New Developments in Aggregate Loss Distributions

1
New Developments in Aggregate Loss Distributions

• Glenn Meyers
• Insurance Services Office, Inc.
• CAS Ratemaking Seminar
• March 12, 1998

2
The Collective Risk Model

• Easiest to view it as a simulation
• For each line of insurance, J
• 1. Pick a random number of claims, N
• 2. For K 1 to N
• Pick a random loss amount Z(J,K)
• Let X Sum of all Z(J,K)s

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Calculating the Distribution of X

• Computer Simulation
• Panjer Recursive Algorithm
• Does not do multiple lines
• Fourier Inversion
• Fast Fourier transform ? Robertson
• Numerical inversion ? Heckman/Meyers

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The Idea Behind Fourier Inversion

• Fourier Transform defined by
• Why use the Fourier Transform?

5
A Side TripDirect Fourier Inversion vs FFT

• FFT - 2N equally spaced discrete Xs
• Direct - piecewise linear CDF
• FFT - Fast closed form solution (but you must get
  the entire distribution)
• Direct - Messy numerical integration

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The Idea Behind Fourier Inversion

• The probability generating function of the claim
  count, N, is given by
• The Fourier transform of the aggregate loss
  distribution is then given by

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The Sum of Losses from Multiple Lines of
Insurance

• Given
• Random claim counts, NJ
• Random claim severities, ZJ
• Lines are independent.

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Problem with the Independence Assumption

• Lines of insurance are often correlated
• Independence
• VarXY VarXVarY
• Correlated
• VarXYVarX 2 ? CovX,Y VarY

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Removing the Independence Assumptionin the
Collective Risk Model

• CAS Committee on the Theory of Risk (COTOR)
  invited proposals for ways to remove the
  independence assumption.
• Shaun Wang was awarded the contract
• Aggregation of Correlated Risk Portfolios Models
  and Algorithms
• His work covers both simulation and Fourier
  inversion

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Available on the CAS Web Site

• FFTCalc - An Excel spreadsheet illustrating
  Shauns methodology
• The report is currently available in the FFTCalc
  portion in the Downloadable Programs section.
• Also available at the COTOR committee home page.

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A Discussion of Shauns report

• Correlations generated by parameter uncertainty
• Use of Fourier methods
• Provide examples illustrating the effect of
  correlated portfolios on insurance contracts.

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Correlations Generated by Parameter Uncertainty
(or Mixtures)

• Let ? index a collection of individual line
  parameters. Then

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Graphical IllustrationPoisson Distribution
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The Fourier Transform of a Mixture

• n lines can be combined in a variety of ways
  (illustrated below)

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The Covariance Generator

• Applies to ENJ for each line J
• Each line is put in a Covariance Group
• The mean of each count distribution in a
  covariance group is multiplied by ?
• Where
• E? 1
• Var? Covariance Generator

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An Illustrative Example

• Four Weibull Severity Distributions
• Weibull-50 ? 0.50
• Weibull-45 ? 0.45
• Weibull-40 ? 0.40
• Weibull-35 ? 0.35
• ? is scaled so that ESeverity 10,000

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An Illustrative Example

• Base count distribution is Poisson (100)
• Case 1
• Each has its own covariance group (CG)
• Negative binomial
• Case 2
• Weibull-50 and Weibull-45 in one CG
• Weibull-40 and Weibull-35 in the other CG
• Case 3
• All are in the same CG

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Count Correlation Matrix 1
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Count Correlation Matrix 2
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Count Correlation Matrix 3
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A Side TripCase 3 with 100,000 per claim
deductible
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Aggregate Loss Summary Statistics
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Excess Pure Premiums
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Retrospective Rating
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Concluding Remarks

• Correlation matters most when the expected loss
  is large.
• It must be used when using the collective risk
  model to model an insurers distribution of
  surplus.
• We now have the computational tools to handle
  correlations.
• BUT
• We need to determine the appropriate correlation
  structure.