Credibility Theory

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					        Introduction
              Basics
 The Bayes Premium




     Credibility Theory

     Ana-Iulia DOMNISAN


           30th April 2009




Ana-Iulia DOMNISAN     Credibility Theory
                            Introduction   Definition & Formula
                                  Basics   A Simple Example
                     The Bayes Premium     Approaches


Introduction



      Origins.

      Definition.

      Example.

      Approaches.




                    Ana-Iulia DOMNISAN     Credibility Theory
                            Introduction   Definition & Formula
                                  Basics   A Simple Example
                     The Bayes Premium     Approaches


Definition




      mathematical tool

      deals with the randomness of data

      helps predict future events or costs




                   Ana-Iulia DOMNISAN      Credibility Theory
                            Introduction   Definition & Formula
                                  Basics   A Simple Example
                     The Bayes Premium     Approaches


The First Formula



      basic formula for credibility weighted estimates

      Estimate = Z × [Observation] +        1 − Z          × [Other Information]


      Z is the credibility assigned to the observation
      1−Z is referred to as the complement of credibility
      0≤Z ≤1




                   Ana-Iulia DOMNISAN      Credibility Theory
                           Introduction   Definition & Formula
                                 Basics   A Simple Example
                    The Bayes Premium     Approaches


A Simple Example



      large population of drivers observed over a 5-year period.
      average driver has an annual frequency of 0.20 accidents per
      year.
      random driver has an annual frequency of 0.60 accidents per
      year.

      Question: What is the estimate of the expected future
      frequency rate for this driver?




                   Ana-Iulia DOMNISAN     Credibility Theory
                            Introduction   Definition & Formula
                                  Basics   A Simple Example
                     The Bayes Premium     Approaches


Solution



      Dilemma! 0.20, 0.60 or something in between?

      Best Solution: this driver’s
      Expected Future Accident Frequency = Z × 0.60 +            1 − Z ) × 0.20



      Attention: anti-selection effect!




                    Ana-Iulia DOMNISAN     Credibility Theory
                           Introduction   Definition & Formula
                                 Basics   A Simple Example
                    The Bayes Premium     Approaches


Approaches


      Classical Credibility:
      − limited fluctuation credibility model.
      − homogenous risk classes.

       u
      B¨hlmann Credibility:
      − least squares credibility model.

      Bayesian Analysis:
      − formulas match those of B¨hlmann credibility estimation.
                                   u
      − linear weightening of current and prior information
                                                 u
      with weights Z and (1 - Z) where Z is the B¨hlmann credibility.



                   Ana-Iulia DOMNISAN     Credibility Theory
                            Introduction   The Model
                                  Basics   Individual Risk
                     The Bayes Premium     Premium Types


The Model

     insurance company
     I insured risks, numbered i = 1, 2, . . . , I
     in a well-defined insurance period, the risk i produces:
      − a number of claims Ni ,
                           (ν)
      − with claim sizes Yi , where ν = 1, 2, . . . , Ni ,
      − which together give the aggregate claim amount

                                             Ni    (ν)
                                 Xi =        ν=1 Yi

     gross premium
     premium volume
     task: determine the pure risk premium Pi = E [Xi ].

                  Ana-Iulia DOMNISAN       Credibility Theory
                             Introduction   The Model
                                   Basics   Individual Risk
                      The Bayes Premium     Premium Types


The Individual Risk


       black box
       Xj (j = 1, 2, . . . , n) denotes the claim amount during the
       time period j
       previous periods: X = (X1 , . . . , Xn )
       present period: Xn+1
       standard assumptions regarding the distribution function F (x)
       of the random variables Xj :
        − stationarity,
        − (conditional) independence.
       parameterisation: ϑ = ”risk profile”, element of the set Θ.


                     Ana-Iulia DOMNISAN     Credibility Theory
                         Introduction   The Model
                               Basics   Individual Risk
                  The Bayes Premium     Premium Types


From the Fictional to the True Premium


      The Two-urn Model

      The individual premium:
                   P ind = µ Θ           = E [ Xn+1 |X ]
      The collective premium:
             P coll = µ0 =         Θ µ(ϑ)   dU(ϑ) = E [ Xn+1 ]
      The Bayes premium:
                  P Bayes = µ Θ           = E [ µ Θ |X ]



                 Ana-Iulia DOMNISAN     Credibility Theory
                              Introduction   Necessary Statistical Elements
                                    Basics   Bayes Statistics
                       The Bayes Premium     Three Special Cases


Statistical Decision Theory


       observation vector X = (X1 , X2 , . . . , Xn )

       distribution function Fϑ (x) = Pϑ [X ≤ x]

       GOAL: the value of a specific functional g (ϑ) of the
       parameter ϑ

       T (X) which:
       − depends only on X ,
       − will estimate g (ϑ) ”as well as possible”.




