A few basics of credibility theory Greg Taylor Director, Taylor - - PowerPoint PPT Presentation

A few basics of credibility theory

Greg Taylor

Director, Taylor Fry Consulting Actuaries Professorial Associate, University of Melbourne Adjunct Professor, University of New South Wales

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General credibility formula

• Consider random variable X with E[X]=µ
• Suppose we have an observation of X and

some collateral information leading to an independent estimate m of µ

• A credibility estimator is an estimator of

the form (1-z)m + zX and z is called the credibility (coefficient) associated with X

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American credibility

• Origins in workers compensation rating

– Mowbray A H (1914). How extensive a payroll exposure is necessary to give a dependable pure premium? PCAS, 1, 24-30

• Asks the question: “How large must the

claims experience be in order to be assigned full credibility?”

• Answer takes the form: Sufficiently large

that Prob[|X-µ|>qµ] < p where p, q are selected constants

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American credibility

Prob[|X-µ|>qµ] < p

• American credibility also called

limited fluctuation credibility

• Example: X~Poisson

Prob[|X-µ| / µ½ > qµ½] < p qµ½ > z1-½p [normal standard score] µ > [z1-½p/q]2 For p=10%, q=5%, µ > 1082

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Problems with American credibility

Prob[|X-µ|>qµ] < p 1. What if X not Poisson? There is then a need to estimate V[X] and include it in the treatment of full credibility 2. The theory gives the sample size for full

• credibility. What treatment of smaller

sample sizes?

• Ad hoc solutions
• Partial credibility: z=[n/nfull]½

where n is actual sample size and nfull is sample size required for full credibility

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European credibility

• Consider a collective of risks 1,2,…
• Risks labelled by some unobservable θ=θ1,θ2,…

[Latent parameter]

• Let the frequency of occurrence of a value θ in

the collective be represented by d.f. U(θ) [Structure function]

• Let Xi = claims experience of risk i
• Let µ(θi) = E[Xi|θi]
• Take a single observation Xi
• How should µ(θi) be estimated?

– Remember that θi determines µ(θi) but we cannot

• bserve it

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European credibility (cont’d)

• Form a measure of the error in any

candidate estimator µ*(Xi) of µ(θi) and then select the estimator with the least error

• Choose error measure

E[ [µ*(Xi) - µ(θi)]2| θi ]

error for given θi

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European credibility (cont’d)

• Form a measure of the error in any

candidate estimator µ*(Xi) of µ(θi) and then select the estimator with the least error

• Choose error measure

∫ E[ [µ*(Xi) - µ(θ)]2| θ ] dU(θ)

error for given θ allowance for unknown θ

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European credibility (cont’d)

• Form a measure of the error in any

candidate estimator µ*(Xi) of µ(θi) and then select the estimator with the least error

• Choose error measure

R(µ*) = ∫ E[ [µ*(Xi) - µ(θ)]2| θ ] dU(θ)

error for given θ allowance for unknown θ

10

European credibility (cont’d)

• Form a measure of the error in any

candidate estimator µ*(Xi) of µ(θi) and then select the estimator with the least error

• Choose error measure

R(µ*) = ∫ E[ [µ*(Xi) - µ(θ)]2| θ ] dU(θ)

error for given θ allowance for unknown θ

[R(µ*) is the risk associated with estimator µ*]

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Derivation of European credibility

R(µ*) = ∫ E[ [µ*(Xi) - µ(θ)]2| θ ] dU(θ)

• Assume that µ*(Xi) is to be linear in Xi

µ*(Xi) = a + zXi

• Choose constants a, z so as to minimise the risk R(µ*)
• Differentiate R(µ*) with respect to a:

∫ E[ 2[µ*(Xi) - µ(θ)] | θ ] dU(θ) = 0 ∫ E[ [a + zXi - µ(θ)] | θ ] dU(θ) = 0 [µ*(Xi) unbiased] ∫ [a + zµ(θ) - µ(θ)] dU(θ) = 0 a = (1- z)m where m = ∫ µ(θ) dU(θ) = portfolio-wide mean claims experience µ*(Xi) = (1- z)m + zXi

