Robustness and bootstrap techniques in portfolio efficiency tests

Robustness and bootstrap techniques in portfolio efficiency tests Dept. of Probability and Mathematical Statistics, Charles University, Prague, Czech Republic July 8, 2013

Motivation

Portfolio selection model Mean risk models max λ Λ m(λ ϱ) νr(λ ϱ) or min λ Λ r(λ ϱ) s.t. m(λ ϱ) µ ϱ is a random vector of assets returns maximizing mean m(λ ϱ) & minimizing risk r(λ ϱ) risk measures (variance, semi variance,...,var, CVaR) risk or return parameter (ν, µ) If we do not know the parametrs efficiency frontier

Portfolio selection model Utility functions approach max λ Λ Eu(λ ϱ) u is an utility function (non-decreasing function) maximizing expected utility choice of utility function - risk attitude If no information about utility function is known stochastic dominance portfolio efficiency with respect to stochastic dominance

Efficient portfolios Crucial question of portfolio efficiency (in the sense of optimality): Is a given portfolio a maximizer of expected utility for at least one considered utility function? Crucial question of portfolio efficiency (in the sense of admissibility): Does there exist a better portfolio (having higher expected utility) than a given portfolio for all considered investors (utility functions) If not - the portfolio is not the optimal choice for any considered investor (reduction of the set of feasible strategies).

Randomness in the model Known returns of investments no randomness deterministic optimization model Returns are random with known probability distributions optimization under risk Returns are random, probability distribution is not precisely known robust optimization Returns are random with no information about their probability distribution optimization under uncertainty

Summary We will consider: maximizing expected utility problems an utility function is not known all admissible utility functions are used returns are random with partially known probability distribution

Outline Introduction to stochastic dominance and portfolio efficiency Robust stochastic dominance portfolio efficiency tests 1 fixed scenario probabilities (Kopa (2010)) 2 fixed scenario values (Kopa (2012)) Robustness via contamination (Dupačová and Kopa (2012,2013)) Bootstrap techniques in portfolio efficiency testing (Post and Kopa (2013A, 2013B))

First order stochastic dominance (FSD) - notation Consider N assets and a random vector of their returns ϱ with distribution P. A decision maker may combine assets into portfolios and all portfolio possibilities are given by Λ = {λ R N 1 λ = 1, λ n 0, n = 1, 2,..., N}. Let F ϱ λ(x) denote the cumulative probability distribution function of returns of portfolio λ. Definition Portfolio λ Λ dominates portfolio τ Λ by the first-order stochastic dominance (ϱ λ FSD ϱ τ ) if F ϱ λ(x) F ϱ τ (x) x R with strict inequality for at least one x R.

First order stochastic dominance (FSD) - interpretation Other equivalent definitions: ϱ λ FSD ϱ τ if Eu(ϱ λ) Eu(ϱ τ ) for all utility functions and strict inequality holds for at least some utility function. No non-satiable decision maker prefers portfolio τ to portfolio λ and at least one prefers λ to τ. F 1 1 ϱ λ (y) Fϱ τ (y) y [0, 1] with strict inequality for at least one y [0, 1]. VaR α ( ϱ λ) VaR α ( ϱ τ ) α [0, 1] with strict inequality for at least one α [0, 1].

Portfolio efficiency with respect to FSD FSD efficiency: admissibility vs. optimality Definition (Admissibility case) A given portfolio τ Λ is FSD inefficient if there exists portfolio λ Λ such that ϱ λ FSD ϱ τ. Otherwise, portfolio τ is FSD efficient. Definition (Optimality case) Portfolio τ Λ is FSD efficient if it is the optimal solution of max λ Λ Eu(ϱ λ) for at least some utility function, i.e., there exists u such that Otherwise, τ is FSD non-optimal. Eu(ϱ τ ) Eu(ϱ λ) 0 λ Λ. See Kuosmanen (2004), Kopa & Post (2009). We focus on the admissibility case.

