Chain ladder with random effects

Greg Fry Consulting Actuaries Sydney Australia Astin Colloquium, The Hague 21-24 May 2013

Overview Fixed effects models Families of chain ladder models Maximum likelihood estimators Random effects models Bayes, linear Bayes and MAP estimators in the EDF Families of chain ladder models Bayes estimators for chain ladder Uninformative priors Bornhuetter-Ferguson estimators 2

Framework and notation Development year 1 2... j.... J Accident year K... k... 2 1 PAST FUTURE Claims trapezoid K rows J columns (J K) Entries Y kj Triangle if K = J Nature of Y kj left unspecified Define cumulative row sums X kj = j i=1 Y ki 3

Families of chain ladder models Two distinct families EDF Mack models Accident periods stochastically independent For each k, the X kj form a Markov chain For each k, j, E X k,j+1 X kj = f j X kj Distribution of Y k,j+1 X kj drawn from exponential dispersion family (EDF) with dispersion parameter φ kj X kj EDF cross-classified (ANOVA) models Y kj stochastically independent For each k, j, Distribution of Y kj drawn from EDF with dispersion parameter φ kj E Y kj = α k β j J j=1 β j = 1 Age-to-age factor (column effect) Row effect Column effect 5

Chain ladder algorithm Define estimators of Mack age-to-age factors f j = K j k=1 X k,j+1 K j k=1 X kj Define forecasts of future claims experience X kj = X k,k k+1 f K k+1 f K k+2 f j 1 Most recent diagonal 6

Maximum likelihood estimators Estimators from chain ladder algorithm generate MLE forecasts for some models subject to certain restrictions on dispersion parameters: EDF Mack model Var Y k,j+1 X kj = σ 2 j+1 X kj EDF cross-classified model φ kj = φ 8

Chain ladder algorithm as MLE EDF Mack model Tweedie Mack model Over-dispersed Poisson (ODP) Mack model Chain ladder algorithm is maximum likelihood estimator of parameters Chain ladder is not MLE EDF crossclassified model Tweedie crossclassified model ODP crossclassified model 9

Bayes estimators in the EDF EDF log-likelihood l(y θ, φ) = [yθ κ(θ)]/a(φ) + λ(y, θ) Natural conjugate prior log-likelihood mθ κ θ l θ; m, ψ = + λ ψ ψ Posterior log-likelihood Location parameter Dispersion parameter l θ y; m, φ, ψ = yφ 1 + mψ 1 φ 1 + ψ 1 φ 1 + ψ 1 1 Location parameter Dispersion parameter 1 θ κ θ NOTE: linear in prior location parameter m and observation y 11

Bayes estimators in the EDF (cont.) Posterior log-likelihood l θ y; m, φ, ψ = yφ 1 + mψ 1 φ 1 + ψ 1 1 θ κ θ φ 1 + ψ 1 1 Bayes estimator E μ θ y; m, φ, ψ = yφ 1 + mψ 1 φ 1 1 1 + ψ Mean value of observation where = zy + 1 z m z = φ 1 φ 1 + ψ 1 = 1 1 + φ/ψ Credibility result Linear combination of prior mean and observation 12

Linear Bayes and MAP estimators in the EDF Bayes estimator is linear in observation Hence same as linear Bayes estimator Maximum a posteriori (MAP) estimator is mode of posterior distribution of observation In the EDF case may be shown to be equal to the Bayes estimator 13

Families of random effects chain ladder models Previously considered models were fixed effects models All parameters were fixed (i.e. non-stochastic) quantities Any fixed effects model can be converted to a random effects model by randomising its parameters Parameters subject to a prior distribution 15

Literature on random effects chain ladder models Chain ladder EDF Mack model EDF crossclassified model Gisler & Muller (2007) Fixed effects model Row effects Column effects Gisler & Wüthrich (2008) Random effects model Verrall (2000, 2004) England & Verrall (2002) Fixed effects Random effects Fixed effects Random effects England, Verrall & Wüthrich (2012) ODP Wüthrich (2007) EDF Wüthrich & Merz (2008) 16

Families of chain ladder models Two distinct families EDF Mack models Accident periods stochastically independent For each k, the X kj form a Markov chain For each k, j, E X k,j+1 X kj = f j X kj Distribution of Y k,j+1 X kj drawn from exponential dispersion family (EDF) with dispersion parameter φ kj X kj Distribution of g j = f j 1 from EDF, conjugate to distribution of Y k,j+1 X kj EDF cross-classified (ANOVA) models Y kj stochastically independent For each k, j, Distribution of Y kj drawn from EDF with dispersion parameter φ kj E Y kj = α k β j J j=1 β j = 1 Distributions of κ 1 α k, κ 1 β j from EDF, conjugate to distribution of Y kj In each model the prior distributions of distinct parameters are stochastically independent 17

Differences from previous models EDF Mack Gisler & Muller (2007), Gisler & Wüthrich (2008) Used variance structure of fixed effects Mack model Var Y k,j+1 X kj = σ 2 j X kj (2013) More general variance structure Var Y k,j+1 X kj = function of X kj 18

Differences from previous models EDF cross-classified Wüthrich (2007), Wüthrich & Merz (2008) Y kj a k b j ~EDF with location parameter θ k θ k a random parameter with natural conjugate distribution and location parameter 1 Only row effects are randomised Tweedie sub-family Location parameter = mean E Y kj = θ k a k b j E θ k =1 (2013) Y kj ~EDF with location parameter a k β j a k, β j random parameters, κ 1 α k, κ 1 β j each with natural conjugate distribution Both row and column parameters randomised Different distribution from Wüthrich & Merz 19

