David Raich Angela Wünsche Bahnhofskolloquium, Zurich 11 February 2013 Application of the credibility principle in reinsurance pricing
David Raich
Angela Wünsche
Bahnhofskolloquium, Zurich 11 February 2013
Application of the credibility principle in reinsurance pricing
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Agenda
1. Introduction into credibility theory
2. Some maths
3. Credibility for reinsurance pricing
4. Application – method used for MTPL
5. Vision
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Introduction
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Portfolio of similiar risks
(collective)
Individual risk within
the collective
Introduction
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Initial situation:
Comprehensive information
available for the collective
(e.g. solid loss history or
more)
Limited data history
available for individual risk
GOAL:
Make use of all (relevant)
available information in order to
get best estimate for the
individual premium
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Introduction Collective vs individual information
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Individual information
Collective information
Statistical significance
Different charateristics
than individual risk
Contains significant random element
Data stems from
individual risk
Credibility premium
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Introduction History of credibility theory
• Limited Fluctuation CT
Based on central limit
theorem
Originally a “full or zero-
credibility” method
Parameters in partial model
introduced later calibrated
according to actuarial
judgement
Stability-oriented approach
• Greatest Accuracy CT
Heavily based on Bayesian
statistics
“Best premium to charge”-
approach
Results in stable and
responsive estimator
Precision-oriented approach
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Some maths
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collective
Individual risk
Mathematical Formulation
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]|[][
]|[][
}|{F
XEEXE
XEH
F
Family of distributions indexed by risk profile ϑ
Individual premium
Collective premium
][)(
premium individual for theestimator good a Find
risk individualan for ,, nsobservatioGiven
1
1
|XEH
Fxx
n
n
GOAL:
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Main results – Bayesian estimator
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dHX n xxX1E
)|( x
A posteriori pdf of risk
profile
)(
A priori density
function of risk profile
)|,...,( 1 nxxf
Conditional density
function of losses
),,...,( 1 nxxf
Joint density function
dxxfxH )|()(
Individual Premium
Individual risk
1x
2x
nx...
)()|(),( BPBAPBAP
i
ii APABPBP )()|()(
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Main results – Bühlmann model
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nnn XZZ 1ˆ
kn
nZn
ak
2
collective observations
Vara
iXVar2
22 E
2
a
more weight assigned
to collective information
more weight assigned
to individual information
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Main results – Bühlmann-Straub model
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collective observations
Vara
iXVar2
22 E
2
a
more weight assigned
to collective information
more weight assigned
to individual information
w
nw
n
w
n
w XZZ 1ˆ
ak
2
kw
wZ w
n
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Credibility for reinsurance pricing
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General comments
What problems do we face in reinsurance pricing?
• Pricing XL business for motor
• Usually data are only given back for the last 10 years
• Need to project losses to ultimate, where development can take much
longer than 10 years
• Data are only available excess a threshold
• Hence scarce data, which may be insufficient to price a client based on
experience
• We want to make use of all available data in market and weight a client
against the market
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naturally a application field of credibility
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General comments
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0
5,000,000
10,000,000
15,000,000
20,000,000
25,000,000
30,000,000
35,000,000
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Layer 5
Layer 4
Layer 3
Layer 2
Layer 1
Retention
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Challenges
Challenges:
• Choice of appropriate portfolio
• Pure niche portfolios still require individual treatment
• Pricing of a layer 3 m xs 2 m is different than pricing ill xs 25 m
• Credibility weight needs to be calculated in dependency of claims size
• Credibility applied to frequency / severity or rate?
• Parameter uncertainty?
• Which is the appropriate exposure measure?
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Requirements for credibility approach
• Produce reasonable results i.e. increase precision
• Ensure stability and responsiveness
• One model for all layers
• Easy to explain
• Ensure consistent approach within one market
• Application still allows for underwriting judgement
“Any credibility procedure requires the actuary to exercise
informed judgment, using relevant information. The use of
credibility procedures is not always a precise mathematical
process” (Actuarial Standards board)
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Portfolio of similiar risks
(collective)
Individual risk
Credibility for reinsurance XoL pricing
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Initial situation:
Comprehensive information
available for the collective
(e.g. solid loss history or
more)
Limited data history
available for individual risk
GOAL:
Make use of all (relevant)
available information in order to
get best estimate for the
individual premium
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market
Individual cedent
Credibility for reinsurance XoL pricing
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Initial situation:
Net Market rate available
(=Collective information)
Limited loss history
available for individual
portfolio
GOAL:
Make use of both information in
order to get good estimate for
the individual net rate
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Application – Method used for MTPL
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Theoretical framework – Compound Gamma-Poisson
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!)(
n
enf
n
N
)()(
1
a
ebf
baa
The unconditional distribution of N is negative binomial with parameter(a, b /(1+b)).
w
ll
Zw
ll NZNZ ,1̂
bw
wZ
l
l
l
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m
l
lm
ba1
1/][E
m
l
lm
ba1
22
1
1/Var
Estimation of parameter b:
m
l
lm
b
1
2
1
1
Estimation of the parameters- Compound Gamma model
Process needs to
be repeated for
different
thresholds
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m
l
lT Tm
TbTa1
)(1
)(/)(][E
m
l
lT TTm
TbTa1
22 )()(1
1)(/)(Var
Estimation of parameter b(T):
m
l
l TTm
TTb
1
2)()(
1
1
)()(
Estimation of the parameters- Compound Gamma model
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Estimation of the parameters- Frequency of cedents
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Expected Frequency of cedents @ different thresholds
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Estimation of the parameters- Fit b(T)
Estimation of b for different thresholds incl. exponential fit
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Application on client example
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Credibility factors for layer:
Limit xs 1m
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Status quo
• Where are we?
• Credibility weight is calculated dependent on claim size and exposure
• Calibrated on frequencies
• Applied to the rate
• Underwriting jugdement is possible, because of the range given for the
weight
• Uncertainty of rate not explicitely taken into account, but within
underwriting judgement
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Vision
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Vision
• Where to go?
• Application for severity
• Incorporation of market rate uncertainty
• Expand application towards loadings (capital intensities)
• Other approaches in actuarial literature:
• application on loss development factors (Pinot/Gogol)
• making use of lower layers for upper layers
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References
Bühlmann, H., and A. Gisler, A Course in Credibility Theory and
its Applications, New York: Springer, 2005.
Cockroft, M., “Bayesian Credibility for Excess-of-Loss Reinsurance,”
paper presented at the GIRO Conference, 2004.
Parodi, P., and S. Bonche, “Uncertainty-Based Credibility and
its Application to Excess-of-Loss Reinsurance,” Casualty Actuarial
Society E-Forum, Winter 2008.
Mashitz, I., and G. Patrik, “Credibility for Treaty Reinsurance Excess Pricing,” Casualty Actuarial Society 1990 Discussion
Paper Program, pp. 317–368.
Credibility for a Tower of Excess Layers – David R. Clark (2011)
http://www.variancejournal.org/issues/05-01/32.pdfubs/dpp/dpp90/90dpp317.pdf
“An Analysis of Excess Loss Development” Pinto & Gogol; Proceedings of the Casualty Actuarial Society (PCAS) 1987; Vol
LXXIV.2; 227-255.
http://www.casact.org/pubs/proceed/proceed87/87227.pdf
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Thank you for your attention