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1 Individual claim loss reserving conditioned by case estimates Greg Taylor Gráinne McGuire James Sullivan Taylor Fry Consulting Actuaries University of New South Wales Actuarial Symposium 9 November 2006

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2 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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3 Acknowledgement Funding of this work is assisted by a research contract from the Institute of Actuaries (UK)

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4 Individual claim modelling generally

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5 Conventional (aggregate) form of loss reserving DataFitted Forecast ModelForecast Individual claim data are aggregated into “triangulations” Interest here will be in individual claim models rather than the above

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6 Individual claim models Model Forecast DataFittedForecast

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7 Individual claim models Model Y=f (β)+ε Y i = f(X i ; β) + ε i Y i = size of i-th completed claim X i = vector of attributes (covariates) of i-th claim β = vector of parameters that apply to all claims ε i = vector of centred stochastic error terms Forecast g ( ) DataFittedForecast

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8 Form of individual claim model Y i = f(X i ; β) + ε i Convenient practical form is Y i = h -1 (X i T β) + ε i [GLM form] h = link functionError distribution from exponential dispersion family Linear predictor = linear function of the parameter vector

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9 Types of covariates

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10 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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11 Classification of covariates by type Covariates Static Dynamic

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12 Classification of covariates by type Covariates Static Dynamic Time Other Accident quarter Report quarter Gender Pre-injury earnings

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13 Classification of covariates by type Covariates Static Dynamic Time Other Time (predictable) Unpredictable Accident quarter Report quarter Gender Pre-injury earnings Finalisation quarter Operational time at finalisation Case estimate Claim severity

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14 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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15 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 Dynamic time covariates Forecasts required May have only second order effects, e.g. finalisation quarter

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16 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 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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17 Form of individual claim model incorporating case estimates

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18 Form of individual claim model incorporating case estimates As before, main model takes the form Y i = h -1 (X i T β) + ε i where X i includes case estimate of incurred loss associated with i-th claim at the valuation date

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19 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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20 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: 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 Model of numbers of claims reported Model of sizes of unreported claims

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21 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 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 Model of numbers of claims reported Model of sizes of unreported claims Model of finalis- ations over time

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22 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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23 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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24 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*Id* U*U* I d+1 * U*U* I d+2 * U*U* :::::: I f-2 * U*U* I f-1 * U*U* Data records for an individual claim: I d+r = estimate of incurred loss at end of development period d+r U = ultimate claim cost

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25 Prediction error

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26 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 U i Model Y i = h -1 (X i T β) + ε i Require forecast of U i = g(Z i T β) + η i ε i, η i ~ F(.) Forecast is U* i = g(Z i T β*) + η* i

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27 Some numerical results

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28 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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29 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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30 Numerical results Expressed per $1Bn of liability from simple paids model ModelForecast Predictive CoV Mack (chain ladder) $888M10.5% Paids: Just time covariates Other static covariates $1,000M $978M 5.3% 5.7% Incurreds $1,040M 5.3%

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31 Combination of models

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32 Combination of models Suppose more than one model e.g. paids and incurreds models L a (s) = estimated liability from model s for accident period a Take blended estimates L a = ∑ s w a (s) L a (s) with weights w a (s) [∑ s w a (s) = 1 for each a] Let L=[L 1,L 2,…] T and M=Cov[L] We wish to choose the weights so as to minimise Var[∑ a L a ] =L T ML

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33 Combination of models (cont’d) Choose the weights w a (s) so as to minimise L T ML 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) =[w 1 (s), w 2 (s),…] T R=[R 1,R 2,…] T where R a = L a /Q a and Q a = case estimate Then minimise L T ML + λ 1 (KR) T (KR) + λ 2 ∑ s (K w (s) ) T (K w (s) ) where K is a matrix which takes 2 nd differences λ 1,λ 2 >0 are constants

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34 Numerical results Smoothing parameters set to λ 1 =λ 2 =10 14 ModelForecast Predictive CoV Mack (chain ladder) $888M10.5% Paids: Just time covariates Other static covariates $1,000M $978M 5.3% 5.7% Incurreds $1,040M 5.3% Blended $1,021M 3.8%

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35 Combination of models (cont’d) “Paids” model took the form Y i = h -1 (X i T β) + ε i with h(.)=ln(.) Similarly for the age-to-ultimate factors in the “incurreds” model Write these two models as Y i = exp [X i (P) β (P) ] + ε i (P) R i = exp [X i (I) β (I) ] + ε i (I) Y i = exp [ln I i + X i (I) β (I) ] + ε i (I) where I i = current case estimate of incurred loss for i-th claim

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36 Combination of models (cont’d) Paids model: Y i = exp [X i (P) β (P) ] + ε i (P) Incurreds model: Y i = exp [ln I i + X i (I) β (I) ] + ε i (I) Blended model: L a = ∑ s w a (s) L a (s) = w L a (P) + (1-w) L a (I) Similar to Y i = exp {X i (P) [wβ (P) ] +(1-w) ln I i + X i (I) [(1-w)β (I) ]} + ε i Generalise to Y i = exp {(1-w) ln I i + X i (U) β (U) } + ε i where X i (U) is the union of X i (P) and X i (I)

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37 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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38 Numerical results ModelForecast Predictive CoV Mack (chain ladder) $888M10.5% Paids: Just time covariates Other static covariates $1,000M $978M 5.3% 5.7% Incurreds $1,040M 5.3% Blended $1,021M 3.8% Unified $1,071M 3.4%

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39 Bootstrap distribution of unified forecast of loss reserve

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40 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

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