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
Castañer 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.