Various Extensions Based on Munich Chain Ladder Method

Transcription

1 Various Extensions Based on Munich Chain Ladder Method etr Jedlička Charles University, Department of Statistics 20th June 2007, 50th Anniversary ASTIN Colloquium etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

2 Context 1 Introduction 2 Robust Regression 3 Addition of MSE calculation to MCL model 4 Multivariate Extensions to Chain Ladder 5 Multivariate MCL 6 Other Approaches to model aid and Incurred data etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

3 Introduction Scope of presentation 1 Introduction 2 Robust Regression 3 Addition of MSE calculation to MCL model 4 Multivariate Extensions to Chain Ladder 5 Multivariate MCL 6 Other Approaches to model aid and Incurred data etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

4 Introduction Munich Chain Ladder - Introduction Derived and presented by Munich Re its name (MCL) (see aper of Quarg 2004) Used Notation etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

5 Introduction Munich Chain Ladder - Introduction Derived and presented by Munich Re its name (MCL) (see aper of Quarg 2004) Analysis of both aid Y a Incurred Y I schemes Used Notation etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

6 Introduction Munich Chain Ladder - Introduction Derived and presented by Munich Re its name (MCL) (see aper of Quarg 2004) Analysis of both aid Y a Incurred Y I schemes Extension of model of Mack (SCL) Used Notation etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

7 Introduction Munich Chain Ladder - Introduction Derived and presented by Munich Re its name (MCL) (see aper of Quarg 2004) Analysis of both aid Y a Incurred Y I schemes Extension of model of Mack (SCL) Significant improvement: If Y0,n I Y 0,n MCL reduces gap between Y i,n I and Ŷ i,n for i 1 Used Notation etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

8 Introduction Munich Chain Ladder - Introduction Derived and presented by Munich Re its name (MCL) (see aper of Quarg 2004) Analysis of both aid Y a Incurred Y I schemes Extension of model of Mack (SCL) Significant improvement: If Y0,n I Y 0,n MCL reduces gap between Y i,n I and Ŷ i,n for i 1 It does not hold for SCL Used Notation etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

9 Introduction Munich Chain Ladder - Introduction Derived and presented by Munich Re its name (MCL) (see aper of Quarg 2004) Analysis of both aid Y a Incurred Y I schemes Extension of model of Mack (SCL) Significant improvement: If Y0,n I Y 0,n MCL reduces gap between Y i,n I and Ŷ i,n for i 1 It does not hold for SCL Used Notation a(i) = n i level of development i etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

10 Introduction Munich Chain Ladder - Introduction Derived and presented by Munich Re its name (MCL) (see aper of Quarg 2004) Analysis of both aid Y a Incurred Y I schemes Extension of model of Mack (SCL) Significant improvement: If Y0,n I Y 0,n MCL reduces gap between Y i,n I and Ŷ i,n for i 1 It does not hold for SCL Used Notation a(i) = n i level of development i Data of aid to Incurred Ratio Q (/I ) Y Y I i = 0,, n i + j n etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

11 Introduction Munich Chain Ladder - Introduction Derived and presented by Munich Re its name (MCL) (see aper of Quarg 2004) Analysis of both aid Y a Incurred Y I schemes Extension of model of Mack (SCL) Significant improvement: If Y0,n I Y 0,n MCL reduces gap between Y i,n I and Ŷ i,n for i 1 It does not hold for SCL Used Notation a(i) = n i level of development i Data of aid to Incurred Ratio Q (/I ) Y Y I i = 0,, n i + j n Joint available information B i (s) = (Y i (s) ; Y i (s) I ) etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

12 Introduction Munich Chain Ladder - aid to Incurred Ratio (I) Average I Estimate j /I j = n i=0 Y n i=0 Y I etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

13 Introduction Munich Chain Ladder - aid to Incurred Ratio (I) Average I Estimate j /I j = n i=0 Y n i=0 Y I If i + j > n I is defined as (/I ) = Ŷ Y I etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

