A Modern Approach to Group Risk Pricing and Credibility

.................................................................................................................................................. 6 1 Introductory Comments .................................................................................................................. 8 1.1 Background ............................................................................................................................. 8 1.2 Purpose .................................................................................................................................... 8 1.3 Grounds of Conclusions .......................................................................................................... 9 1.4 The Scope of This Paper ......................................................................................................... 9 1.4.1 Product Scope ................................................................................................................. 9 1.4.2 Trends and Other Time Effects ....................................................................................... 9 1.4.3 Risk Premium .................................................................................................................. 9 2 An Overview of Key Concepts ..................................................................................................... 10 2.1 Cross Subsidisation and Anti-Selection ................................................................................ 10 2.2 Credibility and Anti-Selection .............................................................................................. 11 2.3 Key Definitions ..................................................................................................................... 11 2.3.1 Actual ............................................................................................................................ 11 2.3.2 Underlying .................................................................................................................... 11 2.3.3 Expected ........................................................................................................................ 11 2.3.4 Underlying Rating Factor (URF) .................................................................................. 12 2.4 The Framework for Credibility Modelling ........................................................................... 12 2.4.1 The Relationship Between the Expected and Underlying Claims ................................ 12 2.4.2 The Relationship Between the Underlying and Actual Claims ..................................... 14 2.4.3 The Credibility Formula ................................................................................................ 15 2.4.4 Concluding Comments on the Framework ................................................................... 16 2.5 Concluding Comments .......................................................................................................... 16 3 An Overview of Existing Credibility Models ............................................................................... 17 3.1 Bayesian Credibility Models ................................................................................................. 17 3.2 The Bühlmann Credibility Models ....................................................................................... 18 3.2.1 Derivation of the Simple Bühlmann Model (SBM) ...................................................... 18 3.2.2 Estimation of the Credibility Factor Components ........................................................ 20 3.2.3 A Discussion of the SBM Credibility Formula ............................................................. 21 3.2.4 A Discussion of the SBM’s Input Parameters ............................................................... 21 3.2.5 Limitations of the BSM and SBM ................................................................................ 23 3.2.6 Concluding Comments on the Limitations of the BSM and SBM ................................ 24 3.3 The Limited Fluctuation Model (LFM) ................................................................................ 25 A Modern Approach to Group Risk Pricing and Credibility Page 3 of 72 3.3.1 Derivation of the Full Credibility Requirement ............................................................ 25 3.3.2 Derivation of the Partial Credibility Formula ............................................................... 26 3.3.3 Limitations of the LFM ................................................................................................. 27 3.3.4 Concluding Comments on the Limitations of the LFM ................................................ 28 3.4 Some Specific Models Used in Group Risk Pricing ............................................................. 29 3.4.1 Model 1 ......................................................................................................................... 29 3.4.2 Model 2 ......................................................................................................................... 29 3.4.3 Model 3 ......................................................................................................................... 30 4 A New Approach to Group Risk Pricing ...................................................................................... 31 4.1 The Credibility Formula........................................................................................................ 31 4.2 The Conditional Actual Claims Distributions ....................................................................... 31 4.3 The Underlying Claims Distribution ..................................................................................... 31 4.4 The Interaction Between the Two Models ............................................................................ 31 4.4.1 The Proposed Credibility Model’s Inputs and Outputs ................................................. 31 4.4.2 The URF Distribution Model’s Inputs and Outputs ...................................................... 31 4.4.3 The Management of the Two Models’ Interactions in Practice .................................... 32 4.5 The Value Added by the Two Models .................................................................................. 34 4.5.1 The Credibility Model ................................................................................................... 34 4.5.2 The URF Distribution Model ........................................................................................ 34 4.6 The Simulation Based Approach and its ‘Fringe Benefits’ .................................................. 35 5 The Proposed Credibility Model for Group Risk .......................................................................... 36 5.1 Assumptions .......................................................................................................................... 36 5.1.1 Assumption 1: Modelling of Individual Member Claim Amounts ............................... 36 5.1.2 Assumption 2: Modelling of Underlying Probabilities of Claim at Member Level ..... 36 5.1.3 Assumption 3: Modelling of the Actual Claims Amount ............................................. 37 5.2 The Proposed Credibility Model ........................................................................................... 38 5.2.1 The Credibility Formula ................................................................................................ 38 5.2.2 Calculation of the Components of the Credibility Formula .......................................... 39 6 The URF Distribution Model ........................................................................................................ 40 7 Limitations of the Proposed Approach ......................................................................................... 41 7.1 The Assumption of Inter-Plan Independence ........................................................................ 41 7.2 The URF Distribution Model’s Results ................................................................................ 41 7.3 The Magnitude of Pricing Accuracy Improvement .............................................................. 41 8 The Credibility Model Assessment Method ................................................................................. 