ is SSD portfolio efficient. SSD dominates x x) = 0 and x 4.3.4 Equivalent DEA-risk test Let s˜ = arg max{s ∈ Γ : CVaRαs (−R0 ) < 0}. We assume that no CVaRαs of the benchmark is equal to zero. Then the diversification-consistent DEA-risk test can be constructed as θDC

I−O II

(R0 ) = min

1 S

S−1 X

θk +

s ˜ X 1 ϕk

k=0

k=˜ s+1

s.t. −CVaRαk (− CVaRαk (−

n X i=1 n X

(17)

Ri xi ) ≥ ϕk · (−CVaRαk (−R0 )), k = 0, . . . , s˜, Ri xi ) ≤ θk · CVaRαk (−R0 ), k = s˜ + 1, . . . , S − 1,

i=1

0 ≤ θk ≤ 1, ϕk ≥ 1, n X i=1

xi = 1, xi ≥ 0, i = 1, . . . , n,


17

where −CVaRα is the reward measure in the place of the outputs. This model corresponds to DEA-risk model (10) with particular choices of the probability levels. Branda and Kopa (2012B) showed that the diversification-consistent DEA-risk modelP (17) is equivalent to SSD portfolio efficiency test (16), that is, a benchmark ¯ is R0 = n ¯i is DEA-risk efficient (θDC (R0 ) = 1) if and only if portfolio x i=1 Ri x SSD efficient (δ K−C (¯ x) = 0). If the number of inputs or outputs is reduced, than the DEA-risk test states only a necessary condition for SSD portfolio efficiency.

5 Numerical study To compare the power of considered efficiency tests empirically, we apply them to historical US stock market data. We consider monthly excess returns from January 1982 to December 2011 (360 observations) of 48 representative industry stock portfolios that serve as the base assets. The industry portfolios are based on four-digit SIC codes and are from Kenneth French library. Preliminary numerical results were proposed by Branda and Kopa (2012C), where the efficient portfolios with respect to SSD were already identified, cf. Table 2. However, only the DEA-risk tests (4), (5), (8) with CVaRs on 6 levels were used. We will employ all DEA-risk models presented in Sections 3 and 4, where CVaRs at levels2 α = 0.5, 0.75, 0.9, 0.95 are used as the inputs and mean return (the most commonly used reward measure) as the output. In particular, we consider eights DEA-risk models: constant return to scale DEA-risk model, four variable return to scale DEA-risk models and three diversification consistent DEA-risk models: (8), (9) and (10). Only 4 portfolios were classified as efficient by at least one DEArisk model. All other portfolios can be seen as inefficient no matter which of the considered DEA-risk models is used. The DEA scores are presented in Table 1. In accordance with the theory, see Branda (2012C), DC DEA-risk models are the most powerful, no matter which of them is used. In our case, they classify only portfolio with the highest mean return as efficient. This we can expect if we realize that our data contain portfolios that represent different industrial sectors. So, by definition, none of them is well diversified. The constant return to scale model identifies only one efficient portfolio, too. However, it is not the same portfolio as in the DC DEA-risk cases. This portfolio has maximal generalized Sharpe ratio where combination of risk measures is used in the denominator, see Branda (2012C) for details. Finally, variable return to scale model classifies 4 portfolios as efficient including two previous ones. Finally, we recall the results of all three types of SSD efficiency tests (see Branda and Kopa (2012C)): pairwise SSD efficiency test, convex SSD efficiency test and (full) SSD portfolio efficiency test. The results are summarized in Table 2 which presents the only 5 portfolios that are classified as efficient using at least one SSD efficiency test. See Branda and Kopa (2012C) for more details. Comparing the results of DEA-risk models and SSD efficiency tests we find three following results: 1. The pairwise SSD efficiency test identifies the largest set of efficient portfolios (5 from 48) and all ten other efficiency sets are its subsets. However, the VRS 2 The levels belong to the set of values considered in the DEA tests equivalent to SSD tests. This was not the case in Branda and Kopa (2012C).