                      Ana-Iulia DOMNISAN     Credibility Theory
                               Introduction   Necessary Statistical Elements
                                     Basics   Bayes Statistics
                        The Bayes Premium     Three Special Cases


Formulation of the Problem



      ϑ ∈ Θ: set of parameters, which contains the true value of ϑ
      T ∈ D: set of functions to which the estimator function must
      belong.
      D = {g (ϑ) : ϑ ∈ θ}
      loss function: L(ϑ, T (x))
      risk function:
            RT (ϑ) := Eϑ [L(ϑ, T )] =                Rn    L(ϑ, T (x))dF(x) .




                       Ana-Iulia DOMNISAN     Credibility Theory
                            Introduction       Necessary Statistical Elements
                                  Basics       Bayes Statistics
                     The Bayes Premium         Three Special Cases


Bayes Risk, Bayes Estimator


      U(ϑ) a priori distribution for Θ.

      Bayes risk:

                        R(T ) :=           Θ     RT (ϑ)dU(ϑ).

      Bayes Estimator T :

                            T := arg min R(T ).
                                           T∈ D




                    Ana-Iulia DOMNISAN         Credibility Theory
                           Introduction     Necessary Statistical Elements
                                 Basics     Bayes Statistics
                    The Bayes Premium       Three Special Cases


The Quadratic Loss


      correct individual premium µ(ϑ)
      Θ = the set of individual risk profiles ϑ.
      quadratic loss function

                     L(ϑ, T (x)) = (µ(ϑ) − T (x))2
      the Bayes estimator with respect to the quadratic loss
      function is given by

                           µ Θ            = E [µ(Θ)|X]




                   Ana-Iulia DOMNISAN       Credibility Theory
                         Introduction   Necessary Statistical Elements
                               Basics   Bayes Statistics
                  The Bayes Premium     Three Special Cases


The Bayes Premium in Three Special Cases




      1. The Poisson−Gamma Case

      2. The Binomial−Beta Case

      3. The Normal−Normal Case




                Ana-Iulia DOMNISAN      Credibility Theory
                          Introduction   Necessary Statistical Elements
                                Basics   Bayes Statistics
                   The Bayes Premium     Three Special Cases


1. The Poisson−Gamma Case



     Motivation: F.Bichsel’s Problem
     − 1960’s in Switzerland,
     − Bonus-Malus System was created,
     − premium level based on horsepower of the car,
      − Task: construct a better risk premium adjusted to the indi-
     viduals risk profile.
     − differences in individual numbers of claim.




                  Ana-Iulia DOMNISAN     Credibility Theory
                            Introduction   Necessary Statistical Elements
                                  Basics   Bayes Statistics
                     The Bayes Premium     Three Special Cases


Mathematical Modelling


      Nj claims during year j

      corresponding aggregate claim amount Xj .

      implicit assumption of Bichsel:

       − given: individual risk profile ϑ,
       − E [Xj |Θ = ϑ] = C E [Nj |Θ = ϑ] holds
         the aggregate claim amount,
       − C - constant depending on the horsepower of the car,
       − E [Nj |Θ = ϑ] depends only on the driver.



                    Ana-Iulia DOMNISAN     Credibility Theory
                            Introduction    Necessary Statistical Elements
                                  Basics    Bayes Statistics
                     The Bayes Premium      Three Special Cases


Model Assumptions


      PG1: Conditionally, given Θ = ϑ, the Nj ’s (j = 1, 2, . . . , n)
      are independent and Poisson distributed with Poisson
      parameter ϑ:
                                                                 ϑk
                       P(Nj = k|Θ = ϑ) = e −ϑ                    k! .


      PG2: Θ has a Gamma distribution with shape parameter γ
      and scale parameter β, the structural function has density:
                                    βγ
                     u(ϑ) =        Γ(γ)    ϑγ−1 e −βϑ , ϑ ≥ 0.




