It may surprise my readers to discover that I am a fan of equity investments. I think they are likely to offer higher returns to a long-term investor than most other asset classes. The issue is that you cannot rely on the returns.
To invest in anything, it’s important to know the risks involved; and I worry that many investors have vast allocations, 50% or more of their portfolios, in an asset class they don’t fully understand. For instance, investors wary of leverage often have few qualms investing in leveraged companies. For this blog though, I’m going to focus on a point of mathematical interest that highlights just how complicated equity risk is.
Previously, I have blogged about the difficulties inherent in quantifying the Equity Risk Premium, and trying to estimate mean equity returns. One common response is that, if you wait long enough, equity returns will stabilize. However, as well as the objections to this raised in my last blog, there are three other objections which, if valid, would render this false:
1. The mean may change over time
2. The volatility (or any higher moment) may change over time
3. There may be no defined mean or variance
The first two are ultimately macro-economic ideas, and I may come back to them later. However I want to focus on the third, because it is both the most subtle and the starkest demonstration of how complex issues can have profound consequences for investment decisions. But how could the mean or variance not exist?
Basically, where there are no limits on how extreme an event can be, the probability of an extreme event goes down as the event becomes more extreme. If this happens slowly enough1, the mean (or variance) will not exist. Intuitively, we can think of such a distribution as having infinite variance. For such a distribution, averages of samples will vary wildly as results extreme enough to alter the sample average dramatically are still relatively likely.
Granted, an infinite variance might seem like the far-fetched nightmare of a deranged mathematician, but it is hardly unfeasible. Many risk models assume equity returns are normally distributed, but in the real world, extreme equity returns are much more likely, and equity returns have “fat tails”. Graphically, if we plot the distribution of daily returns from the S&P500 index against a normal distribution (with parameters estimated by maximum likelihood), we get this:
Now, the Cauchy distribution is famously pathological, as it has no mean or variance; yet if we plot the returns against a Cauchy distribution (again using MLE parameters), we get a better-looking fit:
Granted, this is all theoretical, and it is not an argument against equity outperformance; in fact, it would imply a very attractive upside. The trouble is, some pension funds have flight plans to full funding that rely on earning a substantial risk premium of 4 or 5 percentage points, and there may not even be such a thing as the equity risk premium2
However you look at it, the message remains that high volatility is a problem for investors, and equity volatility is so high it may not even be finite. Equity volatility is a problem; if only there were some way of controlling it...
 Formally, the mean will be undefined if the tails of the probability distribution, f(x), are fat enough that the area under the curve g(x) = x f(x) is infinite.
 It is worth noting that any attempt to quantify these effects has two major obstacles: choosing an appropriate-shaped distribution (model risk), and choosing suitable parameters (calibration risk). While this blog focuses on the former, even if you do find an appropriate distribution, the choice of parameters can make at least as much difference to any values or predictions generated, and thus to the usefulness of the model.
Please note that all opinions expressed in this blog are the author’s own and do not constitute financial legal or investment advice. Click here for full disclaimer