Claims/Agency metrics

1. Claims/Agency metrics Greg Taylor Taylor Fry Consulting Actuaries University of Melbourne University of New South Wales Casualty Actuarial Society Special Interest Seminar on Predictive Modeling Boston, October 4-5 2006

2. Overview • Individual claim models • “Paids” models • “Incurreds” models • Numerical results • Adaptive models

3. Why individual claim models?

4. Example problem • Classical workers compensation cost centre allocation problem • Claim numbers at the leaves of this tree may be small Total claim cost . . . Cost centre 1 Cost centre 2 Cost centre m … … …

5. Measuring claims performance • Consider measuring claims performance in a segment of a long tail portfolio • Likely that adopted metric will require an estimate of the amount of losses incurred but as yet unpaid (loss reserve) • e.g. metric is expected ultimate losses per policy for a specific underwriting period = Paid to date + unpaid losses Number of policy-years of exposure = average PTD per policy-year + average unpaid per policy-year

6. Measuring claims performance in large portfolio segments • Let there be n policy-years of exposure and ui = i-th amount unpaid • Consider the ui to be random drawings from some distribution • Average amount unpaid is ūi = Σ ui /n = Σ {E[ui] + ui - E[ui]}/n = E[ui] + Σ {ui - E[ui]}/n  E[ui] as n∞ by the large of large numbers d

7. Measuring claims performance in large portfolio segments (cont’d) ūi  E[ui] as n∞ • E[ui] = expected size of a randomly drawn claim • This will be the result produced by most conventional actuarial methods, e.g. • Paid chain ladder • Even incurred chain ladder at early development • While E[ui] may be a good approximation to ūi for large sample sizes, it may be very poor for small ones • Leading to a highly distorted cost allocation d

8. Measuring claims performance in small portfolio segments • Effective estimation of small sample average claim cost must somehow take account of the properties of the individual claims

9. There is a need to change from this… Data Fitted Model Forecast Forecast • Conventional actuarial analysis of loss experience • Call such models “aggregate models”

10. …to this Forecast Model Special case of individual claim reserving – statistical case estimation

11. Individual claim models

12. Form of such a model Forecast g() Model Y=f(β)+ε

13. Form of such a model Forecast g() Model Y=f(β)+ε • Yi = f(Xi; β) + εi • Yi = size of i-th completed claim • Xi = vector of attributes (covariates) of i-th claim • β = vector of parameters that apply to all claims • εi = vector of centred stochastic error terms

14. Form of individual claim model Yi = f(Xi; β) + εi • Convenient practical form is Yi = h-1(XiTβ) + εi [GLM form] h = link function Error distribution from exponential dispersion family Linear predictor = linear function of the parameter vector

15. Form of individual claim model – more specifically • How might one create an individual claim model of the “paids” type? • Aggregate paids model usually takes the form Yjk = f(j,k; β) + εjk for j = accident period k= development period • Compare with Yi = f(Xi; β) + εi Not always formulated

16. Form of “paids” individual claim model • Possible to mimic aggregate model by defining individual model as just Yi = h-1(ji,ki; β) + εi

17. Form of “paids” individual claim model • Possible to mimic aggregate model by defining individual model as just Yi = h-1(ji,ki; β) + εi • But often possible to improve on this, e.g. • Replace development period j with operational time ti (proportion of accident period’s incurred claims completed) at completion of i-th claim • Example Yi = exp [β0+β1ti+β2max(0,ti-0.8)] + εi

18. Example of “paids” individual claim model Yi = exp [β0+β1ti+β2max(0,ti-0.8)] + εi E[Yi] = exp [β0+β1ti+β2max(0,ti-0.8)]

19. Example of “paids” individual claim model (cont’d) Yi = exp [β0+β1ti+β2max(0,ti-0.8)] + εi • Include superimposed inflation • Let q=j+k=calendar period of claim completion • Extend model Yi = exp [β0+β1ti+β2max(0,ti-0.8)+β3qi] + εi • Superimposed inflation at rate exp[β3] per period

20. Example of “paids” individual claim model (cont’d) Yi = exp [β0+β1ti+β2max(0,ti-0.8)+β3qi] + εi • We might wish to model superimposed inflation as beginning at period q=q0 Yi = exp [β0+β1ti+β2max(0,ti-0.8)+β3max(0,qi-q0)] + εi

