The expected shortfall, also known as average value-at-risk or conditional value-at-risk, is a coherent risk measure defined as
for for some .
The average value-at-risk can also be computed by the following useful formula
where is the value-at-risk of , that is
We know that is monotone, i.e., whenever (a.s.).
Is it, however, true that whenever (a.s)? This property is known as strict monotonicity.
We know that this holds when is replaced with the expectation operation, whereas, for this not true.
The strict inequality above is because of the strict monotonicity property of the integral and the second inequality is because (a.s.) implies .
Note: and this is because .
Note: We have denoted by the expectation of the random variable conditioned by the event .
Now note that by convention; this is because and . Take such that (a.s.). Then (a.s.). In that case, however, we may have and ; take for instance and
over the domain .