individual claim loss reserving conditioned by case estimates · 2014. 5. 15. · 1 individual...

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Individual claim loss reserving conditioned by case estimates

Greg TaylorGráinne McGuire

James Sullivan

Taylor Fry Consulting Actuaries

University of New South Wales Actuarial Symposium9 November 2006

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Overview • Concerned with individual claim loss reserving

• Therefore individual loss modelling and forecasting• We consider different types of response variates and

covariates that may be included in the model• Some of these lead to “paids” models• Others lead to “incurreds” models

• We examine examples of each• But with particular emphasis on the “incurreds” models

• We examine how a single forecast may be formed from a combination of a “paids” and an “incurreds” model• Including the construction of a single model that unifies the two

• Numerical examples from a single data set are given in all cases• Forecast error is estimated for each model

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Acknowledgement

• Funding of this work is assisted by a research contract from the Institute of Actuaries (UK)

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Individual claim modelling generally

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Conventional (aggregate) form of loss reserving

Data Fitted

Forecast

Model Forecast

Individual claim data are aggregated into “triangulations”

Interest here will be in individual claim models rather than the above

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Individual claim models

Claim 1

Claim 2Claim 3

Claim n

:

:

:

Claim 1

Claim 2Claim 3

Claim n

:

:

:

Claim 1

Claim 2Claim 3

Claim n

:

:

:

Model Forecast

Data Fitted Forecast

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Individual claim models

Claim 1

Claim 2Claim 3

Claim n

:

:

:

Claim 1

Claim 2Claim 3

Claim n

:

:

:

Claim 1

Claim 2Claim 3

Claim n

:

:

:

Model

Y=f(β)+ε

Yi = f(Xi; β) + εi

Yi = size of i-th completed claimXi = vector of attributes (covariates) of i-th claimβ = vector of parameters that apply to all claimsεi = vector of centred stochastic error terms

Forecast g( )β̂

Data Fitted Forecast

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Form of individual claim model

Yi = f(Xi; β) + εi

• Convenient practical form is

Yi = h-1(XiT β) + εi [GLM form]

h = link function Error distribution from exponential dispersion family

Linear predictor = linear function of the parameter vector

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Types of covariates

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Examples of covariates (for individual claims)• Accident quarter• Operational time at finalisation• Finalisation quarter• Gender of claimant• Pre-injury earnings of claimant• Case estimates of incurred loss at various

dates

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Classification of covariates by type

Covariates

Static Dynamic

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Covariates

Static Dynamic

Time Other

•Accident quarter

•Report quarter

•Gender

•Pre-injury earnings

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Covariates

Static Dynamic

Time OtherTime

(predictable) Unpredictable

•Accident quarter

•Report quarter

•Gender

•Pre-injury earnings

•Finalisation quarter

•Operational time at finalisation

•Case estimate

•Claim severity

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Implications of covariate types for forecasting• Static covariates (time or other)

• Just labels for the claims• But each such covariate implies another dissection of

IBNRs

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IBNRs• Dynamic time covariates

• Forecasts required• May have only second order effects, e.g. finalisation quarter

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IBNRs• Dynamic time covariates

• Forecasts required• May have only second order effects, e.g. finalisation quarter

• Unpredictable dynamic covariates• Forecasts required

• May have first order effects, e.g. case estimate

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Form of individual claim model incorporating case estimates

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Form of individual claim model incorporating case estimates• As before, main model takes the form

Yi = h-1(XiT β) + εi

where Xi includes case estimate of incurred loss associated with i-th claim at the valuation date

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Form of individual claim model incorporating case estimates (cont’d)• In fact, the model requires more structure than this because

of claims and estimates for nil cost• Let (for an individual claim)

• U = ultimate incurred (may = 0)• C = current estimate (may = 0)• X = other claim characteristics

Model of Prob[U=0|C,X]

Model of U|U>0,C=0,X

Model of U/C|U>0,C>0,X

Prob[U=0]

Prob[U>0]

If C=0

If C>0

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Form of individual claim model incorporating case estimates (cont’d)• This is just the model for reported claims

• Which, in any case, is likely to depend on estimated numbers of claims incurred (e.g. to calculate operational times)

• So extended model required (including IBNR claims) is:




Prob[U=0]

Prob[U>0]

If C=0

If C>0

Model of numbers of claims reported

Model of sizes of unreported claims

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Form of individual claim model incorporating case estimates (cont’d)• There is also likely to be a need for a model of finalisations

over time• Mapping of operational times to real times




Prob[U=0]

Prob[U>0]

If C=0

If C>0

Model of numbers of claims reported

Model of sizes of unreported claims

Model of finalis-ations over time

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Form of individual claim model incorporating case estimates (cont’d)• Thus a total of 6 models required

• Model of numbers of claims reported• Model of finalisations over time• Model of sizes of unreported claims• Models conditioned by current case estimates

• Probability of zero finalisation• Model of claim sizes of claims with zero current

estimate• Model of age-to-ultimate factor for claims with non-

zero current estimate

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Data records for individual claim model incorporating case estimates

• Development of incurred loss estimates• We do not observe

this from one year-end to the next

• We always observe it from the date of the estimate to ultimate (claim finalisation)

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Data records for individual claim model incorporating case estimates

• Development of incurred loss estimates• We do not observe

this from one year-end to the next

• We always observe it from the date of the estimate to ultimate (claim finalisation)

Id

*U*

Id+1

*U*

Id+2

*U*

:::If-2

*U*

If-1

*U*

Data records for an individual claim:Id+r = estimate of incurred loss at end of development

period d+r

U = ultimate claim cost

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Prediction error

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Prediction error

• In the usual way for GLM reserving

• Form of bootstrapping• Calibrate GLM

• β^ estimates β with covariance C

• Assume β^ ~ N(β,C)• Draw random realisation β*

of β^• Draw random realisation η*i ~ F(.)

