Data Envelopment Analysis in Finance

Transcription

1 Data Envelopment Analysis in Finance Martin Branda Faculty of Mathematics and Physics Charles University in Prague & Institute of Information Theory and Automation Academy of Sciences of the Czech Republic Ostrava, January 10, 2014 M. Branda DEA in Finance / 88

2 Contents 1 Efficiency of investment opportunities 2 Data Envelopment Analysis 3 Diversification-consistent DEA based on general deviation measures General deviation measures Diversification-consistent DEA models Financial indices efficiency empirical study 4 On relations between DEA and stochastic dominance efficiency Second Order Stochastic Dominance Data Envelopment Analysis Numerical comparison 5 References M. Branda DEA in Finance / 88

3 DEA in finance We do not access efficiency of financial institutions (banks, insurance comp.). We access efficiency of investment opportunities 1 on financial markets. 1 Assets, portfolios, mutual funds, financial indices,... M. Branda DEA in Finance / 88

4 Motivation Together with Miloš Kopa (in 2010): Is there a relation between stochastic dominance efficiency and DEA efficiency? Could we benefit from the relation? DEA traditional strong wide area (many applications and theory, Handbooks, papers in highly impacted journals, e.g. Omega, EJOR, JOTA, JORS, EE, JoBF) Stochastic dominance quickly growing area in finance and optimization Branda, Kopa (2012): an empirical study (a bit naive, but necessary step for us:) Branda, Kopa (2014): equivalences (a bridge ) M. Branda DEA in Finance / 88

5 Motivation Together with Miloš Kopa (in 2010): Is there a relation between stochastic dominance efficiency and DEA efficiency? Could we benefit from the relation? DEA traditional strong wide area (many applications and theory, Handbooks, papers in highly impacted journals, e.g. Omega, EJOR, JOTA, JORS, EE, JoBF) Stochastic dominance quickly growing area in finance and optimization Branda, Kopa (2012): an empirical study (a bit naive, but necessary step for us:) Branda, Kopa (2014): equivalences (a bridge ) M. Branda DEA in Finance / 88

6 Motivation Together with Miloš Kopa (in 2010): Is there a relation between stochastic dominance efficiency and DEA efficiency? Could we benefit from the relation? DEA traditional strong wide area (many applications and theory, Handbooks, papers in highly impacted journals, e.g. Omega, EJOR, JOTA, JORS, EE, JoBF) Stochastic dominance quickly growing area in finance and optimization Branda, Kopa (2012): an empirical study (a bit naive, but necessary step for us:) Branda, Kopa (2014): equivalences (a bridge ) M. Branda DEA in Finance / 88

7 M. Branda DEA in Finance / 88

8 M. Branda DEA in Finance / 88

9 M. Branda DEA in Finance / 88

10 Contents Efficiency of investment opportunities 1 Efficiency of investment opportunities 2 Data Envelopment Analysis 3 Diversification-consistent DEA based on general deviation measures General deviation measures Diversification-consistent DEA models Financial indices efficiency empirical study 4 On relations between DEA and stochastic dominance efficiency Second Order Stochastic Dominance Data Envelopment Analysis Numerical comparison 5 References M. Branda DEA in Finance / 88

11 Efficiency of investment opportunities Efficiency of investment opportunities Various approaches how to find an optimal portfolio or how to test efficiency of an investment opportunity: von Neumann and Morgenstern (1944): Utility, expected utility Markowitz (1952): Mean-variance, mean-risk, mean-deviation Hadar and Russell (1969), Hanoch and Levy (1969): Stochastic dominance Murthi et al (1997): Data Envelopment Analysis M. Branda DEA in Finance / 88

12 DEA in finance Efficiency of investment opportunities This presentation contains DEA efficiency in finance Murthi et al. (1997), Briec et al. (2004), Lamb and Tee (2012) Extension of mean-risk efficiency based on multiobjective optimization principles Markowitz (1952) Risk shaping several risk measures (CVaRs) included into one model Rockafellar and Uryasev (2002) Relations to stochastic dominance efficiency Branda and Kopa (2014) M. Branda DEA in Finance / 88

13 Contents Data Envelopment Analysis 1 Efficiency of investment opportunities 2 Data Envelopment Analysis 3 Diversification-consistent DEA based on general deviation measures General deviation measures Diversification-consistent DEA models Financial indices efficiency empirical study 4 On relations between DEA and stochastic dominance efficiency Second Order Stochastic Dominance Data Envelopment Analysis Numerical comparison 5 References M. Branda DEA in Finance / 88

14 Data Envelopment Analysis Data Envelopment Analysis (DEA) Charnes, Cooper and Rhodes (1978): a way how to state efficiency of a decision making unit over all other decision making units with the same structure of inputs and outputs. Let Z 1i,..., Z Ki denote the inputs and Y 1i,..., Y Ji denote the outputs of the unit i from n considered units. DEA efficiency of the unit 0 {1,..., n} is then evaluated using the optimal value of the following program where the weighted inputs are compared with the weighted outputs. All data are assumed to be (semi-)positive. Charnes et al (1978): fractional programming formulation (Constant Returns to Scale CRS or CCR) M. Branda DEA in Finance / 88

15 Data Envelopment Analysis DEA Variable Returns to Scale (VRS) Banker, Charnes and Cooper (1984): DEA model with Variable Returns to Scale (VRS) or BCC: max y j0,w k0 J j=1 y j0y j0 y 0 K k=1 w k0z k0 s.t. J j=1 y j0y ji y 0 K k=1 w 1, i = 1,..., n, k0z ki w k0 0, k = 1,..., K, y j0 0, j = 1,..., J, y 0 R. M. Branda DEA in Finance / 88

16 Data Envelopment Analysis DEA Variable Returns to Scale (VRS) Dual formulation of VRS DEA: min x i,θ s.t. n x i Y ji Y j0, j = 1,..., J, i=1 n x i Z ki θ Z k0, k = 1,..., K, i=1 n x i = 1, x i 0, i = 1,..., n. i=1 M. Branda DEA in Finance / 88

17 Data Envelopment Analysis Data envelopment analysis production theory (production possibility set), returns to scale (CRS, VRS, NIRS,...), radial/slacks-based/directional distance models, fractional/primal/dual formulations, multiobjective opt. strong/weak Pareto efficiency, stochastic data reliability, chance constraints, dynamic (network) DEA, super-efficiency, cross-efficiency,... the most efficient unit... M. Branda DEA in Finance / 88

18 Data Envelopment Analysis DEA and multiobjective optimization DEA efficiency corresponds to multiobjective (Pareto) efficiency where all inputs are minimized and/or all outputs are maximized under some conditions. M. Branda DEA in Finance / 88

