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Statistical Limitations of Catastrophe
Models
CAS Limited Attendance Seminar
New York, NY
18 September 2006
Timothy Aman, FCAS MAAA
Managing Director, Guy Carpenter Miami
Introduction

Given the limited Atlantic hurricane sample size, speakers discuss the
limitations of predictive modeling from three perspectives:
– A frequentist (broker) approach using bootstrapping techniques
– A Bayesian (modeler) approach incorporating new events into a
prior assumption framework
– A practical (insurer) approach reconciling the politics of actual
claims experience with model-based expectations
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Introduction

When cat models first came out, loss estimates at various return
periods AND upper confidence bounds around those loss estimates
were regularly shown as output

Over the course of time, fewer and fewer output summaries have
focused on confidence bounds and uncertainty

This panel attempts to remind us of the magnitude of that uncertainty,
from various perspectives
3
Outline

Definitions

A frequentist approach

An update

Statistical limitations of cat models
4
Definitions
5
Definitions

Frequentist: One who believes that the probability of an event should
be defined as the limit of its relative frequency in a large number of
trials
– Probabilities can be assigned only to events
– Need well-defined random experiment and sample space

Bayesian: Probability can be defined as degree to which a person
believes a proposition
– Probabilities can be applied to statements
– Need a prior opinion (ideally, based on relevant knowledge)
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Definitions

A bootstrap sample is obtained by randomly sampling n times, with
replacement, from the original data points [Efron]

Bootstrap methods are computer-intensive methods of statistical
analysis that use simulation to calculate standard errors, confidence
intervals, and significance tests [Davison and Hinkley]
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Definitions

In statistics bootstrapping is a method for estimating the sampling
distribution of an estimator by resampling with replacement from the
original sample
– Most often with the purpose of deriving robust estimates of
standard errors and confidence intervals of a population parameter

The bootstrap technique assumes that the observed dataset is a
representative subset of potential outcomes from some underlying
distribution
– Random subsamples from the observed dataset are themselves
representative subsets of potential outcomes
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A frequentist approach
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A frequentist approach


David Miller: “Uncertainty in Hurricane Risk Modeling and Implications
for Securitization”, (Guy Carpenter, 1998)
– CAS Forum 1999, Securitization of Risk
David Miller “thought experiment”
– Create multiple catastrophe simulation models, each based on a
simulated historical event set
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A frequentist approach

Miller’s approach
– Frequency is historical number of hurricanes over time period

Assume distributed Poisson
– Conditional severity is based on bootstrap technique

Assume stationary climate

Each bootstrap replication represents an equivalent realization
of the historical record, and consists of random draw, with
replacement, of N hurricanes from the observed record

Confidence intervals can then be determined from the boostrap
replications
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A frequentist approach

Miller’s approach
– Essentially, each bootstrap replication represents a new
catastrophe simulation model, created as if the observed historical
event set had been the replicated rather than the actual event set
– “Blended” approach

Severity distribution is calculated using a given catastrophe
model

This severity distribution is fit to a parametric model (Beta
distribution)

New parametric severity distribution is fit for each bootstrap
replication

Use fitted parametric distribution for severity
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A frequentist approach

Miller’s conclusions for hurricane loss 90% confidence intervals for
three US nationwide portfolios (personal, commercial, and specialty)
– Low return periods (<10 years)

Lower bound is 0

Upper bound diverges (as multiple of mean)
– Remote return periods (>80 years)

Lower bound 0.5 times mean estimate

Upper bound 2.5 times mean estimate
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A frequentist approach
L(.95)
L̂
L(.05)
L̂
Return Period (Years)
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An update
15
An update

With the addition of more years of hurricane data, how have relative
confidence intervals changed?
16
An update

Suppose we want to estimate “100-year loss” to a portfolio

Suppose we have a reliable sample of 100 years of data
– We might have seen a 100-year loss in the sample (63% of
samples, assuming Poisson frequency)
– We might not (37% of samples)

Now suppose we have a reliable sample of 110 years of data
– The above probabilities are revised to 67% and 33%

…and so on…

With a sample of 300 years, the probabilities are 95% and 5%

With a sample of 450 years, the probabilities are 99% and 1%
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An update

Bootstrap from cat model output
– Simulate datasets using cat model event sets
– “Direct” approach

Eliminates need to specify, fit, and re-fit conditional severity
distributions
– Determine relative confidence intervals at various return periods
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An update

For a given return period n…

Mean
– Generate samples of n years
– Identify largest element of each sample year
– Take the average over all sample years of the largest observation
in each year

Confidence intervals
– Capture through repeated experiment the distribution of the above
mean
– Take the 5th and 95th sample percentiles of the maximum value
across all sample years
– Obtain 90% confidence interval around mean estimate
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An update
20
An update
21
An update
22
An update
23
An update

Now a look at the 250-year level…
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An update
25
An update
26
An update
27
An update
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Statistical Limitations of Cat Models
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Statistical Limitations of Cat Models

John Major: “Uncertainty in Catastrophe Models: Part I: What is it
and where does it come from?” and “Part II: How bad is it?”, (Guy
Carpenter, 1999)
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Statistical Limitations of Cat Models

Sources of uncertainty in catastrophe modeling
1. Limited data sample

For example, estimating 250-year EQ losses with only 100
years of detailed data
2. Model specification error

For example, Poisson frequency (iid assumption)
3. Nonsampling error

Identification of all relevant factors

For example, global climate change
4. Approximation error

For example, limited simulations and discrete event sets
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Statistical Limitations of Cat Models

Cat models are collections of event scenarios
– Discrete approximations, with probabilities attached to each
scenario
– Not exhaustive
– Limited perils
– Calibrated using historical experience

Recalibrated as required, based on research and actual event
experience
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Worldwide Property Catastrophe Insured Losses
$90,000
$80,000
$70,000
$60,000
$50,000
$40,000
$30,000
$20,000
$10,000
$0
'85
'87
'89
'91
'95
'93
USA
'97
'99
'01
'03
'05*
Non-US
* Preliminary estimate. Source: Swiss Re Sigma
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Statistical Limitations of Cat Models

Uncertainty factors due to limited sample size are substantial

Data quality can add significantly to uncertainty

Are we capturing all material factors?

Scientific input can be used to reduce uncertainty
– Hazard sciences (meteorology, seismology, vulcanology)
– Engineering studies
34
Statistical Limitations of Cat Models

Factors potentially influencing relative confidence interval widths
– Larger data sample / destabilizing recent experience
– Improvements in science / weakening of stationary climate
assumption
– Improvements in technology
– Differences in modeled portfolios
– Negative Binomial frequency
– Increased awareness of factors contributing to uncertainty

Further exploration of the general factors influencing relative
confidence interval widths is material for another presentation
35
Statistical Limitations of Cat Models

Relative widths of individual company confidence intervals will depend
on specifics
– Geographical scope

e.g., US hurricane, Peru earthquake, UK flood
– Insured portfolio

e.g., Dwellings, Petrochemical facilities, Hotels
– Financial variables

e.g., Excess policies, EQ sublimits, Business interruption

Further exploration of the portfolio-specific factors influencing relative
confidence interval widths is material for another presentation
36
Statistical Limitations of Cat Models

“Don’t believe the cat model point estimates too much, but don’t
believe them too little.”
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