Measuring risk

It’s that time of year when pundits inflict upon us their forecasts for next year. Such predictions are mostly uninteresting and can be dangerous if they don’t come with a margin of error surrounding them to tell us what the risks are.

So, what are they? We can start by looking at the distribution of past returns and use these to measure risk in 2022. My table does this. It shows annualised standard deviations of weekly returns since January 1990.

There’s a simple interpretation of these. Take the All-Share index’s 16.2 percentage points. It tells us there’s a two-thirds chance of returns being within 16.2 percentage points of a reasonable expectation. So, if you expect the All-Share to deliver a total return (including dividends) of 5 per cent next year, you should also believe there’s a two-thirds chance of returns falling into the range of plus 21.2 per cent to minus 11.2 per cent. Which implies a two-thirds chance of the FTSE 100 ending 2022 between 6600 and 9000. And of course there’s a risk of returns being outside even this range – in particular a roughly one-in-six chance of the FTSE 100 ending the year below 6600.

If this sounds like a wide range, that’s precisely the point. Equity returns are so volatile that we should be prepared for some big moves.

And not just in equities. The same maths applies to other assets. These standard deviations tell us that if you expect zero nominal returns on gilts or gold as your central scenario you should believe there’s around a one-in-six chance of losing 6.3 per cent or more on gilts next year and 16.3 per cent or more on gold. And there’s also around a one-in-six chance of sterling falling more almost 10 per cent against the US dollar – or of rising more than 10 per cent.

My table converts these standard deviations into the chances of a 10 per cent loss, assuming our central expectation is for a 5 per cent return on equities and zero on the other assets. These numbers show that there is a significant risk of big losses, especially for gold.

What we cannot do is to say how these risks might materialise. Of course, we can tell stories in which equities fall a lot – either because markets are spooked by rising interest rates or (on the other hand) because they are disappointed by economic growth. And we can tell different stories in which gilts lose a lot. What we cannot do is reliably quantify the probability of those stories.

What we do know, though, is that we often attach too much credibility to narratives, and the worst losses are often the result of events nobody foresaw, such as last year’s pandemic.

Stories, then, tell us little about risk.

Sadly, however, numbers are unreliable too.

One problem is that I have so far assumed that returns are distributed normally – that is, like a bell curve.

For small variations, this is a reasonable approximation, deviations from which needn’t trouble us much. What should worry us, though, is that big losses have been much more common than a normal distribution implies. For example, in the week to March 13th 2020 the All-Share index lost 16.8 per cent. If returns were normally distributed we’d see such a loss only around one week in every 68 trillion years. But in fact, this wasn’t even the worst loss in the last 15: October 2008 saw a bigger one.

It’s not just equities of which this is true. In one week in 2017 gilts lost over 3 per cent. A normal distribution implies such a loss would happen around once every 140 years. But in fact, there have been four worse weeks since 1990 alone.

Big losses, then, are more likely than the bell curve tells us. Harvard University’s Xavier Gabaix says they instead are distributed like a form of cubic power law. Such a law tells us that the 16.8 per cent loss in March 2020 was a once in 46 years event, and that we should expect a 10 per cent weekly loss every 10 years (though not, of course, spaced out so evenly).

There’s another problem: is the distribution of returns since 1990 a reasonable sample of what to expect in future?

You might think not. It contains a bubble and burst, the worst financial crisis since the 1930s and a pandemic. Perhaps, therefore, my numbers over-sample bad events and so exaggerate equity risk (and perhaps understate gilt risk).

Or perhaps not. These episodes remind us that our knowledge of the future is very limited: that’s what risk means. But we often fail to appreciate this fact. As Friedrich Hayek said in his lecture accepting his Nobel prize (which was entitled “The pretence of knowledge”), the factors that determine market outcomes “will hardly ever be fully known or measurable.”

Our estimates of risk, therefore, must be imprecise. Just because something is imprecise, however, does not mean it is unimportant. We must guard against what Will Page in his book Tarzan Economics calls quantification bias, a tendency to underrate what cannot be precisely measured. Risk matters enormously, even if we don’t know exactly what it is.