Multivariate Chain Ladder
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1 Multivariate Chain Ladder Carsten Pröhl and Klaus D Schmidt Lehrstuhl für Versicherungsmathematik Technische Universität Dresden schmidt@mathtu-dresdende Multivariate Chain Ladder p1/44
2 Multivariate Chain Ladder 1 The Univariate Chain Ladder Method Multivariate Chain Ladder p2/44
3 Multivariate Chain Ladder 1 The Univariate Chain Ladder Method 2 The Univariate Model of Schnaus Multivariate Chain Ladder p2/44
4 Multivariate Chain Ladder 1 The Univariate Chain Ladder Method 2 The Univariate Model of Schnaus 3 The Multivariate Model Multivariate Chain Ladder p2/44
5 Multivariate Chain Ladder 1 The Univariate Chain Ladder Method 2 The Univariate Model of Schnaus 3 The Multivariate Model 4 The Multivariate Chain Ladder Method Multivariate Chain Ladder p2/44
6 Multivariate Chain Ladder 1 The Univariate Chain Ladder Method 2 The Univariate Model of Schnaus 3 The Multivariate Model 4 The Multivariate Chain Ladder Method 5 Conclusion Multivariate Chain Ladder p2/44
7 1 The Univariate Chain Ladder Method Multivariate Chain Ladder p3/44
8 An Example Accident Development Year k Year i Multivariate Chain Ladder p4/44
9 An Example Accident Development Year k Year i Multivariate Chain Ladder p4/44
10 An Example Accident Development Year k Year i Multivariate Chain Ladder p4/44
11 An Example Accident Development Year k Year i Multivariate Chain Ladder p4/44
12 An Example Accident Development Year k Year i Multivariate Chain Ladder p4/44
13 An Example Accident Development Year k Year i Multivariate Chain Ladder p4/44
14 An Example Accident Development Year k Year i Multivariate Chain Ladder p4/44
15 An Example Accident Development Year k Year i , Multivariate Chain Ladder p4/44
16 An Example Accident Development Year k Year i , Multivariate Chain Ladder p4/44
17 An Example Accident Development Year k Year i , Multivariate Chain Ladder p4/44
18 An Example Accident Development Year k Year i ,232 1, Multivariate Chain Ladder p4/44
19 An Example Accident Development Year k Year i ,232 1, Multivariate Chain Ladder p4/44
20 An Example Accident Development Year k Year i ,232 1, Multivariate Chain Ladder p4/44
21 An Example Accident Development Year k Year i Multivariate Chain Ladder p4/44
22 Abstract Run Off Triangle Run off triangle of observable aggregate claims: Accident Development Year Year 0 1 k n i n 1 n 0 S 0,0 S 0,1 S 0,k S 0,n i S 0,n 1 S 0,n 1 S 1,0 S 1,1 S 1,k S 1,n i S 1,n 1 i S i,0 S i,1 S i,k S i,n i n k S n k,0 S n k,1 S n k,k n 1 S n 1,0 S n 1,1 n S n,0 Multivariate Chain Ladder p5/44
23 Chain Ladder Estimation Individual Development Factors: F i,k := S i,k S i,k 1 Multivariate Chain Ladder p6/44
24 Chain Ladder Estimation Individual Development Factors: Chain Ladder Factors: F i,k := S i,k S i,k 1 F CL k := = n k j=0 S j,k n k j=0 S j,k 1 n k j=0 S j,k 1 n k h=0 S h,k 1 F j,k (for k 1) Multivariate Chain Ladder p6/44
25 Chain Ladder Estimation Identity for aggregate claims: S i,k = S i,n i k l=n i+1 F i,l Multivariate Chain Ladder p7/44
26 Chain Ladder Estimation Identity for aggregate claims: S i,k = S i,n i k l=n i+1 F i,l Chain Ladder Estimators: Ŝ CL i,k := S i,n i k k=n i+1 F CL l (for k n i) Multivariate Chain Ladder p7/44
