Modelling the Claims Development Result for Solvency Purposes

Transcription

1 Modelling the Claims Development Result for Solvency Purposes Mario V Wüthrich ETH Zurich Financial and Actuarial Mathematics Vienna University of Technology October 6, 2009 wwwmathethzch/ wueth c 2009 (Mario Wüthrich, ETH Zurich) Collaboration Joint work with: Michael Merz (University of Hamburg) Hans Bühlmann (ETH Zurich) Massimo De Felice (University of Rome) Alois Gisler (ETH Zurich) Franco Moriconi (University of Perugia) c 2009 (Mario Wüthrich, ETH Zurich) 1

2 Overview 1 Non-Life (NL) Insurance and Claims Reserving 2 Claims Development Result and Solvency 3 Bayesian Chain-Ladder Method 4 Conclusions c 2009 (Mario Wüthrich, ETH Zurich 1 Non-Life (NL) Insurance and Claims Reserving NL insurance company: accounting year I + 1 = 2009 budget statement at 1/1/2009 profit & loss (P&L) statement at 31/12/2009 Budget P&L 1/1/ /12/2009 premium earned claims incurred current accident year loss experience prior accident years administrative expenses investment income income before taxes c 2009 (Mario Wüthrich, ETH Zurich) 3

3 Questions and Terminology What is the position loss experience prior accident years? Why do we predict it by 0? What is the uncertainty in this prediction? What are (best estimate) claims reserves? Long term view versus the short term view These questions can not be answered with simple concepts c 2009 (Mario Wüthrich, ETH Zurich) 4 NL Claims Settlement Process accident date claims payments reopening reporting date claims closing payments claims closing Premium insurance period time Often it takes several years until a claim is finally settled Reasons: 1 Reporting delay: time lag between accident date and reporting date (notification at insurance company) 2 Settlement delay: time interval between reporting date and final settlement (severity of claim, recovery process, court decisions, etc) 3 Reopenings due to new (unexpected) claim developments c 2009 (Mario Wüthrich, ETH Zurich) 5

4 Conclusions: Claims Reserving Every claim generates a (random) future payment cashflow The claims reserves should suffice to meet this future cashflow = claims reserving is a prediction problem Determine the prediction uncertainty: deterministic claims reserves stochastic claims payments c 2009 (Mario Wüthrich, ETH Zurich) 6 Prediction Uncertainty X future cashflow (random variable) to be predicted D information available at time I Assume X is a D-measurable predictor for X The (conditional) mean square error of prediction (MSEP) is defined by ( ) msep X D X = E [ ( X X ) 2 D] MSEP is the most common uncertainty measure in practice c 2009 (Mario Wüthrich, ETH Zurich) 7

5 2 Claims Development Result and Solvency cumulative payments C_k accounting years incremental payments in accounting year k 0 are denoted by X k cumulative payments are denoted by C j = j k=0 X k ultimate (total) claim is denoted by C J observations at time k are denoted by D k = {C j : j k} c 2009 (Mario Wüthrich, ETH Zurich) 8 Ultimate Claims Prediction Goal: Predict the ultimate claim C J based on D k, k J At time k, we use the (minimum variance) predictor Ĉ (k) J = E [C J D k ] First Consequences: Ĉ(k) J minimizes the conditional MSEP R (k) = Ĉ(k) J C k are the best-estimate reserves at time k Ĉ(0) J, Ĉ(1) J, Ĉ(2) J, is a martingale c 2009 (Mario Wüthrich, ETH Zurich) 9

6 Time Series Ultimate Claims Prediction 1200 ultimate claim prediction premium accounting years cumulative payments best-estimate reserves c 2009 (Mario Wüthrich, ETH Zurich) 10 Classical Uncertainty View Ĉ (k) J minimizes the conditional MSEP: (Ĉ(k) ) msep CJ D k J = E [ ( C J Ĉ(k) J D k ] = Var (C J D k ) Henceforth, we obtain the prediction variance Classical long term view: Study this prediction variance: Eg Mack (1993), England-Verrall (2002), W-Merz (2008) This is not the Solvency View (short term view) for the P&L position loss experience prior accident years c 2009 (Mario Wüthrich, ETH Zurich) 11

