Chain Ladder Bias and Variance Approximations under a Compound Poisson Model

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Chain Ladder Bias and Variance Approximations
under a Compound Poisson Model

• Janagan Yogaranpan

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Overview

• Part 1
• An outline of Yogaranpan, J., Clarke, S., Ferris,
  S. and Pollard, J. Approximating the Bias and
  Variance of Chain Ladder Estimates Under a
  Compound Poisson Model. Journal of Actuarial
  Practice 11 (2004) 147-167.
• Part 2
• Further research
• Consequences for GLMs applied to run-off
  triangles
• Extending the theory - run-offs with iid entries

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Part 1 Yogaranpan et al. (2004)
Known incremental claims and outstanding claims
estimates
OS1
OS2
OSn-2
OSn-1
Accident years and development years are numbered
from 0 to n-1
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The Chain Ladder Method
Chain ladder development ratio
Relevant product of development ratios
O/S claims estimate for accident year i
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Stochastic Assumptions
Claim sizes are required to be iid (here
exponential variables with a mean of 500 are
assumed)
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Simulation Study

• Results for biases of simulated outstanding
  claims

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Why are there biases?

• An analogy an out of the money call option is
  not worthless, there is a time value which
  depends on stock price volatility.
• Similarly with outstanding claims estimation -
  volatility has an impact on the mean.
• Fundamentally the expectation of a ratio does not
  equal the ratio of expectations.

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Development Ratio Biases

• Consider a development ratio m of the form m TA
  / TB

The VBA term is the approximate proportional bias
in m
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Intermediate formulae

• Define Vn-iC as the sum of all V terms
  corresponding to the development ratios that
  constitute Mi
• Define t1 and t2 as the first and second order
  moments about the origin of the claims size
  distribution
• Define g as the ratio of the variance to the mean
  for each of the run-off entries. g is the same
  for all run-off entries
• Define ai as the expected ultimate total claims
  cost for accident year i, so that ai li t1
• Define

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Practical formulae final results
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Simulation Study

• Biases of outstanding claims

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Simulation Study

• Simulated vs. approximate covariance matrices

Overall Variance 4.362108
Overall Variance 4.270108
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Simulation Study

• The approximate bias in the overall outstanding
  claims estimate is quite accurate (854 vs. 870)
• The approximate variance of the overall
  outstanding claims estimate is only 2.1 less
  than the value from the simulations (4.27108 vs.
  4.36108)
• The simulations used highly skewed claims sizes
  (exponential with a mean of 500). With less
  skewed distributions the approximations match the
  simulations even more accurately.

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Features of the Approximations

• The approximate biases are linear in g as has
  been observed in other papers
• An unintuitive result if the portfolio grows
  such that all ais increase by a common factor
  (eg, doubling in size), then (ceteris paribus)
  the approximate biases do not change.
• Approximate covariances between accident year
  claims estimates grow in proportion to portfolio
  size.
• The first two terms of the Var(OSi) approximation
  behave similarly to the covariances, although the
  last term is effectively constant in respect of
  changes in portfolio size (similar to biases).

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Shortcomings of the theory

• Restrictive assumptions
• Skewness statistics cannot be found
• Parameter values are required in advance -
  estimation creates further biases and variances
• Inflation has been ignored
• Biases may be immaterial
• Chain ladder methods are not widely used for
  incurred costs
• Computer simulations are fast and are probably
  more practical than learning the theory

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Part 2 Further Research

• Implications for bias and variance estimates
  based on GLMs
• Extending the theory iid run-off entries

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GLMs and run-off triangles

• Poisson Count Model
• Well known result of the equivalence of maximum
  likelihood estimates (under the Poisson count
  model) and Chain Ladder estimates
• It follows that the biases and variances are also
  equivalent
• The Poisson data assumptions are the same as
  those used in deriving the approximations, albeit
  with claim sizes restricted to a value of 1
• The features of the Poisson count model
  outstanding claims estimates are therefore best
  described by the approximations

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GLMs and run-off triangles

• GLMs are also based on maximum likelihood
• If the simple Poisson count model cannot possibly
  describe its own complex biases, variances and
  covariances, what about more intricate GLMs and
  other MLE methods?
• Serious errors could exist when non trivial GLMs
  are applied to run-off triangles
• We can expect the complexities of the biases and
  variances of the estimates to grow with the
  intricacy of the run-off models

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Extending the theory

• The key to the entire theory is the fact the
  ratio of the variance to the mean (ie, g) is the
  same for every cell of the run-off.
• This is true for the compound Poisson model where
  the claim size distribution is restricted to be
  iid. The same can also be found for compound
  binomial and compound negative binomial models
  under very severe restrictions.
• While the constant g assumption is generally not
  true for all other models, it is satisfied if
  run-off entries are iid.

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iid run-off entries

• If run-off entries are iid with mean m and
  variance s2, then the approximations do not
  simplify appreciably
• Trial and error has lead to the following
  approximations based on the approximations for
  the overall chain ladder outstanding claims
  estimate

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iid run-off entries

• The iid approximations were tested by simulating
  a variety of distributions, and in each case the
  bias and variance of the overall outstanding
  claims estimate were compared to the
  approximations.
• In general there is excellent agreement with the
  iid approximations, except when very small
  denominators are quite likely in the development
  ratios (for example, when entries are
  exponentially distributed)

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Summary

• Reliable bias and variance approximations have
  been found for chain ladder estimates under a
  compound Poisson model
• The results can be linked to maximum likelihood
  methods, raising questions about the reliability
  of GLMs and other MLE methods applied to run-off
  triangles
• The results can be extended, under severe iid
  restrictions, to non Poisson models