A few basics of credibility theory

Transcription

1 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

2 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 2

3 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, 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 3

4 American credibility Prob[ X-µ >qµ] < p American credibility also called limited fluctuation credibility Example: X~Poisson Prob[ X-µ / µ ½ > qµ ½ ] < p qµ ½ > z 1-½p [normal standard score] µ> [z 1-½p /q] 2 For p=10%, q=5%, µ >

5 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/n full ] ½ where n is actual sample size and n full is sample size required for full credibility 5

6 European credibility Consider a collective of risks 1,2, Risks labelled by some unobservable θ=θ 1,θ 2, [Latent parameter] Let the frequency of occurrence of a value θ in the collective be represented by d.f. U(θ) [Structure function] Let X i = claims experience of risk i Let µ(θ i ) = E[X i θ i ] Take a single observation X i How should µ(θ i ) be estimated? Remember that θ i determines µ(θ i ) but we cannot observe it 6

7 European credibility (cont d) Form a measure of the error in any candidate estimator µ*(x i ) of µ(θ i ) and then select the estimator with the least error Choose error measure E[ [µ*(x i ) - µ(θ i )] 2 θ i ] error for given θ i 7

8 European credibility (cont d) Form a measure of the error in any candidate estimator µ*(x i ) of µ(θ i ) and then select the estimator with the least error Choose error measure E[ [µ*(x i ) - µ(θ)] 2 θ ] du(θ) error for given θ allowance for unknown θ 8

9 European credibility (cont d) Form a measure of the error in any candidate estimator µ*(x i ) of µ(θ i ) and then select the estimator with the least error Choose error measure R(µ*) = E[ [µ*(x i ) - µ(θ)] 2 θ ] du(θ) error for given θ allowance for unknown θ 9

10 European credibility (cont d) Form a measure of the error in any candidate estimator µ*(x i ) of µ(θ i ) and then select the estimator with the least error Choose error measure R(µ*) = E[ [µ*(x i ) - µ(θ)] 2 θ ] du(θ) error for given θ allowance for unknown θ [R(µ*) is the risk associated with estimator µ*] 10

11 Derivation of European credibility R(µ*) = E[ [µ*(x i ) - µ(θ)] 2 θ ] du(θ) Assume that µ*(x i ) is to be linear in X i µ*(x i )= a + zx i Choose constants a, z so as to minimise the risk R(µ*) Differentiate R(µ*) with respect to a: E[ 2[µ*(X i ) - µ(θ)] θ ] du(θ) = 0 E[ [a + zx i - µ(θ)] θ ] du(θ) = 0 [µ*(x i ) 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*(x i ) = (1- z)m + zx i 11

12 Derivation of European credibility (cont d) R(µ*) = E[ [µ*(x i ) - µ(θ)] 2 θ ] du(θ) µ*(x i ) = (1- z)m + zx i R(µ*) = E[ [(1- z)m + zx i - µ(θ)] 2 θ ] du(θ) = E[ [(1- z)(m - µ(θ)) + z(x i - 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{(X i - µ(θ)) 2 θ } du(θ) / (µ(θ) m) 2 du(θ) ] -1 z = [1 + E θ V[X i θ] / V θ E[X i θ] ] -1 Classical credibility formula (Bühlmann, 1967) European credibility also called greatest accuracy credibility 12

13 Derivation of European credibility mathematics R(µ*) = E[ [(1- z)(m - µ(θ)) + z(x i - µ(θ))] 2 θ ] du(θ) Differentiate R(µ*) with respect to z and set result to zero: E{2[(X i - µ(θ)) - (m - µ(θ))] [(1- z)(m - µ(θ)) + z(x i - µ(θ))] θ } du(θ) = 0 E{z(X i - µ(θ)) 2 (1-z) (µ(θ) m) 2 (1-2z) (X i - 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{ (X i - µ(θ)) (µ(θ) m) θ } du(θ) = (µ(θ) m) E{ (X i - µ(θ)) θ } du(θ) = 0 [since the expectation is zero] Then z E{(X i - µ(θ)) 2 θ } du(θ) (1-z) (µ(θ) m) 2 du(θ) = 0 z = [1 + E{(X i - µ(θ)) 2 θ } du(θ) / (µ(θ) m) 2 du(θ) ] -1 13

14 Summary µ*(x i )= (1-z)m + zx i m = µ(θ) du(θ) = E θ E[X i θ] z = [1 + E θ V[X i θ] / V θ E[X i θ] ] -1 14

15 Interpretation of credibility coefficient z = [1 + E θ V[X i θ] / V θ E[X i θ] ] -1 Average across risk groups of within-group Between-group group means Within-group variation variance of withinvariances Betweengroup variation Credibility z 0 Fixed, finite z 1 Fixed, finite z 0 Fixed, finite 0 z 0 Fixed, finite z 1 15

16 Bayesian and non- Bayesian approaches Recall error measure R(µ*) = E[ [µ*(x i ) - 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 16

17 Exact credibility Consider the special case in which θ is a drawing from Θ~Gamma(α,β) and X i ~Poisson(θ), i.e. u(θ) = U'(θ) = const. x θ α-1 exp ( βθ), θ>0 Prob[X i =x θ] = const. x θ x exp(-θ) µ(θ) = E[X i θ] = θ, m = E θ E[X i θ] = α/β 17

18 Exact credibility (cont d) u(θ) = U'(θ) = const. x θ α-1 exp ( βθ) Prob[X i =x θ] = const. x θ x 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[X i =x θ] x u(θ) = const. x θ x+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) 18

19 Exact credibility (cont d) p(θ x) = const. x θ x+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 19

20 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 + E θ V[X θ] / V θ E[X θ] ] -1 X θ ~ Poisson (θ), so E[X θ] = V[X θ] = θ Θ~Gamma(α,β), so E θ V[X θ] = α/β, V θ E[X θ] = α/β 2 z = 1/(1+ β) 20

21 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 21

22 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 22

23 Relation between credibility and GLMs (cont d) A GLM is a model of the form Y = [Y 1,Y 2,,Y n ] T Y i ~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 Y i 23

24 Estimation of credibility coefficient z = [1 + E θ V[X i θ] / V θ E[X i θ] ] -1 Between-group variance of within- group means Average across risk groups of within-group variances Consider case in which there are n risk groups, each observed over t time intervals X ij = 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 {X ij } (Zehnwirth) 24

25 Multi-dimensional credibility Consider same data array {Y ij,i=1,,n;j=1,,t} [now Y instead of X] Assume For given i, the Y ij iid d.f. of Y ij characterised by latent parameter θ i {θ i,i=1,,n} an iid sample from df U(.) µ(θ) = [µ(θ 1 ),,µ(θ n )] T = X β [regression structure] nx1 nxq qx1 Find credibility estimator µ*(y) of µ(θ) 25

26 Multi-dimensional credibility (cont d) Earlier 1- dimensional error measure (Bayes risk) R(µ*) = E[ [µ*(x i ) - 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 = E θ [µ(θ)] as before Z is a credibility matrix with a form dependent on betweenand within-group dispersions, as before 26