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PRICING IN GENERAL INSURANCE PRICING IN GENERAL INSURANCE Pietro Parodi CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2015 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20140626 International Standard Book Number-13: 978-1-4665-8148-7 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http:// www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com To my parents, for their unfailing support and patience over the years; to my sister, for her constant encouragement; and to Vincent, for his lasting and undiminishing intellectual influence. Contents Preface.......................................................................................................................................... xxiii Acknowledgements.................................................................................................................... xxv Section I  Introductory Concepts 1. Pricing Process: A Gentle Start.............................................................................................3 1.1 An Elementary Pricing Example................................................................................. 3 1.1.1 Risk Costing (Gross)......................................................................................... 5 1.1.1.1 Adjusting for ‘Incurred but Not Reported’ Claims......................5 1.1.1.2 Adjusting for Claims Inflation........................................................6 1.1.1.3 Adjusting for Exposure Changes.................................................... 7 1.1.1.4 Adjusting for Other Risk-Profile Changes..................................... 9 1.1.1.5 Adjusting for ‘Incurred but Not Enough Reserved’ Claims....... 9 1.1.1.6 Changes for Cover/Legislation..................................................... 10 1.1.1.7 Other Corrections............................................................................ 10 1.1.2 Risk Costing (Ceded)...................................................................................... 11 1.1.3 Determining the Technical Premium.......................................................... 11 1.1.3.1 Loading for Costs............................................................................ 11 1.1.3.2 Discounting for Investment Income............................................. 11 1.1.3.3 Loading for Capital/Profit............................................................. 12 1.1.4 Commercial Considerations and the Actual Premium............................. 12 1.1.5 Limitations of This Elementary Pricing Example...................................... 13 1.2 High-Level Pricing Process........................................................................................ 13 1.3 Questions....................................................................................................................... 14 2. Insurance and Reinsurance Products................................................................................ 17 2.1 Classification of General Insurance Products: General Ideas................................ 17 2.2 Products by Category of Cover.................................................................................. 18 2.2.1 Property Insurance......................................................................................... 18 2.2.1.1 Legal Framework............................................................................. 18 2.2.1.2 Pricing Characteristics.................................................................... 19 2.2.1.3 Examples of Property Policies....................................................... 20 2.2.2 Liability Insurance.......................................................................................... 20 2.2.2.1 Legal Framework............................................................................. 20 2.2.2.2 Pricing Characteristics....................................................................22 2.2.2.3 Examples of Liability Insurance...................................................22 2.2.3 Financial Loss Insurance............................................................................... 23 2.2.3.1 Examples........................................................................................... 23 2.2.3.2 Pricing Characteristics.................................................................... 23 2.2.4 Fixed Benefits Insurance................................................................................ 23 2.2.4.1 Examples........................................................................................... 23 2.2.4.2 Pricing Characteristics.................................................................... 23 2.2.5 ‘Packaged’ Products........................................................................................ 24 vii viii Contents 2.3 Products by Type of Customer................................................................................... 24 2.3.1 Personal Lines................................................................................................. 24 2.3.1.1 Examples........................................................................................... 24 2.3.1.2 Pricing Characteristics.................................................................... 24 2.3.2 Commercial Lines........................................................................................... 25 2.3.2.1 Examples........................................................................................... 25 2.3.2.2 Pricing Characteristics.................................................................... 26 2.3.3 Reinsurance..................................................................................................... 26 2.3.3.1 Examples........................................................................................... 26 2.3.3.2 Pricing Characteristics.................................................................... 27 2.4 Prudential Regulation Authority Classification...................................................... 27 2.5 Other Classification Schemes..................................................................................... 38 2.6 Non-Extant Products................................................................................................... 38 2.7 Questions....................................................................................................................... 38 3. The Policy Structure............................................................................................................. 41 3.1 Personal Lines............................................................................................................... 