[PPT] - Data Envelopment Analysis in Finance Martin Branda Faculty of PowerPoint Presentation

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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

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DEA in finance

We do not access efficiency of financial institutions (banks, insurance comp.). We access efficiency of investment opportunities1 on financial markets.

1Assets, portfolios, mutual funds, financial indices, ...

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Motivation

Together with Miloˇ s 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

ptimization

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 2014 4 / 88

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Motivation

Together with Miloˇ s 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

ptimization

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 2014 4 / 88

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Motivation

Together with Miloˇ s 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

ptimization

Branda, Kopa (2012): an empirical study (a bit “naive”, but necessary step for us:) Branda, Kopa (2014): equivalences (a “bridge”)

M. Branda

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M. Branda

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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

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Efficiency of investment opportunities

DEA in finance

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

ptimization 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)

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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 assumed to be (semi-)positive. Charnes et al (1978): fractional programming formulation (Constant Returns to Scale – CRS or CCR)

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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

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.

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Data Envelopment Analysis

DEA

Variable Returns to Scale (VRS)

Dual formulation of VRS DEA: 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.

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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 ...

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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.

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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

r 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 Guti´ errez (2008B)

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Data Envelopment Analysis

General class of financial DEA tests

Lamb and Tee (2012) – pure return-risk tests2: Inputs: positive parts of coherent risk measures Outputs: return measures (= minus coherent risk measures, e.g. expected return)

2no transactions costs etc.

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DC DEA models based on GDM

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

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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 : 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

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DC DEA models based on GDM General deviation measures

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.

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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 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).

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DC DEA models based on GDM General deviation measures

General deviation measures

According to Proposition 4 in Rockafellar et al (2006 A): if D = λD0 for λ > 0 and a deviation measure D0, then D is a deviation measure. if D1, . . . , DK are deviation measures, then

D = max{D1, . . . , DK} is also deviation measure. D = λ1D1 + · · · + λKDK is also deviation measure, if λk > 0 and K

k=1 λk = 1.

Rockafellar et al (2006 A, B): Duality representation using risk envelopes and risk identifiers, subdifferentiability and optimality conditions.

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DC DEA models based on GDM General deviation measures

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 . 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).

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DC DEA models based on GDM General deviation measures

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 . 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).

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DC DEA models based on GDM General deviation measures

Coherent risk and return 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).

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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 L2(Ω) having the form {X : D(X) ≤ c} for c ∈ R (level sets) are closed; (D5) lower range dominated if D(X) ≤ EX − infω∈Ω X(ω) for all X.

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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 L2(Ω) having the form {X : D(X) ≤ c} for c ∈ R (level sets) are closed; (D5) lower range dominated if D(X) ≤ EX − infω∈Ω X(ω) for all X.

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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.

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DC DEA models based on GDM DC DEA models based on GDM

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 θI

0(X0) = min θ

s.t.

n

i=1

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

n

i=1

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

n

i=1

xi = 1, xi ≥ 0, i = 1, . . . , n.

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DC DEA models based on GDM DC DEA models based on GDM

The most important slide

Traditional DEA model vs. diversification

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

n

i=1

xiDk(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.

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DC DEA models based on GDM DC DEA models based on GDM

The most important slide

Traditional DEA model vs. diversification

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

n

i=1

xiDk(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.

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DC DEA models based on GDM DC DEA models based on GDM

Traditional DEA and diversification frontier

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DC DEA models based on GDM DC DEA models based on GDM

Models with diversification

Efficiency of mutual funds or financial indexes using DEA related models with diversification: Kuosmannen (2007): Second order stochastic dominance consistent DEA models. Lozano and Guti´ errez (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).

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DC DEA models based on GDM DC DEA models based on GDM

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). 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.

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DC DEA models based on GDM DC DEA models based on GDM

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 asset):

X P = {Ri, i = 1, . . . , n},

2 full diversification (diversification across all assets):

X FD = n

i=1

Rixi :

n

i=1

xi = 1, xi ≥ 0

,

3 limited diversification (diversification across limited number of

assets #): X LD = n

i=1

Rixi :

n

i=1

xi = 1, xi ≥ 0, xi ≤ yi, yi ∈ {0, 1},

n

i=1

yi ≤ #

.
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DC DEA models based on GDM DC DEA models based on GDM

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 asset):

X P = {Ri, i = 1, . . . , n},

2 full diversification (diversification across all assets):

X FD = n

i=1

Rixi :

n

i=1

xi = 1, xi ≥ 0

,

3 limited diversification (diversification across limited number of

assets #): X LD = n

i=1

Rixi :

n

i=1

xi = 1, xi ≥ 0, xi ≤ yi, yi ∈ {0, 1},

n

i=1

yi ≤ #

.
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DC DEA models based on GDM DC DEA models based on GDM

