Everyone talks about fat tails in markets, but how do you model it in an effort to estimate what may be lurking in the future? There are many possibilities, along with lots of statistical baggage. The main challenge is that trying to squeeze reality into one distribution to capture how markets actually work is challenging, to say the least. But the head of global multi-asset strategies at T. Rowe Price outlines a simple and arguably better approach in a new book on asset allocation.

First, let’s briefly review the problem. It’s widely recognized that financial markets exhibit fat tails – a tendency to suffer unusually extreme events that run afoul of expected frequency via so-called normal distributions. Consider the US stock market, based on the S&P 500 Index. Most of the time the returns behave and tend to follow a standard bell curve for the performance distribution. That is, extreme results are rare while most of the gains and losses are clustered around the mean. The trouble, of course, is that in the real world the theoretical normal distribution doesn’t always hold and so extreme results occur more often than a basic statistical model predicts.

One solution is to model returns with a distribution that factors in fat tails – the t distribution, for instance. Other options include using what’s known as Extreme Value Theory, which is a relatively robust methodology for modeling fat tails (see this primer, for example).

But throwing out a normal distribution entirely may not be wise because it’s still a useful tool for modeling market behavior most of the time. Enter Sebastien Page, head of global multi-asset strategy at T. Rowe Price. In a new book, he outlines a simple way to use two or more sets of normal distributions to model fat tail behavior in markets.

“Asset allocators can use the concept of risk regimes to model, and perhaps, forecast, latent risks,” Page writes in Beyond Diversification: What Every Investor Needs to Know About Asset Allocation.

If markets oscillate between high- and low-volatility regimes, we should expect fat tails. The idea is that the fat tails belong to another probability distribution altogether – the risk-off regime, which is characterized by investor panics, liquidity events, and flights to safety. And if we blend two normal distributions (risk-on versus risk-off or “quiet” versus “turbulent”), we can get a highly nonnormal distribution. In other words,

Normal distribution + normal distribution = nonnormal distribution

He outlines a simply way to model this concept in Excel, although the effort is more tractable in a coding language such as Python or R. As a toy example, I’ll use R to give Page’s idea a spin. Using the stats in his example, he starts with a simulated normal dataset of 7% annualized return and 9% volatility. For the risk-off regime dataset he also uses a normal distribution but changes the parameters to -10% return and 31% volatility. In other words, a deeply negative return and high standard deviation relative to the “normal” data set.

Next, combine the two return data sets as follows: 95% of the time the “normal” returns prevail with a 5% incidence of the risk-off data. Finally, run the results 100,000 times. The boxplot below compares the results in each step:

The Combined Regime distribution can be used for modeling and forecasting. Note that this result has extreme results – fat tails – vs. the Normal Regime. At the same time, the Combined Regime still captures the Normal Regime’s core profile (per the interquartile range of results via the gray box). In other words, this method attempts to capture the two facets of market behavior without resorting to the distortions that a formal non-normal distribution can introduce into modeling.

What I like about this approach is that we can further customize the modeling by adding several flavors of nonnormal distributions and blend as we see fit and adjusting the incidence frequency of the risk-off regime. Further, we can resample the historical record for an asset and customize from there, adding a deeper level of reality into the mix.

Is this the last word on simulating fat tails? No, but it’s an intriguing addition to the toolkit.      

Previous articles in this series:

Fat Tails Everywhere? Profiling Extreme Returns: Part I
Fat Tails Everywhere? Profiling Extreme Returns: Part II
Fat Tails Everywhere? Profiling Extreme Returns: Part III
Fat Tails Everywhere? Profiling Extreme Returns: Part IV

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Learn To Use R For Portfolio Analysis
Quantitative Investment Portfolio Analytics In R:
An Introduction To R For Modeling Portfolio Risk and Return
By James Picerno

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