19 Data Envelopment Analysis Traditional DEA in finance Efficiency of mutual funds or financial indexes: Murthi et al. (1997): expense ratio, load, turnover, standard deviation and gross return. Basso and Funari (2001, 2003): standard deviation and semideviations, beta coefficient, costs as the inputs, expected return or expected excess return, ethical measure and stochastic dominance criterion as the outputs. Chen and Lin (2006): Value at Risk (VaR) and Conditional Value at Risk (CVaR). Branda and Kopa (2012): VaR, CVaR, sd, lsd, Drawdow measures (DaR, CDaR) as the inputs, gross return as the output; comparison with second-order stochastic dominance. See Table 1 in Lozano and Gutiérrez (2008B) M. Branda DEA in Finance / 88

20 Data Envelopment Analysis General class of financial DEA tests Lamb and Tee (2012) pure return-risk tests 2 : Inputs: positive parts of coherent risk measures Outputs: return measures (= minus coherent risk measures, e.g. expected return) 2 no transactions costs etc. M. Branda DEA in Finance / 88

21 Contents DC DEA models based on GDM 1 Efficiency of investment opportunities 2 Data Envelopment Analysis 3 Diversification-consistent DEA based on general deviation measures General deviation measures Diversification-consistent DEA models Financial indices efficiency empirical study 4 On relations between DEA and stochastic dominance efficiency Second Order Stochastic Dominance Data Envelopment Analysis Numerical comparison 5 References M. Branda DEA in Finance / 88

22 DC DEA models based on GDM General deviation measures General deviation measures Rockafellar, Uryasev and Zabarankin (2006A, 2006B): GDM are introduced as an extension of standard deviation but they need not to be symmetric with respect to upside X E[X ] and downside E[X ] X of a random variable X. Any functional D : L 2 (Ω) [0, ] is called a general deviation measure if it satisfies (D1) D(X + C) = D(X ) for all X and constants C, (D2) D(0) = 0, and D(λX ) = λd(x ) for all X and all λ > 0, (D3) D(X + Y ) D(X ) + D(Y ) for all X and Y, (D4) D(X ) 0 for all X, with D(X ) > 0 for nonconstant X. (D2) & (D3) convexity M. Branda DEA in Finance / 88

23 DC DEA models based on GDM General deviation measures General deviation measures Rockafellar, Uryasev and Zabarankin (2006A, 2006B): GDM are introduced as an extension of standard deviation but they need not to be symmetric with respect to upside X E[X ] and downside E[X ] X of a random variable X. Any functional D : L 2 (Ω) [0, ] is called a general deviation measure if it satisfies (D1) D(X + C) = D(X ) for all X and constants C, (D2) D(0) = 0, and D(λX ) = λd(x ) for all X and all λ > 0, (D3) D(X + Y ) D(X ) + D(Y ) for all X and Y, (D4) D(X ) 0 for all X, with D(X ) > 0 for nonconstant X. (D2) & (D3) convexity M. Branda DEA in Finance / 88

24 Deviation measures DC DEA models based on GDM General deviation measures Standard deviation D(X ) = σ(x ) = Mean absolute deviation D(X ) = E [ X E[X ] ]. E X E[X ] 2 Mean absolute lower and upper semideviation D (X ) = E [ X E[X ] ], D+ (X ) = E [ X E[X ] + ]. Worst-case deviation D(X ) = sup X (ω) E[X ]. ω Ω See Rockafellar et al (2006 A, 2006 B) for another examples. M. Branda DEA in Finance / 88

25 DC DEA models based on GDM General deviation measures Mean absolute deviation from (1 α)-th quantile CVaR deviation For any α (0, 1) a finite, continuous, lower range dominated deviation measure D α (X ) = CVaR α (X E[X ]). (1) The deviation is also called weighted mean absolute deviation from the (1 α)-th quantile, see Ogryczak, Ruszczynski (2002), because it can be expressed as 1 D α (X ) = min E[max{(1 α)(x ξ), α(ξ X )}] (2) ξ R 1 α with the minimum attained at any (1 α)-th quantile. In relation with CVaR minimization formula, see Pflug (2000), Rockafellar and Uryasev (2000, 2002). M. Branda DEA in Finance / 88

26 DC DEA models based on GDM General deviation measures General deviation measures According to Proposition 4 in Rockafellar et al (2006 A): if D = λd 0 for λ > 0 and a deviation measure D 0, then D is a deviation measure. if D 1,..., D K are deviation measures, then D = max{d 1,..., D K } is also deviation measure. D = λ 1 D λ K D K is also deviation measure, if λ k > 0 and K k=1 λ k = 1. Rockafellar et al (2006 A, B): Duality representation using risk envelopes and risk identifiers, subdifferentiability and optimality conditions. M. Branda DEA in Finance / 88

27 DC DEA models based on GDM Coherent risk and return measures General deviation measures CRM: R : L 2 (Ω) (, ] that satisfies (R1) R(X + C) = R(X ) C for all X and constants C, (R2) R(0) = 0, and R(λX ) = λr(x ) for all X and all λ > 0, (R3) R(X + Y ) R(X ) + R(Y ) for all X and Y, (R4) R(X ) R(Y ) when X Y. Strictly expectation bounded risk measures satisfy (R1), (R2), (R3), and (R5) R(X ) > E[ X ] for all nonconstant X, whereas R(X ) = E[ X ] for constant X. Other classes of risk measures and functionals: Follmer and Schied (2002), Pflug and Romisch (2007), Ruszczynski and Shapiro (2006). M. Branda DEA in Finance / 88

28 DC DEA models based on GDM Coherent risk and return measures General deviation measures CRM: R : L 2 (Ω) (, ] that satisfies (R1) R(X + C) = R(X ) C for all X and constants C, (R2) R(0) = 0, and R(λX ) = λr(x ) for all X and all λ > 0, (R3) R(X + Y ) R(X ) + R(Y ) for all X and Y, (R4) R(X ) R(Y ) when X Y. Strictly expectation bounded risk measures satisfy (R1), (R2), (R3), and (R5) R(X ) > E[ X ] for all nonconstant X, whereas R(X ) = E[ X ] for constant X. Other classes of risk measures and functionals: Follmer and Schied (2002), Pflug and Romisch (2007), Ruszczynski and Shapiro (2006). M. Branda DEA in Finance / 88

29 DC DEA models based on GDM Coherent risk and return measures General deviation measures A return measure is defined as a functional E(X ) = R(X ) for a coherent risk measure R. It is obvious that the expectation belongs to this class. The right part of a return distribution described better by return measures derived from CVaR, cf. Lamb and Tee (2012). M. Branda DEA in Finance / 88

