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
You want to fit a
distribution to 250 Claims
• Knee jerk first reaction, plot a histogram.
Histogram of Cotor Data
250
200
Count
150
100
50
0
0
1
2
3
4
Claim Amount
5
6
7
6
x 10
This will not do! Take logs
• And fit some standard distributions.
0.35
0.3
lcotor data
0.25
lognormal
Density
gamma
0.2
Weibull
0.15
0.1
0.05
0
6
7
8
9
10
11
12
Log of Claim Amounts
13
14
15
16
Still looks skewed. Take double logs.
• And fit some standard distributions.
llcotor data
2.5
Lognormal
Gamma
Weibull
Density
2
1.5
1
0.5
0
1.8
2
2.2
2.4
log log of Claim Amounts
2.6
2.8
Still looks skewed. Take triple logs.
• Still some skewness.
• Lognormal and gamma fits look somewhat better.
lllcotor data
Lognormal
5
Gamma
Normal
Density
4
3
2
1
0
0.55
0.6
0.65
0.7
0.75
0.8
0.85
Triple log of Claim Amounts
0.9
0.95
1
Candidate #1
Quadruple lognormal
Distribution:
Log likelihood:
Domain:
Mean:
Variance:
Lognormal
283.496
0 < y < Inf
0.738351
0.006189
Parameter
Mu
sigma
Estimate
Std. Err.
-0.30898
0.00672
0.106252
0.004766
Estimated covariance of parameter estimates:
mu
sigma
Mu
4.52E-05
1.31E-19
Sigma
1.31E-19
2.27E-05
Candidate #2
Triple loggamma
Distribution:
Log likelihood:
Domain:
Mean:
Variance:
Parameter
A
B
Gamma
282.621
0 < y < Inf
0.738355
0.00615
Estimate
88.6454
0.008329
Std. Err.
7.91382
0.000746
Estimated covariance of parameter estimates:
a
b
A
62.6286
-0.00588
B
-0.00588
5.56E-07
Candidate #3
Triple lognormal
Distribution:
Log likelihood:
Domain:
Mean:
Variance:
Parameter
mu
sigma
Normal
279.461
-Inf < y < Inf
0.738355
0.006285
Estimate Std. Err.
0.738355
0.005014
0.079279
0.003556
Estimated covariance of parameter estimates:
mu
sigma
mu
2.51E-05
-1.14E-19
sigma
-1.14E-19
1.26E-05
All three cdf’s are within confidence
interval for the quadruple lognormal.
1
0.9
0.8
Cumulative probability
0.7
0.6
0.5
0.4
lllcotor data
0.3
Lognormal
confidence bounds (Lognormal)
0.2
Gamma
Normal
0.1
0
0.55
0.6
0.65
0.7
0.75
0.8
0.85
Triple log of Claim Amounts
0.9
0.95
1
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
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.
Posterior model|data   Likelihood  data|model   Prior model
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.
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
Numerical Results
Mean
Standard Deviation
Median
Range
Low at 2.5%
High at 97.5%
6,430
3,370
5,780
1,760
14,710
Histogram of
Predictive Pure Premium
Predictive Distribution of the Layer Pure Premium
0.16
0.14
0.12
Density
0.10
0.08
0.06
0.04
0.02
0.00
0
1
2
3
4
5
6
7
8
9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Low End of Amount (000)
A Guess at the “True” Solution
Distribution:
Log likelihood:
Domain:
Mean:
Variance:
Lognormal
283.496
0 < y < Inf
0.738351
0.006189
Parameter
Mu
sigma
Estimate
Std. Err.
-0.30898
0.00672
0.106252
0.004766
Estimated covariance of parameter estimates:
mu
sigma
Mu
4.52E-05
1.31E-19
Sigma
1.31E-19
2.27E-05
• -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.