Data Envelopment Analysis in Finance - Univerzita...

Data Envelopment Analysis in Finance

Martin Branda

Faculty of Mathematics and PhysicsCharles University in Prague

&Institute of Information Theory and Automation

Academy of Sciences of the Czech Republic

Ostrava, January 10, 2014

M. Branda DEA in Finance 2014 1 / 88

Contents

1 Efficiency of investment opportunities

2 Data Envelopment Analysis

3 Diversification-consistent DEA based on general deviation measuresGeneral deviation measuresDiversification-consistent DEA modelsFinancial indices efficiency – empirical study

4 On relations between DEA and stochastic dominance efficiencySecond Order Stochastic DominanceData Envelopment AnalysisNumerical comparison

5 References


DEA in finance

We do not access efficiency of financial institutions (banks, insurancecomp.).

We access efficiency of investment opportunities1 on financialmarkets.

1Assets, portfolios, mutual funds, financial indices, ...M. Branda DEA in Finance 2014 3 / 88

Motivation

Together with Milos Kopa (in 2010):

Is there a relation between stochastic dominance efficiency andDEA 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 andoptimization

Branda, Kopa (2012): an empirical study (a bit “naive”, butnecessary step for us:)

Branda, Kopa (2014): equivalences (a “bridge”)


Efficiency of investment opportunities

Contents





5 References




Various approaches how to find an “optimal” portfolio or how to testefficiency 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): Stochasticdominance

Murthi et al (1997): Data Envelopment Analysis



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 multiobjectiveoptimization principles – Markowitz (1952)

Risk shaping — several risk measures (CVaRs) included into onemodel – Rockafellar and Uryasev (2002)

Relations to stochastic dominance efficiency – Branda and Kopa(2014)


Data Envelopment Analysis

Contents





5 References



Data Envelopment Analysis (DEA)

Charnes, Cooper and Rhodes (1978): a way how to state efficiency of adecision making unit over all other decision making units with the samestructure of inputs and outputs.

Let Z1i , . . . ,ZKi denote the inputs and Y1i , . . . ,YJi denote the outputs ofthe unit i from n considered units. DEA efficiency of the unit0 ∈ 1, . . . , n is then evaluated using the optimal value of the followingprogram where the weighted inputs are compared with the weightedoutputs.

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



DEAVariable Returns to Scale (VRS)

Banker, Charnes and Cooper (1984): DEA model with Variable Returns toScale (VRS) or BCC:

maxyj0,wk0

∑Jj=1 yj0Yj0 − y0∑Kk=1 wk0Zk0

s.t.∑Jj=1 yj0Yji − y0∑Kk=1 wk0Zki

≤ 1, i = 1, . . . , n,

wk0 ≥ 0, k = 1, . . . ,K ,

yj0 ≥ 0, j = 1, . . . , J,

y0 ∈ R.




Dual formulation of VRS DEA:

minxi ,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.



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

...



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, standarddeviation and gross return.

Basso and Funari (2001, 2003): standard deviation andsemideviations, beta coefficient, costs as the inputs, expected returnor expected excess return, ethical measure and stochastic dominancecriterion as the outputs.

Chen and Lin (2006): Value at Risk (VaR) and Conditional Value atRisk (CVaR).

Branda and Kopa (2012): VaR, CVaR, sd, lsd, Drawdow measures(DaR, CDaR) as the inputs, gross return as the output; comparisonwith second-order stochastic dominance.

See Table 1 in Lozano and Gutierrez (2008B)



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.M. Branda DEA in Finance 2014 18 / 88

DC DEA models based on GDM

Contents





5 References


DC DEA models based on GDM General deviation measures

General deviation measures

Rockafellar, Uryasev and Zabarankin (2006A, 2006B): GDM areintroduced as an extension of standard deviation but they need not to besymmetric with respect to upside X − E[X ] and downside E[X ]− X of arandom variable X .

Any functional D : L2(Ω)→ [0,∞] is called a general deviation measure ifit 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



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 ) = supω∈Ω|X (ω)− E[X ]|.

See Rockafellar et al (2006 A, 2006 B) for another examples.



