1. 1 Analyzing Loss Sensitive Treaty Terms 2001 CARe Seminar g ERC ® Jeff Dollinger, FCAS, Employers Reinsurance Co Kari Mrazek, FCAS, GE Reinsurance Co

2. 2 Introduction to Loss Sharing Provision Definition: A reinsurance contract provision that varies the ceded premium, loss, or commission based upon the loss experience of the contract Purpose: Client shares in ceded experience & could be incented to care more about the reinsurer’s results Typical Loss Sharing Provisions –Profit Commission –Sliding Scale Commission –Loss Ratio Corridors –Annual Aggregate Deductibles –Swing Rated Premiums –Reinstatements g

3. 3 Simple Profit Commission Example A property prorata contract has the following profit commission terms –50% Profit Commission after a reinsurer’s margin of 10%. –Key Point: Reinsurer returns 50% of the contractually defined “profit” to the cedant –Profit Commission Paid to Cedant = 50% x (Premium - Loss - Commission - Reinsurers Margin) –If profit is negative, reinsurers do not get any additional money from the cedant. g

4. 4 Simple Profit Commission Example Ceding Commission = 30% Loss ratio must be less than 60% for us to pay a profit commission Contract Expected Loss Ratio = 70% $1 Prem - $0.7 Loss - $0.3 Comm - $0.10 Reins Margin = minus $0.10 Is the expected cost of profit commission zero? g

5. 5 Simple Profit Commission Example Answer: The expected cost of profit commission is not zero Why: Because 70% is the expected loss ratio. –There is a probability distribution of potential outcomes around that 70% expected loss ratio. –It is possible (and may even be likely) that the loss ratio in any year could be less than 60%. g

6. 6 Cost of Profit Commission: Simple Quantification Earthquake exposed California property prorata treaty LR = 40% in all years with no EQ Profit Comm when there is no EQ = 50% x ($1 of Premium - $0.4 Loss - $0.30 Commission - $0.1 Reinsurers Margin) = 10% of premium Cat Loss Ratio = 30%. –10% chance of an EQ costing 300% of premium, 90% chance no EQ loss Cost of Profit Comm = Profit Comm Costs 10% of Prem x 90% Probability of No EQ + 0% Cost of PC x 10% Probability of EQ Occurring = 9% of Prem g

7. 7 Basic Mechanics of Analyzing Loss Sharing Provisions Build aggregate loss distribution Apply loss sharing terms to each point on the loss distribution or to each simulated year Calculate a probability weighted average cost (or saving) of the loss sharing arrangement g

8. 8 Example of Basic Mechanics: PC: 50% after 10%, 30% Commission, 65% Expected LR g

9. 9 Determining an Aggregate Distribution - 2 Methods Fit statistical distribution to on level loss ratios –Reasonable for prorata treaties. Determine an aggregate distribution by modeling frequency and severity –Typically used for excess of loss treaties. g

10. 10 Fitting a Distribution to On Level Loss Ratios Most actuaries use the lognormal distribution –Reflects skewed distribution of loss ratios –Easy to use Lognormal distribution assumes that the natural logs of the loss ratios are distributed normally. g

11. 11 Skewness of Lognormal Distribution g

12. 12 Fitting a Lognormal Distribution to Projected Loss Ratios Fitting the lognormal  ^2 = LN(CV^2 + 1)  = LN(mean) -  ^2/2 Mean = Selected Expected Loss Ratio CV = Standard Deviation over the Mean of the loss ratio (LR) distribution. Prob (LR  X) = Normal Dist(( LN(x) -  )/  ) i.e.. look up (LN(x) -  )/  ) on a standard normal distribution table g

13. 13 Fitting a Lognormal Loss Ratio Distribution Producing a distribution of loss ratios –For a given point i on the CDF, the following Excel command will produce a loss ratio at that CDFi: Exp (  + Normsinv(CDFi) x  ) Key Question: Is the resulting LR distribution reasonable –Analyze that issue by reviewing historical data –Discuss this issue with your underwriter –If the distribution is not reasonable, adjust the CV selection. g

14. 14 Sample Lognormal Loss Ratio Distribution g

15. 15 Additional Considerations for Fitting a Lognormal Loss Ratio Distribution Selected CV should usually be above indicated –5 to 10 years of data does not reflect full range of possibilities Parameter Uncertainty: Do you really know the true mean of the loss ratio distribution for the upcoming year? g

