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Tail Risk and Linear Thinking


Catastrophic events unlikely to occur tommorrow, but very likely to occur over time, require management of tail risk.


If you ever want to lose some time
Just take off, there's no risk
If you ever want to disappear
Just take off, and think of this
--The Who

The probability that the Dow Jones Industrial Average (DJI) will decline 10% tomorrow is small. But the odds of observing a single day market decline of 10%+ over the next 20 years are very good. Large single day market losses often 'wipe out' unprepared participants. Catastrophic market events unlikely to occur tomorrow, but very likely to occur over time requires management of 'tail risk.'

Effective market participants are aware of tail risk and its challenges.

Tail Events

Many processes and their outcomes are reasonably approximated by the normal distribution (a.k.a. the Gaussian distribution or simply the 'bell shaped curve'). Figure 1 displays the normal distribution. The x (horizontal) axis is incremented in standard deviations (or sigma) from the center (the center should represent the mean or average) and the y (vertical) axis represents frequency or probability of occurrence.

Figure 1: The Normal Distribution

Can you see that, for processes and outcomes that are well modeled by the normal distribution, we should expect most observations to fall relatively close to the middle? Handy rules of thumb in this regard are that about two thirds (67%) of all observations should fall within ± 1 sigma, about 95% of all observations should fall within ± 2 sigma, and about 99.7% of all observations should fall within ± 3 sigma. On the far ends of the curve, we should about 0.3% of all observations to fall outside of ± 3 sigma.

For processes that are reasonably approximated by the normal distribution, then, we should expect 'extreme' occurrences very rarely. In fact, using the ± 3 sigma level as the cutoff for 'extreme,' then we should expect a 'tail event' to occur only 0.3% of the time, or three times in 1000 observations.

Fat Tails...

For many of you, the notion of the bell shaped curve is likely not new. The reason is that, because we assume many natural and industrial phenomena are well approximated by the normal distribution, the bell shaped curve is a staple of statistical methods. Versed in these methods, we liberally model many processes and outcomes using the normal distribution. Efficient market theory (Fama, 1970), the capital asset pricing model (Sharpe, 1964; Litner, 1965), and option pricing models (Black and Scholes, 1972, 1973) constitute some of the prominent financial concepts that are grounded in assumptions based on the bell shaped curve.

The problem is that the bell shaped curve is often only a weak approximator of phenomena that we are interested in modeling--particularly with respect to events that rarely occur. Extreme observations are often more common than predicted by the bell-shaped model. For example, weather events like Category 5 hurricanes and their consequences (Fox, 2005), and extreme movements in market prices like those that reflect 'crashes,' occur more frequently than would be expected if the phenomena were normally distributed. Statisticians call this condition leptokurtosis, but we commonly refer to it as 'fat tails' since the actual distribution of observations appears to have thicker tails than what we would 'normally' expect.

A large number of studies suggest that market price behavior exhibits tendency for leptokurtosis (e.g., Focardi & Fabozzi, 2003; Jansen & DeVries, 1991; Kon, 1984; Longin, 1996; Mandelbrot, 1963). The probabilities of extreme positive and negative returns in the S&P 500 (as defined by returns beyond -3 and +3 standard deviations from the mean) are significantly higher than predicted by a normally distributed model. Market prices are 'autocorrelated,' meaning that price movements in one period depend on prices in previous periods. For example, if market prices experience an extreme move lower on one day, there is an increased chance that the next day will be down. The takeaway: Market price movements are fat tail in nature--more outliers are observable than we would expect if market returns were well modeled by the normal distribution.

Linear Thinking and Tail Risk

Take a look at this 100 year graph of the Dow Jones Industrial Average (DJI). It's pretty apparent that the general trend over the last century has been up. In fact, stock market returns have averaged about 10% per year during this period. Human cognitive processes being what they are, we're prone to process data like this as follows: "Hey, I wouldn't mind making 10% annually over the next couple of decades. Because this trend has generally held for the last 100 years, it seems reasonable that it'll continue in the future. I'd better buy stocks..."

