Corporate Risk Management

CHAPTER 8Conditional VaR

This chapter discusses conditional VaR (CVaR), which is needed because VaR presents methodological gaps. The main gaps can be summarized by the following points.¹

• When losses are not distributed continuously (for example, do not follow normal or elliptical distributions), VaR is difficult to use and is often unstable. Since it may underestimate or overestimate risk, it is less appropriate for credit risk, operational risk, and some forms of market risk.
• Analysts can verify the number of times that VaR is exceeded, but this statistic does not specify the loss amounts. It may thus be exceeded by $1 or $1 million.
• VaR is not coherent mathematically. VaR of two assets may exceed the sum of individual VaRs. It thus implies that VaR does not respect subadditivity or the principle of diversification, which is very useful in finance. It may give more diversified portfolios higher maximum capital!

8.1 MOTIVATION FOR CVAR AND COHERENCE IN RISK MEASURES

CVaR is more difficult to calculate and to test, but offers several advantages:

• CVaR is the expected conditional loss greater than VaR. It may also be interpreted as the mean of VaRs that exceed a given VaR.
• The same portfolio maximization results are obtained with relative CVaR as for relative VaR with a normal distribution and for elliptical distributions, because we minimize the portfolio variance in all cases.
• CVaR may be useful for very asymmetric distributions (market risk, credit risk, operational risk). ...