In the following we consider the expected shortfall from
as an alternative to the VaR and develop a backtesting method for this risk measurement.
The expected shortfall, also
called the Tail-VaR, is in the Delta-Normal Model, i.e.
under the assumptions from (15.7) and (15.8), defined
by
Here
represents the
quantile of the standard normal distribution, where
is the
standard normal distribution function.
Under this model (15.7) and (15.8)
has a standard normal distribution. For a
defined threshold value
we obtain
where
is the standard normal density. For given
observations from a forecast distribution and its realizations
we consider (15.14) as the parameter of interest. Replacing
the expected value with a sample mean and the unobservable
with
 |
(16.16) |
where
in (15.8) is estimated with (15.9) or
(15.10), we obtain an estimator for
 |
(16.17) |
is the random number of times that the threshold value
is exceeded:
Inferencing on the expected shortfall, i.e., on the difference
, we obtain the following
asymptotical result:
N |
(16.18) |
(15.18) can be used to check the adequacy of the Delta-Normal
model.