Risk capital allocation by coherent risk measures based on one-sided moments

In this paper, we propose a methodology for constructing uncertainty sets within the framework of robust optimization for linear optimization problems with uncertain parameters. Our approach relies on decision maker risk preferences. Specifically, we utilize the theory of coherent risk measures initiated by Artzner et al. (1999) [Artzner, P., F. Delbaen, J. Eber, D. Heath. 1999. Coherent measures of risk. Math. Finance 9 203-228.], and show that such risk measures, in conjunction with the support of the uncertain parameters, are equivalent to explicit uncertainty sets for robust optimization. We explore the structure of these sets in detail. In particular, we study a class of coherent risk measures, called distortion risk measures, which give rise to polyhedral uncertainty sets of a special structure that is tractable in the context of robust optimization. In the case of discrete distributions with rational probabilities, which is useful in practical settings when we are sampling from data, we show that the class of all distortion risk measures (and their corresponding polyhedral sets) are generated by a finite number of conditional value-at-risk (CVaR) measures. A subclass of the distortion risk measures corresponds to polyhedral uncertainty sets symmetric through the sample mean. We show that this subclass is also finitely generated and can be used to find inner approximations to arbitrary, polyhedral uncertainty sets.

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“…These are coherent risk measures (Fischer 2001). Moreover, they are representable by the family of measures…”

Section: Coherent Risk Measures and Convex Uncertaintymentioning

confidence: 99%

Constructing Uncertainty Sets for Robust Linear Optimization

Bertsimas

Brown

2009

Operations Research

291

167

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“…Currently, more and more research in the theory of coherent risk measures is related to applications to problems of finance rather than to the study of "pure" risk measures. In particular, the problem of capital allocation was considered in [6], [14], [20], [21], [26], [38], [49], [61]; the problem of pricing and hedging was investigated in [8], [11], [12], [14], [18], [20], [33], [35], [42], [48], [54], [56], [59]; the problem of the optimal portfolio choice was studied in [15], [51], [53]; the equilibrium problem was considered in [7], [8], [15], [32], [37], [44]. This list is very far from being complete; for example, on the Gloria Mundi web page over two hundred papers are related to coherent risk measures.…”

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confidence: 99%

Coherent Measurement of Factor Risks

Cherny

Madan

2006

SSRN Journal

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Abstract. We propose a new procedure for the risk measurement of large portfolios. It employs the following objects as the building blocks:• coherent risk measures introduced by Artzner, Delbaen, Eber, and Heath;• factor risk measures introduced in this paper, which assess the risks driven by particular factors like the price of oil, S&P500 index, or the credit spread; • risk contributions and factor risk contributions, which provide a coherent alternative to the sensitivity coefficients. We also propose two particular classes of coherent risk measures called Alpha V@R and Beta V@R, for which all the objects described above admit an extremely simple empirical estimation procedure. This procedure uses no model assumptions on the structure of the price evolution.Moreover, we consider the problem of the risk management on a firm's level. It is shown that if the risk limits are imposed on the risk contributions of the desks to the overall risk of the firm (rather than on their outstanding risks) and the desks are allowed to trade these limits within a firm, then the desks automatically find the globally optimal portfolio.

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“…for all X ∈ L p (Ω, F, P) we arrive at the so called one-sided moment coherent risk measure (see Fischer (2003)). The associated distortion function ψ ρ satisfies ψ ρ (t) = t + a(1 − t)t 1/p .…”

Section: Resultsmentioning

confidence: 99%

Central Limit Theorems for Law-Invariant Coherent Risk Measures

Belomestny

Krätschmer

2012

Journal of Applied Probability

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In this paper we study the asymptotic properties of the canonical plug-in estimates for law-invariant coherent risk measures. Under rather mild conditions not relying on the explicit representation of the risk measure under consideration, we first prove a central limit theorem for independent identically distributed data and then extend it to the case of weakly dependent ones. Finally, a number of illustrating examples is presented.

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Risk capital allocation by coherent risk measures based on one-sided moments

Cited by 120 publications

References 4 publications

Constructing Uncertainty Sets for Robust Linear Optimization

Constructing Uncertainty Sets for Robust Linear Optimization

Coherent Measurement of Factor Risks

Central Limit Theorems for Law-Invariant Coherent Risk Measures

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