Portfolio efficiency test with respect to FSD In order to find a FSD dominating portfolio λ, we may solve the following problem, Dupačová and Kopa (2013): where Theorem ξ P (τ ) = min λ h P(λ, τ ) (1) s.t. H P (λ, τ ) 0 λ Λ. h P (λ, τ ) := min y R (F ϱ λ(y) F ϱ τ (y)) H P (λ, τ ) := max y R (F ϱ λ(y) F ϱ τ (y)) A given portfolio τ is FSD efficient if and only if ξ P (τ ) = 0. If ξ P (τ ) < 0 then the optimal portfolio λ in (1) is FSD efficient and it dominates portfolio τ by FSD.

FSD efficiency testing - practical issues So far we have assumed an arbitrary distribution P. The FSD efficiency test: can be rewritten as: How to solve the problem? ξ P (τ ) = min λ h P(λ, τ ) (2) s.t. H P (λ, τ ) 0 λ Λ. ξ P (τ ) = min λ h P(λ, τ ) (3) s.t. F ϱ λ(y) F ϱ τ (y) 0 y R (4) λ Λ. To assume a specific distribution (normal,elliptical) and rewrite (4) in a few constraints To assume a scenario approach, i.e. P is a discrete probability distribution, and again rewrite (4) in finitely many constraints.

FSD efficiency testing - scenario approach From now on, we assume that ϱ takes S values r s = (r1 s, r 2 s,..., r N s ), called scenarios, with probabilities p 1, p 2,..., p S. Contrary to the former tests, Kuosmanen (2004), Kopa & Post (2009), we do NOT assume equiprobable scenarios. The scenarios are collected in the matrix R = For any portfolio λ Λ, let ( Rλ) [k] be the k-th greatest element of ( Rλ), i.e. ( Rλ) [1] ( Rλ) [2]... ( Rλ) [S] and let I (λ) be a permutation of the index set I = {1, 2,..., S} such that r i(λ) λ = ( Rλ) [i]. Accordingly, we can order the corresponding probabilities and we denote pi λ = p i(λ). Hence, r 1 r 2. r S p λ i = P( ϱλ = ( Rλ) [i] ). Moreover, we consider cumulative probabilities: qs λ = s i=1 pλ i. The same notation is applied for the tested portfolio τ = (τ 1, τ 2,..., τ N ).

FSD efficiency testing - scenario approach, Dupačová and Kopa (2013) Following quantile approach to FSD the inequality of VaRs: VaR α (ϱ λ) VaR α (ϱ τ ) need not be verified in all α (0, 1], but only in at most S particular points: Theorem A portfolio λ dominates portfolio τ with respect to FSD (ϱ λ FSD ϱ τ ) if and only if VaR q τ s ( ϱ λ) VaR q τ s ( ϱ τ ) for all s = 1, 2,..., S with strict inequality for at least one q τ s. A similar result can be shown in terms of cumulative distribution functions instead of VaRs.

FSD efficiency testing - scenario approach Consider the following measure: ξ(τ, R, p) = min b s,λ b s (5) s=1 s.t. VaR q τ s ( ϱ λ) VaR q τ s ( ϱ τ ) b s, s = 1,..., S b s 0, λ Λ. s = 1,..., S The objective function of (5) represents the sum of differences between VaRs of a portfolio λ and VaRs of the tested portfolio τ. The differences are considered in points qs τ s = 1,..., S and must be nonpositive. The other points need not be taken into account, because VaR α is a piecewise constant function of α. Theorem A given portfolio τ is FSD efficient if and only if ξ(τ, R, p) = 0. If ξ(τ, R, p) < 0 then the optimal portfolio λ in (5) is FSD efficient and it dominates portfolio τ by FSD.

