Prediction Uncertainty in the Chain-Ladder Reserving Method

Transcription

1 Prediction Uncertainty in the Chain-Ladder Reserving Method Mario V. Wüthrich RiskLab, ETH Zurich joint work with Michael Merz (University of Hamburg) Insights, May 8, 2015 Institute of Actuaries of Australia

2 Outline Introduction to claims reserving Chain-ladder method Claims development result Examples 1

3 General insurance company s balance sheet assets as of 31/12/2013 mio. CHF debt securities equity securities loans & mortgages real estate 908 participations short term investments 693 other assets 696 total assets liabilities as of 31/12/2013 mio. CHF claims reserves provisions for annuities other liabilities and provisions share capital 169 legal reserve 951 free reserve, forwarded gains total liabilities & equity Claims reserves are the biggest position on the balance sheet of a general insurance company! Source: Annual Report 2013 of AXA-Winterthur Versicherungen AG 2

4 premium General insurance claims cash flows accident date reporting date claims closing period insured (1 year) claims cash flows time Typically, insurance claims cannot be settled immediately (at occurrence): 1. reporting delay (days, weeks, months or even years); 2. settlement delay (months or years). Predict and value all future claims cash flows (of past exposure claims). Build claims reserves to settle these future claims cash flows. 3

5 Claims development triangle accident development year j year i C i,j to be predicted C i,j = cumulative (nominal) claims cash flows for accident year i {1,..., I} and development year j {0,..., J}. Observations at time t are given by D t = {C i,j ; i + j t}. 4

6 Outline Introduction to claims reserving Chain-ladder method Claims development result Examples 5

7 Chain-ladder method accident development year j year i C i,j to be predicted Chain-ladder (CL) algorithm is based on the observation that data often C i,j+1 f j C i,j, for CL factors f j not depending on accident year i. CL algorithm has been used for many decades in the insurance industry. 6

8 Mack s stochastic CL model Model assumptions (Mack 1993). Assume there are fixed positive constants f 0,..., f J 1 and σ 2 0,..., σ 2 J 1 given. (C i,j ) j=0,...,j are strictly positive, independent (in i) Markov processes (in j); for i = 1,..., I and j = 0,..., J 1 E [C i,j+1 C i,j ] = f j C i,j, Var (C i,j+1 C i,j ) = σ 2 j C i,j. Mack introduced this stochastic model for the study of the CL algorithm. Mack s CL model has the CL property: C i,j+1 f j C i,j. 7

9 CL predictor For known CL factors f j and given observations D I E [C i,j D I ] = C i,i i J 1 j=i i f j. We define the CL predictor for unknown CL factors f j at time I by Ĉ (I) i,j = Ê [C i,j D I ] = C i,i i with CL factor estimators at time t J J 1 j=i i f (I) j, f (t) j = (t j 1) I i=1 C i,j+1 (t j 1) I. i=1 C i,j Note that Ĉ(I) i,j and f (I) j share many good properties (unbiased, uncorrelated). 8

10 CL claims prediction accident development year j year i f (I) j What about prediction uncertainty? Consider the conditional mean square error of prediction (MSEP) msep Ci,J D I (Ĉ(I) i,j ) = E [ ( C i,j Ĉ(I) i,j ) ] 2 D I. 9

11 Mack s conditional MSEP formula Mack (1993) gives the following MSEP estimate for single accident years i: msep Mack (Ĉ(I) ) C i,j D I i,j = (Ĉ(I) i,j ) 2 J 1 j=i i ( ) 2 σj 2/ (I) f j Ĉ (I) i,j + ( ) 2 σj 2/ (I) f j I j 1. l=1 C l,j Note that this is an estimate because of unknown parameters f j (and σ j ) need to be estimated, and estimation error cannot be calculated explicitly. Aggregation over accident years i is slightly more involved, but can be done. 10

12 acc.year i Example, revisited R(I) i msep Mack C i,j D I in % R (I) i % % % % % % % % % total % Consider the CL reserves at time I defined by R (I) i = Ĉ(I) i,j C i,i i, and the corresponding prediction uncertainty. 11

13 Outline Introduction to claims reserving Chain-ladder method Claims development result Examples 12

14 Claims development result (1/2) Mack s formula considers the total prediction uncertainty over the entire run-off. Solvency considerations require a dynamic view: possible changes in predictions over the next accounting year (short term view). Define the claims development result of accounting year t + 1 > I by CDR i (t + 1) = Ĉ(t+1) i,j Ĉ(t) i,j. Martingale property of (Ĉ(t) i,j ) t I implies E [CDR i (t + 1) D t ] = E [Ĉ(t+1) i,j Ĉ(t) i,j ] D t = 0. 13

