Bootstrapping of loss reserves

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Bootstrapping of loss reserves
Greg Taylor 5 August 2005
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Overview

• We shall be concerned with loss reserving where
• There are multiple lines of business (LoBs)
• There is dependency (correlation) between them
• We require estimation of uncertainty
• Separately by LoB
• For aggregated LoBs
• Uncertainty is to be estimated by bootstrapping
• Shall discuss these points
• And a number of side issues concerned with
  estimation of uncertainty, risk margins, etc

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Statement of problem

• Consider an insurance portfolio consisting of
  LoBs labelled i1,2,,I
• Let Zi denote some technical liability associated
  with LoB i, e.g.
• Outstanding claims liabilities
• Premiums liabilities
• Insurance liabilities
• The Zi are not necessarily stochastically
  independent
• Let ZSi Zi Total Liability across all LoBs
• Estimate the distribution of Z

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Context of problem

• Insurance Act 1973
• Prudential Standard GPS 210
• Requires that provision associated with Z be
• P(Z) 75th percentile of distribution of Z
• Most practical distributions are of mean-variance
  type (mean µ, variance s2) with
• P(Z) EZ R(µ,s2)
• risk margin
• factor
• with R(µ,s2) as s2

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Properties of Total Liability

• Total Liability ZSi Zi
• Let
• µi EZi, µ EZ
• s2i VarZi, s2 VarZ
• Then
• µ Si µi
• s2 Si s2i 2 Sij ?ij si sj
• where ?ij CorrZi, Zj
• P(Z) EZ R(µ,s2) with R(µ,s2) as s2
• So R(µ,s2) as ?ij
• How to obtain the correlations?
• If they are positive and omitted, then risk
  margin is under-estimated

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Acknowledgement

• This is joint research with Dr Gráinne McGuire of
  Taylor Fry Consulting Actuaries
• Also supported by IAG

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Data set-up (past)

• Claim triangle Yi for each LoB i
• Yi Yiad where
• aaccident period
• ddevelopment period
• Yi µi ei gi(Xi,ßi) ei Regression model
• where
• Yi is written as a vector
• gi is some function
• Xi is a design matrix
• ßi is a parameter vector
• ei is a vector of iid centred stochastic errors

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Liability forecasts (future)

• Future claims experience (the missing triangle)
  denoted Zi Zijk where
• Zi ai (Ui,ßi) ?i
• ai is some function
• Ui is a design matrix
• ?i is a vector of iid centred stochastic errors
• c.f. Yi gi (Xi,ßi) ei

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Sources of correlation
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Sources of correlation

• The nature of the correlation between LoBs will
  influence the way in which they should be
  estimated
• We identify two main sources of correlation
• Correlated noise
• Parameter correlation

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Correlated noise

• Yi µi ei gi(Xi,ßi) ei
• Yj µj ej gj(Xj,ßj) ej
• Cij Corr(Yi,Yj) Corr(ei,ej)

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Correlated noise (contd)

• LoB 1 LoB 2

µ1ad e1ad expected noise
µ2ad e2ad expected noise
CORRELATED
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Parameter correlation

• Assumed model
• Yi µi ei gi(Xi,ßi) ei
• BUT model mis-specified. True model is
• Yi µi ei gi(Xi,ßi ,?i) ei
• where
• ?i is an additional vector of unrecognised
  parameters
• gi , Xi are a function and design matrix that
  accommodate the unrecognised parameters
• ei and ej are independent
• Note that
• ei ei bi where bi gi(Xi,ßi) - gi(Xi,ßi
  ,?i) bias
• Covariance Eei ejT bi bjT
• Mis-specification creates bias and correlation

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Parameter correlation (contd)

• Shared row parameters
• With unrecognised variation between rows
• LoB 1 LoB 2

Level parameter a1
Level parameter a4
Uncorrelated noise If parameter variation by row
modelled, then no correlation If not modelled,
correlation created
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Sources of correlation - conclusions

• Estimation of correlation cannot be viewed in
  isolation
• It is inseparable from the form of modelling
• In practice, correlations are more likely to be
  created by mis-specification of the model than to
  be true correlations between random quantities
• The more precise the model of claims data, the
  smaller the correlations are likely to be
• The more precise the model of claims data, the
  greater the likely diversification credit

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Allowance for dependencies between LoBs in
bootstrap forecasts
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Number of correlations

• If there are I LoBs each with NxN data triangle,
  the number of correlations to be estimated will
  be
• ½I(I-1) x ½N(N-1)
• e.g. I10, N20 ? 8,550 distinct correlations
• Too many to estimate separately
• Can create a separate model of correlations
• Seemingly Unrelated Regression framework
• Usually requires a general linear model
• Here we work with the bootstrap
• Attempt to incorporate correlations in forecasts
  without explicit estimation

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Some side comments on estimation of prediction
error

• Prediction framework

Data
Fitted
Model
Model
Forecast
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Some side comments on estimation of prediction
error

• Prediction framework

Data
Fitted
Model
Model
Forecast
Residuals
Model
Error
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Some side comments on estimation of prediction
error

• Prediction framework
• Prediction error depends on
• Form of model
• Fitting of model
• Estimation of prediction error cannot (validly)
  be dissociated from the forecast (of the central
  estimate)

Data
Fitted
Model
Model
Forecast
Residuals
Model
Error
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Some side comments on estimation of prediction
error (contd)

