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					 Robust Bayesian Analysis
             of
    Loss Reserves Data
           using
 Scale Mixture Distributions



       Boris Choy (boris.choy@uts.edu.au)
      Department of Mathematical Sciences
     University of Technology Sydney, Australia
Collaborators: Udi E. Makov and Jennifer S.K. Chan

              Robust Bayesian Analysis of Loss
                       Reserve Data                  1
                   Outline
   Loss Reserve Data
   Model for Mean Function
   Scale Mixtures Error Distributions and
    Bayesian Inference
   Results
   Further Work


               Robust Bayesian Analysis of Loss
                        Reserve Data              2
               Loss Reserve
   Is the fund reserved by an insurance
    company to cover the outstanding claims
    incurred but not yet settled

   Is the liability of the insurance company
    and must be predicted accurately.


                  Robust Bayesian Analysis of Loss
                           Reserve Data              3
            Our Objectives
   To predict the amount of loss reserves for
    an insurance company more accurately.
   To allow for extreme claims and protect
    inference using robust error distributions.
   To cope with the structural changes of
    the claims.

                 Robust Bayesian Analysis of Loss
                          Reserve Data              4
                         The Data
Claims paid to the insureds of an insurance company from 1978 to 1995.
There are N=171 observations. Those in red and blue are outliers.




                         Robust Bayesian Analysis of Loss
                                  Reserve Data                        5
               The Outliers
   Large claims in red will overestimate the
    loss reserve and hence lower the profitability.
   Small claims in blue will underestimate the
    loss reserve and hence lower the solvency
    and increase the risk of bankruptcy.
   Accurate prediction of the levels of loss
    reserves is important.


                 Robust Bayesian Analysis of Loss
                          Reserve Data                6
Loss reserves – Structural Change
                       Amount of loss across policy year

           18000
           16000
           14000
                                  *         Achieve a max. in 3-4 yrs
           12000
  Amount




           10000
                                                ¤
            8000                                        Achieve a max. in 5-6 yrs
            6000
            4000                                                         * Y=15546
            2000                                                         ¤ Y=11920
               0
                   1    2     3   4     5   6       7   8   9 10 11 12 13 14 15 16 17 18
                                                        Lag year
                       1978              1979               1980         1981        1982
                       1983              1984               1985         1986        1987
                       1988              1989               1990         1991        1992
                       1993              1994
                                      Robust Bayesian Analysis of Loss
                                               Reserve Data                                 7
    Loss reserves – Structural Change
    From 1978 to 1983
       Claims rise up to a maximum in 3-4 years
       Then decline gradually until almost zero.


    From 1984 to 1994
       Claims rise up to a maximum in 5-6 years
       Then decline gently until almost zero.


                     Robust Bayesian Analysis of Loss
                              Reserve Data              8
                 Outline
   Loss Reserve Data
   Model for Mean Function
   Scale Mixtures Error Distributions and
    Bayesian Inference
   Results
   Further Work


                Robust Bayesian Analysis of Loss
                         Reserve Data              9
                     Log-normal Model
Renshaw & Verrall, 1998
   Log-ANOVA
    log(Yij) = θij + εij ,         εij ~N(0, σ2)
         θij = μ + αi + βj ,      i,j=1,2,….n
    Constraint:               i    j  0
                                  i          j



   Log-ANCOVA
    log(Yij) = θij + εij ,             εij ~N(0, σ2)
         θij = μ + α× i + βj ,       i,j=1,2,….n
    Constraint:                j  j  0
                            Robust Bayesian Analysis of Loss
                                     Reserve Data              10
                  Threshold Model
Hazan and Makov, 2001. To cope with structural
changes of the claims
 Log-ANOVA
 θij = μ1 + α1i + β1j      i≤T
 θij = μ2 + α2i + β2j      i>T
 Log-ANCOVA
 θij = μ1 + α1 × i + β1j      i≤T
 θij = μ2 + α2 × i + β2j      i>T
Different T are adopted and they are selected according
to some goodness-of-fit measures.
                           Robust Bayesian Analysis of Loss
                                    Reserve Data              11
                     Outline
   Loss Reserve Data
   Model for Mean Function
   Scale Mixtures Error Distributions and
    Bayesian Inference
   Results
   Further Work



                Robust Bayesian Analysis of Loss
                         Reserve Data              12
    Heavy-tailed Error Distributions
Shapes of the normal, Student-t, Laplace, Cauchy and logistic distributions.
The Laplace and logistic curves are adjusted to have mean 0 and variance 1.
                                            f(x)




            -4      -3    -2         -1       0       1       2        3     4
                                              x

                 normal        t5          Laplace           Cauchy        logistic
                                    Robust Bayesian Analysis of Loss
                                             Reserve Data                             13
Scale Mixtures Distributions
   Scale mixtures of normal (SMN)
    Andrews and Mallows (1974): X = Z λ
    With location and scale parameters:
                           
                                                     
     f X ( x |  ,  )   N x |  ,  ( ) 2   d
                          0

   Scale mixtures of uniform (SMU)
                           
     f X ( x |  ,  )   U x |    (u )  u du
                          0

                   Robust Bayesian Analysis of Loss
                            Reserve Data                  14
    Examples of SMN Distributions

   Student-t (v)                                            2     
                         t ( x |  ,  )         N  x | ,
                                                                 Ga  | , d
                                             0
                                                                 2 2
                                                                  
   Symmetric Stable ()
                                                 
