On Predictive Modeling for Claim Severity

1
On Predictive Modeling for Claim Severity

• Glenn Meyers
• ISO Innovative Analytics
• CARe Seminar
• June 6-7, 2005

2
Problems with Experience Rating for Excess of
Loss Reinsurance

• Use submission claim severity data
• Relevant, but
• Not credible
• Not developed
• Use industry distributions
• Credible, but
• Not relevant (???)

3
General Problems withFitting Claim Severity
Distributions

• Parameter uncertainty
• Fitted parameters of chosen model are estimates
  subject to sampling error.
• Model uncertainty
• We might choose the wrong model. There is no
  particular reason that the models we choose are
  appropriate.
• Loss development
• Complete claim settlement data is not always
  available.

4
Outline of Remainder of Talk

• Quantifying Parameter Uncertainty
• Likelihood ratio test
• Incorporating Model Uncertainty
• Use Bayesian estimation with likelihood functions
• Uncertainty in excess layer loss estimates
• Bayesian estimation with prior models based on
  data reported to a statistical agent
• Reflect insurer heterogeneity
• Develops losses

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How Paper is Organized

• Start with classical hypothesis testing.
• Likelihood ratio test
• Calculate a confidence region for parameters.
• Calculate a confidence interval for a function of
  the parameters.
• For example, the expected loss in a layer
• Introduce a prior distribution of parameters.
• Calculate predictive mean for a function of
  parameters.

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The Likelihood Ratio Test
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The Likelihood Ratio Test
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An Example The Pareto Distribution

• Simulate random sample of size 1000
• a 2.000, q 10,000

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Hypothesis Testing Example

• Significance level 5
• c2 critical value 5.991
• H0 (q,a) (10000, 2)
• H1 (q,a) ? (10000, 2)
• lnLR 2(-10034.660 10035.623) 1.207
• Accept H0

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Hypothesis Testing Example

• Significance level 5
• c2 critical value 5.991
• H0 (q,a) (10000, 1.7)
• H1 (q,a) ? (10000, 1.7)
• lnLR 2(-10034.660 10045.975) 22.631
• Reject H0

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Confidence Region

• X confidence region corresponds to the 1-X
  level hypothesis test.
• The set of all parameters (q,a) that fail to
  reject corresponding H0.
• For the 95 confidence region
• (10000, 2.0) is in.
• (10000, 1.7) out.

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Confidence Region
Outer Ring 95, Inner Ring 50
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Grouped Data

• Data grouped into four intervals
• 562 under 5000
• 181 between 5000 and 10000
• 134 between 10000 and 20000
• 123 over 20000
• Same data as before, only less information is
  given.

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Confidence Region for Grouped Data
Outer Ring 95, Inner Ring 50
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Confidence Region for Ungrouped Data
Outer Ring 95, Inner Ring 50
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Estimation with Model UncertaintyCOTOR Challenge
November 2004

• COTOR published 250 claims
• Distributional form not revealed to participants
• Participants were challenged to estimate the cost
  of a 5M x 5M layer.
• Estimate confidence interval for pure premium

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You want to fit a distribution to 250 Claims

• Knee jerk first reaction, plot a histogram.

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This will not do! Take logs

• And fit some standard distributions.

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Still looks skewed. Take double logs.

• And fit some standard distributions.

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Still looks skewed. Take triple logs.

• Still some skewness.
• Lognormal and gamma fits look somewhat better.

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Candidate 1Quadruple lognormal
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Candidate 2Triple loggamma
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Candidate 3Triple lognormal
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All three cdfs are within confidence interval
for the quadruple lognormal.
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Elements of Solution

• Three candidate models
• Quadruple lognormal
• Triple loggamma
• Triple lognormal
• Parameter uncertainty within each model
• Construct a series of models consisting of
• One of the three models.
• Parameters within a broad confidence interval for
  each model.
• 7803 possible models

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Steps in Solution

• Calculate likelihood (given the data) for each
  model.
• Use Bayes Theorem to calculate posterior
  probability for each model
• Each model has equal prior probability.

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Steps in Solution

• Calculate layer pure premium for 5 x 5 layer for
  each model.
• Expected pure premium is the posterior
  probability weighted average of the model layer
  pure premiums.
• Second moment of pure premium is the posterior
  probability weighted average of the model layer
  pure premiums squared.

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CDF of Layer Pure Premium

• Probability that layer pure premium x
• equals
• Sum of posterior probabilities for which the
• model layer pure premium is x

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Numerical Results
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Histogram of Predictive Pure Premium
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Example with Insurance Data

• Continue with Bayesian Estimation
• Liability insurance claim severity data
• Prior distributions derived from models based on
  individual insurer data
• Prior models reflect the maturity of claim data
  used in the estimation

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Initial Insurer Models

• Selected 20 insurers
• Claim count in the thousands
• Fit mixed exponential distribution to the data of
  each insurer
• Initial fits had volatile tails
• Truncation issues
• Do small claims predict likelihood of large
  claims?

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Initial Insurer Models
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Low Truncation Point
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High Truncation Point
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Selections Made

• Truncation point 100,000
• Family of cdfs that has correct behavior
• Admittedly the definition of correct is
  debatable, but
• The choices are transparent!

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Selected Insurer Models
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Selected Insurer Models
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Each model consists of

1. The claim severity distribution for all claims
   settled within 1 year
2. The claim severity distribution for all claims
   settled within 2 years
3. The claim severity distribution for all claims
   settled within 3 years
4. The ultimate claim severity distribution for all
   claims
5. The ultimate limited average severity curve

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Three Sample Insurers Small, Medium and Large

• Each has three years of data
• Calculate likelihood functions
• Most recent year with 1 on prior slide
• 2nd most recent year with 2 on prior slide
• 3rd most recent year with 3 on prior slide
• Use Bayes theorem to calculate posterior
  probability of each model

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Formulas for Posterior Probabilities
Model (m) Cell Probabilities
Number of claims
Likelihood (m)
Using Bayes Theorem
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ResultsTaken from paper.
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Formulas for Ultimate Layer Pure Premium

• Use 5 on model (3rd previous) slide to calculate
  ultimate layer pure premium

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Results

• All insurers were simulated from same population.
• Posterior standard deviation decreases with
  insurer size.

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Possible Extensions

• Obtain model for individual insurers
• Obtain data for insurer of interest
• Calculate likelihood, Prdatamodel, for each
  insurers model.
• Use Bayes Theorem to calculate posterior
  probability of each model
• Calculate the statistic of choice using models
  and posterior probabilities
• e.g. Loss reserves