Transcript Powerpoint

Proposed Solution to COTOR Challenge,
Round 2
Jonathan Evans, FCAS, MAAA
“Centered Ogive” Sample Density
1
f(x1) =
n( x 2  x1 )
2
f(xi) =
, 1<i<n
n( xi 1  xi 1 )
1
f(xn) =
n( x n  x n 1 )
Log-Log Space Graph
Implied Probability Density From Claim Sample
(Log vs Log Scale)
1.00000000%
0.10000000%
Sample Density
0.01000000%
0.00100000%
0.00010000%
0.00001000%
0.00000100%
0.00000010%
0.00000001%
100
1,000
10,000
100,000
Claim Amount
1,000,000
10,000,000
Leap of Faith Log-Log Linear
Regression Above 10,000
Log vs. Log Renormalized Regression Above 10,000
-8
Logarithm of Probability Density
-10
Claim Sample
Regression Line
-12
Low 95% Confidence
-14
Upper 95% Confidence
-16
-18
-20
-22
-24
9
10
11
12
13
Logarithm of Claim Amount
14
15
16
Pareto Tail Estimates Based on
Regression
Log vs Log Regression Estimates for Alpha Based on Regression Line Slopes
Alpha
5M xs 5M Pure Premium
Point Estimate
-1.676240
12,553
95% Confidence Interval
Lower Bound
Upper Bound
-1.895328
-1.457152
2,976
53,049
Sample
5,640
Some Key Points
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Regression assumptions are only slightly satisfied.
Confidence interval is for alpha statistic and layer
pure premium random variable (not a statistic).
Similar regressions (even non-linear) on sample
densities (or even distribution) have been published
before.
Challenge raises deep questions about “Bayesian”
philosophy and estimation with extremely high
intrinsic uncertainty.