Model Study about the Applicability of the Chain Ladder Method. Magda Schiegl. ASTIN 2011, Madrid

Transcription

1 Model tudy about the Applcablty of the Chan Ladder Method Magda chegl ATIN 20, Madrd ATIN 20 Magda chegl

2 Clam Reservng P&C Insurance Clam reserves must cover all labltes arsng from nsurance contracts wrtten n the presence and the past. Clam reserves are calculated for homogeneous portfolos of nsurance contracts Clam reserves are needed for holstc rs management Realstc modellng of the clam process! Outlne: 3D model 3D vs. classcal models (2D) Comparson: Results of MC smulaton (3D / 2D) ATIN 20 Magda chegl 2

3 Clam Reservng Classcal Models Classcal Models operate on a 2D data structure (cumulated data) Proecton of complex process to a 2D world calculate only expectaton of reserves (Var under restrctve assumptons) No nformaton about the tal of the reserve dstrbuton Typcal data structures for estmaton: occurrence year run off year Known data (past) reportng year run off year Known data (past) reported clams + IBNR Future = Reserves s proected wth ( zoo of) estmaton methods reported clams IBNR (Incurred but not reported) ATIN 20 Magda chegl 3

4 The 3D Model the Clam Process Idea: Development of clam portfolo s modeled from frst prncples: The sngle clam process Tme Dynamcs of (sngle) clam process occurrence reportng payment(s) t 3 dmensonal structure: occurrence () // reportng () // payment(s) () xamne a portfolo of clams stochastc model ATIN 20 Magda chegl 4

5 The 3D Model Modeled quanttes: ) Number of actve clams N occurred n th year reported year after occurrence stll actve years after reportng 2) Total clam payments Z n th year after reportng arsng from all actve clams wth occurrence year reported wth years delay. ATIN 20 Magda chegl 5

6 The 3D Model Vsualsng the model Number of actve clams N (up to now) reported clams Payments Z Boo reserve (nown clams) IBNR clams now actve clams payments up to now IBNR: Incurred but not reported ATIN 20 Magda chegl 6 IBNR reserves (unnown clams)

7 Connecton 3D and 2D Model Up to now reported clams: N 0 wth I max Number of IBNR Clams: IBNR Reserve: Nˆ ˆ IBNR N Rˆ ˆ IBNR Z,, I max, I max I max 0 Total Reserve: Rˆ total Rˆ IBNR,, Zˆ I max The run off trangle payments: occurrence versus run off year : The run off trangle payments: reportng versus run off year : mn Z m, n m I 2 mn Z n, m n I ATIN 20 Magda chegl 7 max max

8 The 3D Model Modelng the number of actve clams N N 0 (number of clams wth occurrence year =) N Posson N Clam numbers multnomally dstrbuted along reportng years wth parameters,2,.., I and Closng the clams along years after reportng (Bnomal process): max,2,.., K max wth and 0 N BNom N, ATIN 20 Magda chegl 8

9 The 3D Model Modelng the total clam payments: Collectve Model Number of sngle payments (r.v.) l Z X l Bnom N, p ze of sngle payment (r.v.) Probablty of payment for an actve clam ATIN 20 Magda chegl 9

10 Chan Ladder and 3D Model Condton for the approprate use of the CL method appled to the 3D model: ) occurrence vs. run off matrx stmate total reserve () () n n n n wth n () n y, Z y 2) reportng vs. run off matrx stmate reserve of nown clams () n () n, Z n (2) (2) 2 xn xn n xn wth n (2) xn, Z x (2) xn (2) xn, Z n x ATIN 20 Magda chegl 0

11 Chan Ladder and 3D Model Condton for the approprate use of the CL method appled to the 3D model: ) occurrence vs. run off matrx stmate total reserve () () n n n n wth n () n y, Z y 2) reportng vs. run off matrx stmate reserve of nown clams () n () n, Z n (2) (2) 2 xn xn n xn wth n (2) xn, Z x tructure of clam data Z s decsve for applcablty of CL (2) xn (2) xn, Z n x ATIN 20 Magda chegl

12 tructure and Applcablty It s a matter of structure of the clam process and ts realsatons n clam data f the CL method s an approprate estmaton method for the reserves. mprcal analyss, MC mulaton ATIN 20 Magda chegl 2

13 Numercal xamples and MC mulaton ystematc of numercal calculatons We compare examples wth three dfferent structures: xample : Z xample 2: Z xample 3: Z ATIN 20 Magda chegl 3

14 Numercal xamples and MC mulaton ystematc of numercal calculatons Model Parameters xpectaton Value Total Reserve / Res. nown clams (3D) Trangles CL Reserves (2D) MC mulaton 000 scenaros of clam portfolo MC smulated Reserves (3D) Total Reserve Res. nown clams Calculaton of 2D trangles as bass for CL Occurrence year trangle Reportng year trangle CL Method Total Reserve CL Method Res. nown clams ATIN 20 Magda chegl 4

15 Numercal xamples and MC mulaton Results Comparson of estmaton and emprcal mean values. Only Model parameter dependent MC smulated Only Model parameter dependent MC smulated 3D xpectaton 3D Mean CL xpectaton CL Mean xample Z Total Reserve xample Reserve of nown clams xample 2 Total Reserve Z xample 2 Reserve of nown clams xample 3 Total Reserve Z xample 3 Reserve of nown clams ATIN 20 Magda chegl 5

16 Numercal xamples and MC mulaton Results Comparson of estmaton and emprcal mean values. Result for mean values: Applcablty of CL depends on data structure Only Model parameter dependent MC smulated Only Model parameter dependent MC smulated 3D xpectaton 3D Mean CL xpectaton CL Mean xample Z Total Reserve xample Reserve of nown clams xample 2 Total Reserve Z xample 2 Reserve of nown clams xample 3 Total Reserve Z xample 3 Reserve of nown clams ATIN 20 Magda chegl 6

17 Numercal xamples and MC mulaton Results Comparson of estmaton and emprcal mean values. Result: 3D emprcal mean fts better than CL estmaton // CL postve bas Only Model parameter dependent MC smulated Only Model parameter dependent MC smulated 3D xpectaton 3D Mean CL xpectaton CL Mean xample Z Total Reserve xample Reserve of nown clams xample 2 Total Reserve Z xample 2 Reserve of nown clams xample 3 Total Reserve Z xample 3 Reserve of nown clams ATIN 20 Magda chegl 7

18 Numercal xamples and MC mulaton Results MC reserve dstrbutons xample Total reserve Chan Ladder D model Result: 3D model dstrbuton s narrow effcent captal use ATIN 20 Magda chegl 8

19 Numercal xamples and MC mulaton Results MC reserve dstrbutons xample 3 Total Reserve Chan Ladder D model ATIN 20 Magda chegl 9

20 Numercal xamples and MC mulaton Results MC reserve dstrbutons xample 3 Reserve of nown clams Chan Ladder D model ATIN 20 Magda chegl 20

21 Than you for your attenton! I loo forward to your questons and our dscusson! ATIN 20 Magda chegl 2

22 Bac up ATIN ATIN 20 Magda 2009 chegl tochastc clams reservng, M. chegl 22

23 tructure and Applcablty What does ths mean for the 3D Model? ) Occurrence run off matrx: () n n (), n y, Z n Z y Independence on CL s approprate forecast for total reserve 2) Reportng run off matrx: (2) xn xn (2), n, ATIN 20 Magda chegl 23 Z n x Z x Independence on x = + CL s approprate forecast for reported clams reserve