  The paper derives saddlepoint expansions for conditional expectations in the
form of $\mathsf{E}[\overline{X} | \overline{\mathbf Y} = {\mathbf a}]$ and
$\mathsf{E}[\overline{X} | \overline{\mathbf Y} \geq {\mathbf a}]$ for the
sample mean of a continuous random vector $(X, {\mathbf Y}^\top)$ whose joint
moment generating function is available. Theses conditional expectations
frequently appear in various applications, particularly in quantitative finance
and risk management. Using the newly developed saddlepoint expansions, we
propose fast and accurate methods to compute the sensitivities of risk measures
such as value-at-risk and conditional value-at-risk, and the sensitivities of
financial options with respect to a market parameter. Numerical studies are
provided for the accuracy verification of the new approximations.