30 DC DEA models based on GDM General deviation measures General deviation measures We say that general deviation measure D is (LSC) lower semicontinuous (lsc) if all the subsets of L 2 (Ω) having the form {X : D(X ) c} for c R (level sets) are closed; (D5) lower range dominated if D(X ) EX inf ω Ω X (ω) for all X. M. Branda DEA in Finance / 88

31 DC DEA models based on GDM General deviation measures General deviation measures We say that general deviation measure D is (LSC) lower semicontinuous (lsc) if all the subsets of L 2 (Ω) having the form {X : D(X ) c} for c R (level sets) are closed; (D5) lower range dominated if D(X ) EX inf ω Ω X (ω) for all X. M. Branda DEA in Finance / 88

32 DC DEA models based on GDM General deviation measures Strictly expectation bounded risk measures Theorem 2 in Rockafellar et al (2006 A): Theorem Deviation measures correspond one-to-one with strictly expectation bounded risk measures under the relations D(X ) = R(X E[X ]) R(X ) = E[ X ] + D(X ) In this correspondence, R is coherent if and only if D is lower range dominated. M. Branda DEA in Finance / 88

33 DC DEA models based on GDM Traditional DEA model Input oriented (VRS) DC DEA models based on GDM We assume that X 0 is not constant, i.e. D k (X 0 ) > 0, for all k = 1,..., K. Input oriented VRS model can be formulated in the dual form θ I 0(X 0 ) = min θ s.t. n x i E j (R i ) E j (X 0 ), j = 1,..., J, (3) i=1 n x i D k (R i ) θ D k (X 0 ), k = 1,..., K, i=1 n x i = 1, x i 0, i = 1,..., n. i=1 M. Branda DEA in Finance / 88

34 DC DEA models based on GDM The most important slide Traditional DEA model vs. diversification DC DEA models based on GDM The model does not take into account portfolio diversification: For any general deviation measure D k it holds ( n n ) x i D k (R i ) D k x i R i i=1 i=1 for nonnegative weights with n i=1 x i = 1. Linear transformation of inputs is only an upper bound for the real portfolio deviation. M. Branda DEA in Finance / 88

35 DC DEA models based on GDM The most important slide Traditional DEA model vs. diversification DC DEA models based on GDM The model does not take into account portfolio diversification: For any general deviation measure D k it holds ( n n ) x i D k (R i ) D k x i R i i=1 i=1 for nonnegative weights with n i=1 x i = 1. Linear transformation of inputs is only an upper bound for the real portfolio deviation. M. Branda DEA in Finance / 88

36 DC DEA models based on GDM DC DEA models based on GDM Traditional DEA and diversification frontier M. Branda DEA in Finance / 88

37 DC DEA models based on GDM Models with diversification DC DEA models based on GDM Efficiency of mutual funds or financial indexes using DEA related models with diversification: Kuosmannen (2007): Second order stochastic dominance consistent DEA models. Lozano and Gutiérrez (2008 A, 2008 B): Second and Third order stochastic dominance consistent DEA models. Kopa (2011): Comparison of several approaches (VaR, CVaR,...). Branda (2011): models consistent with TSD. Lamb and Tee (2012): input-oriented diversification consistent model with several inputs and outputs (positive parts of risk measures used as the inputs). M. Branda DEA in Finance / 88

38 DC DEA models based on GDM DEA tests with diversification DC DEA models based on GDM Efficiency of mutual funds or industry representative portfolios: Briec et al. (2004), Kerstens et al. (2012): directional-distance mean-variance efficiency. Joro and Na (2006), Briec et al. (2007), Kerstens et al. (2011, 2013): directional-distance mean-variance-skewness efficiency. Lozano and Gutiérrez (2008A, 2008B): tests consistent with secondand third-order stochastic dominance (necessary condition). Branda and Kopa (2014): equivalence with second-order stochastic dominance. Lamb and Tee (2012), Branda (2013A, 2013B): general classes of DEA tests with risk/deviation and return measures. M. Branda DEA in Finance / 88

39 DC DEA models based on GDM Set of investment opportunities DC DEA models based on GDM We consider n assets and denote R i L 2 (Ω) the rate of return of i-th asset and the sets of investment opportunities: 1 pairwise efficiency (investment into one single asset): X P = {R i, i = 1,..., n}, 2 full diversification (diversification across all assets): { n } n X FD = R i x i : x i = 1, x i 0, i=1 i=1 3 limited diversification (diversification across limited number of assets #): { n } n n X LD = R i x i : x i = 1, x i 0, x i y i, y i {0, 1}, y i #. i=1 i=1 i=1 M. Branda DEA in Finance / 88

40 DC DEA models based on GDM Set of investment opportunities DC DEA models based on GDM We consider n assets and denote R i L 2 (Ω) the rate of return of i-th asset and the sets of investment opportunities: 1 pairwise efficiency (investment into one single asset): X P = {R i, i = 1,..., n}, 2 full diversification (diversification across all assets): { n } n X FD = R i x i : x i = 1, x i 0, i=1 i=1 3 limited diversification (diversification across limited number of assets #): { n } n n X LD = R i x i : x i = 1, x i 0, x i y i, y i {0, 1}, y i #. i=1 i=1 i=1 M. Branda DEA in Finance / 88

41 DC DEA models based on GDM Set of investment opportunities DC DEA models based on GDM We consider n assets and denote R i L 2 (Ω) the rate of return of i-th asset and the sets of investment opportunities: 1 pairwise efficiency (investment into one single asset): X P = {R i, i = 1,..., n}, 2 full diversification (diversification across all assets): { n } n X FD = R i x i : x i = 1, x i 0, i=1 i=1 3 limited diversification (diversification across limited number of assets #): { n } n n X LD = R i x i : x i = 1, x i 0, x i y i, y i {0, 1}, y i #. i=1 i=1 i=1 M. Branda DEA in Finance / 88

42 DC DEA models based on GDM Partial ordering of vectors DC DEA models based on GDM Definition Let v, z R n. We say that z strictly dominates (weakly) v, denoted v st z if i : v i < z i, v w z if i : v i z i and ĩ : vĩ < zĩ. Definition Let v, z R n. We say that z partially strictly (partially weakly) dominates v with respect to an index set S {1,..., n}, denoted v pst(s) z if v i < z i for all i S and v i z i for all i {1,..., n} \ S, v pw(s) z if v i z i for all i {1,..., n} and there exists at least one ĩ S for which vĩ < zĩ. M. Branda DEA in Finance / 88

43 DEA efficiency DC DEA models based on GDM DC DEA models based on GDM We assume that X 0 X is not constant, i.e. D k (X 0 ) > 0, for all k = 1,..., K. Definition We say that X 0 X is DEA efficient with respect to the set X if the optimal value of the DEA program is equal to 1. Otherwise, X 0 is inefficient and the optimal value measures the inefficiency. For m {0, 1, 2, 3} we denote the sets of efficient opportunities Ψ I m = {X X : θ I m(x ) = 1}, where θ I m(x 0 ) is the optimal value for benchmark X 0. M. Branda DEA in Finance / 88