FSD efficiency - neighborhood robustness-fixed scenario values, Dupačová and Kopa (2013) Another approach to robustness: ɛ-fsd efficiency test. Assume that the probability distribution P of random returns ϱ takes again values r s, s = 1, 2,..., S but with other probabilities p = ( p 1, p 2,..., p S ). We define the distance between P and P as d( P, P) = max i p i p i. Definition A given portfolio τ Λ is ɛ-fsd inefficient if there exists portfolio λ Λ and P such that d( P, P) ɛ with ϱ λ FSD ϱ τ. Otherwise, portfolio τ is ɛ-fsd efficient. The introduced ɛ-fsd efficiency guarantees stability of the FSD efficiency classification with respect to small changes (prescribed by parameter ɛ) in probability vector p. A given portfolio τ is ɛ-fsd efficient if and only if no portfolio λ FSD dominates τ neither for the original probabilities p nor for arbitrary probabilities p from ɛ-neighborhood of the original vector p. Also stability with respect to changes in values of scenarios r s (fixed probabilities) can be analyzed

ɛ-fsd efficiency test For testing ɛ-fsd efficiency of a given portfolio τ we modify (5) in order to introduce a new measure of ɛ-fsd efficiency: ξ ɛ (τ, R, p) = min b s,λ, p, q s=1 b s (6) s.t. VaR q τ s ( ϱ λ) VaR q τ s ( ϱ τ ) b s, s = 1,..., S s q s τ = p i τ, s = 1,..., S i=1 p i = 1, ɛ p i p i ɛ, p i 0, i = 1, 2,..., S i=1 λ Λ b s 0, s = 1,..., S Theorem Portfolio τ Λ is ɛ-fsd efficient if and only if ξ ɛ (τ, R, p) given by (6) is equal to zero.

Second order stochastic dominance definitions Let F r λ(x) denote the cumulative probability distribution function of returns of portfolio λ. The twice cumulative probability distribution function of returns of portfolio λ is defined as F (2) r λ (y) = y F r λ(x)dx. (7) Definition Portfolio λ Λ dominates portfolio τ Λ by the second-order stochastic dominance (r λ SSD r τ ) if and only if F (2) (2) r λ (y) F r τ (y) y R with strict inequality for at least one y R. Definition A given portfolio τ Λ is SSD inefficient if there exists portfolio λ Λ such that r λ SSD r τ. Otherwise, portfolio τ is SSD efficient.

Second order stochastic dominance interpretation Other equivalent definitions of SSD relation: r λ SSD r τ if Eu(r λ) Eu(r τ ) for all concave utility functions and strict inequality holds for at least some concave utility function. No non-satiable and risk averse decision maker prefers portfolio τ to portfolio λ and at least one prefers λ to τ. F 2 2 r λ (y) Fr τ (y) y [0, 1] with strict inequality for at least one y [0, 1], where F 2 r λ is a cumulated quantile function. CVaR α ( r λ) CVaR α ( r τ ) α [0, 1] with strict inequality for at least one α [0, 1], where CVaR α ( r λ) = min v + 1 v R,z t R + 1 α p t z t (8) t=1 s.t. z t x t λ v, t = 1, 2,..., S

Portfolio efficiency testing, Dupačová and Kopa (2012) Let S 1 ξ(τ, X, p) = min a s,λ s=0 a s s.t. CVaR q λ s ( r λ) CVaR q λ s ( r τ ) a s, s = 0, 1,..., S 1 a s 0, s = 0, 1,..., S 1 λ Λ. (9) Theorem A given portfolio τ is SSD efficient if and only if ξ(τ, X, p) = 0. If ξ(τ, X, p) < 0 then the optimal portfolio λ in (9) is SSD efficient and it dominates portfolio τ by SSD.

Nonlinear integer SSD efficiency test Expressions for CVaR: CVaR α ( r λ) = min v + 1 v R,z t R + 1 α p t z t (10) t=1 s.t. z t x t λ v, t = 1, 2,..., S and the similar expression can be considered for portfolio τ. However, for portfolio τ, we will rather use the dual formulation: CVaR α ( r τ ) = max κ t R + 1 1 α κ t ( x t τ ) (11) t=1 s.t. κ t p t, t = 1, 2,..., S κ t = 1 α t=1 See Rockafellar & Uryasev (2000, 2002) for more details.

