To see why it's harder, consider value at risk, a measure widely used by banks. This measures the loss a bank would expect to suffer in bad times. For example, in 2011 Barclays Capital had a daily VaR of £57m. It expected to lose £57m or more in the worst 5 per cent of days - although VaR can also be calculated for the worst 1 or 2 per cent of days, too.
But this is too simple a measure. For one thing, VaR changes not just because banks' change their portfolios but because the volatility and correlations of their assets change. Barclays Capital's VaR varied from £33m to £88m during 2011.
There's another problem. VaR on its own doesn't tell us about the possibility of extreme losses. For example, a portfolio with a 5 per cent chance of losing £10m has a VaR of £10m while a portfolio that has a 0.5 per cent chance of losing £100m has a VaR (at 5 per cent) of zero. But the latter portfolio might be catastrophically risky as such a loss could drive its owner to bankruptcy whereas a £10m loss might be survivable.
VaR can also deter diversification. Take, for example, two loan books both of £2m. One is a single loan with a 4 per cent probability of default. The other comprises two loans with independent 4 per cent probabilities of default. Which loan book is safer? If you look at VaR, it's the single loan, because at 5 per cent probability it won't default, whereas there's a 7.8 per cent probability that one of the two loans will default. A risk manager who looked only at VaR would therefore not diversify his assets.
The problems here aren't because I'm using VaR for 5 per cent probabilities; we could adjust these examples to apply them to VaRs at other probabilities. Instead, the problem is simply that VaR is incomplete. And Barclays Capital now supplements it by reporting other measures, such as expected shortfall (the average of all possible losses in the worst 5 per cent of days) and '3W', the average of the three worst likely daily losses.
But even these measures might not be enough. Dominique Guegan at the Paris School of Economics says we might need five different measures, depending on the precise shape of the tail of probabilities.
Given the difficulties of estimating the volatility of any asset, and its tail risk and its correlations with other assets - all of which change over time - this seems a complicated business.
But it needn't be, says Andy Haldane at the Bank of England. Take the problem of catching a frisbee. If we think of this mathematically, it's complicated. We need to solve equations containing difficult-to-measure factors such as velocity, angle and wind speed. But in fact, even dogs can catch frisbees. You just need a simple rule, such as 'keep your head steady and eye on the frisbee'.
Banks, says Mr Haldane, are like frisbees. How do you assess the risk of one collapsing? It seems a complex problem requiring us to know the risks of its many complicated assets. But it's not. Mr Haldane shows that a very simple measure of risk - the ratio of banks' assets to equity - did better at predicting which banks failed in the crisis than complicated risk-weighted measures. Regulating banks, he says, is a simpler job than it sounds.
You might object that this is because regulators have an easy job anyway. They merely have to decide whether a bank will get into trouble or not. But bank managers, and investors, have to worry about smaller losses as well - and about profits.
But Mr Haldane adds something remarkable -that other simple measures of risk can outperform not just complicated measures, but even true measures.
This is because 'true' measures of risk apply only to entire samples, whereas we invest only in short subsets of that sample, during which 'true' probabilities might not apply. This could be because volatilities and correlations vary over time, or because portions of a large, random sample don't necessarily have the same statistical properties as the whole sample.
For example, there's no point knowing that the 'true' probability of a share falling 50 per cent in a day is (say) one in a thousand if that day happens to be tomorrow. You'll be better off with a simple but false measure of risk that provides a higher estimate.
So, what simple risk model should equity investors have? The obvious candidate is to assume that the market has annualised volatility of 20 per cent and that returns are distributed as a cubic power law. This has clear implications:
■ There's a one-in-six chance the market will lose 20 per cent in a year, 5.8 per cent in a month, 2.8 per cent in a week, or 1.2 per cent in a day.
■ There's a 5 per cent chance the market will lose almost one-third in a year, 9.5 per cent in a month, 4.6 per cent in a week, or almost 2 per cent in a day.
■ There's a 1 per cent chance the market will lose over 50 per cent in a year, 15.2 per cent in a month, 7.4 per cent in a week, or 3.2 per cent in a day. This implies there's a 90 per cent chance that, over a 12-month period, there will be at least one day in which the FTSE 100 drops 200 points.
■ There's a one in a thousand chance the market will fall almost one-third in a month, 15.9 per cent in a week, or 6.8 per cent in a day.
Whether these odds are too high or not is ultimately a matter of taste, But unless you have a very unusual equity portfolio, something like them apply to your shareholdings.