41 3.1.1 Purpose of the Excess Amount.....................................................................42 3.1.2 Purpose of the Limit.......................................................................................43 3.2 Commercial Lines........................................................................................................43 3.2.1 Policy Bases......................................................................................................43 3.2.1.1 Occurrence Basis.............................................................................43 3.2.1.2 Claims-Made Basis..........................................................................43 3.2.1.3 Other Bases......................................................................................44 3.2.2 Basic Policy Structure.....................................................................................44 3.2.2.1 Each-and-Every-Loss Deductible..................................................44 3.2.2.2 Annual Aggregate Deductible......................................................44 3.2.2.3 Each-and-Every-Loss Limit............................................................ 45 3.2.3 Other Coverage Modifiers............................................................................. 49 3.2.3.1 Non-Ranking Each-and-Every-Loss Deductible......................... 49 3.2.3.2 Residual Each-and-Every-Loss Deductible.................................. 49 3.2.3.3 Quota Share...................................................................................... 49 3.2.3.4 Yet-More-Exotic Coverage Modifiers............................................ 50 3.3 Reinsurance.................................................................................................................. 50 3.3.1 Policy Bases...................................................................................................... 50 3.3.1.1 Losses Occurring During.............................................................. 50 3.3.1.2 Risk Attaching During................................................................... 50 3.3.1.3 Claims Made.................................................................................... 51 3.3.2 Non-Proportional Reinsurance..................................................................... 51 3.3.2.1 Risk Excess of Loss.......................................................................... 51 3.3.2.2 Aggregate Excess of Loss............................................................... 55 3.3.2.3 Catastrophe Excess of Loss............................................................ 55 3.3.2.4 Stop Loss Reinsurance.................................................................... 57 3.3.3 Proportional Reinsurance.............................................................................. 58 3.3.3.1 Quota Share...................................................................................... 58 3.3.3.2 Surplus Reinsurance....................................................................... 58 3.4 Questions....................................................................................................................... 59 Contents ix 4. The Insurance Markets......................................................................................................... 61 4.1 Major Participants in the Insurance Market............................................................ 61 4.1.1 Buyers of Insurance........................................................................................ 61 4.1.2 Insurers............................................................................................................63 4.1.2.1 Insurance and Reinsurance Companies......................................63 4.1.2.2 Lloyd’s of London............................................................................64 4.1.2.3 Captives............................................................................................65 4.1.2.4 Pools and Self-Retention Groups (Excluding Captives)............. 67 4.1.2.5 P&I Clubs.......................................................................................... 67 4.1.2.6 Diversification: How Risk Is Spread in the Insurance Market..... 67 4.1.3 Intermediaries................................................................................................. 68 4.1.3.1 Brokers.............................................................................................. 69 4.1.3.2 Tied Agents...................................................................................... 69 4.1.3.3 Independent Financial Advisors and Consultancy Firms........ 70 4.2 Major Insurance Markets............................................................................................ 70 4.2.1 London Market................................................................................................ 70 4.2.1.1 Slip System....................................................................................... 71 4.2.2 Bermuda and Similar Markets...................................................................... 72 4.3 How Is Insurance Sold?............................................................................................... 72 4.3.1 Personal Lines................................................................................................. 72 4.3.2 Commercial Lines and Reinsurance............................................................ 72 4.4 Underwriting Cycle..................................................................................................... 73 4.4.1 How Does the Underwriting Cycle Affect Pricing?.................................. 75 4.5 Questions....................................................................................................................... 76 5. Pricing in Context..................................................................................................................77 5.1 Regulatory Environment............................................................................................77 5.1.1 Product Restrictions.......................................................................................77 5.1.2 Premium Restrictions.....................................................................................77 5.1.3 Information Restriction.................................................................................. 78 5.1.4 Capital Requirements..................................................................................... 78 5.2 Legal Environment...................................................................................................... 78 5.2.1 Changes in the Legal Environment May Increase or Decrease the Number and Severity of Claims................................................................... 78 5.2.2 Court Rulings Filter Down to Out-of-Court Claims.................................. 79 5.2.3 Lost in Translation.......................................................................................... 79 5.2.4 Court Inflation.................................................................................................80 5.2.5 General Trends in Behaviour and Awareness............................................80 5.3 Larger Economy...........................................................................................................80 5.3.1 How Does This Affect Pricing?.................................................................... 