The next principle to be discovered was a complete surprise. We had assumed it would be pretty easy to define the region in which you had plenty of data—in other words, how much things could vary and still be considered normal. After all, you have plenty of data there, by definition. To make this precise, I'm going to jump ahead and steal a concept that wasn't fully fleshed out until 1992, value at risk (VaR). VaR is defined operationally. That means we specify a property VaR is supposed to have, and then try to figure out what number satisfies the property. For a 1 percent one-day VaR, the property is that one day in 100—1 percent of the time—a portfolio will lose more than the VaR amount over one day, assuming normal markets and no position changes in the portfolio. VaR can also be defined at different probability levels and over different time horizons.

The 99 days in 100 in which markets are normal and you make money or lose less than the VaR amount are used as data for mathematical optimization. The two or three trading days a year that you lose more than VaR—called “VaR breaks”—or that have abnormal markets, are analyzed separately.

To make this useful, you have to specify VaR before trading begins. If the VaR algorithm is good, losses will exceed it one day in 100, within statistical error. Moreover, the breaks should be completely unpredictable. You should have the same chance of a VaR break if yesterday was a break or if there has been no ...

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