Casualty Exposure Rating

1. Casualty Exposure Rating CARe Boot Camp 2007 H. Smosna

2. Reinsurance XOL Pricing • Per occurrence XOL treaties provide a limit of coverage in excess of a ceding company’s retention. • Reinsurance pricing actuaries must calculate expected loss and ALAE in the layer • The expected loss & ALAE must be loaded for internal expense, C&B, profit, contingencies, loss sensitive features • This reinsurance ceded premium is usually expressed as percent of the Ceding Company’s Prospective Subject Premium

3. What is Exposure Rating? • Exposure rating estimates expected loss to the layer for a prospective period • Exposure rating uses your client’s limits profile, hazard characteristics, severity curves and gross loss &ALAE ratio to estimate expected loss to the layer • Severity distributions based on industry data are used to calculate LEVs (Limited Expected Values; also know as LASs or Limited Average Severities) • The LEVs are used to estimate losses to the reinsurance layer by spreading ground up loss into the desired layer

4. Why Exposure Rate? • Complement of credibility for experience rate • Price for ‘free cover’ (when the top of your layer exceeds the largest trended loss in your data) • Experience Rate is not credible • Can adjust experience burns for limits drift • Can use exposure burns to determine relativity based burns for higher layers

5. Advantages of Exposure Rating over Experience Rating • The current risk profile is modeled • For a new book of business (so no experience available) pro forma profiles can be used to project expected loss to the layer • The exposure rating exercise is often easy to perform so UWs can determine an exposure rating based reinsurance rate as part of their triage process

6. Exposure Rating – what info do you need? • Prospective Ground Up / Gross Loss Ratio for subject business • Prospective Subject Premium • Limit & Attachment Point Profile with Premium • Prospective limit and attachment profile • i.e. 50% of premium is written at a $1 million per occurrence limit, 25% is written at a $5 million p.o. limit, 25% is written at a $10 million p.o. limit. All limits attach excess of a 100K SIR. • Severity distribution/LEVs for the line of business reflecting hazard level of underlying risks (Table 123ABC, Auto) – see your UW • Submission data is rarely provided in the full detail corresponding to the ISO Table definitions • The layer you are pricing • No loss experience to the layer required

7. The Concept Exposure Factor based on LEVs = Ceded Loss/Gross Loss • Gross Loss Ratio X Exposure Factor = Gross Loss X Ceded Loss Gross Premium Gross Loss = Ceded Loss = Burn, Loss Cost Gross Premium

8. Visualization – limits profile

9. Visualization – Reinsurance layer

10. Visualization – Ceded Loss

11. Visualization – Ceded Loss as % of Gross Loss – Policy A

12. Visualization – Ceded Loss as % of Gross Loss – Policy B

13. Where’s the frequency? • Again - Severity distributions based on industry data are used to calculate LEVs (Limited Expected Values; also know as LASs or Limited Average Severities) • Again - The LEVs are used to estimate losses to the reinsurance layer by spreading ground up loss into the desired layer • Expected loss = frequency*severity • Exposure Factor based on LEVs = Ceded Loss/Gross Loss • Ceded loss = [LEV(top) – LEV(bottom)]* frequency • Gross loss = [LEV(Policy limit) – LEV(0)]* frequency • Frequency cancels out!

14. Limit • GL policies have 2 kinds of limits: • Per occurrence • Annual aggregate • In this presentation we ignore the effects of aggregate limits. • Assume that the primary policy aggregate limit is high enough and ground up frequency low enough that agg limit has minimal effect • Frequency from previous slide would not cancel out if considering aggregate limits

15. LEV vs. ILF • ISO creates severity distributions and from these they generate ILFs (Increased Limits Factors) – Tables 1,2,3,A,B,C for example • ILFs are used to rate primary policies (at 100K basic limit) to higher limits. They represent the ratio of all per occurrence costs at policy limit to all p.o. costs at basic limit • ILFs include all per occurrence costs at a specific policy limit: • Expected limited p.o. loss cost • All ALAE • ULAE • Risk Load

16. LEV vs. ILF – cont’d • Do not use ISO ILFs in exposure rating • Risk load and ULAE get in to your calculations • The expected ALAE per claim for years was assumed to be the same for all policy limits so was loaded 100% into the basic limit. So when taking ratios of ILFs the ALAE would cancel out • ISO has a new ALAE methodology • Sometimes the ceding company’s ILFs are the only info available (ie for an esoteric LOB). May need to use them. • Use LEVs from the severity distributions • If you are pricing to a different average DOL than that underlying the ISO parameters you can adjust the parameters • If you don’t like the parameters you can change them too (make the tail thicker)

