Data Envelopment Analysis in Finance and Energy New Approaches to Efficiency and their Numerical Tractability

Transcription

1 Data Envelopment Analysis in Finance and Energy New Approaches to Efficiency and their Numerical Tractability Martin Branda Faculty of Mathematics and Physics Charles University in Prague EURO Working Group on Commodity and Financial Modelling May 22 24, 2014, Chania, Greece M. Branda (Charles University) DEA in Finance and Energy / 36

2 Contents 1 Efficiency of investment opportunities 2 Data Envelopment Analysis 3 DEA with diversification 4 Representative portfolio efficiency an empirical study M. Branda (Charles University) DEA in Finance and Energy / 36

3 Contents Efficiency of investment opportunities 1 Efficiency of investment opportunities 2 Data Envelopment Analysis 3 DEA with diversification 4 Representative portfolio efficiency an empirical study M. Branda (Charles University) DEA in Finance and Energy / 36

4 Efficiency of investment opportunities Efficiency of investment opportunities Various approaches how to test efficiency of an investment opportunity with a random outcome (profit, loss, etc.): 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 (DEA) in finance M. Branda (Charles University) DEA in Finance and Energy / 36

5 Efficiency of investment opportunities Efficiency of investment opportunities Our approach combines DEA efficiency Murthi et al. (1997), Briec et al. (2004), Lamb and Tee (2012), Branda (2013A, 2013B) Extension of mean-risk efficiency based on multiobjective optimization principles Markowitz (1952) Risk shaping several risk measures included into one model Rockafellar and Uryasev (2002) M. Branda (Charles University) DEA in Finance and Energy / 36

6 Contents Data Envelopment Analysis 1 Efficiency of investment opportunities 2 Data Envelopment Analysis 3 DEA with diversification 4 Representative portfolio efficiency an empirical study M. Branda (Charles University) DEA in Finance and Energy / 36

7 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 usually assumed to be positive. M. Branda (Charles University) DEA in Finance and Energy / 36

8 Data Envelopment Analysis DEA with Variable Return to Scale (VRS) Banker, Charnes and Cooper (1984): DEA model with Variable Return 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 (Charles University) DEA in Finance and Energy / 36

9 Data Envelopment Analysis DEA with Variable Return to Scale (VRS) Dual formulation of VRS DEA (more useful): min x i,θ s.t. x i Y ji Y j0, j = 1,..., J, x i Z ki θ Z k0, k = 1,..., K, x i = 1, x i 0, i = 1,..., n. M. Branda (Charles University) DEA in Finance and Energy / 36

10 Data Envelopment Analysis Data envelopment analysis DEA traditional strong wide area (many applications and theory, Handbooks, papers in highly impacted journals, e.g. Omega, EJOR, JOTA, JORS, EE, JoBF) 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 (Charles University) DEA in Finance and Energy / 36

11 Contents D-C DEA 1 Efficiency of investment opportunities 2 Data Envelopment Analysis 3 DEA with diversification 4 Representative portfolio efficiency an empirical study M. Branda (Charles University) DEA in Finance and Energy / 36

12 Inputs and outputs D-C DEA Efficiency of investment opportunities with random outcomes R 1,..., R n not directly used as inputs or outputs, in general Inputs: characteristics with lower values preferred to higher values, Outputs: characteristics with higher values preferred to lower values. M. Branda (Charles University) DEA in Finance and Energy / 36

13 D-C DEA Set of investment opportunities 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 opportunity): X P = {R i, i = 1,..., n}, 2 full diversification (diversification across all opportunities): { } X FD = R i x i : x i = 1, x i 0, 3 Short sales and margin requirements, limited diversification, etc. M. Branda (Charles University) DEA in Finance and Energy / 36

14 D-C DEA Set of investment opportunities 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 opportunity): X P = {R i, i = 1,..., n}, 2 full diversification (diversification across all opportunities): { } X FD = R i x i : x i = 1, x i 0, 3 Short sales and margin requirements, limited diversification, etc. M. Branda (Charles University) DEA in Finance and Energy / 36

15 D-C DEA Set of investment opportunities 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 opportunity): X P = {R i, i = 1,..., n}, 2 full diversification (diversification across all opportunities): { } X FD = R i x i : x i = 1, x i 0, 3 Short sales and margin requirements, limited diversification, etc. M. Branda (Charles University) DEA in Finance and Energy / 36

16 D-C DEA 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 (Charles University) DEA in Finance and Energy / 36

17 D-C DEA 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 (Charles University) DEA in Finance and Energy / 36

18 Deviation measures D-C DEA 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 (Charles University) DEA in Finance and Energy / 36

19 D-C DEA 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 (Charles University) DEA in Finance and Energy / 36

20 D-C DEA Coherent risk and return 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. Moreover, risk measures multiplied by a negative constant can be used as return functionals, i.e. E(X ) = R(X ). M. Branda (Charles University) DEA in Finance and Energy / 36

21 D-C DEA Coherent risk and return 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. Moreover, risk measures multiplied by a negative constant can be used as return functionals, i.e. E(X ) = R(X ). M. Branda (Charles University) DEA in Finance and Energy / 36

22 Traditional DEA model Input oriented (VRS) D-C DEA 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 T (X 0 ) = min θ s.t. x i E j (R i ) E j (X 0 ), j = 1,..., J, (3) x i D k (R i ) θ D k (X 0 ), k = 1,..., K, x i = 1, x i 0, i = 1,..., n. M. Branda (Charles University) DEA in Finance and Energy / 36

23 D-C DEA Traditional DEA vs. diversification consistent tests The model does not take into account portfolio diversification: For any general deviation measure D k it holds ( ) x i D k (R i ) D k x i R i for nonnegative weights with n x i = 1. Linear transformation of inputs is only an upper bound for the real portfolio deviation. M. Branda (Charles University) DEA in Finance and Energy / 36