81 5.4 Investment Conditions................................................................................................ 81 5.4.1 How Does This Affect Pricing?.................................................................... 81 5.5 Currency Movements.................................................................................................. 82 5.5.1 How Does This Affect Pricing?.................................................................... 82 5.6 Natural Environment..................................................................................................83 5.6.1 Weather.............................................................................................................83 x Contents 5.6.2 Space Weather.................................................................................................83 5.6.3 Natural Catastrophes.....................................................................................84 5.6.4 How Does This Affect Pricing?.................................................................... 85 5.7 Pricing in the Corporate Context............................................................................... 85 5.7.1 Planning Function.......................................................................................... 86 5.7.2 Underwriting Function.................................................................................. 86 5.7.3 Claims Function.............................................................................................. 86 5.7.3.1 Interaction with the Pricing Function.......................................... 86 5.7.4 Actuarial: Reserving....................................................................................... 86 5.7.4.1 Interaction with the Pricing Function.......................................... 87 5.7.5 Actuarial: Capital Modelling........................................................................ 87 5.7.5.1 Interaction with the Pricing Function.......................................... 87 5.7.6 Finance............................................................................................................. 87 5.7.6.1 Interaction with the Pricing Function.......................................... 88 5.7.7 Management Information.............................................................................. 88 5.7.7.1 Interaction with the Pricing Function.......................................... 88 5.7.8 Investment Function....................................................................................... 88 5.7.8.1 Interaction with the Pricing Function.......................................... 88 5.7.9 Reinsurance Function.................................................................................... 88 5.7.9.1 Interaction with the Pricing Function.......................................... 89 5.7.10 Other Functions.............................................................................................. 89 5.8 Other Things You Need to Keep Up With............................................................... 89 5.9 Questions.......................................................................................................................90 Section II  The Core Pricing Process 6. The Scientific Basis for Pricing: Risk Loss Models and the Frequency/ Severity Risk Costing Process............................................................................................ 93 6.1 Aggregate Loss Models............................................................................................... 93 6.1.1 Individual Risk Model................................................................................... 93 6.1.1.1 Example: Horse Insurance............................................................. 94 6.1.1.2 Another Example: Credit Insurance............................................. 94 6.1.2 Collective Risk Model.................................................................................... 95 6.1.2.1 Example: Modelling Motor Fleet Losses...................................... 97 6.2 Applications to Risk Costing...................................................................................... 98 6.3 Risk-Costing Process for the Frequency/Severity Model...................................... 99 6.4 Questions..................................................................................................................... 100 7. Familiarise Yourself with the Risk.................................................................................. 101 7.1 Things to Look Out for (Commercial Lines/Reinsurance).................................. 101 7.1.1 What Does the Client Do, That Is, What Is Its Business?........................ 101 7.1.2 What Are the Client’s Main Locations and the Main Numbers (Such as Number of Employees, Payroll and Turnover)?........................ 102 7.1.3 Have There Been Any Notable Changes in the Risk Profile of the Client over Time?.......................................................................................... 102 7.1.4 Have There Been Any Important Mergers/Acquisitions or Divestitures in the Company over the Years?........................................... 102 xi Contents 7.1.5 Has the Company Been Implicated in Any Large Claims That Have Possibly Not Appeared in Any Report Yet?.................................... 103 7.1.6 Industry-Related Questions........................................................................ 103 7.1.7 Sources of Knowledge on the Risk............................................................. 103 7.2 Things to Look Out for (Personal Lines)................................................................ 104 7.3 Questions..................................................................................................................... 104 8. Data Requirements for Pricing......................................................................................... 105 8.1 Policies and Cover Data............................................................................................ 105 8.2 Claims Data................................................................................................................. 107 8.3 Exposure Data............................................................................................................ 111 8.3.1 What Is Exposure?........................................................................................ 111 8.3.2 Exposure Data for Experience Rating........................................................ 111 8.3.2.1 Criteria for a Good Measure of Exposure................................. 112 8.3.3 Exposure Data for Exposure Rating.......................................................... 113 8.4 Portfolio and Market Data........................................................................................ 113 8.5 Questions..................................................................................................................... 