I have claimed several times that it is very hard to estimate a VaR that can pass rigorous back-testing. Since this is not a technical manual for risk managers, I'm not going to go into detail. I will show a toy example that illustrates the issue. Suppose you held a simple portfolio, $1,000,000 invested in the Standard & Poor's (S&P) 500 stock index from 1930 to the present. Over that period, 19,922 days, your 1 percent one-day VaR algorithm should have produced 199 breaks. That is, on 199 days the S&P 500 should have gone down more than your VaR prediction made at the close of the previous day.

Of course, you wouldn't expect even a perfect algorithm to produce exactly 199 breaks. You could get any number from 173 breaks to 227 just due to random sampling error. Values outside that range would occur for a perfect VaR algorithm only 5 percent of the time.

One simple algorithm that occurs to people is to look back over the past, say, 1,000 days, and set the VaR level halfway between the 10th and 11th worst losses over that period. That VaR would have produced exactly 1 percent (10 out of 1,000) breaks in the past, so maybe it's a good number to use for tomorrow. This method has a name; it's called *historical simulation VaR*. It's a handy number to know; I compute it myself. But it's not a VaR. It never passes a back-test.

The method has one parameter—one number that must be set. That's the number of days you look back. A long look-back period gives more statistical accuracy ...

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