Stochastics and Statistics
Restricted risk measures and robust optimization

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Highlights

  • We study uncertainty sets associated to coherent and distortion risk measures.

  • By focusing on linear and affine random variables we justify the use of new uncertainty sets.

  • These new sets are expansions of classical sets and are related to the expectation correction of a risk measure.

  • These new sets can be used to mitigate estimation errors of the Conditional Value at Risk.

Abstract

In this paper we consider characterizations of the robust uncertainty sets associated with coherent and distortion risk measures. In this context we show that if we are willing to enforce the coherent or distortion axioms only on random variables that are affine or linear functions of the vector of random parameters, we may consider some new variants of the uncertainty sets determined by the classical characterizations. We also show that in the finite probability case these variants are simple transformations of the classical sets. Finally we present results of computational experiments that suggest that the risk measures associated with these new uncertainty sets can help mitigate estimation errors of the Conditional Value-at-Risk.

Introduction

Coherent risk measures and their relation to robust optimization have received significant attention in the literature (Artzner, Delbaen, Eber, Heath, 1999, Bertsimas, Brown, 2009, Natarajan, Pachamanova, Sim, 2009, Shapiro, Dentcheva, Ruszczyński, 2009, Ben-Tal, Ghaoui, Nemirovski, 2009, Wächter, Mazzoni, 2013). It is known that every coherent risk measure is associated with a precisely determined convex uncertainty set with properties that are strongly tied to the axioms characterizing coherent risk measures (e.g. Bertsimas and Brown (2009); Natarajan et al. (2009)). Similar results have also been given for a special class of coherent risk measures known as distortion risk measures, which include the widely used Conditional Value-at-Risk (Bertsimas, Brown, 2009, Pichler, Shapiro, Shapiro, 2013). All these characterizations are based on the restrictions imposed by the coherence or distortion axioms on the actions of the coherent risk measure over all possible random variables. However, in many settings, the random variables considered are either an affine or linear function of a, potentially correlated, vector of random parameters. A classical example is portfolio optimization (see for example Markowitz (1952); Konno and Yamazaki (1991); Black and Litterman (1992); Cvitanić and Karatzas (1992); Krokhmal et al. (2002); Zymler et al. (2011); Lim et al. (2011); Kawas and Thiele (2011); Fertis et al. (2012); Kolm et al. (2014)) where the random return of a portfolio is usually modeled as a weighted linear combination of the random returns of individual assets (with weights equal to the fraction invested in a given asset) plus a possibly null constant representing investment in a riskless asset. In this paper we show that imposing the coherence and distortion axioms only on random variables that are a linear, or affine linear function of a vector of random variables allows the inclusion of uncertainty sets that are deemed invalid by the classical characterizations. In particular, we show that in the finite probability case these additional sets at least include certain expansions of the classical sets. We also show that such expansions are in turn related to the common practice of taking the convex combination of a risk measure with the expected value. More specifically, we show that risk measures associated to these expansions are affine combinations of a risk measure with the expected value.

Finally we present computational experiments that suggest that the risk measures associated with these uncertainty sets can help mitigate estimation errors of the Conditional Value-at-Risk.

The rest of this paper is organized as follows. In Section 2 we give some notation and background on risk measures and robust optimization. In Section 3 we show the existence of uncertainty sets that do not fall in the classical characterizations, but do yield distortion risk measures on the subspace of random variables that are either affine or linear functions of a fixed random vector. In Section 4 we show that the risk measures associated to these uncertainty sets are affine combinations of a risk measure with the expected value. Then, in Section 5 we present some results of computational experiments showing that these uncertainty sets could be useful to mitigate estimation errors. Finally, in Section 6 we present some final remarks.

Section snippets

Notation

Throughout the paper we will use bold letters to denote column vectors, and we will use an apostrophe to denote the transposition operation. Thus, xRd is a column vector and x′ its transpose. We also note e as the vector with a 1 in every component and eN:=1Ne. For a given set SRn we denote by aff(S),conv(S) and error(S) its affine, convex and closed convex hull respectively. We also let lin(S) be the linear space spanned by S and ri(S) the relative interior of S. For a given convex set C we

Distortion risk measures for uniform, discrete random variables in V and Vo

In this section we will prove that, even in the case of P being a finite uniform distribution, there exists distortion risk measures ρ:VR that are not induced by any distortion risk measure ρ:L1(Ω,F,P)R. For this, we will need some previous technical lemmas.

Lemma 3.1

If 0ri(conv(supp(u˜))) then for any v˜Vo,v˜0a.s. implies v˜=0a.s.

Proof

Since v˜Vo, then xRd such that v˜(ω)=xu˜(ω),ωΩ. From this, v˜0a.s. implies that xu˜(ω)0,ωsupp(u˜). If x = 0 the result is direct. By contradiction, assume

Epsilon scaling of a risk measure

From Lemma 3.7 we know that there exists risk measures represented by qΔ^N whose uncertainty sets do not coincide with any risk measure in Δ^+N. However, it is possible to give a different characterization of these uncertainty sets, providing a natural geometrical interpretation of these measures.

Consider for example the finite uniform probability over the N = 5 points in supp(u˜)={ui}i=1n={(8600,5000), (5700, 8100), (1300, 9900), ( − 9600, 3000), (8500, −5200)} and q=(27/100,27/100,27/100,27/

Computational stability of epsilon scalings

In this section we present a computational example that shows that epsilon scalings seem to be less susceptible to estimation errors when approximated using samples. The need for such estimations is common in applications (e.g. Lagos et al. (2011); Vielma et al. (2009)) and, unfortunately, risk measures such as the Conditional Value-at-Risk (CVaR) measure have been shown to be highly susceptible to estimation errors in this setting (Lim et al., 2011). For this reason we study how using the

Conclusions

We have shown that, at least for finite uniform distributions, the family of uncertainty sets associated with distortion risk measures over affine or linear random variables is strictly larger that those associated with distortion risk measures over arbitrary random variables. In particular, we have shown that certain expansions of uncertainty sets associated with distortion risk measures also yield distortion risk measures over affine or linear random variables. This effectively expands the

Acknowledgments

Daniel Espinoza was supported by FONDECYT grant number 1110024 and ICM grant number P10-024-F and Eduardo Moreno was supported by Anillo grant number ACT88 and FONDECYT grant number 1130681. The authors also thank two anonymous referees for their helpful feedback.

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