27 Chain Ladder Estimation: Chain Ladder estimation is recursive since Ŝ CL i,n i = S i,n i and Ŝ CL i,k = ŜCL i,k 1 F CL k for k = n i + 1,, n Multivariate Chain Ladder p8/44
28 Modified Chain Ladder Estimation Chain Ladder Factors: F CL k = n k j=0 S j,k 1 n k h=0 S h,k 1 F j,k Multivariate Chain Ladder p9/44
29 Modified Chain Ladder Estimation Chain Ladder Factors: F CL k = n k j=0 S j,k 1 n k h=0 S h,k 1 F j,k Modified Chain Ladder Factors: F k := n k j=0 W j,k F j,k with n k j=0 W j,k = 1 (for k 1) Multivariate Chain Ladder p9/44
30 Modified Chain Ladder Estimation Chain Ladder Estimators: Ŝ CL i,k := S i,n i k k=n i+1 F CL l Multivariate Chain Ladder p10/44
31 Modified Chain Ladder Estimation Chain Ladder Estimators: Ŝ CL i,k := S i,n i k k=n i+1 F CL l Modified Chain Ladder Estimators: Ŝ i,k := S i,n i k k=n i+1 F l = Competition!!! (for k n i) Multivariate Chain Ladder p10/44
32 2 The Univariate Model of Schnaus Multivariate Chain Ladder p11/44
33 Truncated Run Off Square Accident Development Year Year 0 1 k 1 k 0 S 0,0 S 0,1 S 0,k 1 S 0,k 1 S 1,0 S 1,1 S 1,k 1 S 1,k n k S n k,0 S n k,1 S n k,k 1 S n k,k n k + 1 S n k+1,0 S n k+1,1 S n k+1,k 1 S n k+1,k n 1 S n 1,0 S n 1,1 S n 1,k 1 S n 1,k n S n,0 S n 1,1 S n,k 1 S n,k We denote by G k 1 the information provided by development years 0, 1,, k 1 Multivariate Chain Ladder p12/44
34 The Model of Schnaus Assumption: For each k = 1,, n there exist random variables F k and V k such that E(S i,k G k 1 ) = S i,k 1 F k holds for all i = 0, 1,, n Multivariate Chain Ladder p13/44
35 The Model of Schnaus Assumption: For each k = 1,, n there exist random variables F k and V k such that E(S i,k G k 1 ) = S i,k 1 F k var(s i,k G k 1 ) = S i,k 1 V k holds for all i = 0, 1,, n Multivariate Chain Ladder p13/44
36 The Model of Schnaus Assumption: For each k = 1,, n there exist random variables F k and V k such that E(S i,k G k 1 ) = S i,k 1 F k var(s i,k G k 1 ) = S i,k 1 V k cov(s i,k, S j,k G k 1 ) = 0 holds for all i, j = 0, 1,, n such that j i Multivariate Chain Ladder p13/44
37 The Model of Schnaus Assumption: For each k = 1,, n there exist random variables F k and V k such that E(S i,k G k 1 ) = S i,k 1 F k var(s i,k G k 1 ) = S i,k 1 V k cov(s i,k, S j,k G k 1 ) = 0 holds for all i, j = 0, 1,, n such that j i A slightly less general model is the Model of Mack Multivariate Chain Ladder p14/44
38 The Model of Schnaus Equivalent formulation of the assumption: For each k = 1,, n there exist random variables F k and V k such that E(F i,k G k 1 ) = F k var(f i,k G k 1 ) = V k /S i,k 1 cov(f i,k, F j,k G k 1 ) = 0 holds for all i, j = 0, 1,, n such that j i Multivariate Chain Ladder p15/44
39 Estimation of a non observable F i,k (i+k > n): Admissible estimators: F k := n k j=0 W j,k F j,k with random variables W 0,k, W 1,k,, W n k,k G k 1 satisfying n k j=0 W j,k = 1 Multivariate Chain Ladder p16/44
40 Estimation of a non observable F i,k (i+k > n): The Chain Ladder Factor F CL k := n k j=0 S j,k 1 n k h=0 S h,k 1 F j,k is an admissible estimator Multivariate Chain Ladder p17/44
41 Estimation of a non observable F i,k (i+k > n): The Chain Ladder Factor F CL k := n k j=0 S j,k 1 n k h=0 S h,k 1 F j,k is an admissible estimator Every admissible estimator is a Modified Chain Ladder Factor Multivariate Chain Ladder p17/44
