On relations between stochastic dominance efficiency tests and - - PowerPoint PPT Presentation

▶
on relations between stochastic dominance efficiency
SMART_READER_LITE
LIVE PREVIEW

On relations between stochastic dominance efficiency tests and - - PowerPoint PPT Presentation

Dec 06, 2022 243 likes •577 views

On relations between stochastic dominance efficiency tests and DEA-risk models Martin Branda, Milo s Kopa Charles University in Prague Faculty of Mathematics and Physics Department of Probability and Mathematical Statistics 13th

slide-1
SLIDE 1

On relations between stochastic dominance efficiency tests and DEA-risk models

Martin Branda, Miloˇ s Kopa

Charles University in Prague Faculty of Mathematics and Physics Department of Probability and Mathematical Statistics

13th International Conference on Stochastic Programming 8-12 July 2013, Bergamo

M.Branda, M.Kopa (Charles University) SP 2013 1 / 32

slide-2
SLIDE 2

Contents

1 Motivation 2 Second Order Stochastic Dominance 3 Data Envelopment Analysis 4 Numerical comparison

M.Branda, M.Kopa (Charles University) SP 2013 2 / 32

slide-3
SLIDE 3

Motivation

Contents

1 Motivation 2 Second Order Stochastic Dominance 3 Data Envelopment Analysis 4 Numerical comparison

M.Branda, M.Kopa (Charles University) SP 2013 3 / 32

slide-4
SLIDE 4

Motivation

A bridge

Data envelopment analysis (production theory, returns to scale, radial/slack-based/directional distance models, primal/dual formulations, multiobjective opt. – Pareto efficiency, chance constraints), large literature: handbook 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.

M.Branda, M.Kopa (Charles University) SP 2013 4 / 32

slide-5
SLIDE 5

Second Order Stochastic Dominance

Contents

1 Motivation 2 Second Order Stochastic Dominance 3 Data Envelopment Analysis 4 Numerical comparison

M.Branda, M.Kopa (Charles University) SP 2013 5 / 32

slide-6
SLIDE 6

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.

M.Branda, M.Kopa (Charles University) SP 2013 6 / 32

slide-7
SLIDE 7

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}. (1)

M.Branda, M.Kopa (Charles University) SP 2013 7 / 32

slide-8
SLIDE 8

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)...

M.Branda, M.Kopa (Charles University) SP 2013 8 / 32

slide-9
SLIDE 9

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.

M.Branda, M.Kopa (Charles University) SP 2013 9 / 32

slide-10
SLIDE 10

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 (2) 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 (2) is strictly

  • positive. Otherwise, portfolio x is convex SSD efficient.

M.Branda, M.Kopa (Charles University) SP 2013 10 / 32

slide-11
SLIDE 11

Data Envelopment Analysis

Contents

1 Motivation 2 Second Order Stochastic Dominance 3 Data Envelopment Analysis 4 Numerical comparison

M.Branda, M.Kopa (Charles University) SP 2013 11 / 32

slide-12
SLIDE 12

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?”

M.Branda, M.Kopa (Charles University) SP 2013 12 / 32

slide-13
SLIDE 13

Data Envelopment Analysis

Data Envelopment Analysis (DEA)

Charnes et al (1978): fractional programming formulation max J

j=1 yj0Yj0

K

k=1 wk0Zk0

s.t. J

j=1 yj0Yji

K

k=1 wk0Zki

≤ 1, i = 1, . . . , n, wk0 ≥ 0, k = 1, . . . , K, yj0 ≥ 0, j = 1, . . . , J. The unit 0 is then DEA efficient if the optimal value is equal to 1,

  • therwise it is DEA inefficient.

M.Branda, M.Kopa (Charles University) SP 2013 13 / 32

slide-14
SLIDE 14

Data Envelopment Analysis

DEA

Constant Return to Scale (CRS)

Dual problem to linear programming formulation: min θ s.t.

n

  • i=1

xiYji ≥ Yj0, j = 1, . . . , J,

n

  • i=1

xiZki ≤ θ · Zk0, k = 1, . . . , K, xi ≥ 0, i = 1, . . . , n. Model with Constant Return to Scale (CRS).

M.Branda, M.Kopa (Charles University) SP 2013 14 / 32

slide-15
SLIDE 15

Data Envelopment Analysis

DEA

Variable Return to Scale (VRS)

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

M.Branda, M.Kopa (Charles University) SP 2013 15 / 32

slide-16
SLIDE 16

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.