7 P&L Position Loss Experience Prior Accident Years The Claims Development Result (CDR) at time k is given by CDR(k) =Ĉ(k 1) J Ĉ(k) J (1) This is the incorporation of the latest information D k available at time k, ie update of information D k 1 D k CDR exactly corresponds to loss experience prior accident years Under Solvency 2 we need to study the uncertainty in this position: Question: Do we have sufficient provisions at time k to cover possible shortfalls in CDR? c 2009 (Mario Wüthrich, ETH Zurich) 12 Time Series of Claims Development Results Short term view: changes over the next accounting year: 8'000 7'000 6'000 claims development result 5'000 4'000 3'000 2'000 1'000 0 time I time I+1 payments in (I,I+1] reserves Therefore: study the Claims Development Result (CDR) c 2009 (Mario Wüthrich, ETH Zurich) 13

8 Properties of Claims Development Results We have (martingale property) E [CDR(k) D k 1 ]=0 (2) Moreover, this implies CDR(1), CDR(2), are uncorrelated (not independent) This immediately implies that (Ĉ(0) ) msep CJ D 0 J = Var (C J D 0 ) (3) =Var CDR(k) k 1 D 0 = Var (CDR(k) D 0 ) k 1 c 2009 (Mario Wüthrich, ETH Zurich) 14 Budget and P&L Statement Equation (2) implies prediction 0 of the CDR: Budget P&L 1/1/ /12/2009 premium earned claims incurred current accident year loss experience prior accident years administrative expenses investment income income before taxes Prediction Uncertainty in accounting year k: study msep CDR(k) Dk 1 (0) = Var (CDR(k) D k 1 ) c 2009 (Mario Wüthrich, ETH Zurich) 15

9 3 Bayesian Chain-Ladder Method We use the following notation: accident years are denoted by i {0,,I} development years are denoted by j {0,,J} accounting years are given by i + j = k (constant) incremental payments are denoted by X i,j cumulative payments are denoted by C i,j = j X i,l l=0 c 2009 (Mario Wüthrich, ETH Zurich) 16 Loss Development Triangle at Time I accident development years j year i j J 0 1 observations D 0 ị I 2 I 1 to be predicted D c 0 I observations: D k = {C i,j : i + j I + k} to be predicted: D c k = {C i,j : i + j>i+ k, i I} c 2009 (Mario Wüthrich, ETH Zurich) 17

10 Example 1: Cumulative Payments Observed historical cumulative payments at time I =9 D 0 = {C i,j : i + j 9} c 2009 (Mario Wüthrich, ETH Zurich) 18 Bayesian CL Model Assumptions Conditional on F =(F 0,,F J 1 ) we have different accident years i are independent; {C i,j } j 0 is a Markov process with E [C i,j C i,j 1, F] = F j 1 C i,j 1, for all i, j Var (C i,j C i,j 1, F) = σ 2 j 1(F j 1 ) C i,j 1, for all i, j The components of F are independent Note that the CL factors F are part of the model which allows for Bayesian inference and parameter uncertainty study within the model c 2009 (Mario Wüthrich, ETH Zurich) 19

11 Properties of the Bayesian CL Model Theorem 1 The posteriors of F, given D k,areindependent Theorem 2 The minimum variance predictor for C i,j is given by Ĉ (k) i,j = E [C i,j D k ]=C i,i i+k J 1 j=i i+k E [F j D k ] (Ĉ(0) Theorem 3 The prediction uncertainties msep Ci,J D 0 i,j msep CDRi (1) D 0 (0) can be calculated analytically ) and See Gisler-W (2008) and Bühlmann et al (2009) c 2009 (Mario Wüthrich, ETH Zurich0 Estimation of CL Factors We choose a non-informative prior distribution for F In that case we obtain at time k f (k) j = E [F j D k ]= (I j 1+k) I i=0 C i,j+1 (I j 1+k) I i=0 C i,j Henceforth, the CL predictor for C i,j at time k is given by Ĉ (k) i,j = E [C i,j D k ]=C i,i i+k J 1 j=i i+k f (k) j c 2009 (Mario Wüthrich, ETH Zurich1

