Conditional Expectation as Quantile Derivative

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dc.creatorTasche, Dirk
dc.date2001-04-19
dc.date.accessioned2026-07-07T12:11:07Z
dc.date.available2026-07-07T12:11:07Z
dc.descriptionFor a linear combination of random variables, fix some confidence level and consider the quantile of the combination at this level. We are interested in the partial derivatives of the quantile with respect to the weights of the random variables in the combination. It turns out that under suitable conditions on the joint distribution of the random variables the derivatives exist and coincide with the conditional expectations of the variables given that their combination just equals the quantile. Moreover, using this result, we deduce formulas for the derivatives with respect to the weights of the variables for the so-called expected shortfall (first or higher moments) of the combination. Finally, we study in some more detail the coherence properties of the expected shortfall in case it is defined as a first conditional moment. Key words: quantile; value-at-risk; quantile derivative; conditional expectation; expected shortfall; conditional value-at-risk; coherent risk measure.
dc.description12 pages
dc.identifierhttps://arxiv.org/abs/math/0104190
dc.identifierhttp://arxiv.org/abs/math/0104190
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/210127
dc.subjectProbability
dc.subjectOptimization and Control
dc.subjectPricing of Securities
dc.subject91B82 (Primary); 91B32, 60E10 (Secondary)
dc.titleConditional Expectation as Quantile Derivative
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