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 asset):

X P = {Ri, i = 1, . . . , n},

2 full diversification (diversification across all assets):

X FD = n

i=1

Rixi :

n

i=1

xi = 1, xi ≥ 0

,

3 limited diversification (diversification across limited number of

assets #): X LD = n

i=1

Rixi :

n

i=1

xi = 1, xi ≥ 0, xi ≤ yi, yi ∈ {0, 1},

n

i=1

yi ≤ #

.
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DC DEA models based on GDM DC DEA models based on GDM

Partial ordering of vectors

Definition Let v, z ∈ Rn. We say that z strictly dominates (weakly) v, denoted v ≺st z if ∀i : vi < zi, v ≺w z if ∀i : vi ≤ zi and ∃˜ i : v˜

i < z˜ i.

Definition Let v, z ∈ Rn. 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 vi < zi for all i ∈ S and vi ≤ zi for all i ∈ {1, . . . , n} \ S, v ≺pw(S) z if vi ≤ zi for all i ∈ {1, . . . , n} and there exists at least

ne ˜

i ∈ S for which v˜

i < z˜ i.

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DEA efficiency

We assume that 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. For m ∈ {0, 1, 2, 3} we denote the sets of efficient opportunities ΨI

m = {X ∈ X : θI m(X) = 1},

where θI

m(X0) is the optimal value for benchmark X0.

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DC DEA models based on GDM DC DEA models based on GDM

Diversification consistent DEA - model 1

For a benchmark X0 ∈ X diversification consistent DEA model θI

1(X0) = min θ

s.t. Ej(X) ≥ Ej(X0), j = 1, . . . , J, (4) Dk(X) ≤ θ · Dk(X0), k = 1, . . . , K, X ∈ X.

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DC DEA models based on GDM DC DEA models based on GDM

Set of efficient opportunities

Proposition X0 ∈ ΨI

1

if for all X ∈ X for which Ej(X) ≥ Ej(X0) for all j holds that Dk(X) ≥ Dk(X0) for at least one k, i.e. there is no X ∈ X for which Ej(X) ≥ Ej(X0) for all j and Dk(X) < Dk(X0) for all k. if there is no vector v from the set PPSR = {(D1(X), . . . , DK(X)) : Ej(X) ≥ Ej(X0), j = 1, . . . , J, X ∈ X} for which v ≺st (D1(X0), . . . , DK(X0)).

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Model 1 - equivalent formulation

θI

1(X0) = min

max

k=1,...,K

Dk(X) Dk(X0) s.t. (5) Ej(X) ≥ Ej(X0), j = 1, . . . , J, X ∈ X. D(X) = maxk=1,...,K Dk(X)/Dk(X0) defines a deviation measure.

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Diversification consistent DEA - model 2

For a benchmark X0 ∈ X, we can introduce a generalized model θI

2(X0) = min 1

K

k=1

θk s.t. Ej(X) ≥ Ej(X0), j = 1, . . . , J, (6) Dk(X) ≤ θk · Dk(X0), k = 1, . . . , K, ≤ θk ≤ 1, k = 1, . . . , K, X ∈ X.

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Set of efficient opportunities

Proposition X0 ∈ ΨI

2

if for all X ∈ X for which Ej(X) ≥ Ej(X0) for all j holds that Dk(X) ≥ Dk(X0) for all k, i.e. there is no X ∈ X for which Ej(X) ≥ Ej(X0) for all j and Dk(X) < Dk(X0) for at least one k. if there is no vector v from the set PPSR = {(D1(X), . . . , DK(X)) : Ej(X) ≥ Ej(X0), j = 1, . . . , J, X ∈ X} for which v ≺w (D1(X0), . . . , DK(X0)).

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DC DEA models based on GDM DC DEA models based on GDM

Model 2 - equivalent formulation

The model is obviously equivalent to θI

2(X0) = min 1

K

k=1

Dk(X) Dk(X0) s.t. Ej(X) ≥ Ej(X0), j = 1, . . . , J, (7) Dk(X) ≤ Dk(X0), k = 1, . . . , K, X ∈ X. D(X) = 1

K

k=1 Dk(X)/Dk(X0) defines a deviation measure.

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DC DEA models based on GDM DC DEA models based on GDM

Diversification consistent DEA

Slacks-based model

Additive or slacks-based DEA model θI

3(X0) = min 1

K

k=1

Dk(X0) − s−

k

Dk(X0) s.t. Ej(X) ≥ Ej(X0), j = 1, . . . , J, (8) Dk(X) + s−

k

= Dk(X0), k = 1, . . . , K, s−

k

≥ 0, k = 1, . . . , K, X ∈ X.

