Timothy Aman, FCAS MAAAManaging Director, Guy Carpenter Miami

Statistical Limitations of Catastrophe Models

CAS Limited Attendance SeminarNew York, NY

18 September 2006

2

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

3

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

4

Outline

Definitions

A frequentist approach

An update

Statistical limitations of cat models

5

Definitions

6

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)

7

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

14Return Period (Years)

L̂L )95(.

L̂L )05(.

A frequentist approach

15

An update

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An update

With the addition of more years of hurricane data, how have relative confidence intervals changed?

17

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

19

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

90% confidence intervals around 100-year loss

0.00

0.50

1.00

1.50

2.00

2.50

3.00

Miller Model A Model B Model C

5%ile 95%ile

21

An update

90% confidence intervals around 100-year loss

0.00

0.50

1.00

1.50

2.00

2.50

3.00

Miller Model A Model B Model C

5%ile 95%ile

22

An update

90% confidence intervals around 100-year loss

0.00

0.50

1.00

1.50

2.00

2.50

3.00

Miller Model A Model B Model C

5%ile 95%ile

23

An update

90% confidence intervals around 100-year loss

0.00

0.50

1.00

1.50

2.00

2.50

3.00

Miller Model A Model B Model C

5%ile 95%ile

24

An update

Now a look at the 250-year level…

25

An update

90% confidence intervals around 250-year loss

0.00

0.50

1.00

1.50

2.00

2.50

3.00

Miller Model A Model B Model C

5%ile 95%ile

26

An update

90% confidence intervals around 250-year loss

0.00

0.50

1.00

1.50

2.00

2.50

3.00

Miller Model A Model B Model C

5%ile 95%ile

27

An update

90% confidence intervals around 250-year loss

0.00

0.50

1.00

1.50

2.00

2.50

3.00

Miller Model A Model B Model C

5%ile 95%ile

28

An update

90% confidence intervals around 250-year loss

0.00

0.50

1.00

1.50

2.00

2.50

3.00

Miller Model A Model B Model C

5%ile 95%ile

29

Statistical Limitations of Cat Models

30

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

32

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

$0

$10,000

$20,000

$30,000

$40,000

$50,000

$60,000

$70,000

$80,000

$90,000

'85 '87 '89 '91 '93 '95 '97 '99 '01 '03 '05*

USA Non-US

* Preliminary estimate. Source: Swiss Re Sigma

34

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

35

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

36

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

37

Statistical Limitations of Cat Models

“Don’t believe the cat model point estimates too much, but don’t believe them too little.”