14 Introduction Munich Chain Ladder - aid to Incurred Ratio (I) Average I Estimate j /I j = n i=0 Y n i=0 Y I If i + j > n I is defined as (/I ) = Ŷ It Holds true that /I /I j Y I = /I i,a(i) /I a (i) etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

15 Introduction Munich Chain Ladder - aid to Incurred Ratio (I) Average I Estimate j /I j = n i=0 Y n i=0 Y I If i + j > n I is defined as (/I ) = Ŷ It Holds true that See Quarg 2004 for proof /I /I j Y I = /I i,a(i) /I a (i) etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

16 Introduction Munich Chain Ladder - aid to Incurred Ratio (I) Average I Estimate j /I j = n i=0 Y n i=0 Y I If i + j > n I is defined as (/I ) = Ŷ It Holds true that See Quarg 2004 for proof /I /I j Y I = /I i,a(i) /I a (i) Interpretation explains drawback of SCL method: etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

17 Introduction Munich Chain Ladder - aid to Incurred Ratio (I) Average I Estimate j /I j = n i=0 Y n i=0 Y I If i + j > n I is defined as (/I ) = Ŷ It Holds true that See Quarg 2004 for proof /I /I j Y I = /I i,a(i) /I a (i) Interpretation explains drawback of SCL method: 1 Low (/I ) for known data in diagonal low (/I ) for prediction etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

18 Introduction Munich Chain Ladder - aid to Incurred Ratio (I) Average I Estimate j /I j = n i=0 Y n i=0 Y I If i + j > n I is defined as (/I ) = Ŷ It Holds true that See Quarg 2004 for proof /I /I j Y I = /I i,a(i) /I a (i) Interpretation explains drawback of SCL method: 1 Low (/I ) for known data in diagonal low (/I ) for prediction 2 Disparity between both projection etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

19 Introduction Munich Chain Ladder - aid to Incurred Ratio (I) Average I Estimate j /I j = n i=0 Y n i=0 Y I If i + j > n I is defined as (/I ) = Ŷ It Holds true that See Quarg 2004 for proof /I /I j Y I = /I i,a(i) /I a (i) Interpretation explains drawback of SCL method: 1 Low (/I ) for known data in diagonal low (/I ) for prediction 2 Disparity between both projection 3 Systematic weakness of SCL etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

20 Introduction Fundamental rinciples of MCL Adjustment of development factors according to (/I ) i,a(i) etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

21 Introduction Fundamental rinciples of MCL Adjustment of development factors according to (/I ) i,a(i) If I Ratio is below average for aid data etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

22 Introduction Fundamental rinciples of MCL Adjustment of development factors according to (/I ) i,a(i) If I Ratio is below average for aid data Low level of claim settlement etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

23 Introduction Fundamental rinciples of MCL Adjustment of development factors according to (/I ) i,a(i) If I Ratio is below average for aid data Low level of claim settlement Could be accelerated in future periods etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

24 Introduction Fundamental rinciples of MCL Adjustment of development factors according to (/I ) i,a(i) If I Ratio is below average for aid data Low level of claim settlement Could be accelerated in future periods Above average factor Y+1 /Y etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

25 Introduction Fundamental rinciples of MCL Adjustment of development factors according to (/I ) i,a(i) If I Ratio is below average for aid data Low level of claim settlement Could be accelerated in future periods Above average factor Y+1 /Y Good to increase standard estimate f j etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

26 Introduction Fundamental rinciples of MCL Adjustment of development factors according to (/I ) i,a(i) If I Ratio is below average for aid data Low level of claim settlement Could be accelerated in future periods Above average factor Y+1 /Y Good to increase standard estimate ( ) f j Y corr < 0 Y I ; Y +1 Y etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

27 Introduction Fundamental rinciples of MCL Adjustment of development factors according to (/I ) i,a(i) If I Ratio is below average for aid data Low level of claim settlement Could be accelerated in future periods Above average factor Y+1 /Y Good to increase standard estimate ( ) f j Y corr < 0 Y I ; Y +1 Y If I Ratio is below average for Incurred data etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

