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:




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