Where Risk Models Can Miss the Mark

Editor’s note: Read the latest on how the coronavirus is rattling the markets and what investors can do to navigate it.

The current market volatility sheds light on the strengths and limitations of different risk models.

Given the frequency of market declines like this--in which the market falls 20% or more--and the subsequent recoveries, you might think that investment professionals would use returns distribution models that account for these extreme declines and recoveries to inform their investment decisions.

But, alas, many do not.

Some investment professionals tend to use tools such as portfolio optimizers and Monte Carlo financial-planning models that rely on thin-tailed models. “Thin tails” refer to thin ends of a distribution curve, which indicate a low probability of extreme events--probabilities that are so low that they can easily be ignored or missed.

These are the models that feed into the concept of a market crisis as a once-in-100-years event. However, our analysis of the historical return data has shown that these crises are far from rarities.

Here, I take a closer look at the different risk models.

How Distribution Models Compare to Real Return Data
To see how different distribution models compare, I mapped them onto the chart below, based on the monthly real market returns in the United States from January 1886 to March 2020. The chart includes:

• a lognormal curve, the standard distribution model that most of the industry uses, which is based entirely on the market’s average return and the standard deviation of return,
• a log-stable curve, an alternative to the standard model, and
• a curve based on real historical return data, which does not rely on any predefined model.



From the chart, we can see:

• The lognormal curve resembles a bell curve. This curve has thin tails, meaning that it assigns virtually no probability to extreme returns, either positive or negative.
• The curve labeled “real return data” has fatter tails than the lognormal model, especially on the left side (indicating a higher frequency of extreme negative events). On the right side, the tail has some weight all the way at around 40%, indicating that extreme events occur more often than standard models predict.
• The curved labeled “log-stable” nearly matches the real return data with a high peak near the middle and fat tails at both ends, indicating a higher probability of both extreme positive and negative returns. A zoomed-in look at the left side of the distributions also highlights how well the log-stable distribution fits the left tail of the data-based distribution.

 

Why Didn’t the Log-Stable Distribution Model Catch On?
The log-stable distribution model is based on the stable Paretian distribution, which mathematician Benoît Mandelbrot first proposed in the 1960s as a model of price changes. One reason why this model wasn’t widely adopted is that a random variable following a stable Paretian distribution has infinite variance.

Accepting the premise that financial returns lack finite variance would mean that Markowitz’s mean-variance model of portfolio construction is invalid. So are asset-pricing theories such as the capital asset pricing model and the arbitrage pricing theory. In other words, accepting the log-stable model of returns would mean rejecting nearly all of standard financial economics.

Financial economist Paul H. Cootner wrote of this implication in his 1964 book, "The Random Character of Stock Market Prices," “Surely before consigning centuries of work to the ash pile, we should like to have some assurance that all of our work is truly useless.”

How the Log-Stable Distribution Model Takes Shape Around the World
We can also evaluate the stable Paretian model’s two shape parameters--peakedness and skewness--to show how it applies to asset returns and how the lognormal model falls short.

• The peakedness parameter indicates whether a distribution has finite or infinite variance and how far the fat tails extend out on both sides. A peakedness parameter can fall between 0 and 2, with a parameter of 2 indicating normal distribution and finite variance. A parameter less than 2 signals infinite variance; the more below 2, the higher the middle and the fatter the tails. That is to say, the more below 2, the greater probability of extreme events.
• The skewness parameter indicates the asymmetry of the distribution. The skewness parameter is between negative 1 and positive 1, where a value of 0 means that the distribution is symmetrical. A negative value means that the distribution is skewed to the left (indicating extreme negative events are more likely than extreme positive events), and a positive value means that it is skewed to the right (indicating extreme positive events are more likely than extreme negative events).



To see how well the log-stable model works as a model of stock market index returns of countries around the world, I fitted it to the monthly returns of Morningstar stock market indexes for 33 countries (including the United States and Canada) from July 1998 to February 2020. These results are shown on the chart below as a scatter plot of the skewness and peakedness parameters.

As you can see, the peakedness parameter is below 2 for all countries, indicating that all countries’ returns experienced fat-tailed distributions, or extreme events. And for the skewness parameter, all countries other than South Korea yielded a negative value. Eight countries (France, Japan, Italy, South Africa, U.S., Mexico, Norway, and Poland) had skewness parameters of negative 1, the most negative possible value, while all the others fell between 0 and negative 1. So, all these countries experienced extreme negative events.

This analysis shows that the notion of historical return distributions having experienced extreme events, particularly extreme negative events, applies not just to the U.S. but to country stock market indexes from around the world. This departure from what standard risk models predict suggests there might be some credence to Mandelbrot’s work on the distribution of price changes.

Using Distribution Models That Offer a Robust View of Risk
Extreme market events are everywhere in the historical stock market return data. They’re prevalent not only in the U.S. and Canada, but in markets all over the world. Yet, these extreme events do not exist in the standard statistical models of finance.

This doesn’t mean investors should throw away the standard ways of summarizing risk. Rather, they should understand both the limitations of standard risk models and the potential of the log-stable model. For a more complete assessment of the risks of investing in stocks, investors should use risk models that tackle three key questions:

• How likely might a decline occur?
• How long might it last?
• How bad might it get?

 

Downturns are numerous and can be long and painful. Investors who are prepared for these events will be better able to endure the crisis and benefit from the inevitable recovery.