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On the coherence of Expected Shortfall
Carlo Acerbi,Dirk Tasche +1 more
TL;DR: In this article, the authors 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.
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Abstract: 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.
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R. T. Rockafellar,S Uryasev +1 more
TL;DR: In this paper, a new approach to optimize or hedging a portfolio of financial instruments to reduce risk is presented and tested on applications, which focuses on minimizing Conditional Value-at-Risk (CVaR) rather than minimizing Value at Risk (VaR), but portfolios with low CVaR necessarily have low VaR as well.
Conditional value-at-risk for general loss distributions
TL;DR: Fundamental properties of conditional value-at-risk are derived for loss distributions in finance that can involve discreetness and provides optimization shortcuts which, through linear programming techniques, make practical many large-scale calculations that could otherwise be out of reach.