A special report on the future of finance in The Economist Fallible mathematical models: In Plato’s cave:

… although the normal distribution closely matches the real world in the middle of the curve, where most of the gains or losses lie, it does not work well at the extreme edges, or “tails”. In markets extreme events are surprisingly common—their tails are “fat”. Benoît Mandelbrot, the mathematician who invented fractal theory, calculated that if the Dow Jones Industrial Average followed a normal distribution, it should have moved by more than 3.4% on 58 days between 1916 and 2003; in fact it did so 1,001 times. It should have moved by more than 4.5% on six days; it did so on 366. It should have moved by more than 7% only once in every 300,000 years; in the 20th century it did so 48 times.

In Mr Mandelbrot’s terms the market should have been “mildly” unstable. Instead it was “wildly” unstable. Financial markets are plagued not by “black swans”—seemingly inconceivable events that come up very occasionally—but by vicious snow-white swans that come along a lot more often than expected.

This puts VAR in a quandary. On the one hand, you cannot observe the tails of the VAR curve by studying extreme events, because extreme events are rare by definition. On the other you cannot deduce very much about the frequency of rare extreme events from the shape of the curve in the middle. Mathematically, the two are almost decoupled.

The drawback of failing to measure the tail beyond 99% is that it could leave out some reasonably common but devastating losses. VAR, in other words, is good at predicting small day-to-day losses in the heart of the distribution, but hopeless at predicting severe losses that are much rarer—arguably those that should worry you most.

When David Viniar, chief financial officer of Goldman Sachs, told the Financial Times in 2007 that the bank had seen “25-standard-deviation moves several days in a row”, he was saying that the markets were at the extreme tail of their distribution. The centre of their models did not begin to predict that the tails would move so violently. He meant to show how unstable the markets were. But he also showed how wrong the models were.

Modern finance may well be making the tails fatter, says Daron Acemoglu, an economist at MIT. When you trade away and invest in stocks, all sorts of specific risk, in foreign exchange, interest rates and so forth, you make your portfolio seem safer. But you are in fact swapping everyday risk for the exceptional risk that the worst will happen and your insurer will fail—as AIG did. Even as the predictable centre of the distribution appears less risky, the unobserved tail risk has grown. Your traders and managers will look as if they are earning good returns on lower risk when part of the true risk is hidden. They will want to be paid for their skill when in fact their risk-weighted returns may have fallen.