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DC DEA models based on GDM DC DEA models based on GDM

Units invariance

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

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Test strength

Proposition Let K ≥ 2. Then for a benchmark X0 ∈ X, the following relations between the optimal values (efficiency scores) hold θI

0(X0) ≥ θI 1(X0) ≥ θI 2(X0) = θI 3(X0).

For the sets of efficient portfolios we obtain ΨI

3 = ΨI 2 ⊆ ΨI 1 ⊆ ΨI 0.

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Test strength

We can construct weaker tests if we restrict the number of investment

pportunities in the portfolio leading to the models with the limited

diversification. Proposition For X LD

#

and X LD

#′ with # > #′, it holds θ(X0) ≤ θ′(X0), where θ(X0) and

θ′(X0) denote the efficiency scores with respect to the sets X LD

# , and X LD #′

respectively.

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DC DEA models based on GDM DC DEA models based on GDM

Input-output oriented models

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).
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DC DEA models based on GDM DC DEA models based on GDM

Input-output oriented test 1

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

1

(X0) = min

θ,ϕ,X

θ ϕ s.t. Ej(X) ≥ ϕ · Ej(X0), j = 1, . . . , J, (9) Dk(X) ≤ θ · Dk(X0), k = 1, . . . , K, ≤ θ ≤ 1, ϕ ≥ 1, X ∈ X.

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DC DEA models based on GDM DC DEA models based on GDM

Input-output oriented test 1

Let X = X FD. Setting 1/t = ϕ and substitute ˜ xi = txi, ˜ θ = tθ, and ˜ ϕ = tϕ, results into an input oriented DEA test with nonincreasing return to scale (NIRS): θI−O

1

(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.

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DEA in Finance 2014 48 / 88

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DC DEA models based on GDM DC DEA models based on GDM

Input-output oriented test 1

Proposition X0 ∈ ΨI−O

1

: if there is no X ∈ X for which either (Ej(X) ≥ Ej(X0) for all j and Dk(X) < Dk(X0) for all k) or (Ej(X) > Ej(X0) for all j and Dk(X) ≤ Dk(X0) for all k), or equivalently if there is no vector v from the production possibility set for which either v ≺pst({1,...,K}) (D1(X0), . . . , DK(X0), −E1(X0), . . . , −EJ(X0)),

r

v ≺pst({K+1,...,K+J}) (D1(X0), . . . , DK(X0), −E1(X0), . . . , −EJ(X0)).

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DEA in Finance 2014 49 / 88

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DC DEA models based on GDM DC DEA models based on GDM

Input-output oriented test 2

For a benchmark X0 ∈ X formulated as θI−O

2

(X0) = min

θk,ϕj,X 1 K

K

k=1 θk 1 J

J

j=1 ϕj

s.t. Ej(X) ≥ ϕj · Ej(X0), j = 1, . . . , J, (10) Dk(X) ≤ θk · Dk(X0), k = 1, . . . , K, ≤ θk ≤ 1, ϕk ≥ 1, k = 1, . . . , K, X ∈ X.

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DC DEA models based on GDM DC DEA models based on GDM

Input-output oriented test 2

Let X = X FD. For a benchmark R0 ∈ X formulated as θ2(R0) = min

˜ θk, ˜ ϕj,˜ xi

1 K

K

k=1

˜ θk s.t. 1 J

J

j=1

˜ ϕj = 1, (11) Ej n

i=1

Ri˜ xi

≥

˜ ϕj · Ej(R0), j = 1, . . . , J, Dk n

i=1

Ri˜ xi

≤

˜ θk · Dk(R0), k = 1, . . . , K,

n

i=1

˜ xi ≤ 1, ˜ xi ≥ 0, i = 1, . . . , n, ˜ ϕj ≥ 0, 1 ≥ ˜ θk ≥ 0.

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DEA in Finance 2014 51 / 88

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DC DEA models based on GDM DC DEA models based on GDM

Input-output oriented test 2

Proposition X0 ∈ ΨI−O

2

: if there is no X ∈ X for which Ej(X) ≥ Ej(X0) for all j and Dk(X) ≤ Dk(X0) 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}) (D1(X0), . . . , DK(X0), −E1(X0), . . . , −EJ(X0)).

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DC DEA models based on GDM DC DEA models based on GDM

Input-output oriented test 3

The additive or slacks-based test inspired by Tone (2001): θI−O

3

(X0) = min

s−

k ,s+ j ,X

1 K

K

k=1 Dk(X0)−s−

k

Dk(X0) 1 J

J

j=1 Ej(X0)+s+

j

Ej(X0)

s.t. Ej(X) − s+

j

≥ Ej(X0), j = 1, . . . , J, Dk(X) + s−

k

≤ Dk(X0), 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=1 s− k /Dk(X0)

1 + 1

J

j=1 s+ j /Ej(X0)

.

