More on Stochastic Reserving in General Insurance GIRO Convention, Killarney, October 2004 Peter Eng

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More on Stochastic Reserving in General
InsuranceGIRO Convention, Killarney, October
2004Peter England and Richard Verrall
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Overview
Developments in Stochastic Reserving

• The story so far
• Bootstrapping recursive models (including Mack's
  model)
• Working with incurred data
• Bayesian recursive models
• A comparison between bootstrapping and Bayesian
  methods
• Bootstrapping and the Bornhuetter-Ferguson
  Technique
• Including curve fitting for estimating tail
  factors

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Background
Developments in Stochastic Reserving
England, P and Verrall, R (1999), Analytic and
bootstrap estimates of prediction errors in
claims reserving, Insurance Mathematics and
Economics 25, pp281-293. England, P (2002),
Addendum to Analytic and bootstrap estimates of
prediction errors in claims reserving,
Insurance Mathematics and Economics 31,
pp461-466. England, PD and Verrall, RJ (2002),
Stochastic Claims Reserving in General Insurance,
British Actuarial Journal 8, III, pp443-544.
many other papers
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Conceptual Framework
Developments in Stochastic Reserving
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Example
Developments in Stochastic Reserving
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Prediction Errors Chain Ladder Structure
Developments in Stochastic Reserving
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Over-Dispersed Poisson
Developments in Stochastic Reserving
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Example Predictor Structures
Developments in Stochastic Reserving
Chain Ladder Hoerl Curve Smoother
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Variability in Claims Reserves
Developments in Stochastic Reserving

• Variability of a forecast
• Includes estimation variance and process variance
• Problem reduces to estimating the two components

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Prediction Variance
Developments in Stochastic Reserving
Individual cell
Row/Overall total
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EMBLEM DemoODP Chain Ladder with constant scale
parameter
Developments in Stochastic Reserving
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Macks Model
Developments in Stochastic Reserving
Mack, T (1993), Distribution-free calculation of
the standard error of chain-ladder reserve
estimates. ASTIN Bulletin, 22, 93-109
Specifies first two moments only
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Macks Model
Developments in Stochastic Reserving
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Macks Model
Developments in Stochastic Reserving
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ResQ Demo Macks Model
Developments in Stochastic Reserving
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Parameter Uncertainty - Bootstrapping
Developments in Stochastic Reserving

• Bootstrapping is a simple but effective way of
  obtaining a distribution of parameters
• The method involves creating many new data sets
  from which the parameters are estimated
• The new data sets are created by sampling with
  replacement from the observed data (or residuals)
• The model is re-fitted to each new data set
• Results in a (simulated) distribution of the
  parameters

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Reserving and Bootstrapping
Developments in Stochastic Reserving
Any model that can be clearly defined can be
bootstrapped (see the England and Verrall papers
for bootstrapping the ODP)
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Bootstrapping Macks Model
Developments in Stochastic Reserving
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Bootstrapping Macks Model
Developments in Stochastic Reserving
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Recursive Models Forecasting
Developments in Stochastic Reserving

• With recursive models, forecasting proceeds
  one-step at a time
• Move one-step ahead by multiplying the previous
  cumulative claims by the appropriate bootstrapped
  development factor
• Include the process error by sampling a single
  observation from the underlying process
  distribution, conditional on the mean given by
  the previous step
• Move to the next step
• Note that the process error is included at each
  step before proceeding

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Igloo Demo 1 BootstrappingChain Ladder Model
Only
Developments in Stochastic Reserving
ODP with constant scale parameter
ODP with non-constant scale parameters
Bootstrapping Macks Model
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Negative Binomial Recursive Model
Developments in Stochastic Reserving
This is a recursive equivalent to the ODP model
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Igloo Demo 2 BootstrappingNegative Binomial
Chain Ladder Model only
Developments in Stochastic Reserving
Negative Binomial with constant scale parameter
Negative Binomial with non-constant scale
parameters

• Compare results with ODP and Mack shown earlier
• ODP and Negative Binomial are very close
• Results with non-constant scale parameters are
  close to Macks method

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Bootstrapping Recursive Models Advantages
Developments in Stochastic Reserving

• Consistent with traditional deterministic
  actuarial techniques
• Individual points can be weighted out for n-year
  volume weighted averages, exclude high/low etc
• Curve fitting can be incorporated
• Bootstrap version of Macks model can be used
  where negative incrementals are encountered
• For example Incurred claims
• Bootstrapping incurred claims
• Gives distribution of Ultimates and IBNR
• Can be combined with Paid to Date to give
  distribution of Outstanding claims
• Must be combined with (simulated) Paid to
  Incurred ratios to give distributions of payment
  cash flows

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Reserving and Bayesian Methods
Developments in Stochastic Reserving
Any model that can be clearly defined can be
fitted as a Bayesian model
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Excel Demo Gibbs SamplingODP - Chain Ladder
Model Only
Developments in Stochastic Reserving
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Igloo Demo 3 Bayesian MethodsODP, Negative
Binomial and Macks modelComparison with
bootstrapping
Developments in Stochastic Reserving
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Bayesian Stochastic Reserving Advantages
Developments in Stochastic Reserving

• Overcomes some practical difficulties with
  bootstrapping
• Sets of pseudo-data are not required, therefore
  far less RAM hungry when simulating
• Arguably more statistically rigorous and
  theoretically appealing
• Flexible approach
• Informative priors
• Bayesian Bornhuetter-Ferguson Method (see latest
  NAAJ)
• Model uncertainty
• Individual claims and additional covariates

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The Bornhuetter-Ferguson Method and Bootstrapping
Developments in Stochastic Reserving

• Pseudo-development factors give simulated
  proportion of ultimate to emerge in each
  development year
• BF prior loss ratio gives prior Ultimate
• Adjust pseudo-data to take account of BF prior
  Ultimate and simulated proportions, before
  forecasting
• Forecast based on adjusted pseudo-data
• BUT, simulate the BF prior ultimate making
  assumptions about precision of prior
• Add simulated forecasts to historic Paid-to-Date
  to give distribution of Ultimate.

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A Bayesian Bornhuetter-Ferguson Method
Developments in Stochastic Reserving

• A Bayesian framework is a natural candidate for a
  stochastic BF method
• Bayesian recursive models offer the best way
  forward
• Work in progress!
• Verrall, RJ (2004), A Bayesian Generalised Linear
  Model for the Bornhuetter-Ferguson Method of
  Claims Reserving, NAAJ, July 2004

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Bootstrapping and Curve Fitting
Developments in Stochastic Reserving
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Igloo Demo 4 BootstrappingCurve fitting and
Tail Factors
Developments in Stochastic Reserving
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Summary of Developments
Developments in Stochastic Reserving

• ODP with non-constant scale parameters
• Bootstrap version of Macks model
• Recursive version of ODP Negative Binomial model
• Recursive models allow weighting out of points
  (exclude Max/Min, n-year volume weighted averages
  etc)
• Bootstrap version of the Negative Binomial model
• Curve fitting, tail factors and bootstrapping
• Bayesian stochastic reserving
• Any clearly defined model (ODP, Mack, NB, curve
  fitting etc)
• A stochastic Bornhuetter-Ferguson method