Nonlinear integer SSD efficiency test - cont. For each portfolio λ, we consider a permutation matrix Π = {π i,j } S i,j=1 that arranges the losses of portfolio λ in ascending order, that is, S j=1 π 1,j( x j λ) S j=1 π 2,j( x j λ)... S j=1 π S,j( x j λ) and S j=1 π s,j( x j λ) = ( X λ) [s]. Using the same permutation matrix, we can reorder also the vector of probabilities: S j=1 π s,jp j = p λ s. And cumulative probabilities are: q λ s = s i=1 S j=1 π i,jp j.

Nonlinear integer SSD efficiency test, Kopa (2012) ξ(τ, X, p) = 1 s.t. v s + 1 qs 1 λ min a s,v s,z s,t,κ s,t,q λ s,λ s=1 a s 1 p tz s,t 1 qs 1 λ t=1 z s,t x t λ v s, s, t = 1, 2,..., S κ s,t p t, s, t = 1, 2,..., S κ s,t = 1 qs 1, λ s = 1, 2,..., S t=1 q λ 0 = 0, q λ s = s i=1 π s,j ( x j λ) j=1 π s,j = s=1 κ s,t( x t τ ) a s, t=1 π i,j p j, s = 1, 2,..., S 1 j=1 π s+1,j ( x j λ), s = 1, 2,..., S 1 j=1 π s,j = 1, π s,j {0, 1}, j, s = 1, 2,..., S j=1 a s 0, s = 1, 2,..., S, z s,t, κ s,t 0, s, t = 1, 2,..., S λ Λ. s = 1, 2,..., S

Robustness in SSD portfolio efficiency testing - notation Following Dupačová & Kopa (2012), we consider ɛ-ssd efficiency approach as a robustification of the classical SSD portfolio efficiency. It guarantees stability of the SSD efficiency classification with respect to small changes (prescribed by parameter ɛ > 0) in probability vector p. Assume that the probability distribution P of random returns r takes again values x s, s = 1, 2,..., S but with other probabilities p = ( p 1, p 2,..., p S ). We define the distance between P and P as d( P, P) = max i p i p i. Definition A given portfolio τ Λ is ɛ-ssd inefficient if there exists portfolio λ Λ and P such that d( P, P) ɛ with r λ SSD r τ. Otherwise, portfolio τ is ɛ-ssd efficient. A portfolio τ is ɛ-ssd efficient if and only if no portfolio λ SSD dominates τ neither for the original probabilities p nor for arbitrary probabilities p from ɛ-neighborhood of the original vector p.

ɛ-ssd portfolio efficiency test ξ ɛ (τ, X, p) = 1 s.t. v s + 1 q s 1 λ min a s a s,v s,z s,t,κ s,t, p i q s λ,λ s=1 1 p t z s,t 1 q s 1 λ t=1 z s,t x t λ v s, s, t = 1, 2,..., S κ s,t p t, s, t = 1, 2,..., S κ s,t = 1 q s 1, λ s = 1, 2,..., S t=1 q λ 0 = 0, q λ s = i=1 j=1 κ s,t ( x t τ ) a s, t=1 s π i,j p j, s = 1, 2,..., S 1 s = 1, 2,..., S

ɛ-ssd portfolio efficiency test - cont π s,j ( x j λ) π s+1,j ( x j λ), s = 1, 2,..., S 1 j=1 j=1 π s,j = π s,j = 1, j, s = 1, 2,..., S s=1 j=1 a s 0, s = 1, 2,..., S z s,t, κ s,t 0, s, t = 1, 2,..., S π i,j {0, 1}, i, j = 1, 2,..., S λ Λ p i = 1 i=1 ɛ p i p i ɛ, i = 1, 2,..., S p i 0, i = 1, 2,..., S.