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DEA in Finance 2014 53 / 88

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DC DEA models based on GDM DC DEA models based on GDM

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 θI−O (X0) ≥ θI−O

1

(X0) ≥ θI−O

2

(X0) = θI−O

3

(X0). Then, for the sets of efficient portfolios it can be obtained ΨI−O

3

= ΨI−O

2

⊆ ΨI−O

1

⊆ ΨI−O .

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DEA in Finance 2014 54 / 88

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DC DEA models based on GDM DC DEA models based on GDM

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 θI

1(X0) ≥ θI−O 1

(X0), θI

2(X0) ≥ θI−O 2

(X0). Then, for the sets of efficient portfolios can be obtained ΨI−O

1

⊆ ΨI

1, ΨI−O 2

⊆ ΨI

2.

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DEA in Finance 2014 55 / 88

SLIDE 64

DC DEA models based on GDM Financial indices efficiency – empirical study

DC DEA model with CVaR deviations

We consider CVaR deviations DS

α for αk ∈ (0, 1), k = 1, . . . , K as the

inputs and the expectation as the output, i.e. J = 1 and E1(X) = EX: min θ s.t.

n

i=1

E[Ri]xi ≥ E[R0], Dαk n

i=1

Rixi

≤

θ · Dαk(R0), k = 1, . . . , K,

n

i=1

xi = 1, xi ≥ 0, i = 1, . . . , n. (full diversification)

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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 Dα(X) = CVaRα(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 Dα(X) = min

ξ∈R

1 1 − αE[max{(1 − α)(X − ξ), α(ξ − X)}] (13) with the minimum attained at any (1 − α)-th quantile. In relation with CVaR minimization formula, see Pflug (2000), Rockafellar and Uryasev (2000, 2002).

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DC DEA models based on GDM Financial indices efficiency – empirical study

DC DEA model with CVaR deviations

Input oriented

For discretely distributed returns (ris, s = 1, . . . , S, ps = 1/S) LP: θI

1(R0) = min 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 − ξ, s = 1, . . . , S, k = 1, . . . , K, usk ≥ αk 1 − αk (ξ −

n

i=1

xiris), s = 1, . . . , S, k = 1, . . . , K,

n

i=1

xi = 1, xi ≥ 0, i = 1, . . . , n.

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DEA in Finance 2014 58 / 88

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DC DEA models based on GDM Financial indices efficiency – empirical study

DC DEA model with CVaR deviations

Input-output oriented (not included)

For discretely distributed returns (ris, s = 1, . . . , S, ps = 1/S) LP: θI−O

1

(R0) = min

θ,xi,usk,ξk

θ s.t.

n

i=1

E[Ri]xi ≥ E[R0], 1 S

S

s=1

usk ≤ θ · DS

αk(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.

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DEA in Finance 2014 59 / 88

SLIDE 68

DC DEA models based on GDM Financial indices efficiency – empirical study

Financial indices

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).

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DC DEA models based on GDM Financial indices efficiency – empirical study

Financial indices efficiency

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, 2006 - September 15, 2008 during crises (D): September 16, 2008 - September 20, 2010 whole period (W). CVaR deviation levels: αk ∈ {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...

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DC DEA models based on GDM Financial indices efficiency – empirical study

