The value at risk discussed in the previous chapter is not the single measure of the market risk. In this section we introduce an alternative risk measure. In addition we discuss how to estimate the measure given extremely high loss.
(A1) ![]() ![]() |
(Monotonicity) |
(A2)
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(Subadditivity) |
(A3)
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(Positive homogeneity) |
(A4)
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(Translation equivariance) |
The VaR does not meet condition (A2) in certain situations. Let
and
, for example, be i.i.d. and both can take on the value
0 or 100 with probability
and
Then
can be 0, 100 and 200 with probability
,
and
respectively. For
and
, for
example, for
, it holds that
The expected shortfall, on the other hand, is a coherent risk
measure that always fulfills all four conditions. It also gives a
more intuitive view of the actual risk of extreme losses than the
Value-at-Risk. The VaR only depends on the probability of losses
above the -quantile
, but it doesn't say anything about
whether these losses are always just a little above the threshold
or whether there are also losses that are much larger than
that need to be taken into account. In contrast the expected
shortfall is the expected value of the potential losses from
and depends on the actual size of the losses.
The Value-at-Risk is simply a quantile and can be, for example,
estimated as a sample quantile
, where
is the empirical distribution of a sample of
negative values, i.e., losses, from the past. As was discussed at
the beginning of the chapter, this particular estimator of
, which is for the typical VaR-level of 0.95 and 0.99,
is often too optimistic. An alternative VaR estimator, which has
the possibility of reflecting extreme losses better, is the POT or
the Hill quantile estimator.
Analogous estimators for the expected shortfall are easy to
derive. This risk measure is closely related to the average excess
function when , as immediately can be seen from the
definition: