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Conditional Expectation as Quantile Derivative

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  • Dirk Tasche

Abstract

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.

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File URL: http://arxiv.org/pdf/math/0104190
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Paper provided by arXiv.org in its series Papers with number math/0104190.

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Date of creation: Apr 2001
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Handle: RePEc:arx:papers:math/0104190

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Cited by:
  1. Acerbi, Carlo & Tasche, Dirk, 2002. "On the coherence of expected shortfall," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1487-1503, July.
  2. Tasche, Dirk, 2002. "Expected shortfall and beyond," Journal of Banking & Finance, Elsevier, vol. 26(7), pages 1519-1533, July.
  3. Michael B. Gordy, 2002. "A risk-factor model foundation for ratings-based bank capital rules," Finance and Economics Discussion Series 2002-55, Board of Governors of the Federal Reserve System (U.S.).
  4. Carlo Acerbi & Dirk Tasche, 2001. "Expected Shortfall: a natural coherent alternative to Value at Risk," Papers cond-mat/0105191, arXiv.org.
  5. Rosen, Dan & Saunders, David, 2009. "Analytical methods for hedging systematic credit risk with linear factor portfolios," Journal of Economic Dynamics and Control, Elsevier, vol. 33(1), pages 37-52, January.

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