                    Ana-Iulia DOMNISAN      Credibility Theory
                          Introduction   Necessary Statistical Elements
                                Basics   Bayes Statistics
                   The Bayes Premium     Three Special Cases


The Claim Frequencies



      The individual claim frequency:
                       F ind = E [Nn+1 |Θ] = Θ.

      The collective claim frequency:
                                                         γ
                           F coll = E [Θ] =              β.




                  Ana-Iulia DOMNISAN     Credibility Theory
                            Introduction     Necessary Statistical Elements
                                  Basics     Bayes Statistics
                     The Bayes Premium       Three Special Cases


The Bayes Claim Frequency:


      Claim Frequency:

                                γ + N•                                        γ
                 F Bayes =       β + n      = αN + (1 − α)                    β
                                            n                     1       n
                      where α =            n+β ,      N =         n       j=1 Nj .


      The quadratic loss of F Bayes is:
                         2                                                    2
       E   FBayes − Θ             =        1-α E             F coll − Θ

                                                                  2
                                  =        αE       N −Θ              .



                    Ana-Iulia DOMNISAN       Credibility Theory
                           Introduction   Necessary Statistical Elements
                                 Basics   Bayes Statistics
                    The Bayes Premium     Three Special Cases


Remarks


     The quantities P ind , P coll and P Bayes are obtained by
     multiplication with C .

     The Bayes premium CF Bayes is a linear function of the
     observations (credibility premium) .

     F Bayes is an average of:

      − N = observed individual claim frequency and
                     γ
      − E [Θ] =      β   = a priori expected claim frequency.




                   Ana-Iulia DOMNISAN     Credibility Theory
                              Introduction   Necessary Statistical Elements
                                    Basics   Bayes Statistics
                       The Bayes Premium     Three Special Cases


Remarks


              n
     α =     n+β   − credibility weight.
     great number of obervation years n leads to a large α .
                E Θ
     if β =              large then α small.
               Var Θ
     Explanation of the quatradic loss:

      − the quadratic loss of F Bayes = (1 − α)· quadratic loss of F coll
      − the quadratic loss of F Bayes = α· quadratic loss of N




                      Ana-Iulia DOMNISAN     Credibility Theory
                            Introduction   Necessary Statistical Elements
                                  Basics   Bayes Statistics
                     The Bayes Premium     Three Special Cases


Estimators




      F coll is the estimator based only on the a priori knowledge
      from the collective, neglecting the individual claim experience.

      N is the estimator based only on the individual claims
      experience, neglecting the apriori knowledge.

      F Bayes combines both sources of information.




                   Ana-Iulia DOMNISAN      Credibility Theory
                           Introduction   Necessary Statistical Elements
                                 Basics   Bayes Statistics
                    The Bayes Premium     Three Special Cases


2. The Binomial−Beta Case


     Motivation:
      − group life/accident insurance,
      − interesting: number of disability cases or disability
     frequency for a certain group.

     A few assumptions to simplify matters:
      − each member of the group has the same probability of
     disablement,
      − disabilities occur independently
      − disabled person leaves group.



                   Ana-Iulia DOMNISAN     Credibility Theory
                                Introduction   Necessary Statistical Elements
                                      Basics   Bayes Statistics
                         The Bayes Premium     Three Special Cases


Random Variables


      Nj = number of new disabilities occuring in the group in the
      year j = 1, 2, . . .

      Vj = number of (not disabled) members in the group at the
      beginning of year j = 1, 2, . . .
              Nj
      Xj =    Vj   observed disablement frequency in year j = 1, 2, . . .

      of interest is:
                                                    Nn+1
                                       Xn+1 =       Vn+1




                        Ana-Iulia DOMNISAN     Credibility Theory
                            Introduction     Necessary Statistical Elements
                                  Basics     Bayes Statistics
                     The Bayes Premium       Three Special Cases


Model Assumptions


      BB1: Conditionally, given Θ = ϑ, Nj (j = 1, 2, . . . , n)
      are independent and binomial distributed with :
                                            Vj
                  P Nj = k|Θ =              k
                                                 ϑk (1 − ϑ)Vj −k .

      BB2: Θ has a Beta(a, b) distribution with a, b > 0,
      equivalently, the structural function has density:
                           1
              u(ϑ) =     B(a,b)    ϑa−1 (1 − ϑ)b−1 , 0 ≤ ϑ ≤ 1.
                                                                       Γ(a)Γ(b)
                                           where B(a, b) =              Γ(a+b) .




