# Choy

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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
   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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