When we view that upward sloping line and then forecast its continuation into the future, we are engaging in extrapolation--an operation that you may remember from math class. Extrapolation is taking a line and extending it beyond the data in order to make predictions. Another term for this is 'linear thinking.'

Although linear thinking constitutes a fundamental operation of forecasting, scholars have noted the limits of extrapolation in industrial settings (e.g., Armstrong, 1984).

In the context of financial markets, the danger of linear thinking is that market participants tend to extrapolate normal, everyday occurrences and ignore rare extreme events in their decision calculus. Stated another way, market participants tend to ignore 'tail risk'--the risk associated with events that, while rare, incur substantial negative consequences if the event occurs.

Let's return to our enthusiast above who envisions future stock market returns similar to the past. Individuals tend to assign excessively high probabilities of return on risky assets, mostly due to errors in linear thinking. Minyanville professor John Succo suggests the perils of this approach:

"...the statement that, "stocks have returned 7-8% annually over the past fifty years is pure rhetoric. Advising a "mandatory" allocation to equities based on past historical data, especially over long periods of time, is statistically a weak argument because it is linear in nature. I would argue that markets are actually non-linear in nature: one single event or a series of events can affect markets at first in a small way, but then those effects can build over time. For example, what if the US lost WWII? Or what if Reagan's gambit with the Russians had backfired? Or on a smaller scale, what if quantum physics had not been developed and the micro chip had not been invented? Or maybe it was just that Japan invented them. Do you think stock over the last fifty years would have been the same type of investment?

Because these events turned out positive in the distant or recent past, stocks have performed in a certain manner. But the causes cannot be linearly extrapolated into the future."

Using this 'nonlinear' perspective, stock market performance over the past 100 years can be viewed as related to a string of binary events (win/lose a war, invent/not invent a new product) that, if the event went the other way, would have dramatically influenced financial market returns over the last century.

The Meaning of It All

Let's summarize some key points:

How to manage tail risk is a topic for another day. Right now, you should appreciate the value of factoring tail risk into your decision-making processes. When assigning probability to an extreme event, you should consider the potential time frame and consequences of the event's occurrence. Observers have noted, for example, that the probability of a catastrophic hurricane hitting the city of New Orleans over any short period of time was very low. However, the probability of such an event over longer periods was very high with predictably disastrous consequences.

Indeed, the Hurricane Katrina disaster in New Orleans has quickly become a classic lesson of human tendency to underestimate tail risk (Fox, 2005). Savvy financial market participants need not make the same mistake.


Armstong, J.S. (1984). Forecasting by extrapolation: Conclusions from 25 years of research. Interfaces, 14(6): 52-66.

Black, F. & Scholes, M. (1972). The valuation of option contracts and a test of market efficiency. Journal of Finance, 27: 399-417.

Black, F. & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81: 637-654.

Fama, E.F. (1970). Efficient capital markets: A review of theory and empirical work. Journal of Finance, 25: 383-417.

Focardi, S.M. & Fabozzi, F.J. (2003). Fat tails, scaling, and stable laws: A critical look at modeling extremal events in financial phenomena. Journal of Risk Finance, 5(1): 5-26.

Fox, J. (2005). A mediation on risk. Fortune, 152(7): 50-62.

Jansen, D.W. & DeVries (1991). On the frequency of large stock returns: Putting booms and busts into perspective. Review of Economics & Statistics, 73: 18-24.

Kon, S. (1984). Models of stock returns: A comparison. Journal of Finance, 39: 147-165.

Linter, J. (1965). The valuation of risk assets and the selection of risky investments in stock portfolios and capital budgets. Review of Economics and Statistics, 47: 13-37.

Longin, F.M. (1996). The asymptotic distribution of extreme market returns. Journal of Business, 69: 383-408.

Mandelbrot, B. (1963). The variation of certain speculative prices. Journal of Business, 36: 294-419.

Sharpe, W.F. (1964). Capital asset prices: A theory of market equilibrium and conditions of risk. Journal of Finance, 19: 425-442.

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