Mean absolute deviation from (1− α)-th quantileCVaR deviation

For any α ∈ (0, 1) a finite, continuous, lower range dominated deviationmeasure

Dα(X ) = CVaRα(X − E[X ]). (1)

The deviation is also called weighted mean absolute deviation fromthe (1−α)-th quantile, see Ogryczak, Ruszczynski (2002), because it canbe expressed as

Dα(X ) = minξ∈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 withCVaR minimization formula, see Pflug (2000), Rockafellar and Uryasev(2000, 2002).




According to Proposition 4 in Rockafellar et al (2006 A):

if D = λD0 for λ > 0 and a deviation measure D0, then D is adeviation measure.

if D1, . . . ,DK are deviation measures, then

D = maxD1, . . . ,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 envelopesand risk identifiers, subdifferentiability and optimality conditions.



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 , whereasR(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).



Coherent risk and return measures

A return measure is defined as a functional E(X ) = −R(X ) for acoherent risk measure R. It is obvious that the expectation belongs to thisclass. The right part of a return distribution described better by returnmeasures derived from CVaR, cf. Lamb and Tee (2012).




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) areclosed;

(D5) lower range dominated if D(X ) ≤ EX − infω∈Ω X (ω) forall X .



Strictly expectation bounded risk measures

Theorem 2 in Rockafellar et al (2006 A):

Theorem

Deviation measures correspond one-to-one with strictly expectationbounded 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 rangedominated.


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

Traditional DEA modelInput 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

θI0(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.



The most important slideTraditional DEA model vs. diversification

The model does not take into account portfolio diversification: Forany 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 realportfolio deviation.



Traditional DEA and diversification frontier



Models with diversification

Efficiency of mutual funds or financial indexes using DEA related modelswith diversification:

Kuosmannen (2007): Second order stochastic dominance consistentDEA models.

Lozano and Gutierrez (2008 A, 2008 B): Second and Third orderstochastic 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 modelwith several inputs and outputs (positive parts of risk measures usedas the inputs).



DEA tests with diversification

Efficiency of mutual funds or industry representative portfolios:

Briec et al. (2004), Kerstens et al. (2012): directional-distancemean-variance efficiency.

Joro and Na (2006), Briec et al. (2007), Kerstens et al. (2011,2013): directional-distance mean-variance-skewness efficiency.

Lozano and Gutierrez (2008A, 2008B): tests consistent with second-and third-order stochastic dominance (necessary condition).

Branda and Kopa (2014): equivalence with second-order stochasticdominance.

Lamb and Tee (2012), Branda (2013A, 2013B): general classes ofDEA tests with risk/deviation and return measures.



Set of investment opportunities

We consider n assets and denote Ri ∈ L2(Ω) the rate of return of i-thasset and the sets of investment opportunities:

1 pairwise efficiency (investment into one single asset):

XP = 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 ofassets #):

X LD =

n∑

i=1

Rixi :n∑

i=1

xi = 1, xi ≥ 0, xi ≤ yi , yi ∈ 0, 1,n∑

i=1

yi ≤ #

.



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 : vi < zi .

Definition

Let v , z ∈ Rn. We say that z partially strictly (partially weakly) dominatesv 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

one i ∈ S for which vi < zi .



DEA efficiency

We assume that X0 ∈ X is not constant, i.e. Dk(X0) > 0, for allk = 1, . . . ,K .

Definition

We say that X0 ∈ X is DEA efficient with respect to the set X if theoptimal value of the DEA program is equal to 1. Otherwise, X0 isinefficient and the optimal value measures the inefficiency.

For m ∈ 0, 1, 2, 3 we denote the sets of efficient opportunities

ΨIm = X ∈ X : θIm(X ) = 1,

where θIm(X0) is the optimal value for benchmark X0.



Diversification consistent DEA - model 1

For a benchmark X0 ∈ X diversification consistent DEA model

θI1(X0) = min θ

s.t.

Ej(X ) ≥ Ej(X0), j = 1, . . . , J, (4)

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

X ∈ X .



Set of efficient opportunities

Proposition

X0 ∈ ΨI1

if for all X ∈ X for which Ej(X ) ≥ Ej(X0) for all j holds thatDk(X ) ≥ Dk(X0) for at least one k, i.e. there is no X ∈ X for whichEj(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)).



Model 1 - equivalent formulation

θI1(X0) = min maxk=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.



Diversification consistent DEA - model 2

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

θI2(X0) = min1

K

K∑k=1

θk

s.t.

