2001/04/19 by Dirk Tasche, Tasche, Dirk · 1 citation
Decision Sciences · Economics, Econometrics and Finance · Mathematics · #60E10 (Secondary) #91B32 #91B82 (Primary) #FOS: Economics and business #FOS: Mathematics #Fuzzy Systems and Optimization #Multi-Criteria Decision Making #Optimization and Control (math.OC) #Pricing of Securities (q-fin.PR) #Probability (math.PR) #Risk and Portfolio Optimization #math.OC #math.PR #msc:60E10 #msc:91B32 #msc:91B82 #q-fin.PR
paper · pdf · doi:10.48550/arxiv.math/0104190
12 pages
arxiv created 2001/04/19 · openalex publication_date 2001/04/19 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For 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.