Testing for Parametric Orderings Efficiency Sergio Ortobelli 1 , 3 - - PowerPoint PPT Presentation

Introduction & Aims and Motivations Classification of Choices Testing Methodology Conclusions References

Testing for Parametric Orderings Efficiency

Sergio Ortobelli1,3 Nikolas Topaloglou4 Matteo Malavasi2,1

2Department of Actuarial Studies and Business Analytics, Macquarie University,

Eastern Road, North Ryde, NSW 2109, Australia

1Department of Management, Economics and Quantitative Methods, University

• f Bergamo, Via dei Caniana 2, 24127 Bergamo, Italy

3Department of Finance, VSB-TUO, Sokolska 33, 701 21 Ostrava, Czech Republic 4DIEES, Athens University of Economics and Business, Patision street 76, 10434

Athens, Greece

CMS 2019 and MMEI 2019

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Introduction & Aims and Motivations Classification of Choices Testing Methodology Conclusions References

Summary

Introduction & Aims and Motivations Classification of Choices Testing Methodology Conclusions References

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Introduction & Aims and Motivations Classification of Choices Testing Methodology Conclusions References

Introduction

Expected Utility Framework (Von Neumann and Morgenstern 1944):

• Finite number of axioms identify agents preferences and

provide general rules under which investors take decision.

• Investors order possible outcomes according to Stochastic

Orderings.

• Stochastic Dominance: First Order of Stochastic Dominance

(FSD), Increasing and concave order (ICV or SSD),Increasing and Convex (ICX), ... Nevertheless, several studies on behavioral finance have shown that investors (Levy and Levy (2002), Kahneman and Tversky (1979) and Barberis and Thaler (2003)) :

• prefer “more”to “less”
• are nor risk seeker nor risk averse

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Introduction & Aims and Motivations Classification of Choices Testing Methodology Conclusions References

Introduction: Aim

The aim of this paper is threefold.

• Extend stochastic dominance conditions for FSD, ICV and

ICX , when return distributions depend on a positive homogeneous and translation equivariant reward measure, a positive homogeneous and translation invariant risk measure and other distributional parameters.

• Define a new class of stochastic orderings, coherent with the

preference of non satiable, nor risk averse nor risk seeker investors, that we call λ-Rachev ordering.

• Propose a methodology to test whether a given portfolio is

efficient, with respect to ICV, ICX and λ-Rachev ordering, exploiting estimation function theory (see Godambe and Thompson (1989)). Finally, we propose an empirical analysis by testing whether the Fama and French market portfolio (see Fama and French 1993) can be considered efficient according to the proposed semi-parametric tests. To apply our methodology to a large scale problem, we also test whether the NYSE and the Nasdaq market

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Introduction & Aims and Motivations Classification of Choices Testing Methodology Conclusions References

Classification of Choices: Stochastic Dominance (1)

Definition

Given a pair of random variables W and Y , defined on a probability space (Ω, F, P), with distribution functions FW and FY respectively, we say:

• W FSD Y if and only if FW (λ) ≤ FY (λ) ∀ λ ∈ R.
• W ICV Y if and only if

λ

−∞ FW (t)dt ≤

λ

−∞ FY (t)dt

∀ λ ∈ R.

• .W CV Y if and only if W ICV Y and E(W ) = E(Y )
• W ICX Y if and only if

∞

λ 1 − FW (t)dt ≥

∞

λ 1 − FY (t)dt

∀ λ ∈ R

• W CX Y if and only if W ICX Y and E(W ) = E(Y ).

Where all the above inequalities are strict for at least a real λ.

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Introduction & Aims and Motivations Classification of Choices Testing Methodology Conclusions References

Classification of Choices: Assumptions

Following Ortobelli (2001), assume that all portfolio gross return distributions belong to a scale invariant family σ+

4 (¯

a), with parameters (µ(X), ρ(X), a1(X), a2(X)), with the following properties:

• 1. Every distribution function F ∈ σ+

4 (¯

a) is weakly determined by the set of parameters (µ(X), ρ(X), a1(X), a2(X)), i.e. F, G ∈ σ+

4 (¯

a), then (µ(X), ρ(X), a1(X), a2(X)) = (µ(Y ), ρ(Y ), a1(Y ), a2(Y )) implies X d = Y , but the converse is not necessarily true.

• 2. µ(X) is a reward measure translation invariant, i.e

µ(X + t) = µ(X) + t for all admissible t and positive homogeneous, and ρ(X) is a risk measure consistent with the additive shift, i.e. ρ(X + t) ≤ ρ(X) ∀t ≤ 0 and positive homogeneous.