28 Introduction Fundamental rinciples of MCL Adjustment of development factors according to (/I ) i,a(i) If I Ratio is below average for aid data Low level of claim settlement Could be accelerated in future periods Above average factor Y+1 /Y Good to increase standard estimate ( ) f j Y corr < 0 Y I ; Y +1 Y If I Ratio is below average for Incurred data High level of Claims Reserving etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

29 Introduction Fundamental rinciples of MCL Adjustment of development factors according to (/I ) i,a(i) If I Ratio is below average for aid data Low level of claim settlement Could be accelerated in future periods Above average factor Y+1 /Y Good to increase standard estimate ( ) f j Y corr < 0 Y I ; Y +1 Y If I Ratio is below average for Incurred data High level of Claims Reserving Lower increase of incurred amount is expected etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

30 Introduction Fundamental rinciples of MCL Adjustment of development factors according to (/I ) i,a(i) If I Ratio is below average for aid data Low level of claim settlement Could be accelerated in future periods Above average factor Y+1 /Y Good to increase standard estimate ( ) f j Y corr < 0 Y I ; Y +1 Y If I Ratio is below average for Incurred data High level of Claims Reserving Lower increase of incurred amount is expected Below average factor Y+1 I /Y I etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

31 Introduction Fundamental rinciples of MCL Adjustment of development factors according to (/I ) i,a(i) If I Ratio is below average for aid data Low level of claim settlement Could be accelerated in future periods Above average factor Y+1 /Y Good to increase standard estimate ( ) f j Y corr < 0 Y I ; Y +1 Y If I Ratio is below average for Incurred data High level of Claims Reserving Lower increase of incurred amount is expected Below average factor Y I +1 /Y I Good to decrease standard estimate f I j etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

32 Introduction Fundamental rinciples of MCL Adjustment of development factors according to (/I ) i,a(i) If I Ratio is below average for aid data Low level of claim settlement Could be accelerated in future periods Above average factor Y+1 /Y Good to increase standard estimate ( ) f j Y corr < 0 Y I ; Y +1 Y If I Ratio is below average for Incurred data High level of Claims Reserving Lower increase of incurred amount is expected Below average factor Y+1 I /Y I Good to decrease standard estimate f ( ) j I Y corr > 0 Y I ; Y I +1 Y I etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

33 Introduction Regression Models of MCL method - aid data Variables are standardised conditional residuals Res(X C) = X E(X C) σ(x C) etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

34 Introduction Regression Models of MCL method - aid data Variables are standardised conditional residuals Res(X C) = X E(X C) σ(x C) Dependency structure aid data MCL assumption ( ( ) ) Y i,s+1 E Res Yi,s Y i (s) B i (s) = λ Res(Q 1 i,s Y i(s) ) etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

35 Introduction Regression Models of MCL method - aid data Variables are standardised conditional residuals Res(X C) = X E(X C) σ(x C) Dependency structure aid data MCL assumption ( ( ) ) Y i,s+1 E Res Yi,s Y i (s) B i (s) = λ Res(Q 1 i,s Y i(s) ) Could be transformed onto ( ) Y σ i,s+1 E B i (s) = fs +λ Y i,s ( Y i,s+1 Y i,s Y i (s) ) σ(q 1 i,s Y i(s) ) (Q 1 i,s E(Q 1 i,s Y i(s) )) etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

36 Introduction Regression Models of MCL method - Incurred Data Analogous as for aid etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

37 Introduction Regression Models of MCL method - Incurred Data Analogous as for aid ( ( ) ) Y I i,s+1 E Res Yi,s I Y i (s) I B i (s) = λ I Res(Q i,s Y i (s) I ) etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

38 Introduction Regression Models of MCL method - Incurred Data Analogous as for aid ( ( ) ) Y I i,s+1 E Res Yi,s I Y i (s) I B i (s) = λ I Res(Q i,s Y i (s) I ) Transformation ( ) Y I i,s+1 E B i (s) = f I Y i,s I ( Y σ I s + λ I i,s+1 Y I i,s Y i (s) I ) σ(q i,s Y i (s) I ) (Q i,s E(Q i,s Y i (s) I )) etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