Ej(X ) ≥ Ej(X0), j = 1, . . . , J, (6)

Dk(X ) ≤ θk · Dk(X0), k = 1, . . . ,K ,

0 ≤ θk ≤ 1, k = 1, . . . ,K ,

X ∈ X .



Set of efficient opportunities

Proposition

X0 ∈ ΨI2

if for all X ∈ X for which Ej(X ) ≥ Ej(X0) for all j holds thatDk(X ) ≥ Dk(X0) for all k, i.e. there is no X ∈ X for whichEj(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)).



Model 2 - equivalent formulation

The model is obviously equivalent to

θI2(X0) = min1

K

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 ) = 1K

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



Diversification consistent DEASlacks-based model

Additive or slacks-based DEA model

θI3(X0) = min1

K

K∑k=1

Dk(X0)− s−kDk(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 .



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.



Test strength

Proposition

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

θI0(X0) ≥ θI1(X0) ≥ θI2(X0) = θI3(X0).

For the sets of efficient portfolios we obtain

ΨI3 = ΨI

2 ⊆ ΨI1 ⊆ ΨI

0.



Test strength

We can construct weaker tests if we restrict the number of investmentopportunities in the portfolio leading to the models with the limiteddiversification.

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.



Input-output oriented models

Input-output oriented DC DEA models - (in)efficiency is measured alsowith respect to the outputs (assume Ej(X0) > 0):

optimal values (efficiency scores) and strength can be compared,

input and input-output oriented models can be compared: I-O testsare stronger in general,

cf. Branda (2013A, 2013B).



Input-output oriented test 1

We assume that Ej(X0) is positive for at least one j . An input-outputoriented test where inefficiency is measured with respect to the inputs andoutputs separately can be formulated as follows

θI−O1 (X0) = minθ,ϕ,X

θ

ϕs.t.

Ej(X ) ≥ ϕ · Ej(X0), j = 1, . . . , J, (9)

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

0 ≤ θ ≤ 1, ϕ ≥ 1,

X ∈ X .




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

θI−O1 (R0) = minθ,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 alloutputs Ej are positively homogeneous. We obtained a convexprogramming problem.




Proposition

X0 ∈ ΨI−O1 :

if there is no X ∈ X for which either (Ej(X ) ≥ Ej(X0) for all j andDk(X ) < Dk(X0) for all k) or (Ej(X ) > Ej(X0) for all j andDk(X ) ≤ Dk(X0) for all k), or equivalently

if there is no vector v from the production possibility set for whicheither

v ≺pst(1,...,K) (D1(X0), . . . ,DK (X0),−E1(X0), . . . ,−EJ(X0)),

or

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




For a benchmark X0 ∈ X formulated as

θI−O2 (X0) = minθk ,ϕj ,X

1K

∑Kk=1 θk

1J

∑Jj=1 ϕj

s.t.

Ej(X ) ≥ ϕj · Ej(X0), j = 1, . . . , J, (10)

Dk(X ) ≤ θk · Dk(X0), k = 1, . . . ,K ,

0 ≤ θk ≤ 1, ϕk ≥ 1, k = 1, . . . ,K ,

X ∈ X .




Let X = X FD . For a benchmark R0 ∈ X formulated as

θ2(R0) = minθk ,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.M. Branda DEA in Finance 2014 51 / 88



Proposition

X0 ∈ ΨI−O2 :

if there is no X ∈ X for which Ej(X ) ≥ Ej(X0) for all j andDk(X ) ≤ Dk(X0) for all k with at least one inequality strict, orequivalently

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)).




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

θI−O3 (X0) = mins−k ,s

+j ,X

1K

∑Kk=1

Dk (X0)−s−kDk (X0)

1J

∑Jj=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 themean output inefficiencies. The objective function can be rewritten as

1− 1K

∑Kk=1 s−k /Dk(X0)

1 + 1J

∑Jj=1 s+

j /Ej(X0).



Properties and relations

Proposition

Let maxJ,K ≥ 2. Then for a benchmark X0 ∈ X with Dk(X0) > 0 forall k and Ej(X0) > 0 for all j , the following relations hold

θI−O0 (X0) ≥ θI−O1 (X0) ≥ θI−O2 (X0) = θI−O3 (X0).

Then, for the sets of efficient portfolios it can be obtained

ΨI−O3 = ΨI−O

2 ⊆ ΨI−O1 ⊆ ΨI−O

0 .



