optimal risk selection using cat models

Optimal Risk SelectionOptimal Risk SelectionUsing Cat ModelsUsing Cat Models

Lixin Zeng, Ph.D.Lixin Zeng, Ph.D.CAS Seminar on Funding of Catastrophe RisksCAS Seminar on Funding of Catastrophe Risks

Providence RIProvidence RIOctober 17, 2000October 17, 2000

beyond The BoxThe Box

ThinkingThinking

22

Use and Misuse of Cat ModelUse and Misuse of Cat Model

Optimal Risk SelectionOptimal Risk Selection

ExampleExample

OutlineOutline


33

Loss Probability DistributionLoss Probability Distribution Expected LossExpected Loss Probable Maximum Loss Probable Maximum Loss

(a.k.a. Value at Risk)(a.k.a. Value at Risk)

Relative ValueRelative Value Deal A is riskier than Deal BDeal A is riskier than Deal B Correlation: Constructing a Portfolio Correlation: Constructing a Portfolio

with High Return on Risk Capitalwith High Return on Risk Capital

What a Cat Model Tells UsWhat a Cat Model Tells Us


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Underwriting DecisionsUnderwriting Decisions

Rate MakingRate Making

Reinsurance PurchasingReinsurance Purchasing

SecuritizationSecuritization

Great! Cat Problem Solved?Great! Cat Problem Solved?


55

State-of-the-Art Science State-of-the-Art Science in Meteorology and in Meteorology and SeismologySeismology

Engineering Engineering Experts’ Experts’ OpinionsOpinions for Structure for Structure DamageDamage

Modern Modern SimulationSimulation TechnologyTechnology

What’s Inside a Cat ModelWhat’s Inside a Cat Model


Lack of Consensus in Lack of Consensus in Scientific Community Scientific Community on Key Issueson Key Issues

Best Guesses Given Best Guesses Given Limited Data and Limited Data and ModelingModeling

Computation Hurdles Computation Hurdles vs. Convergencevs. Convergence

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Understand Key AssumptionsUnderstand Key Assumptions

Appreciate Sources of UncertaintyAppreciate Sources of Uncertainty

Independent Model EvaluationIndependent Model Evaluation

Integrate Multiple ModelsIntegrate Multiple Models

User’s ResponsibilitiesUser’s Responsibilities


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Relative (Not Absolute) IndicatorsRelative (Not Absolute) Indicators

Differentiate Good and Bad Differentiate Good and Bad Risks/AreasRisks/Areas

Risk SelectionRisk Selection

Portfolio OptimizationPortfolio Optimization

What’s a Cat Model Good For?What’s a Cat Model Good For?


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Goal of Risk SelectionGoal of Risk Selection


Existing Existing PortfolioPortfolio““Bad” RisksBad” Risks ““Good” RisksGood” Risks

Final PortfolioFinal Portfolio

Maximum Return on Risk CapitalMaximum Return on Risk Capital

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ReturnReturn Cat premium minus expected cat lossCat premium minus expected cat loss

Risk CapitalRisk Capital Probable maximum loss (or value at risk)Probable maximum loss (or value at risk) Expected policy holder deficitExpected policy holder deficit Loss standard deviationLoss standard deviation

Applicable to Both Individual Risks and Applicable to Both Individual Risks and PortfoliosPortfolios

Return on Risk Capital (RORC)Return on Risk Capital (RORC)


1010

A Simple DefinitionA Simple Definition

Cat Premium - Expected Cat LossCat Premium - Expected Cat Loss

Cat X-Year PMLCat X-Year PML

Different DefinitionsDifferent Definitions Financial strengthFinancial strength Risk toleranceRisk tolerance etc.etc.

RORC: DefinitionRORC: Definition


1111

An Individual Risk Is the Worst in a An Individual Risk Is the Worst in a Portfolio ifPortfolio if

(1) It has the lowest RORC among all risks(1) It has the lowest RORC among all risks

(2) Removing it will increase the (2) Removing it will increase the portfolio’s RORC the most vs. removing portfolio’s RORC the most vs. removing any other individual riskany other individual risk

The right answer: The right answer: (1) or (2)?(1) or (2)?