18


Table 1 DEA-risk efficiency scores of efficient portfolios

Model

CRS I (4)

VRS I (5)

Agric Food Soda Beer Smoke Toys Fun Books Hshld Clths Hlth MedEq Drugs Chems Rubbr Txtls BldMt Cnstr Steel FabPr Mach ElcEq Autos Aero Ships Guns Gold Mines Coal Oil Util Telcm PerSv BusSv Comps Chips LabEq Paper Boxes Trans Whlsl Rtail Meals Banks Insur RlEst Fin Other

0.57 1.00 0.63 0.92 0.78 0.42 0.50 0.55 0.74 0.57 0.40 0.64 0.90 0.63 0.58 0.41 0.53 0.43 0.29 0.22 0.47 0.66 0.42 0.59 0.45 0.56 0.31 0.39 0.42 0.70 0.84 0.58 0.44 0.58 0.44 0.41 0.44 0.64 0.58 0.59 0.63 0.78 0.76 0.49 0.59 0.18 0.54 0.29

0.68 1.00 0.65 1.00 1.00 0.58 0.52 0.73 0.89 0.62 0.59 0.77 0.95 0.73 0.67 0.53 0.64 0.57 0.47 0.51 0.59 0.67 0.54 0.62 0.54 0.62 0.41 0.51 0.43 0.80 1.00 0.74 0.64 0.63 0.52 0.49 0.56 0.77 0.65 0.72 0.80 0.80 0.81 0.61 0.73 0.55 0.58 0.59

DEA tests VRS I II VRS I-O II (6) (7) 0.63 1.00 0.62 1.00 1.00 0.54 0.50 0.69 0.86 0.59 0.54 0.74 0.87 0.69 0.65 0.49 0.61 0.52 0.44 0.46 0.56 0.63 0.51 0.61 0.50 0.60 0.36 0.48 0.39 0.76 1.00 0.72 0.60 0.58 0.48 0.45 0.52 0.73 0.62 0.68 0.76 0.75 0.77 0.58 0.69 0.52 0.56 0.55

0.68 1.00 0.69 1.00 1.00 0.58 0.59 0.71 0.87 0.66 0.58 0.77 0.89 0.73 0.70 0.56 0.66 0.57 0.48 0.47 0.61 0.70 0.57 0.68 0.57 0.66 0.44 0.55 0.51 0.79 1.00 0.74 0.62 0.65 0.56 0.54 0.58 0.76 0.68 0.71 0.78 0.80 0.80 0.64 0.73 0.49 0.64 0.55

VRS I-O II bin (12) 0.68 1.00 0.74 1.00 1.00 0.58 0.59 0.71 0.87 0.66 0.58 0.77 0.89 0.73 0.70 0.56 0.66 0.57 0.48 0.47 0.61 0.77 0.57 0.68 0.57 0.66 0.44 0.55 0.54 0.79 1.00 0.74 0.62 0.65 0.56 0.54 0.58 0.76 0.68 0.71 0.78 0.80 0.80 0.64 0.73 0.49 0.64 0.55

Diversification consistent tests DC DC I-O I DC I-O II (8) (9) (10) 0.57 0.85 0.53 0.82 1.00 0.50 0.40 0.62 0.69 0.51 0.48 0.62 0.77 0.59 0.53 0.41 0.50 0.48 0.38 0.40 0.48 0.55 0.45 0.50 0.45 0.50 0.35 0.42 0.35 0.65 0.85 0.63 0.55 0.51 0.41 0.38 0.47 0.64 0.51 0.60 0.65 0.65 0.66 0.50 0.57 0.44 0.48 0.49

0.69 0.92 0.76 0.91 1.00 0.60 0.69 0.68 0.78 0.71 0.58 0.74 0.87 0.74 0.71 0.61 0.68 0.61 0.50 0.42 0.64 0.77 0.61 0.74 0.63 0.71 0.53 0.59 0.67 0.77 0.84 0.70 0.61 0.71 0.64 0.62 0.62 0.73 0.72 0.70 0.73 0.82 0.80 0.65 0.71 0.40 0.69 0.49

0.57 0.86 0.60 0.83 1.00 0.49 0.51 0.59 0.72 0.56 0.49 0.64 0.74 0.62 0.59 0.48 0.56 0.49 0.41 0.38 0.52 0.61 0.48 0.58 0.49 0.56 0.38 0.47 0.46 0.66 0.80 0.61 0.52 0.55 0.48 0.46 0.49 0.63 0.58 0.60 0.65 0.68 0.67 0.54 0.61 0.40 0.54 0.45


