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Application of the credibility principle in reinsurance ...b4ae4834... · its Application to...
Transcript of Application of the credibility principle in reinsurance ...b4ae4834... · its Application to...
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