39 Introduction Regression Models of MCL method - Incurred Data Analogous as for aid ( ( ) ) Y I i,s+1 E Res Yi,s I Y i (s) I B i (s) = λ I Res(Q i,s Y i (s) I ) Transformation ( ) Y I i,s+1 E B i (s) = f I Y i,s I ( Y σ I s + λ I i,s+1 Y I i,s Y i (s) I ) σ(q i,s Y i (s) I ) (Q i,s E(Q i,s Y i (s) I )) Note - differences between models Q is explanatory variable at Incurred Model Q 1 is explanatory variable at aid Model in rational cases should be λ > 0 and λ I > 0 etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

40 Introduction Implementation of Regressions Originally traditional OLS method etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

41 Introduction Implementation of Regressions Originally traditional OLS method Explanatory ower of the model rather weak (especially for Incurred model) etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

42 Introduction Implementation of Regressions Originally traditional OLS method Explanatory ower of the model rather weak (especially for Incurred model) Interpretation of causality relation between aid and Incurred? etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

43 Introduction Implementation of Regressions Originally traditional OLS method Explanatory ower of the model rather weak (especially for Incurred model) Interpretation of causality relation between aid and Incurred? The Best achieved results by standard approach not so appropriate etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

44 Robust Regression Scope of presentation 1 Introduction 2 Robust Regression 3 Addition of MSE calculation to MCL model 4 Multivariate Extensions to Chain Ladder 5 Multivariate MCL 6 Other Approaches to model aid and Incurred data etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

45 Robust Regression Application of Robust Regression Detection of outliers of the model etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

46 Robust Regression Application of Robust Regression Detection of outliers of the model Various available method performed etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

47 Robust Regression Application of Robust Regression Detection of outliers of the model Various available method performed For example Huber, Bi square, etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

48 Robust Regression Application of Robust Regression Detection of outliers of the model Various available method performed For example Huber, Bi square, Lower weight given to outlying observation etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

49 Robust Regression Application of Robust Regression Detection of outliers of the model Various available method performed For example Huber, Bi square, Lower weight given to outlying observation Least Trimmed squares etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

50 Robust Regression Application of Robust Regression Detection of outliers of the model Various available method performed For example Huber, Bi square, Lower weight given to outlying observation Least Trimmed squares Selected portion of outliers is directly cut off the model etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

51 Robust Regression Application of Robust Regression Detection of outliers of the model Various available method performed For example Huber, Bi square, Lower weight given to outlying observation Least Trimmed squares Selected portion of outliers is directly cut off the model Outliers have strong influence onto model etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

52 Robust Regression Least Trimmed Squares LTS estimator ˆβ LTS = arg min β R p+1 h i=1 r 2 [i] (β) etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

53 Robust Regression Least Trimmed Squares LTS estimator ˆβ LTS = arg min β R p+1 h i=1 r 2 [i] (β) r 2 [i] (β) represents i-th smallest value among r 2 1 (β),..., r 2 n (β) etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

54 Robust Regression Least Trimmed Squares LTS estimator ˆβ LTS = arg min β R p+1 h i=1 r 2 [i] (β) r 2 [i] (β) represents i-th smallest value among r 2 1 (β),..., r 2 n (β) r i (β) = y i x i β OLS residuals etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

55 Robust Regression Least Trimmed Squares LTS estimator ˆβ LTS = arg min β R p+1 h i=1 r 2 [i] (β) r 2 [i] (β) represents i-th smallest value among r 2 1 (β),..., r 2 n (β) r i (β) = y i x i β OLS residuals trimming constant h etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

56 Robust Regression Least Trimmed Squares LTS estimator ˆβ LTS = arg min β R p+1 h i=1 r 2 [i] (β) r 2 [i] (β) represents i-th smallest value among r 2 1 (β),..., r 2 n (β) r i (β) = y i x i β OLS residuals trimming constant h n 2 < h n etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

