1. 1 A Linear Model of ULAE Leigh J. Halliwell, FCAS, MAAA Consulting Actuary leigh@lhalliwell.com Casualty Loss Reserve Seminar Atlanta, GA September 11, 2006 ULAE/A&O Estimation and Modeling Leigh J. Halliwell, FCAS, MAAA Consulting Actuary leigh@lhalliwell.com Casualty Loss Reserve Seminar Atlanta, GA September 11, 2006 ULAE/A&O Estimation and Modeling

2. 2 Outline Schedule-P Origin of Simple Method Refined Method Statistical Model Schedule-P Origin of Simple Method Refined Method Statistical Model

3. 3 ULAE/A&O ALAE / ULAE  (Un)allocated loss adjustment expense D&CC / A&O  Defense and cost containment  Adjusting and other expense New categories effective 1/1/1998 ALAE  D&CC; hence, A&O  ULAE ALAE still common in reinsurance ULAE/A&O basically the cost of running a claims department ALAE / ULAE  (Un)allocated loss adjustment expense D&CC / A&O  Defense and cost containment  Adjusting and other expense New categories effective 1/1/1998 ALAE  D&CC; hence, A&O  ULAE ALAE still common in reinsurance ULAE/A&O basically the cost of running a claims department

4. 4 Schedule P Interrogatories “A&O should be allocated to the [AYs] based on number of claims reported, closed, and outstanding in those years.” The only hard number is CY A&O paid No explicit estimation of unpaid A&O  Allocate estimate acc. to projected claims 50/50 rule: half a claim’s ULAE (A&O?) paid when reported, half when closed  Not accurate for multipayment claims, e.g., WC  Outstanding-claims ignored for simplicity “A&O should be allocated to the [AYs] based on number of claims reported, closed, and outstanding in those years.” The only hard number is CY A&O paid No explicit estimation of unpaid A&O  Allocate estimate acc. to projected claims 50/50 rule: half a claim’s ULAE (A&O?) paid when reported, half when closed  Not accurate for multipayment claims, e.g., WC  Outstanding-claims ignored for simplicity

5. 5 Implicit Method

6. 6 Weaknesses of the Method Maybe some claims more difficult to settle Maybe 30/70 or 60/40; why 50/50?  If 50/50 right for ULAE, maybe wrong for A&O Maybe IBNR claims more costly to settle Ignores inflation ($1000/claim over 5+ years) Maybe some claims more difficult to settle Maybe 30/70 or 60/40; why 50/50?  If 50/50 right for ULAE, maybe wrong for A&O Maybe IBNR claims more costly to settle Ignores inflation ($1000/claim over 5+ years)

7. 7 Refined Example

8. 8 Refined Method/Model

9. 9 Refined Estimate Note the start-up inefficiency; ACC (  ) decreasing We can do better: two-moment statistical model better than deterministic method.  Variance proportional to base activity, quadratic to indices. So  = X ∙ TotalCostIndex in slide 8. Note the start-up inefficiency; ACC (  ) decreasing We can do better: two-moment statistical model better than deterministic method.  Variance proportional to base activity, quadratic to indices. So  = X ∙ TotalCostIndex in slide 8.

10. 10 Heteroskedastic Statistical Model To predict from design X p and diagonal  p : Details in my papers, esp. 1996 PCAS and Summer 1997 Forum X p uncertain by treating IBNR counts as stochastic (here Poisson) Details in my papers, esp. 1996 PCAS and Summer 1997 Forum X p uncertain by treating IBNR counts as stochastic (here Poisson)

11. 11 Solution of Statistical Model

12. 12 Final Remarks on Model Prediction-design a form of “model” risk  the least significant of the variances Quantiles obtainable from 2-MoM fit  or simulate with empirical residuals More complexity possible  Regressor for outstanding claims (~opened,~closed)  Prediction for many future periods and discounting  Autocorrelation of a claim’s annual payments In general, why method when you can model? Prediction-design a form of “model” risk  the least significant of the variances Quantiles obtainable from 2-MoM fit  or simulate with empirical residuals More complexity possible  Regressor for outstanding claims (~opened,~closed)  Prediction for many future periods and discounting  Autocorrelation of a claim’s annual payments In general, why method when you can model?