44 DC DEA models based on GDM DC DEA models based on GDM Diversification consistent DEA - model 1 For a benchmark X 0 X diversification consistent DEA model θ1(x I 0 ) = min θ s.t. E j (X ) E j (X 0 ), j = 1,..., J, (4) D k (X ) θ D k (X 0 ), k = 1,..., K, X X. M. Branda DEA in Finance / 88

45 DC DEA models based on GDM Set of efficient opportunities DC DEA models based on GDM Proposition X 0 Ψ I 1 if for all X X for which E j (X ) E j (X 0 ) for all j holds that D k (X ) D k (X 0 ) for at least one k, i.e. there is no X X for which E j (X ) E j (X 0 ) for all j and D k (X ) < D k (X 0 ) for all k. if there is no vector v from the set PPS R = {(D 1 (X ),..., D K (X )) : E j (X ) E j (X 0 ), j = 1,..., J, X X } for which v st (D 1 (X 0 ),..., D K (X 0 )). M. Branda DEA in Finance / 88

46 DC DEA models based on GDM Model 1 - equivalent formulation DC DEA models based on GDM θ I 1(X 0 ) = min max k=1,...,k D k (X ) D k (X 0 ) s.t. (5) E j (X ) E j (X 0 ), j = 1,..., J, X X. D(X ) = max k=1,...,k D k (X )/D k (X 0 ) defines a deviation measure. M. Branda DEA in Finance / 88

47 DC DEA models based on GDM DC DEA models based on GDM Diversification consistent DEA - model 2 For a benchmark X 0 X, we can introduce a generalized model θ I 2(X 0 ) = min 1 K K k=1 θ k s.t. E j (X ) E j (X 0 ), j = 1,..., J, (6) D k (X ) θ k D k (X 0 ), k = 1,..., K, 0 θ k 1, k = 1,..., K, X X. M. Branda DEA in Finance / 88

48 DC DEA models based on GDM Set of efficient opportunities DC DEA models based on GDM Proposition X 0 Ψ I 2 if for all X X for which E j (X ) E j (X 0 ) for all j holds that D k (X ) D k (X 0 ) for all k, i.e. there is no X X for which E j (X ) E j (X 0 ) for all j and D k (X ) < D k (X 0 ) for at least one k. if there is no vector v from the set PPS R = {(D 1 (X ),..., D K (X )) : E j (X ) E j (X 0 ), j = 1,..., J, X X } for which v w (D 1 (X 0 ),..., D K (X 0 )). M. Branda DEA in Finance / 88

49 DC DEA models based on GDM Model 2 - equivalent formulation DC DEA models based on GDM The model is obviously equivalent to θ I 2(X 0 ) = min 1 K K k=1 D k (X ) D k (X 0 ) s.t. E j (X ) E j (X 0 ), j = 1,..., J, (7) D k (X ) D k (X 0 ), k = 1,..., K, X X. D(X ) = 1 K K k=1 D k(x )/D k (X 0 ) defines a deviation measure. M. Branda DEA in Finance / 88

50 DC DEA models based on GDM Diversification consistent DEA Slacks-based model DC DEA models based on GDM Additive or slacks-based DEA model θ I 3(X 0 ) = min 1 K K k=1 D k (X 0 ) s k D k (X 0 ) s.t. E j (X ) E j (X 0 ), j = 1,..., J, (8) D k (X ) + s k = D k (X 0 ), k = 1,..., K, s k 0, k = 1,..., K, X X. M. Branda DEA in Finance / 88

51 Units invariance DC DEA models based on GDM DC DEA models based on GDM Proposition The considered DEA models are units invariant. For arbitrary k and j λd k (X ) = D k (λx ) which implies E j (λx ) = λe j (λx ) for arbitrary X X and λ > 0. M. Branda DEA in Finance / 88

52 Test strength DC DEA models based on GDM DC DEA models based on GDM Proposition Let K 2. Then for a benchmark X 0 X, the following relations between the optimal values (efficiency scores) hold θ I 0(X 0 ) θ I 1(X 0 ) θ I 2(X 0 ) = θ I 3(X 0 ). For the sets of efficient portfolios we obtain Ψ I 3 = Ψ I 2 Ψ I 1 Ψ I 0. M. Branda DEA in Finance / 88

53 Test strength DC DEA models based on GDM DC DEA models based on GDM We can construct weaker tests if we restrict the number of investment opportunities in the portfolio leading to the models with the limited diversification. Proposition For X# LD and X LD # with # > #, it holds θ(x 0 ) θ (X 0 ), where θ(x 0 ) and θ (X 0 ) denote the efficiency scores with respect to the sets X# LD, and X LD # respectively. M. Branda DEA in Finance / 88

54 DC DEA models based on GDM Input-output oriented models DC DEA models based on GDM Input-output oriented DC DEA models - (in)efficiency is measured also with respect to the outputs (assume E j (X 0 ) > 0): optimal values (efficiency scores) and strength can be compared, input and input-output oriented models can be compared: I-O tests are stronger in general, cf. Branda (2013A, 2013B). M. Branda DEA in Finance / 88

55 DC DEA models based on GDM Input-output oriented test 1 DC DEA models based on GDM We assume that E j (X 0 ) is positive for at least one j. An input-output oriented test where inefficiency is measured with respect to the inputs and outputs separately can be formulated as follows θ1 I O θ (X 0 ) = min θ,ϕ,x ϕ s.t. E j (X ) ϕ E j (X 0 ), j = 1,..., J, (9) D k (X ) θ D k (X 0 ), k = 1,..., K, 0 θ 1, ϕ 1, X X. M. Branda DEA in Finance / 88

56 DC DEA models based on GDM Input-output oriented test 1 DC DEA models based on GDM Let X = X FD. Setting 1/t = ϕ and substitute x i = tx i, θ = tθ, and ϕ = tϕ, results into an input oriented DEA test with nonincreasing return to scale (NIRS): θ1 I O (R 0 ) = min θ θ, x i ( n ) s.t. E j R i x i E j (R 0 ), j = 1,..., J, i=1 ( n ) D k R i x i θ D k (R 0 ), k = 1,..., K, i=1 n x i 1, x i 0, 1 θ 0. i=1 Note that it is important for the reformulation that all inputs D k and all outputs E j are positively homogeneous. We obtained a convex programming problem. M. Branda DEA in Finance / 88

