Black Swans: Frequent Once-in-a-Lifetime Crises
Black Swans: Frequent Once-in-a-Lifetime Crises
Randal Allen, John Volpi, Lone Star Analysis, Addison, TX
Nassim Taleb (Taleb 2010) labels black swans with the following attributes: rarity, impact, and
retrospective apparent predictability. However, Benoit Mandlebrot (Wright 2007) claims gray swans
are events of considerable nature, which are predictable and for which one can take precautions.
Admiral Nimitz is quoted as stating, “The war with Japan had been enacted in the game rooms at the
War College by so many people and in so many different ways that nothing that happened during the
war was a surprise—absolutely nothing except the kamikaze tactics toward the end of the war.”
Surely, this may have been a black swan from his perspective.
This paper addresses three questions posed at the “2015 I/ITSEC Black Swan Kickoff.”
1. How do we prepare, organize, train, and equip for Black Swan resiliency?
2. How can Modeling and Simulation be used to analyze and prepare or create a Black Swan?
3. Can we develop complex adaptive models and simulation tools that will enable the analysis?
The authors will follow Dr. Mandelbrot’s assertion to answer the first question. For the second and
third questions, we will outline a procedure to use modeling and simulation with prescriptive
analytics to reduce the potential intractability of black swans, thus demoting their status to
gray. With frequency histograms and curve-fitting, we first show how distributions with thin-tails
don’t fully account for risk, while fat-tail distributions better fit extremely rare event data.
Then, by applying Percent Point Functions and stochastic optimization techniques to a Monte Carlo
simulation of fat-tailed distributions, we show which configuration of input parameters creates a
black swan. Given our approach, we offer an analytics method to evaluate black swan events and
downgrade them to gray swan events.
ABOUT THE AUTHORS
Randal Allen has over 20 years of industry experience and has been with Lone Star Analysis since
2006. As Chief Scientist, he is responsible for applied research and technology development. He is
certified modeling and simulation professional (NTSA). He is the co-author of “Simulation of
Dynamic Systems with MATLAB and Simulink,” 2nd ed. He is also an Associate Fellow of the American
Institute of Aeronautics and Astronautics (AIAA). He holds a Ph.D. in Mechanical Engineering from
the University of Central Florida, an Engineer’s Degree in Aeronautical and Astronautical
Engineering from Stanford University, an M.S. in Applied Mathematics, and a B.S. in Engineering
Physics, both from the University of Illinois (Urbana-Champaign). He also serves as an Adjunct
Professor in the Mechanical and Aerospace Engineering (MAE) department at UCF.
John Volpi is the Chief Technology Officer (CTO) of Lone Star Analysis and has served in this role
since 2004. He is responsible for all technical activities, intellectual property evaluation, and
process development. He began his professional career at Texas Instruments as a theoretical systems
analyst and evolved into Systems Engineering, where he was named a Senior Member of the Technical
Staff. He attended the Illinois Institute of Technology (B.S. Physics) and Michigan State
University (M.S. Physics). He also attended graduate classes at Southern Methodist University and
the University of Southern California. He has over 30 patents awarded or pending. He has been a
member of the IEEE for over 30 years and was named a Senior Member for his work in wireless
technologies and his efforts in the development of Intellectual Property. In 2012, John was
awarded the Tech Titans Award for Corporate CTO by the
DFW Metroplex Technology Business Counsel out of a pool of 4,000 firms.
Before addressing the three questions from the “2015 I/ITSEC Black Swan Kickoff,” a little
background on black swans and gray swans is in order.
A black swan is an improbable event with colossal consequences. It is a metaphor for believing
something is impossible until the belief is disproven. For example, all swans were assumed to be
white, and black swans were thought to be non-existent until discovered in Western Australia. In
The Black Swan, Taleb (2010, p. xxii) defines three black swan attributes: (1) It’s an outlier,
outside the realm of expectation because nothing in the past points to its possibility; (2) It
brings extreme impact; and (3) We concoct explanations for it after the fact, making it seem
predictable. In short, the three attributes are: “rarity, impact, and retrospective apparent
predictability” (Taleb 2010). Also, by symmetry, the non-occurrence of a seemingly certain event is
also a black swan. Furthermore, the lack of evidence of black swans doesn’t mean they do not exist.