Robustness in SSD portfolio efficiency testing - equiprobable scenarios, Kopa (2010) We consider P containing all discrete probability distributions with S equiprobable scenarios and scenario matrix R P from the ɛ-neighbourhood of P 0, that is, satisfying d(p, P 0 ) ɛ. Let matrix Υ = {υ ij } S i,j=1 be defined as Υ = R P R. Moreover, if D(P, P 0 ) = max i,j υ ij then a robust counterpart of the Kuosmanen (2004) SSD efficiency test can be formulated as follows: min 1 a s.t. W (R + Υ)τ (R + Υ)λ a 1 W = 1, W 1 = 1 ɛ υ ij ɛ i, j = 1,..., S x X.

Robustness analysis via Contamination Intorduced for general stochastic programs by Dupačová et. al. (1996, 2000, 2006...) Assume that a problem was solved for original distribution P. Changes in probability distribution P are modeled using contaminated distributions P(t) := (1 t)p + tq, t [0, 1] with Q another fixed probability distribution such that optimal value function ϕ(q) is finite. Via contamination, robustness analysis wrt. changes in P gets reduced to much simpler analysis of parametric program with scalar parameter t. One can compute lower and upper bound for optimal value function ϕ(t). We apply this notion in the easiest manner - the alternative distribution is just one scenario (can be seen as stress test scenario or worst case scenario)

Directional FSD portfolio efficiency with respect to an additional scenario, Dupačová and Kopa (2013) Assume continuous original distribution of returns P. Consider a contamination of the original distribution of returns by additional scenario s: P(t) = (1 t)p + tδ {s}, t [0, 1]. Let ϱ(t) be a random variable with distribution P(t). Similarly to (1) we consider: ξ P(t) (τ ) = min λ h P(t)(λ, τ ) (12) s.t. H P(t) (λ, τ ) 0 λ Λ where H P(t) (λ, τ ) = max y R (F ϱ(t) λ(y) F ϱ(t) τ (y)) and h P(t) (λ, τ ) = min y R (F ϱ(t) λ(y) F ϱ(t) τ (y))

Directional FSD portfolio efficiency with respect to an additional scenario - cont. A robust version of FSD efficiency: Definition A given portfolio τ Λ is directionally FSD inefficient with respect to additional scenario s if for each t exists λ(t) such that ϱ(t) λ(t) FSD ϱ(t) τ. Moreover, a given portfolio τ Λ is directionally FSD efficient with respect to additional scenario s if there is no (t, λ(t)) such that ϱ(t) λ(t) FSD ϱ(t) τ. The definition classifies portfolio τ as directionally FSD efficient (inefficient) with respect to additional scenario s if τ is FSD efficient (inefficient) when using the original distribution P as well as in any contaminated case P(t).

Directional FSD portfolio efficiency with respect to an additional scenario - necessary and sufficient conditions, Dupačová and Kopa (2013) Theorem A given portfolio τ Λ is directionally FSD efficient with respect to additional scenario s if and only if min ξ P(t) (τ ) = 0. t [0,1] Using (12): min ξ P(t) (τ ) = min h P(t)(λ, τ ) (13) t [0,1] λ,t s.t. H P(t) (λ, τ ) 0 λ Λ. Theorem A given portfolio τ Λ is directionally FSD inefficient with respect to additional scenario s if and only if max ξ P(t) (τ ) < 0. t [0,1] It leads to minimax...

Directional FSD portfolio efficiency with respect to an additional scenario - sufficient condition, Dupačová and Kopa (2013) If H P(t) (λ, τ ) is concave in t then ξ P(t) (τ ) is quasiconcave in t and ξ P(t) (τ ) min{ξ P(0) (τ ), ξ P(1) (τ )}. As a consequence we can derive the following sufficient condition for directional FSD efficiency with respect to additional scenario s. Theorem If H P(t) (λ, τ ) is concave in t. τ is FSD efficient when using original probability distribution P τ argmax λ Λ s λ then τ is directionally FSD efficient with respect to s.