57 Robust Regression Least Trimmed Squares LTS estimator ˆβ LTS = arg min β R p+1 h i=1 r 2 [i] (β) r 2 [i] (β) represents i-th smallest value among r 2 1 (β),..., r 2 n (β) r i (β) = y i x i β OLS residuals trimming constant h n 2 < h n Our choices h = 0.6 n and h = 0.75 n etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

58 Robust Regression Least Trimmed Squares LTS estimator ˆβ LTS = arg min β R p+1 h i=1 r 2 [i] (β) r 2 [i] (β) represents i-th smallest value among r 2 1 (β),..., r 2 n (β) r i (β) = y i x i β OLS residuals trimming constant h n 2 < h n Our choices h = 0.6 n and h = 0.75 n Computational algorithm of LTS 1 Randomly select h observation and perform OLS regression for them 2 Compute OLS residuals based on the model for all data and choose h with smallest absolute values of residuals 3 For newly selected h observation compute OLS regression again. Did RSS for selected mode decrease? yes go to 2 no stop etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

59 Numerical Results Robust Regression Estimates of λ a λ I differ across a method relatively a lot No large influence on ultimates and reserves values Numerical Illustration performed etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

60 Robust Regression Derivation of theoretical principles Elasticity of MCL reserve = sensitivity of ultimates with respect to parameters λ. Remark of basic formula f,mcl i,k ( λ ) = f,scl k + λ σ Y i,k I k k ρ Linearity of the function f,mcl i,k k Yi,k f,mcl i,k 1 q k ( λ ) = f,scl + λ ( λ ) So the derivative of development factors could be rewritten to f,mcl i,k ( λ σk Y ) = i,k I 1 q ρ k Yi,k k etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

61 Robust Regression Derivation of theoretical principles Implications to elasticity of projections depending on λ. Standard formula: Ŷ i,n = Y i,a(i) n 1 Rearranging the development factors (Y i,n) = n 1 j=a(i) Y i,a(i) f (Y i,n ) Yi,n Final Result ( ) E(Q 1 i,k ) = q k 1 E f i,k = 0 (f ) f = 1 λ j=a(i) f i,a(i)... f i,n 1 = Ŷ n 1 j=a(i) i,n f j (1 ) f n 1 j=a(i) ( f ) f i,k etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

62 Robust Regression Conclusions to MCL Elasticity ( ) Using formula E f i,k = 0 (Y i,n ) n 1 Y = 1 λ i,n j=a(i) ( f j f ( ) (Yi,n ) holds E Y i,n analogously also E Interpretation = 0 ( (Yi,n I ) Y i,n I ) ) = 0 Systematic influence does not depend on λ Confirming original numerical results Hard to say what is right point estimate of MCL Loss Reserve Computation of Risk margin also needed etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

63 Addition of MSE calculation to MCL model Scope of presentation 1 Introduction 2 Robust Regression 3 Addition of MSE calculation to MCL model 4 Multivariate Extensions to Chain Ladder 5 Multivariate MCL 6 Other Approaches to model aid and Incurred data etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

64 Addition of MSE calculation to MCL model MCL variability MCL provides expectation E Variability formula Var ( Yi,s+1 Yi,s ( Y rocess of Derivation Start( from linear model ) of MCL ( Y Res Yi (s) = λ Res i,s+1 Y i,s i,s+1 Y i,s B i (s) Yi,s I Yi,s ) B i (s) ) =? ) Yi (s) + ε i,s roperties of residuals E(ε i,s B i (s)) = 0 a var(ε i,s B i (s)) = σ 2 R Adjustment of formula var ( Res ( Y i,s+1 Y i,s Yi (s) ) B i (s) ) = σ 2 R i Res 2 ( Y I i,s Y i,s s Res2 ( ) Yi (s) Yi,s I Yi,s Yi (s) ) etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

65 Addition of MSE calculation to MCL model MCL variability - end of derivation We use also formula ( ( Y var Res i,s+1 Y i,s Yi (s) ) B i (s) ) = var(y i,s+1 /Y σ 2 s If we combine both formulae and remind Mack s model ( ) Y i,s+1 Var Yi,s Yi (s) = (σ i ) 2 Yi,s i,s B i(s)) /Y i,s Variability Formula for MCL model could be seen as generalisation of Mack s approach ( ) ( ) ( Y i,s+1 Y Var Yi,s B i (s) = var( λ )σ 2 i,s+1 Y I Yi,s Yi (s) Res 2 i,s Yi,s Yi (s) ) etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