                    Ana-Iulia DOMNISAN       Credibility Theory
                          Introduction   Necessary Statistical Elements
                                Basics   Bayes Statistics
                   The Bayes Premium     Three Special Cases


The Claim Frequencies



      The individual claim frequency:
                       F ind = E [Xn+1 |Θ] = Θ.

      The collective claim frequency:
                                                        a
                         F coll = E [Θ] =              a+b .




                  Ana-Iulia DOMNISAN     Credibility Theory
                            Introduction    Necessary Statistical Elements
                                  Basics    Bayes Statistics
                     The Bayes Premium      Three Special Cases


Bayes Claim Frequency:


      Claim Frequency:

                               a+N•                        a
                F Bayes =     a+b+N•       = αN + (1 − α) a+b
                                             N•                    V•
                             where N =       V• ,      α=        a+b+V• .


      The quadratic loss of F Bayes is:
                         2                                                   2
       E   FBayes − Θ             =        1-α E            F coll − Θ

                                                                 2
                                  =        αE       N −Θ             .



                    Ana-Iulia DOMNISAN      Credibility Theory
                            Introduction   Necessary Statistical Elements
                                  Basics   Bayes Statistics
                     The Bayes Premium     Three Special Cases


The Normal−Normal Case - Remarks



      no practical motivation
      insurance data is mostly not normally distributed
      sometimes useful for large portfolios

      consider an individual risk
      observation vector: X = (X1 , . . . , Xn )
      Xj aggregate claim amount in the j − th year




                    Ana-Iulia DOMNISAN     Credibility Theory
                            Introduction     Necessary Statistical Elements
                                  Basics     Bayes Statistics
                     The Bayes Premium       Three Special Cases


Model Assumptions



      NN1: Conditionally, given Θ = ϑ, the Xj ’s (j = 1, 2, . . . , n)
      are independent and normally distributed with :
                                 Xj ∼ N ϑ, σ 2 .

      NN2: Θ ∼ N µ, τ 2 ,
      and the structural function has the density:
                                                   1 ϑ − µ 2
                        u(ϑ) =         √1
                                        2π
                                              e− 2 (   τ
                                                          )
                                                                   .




                    Ana-Iulia DOMNISAN       Credibility Theory
                         Introduction   Necessary Statistical Elements
                               Basics   Bayes Statistics
                  The Bayes Premium     Three Special Cases


The Premiums




     The individual Premium:
                      P ind = E [Xn+1 |Θ] = Θ.

     The collective claim frequency:
                          P coll = E [Θ] = µ.




                 Ana-Iulia DOMNISAN     Credibility Theory
                            Introduction     Necessary Statistical Elements
                                  Basics     Bayes Statistics
                     The Bayes Premium       Three Special Cases


Bayes Premium:

      Premium:

                             τ 2 µ+σ 2 X•
              P Bayes =        τ 2 +nσ 2
                                            = αX + (1 − α)E [Θ]
                                       1                                n
                             where X = n X• ,              α=             2    .
                                                                      n+ σ2
                                                                           τ



      The quadratic loss of P Bayes is:
                         2                                                         2
       E   PBayes − Θ             =         1-α E            P coll − Θ

                                                                  2
                                  =         αE       X −Θ              .


                    Ana-Iulia DOMNISAN       Credibility Theory
                          Introduction    Necessary Statistical Elements
                                Basics    Bayes Statistics
                   The Bayes Premium      Three Special Cases


Common Features

     the Bayes Premium is a linear function of the observations
     =⇒ a credibility premium
     P Bayes can be expressed as a weighted mean:

                  P Bayes = α X + (1 − α) P coll .
     the weight α is given by:
                    n
            α =    n+κ ,   where κ is an appropiate constant.
     the quadratic loss of the Bayes premium is:
                       2                                                   2
      E PBayes − Θ           = 1 - α E P coll − Θ

                                                               2
                                =        αE       X −Θ             .

                  Ana-Iulia DOMNISAN      Credibility Theory
              Introduction   Necessary Statistical Elements
                    Basics   Bayes Statistics
       The Bayes Premium     Three Special Cases




Thank You for Your Attention!




      Ana-Iulia DOMNISAN     Credibility Theory

				
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