While swans of unusual character are labeled black, Benoit Mandlebrot (Wright 2007) claims we can
predict something of their behavior – and in doing so, they are no longer black, but can be thought
of as gray. They only seem black if we fail to acknowledge their potential existence or we fail to
look. The modeling and simulation tools applied to examine potential black swans are inherently
stochastic, meaning they are based on probabilistic inputs, and outputs are viewed from a
statistical perspective. In this paper, we discuss how such tools can cause seemingly black swans
to fade to gray – and in doing so, we address the second and third questions from the “2015 I/ITSEC
Black Swan Kickoff” – “How can Modeling and Simulation be used to analyze and prepare or create a
Black Swan?” and “Can we develop complex adaptive models and simulation tools that will enable the
An appropriately visualized model architecture may identify the succession of events leading to the
colossal black swan which may be confirmed by examination of Tornado charts (specialized bar
charts) and the Percent Point Function. Additional help with exposing black swans is available
through stochastic optimization, where random variables appear in the formulation of the
optimization problem thus producing a random objective function for which random iterates are
employed to solve the problem. When equipped with these tools and their insights, we can reduce the
surprise of a black swan, rendering it gray, and thus prepare, organize, train, and equip for black
MODELING RARE EVENTS
With a general understanding of black and gray swans, a model is created such that rare events may
be simulated. We need a distribution that shows non-zero probabilities for data lying far from the
mean on either side. In this paper, we use the financial markets to make our points, because most
of us can relate to risk versus reward from a financial perspective. We will see that if thin tail
distributions are used, the risk is modeled too conservatively; whereas a fat tail distribution exposes
a greater degree of risk and the potential for a black swan. The analogy can be generalized to a
portfolio of lines of business, where different business opportunities may be assessed for return
on investment, given its corresponding risk.
A recent example of a black swan event is the financial collapse of 2008. While we’re not
interested in the cause of the collapse, per se, we are interested in one of the many lessons
learned, i.e. observance of the fat tail distribution as a more accurate representation of the
collapse, and thereby categorize it as a black swan event. While implementing a normal distribution
in a Monte Carlo simulation is far superior to simply use average values of risk and return
(Savage 2009), the “thin tail” of the normal distribution assigns a negligible probability to data
far from the mean. Harry Markowitz (Markowitz 1952; Markowitz 1979; Markowitz 1999) consistently
warned that distribution selection was tricky and urged that when moving from theory to practice,
some caution was warranted. Benoit Mandlebrot (Mandlebrot, 1963) found price changes in some
markets (especially cotton futures) were well described by Lévy’s stable distributions. Eugene Fama
(Fama 1963) performed similar research to what is presented here and further demonstrated the
merits of “fat tail” distributions in stocks. Paul Kaplan (Kaplan 2012) shows a log-stable
distribution (see Appendix) captures the non-zero probability of occurrence for rare events far
from the mean. The log-stable is a generalization of the log-normal distribution commonly used to
model investment return. It assumes the logarithm of one plus the decimal form of risk and returns
following what Mandlebrot referred to as a stable Paretian distribution
Figure 1 – S&P 500 data from January 1926 to March 2009 fitted with Normal (left) and Log-Stable
The left pane of Figure 1 shows monthly returns of the S&P 500 stock index from January 1926 to
March 2009 as represented by the frequency histogram. Historical returns over this time period
include maximum monthly losses of 26% in November 1929, 24% in April 1932, 20% in October 2008, and 19% in December 1931; while the maximum gain was +50% in August 1932. The data is fitted with a
normal distribution (dashed line) and a log-stable distribution (solid line). In the right pane, a
closer examination displays the characteristics of the normal distribution’s thin tail (dashed)
versus the log-stable distribution’s fat tail (solid). The normal distribution shows a negligible
probability of losses beyond 16% (-0.16). While the theoretical tail of the normal distribution
extends to infinity, it is clear from this exploded view that the probability of a 16% loss is
practically zero. Using the normal distribution curve fit parameters, the actual probability is
only 1.4%. Contrast this with the log-stable distribution which more accurately shows losses beyond
26% are indeed possible. In fact, using the log-stable curve fit parameters, the probability of a
16% loss is 9.6% – almost seven times more likely to occur.
From this analysis, we conclude that the log-stable distribution is superior to the normal
distribution for modeling rare events of this type. While we have shown this to be true for the S&P
500, if rare events have a non-zero probability of occurrence in any practical application, the
log-stable distribution should certainly be considered as the apparatus of choice.
MEASUREMENT OF PORTFOLIO RISK
In order to show the impact of portfolio risk, we model an aggressive asset allocation with the
percentages shown in Table 1, where each asset class is represented by a corresponding Exchange
Traded Fund (ETF) ticker symbol, e.g. small-cap stocks are represented by the ETF ticker symbol
IWM, etc. Historical monthly return data for each ETF was obtained from the Investools / TD Ameritrade database.