66 Addition of MSE calculation to MCL model Application to Mean Square Error Calculation for Incurred analogously ( ) ( Y I i,s+1 Var Yi,s I B i (s) = var( λ Y I I )σ 2 Application onto MCL mse(ˆr i ) = Ŷ 2 N i,n k=n i σ 2 k f k 2 i,s+1 Yi,s I ( Y I i (s) ) ( Y Res 2 i,s ) Ŷ n k i,k j=1 Y Substitute the theoretical parameters by their estimates 2 = var( λ ) σ,mcl i,s σs,scl2 Y I i,s ( ) Y I Res 2 i,s Y i (s)) Y i,s Joint information leads to decrease of reserve variability See the following illustration Y I i (s) etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44 )

67 Addition of MSE calculation to MCL model Application to Real Data Comparison of MSE calculation between SCL and MCL etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

68 Addition of MSE calculation to MCL model MSE graph MSE is significantly lower in MCL model etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

69 Multivariate Extensions to Chain Ladder Scope of presentation 1 Introduction 2 Robust Regression 3 Addition of MSE calculation to MCL model 4 Multivariate Extensions to Chain Ladder 5 Multivariate MCL 6 Other Approaches to model aid and Incurred data etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

70 Multivariate Extensions to Chain Ladder Recall of approach suggested by Schmidt Column vector Y = (Y 1,..., Y K ) cumulative amount of claims occurred in period i and developed after j period after occurrence K insurance portfolios are analysed simultaneously Useful notation Υ = diag(y ). Thus Y = Υ 1 One dimensional case: Multivariate extension: Y +1 = Y F F = (F 1,..., F K ) Y +1 = Υ F generalisation of individual factor etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

71 Multivariate Extensions to Chain Ladder Multivariate Chain Ladder Stochastic assumptions Corollary Conditional Expectation There exists K-dimensional development factor independent on year of occurrence that holds E (Y +1 Y i (j)) = Υ f j Conditional Variance and inter-row dependance There exists matrix Σ j so that if i = i 1 = i 2 and also otherwise E (F Y i (j)) = f j Cov(Y i1,j+1, Y i2,j+1 Y i1 (j), Y i2 (j)) = Υ 1/2 Σ j Υ 1/2 Cov(Y i1,j+1, Y i2,j+1 Y i1 (j), Y i2 (j)) = 0 Cov(F i1,j+1, F i2,j+1 Y i1 (j), Y i2 (j)) = Υ 1/2 Σ j Υ 1/2, etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

72 Multivariate Extensions to Chain Ladder Estimation in Multivariate Case Univariate Case Multivariate Case estimate of f j was found as f j = n j 1 i=0 w i F unbiased if n j 1 i=0 w i = 1 OLS if w i = Y n j 1 i=0 Y estimator f j as f j = n j 1 i=0 W i F Conditional unbiased if n j 1 i=0 W i = I MSE is minimised if fj = ( n j 1 i=0 Υ 1/2 Σ 1 j Υ 1/2 ) n j 1 i=0 Υ 1/2 Σ 1 j Υ 1/2 F etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

73 Multivariate Extensions to Chain Ladder How to estimate Covariance matrix? It is important for practical purposes It might be defined in a standard way like Σ j = 1 n j 1 n j 1 i=0 ( Υ 1/2 ( F f )) ( j Υ 1/2 Drawback: Σ j is not well defined if j n k Benefit of the method might be limited ( F f j )) etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

74 Multivariate Extensions to Chain Ladder Recall of approach suggested by Kremer Multivariate model j holds Y +1 = Y.f j + ε i = 0,..., n E(ε ) = 0 var(ε ) = σ 2 j.y. Y k +1 = Y k.f k j + ε k i = 0,..., n k = 1,..., K Original linear model is assumed for all of K analysed run-off triangles In addition cov(ε k1, εk2 k1,k2 ) = Ci Y k1 Y k2 and var(ε k ) = σk,2 j. If i 1 i 2 or j 1 j 2 then residuals are assumed to be uncorrelated cov(ε k1 i1,j1, εk2 i2,j2 ) = 0 etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

