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1
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Are law-invariant risk functions concave on distributions?

100%
Acciaio B., Svindland G.
Dependence Modeling
|
2013
|
tom 1
54-64
EN
While it is reasonable to assume that convex combinations on the level of random variables lead to a reduction of risk (diversification effect), this is no more true on the level of distributions. In the latter case, taking convex combinations corresponds to adding a risk factor. Hence, whereas asking for convexity of risk functions defined on random variables makes sense, convexity is not a good property to require on risk functions defined on distributions. In this paper we study the interplay between convexity of law-invariant risk functions on random variables and convexity/concavity of their counterparts on distributions. We show that, given a law-invariant convex risk measure, on the level of distributions, if at all, concavity holds true. In particular, this is always the case under the additional assumption of comonotonicity.
2
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Measuring herd behavior: properties and pitfalls

100%
Lee W., Ahn J. Y.
Dependence Modeling
|
2017
|
tom 5
|
nr 1
316-329
EN
Herd behavior is an important economic phenomenon, especially in the context of the recent financial crises. Prior studies propose several measures to quantify herd behavior. In this paper, we show that these measures reflect different perspectives on this behavior, and hence, their interpretation requires great care. Taking a critical attitude toward existing herd behavior measures, we study their properties and pitfalls in detail.
3
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Seven Proofs for the Subadditivity of Expected Shortfall

75%
Embrechts P., Wang R.
Dependence Modeling
|
2015
|
tom 3
|
nr 1
EN
Subadditivity is the key property which distinguishes the popular risk measures Value-at-Risk and Expected Shortfall (ES). In this paper we offer seven proofs of the subadditivity of ES, some found in the literature and some not. One of the main objectives of this paper is to provide a general guideline for instructors to teach the subadditivity of ES in a course. We discuss the merits and suggest appropriate contexts for each proof.With different proofs, different important properties of ES are revealed, such as its dual representation, optimization properties, continuity, consistency with convex order, and natural estimators.
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