114 9. Setting the Claims Inflation Assumptions.................................................................... 115 9.1 Sources of Inflation Information.............................................................................. 115 9.2 Data-Driven Estimates of Claims Inflation............................................................ 117 9.2.1 Statistical Estimation of Claims Inflation Using Robust Statistics........ 117 9.2.2 Basket Method, Based on Selected Samples............................................. 119 9.2.3 Statistical Approach to Large Losses Inflation......................................... 120 9.3 Questions..................................................................................................................... 121 10. Data Preparation.................................................................................................................. 123 10.1 Data Cleansing........................................................................................................... 123 10.2 Data Transformation.................................................................................................. 124 10.2.1 Claims Revaluation...................................................................................... 124 10.2.1.1 Which Claims Do We Need to Revalue, What Amount Do We Have to Revalue Exactly?................................................ 126 10.2.1.2 Other Considerations.................................................................... 127 10.2.2 Currency Conversion................................................................................... 127 10.2.3 Policy Year Allocation.................................................................................. 128 10.2.4 Individual Loss Adjustments for Exposure Changes............................. 128 10.3 Data Summarisation.................................................................................................. 129 10.4 Questions..................................................................................................................... 130 11. Burning Cost Analysis....................................................................................................... 133 11.1 Burning Cost Methodology...................................................................................... 134 11.1.1 Data and Assumptions................................................................................ 134 11.1.2 Input to the Burning Cost Analysis........................................................... 134 11.1.3 Step A: Revalue Losses and Apply Policy Modifiers to All Claims...... 134 11.1.4 Step B: Aggregate Loss by Policy Year....................................................... 134 11.1.5 Step C: Adjust for Exposure Changes........................................................ 136 11.1.6 Step D: Make Adjustments to IBNR and IBNER...................................... 137 11.1.7 Step E: Make Adjustments for Large Losses............................................. 137 11.1.8 Step F: Make Other Adjustments................................................................ 140 xii Contents 11.1.9 Step G: Impose Aggregate Deductibles/Limits as Necessary............... 140 11.1.10 Step H: Calculate the Burning Cost........................................................... 140 11.1.11 Step I: Premium Calculation Based on the Burning Cost....................... 143 11.2 Adjusting for IBNR and IBNER: An In-Depth Look............................................ 146 11.3 Producing an Aggregate Loss Distribution........................................................... 149 11.4 Limitations.................................................................................................................. 152 11.5 Questions..................................................................................................................... 153 12. What Is This Thing Called Modelling?.......................................................................... 157 12.1 Why Do We Need Modelling at All?...................................................................... 157 12.1.1 Models Allow Us to Make Predictions That Generalise Loss Experience...................................................................................................... 158 12.1.2 Other Uses of Models................................................................................... 160 12.2 Modelling Approach................................................................................................. 160 12.3 How Do You Select a Good Model? A Foray into Statistical Learning.............. 162 13. Frequency Modelling: Adjusting for Claim Count IBNR.......................................... 167 13.1 Input to IBNR Adjustment........................................................................................ 167 13.2 Triangle Development Methods for IBNR Estimation......................................... 168 13.2.1 Method 1: Ignore the Last Diagonal........................................................... 171 13.2.2 Method 2: Gross Up the Last Diagonal..................................................... 172 13.2.3 Method 3: Choose an ’Ad Hoc’ Observation Period................................ 172 13.2.4 Method 4: Shift the Observation Period.................................................... 175 13.2.5 Comparison of Methods 1 through 4......................................................... 176 13.3 Triangle-Free Method for IBNR Estimation........................................................... 177 13.3.1 Estimating the Reporting Delay Distribution.......................................... 177 13.3.2 Projecting Claim Counts to Ultimate......................................................... 180 13.3.3 Uncertainty.................................................................................................... 182 13.3.4 General Case.................................................................................................. 183 13.3.5 Analysis of Zero Claims.............................................................................. 183 13.3.5.1 Approach Based on the Number of All Claims........................ 183 13.3.5.2 Approach Based on the Number of Non-Zero Claims............ 184 13.3.6 Comparison between the Triangle-Free Approach and the Triangle-Based Approach to IBNER.......................................................... 185 13.4 Questions..................................................................................................................... 185 14. Frequency Modelling: Selecting and Calibrating a Frequency Model.................... 187 14.1 Binomial Distribution................................................................................................ 189 14.1.1 Simulating from a Binomial Distribution................................................. 190 14.1.2 Generalisation to the Case of Different Claim Probabilities.................. 190 14.2 Poisson Distribution.................................................................................................. 