A Statistical Approach To Pricing Catastrophic Loss (CAT) Securities

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A Statistical Approach To Pricing Catastrophic
Loss (CAT) Securities

• J. David Cummins
• University of Pennsylvania
• Christopher Lewis
• U.S. Office of Federal Housing Enterprise
  Oversight
• Richard D. Phillips
• Georgia State University

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Number of CAT Losses 1970-98
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Cost of Top 40 CAT Losses 1970-1998 (Cumulative)
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Top 10 CAT Losses 1970-98
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Projected Catastrophes

• 75 billion Florida hurricane
• 21 billion Northeast hurricane
• 72 billion California earthquake
• 100 billion New Madrid earthquake

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Rates on Line CAT Losses
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Failure of Diversification Types of Events

• High-Frequency, Low-Severity
• Auto collision
• Non-CAT homeowners losses
• Low-Frequency, High-Severity
• Property catastrophes
• Failure of Law of Large Numbers

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Why Time-Diversification Fails

• Holding large amounts of capital to finance
  infrequent events is not possible in practice.
• Holding capital is costly due to agency costs and
  other market imperfections
• Underutilized capital attracts raiders
• Tax and accounting rules discourage holding
  excess capital

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Why Securitization Is the Solution

• US Bonds Stocks 25 trillion75 billion lt
  0.5
• CATs uncorrelated with other events that move
  markets (zero-beta securities)
• Markets reveal information -- reduce reinsurance
  price/quantity cycles

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Recent Experience

• Recent industry experience
• Hurricane Andrew (1992)
• 18.4 billion
• 10 insolvencies
• Northridge Earthquake (1994)
• 12.5 billion
• Demographic Shifts
• Continued population movement to the coasts
• 69 increase in coastal property values since
  1988
• Potential for future events?
• Predicted increase in seismic activity, Gray
  (1990)

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CAT Loss Securities

• CBOT CAT Option Spreads
• CAT Bonds
• Federal Excess of Loss (XOL) Reinsurance

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The Argument for a Federal Role

• Catastrophe risks violate independence
  requirement of an insurable risk
• Cross sectional vs. inter-temporal
  diversification
• Constraints on private market solutions
• Limits on insurer capitalization
• Tax limitations
• Accounting limitations
• Vulnerability to raiders
• Prohibitive post-loss cost of capital
• Unstable reinsurance markets
• Inadequate capital markets solutions

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The Argument for a Federal Role II

• Private insurers have difficulty in diversifying
  large losses across time
• Once in 100 year event difficult to fund in
  advance
• Information asymmetries and other market
  imperfections raise the cost of capital following
  a large event (even if the insurer remains
  solvent)
• Government is the borrower of last resort
• Can borrow at the risk-free rate
• Inter-generational financing of large events may
  be desirable
• Contracts could be priced to break-even or make a
  profit in expected values

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The Argument Against a Federal Role

• Government contracts might slow the growth of
  private market CAT securitization
• Mis-pricing could unfairly penalize taxpayers
• The program might be difficult to kill once an
  adequate private market develops

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CAT Loss Contract Payoff Function

• Option spreads seem to be the dominant
  contractual form
• CBOT options
• CAT bonds
• XOL reinsurance
• The payoff function
• C lower strike
• T upper strike
• d coinsurance proportion

P Max0,d(L - C) - Max0,d(L - T)
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Defining the Underlying (L)

• The contracts could pay off based on
• The insurers own losses (XOL reinsurance, CAT
  bonds)
• An industry loss index (CBOT options, CAT bonds)
• National
• Statewide
• Sub-state
• A parametric index (CAT bonds)
• Richter scale reading
• Saffir-Simpson severity class

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Contract Details Federal XOL Contracts

• Underlying (L) Industry-wide property cat
  losses
• As reported by independent statistical agent
• Coverage period - 1 calendar year
• Loss development period - 18 months
• Single event policies
• Renewal provision
• Sold annually
• Authorized purchasers
• Insurance companies
• Reinsurers
• State pools

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Contract Details II Federal XOL Contracts

• Types of contracts and qualifying lines of
  business
• Hurricane contract
• Homeowners, wind policies, commercial
  multi-peril, fire,
• allied, farmowners, commercial inland
  marine
• Earthquake/volcanic activity contract
• Earthquake shake policies, commercial
  multi-peril,
• commercial inland marine
• Trigger to be set above current market capacity,
  e.g.,25 to 50 billion spreads

