1. 1 Chain ladder for Tweedie distributed claims data Greg Taylor Taylor Fry Consulting Actuaries University of New South Wales Actuarial Symposium 9 November 2007

2. 2 Overview Description of chain ladder Previous maximum likelihood (ML) results Tweedie family ML results for Tweedie Convenient approximation for computation Separation method

3. 3 Description of chain ladder

4. 4 Data set-up and notation Familiar claims triangle We are not specific about the nature of the entries, e.g. Counts Notifications Finalisations etc Amounts Paid claims Incurred claims etc

5. 5 Data set-up and notation (cont’d) Accident period i Development period j Incremental triangle Entries Y ij ≥ 0 Assume E[Y ij ] finite

6. 6 Data set-up and notation (cont’d) Accident period i Development period j Incremental triangle Entries Y ij ≥ 0 Assume E[Y ij ] finite Accident period i Development period j Cumulative triangle Entries S ij ≥ 0 S ij = ∑ j k=1 Y ik

7. 7 Data set-up and notation (cont’d) Accident period i Development period j Denote sum over row i by ∑ R(i) Denote sum over column j by ∑ C(j) Any triangle Denote sum over diagonal k by ∑ D(k)

8. 8 Chain ladder formulation Assumption CL1: E[S i,j+1 | S i1,S i2 …,S ij ] = S ij f j, j=1,2,…,n-1, independently of i for some set of parameters f j Assumption CL2: Rows of the data triangle are stochastically independent, i.e. Y ij and Y kl are independent for i≠k CL1 implies that E[Y ij ] = α i β j for parameters α i, β j

9. 9 Parameter estimation E[S i,j+1 | S i1,S i2 …,S ij ] = S ij f j f j estimated by F j = ∑ n-j i=1 S i,j+1 / S ij [age-to-age factor] The F j may be converted to estimates of α i, β j β j = β 1 [F 1 … F j-2 (F j-1 – 1)] [chain age-to- age factors] α i = S i,n-i+1 /∑ R(i) β j ^ ^ ^

10. 10 Previous ML results

11. 11 Over-dispersed Poisson error structure Theorem. Suppose that Y ij are independent, Y ij ~ ODP(μ ij,φ) with μ ij = α i β j. Then ML estimates of α i, β j are given by precisely the chain ladder algorithm. Proved by Hachemeister & Stanard (1975) for simple Poisson Proved by England & Verrall (2002) for ODP

12. 12 Gamma error structure Suppose that Y ij are independent, Y ij ~ Gamma with mean μ ij = α i β j and CoV independent of i, j Mack (1991) derives reasonably simple ML equations However They are not chain ladder They do not lead to closed form solutions

13. 13 Tweedie family

14. 14 Exponential dispersion family Exponential dispersion family (EDF) consists of those log-likelihoods of the form ℓ(y;θ,λ) = c(λ)[yθ – b(θ)] + a(y,λ) for some functions a(.,.), b(.) and c(.) and parameters θ and λ

15. 15 Tweedie sub-family EDF ℓ(y;θ,λ) = c(λ)[yθ – b(θ)] + a(y,λ) Can be shown that μ = E[Y] = b'(θ), Var[Y] = b''(θ)/c(λ) Tweedie sub-family given by restrictions c(λ) = λ Var[Y] = μ p /λ for some p≤0 or p≥1 This restricts b(.) to the form b(θ) = (2-p) -1 [(1-p)θ] (2-p) / (1-p) Denote this Y ~ Tw(μ,λ,p)

16. 16 Special cases of Tweedie Tweedie family ℓ(y;θ,λ) = λ[yθ – b(θ)] + a(y,λ) Var[Y] = μ p /λ for some p≤0 or p≥1 Special cases p=0: Y ~ normal p=1: Y ~ ODP p=2: Y ~ gamma p=3: Y ~ inverse Gaussian

17. 17 ML results for Tweedie

18. 18 ML equations Theorem. Suppose that Y ij are independent, Y ij ~ Tw(μ ij,λ/w ij,p) for given w ij, p with μ ij = α i β j. Then the ML equations for α i, β j are ∑ R(i) w ij μ ij 1-p [y ij – μ ij ] = 0, i=1,…,n ∑ C(j) w ij μ ij 1-p [y ij – μ ij ] = 0, j=1,…,n

19. 19 Special case: Y ~ ODP ML equations ∑ R(i) w ij μ ij 1-p [y ij – μ ij ] = 0, i=1,…,n ∑ C(j) w ij μ ij 1-p [y ij – μ ij ] = 0, j=1,…,n Special case p=1 (Y ~ ODP) with w ij =1 (same over-dispersion in all cells) ∑ R(i) [y ij – μ ij ] = 0, i=1,…,n ∑ C(j) [y ij – μ ij ] = 0, j=1,…,n Equivalently ∑ R(i) y ij = ∑ R(i) μ ij, i=1,…,n ∑ C(j) y ij = ∑ C(j) μ ij, j=1,…,n This is called marginal sum estimation Row and column sums of the observations and the fitted values are set equal It may be shown that the chain ladder is a marginal sum estimator