57 DC DEA models based on GDM Input-output oriented test 1 DC DEA models based on GDM Proposition X 0 Ψ I O 1 : if there is no X X for which either (E j (X ) E j (X 0 ) for all j and D k (X ) < D k (X 0 ) for all k) or (E j (X ) > E j (X 0 ) for all j and D k (X ) D k (X 0 ) for all k), or equivalently if there is no vector v from the production possibility set for which either or v pst({1,...,k}) (D 1 (X 0 ),..., D K (X 0 ), E 1 (X 0 ),..., E J (X 0 )), v pst({k+1,...,k+j}) (D 1 (X 0 ),..., D K (X 0 ), E 1 (X 0 ),..., E J (X 0 )). M. Branda DEA in Finance / 88

58 DC DEA models based on GDM Input-output oriented test 2 DC DEA models based on GDM For a benchmark X 0 X formulated as θ I O 2 (X 0 ) = min θ k,ϕ j,x 1 K K k=1 θ k 1 J J j=1 ϕ j s.t. E j (X ) ϕ j E j (X 0 ), j = 1,..., J, (10) D k (X ) θ k D k (X 0 ), k = 1,..., K, 0 θ k 1, ϕ k 1, k = 1,..., K, X X. M. Branda DEA in Finance / 88

59 DC DEA models based on GDM Input-output oriented test 2 DC DEA models based on GDM Let X = X FD. For a benchmark R 0 X formulated as 1 K θ 2 (R 0 ) = min θ k θ k, ϕ j, x i K s.t. 1 J E j ( n i=1 D k ( n i=1 k=1 J ϕ j = 1, (11) j=1 R i x i ) R i x i ) ϕ j E j (R 0 ), j = 1,..., J, θ k D k (R 0 ), k = 1,..., K, n x i 1, x i 0, i = 1,..., n, i=1 ϕ j 0, 1 θ k 0. M. Branda DEA in Finance / 88

60 DC DEA models based on GDM Input-output oriented test 2 DC DEA models based on GDM Proposition X 0 Ψ I O 2 : if there is no X X for which E j (X ) E j (X 0 ) for all j and D k (X ) D k (X 0 ) for all k with at least one inequality strict, or equivalently if there is no vector v from the production possibility set for which v pw({1,...,k+j}) (D 1 (X 0 ),..., D K (X 0 ), E 1 (X 0 ),..., E J (X 0 )). M. Branda DEA in Finance / 88

61 DC DEA models based on GDM Input-output oriented test 3 DC DEA models based on GDM The additive or slacks-based test inspired by Tone (2001): θ I O 3 (X 0 ) = min s k,s+ j,x 1 K D k (X 0 ) s k K k=1 D k (X 0 ) 1 J E j (X 0 )+s + j J j=1 E j (X 0 ) s.t. E j (X ) s + j E j (X 0 ), j = 1,..., J, D k (X ) + s k D k (X 0 ), k = 1,..., K, s + j 0, s k 0, X X. Thus, the score can be interpreted as the ratio of the mean input and the mean output inefficiencies. The objective function can be rewritten as 1 1 K K k=1 s k /D k(x 0 ) J J j=1 s+ j /E j (X 0 ). M. Branda DEA in Finance / 88

62 DC DEA models based on GDM Properties and relations DC DEA models based on GDM Proposition Let max{j, K} 2. Then for a benchmark X 0 X with D k (X 0 ) > 0 for all k and E j (X 0 ) > 0 for all j, the following relations hold θ I O 0 (X 0 ) θ I O 1 (X 0 ) θ I O 2 (X 0 ) = θ I O 3 (X 0 ). Then, for the sets of efficient portfolios it can be obtained Ψ I O 3 = Ψ I O 2 Ψ I O 1 Ψ I O 0. M. Branda DEA in Finance / 88

63 DC DEA models based on GDM Properties and relations DC DEA models based on GDM Proposition Let max{j, K} 2. Then for a benchmark X 0 X with D k (X 0 ) > 0 for all k and E j (X 0 ) > 0 for all j, the following relations hold θ I 1(X 0 ) θ I O 1 (X 0 ), θ I 2(X 0 ) θ I O 2 (X 0 ). Then, for the sets of efficient portfolios can be obtained Ψ I O 1 Ψ I 1, Ψ I O 2 Ψ I 2. M. Branda DEA in Finance / 88

64 DC DEA models based on GDM DC DEA model with CVaR deviations Financial indices efficiency empirical study We consider CVaR deviations D S α for α k (0, 1), k = 1,..., K as the inputs and the expectation as the output, i.e. J = 1 and E 1 (X ) = EX : min θ s.t. n E[R i ]x i E[R 0 ], i=1 D αk ( n i=1 (full diversification) R i x i ) θ D αk (R 0 ), k = 1,..., K, n x i = 1, x i 0, i = 1,..., n. i=1 M. Branda DEA in Finance / 88

65 DC DEA models based on GDM Financial indices efficiency empirical study Mean absolute deviation from (1 α)-th quantile CVaR deviation For any α (0, 1) a finite, continuous, lower range dominated deviation measure D α (X ) = CVaR α (X E[X ]). (12) The deviation is also called weighted mean absolute deviation from the (1 α)-th quantile, see Ogryczak, Ruszczynski (2002), because it can be expressed as 1 D α (X ) = min E[max{(1 α)(x ξ), α(ξ X )}] (13) ξ R 1 α with the minimum attained at any (1 α)-th quantile. In relation with CVaR minimization formula, see Pflug (2000), Rockafellar and Uryasev (2000, 2002). M. Branda DEA in Finance / 88

66 DC DEA models based on GDM DC DEA model with CVaR deviations Input oriented Financial indices efficiency empirical study For discretely distributed returns (r is, s = 1,..., S, p s = 1/S) LP: θ I 1(R 0 ) = min θ 1 S s.t. n E[R i ]x i E[R 0 ], i=1 S u sk θ D αk (R 0 ), k = 1,..., K, s=1 u sk u sk n x i r is ξ, s = 1,..., S, k = 1,..., K, i=1 α k 1 α k (ξ n x i r is ), s = 1,..., S, k = 1,..., K, i=1 n x i = 1, x i 0, i = 1,..., n. i=1 M. Branda DEA in Finance / 88

67 DC DEA models based on GDM DC DEA model with CVaR deviations Input-output oriented (not included) Financial indices efficiency empirical study For discretely distributed returns (r is, s = 1,..., S, p s = 1/S) LP: n θ1 I O (R 0 ) = min θ s.t. E[R i ]x i E[R 0 ], θ,x i,u sk,ξ k 1 S i=1 S u sk θ Dα S k (R 0 ), k = 1,..., K, s=1 u sk u sk ( n ) x i r is ξ k, s = 1,..., S, k = 1,..., K, i=1 α k 1 α k ( ξ k ) n x i r is, i=1 n x i 1, x i 0, i = 1,..., n. i=1 M. Branda DEA in Finance / 88