Properties and relations

Proposition

Let maxJ,K ≥ 2. Then for a benchmark X0 ∈ X with Dk(X0) > 0 forall k and Ej(X0) > 0 for all j , the following relations hold

θI1(X0) ≥ θI−O1 (X0), θI2(X0) ≥ θI−O2 (X0).

Then, for the sets of efficient portfolios can be obtained

ΨI−O1 ⊆ ΨI

1, ΨI−O2 ⊆ ΨI

2.


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

DC DEA model with CVaR deviations

We consider CVaR deviations DSα 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)



Mean absolute deviation from (1− α)-th quantileCVaR deviation

For any α ∈ (0, 1) a finite, continuous, lower range dominated deviationmeasure

Dα(X ) = CVaRα(X − E[X ]). (12)

The deviation is also called weighted mean absolute deviation fromthe (1−α)-th quantile, see Ogryczak, Ruszczynski (2002), because it canbe expressed as

Dα(X ) = minξ∈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 withCVaR minimization formula, see Pflug (2000), Rockafellar and Uryasev(2000, 2002).



DC DEA model with CVaR deviationsInput oriented

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

θI1(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

xi ris − ξ, s = 1, . . . ,S , k = 1, . . . ,K ,

usk ≥ αk

1− αk(ξ −

n∑i=1

xi ris), s = 1, . . . ,S , k = 1, . . . ,K ,

n∑i=1

xi = 1, xi ≥ 0, i = 1, . . . , n.



DC DEA model with CVaR deviationsInput-output oriented (not included)

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

θI−O1 (R0) = minθ,xi ,usk ,ξk

θ s.t.

n∑i=1

E[Ri ]xi ≥ E[R0],

1

S

S∑s=1

usk ≤ θ · DSαk

(R0), k = 1, . . . ,K ,

usk ≥

(n∑

i=1

xi ris − ξk

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

usk ≥ αk

1− αk

(ξk −

n∑i=1

xi ris

),

n∑i=1

xi ≤ 1, xi ≥ 0, i = 1, . . . , n.



Financial indices

We consider the following 25 world financial indices which are listed onYahoo Finance:

America (5): MERVAL BUENOS AIRES, IBOVESPA, S&P TSXComposite index, S&P 500 INDEX RTH, IPC,

Asia/Pacific (11): ALL ORDINARIES, SSE Composite Index, HANGSENG INDEX, BSE SENSEX, Jakarta Composite Index, FTSE BursaMalaysia KLCI, NIKKEI 225, NZX 50 INDEX GROSS, STRAITSTIMES 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).



Financial indices efficiency

In our analysis we describe each index by its weekly rates of returns. Wedivided 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, 0.99DEA optimal values/scores with # = n...

DEA optimal values/scores with # = 2...

DEA optimal values/scores with # = 1...



Index θI0 θI1 θI2B 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 0.76S&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 0.60IPC 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 0.64SSE 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 0.68Jakarta 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 0.91NIKKEI 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 0.90STRAITS TIMES INDEX 0.77 0.55 0.73 0.64 0.51 0.62 0.61 0.51 0.59KOSPI 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 0.63ATX 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 0.51DAX 0.92 0.46 0.64 0.73 0.42 0.54 0.72 0.41 0.52AEX 0.83 0.45 0.57 0.70 0.40 0.51 0.65 0.38 0.49SMSI 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 0.54SMI 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 0.58TEL AVIV TA-100 IND 0.68 1.00 0.73 0.58 0.68 0.67 0.47 0.65 0.64





S&P 500 INDEX.RTH 1.00 0.53 0.71 0.84 0.50 0.65 0.84 0.48 0.61IPC 1.00 0.52 0.83 0.93 0.49 0.70 0.93 0.49 0.70






CAC 4 0.87 0.45 0.60 0.73 0.41 0.55 0.72 0.41 0.52DAX 0.93 0.46 0.64 0.81 0.42 0.53 0.81 0.41 0.53AEX 0.83 0.45 0.57 0.71 0.39 0.53 0.71 0.38 0.50SMSI 0.95 0.42 0.57 0.79 0.39 0.52 0.76 0.39 0.49







S&P 500 INDEX.RTH 1.00 0.53 0.79 1.00 0.53 0.79 1.00 0.50 0.74IPC 1.00 0.85 0.79 1.00 0.85 0.79 1.00 0.77 0.74






CAC 4 0.87 0.45 0.60 0.87 0.45 0.60 0.82 0.43 0.57DAX 1.00 0.46 0.68 1.00 0.46 0.68 1.00 0.43 0.64AEX 0.83 0.45 0.57 0.83 0.45 0.57 0.80 0.40 0.55SMSI 0.96 0.42 0.64 0.96 0.42 0.64 0.91 0.40 0.60




On relations between DEA and stochastic dominance efficiency

Contents





5 References


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/dualformulations, multiobjective opt. – Pareto efficiency, chanceconstraints),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.