24 D-C DEA Traditional DEA vs. diversification consistent tests The model does not take into account portfolio diversification: For any general deviation measure D k it holds ( ) x i D k (R i ) D k x i R i for nonnegative weights with n x i = 1. Linear transformation of inputs is only an upper bound for the real portfolio deviation. M. Branda (Charles University) DEA in Finance and Energy / 36

25 D-C DEA Traditional DEA and diversification frontier M. Branda (Charles University) DEA in Finance and Energy / 36

26 D-C DEA DEA tests with diversification 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). Lamb and Tee (2012), Branda (2013A, 2013B): general classes of DEA tests with risk/deviation and return measures. Branda and Kopa (2014): equivalence with second-order stochastic dominance. M. Branda (Charles University) DEA in Finance and Energy / 36

27 DEA efficiency D-C DEA We assume that the benchmark 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. Sets of efficient opportunities Ψ I /I O = {X X : θ I /I O (X ) = 1}, where θ I /I O (X 0 ) is the optimal value for benchmark X 0. M. Branda (Charles University) DEA in Finance and Energy / 36

28 D-C DEA Input oriented tests with diversification For a benchmark X 0 X, the input oriented diversification consistent DEA test: θ I (X 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 (Charles University) DEA in Finance and Energy / 36

29 D-C DEA Input-output oriented tests 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 (Charles University) DEA in Finance and Energy / 36

30 D-C DEA Input-output oriented tests 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 θ I O θ (X 0 ) = min θ,ϕ,x ϕ s.t. E j (X ) ϕ E j (X 0 ), j = 1,..., J, (5) D k (X ) θ D k (X 0 ), k = 1,..., K, 0 θ 1, ϕ 1, X X. M. Branda (Charles University) DEA in Finance and Energy / 36

31 D-C DEA Input-output oriented tests Setting 1/t = ϕ, results into an input oriented DEA test with nonincreasing return to scale (NIRS): θ I O (R 0 ) = min θ θ, x i ( ) s.t. E j R i x i ( ) D k R i x i E j (R 0 ), j = 1,..., J, θ D k (R 0 ), k = 1,..., K, x i 1, x i 0, 1 θ 0. 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 (Charles University) DEA in Finance and Energy / 36

32 Properties and relations D-C DEA Proposition The considered DEA models are unit 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 (Charles University) DEA in Finance and Energy / 36

33 Properties and relations D-C DEA 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 T (X 0 ) θ I (X 0 ) θ I O (X 0 ). Then, for the sets of efficient portfolios can be obtained Ψ I O Ψ I Ψ T. M. Branda (Charles University) DEA in Finance and Energy / 36

34 Properties and relations D-C DEA Proposition The optimal solution of the test is efficient with respect to the test. M. Branda (Charles University) DEA in Finance and Energy / 36

35 Representative portfolio efficiency an empirical study Contents 1 Efficiency of investment opportunities 2 Data Envelopment Analysis 3 DEA with diversification 4 Representative portfolio efficiency an empirical study M. Branda (Charles University) DEA in Finance and Energy / 36

36 Representative portfolio efficiency an empirical study Numerical comparison To compare the efficiency tests, we consider historical US stock market data, monthly excess returns from January 2002 to December 2011 (120 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. Portfolio composed from the representative portfolios = interdisciplinary portfolio. M. Branda (Charles University) DEA in Finance and Energy / 36

37 Representative portfolio efficiency an empirical study DC DEA test with CVaR deviations Input oriented For discretely distributed returns (r is, s = 1,..., S, p s = 1/S) LP: θ I (R 0 ) = min θ,x i,u sk,ξ k θ 1 S s.t. E[R i ]x i E[R 0 ], S u sk θ D αk (R 0 ), k = 1,..., K, s=1 u sk u sk x i r is ξ k, s = 1,..., S, k = 1,..., K, α k 1 α k ( ξ k ) x i r is, x i = 1, x i 0, i = 1,..., n. M. Branda (Charles University) DEA in Finance and Energy / 36

38 Representative portfolio efficiency an empirical study DC DEA test with CVaR deviations Input-output oriented For discretely distributed returns (r is, s = 1,..., S, p s = 1/S) LP: θ I O (R 0 ) = min θ,x i,u sk,ξ k θ 1 S s.t. E[R i ]x i E[R 0 ], S u sk θ D αk (R 0 ), k = 1,..., K, s=1 u sk u sk x i r is ξ k, s = 1,..., S, k = 1,..., K, α k 1 α k ( ξ k ) x i r is, x i 1, x i 0, i = 1,..., n. M. Branda (Charles University) DEA in Finance and Energy / 36

39 Representative portfolio efficiency an empirical study Efficient industry representative portfolios and scores Food Smoke Hshld 1 Drugs Mines Coal Meals VRS DC Inp DC I-O Consumer Goods M. Branda (Charles University) DEA in Finance and Energy / 36

40 Representative portfolio efficiency an empirical study Ranking of the industry representative portfolios Agric Food Soda Beer Smoke Toys Fun Hshld Clths VRS DC Inp DC I-O MedEq Drugs Chems Rubbr Txtls BldMt Cnstr Steel FabPr VRS DC Inp DC I-O ElcEq Autos Aero Ships Guns Gold Mines Coal Oil VRS DC Inp DC I-O Telcm PerSv BusSv Comps Chips LabEq Paper Boxes Trans VRS DC Inp DC I-O Rtail Meals Insur RlEst Fin Other VRS DC Inp DC I-O M. Branda (Charles University) DEA in Finance and Energy / 36

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