15 Claims development result (2/2) accident development year j year i Martingale property of (Ĉ(t) i,j ) t I implies E [CDR i (I + 1) D I ] = E [Ĉ(I+1) i,j Ĉ(I) i,j ] D I = 0. Solvency: study the one-year uncertainty of next accounting year t = I + 1 msep CDRi (I+1) D I (0) = E [(CDR i (I + 1) 0) 2 ] D I. 14

16 One-year uncertainty formula Merz-W. (2008) give the following MSEP estimate for single accident years i: msep MW CDR i (I+1) D I (0) = ( ) 2 σi i 2 / (I) f I i + C i,i i (Ĉ(I) i,j ) 2 ) 2 ( σi i 2 / (I) f I i i 1 l=1 C l,i i + J 1 j=i i+1 α (I) j ( ) 2 σj 2/ (I) f j I j 1, l=1 C l,j with (credibility) weight α (I) j = C I j,j I j l=1 C l,j (0, 1). Eric Dal Moro (SCOR) and Joseph Lo (Aspen UK) basically write: the problem is solved for the CL method, see industrial discussion paper, ASTIN Bulletin 44/3, September

17 Total uncertainty vs. one-year uncertainty Mack s formula for total uncertainty: msep Mack (Ĉ(I) ) C i,j D I i,j = (Ĉ(I) i,j ) 2 J 1 j=i i ( ) 2 σj 2/ (I) f j Ĉ (I) i,j + ( ) 2 σj 2/ (I) f j I j 1. l=1 C l,j MW formula for one-year uncertainty: msep MW CDR i (I+1) D I (0) = ( ) 2 σi i 2 / (I) f I i Ĉ (I) i,i i + (Ĉ(I) i,j ) 2 ) 2 ( σi i 2 / (I) f I i i 1 l=1 C l,i i + J 1 j=i i+1 α (I) j ( ) 2 σj 2/ (I) f j I j 1. l=1 C l,j Process uncertainty, parameter estimation uncertainty and its reduction in time. 16

18 Residual uncertainty for t I + 2 This suggests for accounting year t = I + 2: E [ msep MW CDR i (I+2) D I+1 (0) (Ĉ(I) i,j ) 2 + ] D I σ 2 I i+1 / ( f (I) I i+1 (Ĉ(I) i,j Ĉ CL(I) i,i i+1 ) 2 J 1 j=i i+2 ) 2 + α (I) j 1 ( 1 α (I) I i+1 ( 1 α (I) j )σ 2 I i+1 / ( f (I) I i+1 i 2 l=1 C l,i i+1 ) 2 ( ) σj 2/ (I) f j I j 1 l=1 C l,j. ) 2 This can be derived analytically and iterated for t > I + 2! This allocates Mack s MSEP formula across different accounting periods, i.e., this provides a run-off pattern of the total prediction uncertainty. 17

19 Outline Introduction to claims reserving Chain-ladder method Claims development result Examples 18

20 Motor third party liability: CH & US Expected run off, motor third party liability CH Expected run off, motor third party liability US relative run off claims reserves run off MSEP relative run off claims reserves run off MSEP (future) accounting years (future) accounting years Expected run-off of claims reserves is faster than the one of underlying risks. Legal environment is important for run-off. 19

21 Commercial property & general liability (CH) Expected run off, commercial property CH Expected run off, general liability CH relative run off claims reserves run off MSEP relative run off claims reserves run off MSEP (future) accounting years (future) accounting years Different lines of business behave differently (short- and long-tailed business). 20

22 Collective health & building engineering (CH) Expected run off, collective health CH Expected run off, building engineering CH relative run off claims reserves run off MSEP relative run off claims reserves run off MSEP (future) accounting years (future) accounting years Subrogation and recoveries need special care. 21

23 Conclusions The Merz-W. formula was generalized to arbitrary accounting years. This allocates Mack s total uncertainty formula across accounting years. This provides a run-off pattern for risk. This improves and interprets risk margin calculations (Solvency II versus APRA). Standard approximation techniques typically under-estimate run-off risk. Portfolio characteristics and legal environment are important for risk margins. 22

24 References [1] Dal Moro, E., Lo, J. (2014). An industry question: the ultimate and one-year reserving uncertainty for different non-life reserving methodologies. ASTIN Bulletin 44/3, [2] Mack, T. (1993). Distribution-free calculation of the standard error of chain ladder reserve estimates. ASTIN Bulletin 23/2, [3] Merz, M., Wüthrich, M.V. (2008). Modelling the claims development result for solvency purposes. CAS E-Forum Fall 2008, [4] Merz, M., Wüthrich, M.V. (2014). Claims run-off uncertainty: the full picture. SSRN Manuscript ID [5] R package ChainLadder (2015). CRAN package, Package Vignette. 23

25 Workshop at UNSW 24