At high correlations, molehills become mountains.
Correlations also help to explain the “flashes” of volatility that occur from time to time. Stocks themselves encode the risk of the market they inhabit as well as whichever specific day-to-day or structural risks are particular to their business. The idiosyncratic, non-market risk of each individual stock is clearly particular to that stock. However, the aggregate level of all such risks among all the stocks in the market is no doubt of interest to risk managers, and in fact it is arguably an important characteristic of the market environment.
The idiosyncratic risks of every stock aggregate to a measure known as dispersion. Intuitively, one would expect dispersion move in the opposite direction to correlations over time, whereas (as per the table above) one expects market volatility to go up and down together with correlation. Here is a historical snapshot of each for an extended period in the S&P 500:
The fascinating, and important, observation to make from the scatter plots is that correlation and dispersion have been remarkably independent. This has a deeply important consequence. Other things equal, an increase of either correlation or dispersion will increase volatility. But at high correlation levels, any subsequent increase in dispersion will generally convert at an elevated multiple into market volatility. Since dispersion and correlations are somewhat independent, it is quite possible that, even during periods of high correlation, there will be small changes in dispersion – which entail far more dramatic effects in market volatility. And perhaps this explains why, as the “flash crashes” that followed the financial crisis demonstrated, in periods of high correlation, idiosyncratic events can have systemic consequences.
- Calculating such probabilities requires assumptions on how prices move – the most famous of which that price changes roughly follow a Normal (or “Gaussian”) distribution. This distribution is famous for good reason. It turns up all over the place; in fact the theory says that it should turn up most places, eventually. Beginning with Bachelier’s PhD thesis in 1900, through the Black-Scholes’s Nobel-prize winning options formula of 1973 and beyond to the present day, the normal distribution is inextricably linked with the mathematics of finance. Well, more fool us. “Incredibly unlikely” events happen all the time, it’s just that people can have short memories.