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 Ortobelli 1 , 3 Nikolas Topaloglou 4 Matteo Malavasi 2 , 1 2 Department of Actuarial Studies and Business Analytics, Macquarie University, Eastern Road, North Ryde, NSW 2109, Australia 1 Department of Management, Economics and Quantitative Methods, University of Bergamo, Via dei Caniana 2, 24127 Bergamo, Italy 3 Department of Finance, VSB-TUO, Sokolska 33, 701 21 Ostrava, Czech Republic 4 DIEES, Athens University of Economics and Business, Patision street 76, 10434 Athens, Greece CMS 2019 and MMEI 2019 1 / 30

Introduction & Aims and Motivations Classification of Choices Testing Methodology Conclusions References Summary Introduction & Aims and Motivations Classification of Choices Testing Methodology Conclusions References 2 / 30

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 3 / 30

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 4 / 30 problem, we also test whether the NYSE and the Nasdaq market

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 F W and F Y respectively, we say: • W FSD Y if and only if F W ( λ ) ≤ F Y ( λ ) ∀ λ ∈ R . � λ � λ • W ICV Y if and only if −∞ F W ( t ) dt ≤ −∞ F Y ( 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 − F W ( t ) dt ≥ λ 1 − F Y ( 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 . 5 / 30

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 ) , a 1 ( X ) , a 2 ( 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 ) , a 1 ( X ) , a 2 ( X )), i.e. F , G ∈ σ + 4 (¯ a ), then ( µ ( X ) , ρ ( X ) , a 1 ( X ) , a 2 ( X )) = ( µ ( Y ) , ρ ( Y ) , a 1 ( Y ) , a 2 ( 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. 7 / 30

CMS 2019 and MMEI 2019 Theorem A Assume all random admissible portfolios of gross returns belong to a 𝝉𝜐 𝑙 (𝑏) class. Let W=w ’ Z and Y=y ’ Z be random returns of a couple of portfolios determined by the parameters:     and ( ( W ), ( W ), a ( W ),..., a ( W )) ( ( Y ), ( Y ), a ( Y ),..., a ( Y )) k  k  1 2 1 2  for any i=1, … ,k-2. Then the following implications holds: where a ( W ) a ( Y ) i i   ( W ) ( Y )     1) Suppose then W FSD Y if and only if . ( ) ( ) , Y W   ( 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.      q, then the following implications holds: ( ) ( ) . X q X const     2a) W ICV Y iff and ; E ( W ) E ( Y ) ( W ) ( Y )     2b) W CV Y iff and ; E ( W ) E ( Y ) ( W ) ( Y )     2c) W ICX Y iff and ; E ( W ) E ( Y ) ( W ) ( Y )     2d) W CX Y iff and ; E ( W ) E ( Y ) ( W ) ( Y ) 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 � α 0 F − 1 W ( u ) du , where F − 1 CVaR α ( W ) = − 1 /α W is the left inverse of the cumulative distribution function, i.e. F − 1 W ( u ) = inf { x : P [ W ≤ x ] = F W ( x ) ≥ u } . • µ ( W ) = − CVaR α ( W ) is an admissible reward measure and always isotonic with ICV • µ ( W ) = − CVaR α ( − W ) is an admissible reward measure and always isotonic with ICX 15 / 30

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]: γ α,β,λ ( X ) = λ CVaR α ( X ) − (1 − λ ) CVaR β ( − 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 γ α,β,λ ( X ) ≤ γ α,β,λ ( Y ), α, β ∈ [0 , 1]. 16 / 30

Introduction & Aims and Motivations Classification of Choices Testing Methodology Conclusions References Testing Methodology: Estimation Function Let R = ( R 1 , ..., R T ) be a random vector on a probability space and the distribution family of this vector is parametrized by � � ∂ h i ( R t ,ξ ) E � � T n ∂ξ q ξ = ( ξ 1 , ..., ξ p ). Consider l ∗ ξ, q = � � h i ( R t , ξ ) E h 2 i ( R t , ξ ) t =1 i =1 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 � � √ ˆ → N (0 , V − 1 T ξ − ξ EF ) where N is normal distribution with zero mean vector and variance � ∂ l ∗ � matrix V − 1 EF , with V EF = [ v i , j ] i , j =1 ,..., p and v i , j = E l ξ, i i,j =1,... ∂ξ j p . 17 / 30

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 � ( X − E ( X )) 2 � 1 / 2 ( E [ X ] , ρ ( X ) , a 1 ( X ) , a 2 ( X )), where ρ ( X ) = E E [ X − E ( X )) 3 ] E [ ( X − E ( X )) 4 ] a 1 = E [( X − E ( X )) 2 ] 3 / 2 , a 2 = 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: x i , i =1 ,..., N x ′ E [ Z ] max ρ ( x ′ Z ) ≥ ρ ( x ( P ) ′ Z ) a 1 ( x ′ Z ) = s , a 2 ( x ′ Z ) = k N � x i = 1 , x i ≥ 0 , i = 1 , . . . , N i =1 Call x icx the solution vector. 18 / 30