• This is why PS 300 (paragraph 48) requires an
  actuary to state whether a CoV is estimated
• internally, i.e. from the same data set as the
  central estimate or
• externally

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Conventional bootstrap re-visited

Data
Model
Fitted
Standardised Residuals
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Conventional bootstrap re-visited
REAL DATA

Standardised Residuals (permuted)
Data
PSEUDO DATA
Model
New Data (pseudo data)
Fitted
New Model
Standardised Residuals
New fitted
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Conventional bootstrap re-visited
REAL DATA

FUTURE
Standardised Residuals (permuted)
Data
PSEUDO DATA
Forecast mean values
Model
New Data (pseudo data)
Fitted
New Model
Standardised Residuals (re-sampled again)
Standardised Residuals
Process error
New fitted
Bootstrap realisation
Replicate many times
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Conventional bootstrap re-visited
REAL DATA

FUTURE
Standardised Residuals (permuted)
Data
PSEUDO DATA
Forecast mean values
Model
New Data (pseudo data)
Fitted
New Model
Standardised Residuals (re-sampled again)
Standardised Residuals
Process error
New fitted
Bootstrap realisation
Replicate many times
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Conventional bootstrap re-visited

• Example

Standardised Residuals (permuted)
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Conventional bootstrap of two data sets

• DATA SET 1 DATA SET 2
• Any correlation lost due to independent
  permutation of residuals

Standardised Residuals (permuted)
Standardised Residuals (permuted)
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Synchronous bootstrapping

• Conventional (independent) bootstrapping of
  correlated data sets destroys the correlations
• The solution is to synchronise the permutations
  applied to the residuals of the different data
  sets
• The form of synchronisation depends on the
  assumed form of correlation

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Synchronous bootstrap correlated noise

• DATA SET 1 DATA SET 2

Standardised Residuals (permuted)
Standardised Residuals (permuted)
Standardised Residuals (permuted)
Standardised Residuals (permuted)
Standardised Residuals (permuted)

• Assume
• Background correlation between noise terms of
  non-corresponding cells
• Different correlation for corresponding cells

• Assume
• Background correlation between noise terms of
  non-corresponding cells
• Different correlation for corresponding cells

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Synchronous bootstrap correlated (shared) row
parameters

• LoB 1 LoB 2

Level parameter a1
Level parameter a4
Uncorrelated noise Parameter variation by row not
recognised, creating correlation between
corresponding rows of different data sets
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Synchronous bootstrap correlated (shared) row
parameters

• Synchronise by restricting permutations to within
  rows in each data set (specific permutations need
  not be synchronised)
• High (low) residuals in LoB 1 will tend to be
  associated with high (low) residuals in LoB 2
• LoB 1 LoB 2

Level parameter a1
Level parameter a4

• Assume
• Background correlation between observations in
  non-corresponding rows (e.g. zero)
• Different correlation between observations in
  corresponding rows

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Other correlation structures and synchronisations

• Can synchronise according to any choice of
  subsets of a data triangle
• Another obvious choice is diagonal
  synchronisation
• Adapted to diagonal parameter correlation
• Superimposed inflation
• Other more complex correlation structures

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Implementation
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Diversified risk margin

• These forms of synchronised bootstrapping
  preserve correlation between LoBs in the pseudo
  data sets and forecasts
• Sum estimated liabilities across LoBs to obtain
  total liability for each bootstrap replication of
  pseudo-estimates
• This yields the distribution of total liability
  including allowance for correlation
• Hence diversified risk margin
• Undiversified risk margins may be obtained for
  the individual LoBs by bootstrapping them one at
  a time

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Choice of synchronisation

• Each form of synchronisation is adapted to a
  specific correlation structure
• In practice, the correct structure will be
  unknown BUT
• Correct adaptation will preserve correlations
• Incorrect adaptation will destroy them
• So bootstrap according to a variety of
  synchronisations (point-wise, row-wise,
  diagonal-wise) and select the maximum diversified
  risk margin

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Numerical results
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Point-wise bootstrap

• 3 triangles
• Each 20x20
• Triangles have identical expectations
• Rows within triangles have identical expectations
• Each rows expectation follows a Hoerl curve
  (PPCI)
• Individual cells gamma distributed about
  expectations
• Gamma distributions for corresponding cells of
  different triangles are
• Same as marginals
• Subject to correlation of about 80
• Otherwise independent (within and between
  triangles)
• Correlated noise
• Point-wise synchronised bootstrap

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Point-wise bootstrap (contd)

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Row-wise bootstrap

• Same data generation (of 3 data sets) as before
  except that
• No correlated noise
• Expected level of Hoerl curve follows geometric
  random walk through accident periods
• Same level for each data set for given accident
  period
• Model specification deliberately overlooks
  variation in row parameter
• (Row) parameter correlation
• Row-wise synchronised bootstrap

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Row-wise bootstrap examples of data sets

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Row-wise bootstrap efficiency measurement

• Descriptor of sample ratio R
• VarSi Zi / Si VarZi
• 1 for independent Zi
• 3 for fully correlated Zi
• Efficiency measure
• (Estimated R 1)
• (True R 1)

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Row-wise bootstrap (contd)

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Conclusion

• We have constructed various synchronised
  bootstraps which
• Estimate the distribution of loss reserve
  aggregated over multiple LoBs
• Incorporate dependencies between the LoBs without
  needing to make explicit measurements of them
• With efficiency that is greatest when the
  dependencies are strongest

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Questions?