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

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 2014 1 / 36

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 2014 2 / 36

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 2014 3 / 36

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 2014 4 / 36

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 2014 5 / 36

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 2014 6 / 36

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 2014 7 / 36

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 2014 8 / 36

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 2014 9 / 36

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 2014 10 / 36

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 2014 11 / 36

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 2014 12 / 36

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 2014 13 / 36

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 2014 14 / 36

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 2014 15 / 36

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 2014 16 / 36

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 2014 17 / 36

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 2014 18 / 36

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 2014 19 / 36

D-C DEA Traditional DEA and diversification frontier M. Branda (Charles University) DEA in Finance and Energy 2014 20 / 36

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 2014 21 / 36

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 2014 22 / 36

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 2014 23 / 36

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 2014 24 / 36

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 2014 25 / 36

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 2014 26 / 36

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 2014 27 / 36

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 2014 28 / 36

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 2014 29 / 36

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 2014 30 / 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 2014 31 / 36

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 2014 32 / 36

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 2014 33 / 36

Representative portfolio efficiency an empirical study Efficient industry representative portfolios and scores Food Smoke Hshld 1 Drugs Mines Coal Meals VRS 1.00 1.00 1.00 1.00 1.00 1.00 1.00 DC Inp 0.93 0.87 0.87 0.91 0.83 1.00 0.86 DC I-O 0.65 0.87 0.55 0.27 0.83 1.00 0.84 1 Consumer Goods M. Branda (Charles University) DEA in Finance and Energy 2014 34 / 36

Representative portfolio efficiency an empirical study Ranking of the industry representative portfolios Agric Food Soda Beer Smoke Toys Fun Hshld Clths VRS 18 1 17 8 1 30 42 1 13 DC Inp 19 2 21 8 4 32 42 4 13 DC I-O 13 8 11 14 2 37 27 15 7 MedEq Drugs Chems Rubbr Txtls BldMt Cnstr Steel FabPr VRS 21 1 22 36 46 39 38 45 44 DC Inp 15 3 27 34 45 38 39 46 44 DC I-O 29 35 18 25 38 26 30 35 34 ElcEq Autos Aero Ships Guns Gold Mines Coal Oil VRS 31 40 20 10 28 14 1 1 11 DC Inp 33 41 23 18 30 11 7 1 12 DC I-O 21 41 15 9 22 5 4 1 6 Telcm PerSv BusSv Comps Chips LabEq Paper Boxes Trans VRS 24 29 25 35 41 33 23 15 16 DC Inp 19 29 24 36 40 35 22 16 14 DC I-O 40 32 39 23 45 33 28 10 17 Rtail Meals Insur RlEst Fin Other VRS 12 1 34 43 37 32 DC Inp 10 6 28 43 37 25 DC I-O 24 3 43 31 44 46 M. Branda (Charles University) DEA in Finance and Energy 2014 35 / 36

Representative portfolio efficiency an empirical study Artzner, P., Delbaen, F., Eber, J.-M., Heath, D. (1999). Coherent measures of risk. Mathematical Finance 9, 203 228. Banker, R.D., Charnes, A., Cooper, W. (1984). Some models for estimating technical and scale inefficiencies in Data Envelopment Analysis. Man Sci 30 (9), 1078 1092. Branda, M. (2013A). Diversification-consistent data envelopment analysis with general deviation measures. European Journal of Operational Research 226 (3), 626 635. Branda, M. (2013B). Reformulations of input-output oriented DEA tests with diversification. Operations Research Letters 41 (5), 516 520. Branda, M., Kopa, M. (2014). On relations between DEA-risk models and stochastic dominance efficiency tests. Central European Journal of Operations Research 22 (1), 13 35. 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), 1 27. Briec, W., Kerstens, K., Jokung, O. (2007). Mean variance skewness portfolio performance gauging: A general shortage function and dual approach. Management Science 53, 135 149. Charnes, A., Cooper, W., Rhodes, E. (1978). Measuring the efficiency of decision-making units, European Journal of Operational Research 2, 429-444. Joro, T., Na, P. (2006). Portfolio performance evaluation in a mean variance skewness framework. European Journal of Operational Research 175, 446 461. Lamb, J.D., Tee, K-H. (2012). Data envelopment analysis models of investment funds. European Journal of Operational Research 216, No. 3, 687 696. Lozano, S., Gutiérrez, E. (2008A). Data envelopment analysis of mutual funds based on second-order stochastic dominance. European Journal of Operational Research 189, 230 244. Lozano, S., Gutiérrez, E. (2008B). TSD-consistent performance assessment of mutual funds. Journal of the Operational Research Society 59, 1352 1362. Markowitz, H. M. (1952). Portfolio selection. The Journal of Finance 7, No. 1, 77-91. Murthi, B.P.S., Choi, Y.K., Desai, P. (1997). Efficiency of mutual funds and portfolio performance measurement: a non-parametric approach. European Journal of Operational Research 98, No. 2, 408 418. Ogryczak, W., Ruszczynski, A. (2001). On consistency of stochastic dominance and mean-semideviation models. Mathematical Programmming, Ser. B 89, 217 232. Rockafellar, R.T., Uryasev, S., Zabarankin M. (2006A). Generalized Deviations in Risk Analysis. Finance and Stochastics 10, 51 74. Rockafellar, R.T., Uryasev, S., Zabarankin M. (2006B). Optimality Conditions in Portfolio Analysis with General Deviation Measures. Mathematical Programming 108, No. 2-3, 515 540. M. Branda (Charles University) DEA in Finance and Energy 2014 36 / 36