Modeling multiple runoff tables

Transcription

1 Modeling multiple runoff tables Vincent Lous

2 Motivation Why creating a "model" for all lines of business is important

3 Motivation Why creating a "model" for all lines of business is important Predictions are based on more observations

4 Motivation Why creating a "model" for all lines of business is important Predictions are based on more observations Risk margins and capital requirements can be determined more accurately taking diversification into account

5 Outline Practice

6 Outline Practice Theory

7 Outline Practice Theory Example

8 Practice

9 Two step approach

10 Two step approach Step 1 Estimate the parameters of line of business 1 line of business 2. line of business R

11 Two step approach Step 1 Estimate the parameters of Step 2 line of business 1. line of business 2 Estimate the dependencies between different lines of business line of business R

12 Theory

13 Terminology Let Y r (r = 1,..., R) denote all (known and unknown) cells of a runoff table of LOB r in a portfolio. Each LOB can be modeled by: Y r N (µ r, Σ r )

14 Terminology Let Y r (r = 1,..., R) denote all (known and unknown) cells of a runoff table of LOB r in a portfolio. Each LOB can be modeled by: Y r N (µ r, Σ r ) The term runoff table can mean: incremental paid incremental incurred incremental paid and incurred

15 Problem The goal is to find mean µ and covariance Σ of: Y = (Y 1,..., Y R ) N (µ, Σ)

16 Problem The goal is to find mean µ and covariance Σ of: Y = (Y 1,..., Y R ) N (µ, Σ) µ = (µ 1,..., µ R )

17 Problem The goal is to find mean µ and covariance Σ of: Y = (Y 1,..., Y R ) N (µ, Σ) µ = (µ 1,..., µ R ) Σ 1... Σ 1R Σ =..... Σ R1... Σ R

18 Solution

19 Solution For each Σ r there exists an unique symmetric matrix B r such that B r B r = Σ r B r = JD 1 2 J eigendecomposition

20 Solution For each Σ r there exists an unique symmetric matrix B r such that B r B r = Σ r B r = JD 1 2 J eigendecomposition Cross covariance matrix Σ rr = ρ rr B r B r

21 Solution For each Σ r there exists an unique symmetric matrix B r such that B r B r = Σ r B r = JD 1 2 J Cross covariance matrix eigendecomposition Σ rr = ρ rr B r B r ρ rr is an element of the R R correlation matrix (symmetric, positive definite).

22 Solution Σ 1... ρ 1R B 1 B R Σ =..... ρ 1R B R B 1... Σ R

23 Properties The proposed method for modeling a portfolio

24 Properties The proposed method for modeling a portfolio follows a bottom-up approach:

25 Properties The proposed method for modeling a portfolio follows a bottom-up approach: use knowledge about each LOB in modeling

26 Properties The proposed method for modeling a portfolio follows a bottom-up approach: use knowledge about each LOB in modeling allows for determination of the impact of the dependence on e.g. risk margins

27 Properties The proposed method for modeling a portfolio follows a bottom-up approach: use knowledge about each LOB in modeling allows for determination of the impact of the dependence on e.g. risk margins limits the number of parameters to estimate

28 Properties The proposed method for modeling a portfolio follows a bottom-up approach: use knowledge about each LOB in modeling allows for determination of the impact of the dependence on e.g. risk margins limits the number of parameters to estimate is generic for all covariance matrices

29 Properties The proposed method for modeling a portfolio follows a bottom-up approach: use knowledge about each LOB in modeling allows for determination of the impact of the dependence on e.g. risk margins limits the number of parameters to estimate is generic for all covariance matrices marginal distribution for each LOB does not change

30 Properties The proposed method for modeling a portfolio follows a bottom-up approach: use knowledge about each LOB in modeling allows for determination of the impact of the dependence on e.g. risk margins limits the number of parameters to estimate is generic for all covariance matrices marginal distribution for each LOB does not change gives a Gaussian predictive distribution of (any aggregation of) the unknown elements of Y conditional on (any aggregation of) the known elements

31 Example

32 Example Our portfolio: Island Insurance commercial auto personal auto Agway Insurance commercial auto

33 Example Our portfolio: Island Insurance commercial auto personal auto Agway Insurance commercial auto loss periods: obtained from schedule P NAIC dataset

