Value at Risk (VaR)

Value At Risk (VaR) answers a specific and limited question: over a given horizon, what loss will not be exceeded with a given probability? A one-day 95% VaR of $50,000 means that on 95 days out of 100 you expect to lose less than $50,000. Formally, VaR at confidence level $\alpha$ is the $(1-\alpha )$ quantile of the loss distribution:

Three ways to compute it, in increasing order of honesty and effort:

Parametric (variance-covariance)

Assume returns are normal, then ${\text{VaR}}_{\alpha }=-(\mu +z_{\alpha }\sigma )V$, where $z_{0.95}=-1.645$ and $z_{0.99}=-2.326$. Fast, and wrong in exactly the direction that matters: it assumes away the fat tails that produce the losses you are measuring.

Historical simulation

Rank your actual historical returns and read off the relevant percentile. No distributional assumption, but you can only observe crises that happened to occur in your sample window.

Monte Carlo

Simulate from a chosen distribution. Only as good as the distribution you chose, which returns you to the first problem with extra steps and a false sense of rigor.

The three questions VaR does not answer. First and most damning: how bad is it when you do breach? VaR is a threshold, and says nothing whatsoever about the shape of the distribution beyond it. A 95% VaR of $50,000 is consistent with the remaining 5% averaging $60,000 of loss — or $6 million. Second, it is silent about path: a portfolio can respect its VaR every single day and still be down 40% over a quarter. Third, and least appreciated, VaR is not subadditive. Combine two portfolios and their joint VaR can exceed the sum of the individual VaRs, which means the measure can tell you that diversification increased your risk. That is not a rounding error; it is a defect that makes VaR unusable as a constraint in optimization, because an optimizer will happily exploit it.

VaR’s popularity survives because it produces one comfortable number for a board deck. In 2008 a great many institutions were within their VaR limits on the way to insolvency, and the measure performed exactly as designed — it just was not measuring the thing that killed them.