Aon Limited is authorised and regulated by the Financial Services Authority in respect of insurance mediation activities only The role of uncertainties in pricing motor reinsurance A case study 29 th November 2006

Structure Experience rating Motor insurance Reinsurance Rating motor reinsurance

Risk premium Montecarlo simulation Historical losses DB Poisson, neg binomial… Frequency analysis Frequency modelling Lognormal/Pareto/GEV… Severity analysis Aggregate loss Experience rating model – the general case Loadings (expenses, variance, profit…) Actual premium Severity modelling

Reinsurance: insurance for insurers t 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

Excess-of-loss reinsurance – how it works  1 M£ 2 M£ Retained 5 M£ 10 M£ Unlimited xs 10 5 xs 5 3 xs 2 1 xs 1 t Why a layer structure? Which layers do reinsurers like best?

Experience rating – the case of reinsurance X ~ N ~ 1M£ S ~

The fundamental result for R/I experience rating Pickands-Balkema-de Haan theorem: as   ∞, the severity distribution of the losses above  converges (under quite general conditions) to a Generalised Pareto distribution (GPD): Distributions with finite support (Uniform, Beta…)   < 0 Distributions with exponential-like tail (Exp, Gamma…)   = 0 Distributions with power-law-like tail (Pareto, Burr…)   > 0

Experience rating – pricing a layer X ~ N ~ 1M Sk ~Sk ~ 2M 5M £ ukuk dkdk 

Case study – The context (I) – Commercial environment 75% of market 25% of market Direct insurer Broker Reinsurers

Case study – The context (II) – Motor pricing process Revaluation Severity analysis Frequency analysis Aggregate loss analysis Raw losses (client) MLE ,  Estimated claim count Programme structure Pivot table IBNER analysis Chain ladder + BF Regression analysis Revalued losses Raw losses (whole market) s ds d Estimate claim inflation XiXi X i (1+r k ) rkrk Settlement pattern RcRc R c = f ( ,s d ) imim Exposure EjEj

Case study – The problem: process uncertainties Revaluation Severity analysis Frequency analysis Aggregate loss analysis Raw losses (client) MLE  ±  ± ,  Estimated claim count Programme structure Pivot table IBNER analysis Chain ladder + BF Regression analysis Revalued losses Raw losses (whole market) s d ±  s d Estimate claim inflation Xi±XiXi±Xi X i (1+r k ) ±  rk±rkrk±rk Settlement pattern ±  R c ±  C,unc i m ±  i m Exposure EjEj Parameter uncertainty, data uncertainty, model uncertainty

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

Case study – The solution– Credibility (I) Credibility rate =  x R c + (1 -  ) x R m STOCHASTIC MODELLING (Poisson/GPD model) Loss triangle (client experience) All clients’ data (market experience) Estimated CLIENT RATE, R c Estimated MARKET RATE, R m Notes:   [0,1]  is generally different for each layer (as the client rate and the market rate) MARKET ANALYSIS

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

Case study – Output example

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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