  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.