20. 20 Special case: Y ~ ODP: summary ML estimation Marginal sum estimation Chain ladder estimation

21. 21 General Tweedie case ML equations ∑ R(i) w ij μ ij 1-p [y ij – μ ij ] = 0, i=1,…,n ∑ C(j) w ij μ ij 1-p [y ij – μ ij ] = 0, j=1,…,n Suppose weights take the form w ij = u i v j [includes the case w ij =1] Introduce the transformations Z ij = w ij μ ij 1-p Y ij η ij = w ij μ ij 2-p = u i v j (α i β j ) 2-p = a i b j where a i = u i α i 2-p b j = v j β j 2-p

22. 22 General Tweedie case (cont’d) ML equations ∑ R(i) w ij μ ij 1-p [y ij – μ ij ] = 0, i=1,…,n ∑ C(j) w ij μ ij 1-p [y ij – μ ij ] = 0, j=1,…,n Transformations Z ij = w ij μ ij 1-p Y ij η ij = w ij μ ij 2-p = u i v j (α i β j ) 2-p = a i b j ML equations become ∑ R(i) [z ij – η ij ] = 0, i=1,…,n ∑ C(j) [z ij – η ij ] = 0, j=1,…,n

23. 23 General Tweedie case (cont’d) ML equations ∑ R(i) [z ij – η ij ] = 0, i=1,…,n ∑ C(j) [z ij – η ij ] = 0, j=1,…,n c.f. chain ladder ∑ R(i) [y ij – μ ij ] = 0, i=1,…,n ∑ C(j) [y ij – μ ij ] = 0, j=1,…,n So estimation in the general Tweedie case is carried out by applying the chain ladder to “observations” Z ij = w ij μ ij 1-p Y ij This returns estimates of parameters a i = u i α i 2-p b j = v j β j 2-p

24. 24 General Tweedie case (cont’d) Transformation of observations Z ij = w ij μ ij 1-p Y ij requires knowledge of estimand μ ij So ML equations are implicit in parameters No closed form solution Need to be solved iteratively

25. 25 General Tweedie case – iterative algorithm Implicit solution Z ij = w ij μ ij 1-p Y ij w ij μ ij 2-p = a i b j Iterative solution Denote r-th iterate of any quantity by an upper (r), r=0,1,etc. Z (r) ij = w ij μ (r) ij 1-p Y ij Apply chain ladder to data set {Z (r) ij } to obtain estimates a (r+1) i, b (r+1) j of parameters a i, b j Estimate new iterate μ (r+1) ij according to w ij μ (r+1) ij 2-p = = a (r+1) i b (r+1) j Repeat until convergence of {a (r) i,b (r) j } Initiate iteration with a (0) i,b (0) j given by standard chain ladder

26. 26 Special Tweedie cases p = 1, w ij = 1 (ODP) p=0 (normal) p=2 (gamma) Z ij = Y ij Standard chain ladder (known) Z ij = w ij μ ij Y ij Z ij = w ij [Y ij / μ ij ] Z ij = w ij μ ij 1-p Y ij

27. 27 Numerical performance Speed of convergence declines as p increases above 1 For one particular data triangle p =Number of iterations Convergence of loss reserve to accuracy 110 250.05% 2.4240.10%

28. 28 Convenient approximation for computation

29. 29 Approximation Transformed observations for general Tweedie are Z ij = w ij μ ij 1-p Y ij to yield estimates of a i = u i α i 2-p, b i = v i β i 2-p Implicitness of ML equations arises from the fact that Z involves estimand μ Approximation μ ij ← Y ij Z ij ← w ij Y ij 2-p Perform chain ladder on these quantities to estimate Note that approximation become meaningless at p=2

30. 30 Numerical performance p=Error in approximation to outstanding liability 1.7-6 1.9-7 1.99-8 2 2.01-8 2.2-9 2.4-7

31. 31 Separation method

32. 32 Comparison of chain ladder and separation method Chain ladder μ ij = E[Y ij ] = α i β j Marginal sum estimation ∑ R(i) [y ij – μ ij ] = 0 ∑ C(j) [y ij – μ ij ] = 0 Separation method μ ij = E[Y ij ] = α i+j-1 β j Marginal sum estimation ∑ D(k) [y ij – μ ij ] = 0 ∑ C(j) [y ij – μ ij ] = 0 Each chain ladder result has counterpart for separation method