Bayes estimation for random effects EDF Mack model Generic result E μ θ y; m, φ, ψ = zy + 1 z m Bayes estimator for chain ladder (recall that f j is chain ladder age-to-age factor) f j = z j f j + 1 z j m j Observation = conventional chain ladder estimator chain ladder models Prior mean Credibility: depends on dispersion factors as usual EDF cross-classified model Generic result α k = Row sum β j = Column sum R k (α) z kj Y kj β j (α) + 1 z kj C(j) (β) z kj R k Y kj α k (β) + 1 z kj C(j) These equations are implicit Can be solved by iteration 21 a k b j Abstract location parameters of α k, β j

Bayes EDF cross-classified model Generic result α k = R k (α) z kj (α) Y kj β j + 1 z kj R k a k β j = C(j) (β) z kj (β) Y kj α k + 1 z kj C(j) b j Credibilities z kj (α), zkj (β) complicated functions Much simpler results are obtained for EDF subfamilies: Tweedie Over-dispersed Poisson 22

Bayes ODP cross-classified model Generic result α k = β j = R k C(j) (α) z kj (β) z kj (α) Y kj β j + 1 z kj R k (β) Y kj α k + 1 z kj C(j) Consider ODP (Tweedie case is similar and omitted here) α k = z k (α) Yk (α) + β j = z j (β) Yj (β) + Classical credibility form 1 zk (α) 1 zj (β) a k b j a k b j 23

Bayes ODP cross-classified model (cont.) ODP result α k = z k (α) Yk (α) + β j = z j (β) Yj (β) + Interpretation of terms a k, b j now prior means of α k, β j 1 zk (α) 1 zj (β) a k b j Weighted average of terms Y kj β j (which estimate α k ) Y k (α) = 1 Ykj φ kj 1 β j φ kj (α) (α) z 1 k = 1 1 + 1 ψk β j φ kj Similarly for Y k (β), zk (β) Dispersion parameter of Y kj Dispersion parameter ofα k 24

Bayes ODP cross-classified model further simplification Y k (α) = 1 Ykj φ kj 1 β j φ kj (α) (α) z 1 k = 1 1 + 1 ψk β j φ kj Assume φ kj = φ (which justifies chain ladder) Then Y k (α) = Ykj β j Simplest possible estimate of α k z k (α) = 1 1 + φ ψk (α) β j Familiar credibility ratio of dispersions 25

Uninformative priors Consider the previous case of ODP crossclassified model with φ kj = φ (chain ladder) Could have considered others z k (α) = 1 Informative priors Y k (α) = 1 + φ ψk (α) Ykj α k = z k (α) Yk (α) + β j = z j (β) Yj (β) + β j 1 zk (α) 1 zj (β) a k b j β j Uninformative priors Let ψ k (α), ψj (β) Chain ladder estimators for ODP crossclassified model α k = Y k (α) = z k (α), zj (β) 1 Total reliance on data β j = C(j) Y kj Ykj C(j) α k β j MAP = MLE 27

Bornhuetter-Ferguson estimators Still consider ODP cross-classified model with φ kj = φ (chain ladder) But results hold for EDF cross-classified model z k (α) = 1 Y k (α) = General results 1 + φ ψk (α) Ykj α k = z k (α) Yk (α) + β j = z j (β) Yj (β) + β j 1 zk (α) 1 zj (β) a k b j β j Forecast Particular results Y kj = α k β j = z k (α) Yk (α) + 1 zk (α) Special cases: a k ψ k α (uninformative prior) z k (α) = 1 Y kj = Y k (α) β (simple chain ladder forecast) ψ k α 0 (α k = a k certainly) z k (α) = 0 Y kj = a k β (Bornhuetter-Ferguson forecast) β j Random effects forecast is intermediate 29

Conclusions General results Fixed parameters in chain ladder models replaced by random parameters Gives credibility estimator which reduces to conventional chain ladder estimator in the case of uninformative priors EDF Mack model Results previously obtained by Gisler & Wüthrich (2008) for specific variance structure Variance structure generalised here EDF cross-classified model Previous literature introduced randomisation of row effects Randomisation of column effects introduced here Also different prior distribution Giving choice of prior 30

References (1) England P D & Verrall R J (2002). Stochastic claims reserving in general insurance. British Actuarial Journal, 8(iii), 443-518. England P D, Verrall R J & Wüthrich M V (2012). Bayesian over-dispersed Poisson model and the Bornhuetter & Ferguson claims reserving method. Annals of Actuarial Science, 6(2), 258-281. Gisler A & Müller P (2007). Credibility for additive and multiplicative models. Paper presented to the 37 th ASTIN Colloquium, Orlando FL, USA. See http://www.actuaries.org/astin/colloquia/orlando/presentations/gisler2.pdf Gisler A & Wüthrich M V (2008). Credibility for the chain ladder reserving method. Astin Bulletin, 38(2), 565-597. 31

References (2) Verrall RJ (2000). An investigation into stochastic claims reserving models and the chain-ladder technique. Insurance: mathematics and economics, 26(1), 91-99. Verrall R J (2004). A Bayesian generalised linear model for the Bornhuetter-Ferguson method of claims reserving. North American Actuarial Journal, 8(3), 67-89. Wüthrich M V (2007). Using a Bayesian approach for claims reserving. Variance, 1(2), 292-301. Wüthrich M V & Merz M (2008). Stochastic claims reserving methods in insurance, John Wiley & Sons Ltd, Chichester UK. 32