Directional FSD portfolio inefficiency with respect to an additional scenario - sufficient condition, Dupačová and Kopa (2013) Since F ϱ(t) λ(y) is linear in t for all λ Λ and y R we may derive the following sufficient condition: Theorem If there exists λ Λ such that ϱ λ FSD ϱ τ and s λ s τ then τ is directionally FSD inefficient with respect to s. The proof makes use of an upper bound to show that λ Λ satisfying ϱ λ FSD ϱ τ and s λ s τ FSD dominates τ in any contaminated case, i.e. ϱ(t) λ FSD ϱ(t) τ.

Directional SSD portfolio efficiency with respect to the additional scenario, Dupačová and Kopa (2012) For a contamination parameter t [0, 1], we assume that the random return ϱ(t) takes values r 1, r 2,..., r S+1 with probabilities p(t) = ((1 t)p 1, (1 t)p 2,..., (1 t)p S, t). We denote the extended scenario matrix by R, that is, Definition R = ( R r S+1 A given portfolio τ Λ is directionally SSD inefficient with respect to r S+1 if it exists t 0 > 0 such that for every t [0, t 0 ] there is a portfolio λ(t) Λ satisfying ϱ(t) λ(t) SSD ϱ(t) τ. Definition A given portfolio τ Λ is directionally SSD efficient with respect to r S+1 if there does not exist t 0 > 0 such that for every t [0, t 0 ] there is a portfolio λ(t) Λ satisfying ϱ(t) λ(t) SSD ϱ(t) τ. ).

Robust portfolio efficiency con t, Dupačová and Kopa (2012) Using contamination bounds we can derive a sufficient condition for directional SSD efficiency and directional SSD inefficiency. Theorem Let τ Λ be a SSD efficient portfolio for the original distribution P. Let r S+1 τ r S+1 λ for all λ Λ. (14) Then τ Λ is directionally SSD efficient with respect to r S+1. Theorem Let τ Λ be a SSD inefficient portfolio for the original distribution P. If there exists a portfolio λ Λ such that CVaR q λ s ( ϱ λ) CVaR q λ s ( ϱ τ ) < 0, s = 0, 1,..., S 1(15) r S+1 λ min((rτ ) [1], r S+1 τ ) (16) then τ is directionally SSD inefficient with respect to r S+1.

Bootstrap in SD portfolio efficiency tests, Post and Kopa (2013A, 2013B) Idea: generate 10 000 pseudosamples from the original distribution apply a portfolio efficiency test in each pseudosample compute the bootstrap p-value of zero hypothesis - portfolio is NSD efficient to deal with bias - Entire Distance Bootstrap Results on US stock market data: The higher order of stochastic dominance - the smaller bootstrap p-value (Post and Kopa 2013A) EDB gives better results than naive bootstrap (Post and Kopa 2013B)

End of the presentation Thank you for your attention. kopa@karlin.mff.cuni.cz

Main quoted references J. Dupačová & M. Kopa (2013): Robustness of optimal portfolios under risk and stochastic dominance constraints, European Journal of Operational Research, DOI: http://dx.doi.org/10.1016/j.ejor. T. Post & M. Kopa (2013A): General Linear Formulations of Stochastic Dominance Criteria, European Journal of Operational Research, 230, 2, 321-332. T. Post & M. Kopa (2013B): Aggregate Investor Preferences and Beliefs: A comment, forthcomming in Journal of Empirical Finance. J. Dupačová & M. Kopa (2012): Robustness in stochastic programs with risk constraints, Annals of Operations Research 200, 1, 55-74. M. Kopa (2012): Robustness in SSD portfolio efficiency testing, Proceedings of International Conference on Operations Research, pp. 527-532 Springer, Berlin, 2012. M. Kopa (2010): Measuring of second-order stochastic dominance portfolio efficiency, Kybernetika 46, 3, 488 500. M. Kopa & T. Post (2009): A portfolio optimality test based on the first-order stochastic dominance criterion, Journal of Financial and Quantitative Analysis 44, 5, 1103 1124.