Almost as damaging is the hash that banks have made of “value-at-risk” (VAR) calculations, a measure of the potential losses of a portfolio…The mistake was to turn a blind eye to what is known as “tail risk”…In markets extreme events are surprisingly common—their tails are “fat”.

I just can’t believe banks did this as an intellectual error, because it's easy to incorporate tail risk into

VAR. This is because we do have a statistical description of tail risk - the

power law.

To see how this affects VAR, let’s take a very simple example, starting by assuming a normal distribution.

You have a £100m equity portfolio with annualized volatility of 30%. You want to know what daily VAR is at a 1% probability.

All we do is deannualize that 30%, which gives us daily volatility of 1.86% - this is one standard deviation. We then read off from a normal distribution

table that a 1% probability is a 2.32 SD event.

So we multiply 1.86 by 2.32.

This tells us that VAR is £4.32m. We can expect to lose £4.32m or more in one day every hundred.

To incorporate tail risk, we just have to tweak this.

We do so by recognizing that losses of more than two standard deviations are distributed as a

cubic power law. This means that, to find the probability of a rare event - with SD of x - we simply divide x by two and then raise to the power 3. This tells us how much less likely the event is than a two standard deviation event. So, for example, a 10 standard deviation loss is five to the power three = 125 times less likely than a two standard deviation event.

Now, we are interested in 1% probabilities, which are 2.28 times less likely than a two SD event. Our cubic power law tells us these are 2.63 SD events. So, we just multiply 2.63 by 1.86. This gives us £4.89m.

So VAR is 13% greater than under a normal distribution.

This doesn’t seem much of a difference. But this is because 1% events are quite common. It’s when we go further down the extreme of the distribution that tail risk matters. For example, for one day in 1000 losses, a normal distribution gives us a VAR of £5.75m, but our cubic power law gives us a VAR of £10.55m.

And here’s my problem. Any idiot can tweak VAR models to incorporate fat tails - hey, an idiot just did. I just can’t believe that banks’ risk managers have been so stupid as to be unable to do this. It’s certainly not the case that fat tails are something new; we’ve known about them since at least October 1987.

Instead, I suspect that banks’ failure lay elsewhere. One difficulty was that their portfolios were much more complex than mere equities, so it was harder to calculate the distribution of extreme returns. But I would have thought that, given that these portfolios were vulnerable to default risk, liquidity risk and correlation risk, it would have been reasonable to assume an equity-like cubic power law, or an even higher exponent.

Another difficulty, though, was internal politics. Risk managers might have wanted to assume fat tails and a high VAR, but traders, anxious for a freer rein, would have resisted.

Which comes down to an important point. Banks’ failures are more likely to have been due to organizational failures - the difficulty of measuring their portfolio and reconciling competing internal interests - than to the inability to use mathematical models. Which in turn suggests they need more radical reform than merely better risk models.