75 Multivariate Extensions to Chain Ladder Remarks to model Not only the estimate of development factor but also the estimator of variance is stressed Aitken s estimator of f j ossibly time consuming computation of large-dimensional inverse matrix Ψ 1 More useful for multivariate extension of Munich Chain Ladder etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

76 Multivariate Extensions to Chain Ladder Computation of estimates - Algorithm 1 Calculation of estimators of f k j for each triangle separately 2 variability estimator corresponding above mentioned estimates of development factor is derived through standard formulae n j 1 σ 2,k i=1 (Y+1 k j = f k n j 1 i=1 Y and also covariance estimator as n j 1 Ĉ k1,k2 i = i=1 (Y+1 k1 f k1 j n j 1 i=1 Y k1 Y k1 j Y k )2 )(Y k2 +1 Y k2 f k2 j Y k2) 3 Application of these estimates to estimate of development factors f j l+1 based on inverse matrix l 2,k σ j and Ĉ k1,k2 4 Repeat it until the parameters estimates do not converge i l. etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

77 Multivariate MCL Scope of presentation 1 Introduction 2 Robust Regression 3 Addition of MSE calculation to MCL model 4 Multivariate Extensions to Chain Ladder 5 Multivariate MCL 6 Other Approaches to model aid and Incurred data etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

78 Multivariate MCL roposal for multivariate Extensions of Munich Chain Ladder Kremer s approach found more suitable for MMCL linear model with slope parameters λ a λ I vector of parameters of (λ,1,..., λ,k ) is to be estimated simultaneously MCL model assumption holds for all triangles k = 1,..., K ( ) Y,k i,s+1 Res Y,k Y i (s),k B i (s) k = λ,k Res((Q i,s) k 1 Y i (s) )+ε k Y i (s), i,s Recall univariate case E(ε ) = 0 and var(ε ) = σ 2 etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

79 Multivariate MCL roposal for multivariate Extensions of Munich Chain Ladder Multivariate stochastic assumptions if i 1 i 2 and cov(ε k1 i1,j1, ε k2 i2,j2 ) = 0 cov(ε k1 1, ε k2 2 ) = 0 if j 1 j 2 Moreover for equal occurrence and development periods General model specification Y,1 X,1 Y,2. = X,2 Y,K cov(ε k1, ε k2 ) = σ k1,k2 β 1 β 2 β K X,K ε,1 ε,2. ε,k etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

80 Multivariate MCL Variables of the model in the multivariate case Response variable and Explanatory variable Res Y,k Res = Res ( Y,k 0,1 Y I,k ( 0,0 Y,k 0,2 Y I,k 0,0. ( Y,k n 1,1 Y I,k n 1,0 ) ) ) Res X,k Res = Res ( Y I,k 0,0 Y,k ( 0,0 Y I,k 0,1 Y,k 0,1. ( Y I,k n 1,0 Y,k n 1,0 ) ) ) etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

81 Multivariate MCL Multivariate MCL - computation 1 get standard OLS estimator likewise in univariate case λ,k = b k = (X,k X,k ) 1 X,k Y,k 2 Matrix Σ is estimated using following formula σ k1,k2 = ε.,k1 ε.,k2 n (n 1)/2 ε.,k1 vector of OLS calculated residuals of k1th model. 3 Estimator with non constant variance β = λ is derived as β = (Z Ψ 1 Z) 1 Z Ψ 1 Y Ψ = Σ I a Z is block-diagonal matrix X,k, thus Z = diag(x,1,..., X,K ). Notes initial estimator is replaced by that one calculated in the 3th step repeat process stop if parameters converges etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

82 Other Approaches to model aid and Incurred data Scope of presentation 1 Introduction 2 Robust Regression 3 Addition of MSE calculation to MCL model 4 Multivariate Extensions to Chain Ladder 5 Multivariate MCL 6 Other Approaches to model aid and Incurred data etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