42 Estimation of a non observable F i,k (i+k > n): The Chain Ladder Factor F CL k := n k j=0 S j,k 1 n k h=0 S h,k 1 F j,k is an admissible estimator Every admissible estimator is a Modified Chain Ladder Factor Every admissible estimator is unbiased in the sense that ( ) ) E Fk G k 1 = E (F i,k G k 1 Multivariate Chain Ladder p17/44
43 Estimation of a non observable F i,k (i+k > n): Optimization problem: Minimize E( ( Fk F i,k over all admissible estimators ) 2 G k 1 ) Multivariate Chain Ladder p18/44
44 Estimation of a non observable F i,k (i+k > n): Optimization problem: Minimize ) 2 G k 1 ) E( ( Fk F i,k over all admissible estimators Solution: The optimization problem has the unique solution F CL k := n k j=0 S j,k 1 n k h=0 S h,k 1 F j,k Multivariate Chain Ladder p18/44
45 Estimation of S i,n i+1 : Admissible estimators: Ŝ i,n i+1 := S i,n i Fn i+1 with an admissible estimator F n i+1 of F i,n i+1 Multivariate Chain Ladder p19/44
46 Estimation of S i,n i+1 : The Chain Ladder Estimator Ŝ CL i,n i+1 := S i,n i F CL n i+1 is an admissible estimator Multivariate Chain Ladder p20/44
47 Estimation of S i,n i+1 : The Chain Ladder Estimator Ŝ CL i,n i+1 := S i,n i F CL n i+1 is an admissible estimator Every admissible estimator is a Modified Chain Ladder Estimator Multivariate Chain Ladder p20/44
48 Estimation of S i,n i+1 : The Chain Ladder Estimator Ŝ CL i,n i+1 := S i,n i F CL n i+1 is an admissible estimator Every admissible estimator is a Modified Chain Ladder Estimator Every admissible estimator is unbiased in the sense that ) ) E (Ŝi,n i+1 G n i = E (S i,n i+1 G n i Multivariate Chain Ladder p20/44
49 Estimation of S i,n i+1 : Optimization problem: Minimize E( (Ŝi,n i+1 S i,n i+1 over all admissible estimators ) 2 G n i ) Multivariate Chain Ladder p21/44
50 Estimation of S i,n i+1 : Optimization problem: Minimize ) 2 G n i ) E( (Ŝi,n i+1 S i,n i+1 over all admissible estimators Solution: The optimization problem has the unique solution Ŝ CL i,n i+1 := S i,n i F CL n i+1 = Chain Ladder is the Winner!!! Multivariate Chain Ladder p21/44
51 Estimation of S i,n i+1 : Proof: E( (Ŝi,n i+1 S i,n i+1 ) 2 G n i ) = E( ( S i,n i F n i+1 S i,n i F i,n i+1 = S 2 i,n i E( ( Fn i+1 F i,n i+1 ) 2 G n i ) ) 2 G n i ) Multivariate Chain Ladder p22/44
52 Sequential Optimality: As a consequence of the previous result, the chain ladder estimators are optimal in a sequential sense Multivariate Chain Ladder p23/44
53 3 The Multivariate Model Multivariate Chain Ladder p24/44
54 Correlated Lines Of Business In the present section we consider the loss development for m = 2 correlated lines of business Multivariate Chain Ladder p25/44
55 Correlated Lines Of Business In the present section we consider the loss development for m = 2 correlated lines of business All definitions and results can be extended to arbitrary m Multivariate Chain Ladder p25/44
56 Run Off Triangle for Portfolio (1) Accident Development Year Year 0 1 k n i n 1 n 0 S (1) 0,0 S (1) 0,1 S (1) 0,k S (1) 0,n i S (1) 0,n 1 S (1) 0,n 1 S (1) 1,0 S (1) 1,1 S (1) 1,k S (1) 1,n i S (1) 1,n 1 i S (1) i,0 S (1) i,1 S (1) n k S (1) n k,0 S (1) n 1 S (1) n 1,0 S (1) n 1,1 n S (1) n,0 n k,1 S (1) i,n i i,k S (1) n k,k Multivariate Chain Ladder p26/44