17. More on ISO ILFs • The data underlying ISO ILFS comes from 2 sources: • individual loss info from primary polices included in the standard statistical reporting to ISO • Data from a special ‘call’ for individual loss info from Umbrella and excess policies • ISO fits a mixture of exponential distributions to empirical distributions derived from the data • Because the data is sparse in the tail ISO uses a distribution above some point to smooth the empirical distribution • The mixed exponential (ME) curves are less severe than the Pareto curves • Curve history: Truncated pareto, Pareto Soup, ME • ISO data falls off at a relatively low level and is very sparse over 5M

18. Proceed with Caution when using ISO Mixed Exponential curves • Not a lot of data above $1M • M/E fits data closely up to $1M, but excess of $1 million tail appears to thin • ME maxes out at 10M • M/E 4x1 and 5x5 factors are inconsistent, for example, Premops 2 4x1 factors are lower than Premops 1

19. Clash – not captured in exposure rating • Casualty treaties often cover losses for which exposure rating does not provide an answer • A Long-haul trucker collides with an auto • WC loss from trucker • CAL loss from individual in the auto • Treaty will define this as one occurrence • The WC claim is below the treaty retention • The CAL claim is below the treaty retention • Combined the WC &CAL claims pierce the treaty retention – Clash loss • Exposure rating also does not estimate for ECO/XPL

20. What is an LEV? • Limited Expected Value • The average size of loss when all losses are limited to a particular value • The expected loss from ground up to some limit (k) • LEV(k)=∫xf(x)dx+k[1–F(k)] • x is the severity of an individual claim • f(x) is the pdf of the severity • F(x) is the cdf of the severity k 0

21. What is an LEV? • Limited Expected Value • LEV(k)=∫xf(x)dx+k[1–F(k)] • For any random variable x, one of two things can happen: • x is <= the limitation k • x is > the limitation k • The first part of the equation tackles (1) by calculating the expected loss limited to k when x <= k. • The second part of the equation tackles (2). For any x>k, you ‘cap’ x at k. • The sum of (1) and (2) gives you the average ground up loss when all losses are limited to k. k 0

22. Empirical LEV Example

23. The Concept - again • Exposure Factor based on LEVs = Ceded Loss/Gross Loss • Gross Loss Ratio X Exposure Factor = Gross Loss X Ceded Loss Gross Premium Gross Loss = Ceded Loss = Burn, Loss Cost Gross Premium • Credibility weight exposure burn with experience burn and then load up Z weighted burn for expenses, profit, etc to derive reinsurance rate • Reinsurance rate * Subject Premium = Ceded Premium

24. Exposure Rating – what info do you need? • Ground Up Loss Ratio for the subject business • 55%

25. Exposure Rating – what info do you need? • The Layer you are pricing: If ALAE is included in the limit the effective reinsurance limit and retention should be reduced by the ALAE load

26. ALAE • ALAE in reinsurance contracts: • Pro-rata • Included with loss • ALAE in primary policies • Unlimited (ISO) or outside the limit • Included in limit

27. ALAE • Assume the primary policy’s ALAE is unlimited (as per ISO) • When reinsurance treatment of ALAE is prorata: • Estimate ratio of ALAE to loss and load in (use a gross loss & alae ratio when deriving your burn) • When ALAE is included with loss • Correct way is complicated • Can reduce attachment point and limit by dividing both by (1 + XOL ALAE%)

28. Exposure Rating – what info do you need? • Limits Profile

29. Exposure Rating – what info do you need? ….Severity distribution/LEVs for the line of business reflecting hazard level of underlying risks

30. Severity Distributions : Mixed Exponential LEVs • Send ISO a check • For the mixed exponential you will receive the parameters, weights and closed form formula - all you need to build your own exposure model • ME closed form formula: LEV = ∑ wjλj [1 – e –(x/λ)] 8 J=1

31. Severity Distributions : 5 Parameter Pareto LEVs • For the 5 Parameter Pareto you need the parameters and closed form formula – and you can build your own exposure model • 5PP closed form formula: LEV = P*S + [(1-P)/(Q-1)] * [(B+Q*T)-(B+x)*{(B+T)/(B+x)}^Q] for Q>1

