ABSTRACT
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311
Appendix 14.A: The Replica Symmetric Solution . . . . . . . . . . . . . . . . . . . . . 312
A MONG THE SEVERAL EXISTING RISK MEASURES in the context of portfolio optimization,expected shortfall (ES) has certainly gained increasing popularity in recent years. In several practical applications, ES is starting to replace the classical Value-at-Risk (VaR).
There are a number of reasons for this. For a given threshold probability b, the VaR is
defined so that with probability b the loss will be smaller than VaR. This definition only
gives the minimum loss one can reasonably expect but does not tell anything about the
typical value of that loss which can be measured by the conditional value-at-risk (CVaR,
which is the same as ES for the continuous distributions that we consider here).1 We will
be more precise with these definitions below. The point we want to stress here is that the
VaR measure, lacking the mandatory properties of subadditivity and convexity, is not
coherent (Artzner et al. 1999). This means that summing the VaRs of individual portfolios
will not necessarily produce an upper bound for the VaR of the combined portfolio, thus
contradicting the holy principle of diversification in finance. A nice practical example of
the inconsistency of VaR in credit portfolio management is reported by Frey and McNeil
(2002). On the other hand, it has been shown (Acerbi and Tasche 2002) that ES is a
coherent measure with interesting properties (Pflug 2000). Moreover, the optimization of
ES can be reduced to linear programming (Rockafellar and Uryasev 2000) (which allows
for a fast implementation) and leads to a good estimate for the VaR as a byproduct of the
minimization process. To summarize, the intuitive and simple character, together with the
mathematical properties (coherence) and the fast algorithmic implementation (linear
programming), are the main reasons behind the growing importance of ES as a risk
measure.
