# Credibility Theory

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```					        Introduction
Basics

Credibility Theory

Ana-Iulia DOMNISAN

30th April 2009

Ana-Iulia DOMNISAN     Credibility Theory
Introduction   Deﬁnition & Formula
Basics   A Simple Example

Introduction

Origins.

Deﬁnition.

Example.

Approaches.

Ana-Iulia DOMNISAN     Credibility Theory
Introduction   Deﬁnition & Formula
Basics   A Simple Example

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 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

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

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

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 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

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

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

From the Fictional to the True Premium

The Two-urn Model

P ind = µ Θ           = E [ Xn+1 |X ]
P coll = µ0 =         Θ µ(ϑ)   dU(ϑ) = E [ Xn+1 ]
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

Θ = the set of individual risk proﬁles ϑ.

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,
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

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

− 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

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

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
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.
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

Ana-Iulia DOMNISAN     Credibility Theory

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