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218 RISK MANAGEMENT AND VALUE CREATION IN FINANCIAL INSTITUTIONS
Therefore, VaR is the same kind of risk measure as the LPM 0
. 398
LPM 0
(–VaR α
) and (–)VaR α
look at the same point of the cumulative probability
distribution—but from opposite angles. 399 However, since LPM 1
is a
necessary condition for second-order stochastic dominance, 400 VaR is not a
risk measure that is compatible with maximizing the expected utility. Because
VaR only measures the probability, but not the extent (severity), of the
losses when they occur, 401 it shows merely first-order stochastic dominance. 402
VaR is, therefore, a suitable risk measure for risk-neutral investors. 403 Thus,
Schröder comes to the conclusion that only LPM n
measures with n > 0 provide
the basis for the development of a generalized VaR measure that takes into
account risk aversion. 404
Similarly, approaches from extreme value theory 405 are being used to
try to answer the question “how bad is bad?” by considering not only the
probability but also the extent to which losses occur beyond a critical threshold.
These approaches can be defined as:
−VaR
E[–X | X ≤ VaR α
] = LPM ( − VaR ) = ( − 1
VaR − X ) f ( X ) dX
α
∫
−∞
α
α
(5.41)
They measure the average of the future values of the return X of a position
or portfolio, conditional on the fact that the value is below a certain threshold
value or VaR at a certain quantile α (VaR α
) of the value or return distribution.
Therefore, they are also called “tail conditional expectations.” 406
These measures can best be explained in a simulation context. If, for example,
VaR is calculated at the α = 1% level and we have 10,000 simulation
runs, then VaR 1%
is the largest of the 100 smallest realizations in the simulation,
whereas the tail conditional expectation calculates the average of the
100 smallest realizations, 407 thus being more conservative than VaR. 408 Given
that the tail conditional expectation can be related back to the lower partial
moment one (LPM 1
) of the distribution (as was shown in the above equa-
398 VaR can be, therefore, viewed as a special case of the shortfall risk measures; see
Schröder (1996), p. 1.
399 See Guthoff et al. (1998), pp. 32+.
400 See Hirschbeck (1998), p. 271, and his references to the literature. For an extensive
discussion of this point see Guthoff et al. (1998), pp. 24+.
401 See Johanning (1998), p. 57.
402 See Guthoff et al. (1998), p. 33.
403 See Schröder (1996), pp. 1–2.
404 See Schröder (1996), pp. 12+.
405 See, for example, Embrechts et al. (1997).
406 See Artzner et al. (1999), p. 204.
407 See Artzner et al. (1997), p. 68.
408 See Artzner et al. (1999), p. 204.