2007 National Council on Compensation Insurance, Inc. 1 Catastrophe Modeling and Actuarial Applications Jonathan Evans, FCAS, MAAA Actuary Iowa Actuarial.

2007 National Council on Compensation Insurance, Inc. 1

Catastrophe Modeling and Actuarial Applications

Jonathan Evans, FCAS, MAAA

Actuary

Iowa Actuarial Club

February 23, 2007

Source: 2007 National Council on Compensation Insurance, Inc.

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What Are Catastrophes ?

• Single events that produce such a large number of claims – that it is very highly improbable such an event

would actually occur (assuming risk exposures are independent)

• Usually resulting from a special type of peril: hurricane, earthquake, terrorist attack, asbestos, etc.

• More rarely, sometimes extremely large single claims are called “catastrophes”

Source: 2007 National Council on Compensation Insurance, Inc.

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What Applications Make Actuaries Care About Catastrophes ?

• Ratemaking for policies exposed to catastrophes, such as homeowners in Florida

• Reserving for long tail catastrophic losses, such as asbestos in general liability

• Analyses of reinsurance programs• Analyses of insurer catastrophe risk, such as

catastrophic return period loss estimates

Source: 2007 National Council on Compensation Insurance, Inc.

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Why Model Catastrophes ?

• For non-catastrophic perils (such as an automobile accident that independently affects risk exposures or only a few risk exposures simultaneously) it is easy to gather a credible volume of experience

• For catastrophic perils (such as hurricanes) even nationwide experience over many years may not be credible

Source: 2007 National Council on Compensation Insurance, Inc.

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Catastrophic Versus Non-CatastrophicExperience

Hypothetical Example Of Characteristic Non-Catastrophic Experience Pattern

Hypothetical Example Of Characteristic Catastrophic Experience Pattern

Source: 2007 National Council on Compensation Insurance, Inc.

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Who Builds Catastrophe Models ?

• Prior to the 1980s most models were built by insurers for underwriting purposes

• From the 1980s to present most models were built by specialized modeling firms and sold as software, that allows users to input exposure information

• Event databases and estimated frequencies are often prepared by scientists: geophysical for earthquake, atmospheric for hurricanes, social/behavioral for terrorism, etc.

• Loss functions are often determined by engineers• Actuaries are mostly end users of models; except for

reserving models for mass liability, such as asbestos/environmental, where actuaries have compiled databases and built their own models

Source: 2007 National Council on Compensation Insurance, Inc.

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How Are Catastrophes Modeled ?

Catastrophe models generally consist of:

1. A database of individual exposures, such as homeowners policies by location and insured value

2. A database of potential catastrophic events, such as hurricanes by landfall location and wind speed

3. A loss function that uses as input the characteristics of a single exposure from #1 and a single catastrophic event from #2

4. Estimates for the frequency of each catastrophic event in #2

Source: 2007 National Council on Compensation Insurance, Inc.

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Simplified Hypothetical Example Of A Catastrophe Model

Policy Insured Value (IV) Location (L)

A 1,000,000 100B 1,000,000 110C 500,000 130D 5,000,000 500E 7,000,000 600

Event Intensity (I) Center Location (LC)

U 10 300W 5 250X 3 260Y 2 105Z 1 550

Event Frequency Per Year (F)

U 0.001 W 0.005 X 0.010 Y 0.050 Z 0.100

2(L -LC)Loss = IV exp -

2000 I

1

3 4

2

Source: 2007 National Council on Compensation Insurance, Inc.

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Example Of Calculating Loss

For Policy A and Event U the loss is calculated as:

2(100-300)Loss = 1,000,000 exp - 135,335

2000 10

Source: 2007 National Council on Compensation Insurance, Inc.

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Example Model Loss Output

Policy U W X Y Z

A 135,335 105,399 14,028 993,769 0 B 164,474 140,858 23,518 993,769 0 C 117,873 118,464 29,902 427,673 0 D 676,676 9,652 339 0 1,432,524 E 77,763 33 0 0 2,005,534

Event Total 1,172,122 374,407 67,787 2,415,212 3,438,058

Frequency 0.001 0.005 0.010 0.050 0.100

Event

Source: 2007 National Council on Compensation Insurance, Inc.

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Example Ratemaking Application

For Policy AEvent Loss Frequency Expected Loss

U 135,335 0.001 135 W 105,399 0.005 527 X 14,028 0.010 140 Y 993,769 0.050 49,688 Z 0 0.100 0

All Events 50,491

Insured Value 1,000,000

Pure Premium (per 100 value) 5.05

Pure PremiumPolicy Expected Loss (per 100 value)

A 50,491 5.05 B 50,792 5.08 C 22,393 4.48 D 143,981 2.88 E 200,631 2.87

All Policies 468,288 3.23

Source: 2007 National Council on Compensation Insurance, Inc.

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Example Portfolio Risk Application

Frequency Return ExceedanceEvent Loss Frequency at or Above Period Probability

3,438,058 0.100 0.100 10 10%2,415,212 0.050 0.150 7 14%1,172,122 0.001 0.151 7 14%

374,407 0.005 0.156 6 14%67,787 0.010 0.166 6 15%

Source: 2007 National Council on Compensation Insurance, Inc.

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Questions

Source: 2007 National Council on Compensation Insurance, Inc.

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Further Reading

-Bouska, Amy, “From Disability Income to Mega-Risks: Policy-Event Based Loss Estimation”, CAS Forum, Summer 1996.

-Grossi, Patricia and Kunreuther, Howard, Catastrophe Modeling: A New Approach to Managing Risk (Huebner International Series on Risk, Insurance and Economic Security), Springer, 2005.

-Kozlowski, Ronald T. and Mathewson, Stuart B., “Measuring and Managing Catastrophic Risk”, CAS Discussion Paper Program, 1995.

-Woo, Gordon, The Mathematics of Natural Catastrophes, World Scientific Publishing Company, 1999.