83 Other Approaches to model aid and Incurred data Suggestion how to model aid and Incurred Different idea how to predict future payments and Incurred values May work for non finished schemes as well (tail factor) define Y a I Y I incremental value of aid amount in calendar period i + j is signed d = 1 model specification paid amount in the next development period could be explained by the value of reserve in the present R = I linear predictor d +1 = α j R + ε A, var(ε A ) = σ 2 A R respect the key idea of Munich Chain Ladder that one might expect higher future amount of paid compensation in case of higher reserve estimator R necessary for estimators d i + j > n etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

84 Other Approaches to model aid and Incurred data Models for reserve development quite simple model for reserve development R +1 = β j R + ε B, var(ε B ) = σ 2 B R reminds standard chain ladder evolution. However it holds R +1 = R d +1 + RT +1 RR +1 R T +1 shows increase of reserve (if new claims are detected) a RR +1 represents decrease of reserve Run-off model R T R R = γ j R + ε C, var(ε C ) = σ 2 C R derived from R +1 = R d +1 + RT +1 RR +1 = R α j R + R T +1 RR +1 + εa = β jr + ε B β j + α j 1 = γ j and ε C = εa + εb etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

85 Other Approaches to model aid and Incurred data Numerical Illustration Various portfolios analysed using suggested models simple reserve development alternative I run off model alternative II Obtained results compared with SCL and MCL approach Confirmed better fit between aid and Incurred data using alternative models Results on 3 different portfolios presented as follows 1 Example presented in original paper of MCL 2 Not finalised but smooth triangle 3 Not finalised and volatile data with increase in accident year direction etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

86 Other Approaches to model aid and Incurred data Causality for aid data We know that +1 = + d So far we have presented 2 basic models for future aid developments 1 Standard Chain Ladder: +1 = f j + ε 2 Alternative model I: +1 = + α j R + ε Why not try to combine these two approaches? ( +1 R +1 ) ( ) fj α = j. δ j β j ( R ) ( ε ) + ε R Two simple models could be understood as special cases α j = 0 obtain SCL model f j = 1 obtain alternative model 1 We can expect δ j = 0 if paid compensation is not informative for future reserving etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

87 Other Approaches to model aid and Incurred data Estimates of arameters Usual Estimates of matrix parameters used for vector regression models We use notation Y i ( +1 R +1 Π j = ), Π j [ n j ] [ n j Y i X i X i X i i=1 ( fj α j δ j β j Estimate of Variance matrix where ε i = Y i Π X i Σ = i=1 ), X i 1 n j 1 ( R εi. ε i ] 1 ) ( ε ), Σ Var ε R etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

88 Other Approaches to model aid and Incurred data References 1 Cizek,., Robust Estimation in Nonlinear Regression and Limited Dependent Variable Models,. Working aper, CERGE-EI, rague, Hess, T., Schmidt, K.D., Zocher, M., Multivariate loss prediction in the multivariate additive model, Insurance: Mathematics and Economics 39, Jedlicka,., Recent developments in claims reserving, roceedings of Week of doctoral students, Charles University, rague, Kremer, E., The correlated chain ladder method for reserving in case of correlated claims development, Blatter DGVFM 27, etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

89 Other Approaches to model aid and Incurred data References 2 Mack, T., Distribution free Calculation of the Standard Error of Chain Ladder Reserves Estimates, ASTIN Bulletin, Vol. 23, No. 2, rohl, C., Schmidt, K.D., Multivariate Chain ladder, Dresdner Schriften zu Versicherungsmathematik 3/2005, Quarg, G., Mack, T., Munich Chain Ladder, Blatter DGVFM 26, Munich, Verdier, B., Klinger, A., JAB Chain: A model based calculation of paid and incurred developments factors 36th ASTIN Colloquium, etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44

90 Other Approaches to model aid and Incurred data Thank you very much for your attention etr Jedlička (UK MFF) Various Extensions Based on MCL Method 20th June / 44