The Conditional Tail Expectation (or Tail Value-at-Risk) measures the
average of losses above the Value at Risk for some given confidence
level, that is \(E[X|X > \mathrm{VaR}(X)]\) where \(X\) is the loss random
variable.
CTE is a generic function with, currently, only a method for
objects of class "aggregateDist".
For the recursive, convolution and simulation methods of
aggregateDist, the CTE is computed from the definition
using the empirical cdf.
For the normal approximation method, an explicit formula exists:
$$\mu + \frac{\sigma}{(1 - \alpha) \sqrt{2 \pi}}
e^{-\mathrm{VaR}(X)^2/2},$$
where \(\mu\) is the mean, \(\sigma\) the standard
deviation and \(\alpha\) the confidence level.
For the Normal Power approximation, the explicit formula given in
Casta<U+00F1>er et al. (2013) is
$$\mu + \frac{\sigma}{(1 - \alpha) \sqrt{2 \pi}}
e^{-\mathrm{VaR}(X)^2/2}
\left( 1 + \frac{\gamma}{6} \mathrm{VaR}(X) \right),$$
where, as above, \(\mu\) is the mean, \(\sigma\) the standard
deviation, \(\alpha\) the confidence level and \(\gamma\) is
the skewness.
the CTE is computed from the
definition using integrate.