a few basics of credibility theory - actuaries institute · a few basics of credibility theory greg...

A few basics of credibility theoryGreg Taylor

Director, Taylor Fry Consulting ActuariesProfessorial Associate, University of MelbourneAdjunct 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 + zXand 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] < pwhere p, q are selected constants

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American credibilityProb[|X-µ|>qµ] < p

• American credibility also called limited fluctuation credibility

• Example: X~PoissonProb[|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] < p1. 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 observe 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 measureE[ [µ*(Xi) - µ(θi)]2| θi ]

error for given θi

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• Choose error measure∫ E[ [µ*(Xi) - µ(θ)]2| θ ] dU(θ)

error for given θ allowance for unknown θ

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• Choose error measureR(µ*) = ∫ E[ [µ*(Xi) - µ(θ)]2| θ ] dU(θ)

error for given θ allowance for unknown θ

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• Choose error measureR(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) = 0a = (1- z)m

where m = ∫ µ(θ) dU(θ) = portfolio-wide mean claims experienceµ*(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)

= 0Note 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) = 0z = [1 + ∫ E{(Xi - µ(θ))2 | θ } dU(θ) / ∫ (µ(θ) –m)2 dU(θ) ]-1

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Summaryµ*(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-groupgroups of within-group variance of within-variances group means

Within-group variation

Between-group variation

Credibility z

0 Fixed, finite z 1

Fixed, finite z 0

Fixed, finite 0 z 0

Fixed, finite z 1

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Bayesian and non- Bayesian approaches

• Recall error measureR(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 casep(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)• 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 formY = [Y1,Y2,…,Yn]T

Yi~EDFE[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-groupgroups of within-group variance of within-variances group means

• Consider case in which there are n risk groups, each observed 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 versionR(µ*) = ∫ E[ [µ*(Y) - µ(θ)]2| θ ] dU(θ)

= ∫ E[ [µ*(Y) - Xβ]T [µ*(Y) - Xβ] | θ ] dU(θ)• Resultµ*(Y) = (1-Z)m + Z Y

nxn

with m = Eθ[µ(θ)] as beforeZ is a credibility matrix with a form dependent on between-

and within-group dispersions, as before

a few basics of credibility theory - actuaries institute · a few basics of credibility theory greg...

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