Table 1 – Portfolio Asset Allocations and ETFs
Furthermore, we set up two portfolios according to this asset allocation: one called the Normal
Portfolio where historical monthly returns of each asset class are fitted with normal distributions
and the other called the Log-Stable Portfolio, where the historical returns are fitted with
log-stable distributions. The following plots were generated from
10,000 Monte Carlo trials.
Figure 2 – Portfolios of Asset Allocations Fitted with Normal (left) and Log-Stable (right)
From Figure 2, it is difficult to tell the difference between the Normal Portfolio and the
Figure 3 – Zoom of Asset Allocations Fitted with Normal (left) and Log-Stable (right) Distributions
However, upon zooming-in, we see (left pane of Figure 3) the maximum loss for the Normal Portfolio
is 7% (-0.07) with a maximum gain of 7% (not shown). One may recall the normal distribution is
characterized by the mean plus or minus the standard deviation and is therefore symmetric about the
mean. The Log-Stable Portfolio (right pane of Figure 3) shows a maximum loss of 71% (-0.71) with a
maximum gain of 16% (not shown). This left-skew (-71% versus +16%) is based on the data and the
characteristic parameters of the log-stable distribution: alpha, beta, gamma, and delta (see
For the S&P 500 data from January 1926 to March 2009, alpha = 1.5901, beta = -0.5586, gamma =
0.0219, and delta
= 0.0023. Here, we see the negative value of beta as representing the left skew (see Appendix)
corresponding to more risk than reward. If the data had been such that beta was positive, the
distribution would have been right-skewed with returns being greater than risk (an elusive
investment). Incidentally, fat tails can occur on either or both sides of the distribution,
depending on the data being fitted to the log-stable distribution.
Importantly, the results illustrate how the log-stable distribution more accurately predicts high
levels of risk, ten-fold. Both normal and log-stable distributions are fitted to the same
historical return data. Yet, the normal distribution models risk at only 7%, while the log-stable
more accurately characterizes the risk at 71%, for the portfolio. This sheds a little light on the
2008 financial collapse from a risk versus reward perspective. There was actually a higher degree
of risk present than otherwise indicated by normal distributions.
Again, while a financial portfolio has been used to show how the log-stable distribution is
superior to the normal distribution, if rare events (e.g., earthquake magnitudes, city populations,
sizes of power outages, etc.) have a non-zero probability of occurrence (either left-skewed or
right-skewed), fitting the event data to a log-stable distribution will model these
characteristics, raising our awareness and allowing us to prepare, organize, train, and equip for black
EXPOSING BLACK SWANS
This is all rudimentary with simple portfolios of historical returns separated into normal and
log-stable distributions. What if your model of influential architecture contains thousands of
inputs, including normally distributed data as well as (a rare event) log-stable distributed data?
In this section, we model a portfolio with all asset classes fitted to a normal distribution except
for one asset class in order to see which tools are useful for finding which input is causing the
downside risk. The tools available are Tornado charts, Percent Point Functions, and stochastic
Tornado charts are specialized bar charts that show how varying an input impacts the output. In
our case, Tornado charts (Figures 4A and 4B below) show the impact of each asset class variation on
the overall portfolios. The asset class which has the greatest impact on both portfolios is IWM or
small-cap stocks. Changes in the return of IWM can lower the nominal (Normal Portfolio) return by
as much as 178% or raise it by 125%. Similarly, changes in the return of IWM can lower the nominal
(Log-Stable Portfolio) return by as much as 164% or raise it by 127%. Small-cap stocks (weighted at
20%) are more volatile. The asset class with the least impact is AGZ or government bonds. Changes
in the return of AGZ have minimal impact on the nominal (Normal Portfolio) return, lowering it by
7% or raising it by 8%. Similarly, changes in the return of AGZ have minimal impact on the nominal
(Log-Stable Portfolio) return, lowering it by 5% or raising it by 9%. Government bonds (weighted at
10%) are less volatile. If the reader is familiar with asset class returns, this result is not
surprising. Government bonds have much smaller returns (and less risk) than small-cap stocks.
However, the Tornado charts only look at the span between the 10th and 90th percentiles. In this
sense, they fall short from inspecting the tails of distributions – which is where we want to look
to see if risks, in the form of black swans, are hiding.
Figure 4A – Tornado Charts for Normal Portfolio
Figure 4B – Tornado Charts for Log-Stable Portfolio
Percent Point Functions (PPFs)
A PPF shows the probability of a random number being less than or equal to a particular point on
the plot. For example in Figure 5 (left pane), there is a 50% probability the return will be 1% or
less and there is a 90% probability the return will be 3% or less. The latter statement could be
interpreted as a 10% probability the return will be greater than 3%.
