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

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Chain ladder for Tweedie distributed claims data

Greg TaylorTaylor Fry Consulting Actuaries

University of New South Wales Actuarial Symposium

9 November 2007

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Overview

• Description of chain ladder

• Previous maximum likelihood (ML) results

• Tweedie family

• ML results for Tweedie

• Convenient approximation for computation

• Separation method

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Description of chain ladder

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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

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Data set-up and notation (cont’d)

Accident period i

Development period j

Incremental triangle

Entries Yij ≥ 0

Assume E[Yij] finite

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Accident period i


Incremental triangle

Entries Yij ≥ 0

Assume E[Yij] finite

Accident period i


Cumulative triangle

Entries Sij ≥ 0

Sij = ∑jk=1 Yik

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Accident period i


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)

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Chain ladder formulation

• Assumption CL1:

E[Si,j+1 | Si1,Si2…,Sij] = Sij fj, j=1,2,…,n-1, independently of i for some set of parameters fj

• Assumption CL2: Rows of the data triangle are stochastically

independent, i.e. Yij and Ykl are independent for i≠k

• CL1 implies that E[Yij] = αiβj for parameters αi, βj

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Parameter estimation

• E[Si,j+1 | Si1,Si2…,Sij] = Sij fj

• fj estimated by

Fj = ∑n-ji=1 Si,j+1 / Sij [age-to-age factor]

• The Fj may be converted to estimates of αi, βj

βj = β1 [F1… Fj-2 (Fj-1 – 1)] [chain age-to-age factors]

αi = Si,n-i+1/∑R(i) βj

^

^^

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Previous ML results

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Over-dispersed Poisson error structure

Theorem. Suppose that Yij are independent, Yij ~ ODP(μij,φ) with μij = αiβj. Then ML estimates of αi, βj are given by precisely the chain ladder algorithm.

• Proved by Hachemeister & Stanard (1975) for simple Poisson

• Proved by England & Verrall (2002) for ODP

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Gamma error structure

• Suppose that Yij are independent, Yij ~ Gamma with mean μij = αiβj 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

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Tweedie family

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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

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Tweedie sub-family

• EDFℓ(y;θ,λ) = c(λ)[yθ – b(θ)] + a(y,λ)

• Can be shown thatμ = E[Y] = b'(θ), Var[Y] = b''(θ)/c(λ)

• Tweedie sub-family given by restrictionsc(λ) = λ

Var[Y] = μp/λ for some p≤0 or p≥1This restricts b(.) to the form

b(θ) = (2-p)-1[(1-p)θ](2-p) / (1-p) • Denote this Y ~ Tw(μ,λ,p)

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Special cases of Tweedie

• Tweedie family

ℓ(y;θ,λ) = λ[yθ – b(θ)] + a(y,λ)

Var[Y] = μp/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

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ML results for Tweedie

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ML equations

Theorem. Suppose that Yij are independent, Yij ~ Tw(μij,λ/wij,p) for given wij, p with μij = αiβj. Then the ML equations for αi, βj are

∑R(i) wij μij1-p [yij – μij] = 0, i=1,…,n

∑C(j) wij μij1-p [yij – μij] = 0, j=1,…,n

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Special case: Y ~ ODP

• ML equations∑R(i) wij μij

1-p [yij – μij] = 0, i=1,…,n∑C(j) wij μij

1-p [yij – μij] = 0, j=1,…,n• Special case p=1 (Y ~ ODP) with wij=1 (same over-dispersion in all

cells)∑R(i) [yij – μij] = 0, i=1,…,n∑C(j) [yij – μij] = 0, j=1,…,n

• Equivalently ∑R(i) yij = ∑R(i) μij, i=1,…,n∑C(j) yij = ∑C(j) μij, 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

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Special case: Y ~ ODP: summary

ML estimation

Marginal sum estimation

Chain ladder estimation

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General Tweedie case

• ML equations∑R(i) wij μij

1-p [yij – μij] = 0, i=1,…,n∑C(j) wij μij

1-p [yij – μij] = 0, j=1,…,n• Suppose weights take the form

wij = ui vj [includes the case wij=1]• Introduce the transformations

Zij = wij μij1-p Yij

ηij = wij μij2-p = uivj (αiβj)2-p = aibj

whereai = ui αi

2-p

bj = vj βj2-p

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General Tweedie case (cont’d)

• ML equations

∑R(i) wij μij1-p [yij – μij] = 0, i=1,…,n

∑C(j) wij μij1-p [yij – μij] = 0, j=1,…,n

• Transformations


ηij = wij μij2-p = uivj (αiβj)2-p = aibj

• ML equations become

∑R(i) [zij – ηij] = 0, i=1,…,n

∑C(j) [zij – ηij] = 0, j=1,…,n

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• ML equations∑R(i) [zij – ηij] = 0, i=1,…,n∑C(j) [zij – ηij] = 0, j=1,…,n

• c.f. chain ladder∑R(i) [yij – μij] = 0, i=1,…,n∑C(j) [yij – μij] = 0, j=1,…,n

• So estimation in the general Tweedie case is carried out by applying the chain ladder to “observations” Zij = wij μij

1-p Yij

• This returns estimates of parametersai = ui αi

2-p

bj = vj βj2-p

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• Transformation of observations

Zij = wij μij1-p Yij requires knowledge of

estimand μij

• So ML equations are implicit in parameters

• No closed form solution

• Need to be solved iteratively

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General Tweedie case – iterative algorithm

• Implicit solution


wij μij2-p = aibj

• Iterative solution• Denote r-th iterate of any quantity by

an upper (r), r=0,1,etc.

Z(r)ij = wij μ(r)

ij1-p Yij

• Apply chain ladder to data set {Z(r)ij}

to obtain estimates a(r+1)i, b (r+1)

j of parameters ai, bj

• Estimate new iterate μ(r+1)ij according

to

wij μ(r+1)ij

2-p = = a(r+1)ib(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

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Special Tweedie cases

• p = 1, wij = 1 (ODP)

• p=0 (normal)

• p=2 (gamma)

• Zij = Yij

• Standard chain ladder (known)

Zij = wij μij Yij

Zij = wij [Yij / μij]


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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

1 1 0

2 5 0.05%

2.4 24 0.10%

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Convenient approximation for computation

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Approximation

• Transformed observations for general Tweedie are Zij = wij μij

1-p Yij to yield estimates of ai = ui αi2-p, bi =

vi βi2-p

• Implicitness of ML equations arises from the fact that Z involves estimand μ

• Approximation

μij ← Yij

Zij ← wij Yij2-p

• Perform chain ladder on these quantities to estimate• Note that approximation become meaningless at p=2

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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

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Separation method

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Comparison of chain ladder and separation method

Chain ladder

μij = E[Yij] = αiβj

• Marginal sum estimation

∑R(i) [yij – μij] = 0

∑C(j) [yij – μij] = 0

Separation method

μij = E[Yij] = αi+j-1βj

• Marginal sum estimation

∑D(k) [yij – μij] = 0

∑C(j) [yij – μij] = 0

• Each chain ladder result has counterpart for separation method

Documents

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