1. Casualty Exposure Rating Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp 7-28-05

2. Page 2 Outline Basics of Exposure-Rating Practical Issues Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp 7-28-05

3. Page 3 Basics of Exposure-Rating What is exposure-rating? Why do it? How Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp 7-28-05

4. Page 4 What is Exposure Rating? An estimate of the covered Loss + ALAE in an excess layer Based on reinsured information – Policy Limits – Insured characteristics (hazard) But not – Reinsured loss experience Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp 7-28-05

5. Page 5 Typical Submission Data Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp 7-28-05 Policy Limit Range BottomtoTop Policy Count Premium ($000) 10250,000 7 20 2250,001500,000 100 250 3500,0011,000,000 200 700 41,000,0012,000,000 700 3,500 52,000,0015,000,000 30 500

6. Page 6 Typical output of exposure rating Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp 7-28-05 Policy Limit Range Layer = 500x500 BottomtoTop Loss To Layer ($000) 10250,000 0 2250,001500,000 0 3500,0011,000,000 200 41,000,0012,000,000 700 52,000,0015,000,000 100

7. Page 7 Why Exposure-Rate Provide complement of credibility for experience rate Relate higher layer to lower (credible) layer Price “Free Cover” – Layer higher than largest trended claim in the experience Adjust experience rate for shift in limits/deductibles “Use eclectic methods” – Armstrong, Long-Range Forecasting Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp 7-28-05

8. Page 8 Why Exposure-Rate Why experience rate not 100% credible/relevant – Volume – Shift in business – Difficulties in estimating - Trend, Development, On-leveling, Data problems Pricing “Free Cover” is a case where experience rate given 0% credibility Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp 7-28-05

9. Page 9 How to exposure-rate How – Estimate fgu loss – Use severity distribution to allocate to layers Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp 7-28-05

10. Page 10 Estimate fgu loss Fac: Extend exposures – (Projected Exposures) x (Manual Rate) Treaty: Loss ratio approach – (Projected Subject Premium) x (Estimated Loss Ratio) Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp 7-28-05

11. Page 11 Estimate fgu loss Fac: Extend exposures Manual Rate contains provisons for – ALAE – ULAE – Profit – Internal and external expense which must be stripped out. Should you INCLUDE (primary) experience mod? Some LOB’s and some classes of business must be judgementally estimated – E.g., “a-rated” Products Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp 7-28-05

12. Page 12 Estimate fgu loss Treaty: Projected subject premium x loss ratio Loss ratio sources – Client internal data – Client Annual Statement data – Peer group AS data – Industry AS data – Rate filings? – Underwriting audit Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp 7-28-05

13. Page 13 Estimating Loss Ratios-Issues Problems with Annual Statement data – AS LOB, not program – Net (mostly) Reliability of data – Relevance – Predicted vs achieved rate-level changes – As-if data Is the loss ratio the same for all policy limits? (ditto) tables Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp 7-28-05

14. Page 14 Use severity distribution to allocate fgu expected loss to layers What exposes the layer? Using the “LEV” function Why not use ILFs? Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp 7-28-05

15. Page 15 What exposes the layer? Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp 7-28-05 A B C Policy Limit A Policy Limit B Policy Limit C

16. Page 16 What exposes the layer? Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp 7-28-05 A B C Top of Layer Bottom of Layer Policy Limit A Policy Limit B Policy Limit C

17. Page 17 What exposes the layer? Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp 7-28-05 A B C Top of Layer Bottom of Layer Policy Limit A Policy Limit B Policy Limit C

18. Page 18 What exposes the layer? Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp 7-28-05 A B C Top of Layer Bottom of Layer Policy Limit A Policy Limit B Policy Limit C “Top”

19. Page 19 What exposes the layer? Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp 7-28-05 A B C Top of Layer Bottom of Layer Policy Limit A Policy Limit B Policy Limit C “Bottom”

20. Page 20 Contribution to expected losses Contribution to expected layer losses from policy = FGU losses up to “Top” – FGU losses up to “Bottom” Contribution to all expected losses from policy = FGU losses up to policy limit* ______________________________________________ *assuming no deductible, SIR Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp 7-28-05

21. Page 21 Contribution to expected losses Contribution to expected layer losses from policy = {LEV(“Top”) – LEV(“Bottom”)} x frequency Contribution to all expected losses from policy = {LEV(Policy Limit) – LEV(0*)} x frequency ______________________________________________ *assuming no deductible, SIR Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp 7-28-05

22. Page 22 What is a LEV? Limited Expected Value function – LEV(k) - = “Losses up to k” - Expected loss fgu, up to the value k, per ground-up claim - Also written E(X; k) - Also written LAS(k) - (Limited Avg Sev) Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp 7-28-05

