The Logjam Session

1
The Logjam Session

• Glenn Meyers
• Insurance Services Office
• CAS Annual Meeting
• November, 2004
• Full write-up and spreadsheet at
• http//www.casact.org/cotor/round2.htm

2
You want to fit a distribution to 250 Claims

• Knee jerk first reaction, plot a histogram.

3
This will not do! Take logs

• And fit some standard distributions.

4
Still looks skewed. Take double logs.

• And fit some standard distributions.

5
Still looks skewed. Take triple logs.

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

6
Candidate 1Quadruple lognormal
7
Candidate 2Triple loggamma
8
Candidate 3Triple lognormal
9
All three cdfs are within confidence interval
for the quadruple lognormal.
10
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

11
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.

12
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.

13
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

14
Numerical Results
15
Histogram of Predictive Pure Premium
16
A Guess at the True Solution

• -0.3 within two standard errors of mu.
• 0.1 within two standard errors of sigma.
• Stop at quadruple logs.
• Guess is quadruple lognormal mu-0.3 and sigma
  0.1.
• Layer cost 4081.