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# On the Feasibility of Portfolio Optimization under Expected Shortfall

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On the Feasibility of Portfolio Optimization under Expected Shortfall book

# On the Feasibility of Portfolio Optimization under Expected Shortfall

DOI link for On the Feasibility of Portfolio Optimization under Expected Shortfall

On the Feasibility of Portfolio Optimization under Expected Shortfall book

## 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.