Index θI θI

1

θI

2

B D W B D W B D W IBOVESPA 1.00 0.47 0.90 1.00 0.44 0.78 1.00 0.40 0.76 S&PTSX Composite index 1.00 0.52 0.70 0.89 0.50 0.60 0.84 0.47 0.57 S&P 500 INDEX.RTH 1.00 0.53 0.71 0.82 0.48 0.63 0.78 0.48 0.60 IPC 1.00 0.52 0.83 0.82 0.48 0.71 0.81 0.47 0.66 ALL ORDINARIES 1.00 0.57 0.76 0.78 0.53 0.67 0.75 0.52 0.64 SSE Composite Index 0.67 0.98 0.88 0.56 0.80 0.68 0.54 0.66 0.62 BSE SENSEX 0.86 0.77 0.87 0.69 0.56 0.73 0.66 0.53 0.68 Jakarta Composite Index 0.85 1.00 1.00 0.72 1.00 1.00 0.61 1.00 1.00 FTSE Bursa Malaysia KLCI 0.78 1.00 1.00 0.68 0.97 0.94 0.63 0.95 0.91 NIKKEI 225 0.77 0.48 0.61 0.68 0.43 0.55 0.60 0.41 0.52 NZX 50 INDEX GROSS 1.00 0.98 1.00 0.89 0.87 0.92 0.82 0.81 0.90 STRAITS TIMES INDEX 0.77 0.55 0.73 0.64 0.51 0.62 0.61 0.51 0.59 KOSPI Composite Index 0.76 0.51 0.67 0.62 0.48 0.63 0.60 0.44 0.57 TSEC weighted index 0.61 0.80 0.83 0.53 0.67 0.67 0.47 0.61 0.63 ATX 0.82 0.35 0.47 0.61 0.31 0.42 0.61 0.30 0.40 CAC 4 0.87 0.45 0.60 0.68 0.41 0.54 0.66 0.40 0.51 DAX 0.92 0.46 0.64 0.73 0.42 0.54 0.72 0.41 0.52 AEX 0.83 0.45 0.57 0.70 0.40 0.51 0.65 0.38 0.49 SMSI 0.95 0.42 0.57 0.71 0.40 0.51 0.70 0.38 0.49 OMX Stockholm PI 0.80 0.50 0.65 0.63 0.48 0.56 0.61 0.43 0.54 SMI 0.97 0.59 0.70 0.76 0.53 0.64 0.74 0.47 0.59 FTSE 100 0.93 0.54 0.71 0.76 0.49 0.62 0.71 0.46 0.58 TEL AVIV TA-100 IND 0.68 1.00 0.73 0.58 0.68 0.67 0.47 0.65 0.64

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DC DEA models based on GDM Financial indices efficiency – empirical study

Index θI θI

1

θI

2

B D W B D W B D W IBOVESPA 1.00 0.47 0.90 1.00 0.44 0.83 1.00 0.42 0.79 S&PTSX Composite index 1.00 0.52 0.70 0.93 0.50 0.58 0.98 0.47 0.58 S&P 500 INDEX.RTH 1.00 0.53 0.71 0.84 0.50 0.65 0.84 0.48 0.61 IPC 1.00 0.52 0.83 0.93 0.49 0.70 0.93 0.49 0.70 ALL ORDINARIES 1.00 0.57 0.76 0.84 0.55 0.69 0.84 0.53 0.64 SSE Composite Index 0.67 0.98 0.88 0.60 0.68 0.67 0.60 0.66 0.67 BSE SENSEX 0.86 0.77 0.87 0.75 0.53 0.71 0.75 0.53 0.71 Jakarta Composite Index 0.85 1.00 1.00 0.72 1.00 1.00 0.68 1.00 1.00 FTSE Bursa Malaysia KLCI 0.78 1.00 1.00 0.72 1.00 0.98 0.72 1.00 0.98 NIKKEI 225 0.77 0.48 0.61 0.65 0.42 0.56 0.64 0.42 0.52 NZX 50 INDEX GROSS 1.00 0.98 1.00 0.89 0.84 0.97 0.89 0.82 0.91 STRAITS TIMES INDEX 0.78 0.55 0.73 0.69 0.52 0.61 0.69 0.51 0.61 KOSPI Composite Index 0.76 0.51 0.67 0.67 0.45 0.62 0.67 0.44 0.62 TSEC weighted index 0.61 0.80 0.83 0.51 0.64 0.65 0.51 0.64 0.65 ATX 0.82 0.35 0.47 0.66 0.31 0.43 0.66 0.31 0.40 CAC 4 0.87 0.45 0.60 0.73 0.41 0.55 0.72 0.41 0.52 DAX 0.93 0.46 0.64 0.81 0.42 0.53 0.81 0.41 0.53 AEX 0.83 0.45 0.57 0.71 0.39 0.53 0.71 0.38 0.50 SMSI 0.95 0.42 0.57 0.79 0.39 0.52 0.76 0.39 0.49 OMX Stockholm PI 0.80 0.50 0.65 0.67 0.45 0.54 0.66 0.43 0.54 SMI 0.97 0.59 0.70 0.81 0.48 0.64 0.80 0.47 0.60 FTSE 100 0.93 0.54 0.71 0.77 0.47 0.63 0.77 0.46 0.59 TEL AVIV TA-100 IND 0.68 1.00 0.73 0.52 0.65 0.72 0.52 0.65 0.70

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DEA in Finance 2014 63 / 88

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DC DEA models based on GDM Financial indices efficiency – empirical study