21. Example of “paids” individual claim model (cont’d) Yi = exp [β0+β1ti+β2max(0,ti-0.8)+β3qi] + εi • We might wish to model superimposed inflation as beginning at period q=q0 Yi = exp [β0+β1ti+β2max(0,ti-0.8)+β3max(0,qi-q0)] + εi • …and we might wish to model superimposed inflation with a rate that decreases with increasing operational time Yi = exp [β0+β1ti+β2max(0,ti-0.8)+(β3-β4ti) max(0,qi-q0)] + εi etc etc

22. “Paids” estimate of loss reserve scaled to baseline $1,000M Prediction CoV = 5.3% Mack (incurreds) estimate is $887M with CoV = 10.5% Mack estimate produces negative reserves for the old years of origin “Paids” chain ladder fails completely Example of “paids” individual claim model (cont’d)

23. This model is very economical Contains only 9 parameters to represent many thousands of claims Example of “paids” individual claim model (cont’d)

24. Further extension of “paids” individual claim model Yi = exp [β0+β1ti+β2max(0,ti-0.8)+(β3-β4ti) max(0,qi-q0)] + εi • May include claim characteristics other than time-related, e.g. • Nature of injury • Claim severity (MAIS scale) • Pre-injury earnings Yi = exp [β0+β1ti+β2max(0,ti-0.8)+(β3-β4ti) max(0,qi-q0) + more terms] + εi

25. Example of “incurreds” individual claim model • Similar to “paids” model • Basic set-up is still Yi = h-1(ji,ki,qi,ti,other;β) + εi • Example Yi = exp(Ci,ji,ki,qi,ti,other;β) + εi where Ci = current manual estimate of incurred cost of i-th claim

26. Example of “incurreds” individual claim model (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] Prob[U=0] Prob[U>0] Model of U|U>0,C=0,X Model of U/C|U>0,C>0,X If C=0 If C>0

27. “Paids” estimate of loss reserve = $1,000M Prediction CoV = 5.3% “Incurreds” estimate of loss reserve = $1,040M Prediction CoV = 5.3% Example of “incurreds” individual claim model (cont’d)

28. Adaptive reserving

29. Static and dynamic models • Return for a while to models based on aggregate (not individual claim) data • Model form is still Y=f(β)+ε • Example • j = accident quarter • k = development quarter • E[Yjk] = a kb exp(-ck) = exp [α+βln k - γk] • (Hoerl curve for each accident period)

30. Static and dynamic models (cont’d) • Example E[Yjk] = a kb exp(-ck) = exp [α+βln k - γk] • Parameters are fixed • This is a static model But parameters α, β,γ may vary (evolve) over time, e.g. with accident period Then • E[Yjk] = exp [α(j)+β(j) ln k - γ(j) k] • This is a dynamic model, or adaptive model

31. Illustrative example of evolving parameters

32. Formal statement of dynamic model • Suppose parameter evolution takes place over accident periods • Y(j)=f(β(j)) +ε(j) [observation equation] • β(j) = u(β(j-1)) + ξ(j) [system equation] • Let (j|s) denote an estimate of β(j) based on only information up to time s Some function Centred stochastic perturbation

33. Adaptive reserving q-th diagonal (1|q) (2|q) Forecast at valuation date q (q|q)

34. Adaptive reserving (cont’d) • Reserving by means of an adaptive model is adaptive reserving • Parameter estimates evolve over time • Fitted model evolves over time • The objective here is “robotic reserving” in which the fitted model changes to match changes in the data • This would replace the famous actuarial “judgmental selection” of model

35. Special case of dynamic model: DGLM • Y(j)=f(β(j)) +ε(j) [observation equation] • β(j) = u(β(j-1)) + ξ(j) [system equation] • Special case: • f(β(j)) = h-1(X(j) β(j)) for matrix X(j) • ε(j) has a distribution from the exponential dispersion family • Each observation equation denotes a GLM • Link function h • Design matrix X(j) • Whole system called a Dynamic Generalised Linear Model(DGLM)

36. Adaptive form of individual claim models • Individual claim models can also be converted to adaptive form • Just subject parameters to evolutionary model • We have experimented with this type of model and adaptive reserving • Moderately successful

37. Conclusions • Effective forecast of costs of small samples of claims requires individual claim models • Such models condition the forecasts on much more information than aggregate models • Even for large samples, individual claim models may yield considerably more efficient forecasts • Lower coefficient of variation • This may save real money • Lower uncertainty implies lower capitalisation • Adaptive forms of individual claim models may further improve the tracking of claims experience over time