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Derivation of European credibility (cont’d)

R(µ*) = ∫ E[ [µ*(Xi) - µ(θ)]2| θ ] dU(θ) µ*(Xi) = (1- z)m + zXi R(µ*) = ∫ E[ [(1- z)m + zXi - µ(θ)]2| θ ] dU(θ) = ∫ E[ [(1- z)(m - µ(θ)) + z(Xi - µ(θ))]2| θ ] dU(θ)

• Differentiate R(µ*) with respect to z and set result

to zero:

– Mathematics can be found on next slide

z = [1 + ∫ E{(Xi - µ(θ))2 | θ } dU(θ) / ∫ (µ(θ) –m)2 dU(θ) ]-1

z = [1 + EθV[Xi | θ] / VθE[Xi | θ] ]-1 Classical credibility formula (Bühlmann, 1967) European credibility also called greatest accuracy credibility

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Derivation of European credibility – mathematics

R(µ*) = ∫ E[ [(1- z)(m - µ(θ)) + z(Xi - µ(θ))]2| θ ] dU(θ)

• Differentiate R(µ*) with respect to z and set result

to zero:

∫ E{2[(Xi - µ(θ)) - (m - µ(θ))] [(1- z)(m - µ(θ)) + z(Xi - µ(θ))] | θ } dU(θ) = 0 ∫ E{z(Xi - µ(θ))2 – (1-z) (µ(θ) –m)2 – (1-2z) (Xi - µ(θ)) (µ(θ) –m) | θ } dU(θ) = 0 Note that µ(θ) and m are constants for given θ. So ∫ E{ (µ(θ) –m)2 | θ } dU(θ) = ∫ (µ(θ) –m)2 dU(θ) ∫ E{ (Xi - µ(θ)) (µ(θ) –m) | θ } dU(θ) = ∫ (µ(θ) –m) E{ (Xi - µ(θ)) | θ } dU(θ) = 0 [since the expectation is zero] Then z ∫ E{(Xi - µ(θ))2 | θ } dU(θ) – (1-z) ∫ (µ(θ) –m)2 dU(θ) = 0 z = [1 + ∫ E{(Xi - µ(θ))2 | θ } dU(θ) / ∫ (µ(θ) –m)2 dU(θ) ]-1

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Summary

µ*(Xi) = (1- z)m + zXi m = ∫ µ(θ) dU(θ) = EθE[Xi | θ] z = [1 + EθV[Xi | θ] / VθE[Xi | θ] ]-1

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Interpretation of credibility coefficient

z = [1 + EθV[Xi | θ] / VθE[Xi | θ] ]-1

Average across risk Between-group groups of within-group variance of within- variances group means

Within-group variation Between- group variation Credibility z

Fixed, finite z 1 Fixed, finite z 0 Fixed, finite z 0 Fixed, finite z 1

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Bayesian and non- Bayesian approaches

• Recall error measure

R(µ*) = ∫ E[ [µ*(Xi) - µ(θ)]2| θ ] dU(θ)

• U(.) was the d.f. of risks in the collective under

consideration

• Alternatively, U(.) might be a Bayesian prior

– Then R(µ*) is the risk integrated over the prior – It may apply to a single risk whose θ is a single drawing from the prior – Now R(µ*) is called the Bayes risk

• Credibility estimator is linear Bayes estimator of

µ(θ)

• Mathematics all works exactly as before, just

interpreted differently

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Exact credibility

• Consider the special case in which θ is a

drawing from Θ~Gamma(α,β) and Xi~Poisson(θ), i.e. u(θ) = U'(θ) = const. x θα-1 exp (– βθ), θ>0 Prob[Xi=x|θ] = const. x θx exp(-θ) µ(θ) = E[Xi|θ] = θ, m = EθE[Xi | θ] = α/β

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Exact credibility (cont’d)

u(θ) = U'(θ) = const. x θα-1 exp (– βθ) Prob[Xi=x|θ] = const. x θx exp(-θ)

