Central Limit Theorems for Law-Invariant Coherent Risk Measures

In this paper we study the asymptotic properties of the canonical plugin 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 and identically distributed data, and then extend it to the case of weakly dependent data. Finally, a number of illustrating examples is presented.

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INSTABILITY OF PORTFOLIO OPTIMIZATION UNDER COHERENT RISK MEASURES

Advances in Complex Systems ◽

10.1142/s0219525910002591 ◽

2010 ◽

Vol 13 (03) ◽

pp. 425-437 ◽

Cited By ~ 10

Author(s):

IMRE KONDOR ◽

ISTVÁN VARGA-HASZONITS

Keyword(s):

Portfolio Optimization ◽

Risk Measures ◽

Risk Measure ◽

Expected Shortfall ◽

Optimal Portfolio ◽

The Other ◽

Coherent Risk Measures ◽

Coherent Risk ◽

Large Samples

It is shown that the axioms for coherent risk measures imply that whenever there is a pair of portfolios such that one of them dominates the other in a given sample (which happens with finite probability even for large samples), then there is no optimal portfolio under any coherent measure on that sample, and the risk measure diverges to minus infinity. This instability was first discovered in the special example of Expected Shortfall which is used here both as an illustration and as a springboard for generalization.

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Bigger Is Not Always Safer: A Critical Analysis of the Subadditivity Assumption for Coherent Risk Measures

Risks ◽

10.3390/risks7030091 ◽

2019 ◽

Vol 7 (3) ◽

pp. 91

Author(s):

Hans Rau-Bredow

Keyword(s):

Critical Analysis ◽

Deposit Insurance ◽

Risk Measures ◽

Risk Measure ◽

Coherent Risk Measures ◽

Bank Mergers ◽

Insurance Scheme ◽

Bank Merger ◽

Coherent Risk

This paper provides a critical analysis of the subadditivity axiom, which is the key condition for coherent risk measures. Contrary to the subadditivity assumption, bank mergers can create extra risk. We begin with an analysis how a merger affects depositors, junior or senior bank creditors, and bank owners. Next it is shown that bank mergers can result in higher payouts having to be made by the deposit insurance scheme. Finally, we demonstrate that if banks are interconnected via interbank loans, a bank merger could lead to additional contagion risks. We conclude that the subadditivity assumption should be rejected, since a subadditive risk measure, by definition, cannot account for such increased risks.

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LIQUIDITY RISK AND INSTABILITIES IN PORTFOLIO OPTIMIZATION

International Journal of Theoretical and Applied Finance ◽

10.1142/s0219024916500357 ◽

2016 ◽

Vol 19 (05) ◽

pp. 1650035 ◽

Cited By ~ 12

Author(s):

FABIO CACCIOLI ◽

IMRE KONDOR ◽

MATTEO MARSILI ◽

SUSANNE STILL

Keyword(s):

Portfolio Optimization ◽

Risk Measures ◽

Risk Measure ◽

Estimation Error ◽

Coherent Risk Measures ◽

Important Special Case ◽

Coherent Risk ◽

Finite Samples ◽

Impact Function ◽

The Impact

We show that including a term which accounts for finite liquidity in portfolio optimization naturally mitigates the instabilities that arise in the estimation of coherent risk measures on finite samples. This is because taking into account the impact of trading in the market is mathematically equivalent to introducing a regularization on the risk measure. We show here that the impact function determines which regularizer is to be used. We also show that any regularizer based on the norm [Formula: see text] with [Formula: see text] makes the sensitivity of coherent risk measures to estimation error disappear, while regularizers with [Formula: see text] do not. The [Formula: see text] norm represents a border case: its “soft” implementation does not remove the instability, but rather shifts its locus, whereas its “hard” implementation (including hard limits or a ban on short selling) eliminates it. We demonstrate these effects on the important special case of expected shortfall (ES) which has recently become the global regulatory market risk measure.