68 Financial indices DC DEA models based on GDM Financial indices efficiency empirical study We consider the following 25 world financial indices which are listed on Yahoo Finance: America (5): MERVAL BUENOS AIRES, IBOVESPA, S&P TSX Composite index, S&P 500 INDEX RTH, IPC, Asia/Pacific (11): ALL ORDINARIES, SSE Composite Index, HANG SENG INDEX, BSE SENSEX, Jakarta Composite Index, FTSE Bursa Malaysia KLCI, NIKKEI 225, NZX 50 INDEX GROSS, STRAITS TIMES INDEX, KOSPI Composite Index, TSEC weighted index, Europe (8): ATX, CAC 4, DAX, AEX, SMSI, OMX Stockholm PI, SMI, FTSE 100, Middle East (1): TEL AVIV TA-100 IND. The same dataset analyzed by Branda and Kopa (2010, 2012). M. Branda DEA in Finance / 88

69 DC DEA models based on GDM Financial indices efficiency Financial indices efficiency empirical study In our analysis we describe each index by its weekly rates of returns. We divided the returns into three datasets: before crises (B): September 11, September 15, 2008 during crises (D): September 16, September 20, 2010 whole period (W). CVaR deviation levels: α k {0.75, 0.9, 0.95, 0.99} DEA optimal values/scores with # = n... DEA optimal values/scores with # = 2... DEA optimal values/scores with # = 1... M. Branda DEA in Finance / 88

70 DC DEA models based on GDM Financial indices efficiency empirical study Index θ0 I θ1 I θ2 I B D W B D W B D W IBOVESPA S&PTSX Composite index S&P 500 INDEX.RTH IPC ALL ORDINARIES SSE Composite Index BSE SENSEX Jakarta Composite Index FTSE Bursa Malaysia KLCI NIKKEI NZX 50 INDEX GROSS STRAITS TIMES INDEX KOSPI Composite Index TSEC weighted index ATX CAC DAX AEX SMSI OMX Stockholm PI SMI FTSE TEL AVIV TA-100 IND M. Branda DEA in Finance / 88

71 DC DEA models based on GDM Financial indices efficiency empirical study Index θ0 I θ1 I θ2 I B D W B D W B D W IBOVESPA S&PTSX Composite index S&P 500 INDEX.RTH IPC ALL ORDINARIES SSE Composite Index BSE SENSEX Jakarta Composite Index FTSE Bursa Malaysia KLCI NIKKEI NZX 50 INDEX GROSS STRAITS TIMES INDEX KOSPI Composite Index TSEC weighted index ATX CAC DAX AEX SMSI OMX Stockholm PI SMI FTSE TEL AVIV TA-100 IND M. Branda DEA in Finance / 88

72 DC DEA models based on GDM Financial indices efficiency empirical study Index θ0 I θ1 I θ2 I B D W B D W B D W IBOVESPA S&PTSX Composite index S&P 500 INDEX.RTH IPC ALL ORDINARIES SSE Composite Index BSE SENSEX Jakarta Composite Index FTSE Bursa Malaysia KLCI NIKKEI NZX 50 INDEX GROSS STRAITS TIMES INDEX KOSPI Composite Index TSEC weighted index ATX CAC DAX AEX SMSI OMX Stockholm PI SMI FTSE TEL AVIV TA-100 IND M. Branda DEA in Finance / 88

73 On relations between DEA and stochastic dominance efficiency Contents 1 Efficiency of investment opportunities 2 Data Envelopment Analysis 3 Diversification-consistent DEA based on general deviation measures General deviation measures Diversification-consistent DEA models Financial indices efficiency empirical study 4 On relations between DEA and stochastic dominance efficiency Second Order Stochastic Dominance Data Envelopment Analysis Numerical comparison 5 References M. Branda DEA in Finance / 88

74 On relations between DEA and stochastic dominance efficiency A bridge Data envelopment analysis (production theory, returns to scale, radial/slacks-based/directional distance models, primal/dual formulations, multiobjective opt. Pareto efficiency, chance constraints), large literature: handbooks on DEA, Omega, EJOR, JORS,... Stochastic dominance efficiency (pairwise, convex, portfolio): Is there R such that R SSD R 0? No R 0 is efficient. Compare with the problem max f (R) : s.t. R SSD R 0. M. Branda DEA in Finance / 88

75 On relations between DEA and stochastic dominance efficiency Second order stochastic dominance Second Order Stochastic Dominance F 1, F 2... cumulative probability distributions functions of random variables X 1, X 2. Second order (strict) stochastic dominance (SSD): X 1 SSD X 2 iff E F1 u(x) E F2 u(x) 0 for every concave utility function u with at least one strict inequality. Consider twice cumulated probability distributions functions: t F (2) i (t) = F i (x)dx i = 1, 2. Theorem (Hanoch & Levy (1969)): X 1 SSD X 2 F (2) 1 (t) F (2) 2 (t) t R with at least one strict inequality. M. Branda DEA in Finance / 88

76 On relations between DEA and stochastic dominance efficiency SSD portfolio efficiency Second Order Stochastic Dominance We consider n assets and we denote R i the rate of return of i-th asset with finite mean value, r = {R 1,..., R n }. a discrete probability distribution of rate of returns described by scenarios r i,s, s = 1,..., S that are taken with equal probabilities p s = 1/S. a decision maker that may combine the assets into portfolios represented by weights x = {x 1,..., x n }. the set of feasible weights (no short sales allowed): X = {x R n x i = 1, x i 0, i = 1,..., n}. (14) i=1 M. Branda DEA in Finance / 88

77 On relations between DEA and stochastic dominance efficiency SSD portfolio efficiency Second Order Stochastic Dominance A given portfolio τ X is SSD portfolio efficient if and only if there exists no portfolio λ X such that r λ SSD r τ. Otherwise, portfolio τ is SSD inefficient. SSD portfolio efficiency tests: Post (2003), Kuosmanen (2004), Kopa and Chovanec (2008)... M. Branda DEA in Finance / 88

78 On relations between DEA and stochastic dominance efficiency Second Order Stochastic Dominance Convex second-order stochastic dominance Fishburn (1974) defines a concept of convex stochastic dominance: We say that portfolio x is convex SSD inefficient if every investor prefers some of the assets to portfolio x. Formal definition: A given portfolio τ is convex SSD efficient if there exists at least some nondecreasing concave u such that Eu(r τ ) > Eu(r i ) for all i = 1, 2,..., n. M. Branda DEA in Finance / 88