16. 16 Modeling Parameter Uncertainty: One Possible Method Select 3 equally likely expected loss ratios Assign weight to each loss ratio so that the weighted average ties to your selected expected loss ratio –Example: Expected LR is 65%, assume 1/3 probability that true mean LR is 60%, 1/3 probability that it is 65%, and 1/3 probability that it is 70%. –Simulate the “true” expected loss ratio Simulate the loss ratio for the year modeled using the lognormal based on simulated expected loss ratio & your selected CV g

17. 17 Example of Modeling Parameter Uncertainty g

18. 18 Common Loss Sharing Provisions for Prorata Treaties Profit Commissions –Already covered Sliding Scale Commission Loss Ratio Corridor Loss Ratio Cap g

19. 19 Sliding Scale Comm Commission initially set at Provisional amount Ceding commission increases if loss ratios are lower than expected Ceding commission decreases if losses are higher than expected g

20. 20 Sliding Scale Commission Example Provisional Commission: 30% If the loss ratio is less than 65%, then the commission increases by 1 point for each point decrease in loss ratio up to a maximum commission of 35% at a 60% loss ratio If the loss ratio is greater than 65%, the commission decreases by 0.5 for each 1 point increase in LR down to a minimum comm of 25% at a 75% loss ratio g

21. 21 Sliding Scale Commission - Solution g

22. 22 Loss Ratio Corridors A loss ratio corridor is a provision that forces the ceding company to retain losses that would be otherwise ceded to the reinsurance treaty Loss ratio corridor of 100% of the losses between a 75% and 85% LR –If gross LR equals 75%, then ceded LR is 75% –If gross LR equals 80%, then ceded LR is 75% –If gross LR equals 85%, then ceded LR is 75% –If gross LR equals 100%, then ceded LR is ??? g

23. 23 Loss Ratio Cap This is the maximum loss ratio that could be ceded to the treaty. Example: 200% Loss Ratio Cap –If LR before cap is 150%, then ceded LR is 150% –If LR before cap is 250%, then ceded LR is 200% g

24. 24 Loss Ratio Corridor Example Reinsurance treaty has a loss ratio corridor of 50% of the losses between a loss ratio of 70% and 80%. Use the aggregate distribution to your right to estimate the ceded LR net of the corridor g

25. 25 Loss Ratio Corridor Example - Solution g

26. 26 Modeling Property Treaties with Significant Cat Exposure Model noncat & cat LR’s separately –Non Cat LR’s fit to a lognormal curve –Cat LR distribution produced by commercial catastrophe model Combine (convolute) the noncat & cat loss ratio distributions g

27. 27 Convoluting Noncat & Cat LR’s - Example g

28. 28 Truncated Loss Ratio Distributions Problem: To reasonably model the possibility of high LR requires a high lognormal CV High lognormal CV often leads to unrealistically high probabilities of low LR’s, which overstates cost of PC Solution: Don’t allow LR to go below selected minimum, e.g.. 0% probability of LR<30% –Adjust lognormal mean so that aggregate distribution will probability weight back to initial expected LR g

29. 29 Summary of Loss Ratio Distribution Method Advantage: –Easier and quicker than separately modeling frequency and severity –Reasonable for most prorata treaties Usually inappropriate for excess of loss contracts –Continuous distribution around the mean does not reflect the hit or miss nature of many excess of loss contracts –Understates probability of zero loss –Understates potential of losses much greater than the expected loss g

30. 30 Excess of Loss Contracts: Separate Modeling of Frequency and Severity Used mainly for modeling excess of loss contracts A detailed mathematical explanation is beyond the scope of this session Software that can be Used to do the above modeling –Crystal Ball –Crimcalc –@Risk –Excel Most aggregate distribution approaches assume that frequency and severity are independent g

31. 31 Common Frequency Distributions Poisson f(x| ) = exp(- ) ^x / x! where = mean of the claim count distribution and x = claim count = 0,1,2,... g

32. 32 Fitting a Poisson Claim Count Distribution Estimate ultimate claim counts by year Multiply ultimate claim counts by frequency trend factor to bring them to the frequency level of the upcoming treaty year Adjust for change in exposure levels, ie. Adjusted Claim Count year i = Trended Ultimate Claim Count i x (SPI for upcoming treaty year / On Level SPI year i) Poisson parameter equals the mean of the ultimate, trended, adjusted claim counts from above g

33. 33 Example of Simulated Claim Count g

34. 34 Modeling Frequency- Negative Binomial Negative Binomial: Same form as the poisson distribution, except that it assumes that is not fixed, but rather has a gamma distribution around the selected –Claim count distribution is negative binomial if the variance of the count distribution is greater than the mean –The gamma distribution around has a mean of 1 Negative Binomial is the preferred distribution –Reflects parameter uncertainty regarding the true mean claim count –The extra variability of the Negative Binomial is more in line with historical experience g

35. 35 Algorithm for Simulating Claim Counts Poisson –Manually create a poisson cumulative distribution table –Simulate the CDF (a number between 0 and 1) and lookup the number of claims corresponding to that CDF. This is your simulated claim count for year 1 –Repeat the above two steps for however many years that you want to simulate g

36. 36 Additional Steps for Simulating Claim Counts using Negative Binomial Determine contagion parameter, c, of claim count distribution: (  ^2 /  ) = 1 + c  If the claim count distribution is poison, then c=0 If it is negative binomial, then c>0 Solve for the contagion parameter: c = [(  ^2 /  ) - 1] /  g

37. 37 Additional Steps for Simulating Claim Counts using Negative Binomial Simulate gamma random variable with a mean of 1 –Gamma distribution has two parameters:  and   = 1/c;  = c –Using Excel, simulate gamma random variable as follows Gammainv(Simulated CDF, ,  ) –Simulated Poisson parameter = = x Simulated Gamma Random Variable Above –Use the poisson distribution algorithm using the above simulated poisson parameter,, to simulate the claim count for the year g

38. 38 Year 1 Simulated Negative Binomial Claim Count g

39. 39 Year 1 Simulated Negative Binomial Claim Count g

40. 40 Year 2 Simulated Negative Binomial Claim Count g

41. 41 Year 2 Simulated Negative Binomial Claim Count g

42. 42 Modeling Severity Common Severity Distributions –Lognormal –Pareto –Mixed Exponential (currently used by ISO) –Truncated Pareto. This curve was used by ISO before moving to the Mixed Exponential and will be the focus of this presentation. Key Point: The ISO Truncated Pareto focused on modeling the larger claims. Typically those over $50,000 g

43. 43 Truncated Pareto Truncated Pareto Parameters t = truncation point. s = average claim size of losses below truncation point p = probability claims are smaller than truncation point b = pareto scale parameter q = pareto shape parameter Cumulative Distribution Function F(x) = 1 - (1-p) ((t+ b)/(x+ b))^q Where x>t g

44. 44 Algorithm for Simulating Severity to the Layer For each loss to be simulated, choose a random number between 0 and 1. This is the simulated CDF Transformed CDF for losses hitting layer (TCDF) = Prob(Loss<Reins Att Pt) + Simulated CDF x (1 - Prob(Loss<Reins Att Pt)) Find simulated ground up loss, x, that corresponds to simulated TCDF Doing some algebra, find x using the following formula: x = Exp{ln(t+b) - [ln(1-TCDF) - ln(1-p)]/Q} - b From simulated ground up loss calculate loss to the layer g

45. 45 Year 1 Loss # 1 Simulated Severity to the Layer g

46. 46 Year 1 Loss # 2 Simulated Severity to the Layer g

47. 47 Simulation Summary g

48. 48 Common Loss Sharing Provisions for Excess of Loss Treaties Profit Commissions –Already covered Swing Rated Premium Annual Aggregate Deductibles Limited Reinstatements g

49. 49 Swing Rated Premium Ceded premium is dependent on loss experience Typical Swing Rating Terms –Provisional Rate: 10% of subject premium –Ceded premium is adjusted to equal ceded loss times 100/80 loading factor, subject to a minimum rate of 5% and a maximum rate of 15% g

50. 50 Swing Rated Premium - Example Burn (ceded loss / SPI) = 10%. Rate = 10% x 100/80 = 12.5% Burn = 2%. Calculated Rate = 2% x 100/80 = 2.5%. Rate = 5% minimum rate Burn = 14%. Calculated Rate = 14% x 100/80 = 17.5%. Rate = 15% maximum rate g

51. 51 Swing Rated Premium Example Swing Rating Terms: Ceded premium is adjusted to equal ceded loss times 100/80 loading factor, subject to a minimum rate of 5% and a maximum rate of 15% Use the aggregate distribution to your right to calculate the ceded loss ratio under the treaty g

52. 52 Swing Rated Premium Example - Solution g

53. 53 Annual Aggregate Deductible The annual aggregate deductible (AAD) refers to a retention by the cedant of losses that would be otherwise ceded to the treaty Example: Reinsurer provides a $500,000 xs $500,000 excess of loss contract. Cedant retains an AAD of $750,000 –Total Loss to Layer = $500,000. Cedant retains all $500,000. No loss ceded to reinsurers –Total Loss to Layer = $1 mil. Cedant retains $750,000. Reinsurer pays $250,000. –Total Loss to Layer =$1.5 mil. Cedant retains? Reinsurer pays? g

54. 54 Annual Aggregate Deductible Discussion Question: Reinsurer writes a $500,000 xs $500,000 excess of loss treaty. –Expected Loss to the Layer is $1 million (before AAD) –Cedant retains a $500,000 annual aggregate deductible. –Cedant says, “I assume that you will decrease your expected loss by $500,000.” –How do you respond? g

55. 55 Annual Aggregate Deductible Example Your expected burn to a $500K xs $500K reinsurance layer is 11.1%. Cedant adds an AAD of 5% of subject premium Using the aggregate distribution of burns to your right, calculate the burn net of the AAD. g

56. 56 Annual Aggregate Deductible Example - Solution g

57. 57 Limited Reinstatements Limited reinstatements refers to the number of times that the occurrence or risk limit of an excess can be reused. Example: $1 million xs $1 million layer –1 reinstatement: It means that after the cedant uses up the first limit, they also get a second occurrence limit Treaty Aggregate Limit = = Occurrence Limit x (1 + number of Reinstatements) g

58. 58 Limited Reinstatements Example g

59. 59 Reinstatement Premium In many cases to “reinstate” the limit, the cedant is required to pay an additional premium Choosing to reinstate the limit is almost always mandatory –Reinstatement premium can simply be viewed as additional premium that reinsurers receive depending on loss experience g

60. 60 Reinstatement Premium Example 1 g

61. 61 Reinstatement Premium Example 2 g

62. 62 Reinstatement Example 3 Reinsurance Treaty: $1 mil xs $1 mil Upfront Prem = 400K 2 Reinstatements: 1st at 50%, 2nd at 100% Using the aggregate distribution to the right, calculate our expected ultimate loss, premium, and loss ratio g

63. 63 Reinstatement Example 3 - Solution g

64. 64 Reinstatement Example 4 Note: Reinstatement provisions are typically found on high excess layers, where loss tends to be either 0 or a full limit loss. Assume: Layer = 10M xs 10M, Expected Loss = 1M, Poisson Frequency with mean =.1 g

65. 65 Deficit Carryforward Treaty terms may include Deficit Carryforward Provisions, in which some losses are carried forward to next year’s contract in determining the commission paid. Example: - Provisional Commission: 30% - Min Comm 25% at 75% LR - Sliding.5-to-1 to a 30% comm at a 65% LR - Sliding 1-to-1 to a max comm of 35% at 60% LR g

66. 66 Deficit Carryforward Example Option 1 - Add Deficit Carryforward % to each simulated LR and recalculate average commission. g

67. 67 Deficit Carryforward Example Option 2 - Shift Sliding Scale Commission terms. g

68. 68 DCF/Multi-Year Block Question: How much credit do you give an account for Deficit Carryforwards, other than using the CF from the previous year (e.g. unlimited CFs)? Can estimate using an average of simulated “years”, but this method should be used with caution: – Assumes independence (probably unrealistic) – Accounts for both Deficit and Credit carryforwards – Deficits are often forgiven, treaty terms may change, or treaty may be terminated before the benefit of the deficit carryforward is felt by the reinsurer. g

69. 69 DCF/Multi-Year Block - Example g

70. 70 Loss Sharing Summary Modeling loss sharing provisions is easy. Selecting your expected loss and aggregate distribution is hard Steps to analyzing loss sharing provisions –Build aggregate loss distribution –Apply loss sharing terms to each point on the loss distribution or to each simulated year –Calculate probability weighted average of treaty results g

71. 71 Additional Issues & Uses of Aggregate Distributions Correlation between lines of business Aggregate distributions are just a guess Reserving for loss sensitive treaty terms Using aggregate distributions to measure risk & allocate capital Capital = 99th percentile Discounted Loss x Correlation Factor Fitting Severity Curves: Don’t Ignore Loss Development –Increases average severity –Increases variance. –See “Survey of Methods Used to Reflect Development in Excess Ratemaking” by Stephen Philbrick, CAS 1996 Winter Forum g