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

Second order stochastic dominance

F1,F2... cumulative probability distributions functions of random variablesX1,X2.

Second order (strict) stochastic dominance (SSD):X1 SSD X2 iff EF1u(x)− EF2u(x) ≥ 0 for every concave utilityfunction 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.



SSD portfolio efficiency

We consider

n assets and we denote Ri the rate of return of i-th asset with finitemean value, r = R1, . . . ,Rn.a discrete probability distribution of rate of returns described byscenarios ri ,s , s = 1, . . . ,S that are taken with equal probabilitiesps = 1/S .

a decision maker that may combine the assets into portfoliosrepresented 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)



SSD portfolio efficiency

A given portfolio τ ∈ X is SSD portfolio efficient if and only if thereexists no portfolio λ ∈ X such that r′λ SSD r′τ . Otherwise, portfolio τis SSD inefficient.

SSD portfolio efficiency tests: Post (2003), Kuosmanen (2004), Kopaand Chovanec (2008)...



Convex second-order stochastic dominance

Fishburn (1974) defines a concept of convex stochastic dominance: Wesay that portfolio x is convex SSD inefficient if every investor prefers someof the assets to portfolio x.

Formal definition:A given portfolio τ is convex SSD efficient if there exists at least somenondecreasing concave u such that Eu(r′τ ) > Eu(ri ) for all i = 1, 2, ..., n.



Convex SSD efficiency test

Bawa et al. (1985): Let D = d1, d2, ..., d(n+1)S be the set of all scenarioreturns of the assets and portfolio x, that is, for every i ∈ 1, 2, ..., n + 1and s ∈ 1, 2, ...,S exists k ∈ 1, 2, ..., (n + 1)S such that ri ,s = dk andvice versa, where r(n+1),s =

∑ni=1 ri ,sxi .

Convex SSD efficiency test of portfolio x:

δ∗(x) = maxδk ,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) isstrictly positive. Otherwise, portfolio x is convex SSD efficient.


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 adecision making unit over all other decision making units with the samestructure of inputs and outputs.

Let Z1i , . . . ,ZKi denote the inputs and Y1i , . . . ,YJi denote the outputs ofthe unit i from n considered units. DEA efficiency of the unit0 ∈ 1, . . . , n is then evaluated using the optimal value of the followingprogram where weighted inputs are compared with the weighted outputs.

“Are inputs transformed into outputs in an efficient way?”




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

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.



DEA in finance

Efficiency of mutual funds or financial indexes:

Murthi et al (1997): expense ratio, load, turnover, standard deviationand gross return.

Basso and Funari (2001, 2003): standard deviation andsemideviations, beta coefficient, costs as the inputs, expected returnor expected excess return, ethical measure and stochastic dominancecriterion as the outputs.

Chen and Lin (2006): Value at Risk and Conditional Value at Risk.

Lozano and Gutierrez (2008): tests consistent with SSD (necessarycondition).

B. and Kopa (2010, 2012A): VaR, CVaR, sd, lsd, drawdown measures(DaR, CDaR) as the inputs, gross return as the output; comparisonwith SSD.

Lamb and Tee (2012), B. (2013): DEA tests with diversification.



Equivalent DEA test to convex SSD efficiency test

Let D = d1, d2, ..., d(n+1)S be the set of all sorted scenarioreturns of the assets Ri and portfolio x.

The test can be rewritten using lower partial momentsLi (d) = 1/S

∑Ss=1 [d − ri ,s ]+.

