Data Envelopment Analysis in Finance and Energy New Approaches to - - PowerPoint PPT Presentation

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

Efficiency of investment opportunities

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 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

• ptimization 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

Data Envelopment Analysis

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 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 Z1i, . . . , ZKi denote the inputs and Y1i, . . . , YJi 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

• utputs.

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

yj0,wk0

J

j=1 yj0Yj0 − y0

K

k=1 wk0Zk0

s.t. J

j=1 yj0Yji − y0

K

k=1 wk0Zki

≤ 1, i = 1, . . . , n, wk0 ≥ 0, k = 1, . . . , K, yj0 ≥ 0, j = 1, . . . , J, y0 ∈ 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

xi,θ θ

s.t.

n

• i=1

xiYji ≥ Yj0, j = 1, . . . , J,

n

• i=1

xiZki ≤ θ · Zk0, k = 1, . . . , K,

n

• i=1

xi = 1, xi ≥ 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

D-C DEA

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 11 / 36

D-C DEA

Inputs and outputs

Efficiency of investment opportunities with random outcomes R1, . . . , Rn – 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 Ri ∈ L2(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 = {Ri, i = 1, . . . , n},

2 full diversification (diversification across all opportunities):

X FD = n

• i=1

Rixi :

n

• i=1

xi = 1, xi ≥ 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

Set of investment opportunities

We consider n assets and denote Ri ∈ L2(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 = {Ri, i = 1, . . . , n},

2 full diversification (diversification across all opportunities):

X FD = n

• i=1

Rixi :

n

• i=1

xi = 1, xi ≥ 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

Set of investment opportunities

We consider n assets and denote Ri ∈ L2(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 = {Ri, i = 1, . . . , n},

2 full diversification (diversification across all opportunities):

X FD = n

• i=1

Rixi :

n

• i=1

xi = 1, xi ≥ 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 : L2(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

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 : L2(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

D-C DEA

Deviation measures

Standard deviation D(X) = σ(X) =

• E X − E[X]2

Mean absolute deviation D(X) = E

• |X − E[X]|
• .

Mean absolute lower and upper semideviation D−(X) = E

• |X − E[X]|−
• , D+(X) = E
• |X − E[X]|+
• .

Worst-case deviation D(X) = sup

ω∈Ω

|X(ω) − E[X]|. 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 Dα(X) = CVaRα(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 Dα(X) = min

ξ∈R

1 1 − αE[max{(1 − α)(X − ξ), α(ξ − X)}] (2) 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 : L2(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

D-C DEA

Coherent risk and return measures

CRM: R : L2(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

D-C DEA

Traditional DEA model

Input oriented (VRS)

We assume that X0 is not constant, i.e. Dk(X0) > 0, for all k = 1, . . . , K. Input oriented VRS model can be formulated in the dual form θT(X0) = min θ s.t.

n

• i=1

xi · Ej(Ri) ≥ Ej(X0), j = 1, . . . , J, (3)

n

• i=1

xi · Dk(Ri) ≤ θ · Dk(X0), k = 1, . . . , K,

n

• i=1

xi = 1, xi ≥ 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 Dk it holds

n

• i=1

xi · Dk(Ri) ≥ Dk n

• i=1

xiRi

• for nonnegative weights with n

i=1 xi = 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 vs. diversification consistent tests

The model does not take into account portfolio diversification: For any general deviation measure Dk it holds

n

• i=1

xi · Dk(Ri) ≥ Dk n

• i=1

xiRi

• for nonnegative weights with n

i=1 xi = 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 Guti´ errez (2008A, 2008B): tests consistent with second- and 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

D-C DEA

DEA efficiency

We assume that the benchmark X0 ∈ X is not constant, i.e. Dk(X0) > 0, for all k = 1, . . . , K. Definition We say that X0 ∈ X is DEA efficient with respect to the set X if the

• ptimal value of the DEA program is equal to 1. Otherwise, X0 is

inefficient and the optimal value measures the inefficiency. Sets of efficient opportunities ΨI/I−O = {X ∈ X : θI/I−O(X) = 1}, where θI/I−O(X0) is the optimal value for benchmark X0.