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CMS 2019 and MMEI 2019

Theorem A Assume all random admissible portfolios of gross returns belong to a 𝝉𝜐𝑙(𝑏)

• class. Let W=w’Z and Y=y’Z be random returns of a couple of portfolios determined

by the parameters:

)) ( ),..., ( ), ( ), ( (

2 1

W a W a W W

k 

 

and

)) ( ),..., ( ), ( ), ( (

2 1

Y a Y a Y Y

k 

 

where

) ( ) ( Y a W a

i i



for any i=1,…,k-2. Then the following implications holds: 1) Suppose

, ) ( ) ( ) ( ) ( Y Y W W     

then W FSD Y if and only if

) ( ) ( W Y   

. 2) Suppose W does not FSD Y and the reward measure

(.) 

is the mean (i.e.

(.) 

=E(.)) and the scalar measure (.)



is translation invariant i.e.

. ) ( ) ( const X q X     

q, then the following implications holds: 2a) W ICV Y iff

) ( ) ( Y E W E 

and

) ( ) ( Y W   

; 2b) W CV Y iff

) ( ) ( Y E W E 

and

) ( ) ( Y W   

; 2c) W ICX Y iff

) ( ) ( Y E W E 

and

) ( ) ( Y W   

; 2d) W CX Y iff

) ( ) ( Y E W E 

and

) ( ) ( Y W   

; Definition We say that a portfolio W is efficient with respect to a given ordering Rel if it does

not exist another portfolio Y such that Y dominate W with respect to order Rel.

Introduction & Aims and Motivations Classification of Choices Testing Methodology Conclusions References

Classification of Choices: non satiable, nor risk averse nor risk seeker(1)

Consider the Conditional Value at Risk (CVaR), defined as CVaRα(W ) = −1/α α

0 F −1 W (u)du, where F −1 W is the left inverse of

the cumulative distribution function, i.e. F −1

W (u) = inf{x : P [W ≤ x] = FW (x) ≥ u}.

• µ(W ) = −CVaRα(W ) is an admissible reward measure and

always isotonic with ICV

• µ(W ) = −CVaRα(−W ) is an admissible reward measure and

always isotonic with ICX

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Introduction & Aims and Motivations Classification of Choices Testing Methodology Conclusions References

Classification of Choices: non satiable, nor risk averse nor risk seeker(2)

Therefore functionals, coherent with the preference of a non satiable, nor risk seeker nor risk averse investors behavior can be defined as difference of CVaRs, i.e. ∀, α, β ∈ [0, 1] and λ ∈ [0, 1]: γα,β,λ(X) = λCVaRα(X) − (1 − λ)CVaRβ(−X)

Definition

Given two real-valued random variables X, Y defined on (Ω, F, P), we say that X dominates Y in the sense of λ-Rachev ordering with parameters α, β, λ ∈ [0, 1] (i.e. X ≥R

α,β,λ Y ) if and only if

γα,β,λ(X) ≤ γα,β,λ(Y ), α, β ∈ [0, 1].

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Introduction & Aims and Motivations Classification of Choices Testing Methodology Conclusions References

Testing Methodology: Estimation Function

Let R = (R1, ..., RT) be a random vector on a probability space and the distribution family of this vector is parametrized by ξ = (ξ1, ..., ξp). Consider l∗

ξ,q = T

• t=1

n

• i=1

E

• ∂hi(Rt,ξ)

∂ξq

• E
• h2

i (Rt, ξ)

hi(Rt, ξ) q=1,. . . ,p as the optimal EFs. Then, an estimate ˆ ξ of ξ is obtained by solving the system of equations l∗

ξ,q = 0, q=1,. . . ,p. According

to the estimating function theory the optimal EFs obtained as consistent solution of equations l∗

ξ,q = 0, have the property

√ T

• ˆ

ξ − ξ

• → N(0, V −1

EF )

where N is normal distribution with zero mean vector and variance matrix V −1

EF , with VEF = [vi,j]i,j=1,...,p and vi,j = El

∂l∗

ξ,i

∂ξj

• i,j=1,...

p.

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Introduction & Aims and Motivations Classification of Choices Testing Methodology Conclusions References

Testing Methodology: testing for ICX, Step 1

Assume that all gross portfolio returns belong to the scale invariant family στ +

q (¯

a) weakly determined by the first four moments (E[X], ρ(X), a1(X), a2(X)), where ρ(X) = E

• (X − E(X))21/2

a1 =

E[X−E(X))3] E[(X−E(X))2]3/2 , a2 = E[(X−E(X))4] E[(X−E(X))2]2 . Let P be a portfolio with

parameters given by µ(P), ρ(P), s, k and the risk measure ρ(X) is translation invariant. Solve the following optimization problem: max

xi,i=1,...,N x′E [Z]

ρ(x′Z) ≥ ρ(x(P)′Z) a1(x′Z) = s, a2(x′Z) = k

N

• i=1

xi = 1, xi ≥ 0, i = 1, . . . , N Call xicx the solution vector.