12 Prediction Uncertainty (Linear Approximation) msep Ci,J D 0 (Ĉ(0) i,j [ J 1 j=i i ) = σ 2 j / ( f (0) j Ĉ (0) i,j (Ĉ(0) i,j + J 1 j=i i ( σ j 2/ (0) ] f j I j 1, l=0 C l,j msep CDRi (1) D 0 (0) = [ σ 2 I i / ( f (0) I i Ĉ (0) i,i i (Ĉ(0) i,j ( σ I i 2 / (0) f I i + i 1 k=0 C k,j + J 1 j=i i+1 C I j,j I j l=0 C l,j ( σ j 2/ (0) ] f j I j 1 l=0 C l,j See Gisler-W (2008), Merz-W (2008) and Bühlmann et al (2009) c 2009 (Mario Wüthrich, ETH Zurich2 Example 1, revisited i CL reserves b R (0) i msep CDRi (1) D (0) 1/2 msep bc (0) 1/2 0 Ci,J D 0 i,j % % % % % % % % % % % % % % % % % % cov Total % % We see that the ratio is around 90% c 2009 (Mario Wüthrich, ETH Zurich3

13 More Examples CL reserves R b (0) msep CDR(1) D0 (0) 1/2 msep bc (0) 1/2 CJ D 0 J Example 2 (commercial liability) % % Example 3 (Merz-W (2008) % % We see that the ratio is around 60% in Example 2 We see that the ratio is around 75% in Example 3 c 2009 (Mario Wüthrich, ETH Zurich4 Example: Italian MTPL (37 Companies) company business msep total runoff msep CDR(1) msep CDR(1) msep total runoff volume (in % reserves) (in % reserves) (in %) Total For more explanation see Bühlmann et al (2009) c 2009 (Mario Wüthrich, ETH Zurich5

14 4 Conclusions In all examples considered: the ratio between one-year CDR risk and full run-off risk was within the intervall [50%, 95%] (range between liability insurance and property insurance) This is also supported by the AISAM-ACME field study 2007 We have measured risk with the help of the conditional MSEP For Value-at-Risk or Expected Shortfall considerations fit distribution with appropriate moments (= proxy) A full distributional approach can only be solved numerically, eg Markov chain Monte Carlo (MCMC) methods (Bayesian models) c 2009 (Mario Wüthrich, ETH Zurich6 Conclusions Dependence is not appropriately modelled Especially, accounting year dependence and claims inflation needs special care (MCMC methods) The one-year CDR view needs a Cost-of-Capital charge (risk margin) for the risk that is beyond the one-year time horizon, market-value margin, see Salzmann-W (2009) Discounting and financial risk is not considered First results on this topic are obtained in W-Bühlmann (2009) c 2009 (Mario Wüthrich, ETH Zurich7

15 References [1] Bühlmann, H, De Felice, M, Gisler, A, Moriconi, F, Wüthrich, MV (2009) Recursive credibility formula for chain ladder factors and the claims development result Astin Bulletin 39/1, [2] Gisler, A, Wüthrich, MV (2008) Credibility for the chain ladder reserving method Astin Bulletin 38/2, [3] Merz, M, Wüthrich, MV (2008) Modelling the claims development result for solvency purposes CAS E-Forum, Fall 2008, [4] Salzmann, R, Wüthrich, MV (2009) Cost-of-capital margin for a general insurance runoff Submitted preprint [5] Wüthrich, MV, Bühlmann, H (2009) The one-year runoff uncertainty for discounted claims reserves To appear in Giornale dell Istituto Italiano degli Attuari [6] Wüthrich, MV, Merz, M (2008) Stochastic Claims Rerserving Methods in Insurance Wiley Finance c 2009 (Mario Wüthrich, ETH Zurich8