Figure 5 – PPFs for Normal and Log-Stable Portfolios
By examining Figure 5 (left pane), it appears as if the Normal and Log-Stable PPFs are identical.
However, upon closer inspection (right pane), we see the Normal Portfolio tail stops near -7%,
while the Log-Stable Portfolio tail continues down to -71%. Similarly, but not shown, the positive
tails for the Normal and Log-Stable PPF returns are 7% and 16%, respectively. Once more, we’ve
shown the log-stable distribution reveals a larger risk, ten-fold for the portfolios. Therefore, the
method of “PPF tail inspection” is a viable method to see if potential rare events (black swans)
might be lurking in the data.
Rather than comparing Normal and Log-Stable Portfolios, we now blend two portfolios – one with
historical monthly returns for all asset classes fit to normal distributions except for government
bond returns (AGZ) which are fit with the log-stable distribution; the other portfolio will fit
only real estate returns (IYR) to the log-stable distribution. These two (separate, but mixed)
portfolios are selected knowing ahead of time government bonds have the lowest (historical) spread
between risk and reward (-2% to 4%), while real estate has the highest (historical) spread (-31% to
29%). The reason for this is to examine the tails to see if any black swans might be identified,
independent of risk and reward spread.
Figure 6 – PPFs for AGZ (left) and IYR (right) Portfolios
In the case of the AGZ Mixed Portfolio (Figure 6, left pane), we see it has significantly more risk
(-27%) than the Normal Portfolio, which had 7% to the downside (Figure 3, left pane). Even by
modeling a less volatile asset class with the log-stable distribution, we are able to see
additional risk through the lens of the PPF tail. Of course, for the IYR Mixed Portfolio (Figure 6,
right pane) the risk is more pronounced (-48%) due to its higher volatility and by virtue of being
modeled with a log-stable distribution. The risk of IYR alone is driving the risk of the entire
portfolio. In sum, the PPF tail exposes the possibility of the occurrence of a rare event.
Finally, we come to the method of stochastic optimization, where random variables appear in the
formulation of the optimization problem thus producing a random objective function for which random
iterates are employed to solve the problem. When optimizing (maximizing or minimizing) an objective
function (portfolio), stochastic optimization assesses the range of each probabilistic input and
selects whatever values are necessary to yield the desired result. For example, to find the minimum
portfolio value, the minimum historical return of each asset class will be chosen.
For a small portfolio of only nine asset classes, it is rather simple to calculate deterministic
minimum and maximum portfolio returns. Table 2 (below) shows minimum and maximum historical returns for each ETF. Based on the asset allocation we’ve been using (repeated in the table), the minimum and maximum portfolio returns are -22% and 14%, respectively.
Table 2 – Allocated Portfolio Minimum and Maximum Returns
The results of running stochastic optimization on this small portfolio were identical with the
deterministic case, i.e. minimum and maximum returns of -22% and 14%, respectively. Using
stochastic optimization for this sized problem is excessive. But, if a portfolio contains thousands
of random inputs, including complex interconnections, it soon becomes intractable to perform these
calculations with a spreadsheet, let alone by hand. The result of stochastic optimization is a list
of all the inputs and the values that have been chosen so as to achieve either the minimum or
The astute reader will wonder how the stochastic optimization risk and return range (-22% to 14%)
relates to the prior results of the Normal (-7% to 7%) and Log-Stable (-71% to 16%) Portfolios.
We’ve already discussed the normal distribution and how it naïvely characterizes risk. This
explains why the risk and return range is lower than either of the other results. To explain the
difference between the stochastic optimization and the Log-Stable Portfolio results, we recognize
the historic lows and highs from Table 2 are bounded. For example, when stochastic optimization
seeks a minimum, the algorithm selects minimal values of the inputs, which are the lower bounds.
Likewise, for the maximum portfolio value, upper bounds are chosen. Stochastic optimization is
dependent on the bounded values of the inputs. Compare this to the log-stable distribution which
can return values beyond these limits, albeit with small (but non-zero) probability. Even though
the historical data is bounded, the log-stable distribution fits the data with parameters which
allows random numbers to be drawn in excess of these bounds – again, with small, but non-zero
probability. Think of it this way, while the largest earthquake on record is magnitude 9.5, the
possibility exists for a larger earthquake to occur – we just haven’t experienced it, yet.
Tornado charts can show how varying an input impacts the output. This is useful to identify which
inputs are influencing the output, but only between the 10th and 90th percentile. Examining the
tails of the PPF shows how much impact rare events could have. Stochastic optimization displays the
values of each of the inputs to achieve output extrema. Together, these three tools and their
insights reduce the surprise of a black swan, rendering it gray.
