The Mathematics of Financial Modelingand Investment Management

23-RiskManagement Page 749 Wednesday, February 4, 2004 1:13 PM

Risk Management 749

ent. A coherent risk measure must satisfy a number of properties includ-

ing sub-additivity conditions, monotonicity conditions, risk-free

conditions, and diversification conditions. To solve the problems inher-

ent in the noncoherence of VaR, Artzner et al. proposed a coherent mea-

sure of risk known as expected shortfall (ES); Rockafellar and Uryasev^9

call the measure conditional VaR (CVaR). The ES at a given confidence

level αis defined as the expected loss given that the loss exceeds VaR at

the confidence level α. If the loss distribution is continuous, the VaR and

the ES or CVaR can be written as follows.

VaR at the 100(1 – α) percent confidence level is the upper

100 αpercentile of the loss distribution. If we denote the

VaR at the 100(1 – α) percent confidence level as VaRα(L),

where L is the random variable of loss, then the expected

shortfall at the 100(1 – α) percent confidence level ESα(L)

is defined by the following equation:

ESα()L = EL[ L ≥VaRα()L]

If the distribution is not continuous, the definition of ES is slightly

more complicated. Acerbi and Tasche,^10 and Rockafellar and Uryasev

provide a thorough discussion of the definitions of ES and CVaR under

different distributional assumptions. It can be demonstrated that at the

same confidence levels, the ES and VaR are equivalent measures for nor-

mal distributions in the sense that ES can be inferred from VaR and vice

versa. However other distributions, and in particular those with fat-

tails, might exhibit the same VaR but different ES and vice versa. It has

been demonstrated that ES is a coherent risk measure while VaR is not.

Yamai and Yoshiba^11 offer a comparison of ES and VaR under a number

of assumptions.

(^9) Tyrrell R. Rockafellar and Stanislav Uryasev, “Optimization of Conditional Value-
at-Risk,” Journal of Risk 2, no. 3 (2000), pp. 21–41.
(^10) Carlo Acerbi and Dirk Tasche, “On the Coherence of Expected Shortfall,” work -
ing Paper, Center for Mathematical Sciences, Munich University of Technology,
2001.
(^11) Yasuhiro Yamai and Toshinao Yoshiba, “On the Validity of Value-at-Risk: Com -
parative Analyses with Expected Shortfall,” Monetary and Economic Studies 20, no.
1 (published by Institute for Monetary and Economic Studies, Bank of Japan, 2002),
pp. 57–86. A number of papers discuss the use of ES as a risk measure in portfolio
optimization. See, for example, Rockafellar and Uryasev, “Optimization of Condi -
tional Value-at-Risk.”