• M. Branda (Charles University)

DEA in Finance and Energy 2014 22 / 36

D-C DEA

Input oriented tests with diversification

For a benchmark X0 ∈ X, the input oriented diversification consistent DEA test: θI(X0) = min θ s.t. Ej(X) ≥ Ej(X0), j = 1, . . . , J, (4) Dk(X) ≤ θ · Dk(X0), 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 Ej(X0) > 0):

• ptimal 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 Ej(X0) is positive for at least one j. An input-output

• riented test where inefficiency is measured with respect to the inputs and
• utputs separately can be formulated as follows

θI−O(X0) = min

θ,ϕ,X

θ ϕ s.t. Ej(X) ≥ ϕ · Ej(X0), j = 1, . . . , J, (5) Dk(X) ≤ θ · Dk(X0), k = 1, . . . , K, ≤ 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): θI−O(R0) = min

˜ θ,˜ xi

˜ θ s.t. Ej n

• i=1

Ri˜ xi

• ≥

Ej(R0), j = 1, . . . , J, Dk n

• i=1

Ri˜ xi

• ≤

˜ θ · Dk(R0), k = 1, . . . , K,

n

• i=1

˜ xi ≤ 1, ˜ xi ≥ 0, 1 ≥ ˜ θ ≥ 0. Note that it is important for the reformulation that all inputs Dk and all

• utputs Ej are positively homogeneous. We obtained a convex

programming problem.

• M. Branda (Charles University)

DEA in Finance and Energy 2014 26 / 36

D-C DEA

Properties and relations

Proposition The considered DEA models are unit invariant. For arbitrary k and j λDk(X) = Dk(λX) which implies Ej(λX) = λEj(λX) for arbitrary X ∈ X and λ > 0.

• M. Branda (Charles University)

DEA in Finance and Energy 2014 27 / 36

D-C DEA

Properties and relations

Proposition Let max{J, K} ≥ 2. Then for a benchmark X0 ∈ X with Dk(X0) > 0 for all k and Ej(X0) > 0 for all j, the following relations hold θT(X0) ≥ θI(X0) ≥ θI−O(X0). Then, for the sets of efficient portfolios can be obtained ΨI−O ⊆ ΨI ⊆ ΨT.

• M. Branda (Charles University)

DEA in Finance and Energy 2014 28 / 36

D-C DEA

Properties and relations

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 (ris, s = 1, . . . , S, ps = 1/S) LP: θI(R0) = min

θ,xi,usk,ξk

θ s.t.

n

• i=1

E[Ri]xi ≥ E[R0], 1 S

S

• s=1

usk ≤ θ · Dαk(R0), k = 1, . . . , K, usk ≥

n

• i=1

xiris − ξk, s = 1, . . . , S, k = 1, . . . , K, usk ≥ αk 1 − αk

• ξk −

n

• i=1

xiris

• ,

n

• i=1

xi = 1, xi ≥ 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 (ris, s = 1, . . . , S, ps = 1/S) LP: θI−O(R0) = min

θ,xi,usk,ξk

θ s.t.

n

• i=1

E[Ri]xi ≥ E[R0], 1 S

S

• s=1

usk ≤ θ · Dαk(R0), k = 1, . . . , K, usk ≥

n

• i=1

xiris − ξk, s = 1, . . . , S, k = 1, . . . , K, usk ≥ αk 1 − αk

• ξk −

n

• i=1

xiris

• ,

n

• i=1

xi ≤ 1, xi ≥ 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 Hshld1 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

1Consumer 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

• f 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., Guti´ errez, E. (2008A). Data envelopment analysis of mutual funds based on second-order stochastic

• dominance. European Journal of Operational Research 189, 230–244.

Lozano, S., Guti´ errez, 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