Identify “Bad” Individual RisksIdentify “Bad” Individual Risks


1212

Prob. of Non-exceedance

PM

L

0.90 0.92 0.94 0.96 0.98 1.00

10

20

30

40

Policy 1Policy 2Policy 3Portfolio

A Sample PortfolioA Sample Portfolio


1313

RORCRORC


Return Period (year) 100 250 500Policy 1 15.48% 11.50% 9.42%Policy 2 7.09% 5.25% 4.34%Policy 3 7.78% 5.93% 4.76%Portfolio 15.77% 11.85% 10.03%

1414

A Sample PortfolioA Sample Portfolio


Prob. of Non-exceedance

PM

L

0.90 0.92 0.94 0.96 0.98 1.00

10

20

30

40

PortfolioPolicy 1 removedPolicy 2 removedPolicy 3 removed

1515

RORCRORC


Return Period (year) 100 250 500Portfolio 15.77% 11.85% 10.03%Policy 1 removed 11.53% 8.69% 7.44%Policy 2 removed 12.22% 8.98% 7.28%Policy 3 removed 15.93% 12.30% 10.26%

1616

An Individual Risk Is the Best Prospect for a An Individual Risk Is the Best Prospect for a Portfolio ifPortfolio if

(1) It has the highest RORC among all (1) It has the highest RORC among all prospectsprospects

(2) Including it in the portfolio will increase (2) Including it in the portfolio will increase the portfolio’s RORC the most vs. the portfolio’s RORC the most vs. including any other prospectincluding any other prospect

The right answer: The right answer: (1) or (2)?(1) or (2)?

Identify “Good” Prospective Risks: Same IdeaIdentify “Good” Prospective Risks: Same Idea


1717

Real World: Computational IssuesReal World: Computational Issues


Finding the X Worst (or Best) from N Finding the X Worst (or Best) from N RisksRisks Requires CRequires CNN

XX calculations calculations E.g. requires ~ 17,000,000,000,000 E.g. requires ~ 17,000,000,000,000

calculations to pick 10 worst (or best) out calculations to pick 10 worst (or best) out of 100 risksof 100 risks

Need a Faster, More Practical ApproachNeed a Faster, More Practical Approach

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A Real Solution: Discrete Steepest DescentA Real Solution: Discrete Steepest Descent


Existing Existing PortfolioPortfolio

Remove #1 onlyRemove #1 only


………………..

…………......

Remove #N-1 onlyRemove #N-1 only

Remove #N onlyRemove #N only

Portfolio w/o Portfolio w/o worst riskworst risk



…………......

…………......

Remove #N-1 onlyRemove #N-1 only


………………....

Remove #N-X onlyRemove #N-X only

Portfolio w/o Portfolio w/o X worst risksX worst risks

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Finding the X Worst (or Best) from N RisksFinding the X Worst (or Best) from N Risks


Requires O(NRequires O(N22) Calculations) Calculations E.g. requires 1,000 calculations to pick 10 E.g. requires 1,000 calculations to pick 10

worst (or best) out of 100 risksworst (or best) out of 100 risks Innovative algorithm to handle large Innovative algorithm to handle large

portfoliosportfolios

Stochastic Perturbation to Avoid Local Stochastic Perturbation to Avoid Local MinimumMinimum

2020number of policies removed

RO

RC

0 50 100 150 200 250 300

0.1

00

.12

0.1

40

.16

Real-World Example: Portfolio of 1500 RisksReal-World Example: Portfolio of 1500 Risks


Optimal Risk Optimal Risk SelectionSelection

BenchmarkBenchmark

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CautionsCautions


Cat Model Relative BiasCat Model Relative Bias Geographical and structuralGeographical and structural Usually less than absolute biasUsually less than absolute bias But cannot be ignoredBut cannot be ignored

Use of a Single Point on the PML CurveUse of a Single Point on the PML Curve Potentially misleadingPotentially misleading

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ConclusionsConclusions


Cat ModelCat Model Relative indications more credible than Relative indications more credible than

absolute valuesabsolute values

Portfolio OptimizationPortfolio Optimization One of the best uses of cat modelsOne of the best uses of cat models Cat model relative bias must be evaluated Cat model relative bias must be evaluated

and understoodand understood

optimal risk selection using cat models

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