• U*i provide bootstrap replicates of Ui

• ModelYi = h-1(Xi

T β) + εi

• Require forecast ofUi = g(Zi

T β) + ηi

εi, ηi ~ F(.)• Forecast is

U*i = g(ZiT β*) + η*i

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Some numerical results

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Data set

• Data set from a Liability class• About 30 accident years• Both Occurrence and Claims-made coverages

• Individual claims data• More than 20,000 claims

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Models used

• Chain ladder with Mack estimate of CoV• Just used as benchmark• Applied blindly over whole data triangle

• Paids models• Including just time covariates

• Very parsimonious model – 9 parameters• Including other static covariates also

• Incurreds model• Much more complex model• Many more parameters

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Numerical results

• Expressed per $1Bn of liability from simple paids model

Model Forecast Predictive CoV

Mack (chain ladder) $888M 10.5%

Paids:Just time covariatesOther static covariates

$1,000M$978M

5.3%5.7%

Incurreds $1,040M 5.3%

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Combination of models

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Combination of models

• Suppose more than one model• e.g. paids and incurreds models

• La(s) = estimated liability from model s for

accident period a• Take blended estimates La = ∑s wa

(s) La(s)

with weights wa(s) [∑s wa

(s) = 1 for each a]• Let L=[L1,L2,…]T and M=Cov[L]• We wish to choose the weights so as to

minimise Var[∑a La] =LTML

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Combination of models (cont’d)• Choose the weights wa

(s) so as to minimise LTML• BUT we also require that

• Each set of weights is smooth over a• The blended estimates are smoothly related to case estimates of

outstanding liability• Let

• w (s)=[w1(s), w2

(s),…]T

• R=[R1,R2,…]T where Ra = La/Qa and Qa = case estimate• Then minimise

LTML + λ1 (KR)T(KR) + λ2 ∑s (K w (s))T(K w (s))where

• K is a matrix which takes 2nd differences• λ1 ,λ2>0 are constants

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Numerical results

• Smoothing parameters set to λ1=λ2=1014




$1,000M$978M

5.3%5.7%


Blended $1,021M 3.8%

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Combination of models (cont’d)• “Paids” model took the form

Yi = h-1(XiT β) + εi with h(.)=ln(.)

• Similarly for the age-to-ultimate factors in the “incurreds” model

• Write these two models asYi = exp [Xi

(P) β(P)] + εi(P)

Ri = exp [Xi(I) β(I)] + εi

(I)

Yi = exp [ln Ii + Xi(I) β(I)] + εi

(I)

where Ii = current case estimate of incurred loss for i-th claim

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Combination of models (cont’d)

• Paids model: Yi = exp [Xi(P) β(P)] + εi

(P)

• Incurreds model: Yi = exp [ln Ii + Xi(I) β(I)] + εi

(I)

• Blended model: La = ∑s wa(s) La

(s)

= w La(P) + (1-w) La

(I)

• Similar toYi = exp {Xi

(P) [wβ(P)] +(1-w) ln Ii + Xi(I) [(1-w)β(I)]} + εi

• Generalise toYi = exp {(1-w) ln Ii + Xi

(U) β(U)} + εiwhere Xi

(U) is the union of Xi(P) and Xi

(I)

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Numerical results

• w=20% • A number of the coefficients of the more

influential covariates in the incurreds model are reduced by factors not too different from 80%

• roughly 20:80 mixture of paids and incurreds

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Numerical results




$1,000M$978M

5.3%5.7%


Blended $1,021M 3.8%

Unified $1,071M 3.4%

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Bootstrap distribution of unified forecast of loss reserve

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Conclusions • Individual claim models are likely to have greater predictive efficiency

than aggregate models• It is often possible to achieve high efficiency with individual claim

“paids” models that have relatively few parameters• Often fewer than the aggregate models

• The inclusion of static covariates other than time covariates will not necessarily improve predictive efficiency

• An “incurreds” model involves more component sub-models than a “paids” model, and is likely to be considerably more complex. However, it may have greater or less predictive efficiency

• Paids and incurreds models can be blended. Even if the incurreds model has no pay-off in terms of greater efficiency, it is to be expected that the blended model will be mor efficient than either of its components

• A “unified” model, as developed here, is a generalisation of the blended model that is likely to retain some of its features. It is likely to improve on the blended model’s efficiency

individual claim loss reserving conditioned by case estimates · 2014. 5. 15. · 1 individual...

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