(2003). Risk Management: Value at Risk and Beyond. Journal of the American Statistical Association: Vol. 98, No. 462, pp. 494-494.

Risk Management: Value at Risk and Beyond

When estimating the risk of a P&L from historical data or Monte Carlo simulation, the robustness of the estimate is important. We argue here that Hampel’s classical notion of qualitative robustness is not suitable for risk measurement, and we propose and analyze a refined notion of robustness that applies to tail-dependent law-invariant convex risk measures on Orlicz spaces. This concept captures the tradeoff between robustness and sensitivity and can be quantified by an index of qualitative robustness. By means of this index, we can compare various risk measures, such as distortion risk measures, in regard to their degree of robustness. Our analysis also yields results of independent interest such as continuity properties and consistency of estimators for risk measures, or a Skorohod representation theorem for ψ-weak convergence.

Comparative and qualitative robustness for law-invariant risk measures

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Non Additive Measure And Integral

We address the problem of risk sharing among agents using a two-parameter class of quantile-based risk measures, the so-called Range-Value-at-Risk (RVaR), as their preferences. The family of RVaR includes the Value-at-Risk (VaR) and the Expected Shortfall (ES), the two popular and competing regulatory risk measures, as special cases. We first establish an inequality for RVaR-based risk aggregation, showing that RVaR satisfies a special form of subadditivity. Then, the risk sharing problem is solved through explicit construction. Three relevant issues on optimal allocation are investigated: extra sources of randomness, comonotonicty, and model uncertainty. We show that, in general, a robust optimal allocation exists if and only if none of the underlying risk measures is a VaR. Practical implications of our main results for risk management and policy makers are discussed. In particular, in the context of the calculation of regulatory capital, we provide some general guidelines on how a regulatory risk measure can lead to certain desirable or undesirable properties of risk sharing among firms. Several novel advantages of ES over VaR from the perspective of a regulator are thereby revealed.

Quantile-Based Risk Sharing

Recent developments in set optimization are surveyed and extended including various set relations as well as fundamental constructions of a convex analysis for set- and vector-valued functions, and duality for set optimization problems Extensive sections with bibliographical comments summarize the state of the art Applications to vector optimization and financial risk measures are discussed along with algorithmic approaches to set optimization problems

Set optimization: A rather short introduction

We provide a variety of results for quasiconvex, law-invariant functionals defined on a general Orlicz space, which extend well-known results from the setting of bounded random variables. First, we show that Delbaen’s representation of convex functionals with the Fatou property, which fails in a general Orlicz space, can always be achieved under the assumption of law-invariance. Second, we identify the class of Orlicz spaces where the characterization of the Fatou property in terms of norm-lower semicontinuity by Jouini, Schachermayer and Touzi continues to hold. Third, we extend Kusuoka’s representation to a general Orlicz space. Finally, we prove a version of the extension result by Filipovic and Svindland by replacing norm-lower semicontinuity with the (generally non-equivalent) Fatou property. Our results have natural applications to the theory of risk measures.

Fatou property, representations, and extensions of law-invariant risk measures on general Orlicz spaces

The risk of financial positions is measured by the minimum amount of capital to raise and invest in eligible portfolios of traded assets in order to meet a prescribed acceptability constraint. We investigate nondegeneracy, finiteness and continuity properties of these risk measures with respect to multiple eligible assets. Our finiteness and continuity results highlight the interplay between the acceptance set and the class of eligible portfolios. We present a simple, alternative approach to the dual representation of convex risk measures by directly applying to the acceptance set the external characterization of closed, convex sets. We prove that risk measures are nondegenerate if and only if the pricing functional admits a positive extension which is a supporting functional for the underlying acceptance set, and provide a characterization of when such extensions exist. Finally, we discuss applications to set-valued risk measures, superhedging with shortfall risk, and optimal risk sharing.

Measuring risk with multiple eligible assets

We discuss risk measures representing the minimum amount of capital a financial institution needs to raise and invest in a pre-specified eligible asset to ensure it is adequately capitalized. Most of the literature has focused on cash-additive risk measures, for which the eligible asset is a risk-free bond, on the grounds that the general case can be reduced to the cash-additive case by a change of numeraire. However, discounting does not work in all financially relevant situations, especially when the eligible asset is a defaultable bond. In this paper, we fill this gap by allowing general eligible assets. We provide a variety of finiteness and continuity results for the corresponding risk measures and apply them to risk measures based on value-at-risk and tail value-at-risk on L p spaces, as well as to shortfall risk measures on Orlicz spaces. We pay special attention to the property of cash subadditivity, which has been recently proposed as an alternative to cash additivity to deal with defaultable bonds. For important examples, we provide characterizations of cash subadditivity and show that when the eligible asset is a defaultable bond, cash subadditivity is the exception rather than the rule. Finally, we consider the situation where the eligible asset is not liquidly traded and the pricing rule is no longer linear. We establish when the resulting risk measures are quasiconvex and show that cash subadditivity is only compatible with continuous pricing rules.

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Beyond cash-additive risk measures: when changing the numéraire fails

The purpose of this paper is to dispel some common misunderstandings about capital adequacy rules based on Expected Shortfall. We establish that, from a theoretical perspective, Expected Shortfall based regulation can provide a misleading assessment of tail behaviour, does not necessarily protect liability holders' interests much better than Value-at-Risk based regulation, and may also allow for regulatory arbitrage when used as a global solvency measure. We also show that, for a value-maximizing financial institution, the benefits derived from protecting its franchise may not be sufficient to disincentivize excessive risk taking. We further interpret our results in the context of portfolio risk measurement. Our results do not invalidate the possible merits of Expected Shortfall as a risk measure but instead highlight the need for its cautious use in the context of capital adequacy regimes and of portfolio risk control.

https://papers.ssrn.com/sol3/Delivery.cfm/SSRN_ID2661737_code1754159.pdf?abstractid=2536549&mirid=1

Cosimo Munari

Papers

Fatou property, representations, and extensions of law-invariant risk measures on general Orlicz spaces

Measuring risk with multiple eligible assets

Measuring risk with multiple eligible assets

Beyond cash-additive risk measures: when changing the numéraire fails

Unexpected Shortfalls of Expected Shortfall: Extreme Default Profiles and Regulatory Arbitrage