CARe Seminar, NYC

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Uncertainty AndProperty Cat Pricing

• CARe Seminar, NYC
• February 28, 2002
• Jonathan Hayes, ACAS, MAAA

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Agenda

• Models
• Model Results
• Confidence Bands
• Data
• Issues with Data
• Issues with Inputs
• Model Outputs
• Pricing Methods
• Standard Deviation
• Downside Risk
• Role of Judgment
• Still Needed

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The Search For Truth
A Nixon-Agnew administration will abolish the
credibility gap and reestablish the truth the
whole truth as its policy. Spiro T. Agnew,
Sept. 21, 1973
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Florida Hurricane
Amounts in Millions USD
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Florida Hurricane
Amounts in Millions USD
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Modeled Event LossSample Portfolio, Total Event
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Modeled Event LossBy State Distribution
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Modeled Event LossBy County Distribution, State S
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Why Dont The Models Agree?
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Types Of Uncertainty(In Frequency Severity)

• Uncertainty (not randomness)
• Sampling Error
• 100 years for hurricane
• Specification Error
• FCHLPM sample dataset (1996) 1 in 100 OEP of 31m,
  38m, 40m 57m w/ 4 models
• Non-sampling Error
• El Nino Southern Oscillation
• Knowledge Uncertainty
• Time dependence, cascading, aseismic shift,
  poisson/negative binomial
• Approximation Error
• Res Re cat bond 90 confidence interval,
  process risk only, of /- 20, per modeling firm

Source Major, Op. Cit..
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Frequency-Severity UncertaintyFrequency
Uncertainty (Miller)

• Frequency Uncertainty
• Historical set 96 years, 207 hurricanes
• Sample mean is 2.16
• What is range for true mean?
• Bootstrap method
• New 96-yr sample sets Each sample set is 96
  draws, with replacement, from original
• Review Results

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Frequency Bootstrapping

• Run 500 resamplings and graph relative to
  theoretical t-distribution

Source Miller, Op. Cit.
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Frequency Uncertainty Stats

• Standard error (SE) of the mean
• 0.159 historical SE
• 0.150 theoretical SE, assuming Poisson, i.e.,
  (lambda/n)0.5

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Hurricane Freq. UncertaintyBack of the Envelope

• Frequency Uncertainty Only
• 96 Years, 207 Events, 3100 coast miles
• 200 mile hurricane damage diameter
• 0.139 is avg annl storms to site
• SE 0.038, assuming Poisson frequency
• 90 CI is loss /- 45
• i.e., (1.645 0.038) / 0.139

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Frequency-Severity UncertaintySeverity
Uncertainty (Miller)

• Parametric bootstrap
• Cat model severity for some portfolio
• Fit cat model severity to parametric model
• Perform X draws of Y severities, where X is
  number of frequency resamplings and Y is number
  of historical hurricanes in set
• Parameterize the new sampled severities
• Compound with frequency uncertainty
• Review confidence bands

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OEP Confidence Bands
Source Miller, Op. Cit.
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OEP Confidence Bands
Source Miller, Op. Cit.
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OEP Confidence Bands

• At 80-1,000 year return, range fixes to 50 to
  250 of best estimate OEP
• Confidence band grow exponentially at frequent
  OEP points because expected loss goes to zero
• Notes
• Assumed stationary climate
• Severity parameterization may introduce error
• Modelers secondary uncertainty may overlap
  here, thus reducing range
• Modelers severity distributions based on more
  than just historical data set

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The Building BlocksPolicy Records/TIV
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Data Collection/Inputs

• Is this all the subject data?
• All/coastal states
• Inland Marine, Builders Risk, APD, Dwelling Fire
• Manual policies
• General level of detail
• County/zip/street
• Aggregated data
• Is this all the needed policy detail?
• Building location/billing location
• Multi-location policies/bulk data
• Statistical Record vs. policy systems
• Coding of endorsements
• Sublimits, wind exclusions, IM
• Replacement cost vs. limit

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More Data Issues

• Deductible issues
• Inuring/facultative reinsurance
• Extrapolations Defaults
• Blanket policies
• HPR
• Excess policies

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Model Output

• Data Imported/Not Imported
• Geocoded/Not Geocoded
• Version
• Perils Run
• Demand Surge
• Storm Surge
• Fire Following
• Defaults
• Construction Mappings
• Secondary Characteristics
• Secondary Uncertainty
• Deductibles

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Synthesis/Pricing
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SD Pricing Basics

• Surplus Allocation
• v z sL r
• v is contract surplus allocation
• r is contract risk load (expected profit)
• Price
• P E(L) Â sL expenses
• Risk Load or Profit
• Â y z/(1y) (C sL/2S)
• y is target return on surplus
• z is unit normal measure
• C is correlation of contract with portfolio
• S is portfolio sd (generally of loss)

With large enough portfolio this term goes to zero
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SD Pricing with Variable Premiums

• Â Deposit(1-Expensed) E(reinstatement)(1-E
  xpenser)-EL/ sL
• E(Reinstatement) Deposit/Limit E(1st limit
  loss) Time Factor
• 2 or 3 figures define (info-blind) price
• Aggregate expected loss
• Expected loss with first limit (can be
  approximated)
• Standard deviation of loss

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Â-Values (No Tax, C1)
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Tax Inv. Income Adjustments

• Surplus Allocation
• Perfect Correlation v z sL r
• Imperfect Correlation v zC sL r
• After-tax ROE
• Start  yz/(1y)C
• Solve for y y  /(zC Â)
• Conclude
• ya y(1-T) Â (1-T)/zC-r(1-T) if
• T tax rate
• ya after tax return
• if after tax risk free return on allocated
  surplus