79 On relations between DEA and stochastic dominance efficiency Convex SSD efficiency test Second Order Stochastic Dominance Bawa et al. (1985): Let D = {d 1, d 2,..., d (n+1)s } be the set of all scenario returns of the assets and portfolio x, that is, for every i {1, 2,..., n + 1} and s {1, 2,..., S} exists k {1, 2,..., (n + 1)S} such that r i,s = d k and vice versa, where r (n+1),s = n i=1 r i,sx i. Convex SSD efficiency test of portfolio x: δ (x) = max δ k,x i s.t. F x (2) (d k ) n i=1 (n+1)s k=1 δ k (15) x i F (2) i (d k ) δ k, k = 1, 2,..., (n + 1)S, δ k 0, k = 1, 2,..., (n + 1)S, x X. A given portfolio x is convex SSD inefficient if δ (x) given by (15) is strictly positive. Otherwise, portfolio x is convex SSD efficient. M. Branda DEA in Finance / 88

80 On relations between DEA and stochastic dominance efficiency Data Envelopment Analysis (DEA) Data Envelopment Analysis Charnes, Cooper and Rhodes (1978): a way how to state efficiency of a decision making unit over all other decision making units with the same structure of inputs and outputs. Let Z 1i,..., Z Ki denote the inputs and Y 1i,..., Y Ji denote the outputs of the unit i from n considered units. DEA efficiency of the unit 0 {1,..., n} is then evaluated using the optimal value of the following program where weighted inputs are compared with the weighted outputs. Are inputs transformed into outputs in an efficient way? M. Branda DEA in Finance / 88

81 On relations between DEA and stochastic dominance efficiency DEA Variable Returns to Scale (VRS) Data Envelopment Analysis Banker, Charnes and Cooper (1984): DEA model with Variable Returns to Scale: min θ s.t. n x i Y ji Y j0, j = 1,..., J, i=1 n x i Z ki θ Z k0, k = 1,..., K, i=1 n x i = 1, x i 0, i = 1,..., n. i=1 M. Branda DEA in Finance / 88

82 On relations between DEA and stochastic dominance efficiency DEA in finance Data Envelopment Analysis Efficiency of mutual funds or financial indexes: Murthi et al (1997): expense ratio, load, turnover, standard deviation and gross return. Basso and Funari (2001, 2003): standard deviation and semideviations, beta coefficient, costs as the inputs, expected return or expected excess return, ethical measure and stochastic dominance criterion as the outputs. Chen and Lin (2006): Value at Risk and Conditional Value at Risk. Lozano and Gutiérrez (2008): tests consistent with SSD (necessary condition). B. and Kopa (2010, 2012A): VaR, CVaR, sd, lsd, drawdown measures (DaR, CDaR) as the inputs, gross return as the output; comparison with SSD. Lamb and Tee (2012), B. (2013): DEA tests with diversification. M. Branda DEA in Finance / 88

83 On relations between DEA and stochastic dominance efficiency Data Envelopment Analysis Equivalent DEA test to convex SSD efficiency test Let D = {d 1, d 2,..., d (n+1)s } be the set of all sorted scenario returns of the assets R i and portfolio x. The test can be rewritten using lower partial moments L i (d) = 1/S S s=1 [d r i,s] +. Based on the results proposed by Bawa et al. (1985) M. Branda DEA in Finance / 88

84 On relations between DEA and stochastic dominance efficiency Data Envelopment Analysis Equivalent DEA test to convex SSD efficiency test B. and Kopa (2013): Benchmark portfolio x with return R 0 = n i=1 R ix i. Find the index k = arg min{k : L 0 (d k ) > 0}. Then the DEA-risk model with variable return to scale and K = (n + 1)S 1 inputs δ CDEA 1 K (R 0 ) = min θ k + 1 x i,ϕ,θ k K k + 2 ϕ n x i E[R i ] ϕ E[R 0 ], i=1 k= k n x i L i (d k ) θ k L 0 (d k ), k = 1,..., K, (16) i=1 0 θ k 1, ϕ 1, n x i = 1, x i 0, i = 1,..., n. i=1 is equivalent to convex SSD efficiency test. M. Branda DEA in Finance / 88

85 On relations between DEA and stochastic dominance efficiency Diversification-consistent DEA tests Input oriented Data Envelopment Analysis Recently, general DEA tests with diversification effect were introduced by Lamb and Tee (2012) for a benchmark with return R 0 : θ DC (R 0 ) = min θ θ,x i ) n CVaR εj ( R i x i CVaR + α k ( i=1 ) n R i x i i=1 CVaR εj ( R 0 ), j = 1,..., J, (17) θ CVaR + α k ( R 0 ), k = 1,..., K, n x i = 1, x i 0, i = 1,..., n, i=1 where CVaR + α = max{cvar α, 0}, and α k, ε j are different levels, the positive parts of CVaRs serve as the inputs and expected return as the output. M. Branda DEA in Finance / 88

86 On relations between DEA and stochastic dominance efficiency Diversification-consistent test Input-output oriented I Data Envelopment Analysis B. and Kopa (2013): Let CVaR α ( R 0 ) > 0 for α {α 1,..., α K } and CVaR ε ( R 0 ) < 0 for ε {ε 1,..., ε J } [ θ DC I O I 1 (R 0 ) = min θ + 1 ] θ,ϕ,x i K + J ϕ ) n CVaR εj ( R i x i ϕ ( CVaR εj ( R 0 )), j = 1,..., J, (18) CVaR αk ( i=1 ) n R i x i i=1 θ CVaR αk ( R 0 ), k = 1,..., K, 0 θ 1, ϕ 1, n x i = 1, x i 0, i = 1,..., n. i=1 Note that CVaR 0 ( R 0 ) = E[ R 0 ], i.e. expected loss can be also included into this model without any changes in its formulation. M. Branda DEA in Finance / 88

87 On relations between DEA and stochastic dominance efficiency Diversification-consistent test Input-output oriented II Data Envelopment Analysis B. and Kopa (2013): θ DC I O II 1 K J (R 0 ) = min 1 θ k + θ k,ϕ j,x i K + J ϕ j k=1 j=1 ) n CVaR εj ( R i x i ϕ j ( CVaR εj ( R 0 )), j = 1,..., J,(19) CVaR αk ( i=1 ) n R i x i i=1 θ k CVaR αk ( R 0 ), k = 1,..., K, 0 θ k 1, ϕ j 1, n x i = 1, x i 0, i = 1,..., n. i=1 These DEA-risk models can be seen as the extension of Russel measure DEA model (see Cook and Seiford (2009)). M. Branda DEA in Finance / 88

88 On relations between DEA and stochastic dominance efficiency Data Envelopment Analysis Equivalent DEA test to portfolio SSD efficiency test B. and Kopa (2013): We assume that no CVaR αs of the benchmark is equal to zero for α s = s/s, s Γ = {0, 1,..., S 1} Let s = arg max{s Γ : CVaR αs ( R 0 ) < 0}, ε j {0/S, 1/S,..., s/s}, J = s + 1 α k {( s + 1)/S,..., (S 1)/S}, K = S s 1. Then the corresponding diversification-consistent DEA-risk model (I-O II) is equivalent to SSD portfolio efficiency test, that is, a benchmark R 0 = n i=1 R ix i is DEA-risk efficient if and only if portfolio x is SSD portfolio efficient. Proof based on Kopa and Chovanec (2008), Ogryczak and Ruszczynski (2002). M. Branda DEA in Finance / 88