32. Appropriate LEVs – not that simple. • Rare to receive data from the ceding company that maps perfectly to ISO severity distributions • In the example below we are deriving LEVs for an Umbrella profile • Umbrella policies are exposed by underlying GL, AL, EL policies • In exposure rating umbrella we want our LEVs based on a weighting of the underlying , varied exposures • Your exposure rating model should have the capability of deriving LEVs based on weighting together the exposure factors from the various severity distributions

33. An Example: Select a Limit band from the sample profile • Underlying Policy = 500,000 • Policy Limit = 2,000,000 • This represents 7.8% of the premium • Reinsurance Attachment Point = 750,000 • Reinsurance Limit = 1,500,000 You will use LEVs from a severity distribution to allocate ground up expected loss to layers • LEV @ 500,000 = 10,518 • LEV @ 2,000,000 + 500,000 = 12,630 • LEV @ 750,000 = 11,149 • LEV @ 1,500,000 + 750,000 = 12,527

34. How does the reinsurance apply?STOP: Read your contract! • Who wrote the underlying policy? • Is the Umbrella supported (over the ceding company’s own primary)? • Is the Umbrella unsupported (over another company’s primary policy)? • How does the reinsurance respond? • If supported (over their own) typically the reinsurance attaches from the ground up • If unsupported (over other’s primary) the reinsurance attaches on top of the underlying • How is Subject Premium defined?

35. First some simple examples: Umbrella Policy

36. First some simple examples: Umbrella policy with a 4M ground up loss

37. First some simple examples: Reinsurance payment when over own

38. First some simple examples: Reinsurance payment when over other underlying

39. A bit trickier: The Policy

40. A bit trickier: The Reinsurance Layer

41. The Policy & the Reinsurance Layer – OVER THEIR OWN/reinsurance attaches from the ground up

42. Your exposure factor = ceded loss/gross loss Your ceded loss is the expected ground up loss at 2.25M less the expected ground up loss at 750K Your gross loss is the expected ground up loss at 2.5M less the expected loss at the 500K underlying Using LEVs the exposure factor = [LEV(2.25M) – LEV (0.75M)]/[LEV(2.5M) – LEV(0.5M)] Exposure Factor/OVER THEIR OWN

43. The Policy & the Reinsurance Layer – OVER OTHER’S/reinsurance attaches on top of the underlying

44. Your exposure factor = ceded loss/gross loss Your ceded loss is the expected ground up loss at 2.5M less the expected ground up loss at 1.25M Your gross loss is the expected ground up loss at 2.5M less the expected loss at the 500K underlying Using LEVs the exposure factor = [LEV(2.5M) – LEV (1.25M)]/[LEV(2.5M) – LEV(0.5M)] Exposure Factor/OVER OTHER’S

45. Pull it all together In the “over their own” scenario; the exposure factor = [LEV(2.25M) – LEV(0.75M)]/[LEV(2.5M) – LEV(0.5M)] = (12,527 – 11,149)/(12,630 – 10,518) = 65% • LEV @ 500,000 = 10,518 • LEV @ 2,000,000 + 500,000 = 12,630 • LEV @ 750,000 = 11,149 • LEV @ 1,500,000 + 750,000 = 12,527

46. Pull it all together For this 1.5 x 0.75M layer the weighted exposure factor is 88%

47. Pull it all together • Gross Loss Ratio X Exposure Factor = Gross Loss X Ceded Loss Gross Premium Gross Loss = Ceded Loss = Burn Gross Premium = 55% X 88% = 49% IF Subject Premium = 10M the expected loss to the 1.5 x 0.75M layer is $490,000 Gross up for expense, profit, et al and you have your reinsurance ceded premium

48. Data Issues • Limits profile • Broad ranges • Limits profile separate from attachment point profile • Count based (to fix, multiply by the LEV at corresponding policy limit) • Premium from profile doesn’t reconcile well with historical or projected premium • Do you have all business units, companies? • PPA split limits (10/20 per person/per occ) • ILF table breakdown

49. Free Cover • In your experience rating no losses have trended into the highest portion of the layer you are pricing • Example: • You are pricing a 750x250K layer • Largest trended loss is 500K from ground up • Your experience burn will be the same for your 750x250K layer as for a 250x250K layer

50. Free Cover – cont’d • One solution: • Split the layer you are pricing (750x250K) into 2 pieces: • 250x250K • 500x500K • If deemed credible, use the experience burn as your selected burn for the lower part of your layer OR credibility weight your experience and exposure burns to derive your selected burn • Then use exposure burn relativities to derive the burn for the upper part of your layer • Sum the selected burns for the two parts of the layer you are pricing to derive the full layer selected burn