Index θI θI

1

θI

2

B D W B D W B D W IBOVESPA 1.00 0.75 1.00 1.00 0.75 1.00 1.00 0.69 1.00 S&PTSX Composite index 1.00 0.52 0.77 1.00 0.52 0.77 1.00 0.49 0.70 S&P 500 INDEX.RTH 1.00 0.53 0.79 1.00 0.53 0.79 1.00 0.50 0.74 IPC 1.00 0.85 0.79 1.00 0.85 0.79 1.00 0.77 0.74 ALL ORDINARIES 1.00 0.57 0.84 1.00 0.57 0.84 1.00 0.55 0.79 SSE Composite Index 0.78 0.98 1.00 0.78 0.98 1.00 0.73 0.69 1.00 BSE SENSEX 1.00 0.51 1.00 1.00 0.51 1.00 1.00 0.50 1.00 Jakarta Composite Index 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 1.00 FTSE Bursa Malaysia KLCI 0.79 1.00 1.00 0.79 1.00 1.00 0.77 1.00 1.00 NIKKEI 225 0.77 0.48 0.61 0.77 0.48 0.61 0.72 0.44 0.57 NZX 50 INDEX GROSS 1.00 0.98 1.00 1.00 0.98 1.00 1.00 0.86 1.00 STRAITS TIMES INDEX 0.87 0.55 0.75 0.87 0.55 0.75 0.80 0.53 0.70 KOSPI Composite Index 0.77 0.51 0.67 0.77 0.51 0.67 0.72 0.46 0.63 TSEC weighted index 0.66 1.00 0.83 0.66 1.00 0.83 0.61 1.00 0.75 ATX 0.82 0.35 0.47 0.82 0.35 0.47 0.74 0.32 0.44 CAC 4 0.87 0.45 0.60 0.87 0.45 0.60 0.82 0.43 0.57 DAX 1.00 0.46 0.68 1.00 0.46 0.68 1.00 0.43 0.64 AEX 0.83 0.45 0.57 0.83 0.45 0.57 0.80 0.40 0.55 SMSI 0.96 0.42 0.64 0.96 0.42 0.64 0.91 0.40 0.60 OMX Stockholm PI 0.80 0.50 0.70 0.80 0.50 0.70 0.75 0.46 0.66 SMI 0.97 0.61 0.70 0.97 0.61 0.70 0.91 0.50 0.66 FTSE 100 0.99 0.54 0.79 0.99 0.54 0.79 0.91 0.48 0.72 TEL AVIV TA-100 IND 0.72 0.54 0.74 0.72 0.54 0.74 0.64 0.48 0.73

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DEA in Finance 2014 64 / 88

SLIDE 73

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 R0? No – R0 is efficient. Compare with the problem max f (R) : s.t. R SSD R0.

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SLIDE 75

On relations between DEA and stochastic dominance efficiency Second Order Stochastic Dominance

Second order stochastic dominance

F1, F2... cumulative probability distributions functions of random variables X1, X2. Second order (strict) stochastic dominance (SSD): X1 ≻SSD X2 iff EF1u(x) − EF2u(x) ≥ 0 for every concave utility function u with at least one strict inequality. Consider twice cumulated probability distributions functions: F (2)

i

(t) = t

−∞

Fi(x)dx i = 1, 2. Theorem (Hanoch & Levy (1969)): X1 ≻SSD X2 ⇔ F (2)

1 (t) ≤ F (2) 2 (t)

∀t ∈ R with at least one strict inequality.

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DEA in Finance 2014 67 / 88

SLIDE 76

On relations between DEA and stochastic dominance efficiency Second Order Stochastic Dominance

SSD portfolio efficiency

We consider n assets and we denote Ri the rate of return of i-th asset with finite mean value, r = {R1, . . . , Rn}. a discrete probability distribution of rate of returns described by scenarios ri,s, s = 1, . . . , S that are taken with equal probabilities ps = 1/S. a decision maker that may combine the assets into portfolios represented by weights x = {x1, . . . , xn}. the set of feasible weights (no short sales allowed): X = {x ∈ R|

n

i=1

xi = 1, xi ≥ 0, i = 1, . . . , n}. (14)

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SLIDE 77

On relations between DEA and stochastic dominance efficiency 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)...

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SLIDE 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

f 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(ri) for all i = 1, 2, ..., n.

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DEA in Finance 2014 70 / 88

SLIDE 79

On relations between DEA and stochastic dominance efficiency Second Order Stochastic Dominance

Convex SSD efficiency test

Bawa et al. (1985): Let D = {d1, d2, ..., 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 ri,s = dk and vice versa, where r(n+1),s = n

i=1 ri,sxi.

Convex SSD efficiency test of portfolio x: δ∗(x) = max

δk,xi (n+1)S

k=1

δk (15) s.t. F (2)

x

(dk) −

n

i=1

¯ xiF (2)

i

(dk) ≥ δk, k = 1, 2, ..., (n + 1)S, δk ≥ 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.

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DEA in Finance 2014 71 / 88

SLIDE 80

On relations between DEA and stochastic dominance efficiency 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 weighted inputs are compared with the weighted outputs. “Are inputs transformed into outputs in an efficient way?”

