Catastrophe models and extreme events

1. We use catastrophe (Cat) models to compute potential losses for
future events based on physics and the statistical distribution of
historical natural catastrophes. But are these the best tools to capture
extreme events such as hurricane Sandy or the Tohoku earthquake?
Introduction
Cat models are broadly used in the insurance industry as tools to predict potential losses for future events. Cat
models draw knowledge from historical events and from the physical characteristics of the modeled perils
(earthquakes, floods, etc). Some recent theories refer to the existence of "Dragon-kings”, namely extreme
events such as earthquakes that can cause devastating effects in complex systems (1). The term “Dragon-
king” refers to an event that is both extremely large (a “king'”), and of unique origins (“dragon”) relative to
its peers. Both hurricane Sandy and the Tohoku earthquake could be seen as Dragon-kings. According to the
theory these events are not found at the tail of the statistical distribution describing smaller events and
therefore a pure statistical analysis of past events would not capture them.
The theory of Dragon-kings
Rare catastrophic events are often associated with heavy-tailed severity statistical distributions such as the
Pareto distribution (bold line in Fig. 1).
Fig 1: The Pareto statistical distribution, also known as a power law distribution, is heavy tailed and appropriate to
represent the 80/20 rule, such as “80% of the wealth is owned by 20% of the population “.
RiskTopics
Are catastrophe models able to capture extreme events?
January 2016

2. 2
But according to the theory of Dragon-kings this approach is flawed because extreme events do not follow
the same power law distribution of their smaller siblings, but rather appear like outliers in the statistical
analysis.
Figure 2 shows an example of this concept where the Dragon-kings would be associated with the clusters
above the extrapolation of the Gutenberg-Richter distribution calibrated on the smaller events. In these two
examples it appears that the power law distribution linking the magnitude and frequency of earthquakes
coexists with the Dragon-king effect. For a Dragon-king type of event to take place additional amplification
mechanisms with cascading effects are required (2).
Fig 2: Distribution of earthquake magnitudes in a thin strip around the Newport-Inglewood-Rose Canyon fault and the
Whittier-Elsinore fault, respectively (2).
Dragon-kings and Cat modeling
Two very important parameters play a role in whether an event can be categorized as a Dragon-king, namely
the degree of coupling and heterogeneity. Dragon-kings are characterized by a high level of coupling and low
heterogeneity (i.e. low diversification). The Tohoku earthquake followed by the tsunami and nuclear accident
could be seen as an example of such conditions. Although the single underlying risks (earthquake and
tsunami hazard, nuclear hazard) were known, the coupling of the different systems was poorly understood.
The cascading effects were key to the level of destruction caused by the event.
Assuming that the Dragon-kings theory holds, the question is whether these events can be predicted and
more so whether they can be predicted in a way that is appropriate for the insurance industry where risk is
typically estimated on a yearly basis (pricing of policies, reinsurance purchase, etc.). The predictability of
Dragon-kings is first of all subject to the availability of a set of observable variables defining the system under
scrutiny. The system would then evolve towards a state of instability or higher susceptibility to external shocks
before an extreme event takes place. For natural catastrophes it is not trivial to determine an unambiguous
spatial domain (scope) to observe. For earthquakes for instance the spatial domain of influence of a fault
should be defined and for hurricanes it would be necessary to integrate variables such as sea surface
temperature, large scale wind patterns, evolution of the storm trajectory / strength and social systems with
their infrastructure. In the case of hurricanes the prediction would not give any more than a few days lead
time and would therefore not be compatible with an insurer’s risk estimation requirements, although loss
prevention and mitigation as emergency response could still be possible.
Fortunately the more traditional Cat modeling approach is not obsolete, although it can certainly be
improved. Cat models are in fact not purely based on statistical analyses of past events. One important

3. 3
component is the underlying physical model that coupled with the statistical one moves away from the simple
extrapolation based on a power law distribution as seen in figure 2, and provides a better fit for extreme
events. Nicolas Taleb (5) developed the theory of “Black swans” as events characterized by rarity, extreme
impact, and retrospective (though not prospective) predictability. Lin and Emanuel (4) define the term “Grey
swan” as a rare tropical cyclone that would not be predicted based on history, but may be foreseeable using
physical knowledge together with historical data. Moreover, the size of the simulation dataset also plays a key
role for the distribution to be able to capture more extreme events as was shown by Hall and Sobel (3) in their
study of hurricane Sandy’s impact angle at landfall.
One of the most important aspects that was highlighted above is the cascading effect taking place during
catastrophic events. These are indeed only marginally included in Cat models and certainly need to be more
clearly understood. The impact of interdependencies and indirect losses (contingent business interruption)
need to be better integrated into Cat modeling results.
A critical approach to Cat modeling like that of Zurich is also of outmost importance. The so-called “Zurich
View” of Cat risk is the internally developed approach that allows us to appropriately adjust vendor models'
results to a more accurate representation of the risk, based on our understanding of the models and
expertise in natural catastrophe risk assessment. The “Zurich View” of risk also includes the assessment of
non-modeled items (for example demand surge, fire following earthquake, sprinkler leakage, storm surge,
inland flooding, tsunami) where applicable.
Finally, modeling of uncertainties is key to build a buffer for the unknown and for the amplification effects in
extreme events therefore leading to more reliable risk estimates.
Conclusions
The purpose of this paper was to discuss the common methodology applied in the insurance industry to
predict extreme events in the context of natural hazards. This methodology is based on the use of Cat models
are tools to merge the physics of natural catastrophes and the statistical properties of historical occurrences.
The theory of Dragon-kings challenges the Cat modeling approach, but has limited applicability for natural
catastrophes due to a series of caveats, among which the difficulty of defining an unambiguous spatial
domain to observe and the short time window for prediction. Conversely, the combination of a physical and a
statistical approach, both part of a standard Cat model, can overcome the power law distribution limitations
and therefore allow the capturing of more extreme events.
Cat modeling results can still be improved by integrating structural, organizational and socio-economic
vulnerabilities, like the impact of interdependencies, indirect losses as well as by building a buffer for
uncertainties.
Zurich is one of the few players in the industry that takes a critical approach to Cat modeling and builds its
own view of the risk. The same view is then passed on to its customers in an effort to share knowledge and
provide expertise in natural catastrophes risk assessment.

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References
1) “Predictability and Suppression of Extreme Events in a Chaotic System”, Hugo L. D. de S. Cavalcante,
Marcos Oriá, Didier Sornette, Edward Ott, and Daniel J. Gauthier, Phys. Rev. Lett. 111, 198701 – Published 4
November 2013
2) “Dragon-Kings, Black Swans and the Prediction of Crises”, Didier Sornette, International Journal of
Terraspace Science and Engineering, 2(1), 1-18 (2009)
3) “On the impact angle of Hurricane Sandy’s New Jersey landfall”, T. M. Hall and A. H. Sobel, Geophys. Res.
Lett., 40, 2312–2315, doi:10.1002/grl.50395 (2013)
4) Ning Lin & Kerry Emanuel, “Grey swan tropical cyclones,” Nature Climate Change,
doi:10.1038/nclimate2777 (2015)
5) Taleb, N. N., “The Black Swan: The Impact of the Highly Improbable”, Random House (2007)

5. RiskTopicsDragonKingsfinalv4.1_final.docx
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