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Introduction & Aims and Motivations Classification of Choices Testing Methodology Conclusions References

Testing Methodology: testing for ICX, Step 2 (1)

Perform the following hypothesis test: H1

0 : P is not ICX dominated by xicx′Z

H1

1 : E

• xicx′Z
• − E [P] > 0

ρ

• xicx′Z
• − ρ (P) > 0

Call Ricx

t

=

• Ricx

1 , . . . , Ricx t

• be historical observations of portfolio

gross returns of xicx′Z and let: θ1 = E [f (Rt)] θ2

2 = E

• (f (Rt) − θ1)2

θ3 = E

• (f (Rt) − θ1)3

θ2

2

θ4 = E

• (f (Rt) − θ1)4

θ4

2

where f (Rt) = Rt.

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Introduction & Aims and Motivations Classification of Choices Testing Methodology Conclusions References

Testing Methodology: testing for ICX, Step 2 (2)

The optimal estimators θ = [ θ1, θ2] are the roots of the following optimal EFs: lθ1 =

T

• t=1

−1 θ2

2

(f (Rt) − θ1) + θ3 θ3

2

• θ4 − 1 − θ2

3

• (f (Rt) − θ1)2 − θ2

2 − θ3θ2 (f (Rt) − θ1)

• lθ2 =

T

• t=1

−2 θ3

2(θ4 − 1 − θ2 3)

• (f (Rt) − θ1)2 − θ2

2 − θ3θ2 (f (Rt) − θ1)

• The distribution of the optimal estimator then satisfies the following :

√ T

• θ1
• θ2
• −

θ1 θ2

• →d N
• 0, V −1

where V −1 = θ2

2

4

• 4

2θ3 2θ3 θ4 − 1

• .

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Introduction & Aims and Motivations Classification of Choices Testing Methodology Conclusions References

Testing Methodology: testing for ICX, Step 2 (3)

Then the rejection region for the hypothesis test in (??), is given by C 1 = { √ T

• θ1 − E [P]
• > c1

1,

√ T

• θ2 − ρ(P)
• > c1

2} where

c1

1 and c1 2 are non negative real numbers. For a given test size

α ≤ 0.5, c1

1 and c1 2 are chosen such that

∞

c1

1

∞

c1

2 φ(x, y)dxdy = α,

so that: lim

T→∞ P[reject H1 0|H1 0 is true] ≤

∞

c1

1

∞

c1

2

φ(x, y)dxdy = α

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Introduction & Aims and Motivations Classification of Choices Testing Methodology Conclusions References

Testing Methodology: testing for ICV

The test for ICV efficiency follows the same steps, but with a different optimization problem: max

xi,i=1,...,N x′E [Z]

ρ(x′Z) ≤ ρ(x(P)′Z) a1(x′Z) = s, a2(x′Z) = k

N

• i=1

xi = 1, xi ≥ 0, i = 1, . . . , N The null and alternative hypothesis, test statistic and rejection region are similar to the ICX case.

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Introduction & Aims and Motivations Classification of Choices Testing Methodology Conclusions References

Testing Methodology: testing for λ-Rachev ordering, Step 1 (1)

Assume that all portfolios gross returns belong to a scale invariant family στ +

4 (¯

a) of positive random variables, with parameters: µ(X) = E (f (X)) ρ(X) = E

• (f (X) − µ(X))20.5

a1(X) = E

• (f (X) − µ(X))3

ρ3(X) a2(X) = E

• (f (X) − µ(X))4

ρ4(X) with tα(X) = F −1

X (α) and

f (X) = 0.5 I[X≥tβ(X)]X (1 − β) + 0.5I[X≤tα(X)]X α .

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Introduction & Aims and Motivations Classification of Choices Testing Methodology Conclusions References

Testing Methodology: testing for λ-Rachev ordering, Step 1 (2)

A portfolio P is λ-Rachev efficient, if a dominant portfolio X in the sense of λ-Rachev ordering doesn’t exist, i.e. ∄X = x′Z such that infα,β (γα,β,λ(P) − γα,β,λ(X)) ≥ 0. Thus, we can define a testing methodology for λ-Rachev efficiency. In this case, we propose to solve first the following optimization problem: max

x

min

i,j=1,...,T γαi,βj,λ(P) − γαi,βj,λ(x′Z) N

• k=1

xk = 1, xk ≥ 0, k = 1, . . . , N where T is the number of available observations, αi = i

T and

βj = j

T with i, j = 1, . . . , T.

