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CARe Seminar
ILF estimation
Oliver Bettis
15th September 2009
What is an Increased Limit Factor?
The increased limit factor allows excess layer premiums to be calculated
from underlying layer premiums.
i.e. ILF = LEV(increased limit)/LEV(base limit)
ILF does not depend on the frequencyTITLE
of losses.
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Example ILF Curve
ILFs must always:
• Monotonically increase
• Increase at a decreasing rate. (The difference between successive ILFs
approaches 0.)
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Empirical ILF versus ILF from fitting a probability
distribution to the claim sizes
• On higher excess layers there may be no observed losses. Empirical ILF
gives zero premium.
• Smoothing. Empirical ILF is subject to random variation. It might give same
premium for many different layers
e.g. 10m xs 50m and 10m xs 60m might pay the same amount using an
empirical ILF.
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Example of claim size pdfs that can be used to use for
ILFs
Type
Thickness of Result
tail
Pareto
Thick
Layer with double limit and attachment of
underlying pays fixed factor of underlying premium.
e.g. 20x20 pays 1.5 times 10x10 layer
40x40 pays 1.5 times 20x20 layer
Lognormal
Medium
Faster drop off in premium with increasing
attachment point than Pareto. Two parameters so
quite flexible.
Exponential
Thin
Increase attachment by fixed amount and layer
pays fixed factor of underlying.
e.g. 10x20 pays 75% of 10x10 layer
10x30 pays 75% of 10x20 layer
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Example method for ILF estimation: T/P liability claims
First step - collect claim data & categorise claims by
business sector
Circa 150 claims in this data set, collected over 1 year.
Sample of claims below:
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Group business sectors by severity category
E.g. by quartile of mean claim size
Category 1 = business sectors least exposed to large claims
Category 4 = business sectors most exposed to large claims
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Analyse claims within each severity category
Arrange claims in order within each category to get
empirical percentiles
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Decide on pdf to use for loss distribution
Lognormal may be useful for casualty lines
Obtain best fit curve e.g. by MLE method
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Calculate percentiles from best fit lognormal curve
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Compare percentiles from claim data to percentiles from
best fit distribution
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Use different fitting methods e.g. percentile fitting
Or judgementally adjust parameters to get a better fit ILF
curve on part of curve most at interest
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Calculate empirical risk premium to layer by slicing FGU
losses into layers
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Plot premiums obtained from fitted ILFs vs empirical
ILFs to show difference between excess layer premiums
Chart of risk premium for a $10 million layer at various attachment points, where
primary $10 million layer pays $100
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Plot relative premiums for various size primary layers
Compare fitted ILFs vs empirical ILFs
Chart of risk premium for various primary layers ($2.5m, $5m, $7.5m etc), where primary $10
million layer pays $100.
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Thank you very much for your attention.