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
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
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 3
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 > 1082 4
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
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
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
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
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
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
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
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
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
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
Interpretation of credibility coefficient z = [1 + E θ V[ X i | θ ] / V θ E[ X i | θ ] ] -1 Average across risk Between-group groups of within-group variance of within- variances group means Within-group Between- Credibility z variation group variation � 0 Fixed, finite z � 1 � � Fixed, finite z � 0 Fixed, finite � 0 z � 0 Fixed, finite � � z � 1 15
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
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
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 • 18 Bayesian revision (of the Poisson)
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
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
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
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 {[x θ – b( θ )] / φ } p( θ ) = const. x exp {[n θ – b( θ )] / ψ } • It is well known that the EDF includes 22 Poisson, gamma, normal
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 (X β ) [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
Estimation of credibility coefficient z = [1 + E θ V[ X i | θ ] / V θ E[ X i | 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 • 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
Recommend
More recommend