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Â-Values (adjusted for tax, inv. income)
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Cat Pricing Loss On Line Risk Load
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Select 2000 Cat PricingRisk Load Loss on Line
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Loss On Line vs. Layer CV
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Select 2000 Cat PricingRisk Load CV
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SD Pricing Issues

• Issues with C
• Limiting case is C1
• If marginal, order of entry problems for renewals
• Perhaps sbook/Sscontract
• Need to define book of business
• Anecdotally,C0.50 for reasonably diversified US
  cat book
• Adjust up for parameter risk, down for non-US cat
  business and non-cat business
• Is it correlation or downside that matters?
• Issues with Â
• Assumption of normality
• On cat book, error is compressed
• Further offsets when book includes non-cat
• Or move to varying SD risk loads
• Adjust to reflect zone and layer

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SD Pricing Issues (Cont.)

• Issues with sL
• Measure variability Loss or result?
• Variable premium terms
• Reinstatements at 100 vs. 200
• Variable contract expiration terms
• Contingent multi-year contracts with kickers
• sL Downside proxy can we get precise?

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Investment Equivalent Pricing (IERP)

• Allocated capital for ruin protection
• Terminal funds gt X with prob gt Y (VaR)
• Prefer selling reinsurance to traditional
  investment
• Expected return and volatility on reinsurance
  contract should meet benchmark alternative

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IERP Cash Flows
Cedant
Premium Risk Load
Discounted Expected Losses
Actual Losses
Reinsurer
Fund Premium Allocated Surplus
Return
Fund
Net to Reinsurer
Allocated Surplus
Fund Return - Actual Losses
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IERP - Fully Funded Version
Cedant
P R EL/(1f)
L
Reinsurer
F P A
(1rf)F
Fund
Expected return criterion (1rf)F - EL
(1y)A
Variance criterion VarL lt sy2A2
Safety criterion (1rf)F gt S
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IERP, QD Example
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Comparative Risk Loads

• SD sLyz/(1y)
• IERP (y-rf)(S-L)/(1rf)(1y)
• S is safety level of loss distribution
• L is expected loss

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SD vs IERP PricingPrice By Layer
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SD vs IERP PricingLoss Ratio By Layer
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SD vs IERP PricingRisk Load By Layer
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Conclusions

• Cat Model Distributions Vary
• More than one point estimate useful
• Point estimates may not be significantly
  different
• Uncertainty not insignificant but not
  insurmountable
• What about uncertainty before cat models?
• Data Inputs Matter
• Not mechanical process
• Creating model inputs requires many decisions
• User knowledge and expertise critical
• Pricing Methodology Matters
• But market price not always technical price
• Judgment Unavoidable
• Actuaries already well-versed in its use

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References

• Bove, Mark C. et al.., Effect of El Nino on US
  Landfalling Hurricanes, Revisited, Bulletin of
  the American Meteorological Society, June 1998.
• Efron, Bradley and Robert Tibshirani, An
  Introduction to the Bootstrap, New York Chapman
  Hall, 1993.
• Kreps, Rodney E., Risk Loads from Marginal
  Surplus Requirements, PCAS LXXVII, 1990.
• Kreps, Rodney E., Investment-equivalent Risk
  Pricing, PCAS LXXXV, 1998.
• Major, John A., Uncertainty in Catastrophe
  Models, Financing Risk and Reinsurance,
  International Risk Management Institute, Feb/Mar
  1999.
• Mango, Donald F., Application of Game Theory
  Property Catastrophe Risk Load, PCAS LXXXV,
  1998.
• Miller, David, Uncertainty in Hurricane Risk
  Modeling and Implications for Securitization,
  CAS Forum, Spring 1999.
• Moore, James F., Tail Estimation and Catastrophe
  Security Pricing Cat We Tell What Target We Hit
  If We Are Shooting in the Dark, Wharton
  Financial Institutions Center, 99-14.

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QA
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APPENDIX A STANDARD DEVIATION PRICING Derivation
Of Formulas
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Risk Load As Variance Concept
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The Basic Formulas

• P m Âs E
• P Premium
• m Expected Losses
• Â Reluctance Measure
• s Standard Deviation of Contract Loss
  Outcomes
• E Expenses
• Â y z / (1 y)
• y Target Return on Surplus
• z Unit Normal Measure

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Initial Definitions

• V z S - R (1.1)
• given, per Brubaker, where V is that part of
  surplus required to support variability of a book
  of business with expected return R and standard
  deviation S
• R R r (1.2)
• where R is expected return after addition
  of new contract with expected return r
• V z S - R (1.3)
• required surplus with new contract, as per
  (1.1)

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Required Contract Marginal Surplus

• V - V z (S - S) - r (1.4)
• Proof , from (1.1) and (1.3)
• V - V zS - R - (zS - R)
• z(S - S) - (R - R)
• z(S - S) - r

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Required Rate of Return

• r y (V- V) (1.5)
• Given, but intuitively, required yield rate
  y times needed allocated surplus, V - V, given
  required return dollars
• r y z / (1 y) (S - S) (1.6)
• Proof
• r/y (V - V) from (1.5)
• r/y z(S - S) - r from (1.4)
• r/y r z(S - S)
• r(1y)/y z(S - S)
• r yz/(1y)(S-S)

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Marginal Standard Deviation
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Reinsurer Reluctance (Â)
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Risk Load Simplification