On relations between stochastic dominance efficiency tests and DEA-risk models Martin Branda, Miloˇ s Kopa Charles University in Prague Faculty of Mathematics and Physics Department of Probability and Mathematical Statistics 13th International Conference on Stochastic Programming 8-12 July 2013, Bergamo M.Branda, M.Kopa (Charles University) SP 2013 1 / 32
Contents 1 Motivation 2 Second Order Stochastic Dominance 3 Data Envelopment Analysis 4 Numerical comparison M.Branda, M.Kopa (Charles University) SP 2013 2 / 32
Motivation Contents 1 Motivation 2 Second Order Stochastic Dominance 3 Data Envelopment Analysis 4 Numerical comparison M.Branda, M.Kopa (Charles University) SP 2013 3 / 32
Motivation A bridge Data envelopment analysis (production theory, returns to scale, radial/slack-based/directional distance models, primal/dual formulations, multiobjective opt. – Pareto efficiency, chance constraints), large literature: handbook 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, M.Kopa (Charles University) SP 2013 4 / 32
Second Order Stochastic Dominance Contents 1 Motivation 2 Second Order Stochastic Dominance 3 Data Envelopment Analysis 4 Numerical comparison M.Branda, M.Kopa (Charles University) SP 2013 5 / 32
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 F 1 u ( x ) − E F 2 u ( x ) ≥ 0 for every concave utility function u with at least one strict inequality. Consider twice cumulated probability distributions functions: � t F (2) ( t ) = F i ( x ) dx i = 1 , 2 . i −∞ Theorem (Hanoch & Levy (1969)): F (2) 1 ( t ) ≤ F (2) X 1 ≻ SSD X 2 ⇔ 2 ( t ) ∀ t ∈ R with at least one strict inequality. M.Branda, M.Kopa (Charles University) SP 2013 6 / 32
Second Order Stochastic Dominance SSD portfolio efficiency 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): n � X = { x ∈ R | x i = 1 , x i ≥ 0 , i = 1 , . . . , n } . (1) i =1 M.Branda, M.Kopa (Charles University) SP 2013 7 / 32
Second Order Stochastic Dominance SSD portfolio efficiency 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, M.Kopa (Charles University) SP 2013 8 / 32
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 E u ( r ′ τ ) > E u ( r i ) for all i = 1 , 2 , ..., n . M.Branda, M.Kopa (Charles University) SP 2013 9 / 32
Second Order Stochastic Dominance Convex SSD efficiency test 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 , s x i . Convex SSD efficiency test of portfolio x : ( n +1) S � δ ∗ ( x ) = max δ k (2) δ k , x i k =1 n s . t . F (2) x i F (2) � ( d k ) − ¯ ( d k ) ≥ δ k , k = 1 , 2 , ..., ( n + 1) S , x i i =1 δ k ≥ 0 , k = 1 , 2 , ..., ( n + 1) S , ¯ x ∈ X . A given portfolio x is convex SSD inefficient if δ ∗ ( x ) given by (2) is strictly positive. Otherwise, portfolio x is convex SSD efficient . M.Branda, M.Kopa (Charles University) SP 2013 10 / 32
Data Envelopment Analysis Contents 1 Motivation 2 Second Order Stochastic Dominance 3 Data Envelopment Analysis 4 Numerical comparison M.Branda, M.Kopa (Charles University) SP 2013 11 / 32
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 1 i , . . . , Z Ki denote the inputs and Y 1 i , . . . , 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, M.Kopa (Charles University) SP 2013 12 / 32
Data Envelopment Analysis Data Envelopment Analysis (DEA) Charnes et al (1978): fractional programming formulation � J j =1 y j 0 Y j 0 max � K k =1 w k 0 Z k 0 s . t . � J j =1 y j 0 Y ji ≤ 1 , i = 1 , . . . , n , � K k =1 w k 0 Z ki ≥ 0 , k = 1 , . . . , K , w k 0 y j 0 ≥ 0 , j = 1 , . . . , J . The unit 0 is then DEA efficient if the optimal value is equal to 1, otherwise it is DEA inefficient . M.Branda, M.Kopa (Charles University) SP 2013 13 / 32
Data Envelopment Analysis DEA Constant Return to Scale (CRS) Dual problem to linear programming formulation: min θ s . t . n � ≥ Y j 0 , j = 1 , . . . , J , x i Y ji i =1 n � x i Z ki ≤ θ · Z k 0 , k = 1 , . . . , K , i =1 x i ≥ 0 , i = 1 , . . . , n . Model with Constant Return to Scale (CRS). M.Branda, M.Kopa (Charles University) SP 2013 14 / 32
Data Envelopment Analysis DEA Variable Return to Scale (VRS) Banker, Charnes and Cooper (1984): DEA model with Variable Return to Scale: min θ s . t . n � x i Y ji ≥ Y j 0 , j = 1 , . . . , J , i =1 n � ≤ θ · Z k 0 , k = 1 , . . . , K , x i Z ki i =1 n � x i = 1 , x i ≥ 0 , i = 1 , . . . , n . i =1 M.Branda, M.Kopa (Charles University) SP 2013 15 / 32
Data Envelopment Analysis 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 and Conditional Value at Risk. Lozano and Guti´ errez (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, M.Kopa (Charles University) SP 2013 16 / 32
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, M.Kopa (Charles University) SP 2013 17 / 32
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 i x 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 K 1 θ k + 1 δ CDEA ( R 0 ) = min � K − ˜ ϕ k + 2 ¯ x i ,ϕ,θ k k =˜ k n � x i E [ R i ] ¯ ≥ ϕ · E [ R 0 ] , i =1 n � ¯ x i L i ( d k ) ≤ θ k · L 0 ( d k ) , k = 1 , . . . , K , (3) i =1 0 ≤ θ k ≤ 1 , ϕ ≥ 1 , n � ¯ = 1 , ¯ x i ≥ 0 , i = 1 , . . . , n . x i i =1 is equivalent to convex SSD efficiency test. M.Branda, M.Kopa (Charles University) SP 2013 18 / 32
Data Envelopment Analysis Diversification-consistent DEA tests Input oriented 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, x i θ n � � � − CVaR ε j − ≥ − CVaR ε j ( − R 0 ) , j = 1 , . . . , J , (4) R i x i i =1 � n � � CVaR + θ · CVaR + − R i x i ≤ α k ( − R 0 ) , k = 1 , . . . , K , α k i =1 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, M.Kopa (Charles University) SP 2013 19 / 32
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