Identifying a formula for the (lower)
conditional tail expectation (CTE) of a normal distribution that does not
explicitly include integral signs but instead refers to the unit normal density
function and the unit normal cumulative distribution function
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The (lower) conditional tail expectation (CTE) of a random
variable for a given cut-off, , is
defined as the expected value of the random given that it is below the relevant
cut-off. It is thus very closely aligned with the Tail Value-at-Risk
(TVaR) of the distribution, since the TVaR is the CTE for a specific cut-off.
For a normal
distribution the CTE is thus:
The unit normal distribution function, , and
the unit Normal cumulative distribution function, , are
defined as follows:
Substituting in the
above formula for the CTE we obtain, where ):
We can re-express this in a form that relies only on and by
noting the following:
where is
constant and furthermore as and decays
sufficiently fast to zero as we have
. Hence: