1. Transition Matrix Theory and Loss Development John B. Mahon CARe Meeting June 6, 2005 Instrat

2. 2 Ovewview Transition Matrix Theory TMT Applied to GC data Distributional Model of Loss Development Independence and variation Effects on Expected Values and Increased Limits Factors

3. 3 Instrat

4. 4

5. 5 Markov Chains Andrey Markov 1856 – 1922 Best known for work on stochastic process theory Markov Chains Two essential parts – Initial distribution – Transition matrix Need to define the array of available states Transition matrix has two dimensions – Both are defined as the available states Use standard matrix multiplication Markov Property – Any information dating from before the last step for which the state of the process is known is irrelevant in predicting its state at a later step.

6. 6 Instrat

7. 7

8. 8

9. 9

10. 10 Instrat

11. 11 Instrat

12. 12 Instrat

13. 13 Instrat

14. 14 Data Source Claims database of large reinsurance intermediary DB prepared manually from submission for recovery All values entered ground-up at 100% Isolated claims coded GL Eliminated claims based on loss cause description – In order to eliminate “non-standard” losses Losses trended with Bests/Masterson’s GL BI trend

15. 15 Instrat

16. 16 Instrat

17. 17 Instrat

18. 18 Instrat

19. 19 Instrat

20. 20 Instrat

21. 21 Instrat

22. 22 Instrat

23. 23 Instrat

24. 24 Instrat

25. 25 Instrat

26. 26 Instrat

27. 27 Instrat

28. 28 Instrat

29. 29 Formula for estimating mu mu= 1.005 * ln(x), Ingnoring the class 001 data

30. 30 Instrat

31. 31 Instrat

32. 32 Instrat

33. 33 Instrat

34. 34 Instrat

35. 35 Function to forecast sigma values Sigma = 1/(maturity *0.001205+ln(loss size)*0.078874-0.34447

36. 36 Instrat

37. 37 Instrat

38. 38 Instrat

39. 39 Instrat

40. 40 Instrat

41. 41 Comparison of Transition Matrix results to Direct Transitions Transition Matrix method assumes independence Real Life claims are not independent – More mature transitions depend on earlier transitions TM has no memory as to where it came from, real claims do The TM method may introduce excessive variation

42. 42 Comparison of Transition Matrix results to Direct Transitions Create initial to “final” transitions to compare with TM results Initial is the first evaluation of a claim “Final” is the latest evaluation of a claim – Assumed to be closed Prepared a series of transition matrices representing various initial maturities. Data initially as counts Division by the initial total produces probability

43. 43 Instrat

44. 44 Instrat

45. 45 Instrat

46. 46 Instrat

47. 47 Instrat

48. 48 Instrat

49. 49 Instrat

50. 50 Instrat

51. 51 Instrat Test the relationship between the TM sigmas and the emperical sigmas Take ratio emperical sigma / TM sigma Select the average of the ratio as an adjustment to TM sigmas

52. 52 Instrat

53. 53 Instrat

54. 54 Instrat

55. 55 Instrat

56. 56 Instrat Final Test – Compare fitted adjusted sigma surface to emperical sigma surface Difference is TM minus the emperical Take difference and plot

57. 57 Instrat

58. 58 Instrat

59. 59 Instrat Formulation of a working model Open claims are a distribution at ultimate The lognormal distribution is a good model for this distribution It is possible to model ultimate losses with four parameters and two variables Transition matrix introduces additional variation due to its independent nature This additional variation can be removed by simple adjustment

60. 60 Effect of Distributional Loss Development Devise a method to measure the effect of distributional loss development Synthesize a set of losses Lognormal distribution Mu =13, sigma =1 Used stratified sampling

61. 61 Simulated losses

62. 62 Simulated losses

63. 63 Application of distributional development

64. 64 Original Losses Emperical Cumulative Probability

65. 65 LDF Losses Emperical Cumulative Probability

66. 66 Distributional Loss Development Emperical Cumulative Probability

67. 67 Cumulative probability of simulated data

68. 68 Cumulative probability of simulated data

69. 69 Cumulative Distributions Can enter into fitting routines to get parametized distributions Can use directly for calculating limited expected values and ILF’s – Limited to the largest size of loss Can use directly for simulation – Lacks ability to estimate parameter uncertainty

70. 70 Limited Expected Value of simulated losses

71. 71 Limited Expected Value of simulated losses

72. 72 Limited Expected Value of simulated losses

73. 73 Effects on limited average severity LDF is higher at first Distributional method rises to meet LDF values Distributional and unadjusted data are similar for 0 - $1M range

74. 74 Increased Limits Factors Basic Limits =$100,000

75. 75 Increased Limits Factors Basic Limits =$100,000

76. 76 Increased Limits Factors Basic Limits =$100,000

77. 77 Increased Limits Factors Basic Limits =$1,000,000

78. 78 Increased Limits Factors Basic Limits =$1,000,000

79. 79 Effects on increased limits factors The results vary as the basic limit is changed For a $100,000 basic limit, the developed losses resemble each other and are 50% higher than unadjusted For a $1,000,000 basic limit the distributional adjusted is much higher than the LDF adjusted

80. 80 Conclusions Distributional Loss Development shifts the ultimate severity distribution differently than application of loss development factors This can result in differences in increased limits factors up to 30% Increase limits factors are sensitive to the selection of basic limits The differences shown here are upper limits – This study assumed all losses to be developed – In a real life collection of claims, the majority would be closed, and the minority would be developed Development errors could result in 10 – 15% errors