                                                                  1 
                         S ( x |  ,  ,  )   N x |  ,2 2 PS   | ,1d
                                                0
                                                                    2 
   Logistic
                            
                                                     
        log( x |  ,  )   N x |  ,4 2 Kolmogorov density d
                           0

   EP(β)
                                              2  1       1 
      EP( x |  ,  ,  )         N  x | ,
                                                    
                                                       PS   |  ,1d
                                                                     
                               0
                                              2co                 
                          Robust Bayesian Analysis of Loss
                                   Reserve Data                                    15
    Examples of SMU Distributions
   Normal
                    N ( x |  , )   2
                                                  
                                                                                 
                                                                               3 1
                                                      U x    u ,    u Ga u , du
                                                                               2 2
                                                 0
                                                                                   
   EP
                                                                      
    EP( x |  ,  ,  )  
                    2
                              
                                   U  x    u  / 2 ,    u  / 2  Ga u 1   ,2 1/  du
                                                                                             
                              0             2c0             2c0                2          
                                                                                              
                                                                      

   GT
      
     
      0
          

          0
                         1
                          p       1
                                   2
                                           1
                                           p
                                                
                                                
                                                
                                                         1  
                                                           1
                                                           p

                                                         p  
                                                            
                                                               1
                                                                2
                                                                    1
                                                                    p
                                                                        p
              U x |   q s u  ,   q s u  Ga u | 1  ,1GG s | q,1, duds
                                                                         2


                                               Robust Bayesian Analysis of Loss
                                                        Reserve Data                            16
    Bayesian Inference
    Use Bayesian analysis for modelling the loss
     reserves
    Use Markov chain Monte Carlo (MCMC)
     algorithms to simulate the posterior realizations.
    Use Bayesian software “WinBUGS” to obtain the
     posterior functionals.
    Use scale mixture form for the well-known
     distributions to speed up the MCMC algorithms.
    Use the mixing parameters from the scale mixture
     form to identify outliers.

                    Robust Bayesian Analysis of Loss
                             Reserve Data                 17
       SM Error Distributions and
          Prior Specification
   The normal error distribution for  ij is replaced
    by a heavy-tailed scale mixture distribution.

   Vague and non-informative prior distributions are
    adopted to express ignorance of the model
    parameters.



                   Robust Bayesian Analysis of Loss
                            Reserve Data                 18
                 Outline
   Loss Reserve Data
   Scale Mixtures Error Distributions and
    Bayesian Inference
   Model for Mean Function
   Results
   Further Work


                Robust Bayesian Analysis of Loss
                         Reserve Data              19
Result - goodness-of-fit
   Mean-Square of Error (MSE)
       Model with the smallest MSE is preferred.
               1 m n
         MSE    ( yij  yij ) 2
                             ˆ
               N i 1 j 1
   Posterior expected utility (U)
       Model with the largest U is preferred

          1 m       n
        U         ln f ( y   ij   )
          N i 1   j 1



                   Robust Bayesian Analysis of Loss
                            Reserve Data              20
                                        Result
 Mean                                                                         Log-ANOVA
function            Log-ANOVA                   Log-ANCOVA                  (with Threshold)
Error
distribution   Normal     t        EP     Normal       t       EP                   t

Mu1   μ1       6.430    6.703     6.501    6.304     6.845     6.741      6.84     6.73      6.88

mu2   μ2          NA       NA       NA        NA       NA           NA    8.32     8.19      7.97

sigma sq. σ2   1.299    0.097     0.081    1.301     0.091     0.081     0.044    0.035     0.035

MSE            3.413    2.887      3.09    3.426     3.084     3.097     1.776    1.635     1.668

U              -1.441   -0.799   -1.004    -1.489   -0.860    -1.098     -0.526   -0.497    -0.539

Shape par.      NA     1.680     1.968       NA     1.600     1.972     1.315    1.241     1.194

Threshold T       NA       NA       NA        NA       NA           NA       5          6       7

MSE is in the scale of 1,000,000
Threshold ANOVA model with t errors is chosen to be the best model
                                 Robust Bayesian Analysis of Loss
                                          Reserve Data                                               21
    Prediction of the Lower Triangle
Fitted & predicted outstanding claims




                     Robust Bayesian Analysis of Loss
                              Reserve Data              22
    Comparison with Chain Ladder
   Chain-ladder (CL) method is a very common model for loss reserve
   Plot observed against predicted claims using CL method and best
    model shows that errors on large claims are lower for the best model
                            17500

                            15000

                            12500
          predicted claim




                            10000                                                                Y=15546

                            7500

                            5000                                                              Y=11290
                            2500

                               0
                                    0   2500    5000      7500   10000    12500     15000
                            -2500
                                                       observed claim
                                        Chain ladder       Best model      line of equality
                                               Robust Bayesian Analysis of Loss
                                                        Reserve Data                                       23
    Detection of Extremely Claims
Unusual claims are detected using the scale parameter                         
                     Outliers for the threshold ANOVA model with t
                                          errors

               160
               140              33
               120
                               29
      lambda




               100
               80
                                             64
               60                            65
               40        11
               20
                0
                     0               50                  100            150

                                           Observations
                                     Robust Bayesian Analysis of Loss
                                              Reserve Data                        24
                Future Work
   More complicated structure for the mean
    function.
   Develop asymmetric scale mixture
    distributions that can protect statistical
    inference as well as identify extremely
    large claims.


                Robust Bayesian Analysis of Loss
                         Reserve Data              25
Thank You




Robust Bayesian Analysis of Loss
         Reserve Data              26

				
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