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On relations between DEA and stochastic dominance efficiency Data Envelopment Analysis

DEA

Variable Returns to Scale (VRS)

Banker, Charnes and Cooper (1984): DEA model with Variable Returns to Scale: min 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.

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DEA in Finance 2014 73 / 88

SLIDE 82

On relations between DEA and stochastic dominance efficiency 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

r 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.

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DEA in Finance 2014 74 / 88

SLIDE 83

On relations between DEA and stochastic dominance efficiency Data Envelopment Analysis

Equivalent DEA test to convex SSD efficiency test

Let ˜ D = {d1, d2, ..., d(n+1)S} be the set of all sorted scenario returns of the assets Ri and portfolio x. The test can be rewritten using lower partial moments Li(d) = 1/S S

s=1 [d − ri,s]+.

Based on the results proposed by Bawa et al. (1985)

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DEA in Finance 2014 75 / 88

SLIDE 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 R0 = n

i=1 Rixi.

Find the index ˜ k = arg min{k : L0(dk) > 0}. Then the DEA-risk model with variable return to scale and K = (n + 1)S − 1 inputs δCDEA(R0) = min

¯ xi,ϕ,θk

1 K − ˜ k + 2  

K

k=˜

k

θk + 1 ϕ  

n

i=1

¯ xiE[Ri] ≥ ϕ · E[R0],

n

i=1

¯ xiLi(dk) ≤ θk · L0(dk), k = 1, . . . , K, (16) ≤ θk ≤ 1, ϕ ≥ 1,

n

i=1

¯ xi = 1, ¯ xi ≥ 0, i = 1, . . . , n. is equivalent to convex SSD efficiency test.

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SLIDE 85

On relations between DEA and stochastic dominance efficiency 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 R0: θDC(R0) = min

θ,xi θ

−CVaRεj

−

n

i=1

Rixi

≥

−CVaRεj(−R0), j = 1, . . . , J, (17) CVaR+

αk

−

n

i=1

Rixi

≤

θ · CVaR+

αk(−R0), k = 1, . . . , K, n

i=1

xi = 1, xi ≥ 0, i = 1, . . . , n, where CVaR+

α = max{CVaRα, 0}, and αk, εj are different levels, the

positive parts of CVaRs serve as the inputs and expected return as the

utput.
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On relations between DEA and stochastic dominance efficiency Data Envelopment Analysis

Diversification-consistent test

Input-output oriented I

B. and Kopa (2013): Let CVaRα(−R0) > 0 for α ∈ {α1, . . . , αK} and

CVaRε(−R0) < 0 for ε ∈ {ε1, . . . , εJ} θDC I−O I(R0) = min

θ,ϕ,xi

1 K + J

θ + 1

ϕ

−CVaRεj
−

n

i=1

Rixi

≥

ϕ · (−CVaRεj(−R0)), j = 1, . . . , J, (18) CVaRαk

−

n

i=1

Rixi

≤

θ · CVaRαk(−R0), k = 1, . . . , K, ≤ θ ≤ 1, ϕ ≥ 1,

n

i=1

xi = 1, xi ≥ 0, i = 1, . . . , n. Note that CVaR0(−R0) = E[−R0], i.e. expected loss can be also included into this model without any changes in its formulation.

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Diversification-consistent test

Input-output oriented II

B. and Kopa (2013):

θDC I−O II(R0) = min

θk,ϕj,xi

1 K + J  

K

k=1

θk +

J

j=1

1 ϕj   −CVaRεj

−

n

i=1

Rixi

≥

ϕj · (−CVaRεj(−R0)), j = 1, . . . , J,(19) CVaRαk

−

n

i=1

Rixi

≤

θk · CVaRαk(−R0), k = 1, . . . , K, ≤ θk ≤ 1, ϕj ≥ 1,

n

i=1

xi = 1, xi ≥ 0, i = 1, . . . , n. These DEA-risk models can be seen as the extension of Russel measure DEA model (see Cook and Seiford (2009)).

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Equivalent DEA test to portfolio SSD efficiency test

B. and Kopa (2013): We assume that no CVaRαs of the benchmark is

equal to zero for αs = s/S, s ∈ Γ = {0, 1, . . . , S − 1} Let ˜ s = arg max{s ∈ Γ : CVaRαs(−R0) < 0}, εj ∈ {0/S, 1/S, . . . ,˜ s/S}, J = ˜ s + 1 αk ∈ {(˜ 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 R0 = n

i=1 Rixi 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).