• Recall Bayes theorem

p(θ|x) = p(x|θ) p(θ) / p(x) = p(x|θ) p(θ) x normalising constant In our case p(θ|x) = const. x Prob[Xi=x|θ] x u(θ) = const. x θx+α-1 exp(-(1+β)θ)

• Posterior p(θ|x) is gamma, just as prior p(θ) was
• The prior is then called the natural conjugate prior of the

Poisson conditional likelihood p(x| θ)

• The gamma family of priors is said to be closed under

Bayesian revision (of the Poisson)

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Exact credibility (cont’d)

p(θ|x) = const. x θx+α-1 exp(-(1+β)θ) E(µ(θ)|x) = E(θ|x) = (x+α)/(1+β) which is linear in x

• Recall that credibility estimator was the

best linear approximation to µ(θ)|x

• So the linear approximation is exact in this

case

• Credibility estimator is exact for Gamma-

Poisson

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Exact credibility (cont’d)

E(µ(θ)|x) = E(θ|x) = (x+α)/(1+β) =(1-z)m + zx with z = 1/(1+ β) [since m = α/β]

• Can check that this agrees with earlier

credibility coefficient z = [1 + EθV[X | θ] / VθE[X | θ] ]-1

• X | θ ~ Poisson (θ), so E[X | θ] = V[X | θ] = θ
• Θ~Gamma(α,β), so EθV[X | θ] = α/β, VθE[X | θ] =

α/β2 z = 1/(1+ β)

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Exact credibility (cont’d)

• Credibility estimator is exact for

Gamma-Poisson

• This result may also be checked for

certain other conjugate pairs, e.g

– Gamma-gamma – Normal-normal

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Relation between credibility and GLMs

• In fact credibility estimator is exact (with a

minor regularity condition) for all conditional likelihoods from the exponential dispersion family (EDF) with natural conjugate priors (Jewell, 1974, 1975), i.e. p(x|θ) = const. x exp {[xθ – b(θ)] / φ} p(θ) = const. x exp {[nθ – b(θ)] / ψ}

• It is well known that the EDF includes

Poisson, gamma, normal

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Relation between credibility and GLMs (cont’d)

• A GLM is a model of the form

Y = [Y1,Y2,…,Yn]T Yi~EDF E[Y] = h-1(Xβ) [h is link function]

• The error term is such as to produce exact

credibility if a natural conjugate prior is associated with each Yi

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Estimation of credibility coefficient

z = [1 + EθV[Xi | θ] / VθE[Xi | θ] ]-1

Average across risk Between-group groups of within-group variance of within- variances group means

• Consider case in which there are n risk groups, each
• bserved over t time intervals
• Xij = claims experience of i-th group in interval j
• Above description of credibility coefficient suggests

analysis of variance

• In fact estimate z by [1+1/F]-1 where F is ANOVA F-statistic

for array {Xij} (Zehnwirth)

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Multi-dimensional credibility

• Consider same data array

{Yij,i=1,…,n;j=1,…,t}

[now Y instead of X]

• Assume

– For given i, the Yij iid – d.f. of Yij characterised by latent parameter θi – {θi,i=1,…,n} an iid sample from df U(.)

– µ(θ) = [µ(θ1),…,µ(θn)]T = X β

[regression structure]

nx1 nxq qx1

Find credibility estimator µ*(Y) of µ(θ)

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Multi-dimensional credibility (cont’d)

• Earlier 1- dimensional error measure (Bayes risk)

R(µ*) = ∫ E[ [µ*(Xi) - µ(θ)]2| θ ] dU(θ)

• Multi-dimensional version

R(µ*) = ∫ E[ [µ*(Y) - µ(θ)]2| θ ] dU(θ) = ∫ E[ [µ*(Y) - Xβ]T [µ*(Y) - Xβ] | θ ] dU(θ)

• Result

µ*(Y) = (1-Z)m + Z Y

nxn

with m = Eθ[µ(θ)] as before Z is a credibility matrix with a form dependent on between- and within-group dispersions, as before