M.Branda, M.Kopa (Charles University) SP 2013 16 / 32

slide-17
SLIDE 17

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)

M.Branda, M.Kopa (Charles University) SP 2013 17 / 32

slide-18
SLIDE 18

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, (3) ≤ θk ≤ 1, ϕ ≥ 1,

n

  • i=1

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

M.Branda, M.Kopa (Charles University) SP 2013 18 / 32

slide-19
SLIDE 19

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, (4) 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.

M.Branda, M.Kopa (Charles University) SP 2013 19 / 32

slide-20
SLIDE 20

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, (5) 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.

M.Branda, M.Kopa (Charles University) SP 2013 20 / 32

slide-21
SLIDE 21

Data Envelopment Analysis

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, (6) 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)).

M.Branda, M.Kopa (Charles University) SP 2013 21 / 32

slide-22
SLIDE 22

Data Envelopment Analysis

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).

M.Branda, M.Kopa (Charles University) SP 2013 22 / 32

slide-23
SLIDE 23

Numerical comparison

Contents

1 Motivation 2 Second Order Stochastic Dominance 3 Data Envelopment Analysis 4 Numerical comparison

M.Branda, M.Kopa (Charles University) SP 2013 23 / 32

slide-24
SLIDE 24

Numerical comparison

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.

M.Branda, M.Kopa (Charles University) SP 2013 24 / 32

slide-25
SLIDE 25

Numerical comparison

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

M.Branda, M.Kopa (Charles University) SP 2013 25 / 32

slide-26
SLIDE 26

Numerical comparison

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.

M.Branda, M.Kopa (Charles University) SP 2013 26 / 32

slide-27
SLIDE 27

Numerical comparison M.Branda, M.Kopa (Charles University) SP 2013 27 / 32

slide-28
SLIDE 28

Numerical comparison M.Branda, M.Kopa (Charles University) SP 2013 28 / 32

slide-29
SLIDE 29

Numerical comparison M.Branda, M.Kopa (Charles University) SP 2013 29 / 32

slide-30
SLIDE 30

Numerical comparison

References

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. Branda, M. (2013). Diversification-consistent data envelopment analysis with general deviation measures, European Journal of Operational Research 226 (3), 626–635. Branda, M., Kopa, M. (2010). DEA-risk efficiency of stock indices. Proceedings of 47th EWGFM meeting, T. Tich´ y and M. Kopa eds., Ostrava: Vˇ SB - Technical University of Ostrava. Branda, M., Kopa, M. (2012A). DEA-Risk Efficiency and Stochastic Dominance Efficiency of Stock Indices, Czech Journal of Economics and Finance 62(2), 106–124. Branda, M., Kopa, M. (2012B). From stochastic dominance to DEA-risk models: portfolio efficiency analysis, In proceeding 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. (2013). On relations between DEA-risk models and stochastic dominance efficiency tests, Central European Journal of Operations Research, available

  • nline. DOI: 10.1007/s10100-012-0283-2

Charnes, A., Cooper, W., Rhodes, E. (1978). Measuring the efficiency of decision-making units, European Journal of Operational Research 2, 429-–444. Chekhlov, A., Uryasev, S., Zabarankin. M. (2003). Portfolio Optimization With Drawdown Constraints, B. Scherer (Ed.) Asset and Liability Management Tools, Risk Books, London.

M.Branda, M.Kopa (Charles University) SP 2013 30 / 32

slide-31
SLIDE 31

Numerical comparison

References

Fishburn, P.C. (1974). Convex stochastic dominance with continuous distribution functions, Journal of Economic Theory 7, 143–158. Hanoch, G., Levy, H. (1969). The Efficient Analysis of Choices Involving Risk, Review of Economic Studies, 335–346. Kopa, M., Chovanec, P. (2008). A Second-order Stochastic Dominance Portfolio Efficiency Measure, Kybernetika 44(2), 243–258. Kuosmanen, T. (2004). Efficient diversification according to stochastic dominance criteria, Management Science 50(10), 1390–1406. 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. (2008). Data envelopment analysis of mutual funds based on second-order stochastic dominance, European Journal of Operational Research 189, 230–244. Ogryczak, W., Ruszczynski, A. (2002). Dual stochastic dominance and related mean-risk models, SIAM Journal on Optimization 13, 60–78. 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, J. of Banking and Finance 26, 1443–1471.

M.Branda, M.Kopa (Charles University) SP 2013 31 / 32

slide-32
SLIDE 32

Numerical comparison

Thank you for your attention.

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

M.Branda, M.Kopa (Charles University) SP 2013 32 / 32