42 8.1 An Overview of the Credibility Model Assessment Method ................................................ 42 A Modern Approach to Group Risk Pricing and Credibility Page 4 of 72 8.2 The Hypothetical Portfolio.................................................................................................... 44 8.3 The Generated Claims ........................................................................................................... 45 8.3.1 ‘Claims 1’ ...................................................................................................................... 45 8.3.2 ‘Claims 2’ ...................................................................................................................... 45 8.3.3 ‘Claims 3’ ...................................................................................................................... 46 9 Model Results Without Competition ............................................................................................ 47 9.1 Results of the Proposed Model ............................................................................................. 47 9.1.1 The URF Distribution ................................................................................................... 47 9.1.2 Premiums by Plan ......................................................................................................... 48 9.1.3 Premiums by Claims Event ........................................................................................... 49 9.2 Results of Model 1 ................................................................................................................ 51 9.2.1 Premiums by Plan ......................................................................................................... 51 9.2.2 Premiums by Claims Event ........................................................................................... 52 9.3 Results of Model 2 ................................................................................................................ 53 9.3.1 Premiums by Plan ......................................................................................................... 53 9.3.2 Premiums by Claims Event ........................................................................................... 54 9.4 Results of Model 3 ................................................................................................................ 55 9.4.1 Premiums by Plan ......................................................................................................... 55 9.4.2 Premiums by Claims Event ........................................................................................... 56 10 Existing Models and Proposed Model in Competition ............................................................. 57 10.1 No Competition ..................................................................................................................... 57 10.2 In Competition ...................................................................................................................... 57 10.2.1 Proposed Model versus Model 1 ................................................................................... 57 10.2.2 Proposed Model versus Model 2 ................................................................................... 57 10.2.3 Proposed Model versus Model 3 ................................................................................... 58 10.2.4 All Four Models in Competition ................................................................................... 58 10.3 The Premium Differences ..................................................................................................... 58 11 Interpretation and Limitations of the Results ............................................................................ 59 12 Summary of Key Points ............................................................................................................ 60 13 Appendix ................................................................................................................................... 61 13.1 Single Factor Bayesian Approach ......................................................................................... 61 13.1.1 The Two-Urn Model ..................................................................................................... 61 13.1.2 The Bayesian Approach ................................................................................................ 61 13.2 Derivation of the LFM .......................................................................................................... 62 13.2.1 Derivation of the Mean and Variance of Total Claims ................................................. 62 A Modern Approach to Group Risk Pricing and Credibility Page 5 of 72 13.2.2 Derivation of the Full Credibility Requirement ............................................................ 63 13.2.3 Derivation of the Partial Credibility Formula ............................................................... 64 13.3 Derivation of the Proposed Model ........................................................................................ 65 13.4 The Determination of the Conditional Actual Claims Distribution ...................................... 66 13.5 The Hypothetical Portfolio’s Assumptions ........................................................................... 68 14 Acknowledgements ................................................................................................................... 72 15 References ................................................................................................................................. 72 A Modern Approach to Group Risk Pricing and Credibility Page 6 of 72 Abstract Purpose 1. Increase understanding of the core principles of credibility and its significance to group risk pricing The purpose of this paper is to: 2. Increase understanding of the assumptions and limitation of existing credibility models used in group risk pricing 3. Introduce an alternative approach to group risk pricing and credibility to discuss, refine and possibly implement in the medium to long term • Group risk lump sum products Scope The scope of this paper is limited to: • The analysis of trend adjusted claims experience • The determination of risk premium net of the catastrophe risk premium 1. Credibility models are a requirement of competition and should be tested in competition Key Points 2. Credibility models are rating mechanisms; in competition, inaccurate credibility models can lead to anti-selection and portfolio level underpricing 3. Different credibility models can lead to very different estimates of risk premium. The choice of credibility model can have a significant impact on an insurer’s portfolio composition and profitability 4. The determination of a theoretically sound credibility weighted risk premium involves: • Sufficient consideration for and responsiveness to the effectiveness of the insurer’s rating classes and the accuracy of its base rates and loading factors • Sufficient consideration for and responsiveness to the variability and reliability of the group risk plan’s historical claims experience • The use of a theoretically accurate method of calculating the credibility adjusted risk premium on the basis of the two abovementioned inputs 5. On theoretical grounds, both the Bühlmann-Straub model and the Limited Fluctuation model have significant limitations in addressing each of the three abovementioned steps 6. The proposed credibility model theoretically accounts for all three steps 7. The results of a simulation of a hypothetical group risk portfolio support the theoretical arguments posed above Model Proposed Model Model 1 Model 2 Model 3 Prem Won/28,000 $948,063 $295,025 $0 $308,917 Expected Future Claims/28,000 $922,489 $632,815 $0 $603,694 Premium/Expected Future Claims 103% 47% NA 51% Proportion of Business Won 61% 19% 0% 20% Table 1: The figures in the table are based on a simulation of 28,000 hypothetical group risk plans over a period of 5 years. The simulations are used to produce claims data for each plan. The plan and claims information is inserted into four competing credibility models to produce four quotations for each hypothetical plan. The winning model is determined on the basis of cheapest premium. Models 1 and 2 are variations of the Bühlmann-Straub model, Model 3 is a Limited Fluctuation model. A Modern Approach to Group Risk Pricing and Credibility Page 7 of 72