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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, 0.995 are used as the inputs and the expected return (the most commonly used reward measure) as the

utput.
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Traditional Div.-consistent SSD efficiency DEA tests DEA tests VRS CRS I I-O I I-O II convex portfolio Food 1.00 1.00 0.89 0.94 0.89 yes no Beer 1.00 0.92 0.82 0.91 0.82 yes no Smoke 1.00 0.82 1.00 1.00 1.00 yes yes Util 1.00 0.96 0.89 0.88 0.86 yes no

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Conclusions

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.

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References

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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, 81–94. Kopa, M. (2011). Comparison of various approaches to portfolio efficiency. Proceedings of the 29th International Conference on Mathematical Methods in Economics 2011, M. Dlouh´ y, V. Skoˇ cdopolov´ a eds., University of Economics in Prague, J´ ansk´ a Dolina, Slovakia, 64–69. Kopa, M., Chovanec, P. (2008). A Second-order Stochastic Dominance Portfolio Efficiency Measure, Kybernetika 44(2), 243–258.

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References Kuosmanen, T. (2004). Efficient diversification according to stochastic dominance criteria, Management Science 50(10), 1390–1406. Kuosmannen, T. (2007). Performance measurement and best-practice benchmarking of mutual funds: combining stochastic dominance criteria with data envelopment analysis. Journal of Productivity Analysis 28, 71–86. 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. von Neumann, J., Morgenstern, O.: Theory of games and economic behavior. Princeton University Press, 1944. Ogryczak, W., Ruszczynski, A. (2001). On consistency of stochastic dominance and mean-semideviation models. Mathematical Programmming, Ser. B 89, 217–232. Ogryczak, W., Ruszczynski, A. (2002). Dual stochastic dominance and related mean-risk models. SIAM Journal on Optimization 13, 60–78. Pflug, G. (2000). Some Remarks on the value-at-risk and the Conditional value-at-risk. In: Probabilistic Constrained Optimization (S.P. Uryasev, ed.), Kluwer, Dordrecht, 272–281. Pflug, G.Ch., Romisch, W.: Modeling, measuring and managing risk. World Scientific Publishing, Singapore, 2007. Post, T. (2003). Empirical tests for stochastic dominance efficiency, Journal of Finance 58, 1905–1932. Rockafellar, R.T., Uryasev, S. (2000). Optimization of Conditional Value-at-Risk. Journal of Risk, 2, 21–41. Rockafellar, R.T., Uryasev, S. (2002). Conditional Value-at-Risk for General Loss Distributions, Journal of Banking and Finance 26, 1443–1471. 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. Ruszczynski, A., Shapiro, A. (2006). Optimization of convex risk functions. Mathematics of Operations Research 31, 433–452. Tone, K. (2001). A slacks-based measure of efficiency in data envelopment analysis. European Journal of Operations Research 130, 498–509.

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References

Thank you for your attention.

e-mail: branda@karlin.mff.cuni.cz

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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

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

DEA in finance

We do not access efficiency of financial institutions (banks, insurance comp.). We access efficiency of investment opportunities1 on financial markets.

Motivation

Branda, Kopa (2012): an empirical study (a bit “naive”, but necessary step for us:) Branda, Kopa (2014): equivalences (a “bridge”)

Motivation

Branda, Kopa (2012): an empirical study (a bit “naive”, but necessary step for us:) Branda, Kopa (2014): equivalences (a “bridge”)

Motivation

Branda, Kopa (2012): an empirical study (a bit “naive”, but necessary step for us:) Branda, Kopa (2014): equivalences (a “bridge”)

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

Efficiency of investment opportunities

DEA in finance

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

Risk shaping — several risk measures (CVaRs) included into one model – Rockafellar and Uryasev (2002) Relations to stochastic dominance efficiency – Branda and Kopa (2014)

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

Data Envelopment Analysis (DEA)

All data are assumed to be (semi-)positive. Charnes et al (1978): fractional programming formulation (Constant Returns to Scale – CRS or CCR)

DEA

Variable Returns to Scale (VRS)

Banker, Charnes and Cooper (1984): DEA model with Variable Returns 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.

DEA

Variable Returns to Scale (VRS)

Dual formulation of VRS DEA: min

xi,θ θ

s.t.

n

xiYji ≥ Yj0, j = 1, . . . , J,

n

xiZki ≤ θ · Zk0, k = 1, . . . , K,

n

xi = 1, xi ≥ 0, i = 1, . . . , n.

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.

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

General class of financial DEA tests

Lamb and Tee (2012) – pure return-risk tests2: Inputs: positive parts of coherent risk measures Outputs: return measures (= minus coherent risk measures, e.g. expected return)

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

General deviation measures

General deviation measures

Deviation measures

Standard deviation D(X) = σ(X) =

Mean absolute deviation D(X) = E

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

Worst-case deviation D(X) = sup