19

Table 2 Stochastic dominance efficiency pairwise Food Beer Smoke Drugs Util

yes yes yes yes yes

SSD efficiency convex (full) portfolio yes yes yes no yes

no no yes no no

model with binary weights and reduced inputs identified only four efficient portfolios, thus the equivalence is not preserved. 2. The efficiency classification given by all VRS DEA-risk models coincides with that of convex SSD efficiency test even in the case of reduced number of inputs compared with Branda and Kopa (2012C), where six CVaRs were considered. 3. All diversification consistent DEA-risk models as well as SSD portfolio efficiency test classify only the portfolio with the highest mean as efficient.

6 Conclusions In this paper we compared the efficiency of portfolios using two various approaches: stochastic dominance efficiency and DEA-risk efficiency. We identified efficient portfolios with respect to: constants return to scale DEA-risk model, variable return to scale DEA-risk models, diversification consistent DEA-risk models as well as pairwise SSD efficiency criterion, convex SSD efficiency criterion and (full) SSD portfolio efficiency criterion. Following Branda and Kopa (2012A), we constructed DEA-risk models based on classical DEA model of Charnes, Cooper and Rhodes (1978), where CVaR at some levels are used as inputs or outputs. The CVaR levels are given by the cumulative probabilities of the scenarios. Comparing to Branda and Kopa (2012A) we enrich the models by more reward measures as the outputs. Moreover, we introduced a new input-output oriented variable return to scale DEA-risk model that is equivalent to pairwise SSD portfolio efficiency test. Contrary to equivalent DEA-risk models to convex SSD efficiency and SSD portfolio efficiency presented in Branda and Kopa (2012B), the new DEA-risk model allows only the binary weights. Omitting this limitation, variable return to scale DEA-risk models with properly selected outputs and inputs would be equivalent to the convex SSD efficiency test instead of the pairwise SSD efficiency test. In the empirical study we followed Branda and Kopa (2012C) in analyzing 48 US representative industry portfolios. Comparing to Branda and Kopa (2012C) we modified the structure of models and we considered only 4 CVaR inputs and only one output (mean return). The selection of CVaR inputs was made in concordance with the theoretical equivalence results. Similarly as in Branda and Kopa (2012C) we found that all VRS DEA-risk models identified 4 efficient portfolios while the CRS and DC DEA-risk models classify as efficient only one of them. While DC DEA-risk models simply identified the portfolio with the highest mean, the CRS DEA-risk model classifies as efficient another portfolio that has the smallest risk compared to the mean return. Comparing it to SSD efficiency results obtained

20


in Branda and Kopa (2012C) we found that the equivalence between the new VRS DEA-risk model with binary weights and pairwise SSD efficiency test is not preserved. However, the other two equivalences, presented in Branda and Kopa (2012B), remain valid even for the reduced model. It shows that the new VRS DEA-risk model with binary weights is more sensitive to the choice of inputs then the other considered DEA-risk models. It probably follows from the integer feature of the model. All the presented results were derived under assumption of equiprobable discrete distribution of returns. This scenario assumption is very important because it allows to express all efficiency tests in terms of linear or nonlinear convex programs. The disadvantage of this approach may be a high sensitivity to scenario values, that is, a small error in the scenarios can completely change the efficiency classification: the inefficient portfolio may become efficient and vice versa. Therefore, one can enrich these testings by the robustness and stability analysis following e.g. Kopa (2010), Branda and Dupaˇcov´ a (2012) or Dupaˇcov´ a and Kopa (2012). Unfortunately, any of these improvements can be done only at costs of significantly higher computational load. Moreover, one can similarly use also first-order stochastic dominance (see Bawa et al. (1985), Kopa and Post (2009)) or third-order stochastic dominance (see Whitmore (1970), Gotoh and Konno (2000)) instead of considered second-order one. However, it would again increase the complexity of the problems. Finally, relaxing the assumption of discrete distribution of returns, one may address sample approximation technique, see Branda (2012A, 2012B), to handle the tests with a multivariate continuous distribution of returns. Acknowledgements The research was supported by the Czech Science Foundation under the Grants P402/10/1610, P402/12/0558. We would like to express our gratitude to the anonymous referees, whose comments have greatly improved the paper.

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