34 Step 1: Paid-incurred loss reserving model Per line of business

35 Step 1: Paid-incurred loss reserving model Per line of business Y r N (µ r, Σ r ) is an incremental paid-incurred runoff table

36 Step 1: Paid-incurred loss reserving model Per line of business Y r N (µ r, Σ r ) is an incremental paid-incurred runoff table conditions on paid = incurred for each loss period

37 Step 1: Paid-incurred loss reserving model Per line of business Y r N (µ r, Σ r ) is an incremental paid-incurred runoff table conditions on paid = incurred for each loss period mean and covariance are functions of:

38 Step 1: Paid-incurred loss reserving model Per line of business Y r N (µ r, Σ r ) is an incremental paid-incurred runoff table conditions on paid = incurred for each loss period mean and covariance are functions of: given exposure for each loss period (also future)

39 Step 1: Paid-incurred loss reserving model Per line of business Y r N (µ r, Σ r ) is an incremental paid-incurred runoff table conditions on paid = incurred for each loss period mean and covariance are functions of: given exposure for each loss period (also future) ultimate loss ratio for each loss period

40 Step 1: Paid-incurred loss reserving model Per line of business Y r N (µ r, Σ r ) is an incremental paid-incurred runoff table conditions on paid = incurred for each loss period mean and covariance are functions of: given exposure for each loss period (also future) ultimate loss ratio for each loss period development fraction for each development period

41 Step 1: Estimation results Ultimate loss ratio Incremental development curves Figure : Island Insurance commercial auto

42 Step 2: Estimation results The estimated ρ r,r s: Island Commercial Island Personal Agway Commercial Island Commercial Island Personal Agway Commercial

43 Realizations versus predictions ultimate losses (predicted in january 1998 versus realized)

44 Realizations versus predictions ultimate losses (predicted in january 1998 versus realized)

45 Realizations versus predictions ultimate losses (predicted in january 1998 versus realized)

46 Realizations versus predictions ultimate losses (predicted in january 1998 versus realized)

47 Backtesting

48 Backtesting 1. cut off the last diagonal of the different triangles

49 Backtesting 1. cut off the last diagonal of the different triangles 2. repeat estimation process on these old triangles

50 Backtesting 1. cut off the last diagonal of the different triangles 2. repeat estimation process on these old triangles 3. compare the predictions with the realizations

51 Backtesting 1. cut off the last diagonal of the different triangles 2. repeat estimation process on these old triangles 3. compare the predictions with the realizations The estimated ρ r,r s: Island Commercial Island Personal Agway Commercial Island Commercial Island Personal Agway Commercial

52 Backtesting

53 Backtesting observation expected value 80th percentile loss upon period

54 Should Island Insurance acquire Agway s commercial auto LOB?

55 Should Island Insurance acquire Agway s commercial auto LOB? compare the increase in premium with the increase in fair value

56 Should Island Insurance acquire Agway s commercial auto LOB? compare the increase in premium with the increase in fair value next year s premium is the same as this year s premium for each LOB

57 Should Island Insurance acquire Agway s commercial auto LOB? compare the increase in premium with the increase in fair value next year s premium is the same as this year s premium for each LOB premium received for 1998 without Agway with Agway increase 36,520 53,040 16,520

58 Should Island Insurance acquire Agway s commercial auto LOB? compare the increase in premium with the increase in fair value next year s premium is the same as this year s premium for each LOB premium received for 1998 without Agway with Agway increase 36,520 53,040 16,520 fair value based on Cost of Capital - method (6%) (1998) without Agway with Agway increase 26,223 35,620 9,397

59 Impact of merging the two portfolios change in fair value per LOB per loss period

60 Impact of merging the two portfolios change in fair value per LOB per loss period

61 Impact of merging the two portfolios change in fair value per LOB per loss period

62 Conclusion

63 Conclusion A two step approach to modeling multiple lines of business

64 Conclusion A two step approach to modeling multiple lines of business where the dependencies between different lines of business are modeled with: Σ rr = ρ rr B r B r

65 Conclusion A two step approach to modeling multiple lines of business where the dependencies between different lines of business are modeled with: Σ rr = ρ rr B r B r The paid-incurred and the portfolio models are implemented in the IFM loss reserving software.

66 Conclusion A two step approach to modeling multiple lines of business where the dependencies between different lines of business are modeled with: Σ rr = ρ rr B r B r The paid-incurred and the portfolio models are implemented in the IFM loss reserving software.