89 On relations between DEA and stochastic dominance efficiency Numerical comparison Numerical comparison B. and Kopa (2012B,2013): To compare the power of considered efficiency tests, we consider historical US stock market data, 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. five DEA-risk models where CVaRs at levels α = 0.5, 0.75, 0.9, 0.95, 0.99, are used as the inputs and the expected return (the most commonly used reward measure) as the output. M. Branda DEA in Finance / 88

90 On relations between DEA and stochastic dominance efficiency Numerical comparison Traditional Div.-consistent SSD efficiency DEA tests DEA tests VRS CRS I I-O I I-O II convex portfolio Food yes no Beer yes no Smoke yes yes Util yes no M. Branda DEA in Finance / 88

91 On relations between DEA and stochastic dominance efficiency Conclusions Numerical comparison Standard DEA tests can be equivalent to convex stochastic dominance tests. DEA tests equivalent to portfolio stochastic dominance efficiency should take into account diversification effect leading diversification-consistent DEA tests. M. Branda DEA in Finance / 88

92 Contents References 1 Efficiency of investment opportunities 2 Data Envelopment Analysis 3 Diversification-consistent DEA based on general deviation measures General deviation measures Diversification-consistent DEA models Financial indices efficiency empirical study 4 On relations between DEA and stochastic dominance efficiency Second Order Stochastic Dominance Data Envelopment Analysis Numerical comparison 5 References M. Branda DEA in Finance / 88

93 References Artzner, P., Delbaen, F., Eber, J.-M., Heath, D. (1999). Coherent measures of risk. Mathematical Finance 9, Banker, R.D., Charnes, A., Cooper, W. (1984). Some models for estimating technical and scale inefficiencies in Data Envelopment Analysis. Man Sci 30 (9), Basso, A., Funari, S. (2001). A data envelopment analysis approach to measure the mutual fund performance. European Journal of Operations Research 135, No. 3, Basso, A., Funari, S., (2003). Measuring the performance of ethical mutual funds: A DEA approach. Journal of the Operational Research Society 54, Bawa, V.S., Bodurtha, J.N., Rao, M.R., Suri, H.L. (1985). On Determination of Stochastic Dominance Optimal Sets. Journal of Finance 40, Branda, M. (2011). Third-degree stochastic dominance and DEA efficiency - relations and numerical comparison. Proceedings of the 29th International Conference on Mathematical Methods in Economics 2011, M. Dlouhý, V. Skočdopolová eds., University of Economics in Prague, Jánská Dolina, Slovakia, Branda, M. (2012). Stochastic programming problems with generalized integrated chance constraints. Optimization 61 (8), Branda, M. (2013A). Diversification-consistent data envelopment analysis with general deviation measures. European Journal of Operational Research 226 (3), Branda, M. (2013B). Reformulations of input-output oriented DEA tests with diversification. Operations Research Letters 41 (5), Branda, M. (2013C). Sample approximation technique for mixed-integer stochastic programming problems with expected value constraints. Accepted to Optimization Letters, DOI: /s Branda, M., Kopa, M. (2010). DEA-risk efficiency of stock indices. Proceedings of 47th EWGFM meeting, T. Tichý and M. Kopa eds., Ostrava: VŠB - Technical University of Ostrava. Branda, M., Kopa, M. (2012). DEA-Risk Efficiency and Stochastic Dominance Efficiency of Stock Indices. Czech Journal of Economics and Finance 62 (2), Branda, M., Kopa, M. (2012B). From stochastic dominance to DEA-risk models: portfolio efficiency analysis. In proceeding of international workshop on Stochastic Programming for Implementation and Advanced Applications, L. Sakalauskas, A. Tomasgard, S.W. Wallace (Eds.), Vilnius, Branda, M., Kopa, M. (2014). On relations between DEA-risk models and stochastic dominance efficiency tests. Central European Journal of Operations Research 22 (1), M. Branda DEA in Finance / 88

94 References Briec, W., Kerstens, K., Lesourd, J.-B. (2004). Single period Markowitz portfolio selection, performance gauging and duality: a variation on the Luenberger shortage function. Journal of Optimization Theory and Applications 120 (1), Briec, W., Kerstens, K., Jokung, O. (2007). Mean variance skewness portfolio performance gauging: A general shortage function and dual approach. Management Science 53, Chekhlov, A., Uryasev, S., Zabarankin. M. (2003). Portfolio Optimization With Drawdown Constraints, B. Scherer (Ed.) Asset and Liability Management Tools, Risk Books, London. Chen, Z., Lin, R. (2006). Mutual fund performance evaluation using data envelopment analysis with new risk measures. OR Spectrum 28, Charnes, A., Cooper, W. (1962). Programming with Linear Fractional Functionals. Naval Research Logistics Quarterly 9, Charnes, A., Cooper, W., Rhodes, E. (1978). Measuring the efficiency of decision-making units, European Journal of Operational Research 2, Cook, W.D., Seiford, L.M. (2009). Data envelopment analysis (DEA) Thirty years on. European Journal of Operations Research 192, Cooper, W.W., Seiford, L.M., Tone, K.: Data Envelopment Analysis, Springer Fishburn, P.C. (1974). Convex stochastic dominance with continuous distribution functions. Journal of Economic Theory 7, Follmer, H., Schied, A.: Stochastic Finance: An Introduction In Discrete Time. Walter de Gruyter, Berlin, Hadar, J., Russell W.R. (1969). Rules for ordering uncertain prospects. American Economic Review 9, Hanoch, G., Levy, H. (1969). The Efficient Analysis of Choices Involving Risk. Review of Economic Studies, Joro, T., Na, P. (2006). Portfolio performance evaluation in a mean variance skewness framework. European Journal of Operational Research 175, Kerstens, K., Mounir, A., Van de Woestyne, I. (2011). Geometric representation of the mean variance skewness portfolio frontier based upon the shortage function. European Journal of Operational Research 210, Kopa, M. (2011). Comparison of various approaches to portfolio efficiency. Proceedings of the 29th International Conference on Mathematical Methods in Economics 2011, M. Dlouhý, V. Skočdopolová eds., University of Economics in Prague, Jánská Dolina, Slovakia, Kopa, M., Chovanec, P. (2008). A Second-order Stochastic Dominance Portfolio Efficiency Measure, Kybernetika 44(2), M. Branda DEA in Finance / 88