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



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 mink : L0(dk) > 0. Then the DEA-risk modelwith variable return to scale and K = (n + 1)S − 1 inputs

δCDEA(R0) = minxi ,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)

0 ≤ θk ≤ 1, ϕ ≥ 1,n∑

i=1

xi = 1, xi ≥ 0, i = 1, . . . , n.

is equivalent to convex SSD efficiency test.M. Branda DEA in Finance 2014 76 / 88


Diversification-consistent DEA testsInput oriented

Recently, general DEA tests with diversification effect were introduced byLamb and Tee (2012) for a benchmark with return R0:

θDC (R0) = minθ,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 = maxCVaRα, 0, and αk , εj are different levels, the

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



Diversification-consistent testInput-output oriented I

B. and Kopa (2013): Let CVaRα(−R0) > 0 for α ∈ α1, . . . , αK andCVaRε(−R0) < 0 for ε ∈ ε1, . . . , εJ

θDC I−O I (R0) = minθ,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 ,

0 ≤ 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 includedinto this model without any changes in its formulation.



Diversification-consistent testInput-output oriented II

B. and Kopa (2013):

θDC I−O II (R0) = minθk ,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 ,

0 ≤ 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 measureDEA model (see Cook and Seiford (2009)).



Equivalent DEA test to portfolio SSD efficiency test

B. and Kopa (2013): We assume that no CVaRαs of the benchmark isequal to zero for αs = s/S , s ∈ Γ = 0, 1, . . . ,S − 1

Let s = arg maxs ∈ 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 benchmarkR0 =

∑ni=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).


On relations between DEA and stochastic dominance efficiency Numerical comparison

Numerical comparison

B. and Kopa (2012B,2013): To compare the power of considered efficiencytests, we consider

historical US stock market data, monthly excess returns from January1982 to December 2011 (360 observations) of 48 representativeindustry stock portfolios that serve as the base assets. The industryportfolios are based on four-digit SIC codes and are from KennethFrench library.

five DEA-risk models where CVaRs at levelsα = 0.5, 0.75, 0.9, 0.95, 0.99, 0.995 are used as the inputs and theexpected return (the most commonly used reward measure) as theoutput.



Traditional Div.-consistent SSD efficiencyDEA tests DEA tests

VRS CRS I I-O I I-O II convex portfolioFood 1.00 1.00 0.89 0.94 0.89 yes noBeer 1.00 0.92 0.82 0.91 0.82 yes no

Smoke 1.00 0.82 1.00 1.00 1.00 yes yesUtil 1.00 0.96 0.89 0.88 0.86 yes no



Conclusions

Standard DEA tests can be equivalent to convex stochasticdominance tests.

DEA tests equivalent to portfolio stochastic dominance efficiencyshould take into account diversification effect leadingdiversification-consistent DEA tests.


References

Contents





5 References


References

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Banker, R.D., Charnes, A., Cooper, W. (1984). Some models for estimating technical and scale inefficiencies in DataEnvelopment Analysis. Man Sci 30 (9), 1078–1092.

Basso, A., Funari, S. (2001). A data envelopment analysis approach to measure the mutual fund performance.European Journal of Operations Research 135, No. 3, 477–492.

Basso, A., Funari, S., (2003). Measuring the performance of ethical mutual funds: A DEA approach. Journal of theOperational Research Society 54, 521-531.

Bawa, V.S., Bodurtha, J.N., Rao, M.R., Suri, H.L. (1985). On Determination of Stochastic Dominance Optimal Sets.Journal of Finance 40, 417–431.

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Branda, M. (2013A). Diversification-consistent data envelopment analysis with general deviation measures. EuropeanJournal of Operational Research 226 (3), 626–635.

Branda, M. (2013B). Reformulations of input-output oriented DEA tests with diversification. Operations ResearchLetters 41 (5), 516–520.

Branda, M. (2013C). Sample approximation technique for mixed-integer stochastic programming problems withexpected value constraints. Accepted to Optimization Letters, DOI: 10.1007/s11590-013-0642-5

Branda, M., Kopa, M. (2010). DEA-risk efficiency of stock indices. Proceedings of 47th EWGFM meeting, T. Tichyand M. Kopa eds., Ostrava: VSB - Technical University of Ostrava.

Branda, M., Kopa, M. (2012). DEA-Risk Efficiency and Stochastic Dominance Efficiency of Stock Indices. CzechJournal of Economics and Finance 62 (2), 106–124.

Branda, M., Kopa, M. (2012B). From stochastic dominance to DEA-risk models: portfolio efficiency analysis. Inproceeding of international workshop on Stochastic Programming for Implementation and Advanced Applications, L.Sakalauskas, A. Tomasgard, S.W. Wallace (Eds.), Vilnius, 13–18.

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.


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References

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