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Coherent Risk Measures

Emerging Trends in Smart Banking ◽

10.4018/978-1-4666-5950-6.ch007 ◽

2014 ◽

pp. 115-142

Keyword(s):

At Risk ◽

Extreme Value Theory ◽

Risk Measures ◽

Risk Measure ◽

Value Theory ◽

Extreme Value ◽

Expected Shortfall ◽

Coherent Risk Measures ◽

Coherent Risk ◽

Distortion Risk Measure

This chapter introduces some alternative risk measures to Vale-At-Risk (VaR) calculations: Extreme Value Theory (EVT), Expected Shortfall (ES) and distortion risk measure. It also discusses their more coherent characteristics useful for shoring up the weaknesses of VaR.

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REPRESENTATION OF BSDE-BASED DYNAMIC RISK MEASURES AND DYNAMIC CAPITAL ALLOCATIONS

International Journal of Theoretical and Applied Finance ◽

10.1142/s0219024914500320 ◽

2014 ◽

Vol 17 (05) ◽

pp. 1450032 ◽

Cited By ~ 8

Author(s):

EDUARD KROMER ◽

LUDGER OVERBECK

Keyword(s):

Risk Measures ◽

Risk Measure ◽

Static Case ◽

Coherent Risk Measures ◽

Dynamic Risk Measures ◽

Convex Risk Measures ◽

Coherent Risk ◽

Dynamic Risk ◽

Entropic Risk Measure ◽

Representation Result

In this paper, we provide a new representation result for dynamic capital allocations and dynamic convex risk measures that are based on backward stochastic differential equations (BSDEs). We derive this representation from a classical differentiability result for BSDEs and the full allocation property of the Aumann–Shapley allocation. The representation covers BSDE-based dynamic convex and dynamic coherent risk measures. The results are applied to derive a representation for the dynamic entropic risk measure. Our results are also applicable in a specific way to the static case, where we are able to derive a new representation result for static convex risk measures that are Gâteaux-differentiable.

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Quasi-Hadamard differentiability of general risk functionals and its application

Statistics & Risk Modeling ◽

10.1515/strm-2014-1174 ◽

2015 ◽

Vol 32 (1) ◽

Cited By ~ 1

Author(s):

Volker Krätschmer ◽

Alexander Schied ◽

Henryk Zähle

Keyword(s):

Limit Theorems ◽

Risk Measures ◽

Delta Method ◽

Fine Tuning ◽

Coherent Risk Measures ◽

Coherent Risk ◽

Hadamard Differentiability ◽

Functional Delta Method ◽

Statistical Functionals

AbstractWe apply a suitable modification of the functional delta method to statistical functionals that arise from law-invariant coherent risk measures. To this end we establish differentiability of the statistical functional in a relaxed Hadamard sense, namely with respect to a suitably chosen norm and in the directions of a specifically chosen “tangent space”. We show that this notion of quasi-Hadamard differentiability yields both strong laws and limit theorems for the asymptotic distribution of the plug-in estimators. Our results can be regarded as a contribution to the statistics and numerics of risk measurement and as a case study for possible refinements of the functional delta method through fine-tuning the underlying notion of differentiability.

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Coherent Risk Measures and Convex Combinations of the Conditional Value at Risk (C V@R)

Austrian Journal of Statistics ◽

10.17713/ajs.v31i1.471 ◽

2016 ◽

Vol 31 (1) ◽

pp. 73-75 ◽

Cited By ~ 2

Author(s):

Georg Ch. Pflug

Keyword(s):

At Risk ◽

Value At Risk ◽

Risk Measures ◽

Risk Measure ◽

Conditional Value At Risk ◽

Coherent Risk Measures ◽

Coherent Risk

The conditional-value-at-risk (C V@R) has been widely used as a risk measure. It is well known, that C V@R is coherent in the sense of Artzner, Delbaen, Eber, Heath (1999). The class of coherent risk measures is convex. It was conjectured, that all coherent risk measures can be represented as convex combinations of C V@R’s. In this note we show that this conjecture is wrong.

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