Investment Theory and Risk Management, + Website

The Failure of VaR

Reactions from risk managers are varied. Before the crisis, Basel II capital requirements focused on value at risk (VaR) methodologies. Standard VaR assumes returns are normally distributed, when, in fact, they generally are not. Moreover, during financial turmoil, the tails of returns distributions are much fatter than indicated by normality; hence, losses are underestimated during these periods. Countless empirical studies find the normality assumption applied to asset returns to be excessively restrictive. Heavy-tailed distributions (distributions with more mass under the tails attach higher likelihoods to extreme events. Examples include the T-distribution, Pareto, and lognormal) have been used as an alternative.

VaR debuted as a risk measure by informing us about what proportion of the portfolio is at risk during any given time interval. If, for example, returns were assumed normally distributed with mean return 7 percent and volatility equal to 15 percent, then 95 percent of the time the portfolio would lose less than (7% – 1.645^∗15% =) –17.175 percent. Depending on how we want to frame risk, an equivalent statement would be that returns would be even less than –17.175 percent, but only 5 percent of the time. That's a lower bound. Unfortunately, it leaves open the question of how large losses may be, which is precisely the question of interest during a financial crisis. In any case, VaR is not a coherent risk measure, that is, one cannot cap-weight the ...