In a risk assessment paradigm, several useful risk evaluation measures have been suggested, such as the value at risk (VaR), the tail conditional expectation (TCE), the distorted risk measures (DRM), and distortion risk measures based on copula (DRMC, in short)—for pertinent reference in this context, see [
8] and the references cited therein. For a real number
in
the TCE of a risk
Y will be:
where
is the
th order quantile corresponding to the cumulative distribution function (c.d.f.)
In practice, the expectation of
Y is computed when the conditional event
is fixed (for example, to be equal to 90% or 95%). Then, let us assume that we encounter with a bivariate random risk (or losses) represented by
It is quite obvious that the TCE of
is unrelated to
Consequently, if we want to control the overflow of the two risks
and
at the same time, the above formula of TCE does not provide a satisfactory remedy to this problem; therefore, one might require a separate formula of TCE which takes into account the excess of the two risks
and
Then, we deal with the amount:
If the bivariate random risks
are independent in nature, then the expression in (
23) only defined the TCE of a univariate risk,
for a fixed conditional event
Therefore, the case of independence is of much importance. Recently, dependence is beginning to play a vital role in portfolio risk modeling. For relative merits and demerits between the assumption(s) of independent and dependent risks, see, [
9] and the references cited therein. However, in reality, the dependence assumption appears to be more reasonable. The above risk measure in (
24) is known as the copula conditional tail expectation (CCTE); for details, see [
10]. Let
be the market-determined values of a portfolio of assets over
m periods, and
be the negative log return (loss) over the
t-th period. Then, given a positive number
, a very small quantity (almost close to 0), the VaR of
Y at the confidence level
will be:
For a detailed study on the computation of VaR used in the pure copula method, we refer the interested reader to [
11]. Then, we state the following proposition (according to Proposition
(p. 85, [
9]) which represents a useful formula to compute the copula-based CCTE (equivalently, TCE in our terminology) given as follows. However, we slightly modified the original proposition (which was not clearly mentioned in [
9]) to make it a correct one.