Stochastics and StatisticsRestricted risk measures and robust optimization
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, is a column vector and x′ its transpose. We also note e as the vector with a 1 in every component and . For a given set we denote by and its affine, convex and closed convex hull respectively. We also let be the linear space spanned by S and the relative interior of S. For a given convex set C we
Distortion risk measures for uniform, discrete random variables in and
In this section we will prove that, even in the case of being a finite uniform distribution, there exists distortion risk measures that are not induced by any distortion risk measure . For this, we will need some previous technical lemmas.
Lemma 3.1 If then for any implies Proof Since then such that . From this, implies that . 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 whose uncertainty sets do not coincide with any risk measure in . 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 (5700, 8100), (1300, 9900), ( − 9600, 3000), (8500, −5200)} and
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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