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Introduction & Aims and Motivations Classification of Choices Testing Methodology Conclusions References

Testing Methodology: testing for λ-Rachev ordering, Step 1 (2)

call xR, i∗, and j∗ the portfolio and indexes solution of the latter

• problem. Then, the test for λ-Rachev ordering efficiency can be

formulated as: H0 : µ(xR′Z) − µ(P) ≤ 0 H1 : µ(xR′Z) − µ(P) > 0 The test statistic is computed similarly to the ICX case where f (X) = λ I[X≥tβj∗ (X)]X (1 − βj∗) + (1 − λ) I[X≤tαi∗ (X)]X αi∗ and tαi∗(X) = F −1

X (αi∗).

The rejection region is given by C 3 = { √ T

• θ1 − µ(P)
• > c3}.

Observe that for λ = 1 and λ = 0 we are proposing alternative tests for ICX and ICV orderings respectively.

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Introduction & Aims and Motivations Classification of Choices Testing Methodology Conclusions References

Empirical Results: Fama and Fench

Descriptive Statistics Benchmark portfolios Mean

• Std. Dev.

Skewness Kurtosis 1 0.316 7.070

• 0.337
• 1.033

2 0.726 5.378

• 0.512

0.570 3 0.885 5.385

• 0.298

1.628 4 0.323 4.812

• 0.291
• 1.135

5 0.399 4.269

• 0.247
• 0.706

6 0.581 4.382

• 0.069
• 0.929

FF 0.462 4.461

• 0.498

2.176

Table: Descriptive statistics of monthly returns in percentage from July 1963 to October 2001 of the Fama and French market portfolio and the six Fama and French benchmark portfolios formed on size and book-to-market equity ratio. Portfolio 1 has low BE/ME and small size, portfolio 2 has medium BE/ME and small size, etc.

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CMS 2019 and MMEI 2019

Ordering % Dominance % Dominance Orderings

• Conf. Level

NYSE Nasdaq ICX 95% 89.286% 82.142% ICV 95% 87.857% 77.857% 0-Rachev=ICV 95% 90.714% 83.571% 0.1-Rachev 95% 79.286% 76.429% 0.2-Rachev 95% 64.286% 62.143% 0.3-Rachev 95% 54.286% 56.429% 0.4-Rachev 95% 59.286% 52.857% 0.5-Rachev 95% 66.429% 57.857% 0.6-Rachev 95% 75% 65.714% 0.7-Rachev 95% 80.714% 74.286% 0.8-Rachev 95% 86.429% 80% 0.9-Rachev 95% 90.714% 84.286% 1-Rachev=ICX 95% 93.571% 87.143%

Table: Percentage of times we get the dominance (λ-Rachev, ICV and ICX dominance) from June 2006 till May 2017 for a total of 140 optimizations.

Introduction & Aims and Motivations Classification of Choices Testing Methodology Conclusions References

Conclusions

• Methodology to asses the efficiency of a given portfolio able

to consider a wide class of investors

• Stochastic dominance conditions, when the return

distributions depend on a reward measure, a risk measure and

• ther distributional parameters, satisfying a minimal set of

assumptions.

• Non satiable, nor risk averse nor risk seeker investors, adjust

their risk attitude according to market conditions.

• Efficiency of the Fama and French market portfolio
• Nasdaq and NYSE composite indexes re often dominated for

all the three stochastic orderings.

• The proposed methodology is general and can be applied to

any stochastic ordering defined by a positive functional, satisfying positive homogeneity and translation equivariance

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Introduction & Aims and Motivations Classification of Choices Testing Methodology Conclusions References

References

Artzner, P. and Delbaen, F. and Eber, J. and Heath, D. (1999). Coherent measures of risk. Mathematical finance, 3, p.203. Barberis, N. and Thaler, R. (2003). A survey of behavioral finance, in Handbook of the Economics of Finance, Elsevier. Biglova, A., and Ortobelli, S., and Rachev, S. T., and Stoyanov, S. (2004). Different approaches to risk estimation in portfolio theory. The Journal of Portfolio Management, 1, p.103. Crowder, M. (1986). On consistency and inconsistency of estimating

• equations. Econometric Theory, 2(3), p.330.

Godambe, V. P. and Thompson, M. E. (1989). An extension of quasi-likelihood estimation. Journal of Statistical Planning and Inference, 22(2) p.137.

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Introduction & Aims and Motivations Classification of Choices Testing Methodology Conclusions References

References

Kahneman D., and Tversky A. (1979). Prospect theory: An analysis of decision under risk. Econometrica: Journal of the econometric society, 47(2), p. 263 Levy, M. and Levy H. (2002). Prospect theory: much ado about nothing?. Management Science, 48(10), p. 1334. Ortobelli, S. (2001). The classification of parametric choices under uncertainty: analysis of the portfolio choice problem. Theory and Decision, 51(2-4), p. 297. Shalit, H. and Yitzhaki, S. (1994). Marginal conditional stochastic

• dominance. Management Science, 5, p.670.

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