57 Run Off Triangle for Portfolio (2) Accident Development Year Year 0 1 k n i n 1 n 0 S (2) 0,0 S (2) 0,1 S (2) 0,k S (2) 0,n i S (2) 0,n 1 S (2) 0,n 1 S (2) 1,0 S (2) 1,1 S (2) 1,k S (2) 1,n i S (2) 1,n 1 i S (2) i,0 S (2) i,1 S (2) n k S (2) n k,0 S (2) n 1 S (2) n 1,0 S (2) n 1,1 n S (2) n,0 n k,1 S (2) i,n i i,k S (2) n k,k Multivariate Chain Ladder p27/44
58 Run Off Triangle for Both Portfolios Accident Development Year Year 0 1 k n i n 1 n 0 S 0,0 S 0,1 S 0,k S 0,n i S 0,n 1 S 0,n 1 S 1,0 S 1,1 S 1,k S 1,n i S 1,n 1 i S i,0 S i,1 S i,k S i,n i n k S n k,0 S n k,1 S n k,k n 1 S n 1,0 S n 1,1 n S n,0 Multivariate Chain Ladder p28/44
59 Notation S i,k := T i,k := F i,k := S(1) i,k S (2) i,k S(1) i,k 0 0 S (2) i,k F (1) i,k F (2) i,k Multivariate Chain Ladder p29/44
60 Truncated Run Off Square Accident Development Year Year 0 1 k 1 k 0 S 0,0 S 0,1 S 0,k 1 S 0,k 1 S 1,0 S 1,1 S 1,k 1 S 1,k n k S n k,0 S n k,1 S n k,k 1 S n k,k n k + 1 S n k+1,0 S n k+1,1 S n k+1,k 1 S n k+1,k n 1 S n 1,0 S n 1,1 S n 1,k 1 S n 1,k n S n,0 S n 1,1 S n,k 1 S n,k We denote by G k 1 the information provided by development years 0, 1,, k 1 Multivariate Chain Ladder p30/44
61 The Multivariate Model Assumption: For each k = 1,, n there exists a random vector F k and a random matrix V k such that E(S i,k G k 1 ) = S i,k 1 F k holds for all i = 0, 1,, n Multivariate Chain Ladder p31/44
62 The Multivariate Model Assumption: For each k = 1,, n there exists a random vector F k and a random matrix V k such that E(S i,k G k 1 ) = T i,k 1 F k var(s i,k G k 1 ) = T 1/2 i,k 1 V kt 1/2 i,k 1 holds for all i = 0, 1,, n Multivariate Chain Ladder p31/44
63 The Multivariate Model Assumption: For each k = 1,, n there exists a random vector F k and a random matrix V k such that E(S i,k G k 1 ) = T i,k 1 F k var(s i,k G k 1 ) = T 1/2 i,k 1 V kt 1/2 i,k 1 cov(s i,k, S j,k G k 1 ) = O holds for all i, j = 0, 1,, n such that j i Multivariate Chain Ladder p31/44
64 The Multivariate Model Assumption: For each k = 1,, n there exists a random vector F k and a random matrix V k such that E(S i,k G k 1 ) = T i,k 1 F k var(s i,k G k 1 ) = T 1/2 i,k 1 V kt 1/2 i,k 1 cov(s i,k, S j,k G k 1 ) = O holds for all i, j = 0, 1,, n such that j i A slightly less general model is the Model of Braun [2004] Multivariate Chain Ladder p32/44
65 The Multivariate Model Equivalent formulation of the assumption: For each k = 1,, n there exists a random vector F k and a random matrix V k such that E(F i,k G k 1 ) = F k var(f i,k G k 1 ) = T 1/2 i,k 1 V kt 1/2 i,k 1 cov(f i,k, F j,k G k 1 ) = O holds for all i, j = 0, 1,, n such that j i Multivariate Chain Ladder p33/44
66 4 The Multivariate Chain Ladder Method Multivariate Chain Ladder p34/44
67 Estimation of a non observable F i,k (i+k > n): Admissible estimators: F k := n k j=0 W j,k F j,k with random matrices W 0,k, W 1,k,, W n k,k G k 1 satisfying n k j=0 W j,k = I Multivariate Chain Ladder p35/44
68 Estimation of a non observable F i,k (i+k > n): Admissible estimators: F k := n k j=0 W j,k F j,k with random matrices W 0,k, W 1,k,, W n k,k G k 1 satisfying n k j=0 W j,k = I Every admissible estimator is unbiased in the sense that ) ) E ( Fk G n i = E (F i,k G n i Multivariate Chain Ladder p35/44
