The role of uncertainties in pricing motor reinsurance

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The role of uncertainties in pricing motor
reinsurance

• A case study

29th November 2006
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Structure
Experience rating
Reinsurance
Rating motor reinsurance
Motor insurance
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Experience rating model the general case
Frequency analysis
Loadings (expenses, variance, profit)
Frequency modelling
Montecarlo simulation
Poisson, neg binomial
Historical losses DB
Aggregate loss
Risk premium
Actual premium
Severity analysis
Severity modelling
Lognormal/Pareto/GEV
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Reinsurance insurance for insurers

• Why is reinsurance bought?
• Decrease probability of ruin after one or more
  large losses
• Increase financial stability
• Tap into reinsurers expertise
• Free capital to write more business
• Provides cheaper capital
• Types of reinsurance
• Treaties vs facultative
• Excess-of-loss vs proportional
• Retrocession

t
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Excess-of-loss reinsurance how it works

• Why a layer structure?
• Which layers do reinsurers like best?

t
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Experience rating the case of reinsurance
X
1M
S
N
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The fundamental result for R/I experience rating

• Pickands-Balkema-de Haan theorem as m ? 8, the
  severity distribution of the losses above m
  converges (under quite general conditions) to a
  Generalised Pareto distribution (GPD)

Distributions with finite support (Uniform,
Beta) ? x lt 0 Distributions with
exponential-like tail (Exp, Gamma) ?
x 0 Distributions with power-law-like tail
(Pareto, Burr) ? x gt 0
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Experience rating pricing a layer
X
5M
1M
2M

uk
dk
m
Sk
N
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Case study The context (I) Commercial
environment
Reinsurers
Direct insurer
25 of market
Direct insurer
75 of market
Broker
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Case study The context (II) Motor pricing
process
Ej
Exposure
Raw losses (whole market)
Programme structure
Estimated claim count
s d
l
Frequency analysis
Settlement pattern
Estimate claim inflation

• Chain ladder BF
• Regression analysis

im
rk
Rc
Aggregate loss analysis
IBNER analysis
Revaluation
Revalued losses
Raw losses (client)
Severity analysis
Xi(1rk)
x,s,m
Xi

• MLE

Rc f (x,s,m,l,sd)
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Case study The problem process uncertainties
Ej
Exposure
Raw losses (whole market)
Estimated claim count
Programme structure
lD l
s dD s d
Frequency analysis
Settlement pattern
Estimate claim inflation

• Chain ladder BF
• Regression analysis

imD im
rkDrk
Aggregate loss analysis
IBNER analysis
Revaluation
RcsC,unc
Revalued losses
Raw losses (client)
Severity analysis
Xi(1rk) ...
x Dx,sDs,m
XiDXi

• MLE

• Parameter uncertainty, data uncertainty, model
  uncertainty

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Case study Sources of uncertainty

• Claim inflation estimated by assuming no of
  claims constant over the years (? Poisson
  variations)
• Loss data are incurred, not final paid amounts
• IBNER factors estimated on very limited samples
  (related to point above)
• Claim count affected by errors on projections
  Poisson variations regression analysis errors
• GPD parameters affected by MLE errors ?
• Settlement pattern based on limited data
  regression errors
• Simulation uncertainties
• Models are themselves uncertain

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Case study The solution Credibility (I)
Loss triangle (client experience)
All clients data (market experience)
STOCHASTIC MODELLING (Poisson/GPD model)
MARKET ANALYSIS
Estimated CLIENT RATE, Rc
Estimated MARKET RATE, Rm
Credibility rate z x Rc (1- z) x Rm

• Notes
• z ? 0,1
• z is generally different for each layer (as the
  client rate and the market rate)

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Case study The solution Credibility (II)

• Uncertainty-based credibility
• The key idea the credibility of the price is
  driven by
• the uncertainty on the client rate
• the uncertainty of the market rate
• the relevance of the market price.
• This can be shown to lead to the optimal choice
  for the credibility price

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Case study Output example
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Case study Morals

• Reinsurance pricing is afflicted by two sources
  of uncertainty
• Lack of accuracy (data, parameter and model)
• Lack of relevance (old data, external data...)
• Be wary of spurious accuracy...
• ...in presentation of results...
• ... and in models!
• Uncertainty itself can be a guide in weighing
  different pieces of information on the client
  price and in coming up with the best estimate

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