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Hedging with Federal XOL Catastrophe Contracts -
Case Data
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Hedging with Federal XOL Catastrophe Contracts

• Loss ratio w/o XOL contracts
• Loss ratio with N XOL contracts

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Case 1 - Cap the companys loss ratio at 25 - no
basis risk
Purchase
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Hedging Catastrophe Losses with Federal XOL
Contracts with Basis Risk
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Hedging Catastrophe Losses with Federal XOL
Contracts when Industry Losses Exceed the Cap
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Approaches To CAT Risk Modeling

• Engineering/actuarial simulation modeling AIR,
  RMS
• Statistical modeling using realized CAT losses

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Pricing Model The Loss Distribution Function
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Contracts Covering a Single EventFrequency
Distribution
let Plt Prob(LltT) Pgt 1- Plt ,
Taking the expectation over N yields and
assuming Poisson arrival rate l, yields
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Contracts Covering a Single EventSeverity
Distribution
Pareto
Lognormal
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Contracts Covering a Single EventSeverity
Distribution
Burr12
GB2
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Loss Estimates - Historical Data

• Database
• Compiled by Property Claims Service (PCS)
• Covers all insured CAT losses since 1949
• CAT single event losses gt 5M
• Catastrophes included
• Hurricanes
• Tornadoes
• Windstorms
• Hail
• Fire and Explosions
• Riots
• Brush fires
• Floods

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Adjusting Historical Data

• Need to adjust for both
• Changes in exposure levels
• Price levels
• Adjustment method 1 - PA
• Exposure - State Population Index
• Price Levels - State Construction Cost Index
• Adjustment method 2 - VA
• Exposure and price levels
• U.S. Census of Housing, Series HC80-1-A

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Property Catastrophe Loss Statistics Since 1949
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Estimating Severity DistributionsHurricanes and
Earthquakes
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Severity of Loss Distribution FunctionsPCS-VA
Hurricanes and Earthquakes
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Severity of Loss Distribution Function Tails
PCS-VA Hurricanes and Earthquakes
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Expected Loss for the 25-50B LayerPCS
Historical Data
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Summary Statistics PCS Reported Losses Vs. RMS
Simulated Losses
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Estimating Severity Distributions Historical
Losses vs. RMS Simulated Losses
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Fitting Severity Distributions PCS-VA Reported
Losses Vs. RMS Simulated Losses
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Fitting Severity Distributions Tails PCS-VA
Losses Vs. RMS Simulated Losses
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Total Expected Loss for 25-50B Layers PCS
Losses Vs. RMS Simulated Losses
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Price Estimates of Federal XOL Contracts
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Average Prices and Rates on Line for Federal XOL
Contracts
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Risk Loadings Problem and Solutions

• Problem Market incompleteness difficult to
  hedge jump risk
• Solutions
• Asset pricing model with unsystematic jump risk
  (Merton 1976)
• Option pricing with assumption about investor
  preferences (e.g., Chang 1995)

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Is CAT Risk Really Zero-Beta?

• CATs to date are zero beta but
• We have not observed a 100 billion event
• Could cause a solvency crisis in insurance
  markets
• Could be spillovers to other parts of the
  economy, e.g., Federal or private borrowing could
  raise interest rates, etc.

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Selected CAT Bond Issues
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Why Are CAT Bond Spreads So High?

• Lack of liquidity few issues/limited secondary
  market
• Investor unfamiliarity with CAT securities
• Parameter uncertainty

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Insurance Linked Securities The Future I

• Extension to other types of insurance
• Liability insurance
• Health insurance
• Life insurance and annuities
• Automobile insurance

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Insurance Linked Securities The Future II

• Increasing geographical diversification
• US states and regions
• Asian countries and regions
• European countries and regions
• Australia
• Added liquidity will undercut the reinsurance
  price cycle stabilize markets

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Insurance Linked Securities The Future III

• Reinsurers
• Perform underwriting function
• Manage basis risk
• Bear less risk directly
• Convergence of reinsurance investment banking
• Continued role for OTC contracts

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Insurance Linked Securities The Future IV

• Moving towards a public market
• Increasing standardization
• Better indices
• Reducing regulatory barriers
• Educating insurers and investors
• Corporate CAT derivatives industrial firms
  bypass insurers go direct to capital markets

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Conclusions

• CAT securities can be priced using statistical
  modeling and/or engineering/actuarial simulation
• Prices remain high due to illiquidity, investor
  unfamiliarity, and parameter uncertainty
• Significant potential for development of
  world-wide market