69 Estimation of a non observable F i,k (i+k > n): Optimization problem: Minimize E( ( Fk F i,k over all admissible estimators ) ( Fk F i,k ) G k 1 ) Multivariate Chain Ladder p36/44
70 Estimation of a non observable F i,k (i+k > n): Solution: The optimization problem has the unique solution F CL k := ( n k j=0 n k j=0 T 1/2 j,k 1 V 1 k T1/2 j,k 1) 1 T 1/2 j,k 1 V 1 k T1/2 j,k 1 F j,k The unique solution F CL k is said to be the Multivariate Chain Ladder Factor of development year k Multivariate Chain Ladder p37/44
71 Estimation of S i,n i+1 : Admissible estimators: Ŝ i,n i+1 := T i,n i Fn i+1 with an admissible estimator F n i+1 of F i,n i+1 Multivariate Chain Ladder p38/44
72 Estimation of S i,n i+1 : Admissible estimators: Ŝ i,n i+1 := T i,n i Fn i+1 with an admissible estimator F n i+1 of F i,n i+1 Every admissible estimator is unbiased in the sense that ) ) E (Ŝi,n i+1 G n i = E (S i,n i+1 G n i Multivariate Chain Ladder p38/44
73 Estimation of S i,n i+1 : Optimization problem: Minimize E( (Ŝi,n i+1 S i,n i+1 over all admissible estimators ) 2 G n i ) Multivariate Chain Ladder p39/44
74 Estimation of S i,n i+1 : Optimization problem: Minimize ) 2 G n i ) E( (Ŝi,n i+1 S i,n i+1 over all admissible estimators Solution: The optimization problem has the unique solution Ŝ CL i,n i+1 := T i,n i FCL n i+1 Multivariate Chain Ladder p39/44
75 Estimation of S i,n i+1 : The unique solution ŜCL k is said to be the Multivariate Chain Ladder Estimator of S i,n i+1 Multivariate Chain Ladder p40/44
76 Recursive Chain Ladder Estimation: Define Ŝ CL i,n i := S i,n i and for k = n i + 1,, n Then ŜCL i,k Estimator of S i,k T CL i,k 1 Ŝ CL i,k := := diag(ŝcl i,k 1) T CL i,k 1 F CL k is said to be the Multivariate Chain Ladder Multivariate Chain Ladder p41/44
77 Sequential Optimality: Sequential optimality obtains as in the univariate case Multivariate Chain Ladder p42/44
78 5 Conclusion Multivariate Chain Ladder p43/44
79 Conclusion In the univariate case, the method is there and a stochastic model had to be found for its justification Multivariate Chain Ladder p44/44
80 Conclusion In the univariate case, the method is there and a stochastic model had to be found for its justification In the multivariate case, an extension of the stochastic model of Schnaus leads to an extension of the method Multivariate Chain Ladder p44/44
81 Conclusion In the univariate case, the method is there and a stochastic model had to be found for its justification In the multivariate case, an extension of the stochastic model of Schnaus leads to an extension of the method The coordinates of the Multivariate Chain Ladder Estimators usually differ from the Univariate Chain Ladder Estimators of the Coordinates Multivariate Chain Ladder p44/44
82 Conclusion In the univariate case, the method is there and a stochastic model had to be found for its justification In the multivariate case, an extension of the stochastic model of Schnaus leads to an extension of the method The coordinates of the Multivariate Chain Ladder Estimators usually differ from the Univariate Chain Ladder Estimators of the Coordinates The multivariate approach resolves the problem of additivity Multivariate Chain Ladder p44/44
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