23. Page 23 Contribution to expected losses Contribution to expected layer losses from policy = {LEV(“Top”) – LEV(“Bottom”)} x frequency Contribution to all expected losses from policy = {LEV(Policy Limit) – LEV(0*)} x frequency ______________________________________________ The FRACTION of fgu losses that gets into the layer does not depend on the frequency. When you divide, it drops out. Only need the LEV. Note that aggregate limits could make this statement false. Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp 7-28-05

24. Page 24 LEV(k) LEV(k) = X = random severity amount x = realization of X F(x) = cdf of x = Prob (X <=x) f(x) = pdf of x k = “limit” Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp 7-28-05

25. Page 25 LEV(k) LEV(k) = Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp 7-28-05 Limit Loss amount below limit “Probability” of loss amount below limit Limit Probability of loss amount at (or above) limit

26. Page 26 Why use LEV function Difference LEVs to get losses in layer If expect λ losses from ground-up (fgu) then loss to layer A xs B is Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp 7-28-05

27. Page 27 Why use LEV function (cont) Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp 7-28-05

28. Page 28 LEV(k) shortcut LEV(k) = Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp 7-28-05 Use integration by parts Good for Pareto, etc. where F(x) = 1 - ….?... and “?” is easily integrated.

29. Page 29 Why not use ILFs? ILFs are meant for all costs, not just losses + ALAE – ISO ILF = Ratio (all costs at Policy Limit) to (all costs at Basic Limit) Using ISO ILFs: – ISO Risk Load gets in – ULAE creeps in – Aggregate limits distort values (depending on which ILFs you use) – ALAE drops out (because is loaded 100% in basic limit) – Reinsurance treatment of ALAE not reflected

30. Page 30 Practical Issues Submission data issues Trend in limits Treatment of ALAE Gaps in exposure-rating How good is the severity distribution? Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp 7-28-05

31. Page 31 Submission Data Typical Submission Data Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp 7-28-05 Policy Limit Range BottomtoTop Policy CountPrem % 10250,000 7 0.1% 2250,001500,000 100 2% 3500,0011,000,000 200 10% 41,000,0012,000,000 700 80% 52,000,0015,000,000 30 2%

32. Page 32 Submission Data Things to look out for Policy Limit Ranges – But limits are round #’s Numbers do not add up – Written premium vs. Earned – Historical vs. projected Missing data – Post-merger Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp 7-28-05

33. Page 33 Submission Data More things to look out for Policy Limit Warrant No information about ILF table breakdown – Or, worse, underwriter’s vague feeling Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp 7-28-05

34. Page 34 ISO Increase Limits Factors By Table – Prem/Ops - 1, 2, 3 x State Group – Products - A, B, C – C. Auto - Wt Class x State Group Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp 7-28-05

35. Page 35 Policy Limit Trend “Policy limits trend with inflation” How trend round numbers? Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp 7-28-05

36. Question You are covering the 1 x 1 layer: By how much will your exposure change if policy limits are trending at 5% per annum?

37. Page 37 Avg Policy Limit over time (fake ISO data)

38. Page 38 Simple Example

39. Page 39 Treatment of ALAE Ways reinsurance can cover ALAE – Pro-rata – “Added To” – other How to handle

40. Page 40 Joint model Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp 7-28-05 g(I,A) = reinsurance payment given indemnity I and ALAE A. Expected reinsurance payment per fgu claim is Good Luck.

41. Page 41 Simpler (but less accurate) ways to handle ALAE Pro-rata: Apply appropriate ALAE/Indmenity ratio Added to: – Twiddle the severity distribution, and (possibly) limits, retentions, etc. – DOES NOT pick up all exposure to highest layers Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp 7-28-05

42. Page 42 Gaps in traditional Exposure- Rating Does not estimate – Clash – Extra-Contractual Obligations (ECO) – Declatory Judgement (DJ) expense - Is not ALAE? – “Cat” potential? - Asbestos, DES,... Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp 7-28-05

43. Page 43 How good is the severity distribution? ISO data is sparse above (say) $5,000,000 – Umbrella/XS data is often not reported to ISO Mixed exponential maxes out at $10,000,000 Big swings in exposure rate for many tables 2004  2005 Non-bureau: – No distribution – ILF’s from client: do they care? Are they ept? - Is “ept” a word? Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp 7-28-05

44. Page 44 How good is the severity distribution? How reflect parameter risk? How reflect possible “anti-selection” effects? – They buy big limits because they need to...? Chris Svendsgaard, Swiss Re Casualty Exposure Rating CARe Boot Camp 7-28-05