TRAINING AND SIMULATION APPLICATIONS
The triple constraint in the acquisition is cost, schedule, and performance. Any probabilistic inputs
for these models should be fitted with fat tail log-stable distributions to help understand any
exorbitant costs and extreme impacts of schedule slippage and poor performance.
Training proficiency is measured by its baseline effectiveness and the time and number of
iterations invested in the exercise. Proficiency is augmented by media factors and instructional
quality factors but diminished by skill decay due to lack of training. Any or all of these
stochastic inputs should be considered for being fitted with fat tail log-stable distributions.
Strategic Multi-Layer Assessment
Organizational models, social network models, time influence network models, information diffusion
models, and text analysis models comprising strategic multi-layer assessment have numerous inputs,
including event probabilities and event frequencies. Given the purpose of these models, one should
most assuredly use fat tail log-stable distributions to expose any black swan events like the rise
of Al Qaeda, Hezbollah, and most recently, ISIL.
In this paper, we briefly discussed black swans and their attributes. More appropriately answering
the first question from the “2015 I/ITSEC Black Swan Kickoff,” we showed it’s possible to identify
potential black swans and in doing so, render them gray. Thus we can prepare, organize, train and
equip for black swan resiliency.
Furthermore, we showed the log-stable distribution is preferred to the normal distribution when it
comes to modeling data that includes rare events lying far from the mean. The log-stable achieves
this by assigning a non-zero probability of occurrence with its fat tail, whereas the normal
distribution assigns a negligible probability due to its thin tail. We showed practical (financial)
applications of log-stable modeling for both individual data sets (S&P 500) as well as a portfolio
comprised of data sets (ETFs).
Finally, we discussed how three tools (Tornado charts, PPFs, and stochastic unconstrained
optimization) and their insights can reduce the surprise of a black swan, rendering it gray.
In the end, we have shown how modeling and simulation can be used to analyze and prepare or create
a black swan and in practice, we’ve developed models and simulation tools that enable the analysis.
We wish to thank Dr. Paul Kaplan (Morningstar) and Dr. John Nolan (University of Virginia) for
personal email correspondence. We wish to recognize Investools / TD Ameritrade as the database from
which we were able to obtain historical monthly data for each asset class.
Fama, E. (1963). Mandelbrot and the Stable Paretian Hypothesis. Journal of Business, Vol. 36, No.
4, 420-429. Kaplan, P. (2012). Frontiers of Modern Asset Allocation. Hoboken, NJ: John Wiley &
Kaplan, P. (2008). Using Fat Tails to Model Gray Swans. Retrieved from
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No. 4, 394-419. Markowitz, H. (1952). Portfolio Selection. Journal of Finance, Vol. 7, No. 1,
Markowitz, H. (1979). Approximating Expected Utility by a Function of Mean and Variance. The
Economic Review, Vol. 69, No. 3, 308-317.
Markowitz, H. (1999). The Early History of Portfolio Theory: 1600–1960. Financial Analysts Journal,
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Nolan, J. (2009b). User Manual for STABLE 5.1: Matlab Version. Retrieved from
Savage, S. (2009). The Flaw of Averages. Hoboken, NJ: John Wiley & Sons, Inc.
Taleb, N. (2010). The Black Swan – The Impact of the Highly Impossible. NY: Random House Inc.
Veillette, M. (2015). STBL: Alpha stable distributions for MATLAB. Retrieved from
Wright, C. (2007, March/April). Tail Tales. Chartered Financial Analyst Institute Magazine.
APPENDIX – MATHEMATICAL PROPERTIES OF THE LOG-STABLE DISTRIBUTION
The log-stable distribution is frequently used to model investment returns. Returns are expressed
in decimal form, where negative returns represent losses and positive returns represent profit. We
then normalize the returns by adding one and taking the natural log of the result. Once in this
form, the returns conform to a stable distribution. The probability density function (pdf) for a (fat-tail) stable distribution is
α represents the “fatness” of the tails and is in the range between 0 and 2, with 2 being a normal
distribution. Also, if alpha is less than 1, then the mean of the distribution is infinite.
β represents the skewness of the distribution and lies within the range of -1 to 1, where -1
signifies fully left-skewed and +1 signifies fully right-skewed. If beta is 0, the distribution is symmetric.
represents the scale of the distribution and is positive. If alpha = 2 (normal), then gamma
squared is one-half the variance.
represents the location of the distribution. If alpha > 1, then delta is the mean of
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