3,843 research outputs found
On the coherence of Expected Shortfall
Expected Shortfall (ES) in several variants has been proposed as remedy for
the defi-ciencies of Value-at-Risk (VaR) which in general is not a coherent
risk measure. In fact, most definitions of ES lead to the same results when
applied to continuous loss distributions. Differences may appear when the
underlying loss distributions have discontinuities. In this case even the
coherence property of ES can get lost unless one took care of the details in
its definition. We compare some of the definitions of Expected Shortfall,
pointing out that there is one which is robust in the sense of yielding a
coherent risk measure regardless of the underlying distributions. Moreover,
this Expected Shortfall can be estimated effectively even in cases where the
usual estimators for VaR fail.
Key words: Expected Shortfall; Risk measure; worst conditional expectation;
tail con-ditional expectation; value-at-risk (VaR); conditional value-at-risk
(CVaR); tail mean; co-herence; quantile; sub-additivity.Comment: 18 pages, LaTeX + pdfLaTeX, appendix adde
Expected Shortfall: a natural coherent alternative to Value at Risk
We discuss the coherence properties of Expected Shortfall (ES) as a financial
risk measure. This statistic arises in a natural way from the estimation of the
"average of the 100p % worst losses" in a sample of returns to a portfolio.
Here p is some fixed confidence level. We also compare several alternative
representations of ES which turn out to be more appropriate for certain
purposes.Comment: to be published on "Wilmott Magazine" (http://www.wilmott.com
Minimality via second variation for a nonlocal isoperimetric problem
We discuss the local minimality of certain configurations for a nonlocal
isoperimetric problem used to model microphase separation in diblock copolymer
melts. We show that critical configurations with positive second variation are
local minimizers of the nonlocal area functional and, in fact, satisfy a
quantitative isoperimetric inequality with respect to sets that are
-close. The link with local minimizers for the diffuse-interface
Ohta-Kawasaki energy is also discussed. As a byproduct of the quantitative
estimate, we get new results concerning periodic local minimizers of the area
functional and a proof, via second variation, of the sharp quantitative
isoperimetric inequality in the standard Euclidean case. As a further
application, we address the global and local minimality of certain lamellar
configurations.Comment: 35 page
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