190 14.2.1 Facts about the Poisson Distribution......................................................... 191 14.2.2 Simulating from a Poisson Distribution.................................................... 192 14.2.2.1 Generating Random Poisson Variates in Excel......................... 192 14.2.2.2 Generating Random Poisson Variates in R................................ 193 14.2.3 Practical Considerations.............................................................................. 193 14.2.4 How Do You Calculate λ?............................................................................ 195 14.3 Negative Binomial Distribution............................................................................... 196 14.3.1 Technical Generalities.................................................................................. 196 Contents xiii 14.3.2 Simulating from a Negative Binomial Distribution................................ 196 14.3.2.1 Sampling from the Negative Binomial Distribution................ 197 14.3.3 Practical Considerations.............................................................................. 197 14.3.4 Fitting a Negative Binomial (i.e. Calculating r and β) to Past Claim Counts............................................................................................................. 197 14.3.4.1 Method of Moments...................................................................... 197 14.3.4.2 Maximum Likelihood Estimation.............................................. 198 14.4 Choosing and Calibrating a Frequency Model..................................................... 199 14.4.1 Using the Variance/Mean Ratio as a Selection Criterion........................ 200 14.4.2 Theoretical Arguments That Should Drive Model Selection................. 202 14.4.2.1 Inadequacies of Underdispersed Models.................................. 202 14.4.2.2 Inadequacies of the Poisson Model............................................ 203 14.4.2.3 Modelling Recommendations..................................................... 203 14.5 Results of the Frequency Analysis for Our Case Study....................................... 205 14.6 Questions..................................................................................................................... 207 15. Severity Modelling: Adjusting for IBNER and Other Factors................................... 209 15.1 Introduction................................................................................................................ 209 15.2 Methods for Identifying and Dealing with IBNER.............................................. 211 15.2.1 Method 1: Ignore IBNER.............................................................................. 211 15.2.2 Method 2: Just Use Closed Claims............................................................. 211 15.2.3 Method 3: Identify Trends and Adjust for IBNER................................... 211 15.2.3.1 Method 3a: Identify Trends Using Development Triangles.... 212 15.2.3.2 Method 3b: Identify Trends Using Multivariate Analysis...... 216 15.3 Comparison of Different Methods.......................................................................... 217 15.3.1 Method 1: Ignore IBNER.............................................................................. 217 15.3.2 Method 2: Just Use Closed Claims............................................................. 218 15.3.3 Method 3a: Identify Trends and Adjust for IBNER by Using Development Triangles................................................................................ 218 15.3.4 Method 3b: Identify Trends and Adjust for IBNER by Using GLM-Like Techniques.................................................................................. 218 15.4 Questions..................................................................................................................... 218 16. Severity Modelling: Selecting and Calibrating a Severity Model............................ 221 16.1 Input to Severity Modelling..................................................................................... 221 16.2 Choosing the Right Severity Model: A Subtle Problem.......................................223 16.2.1 General Rules for Model Selection.............................................................225 16.2.2 Experiment Revisited...................................................................................225 16.3 Modelling Large (‘Tail’) Losses: Extreme Value Theory...................................... 226 16.3.1 Pickand–Balkema–de Haan Theorem....................................................... 227 16.3.2 Identifying the Extreme Value Region...................................................... 228 16.3.3 Creating a Spliced Small/Large Loss Distribution.................................. 229 16.3.4 Calculating the Values of the Parameters................................................. 230 16.3.5 Extreme Value Theory: Actual Practice..................................................... 230 16.3.6 Limitations of Extreme Value Theory........................................................ 230 16.4 A Simple Strategy for Modelling Ground-Up Losses........................................... 231 16.4.1 Application to a Real-World Case Study................................................... 232 16.4.2 Kolmogorov–Smirnov Test.......................................................................... 232 16.4.3 Moving beyond the Lognormal Model.....................................................234 xiv Contents 16.5 Using Portfolio/Market Data.................................................................................... 235 16.5.1 Advantages of Using Market Data for Tail Modelling............................ 236 16.5.2 Disadvantages of Using External Data for Tail Modelling..................... 236 16.6 Appendix A: Kolmogorov–Smirnov Test............................................................... 236 16.7 Questions..................................................................................................................... 237 17. Aggregate Loss Modelling................................................................................................ 239 17.1 ‘Exact’ Solution for the Collective Risk Model...................................................... 239 17.2 Parametric Approximations..................................................................................... 241 17.2.1 Gaussian Approximation............................................................................. 241 17.2.2 Translated Gamma Approximation........................................................... 243 17.3 Numerical Quasi-Exact Methods............................................................................. 244 17.3.1 Fast Fourier Transform Method.................................................................. 245 17.3.1.1 Discrete Fourier Transform and Fast Fourier Transform........ 247 17.3.1.2 Practical Issues............................................................................... 248 17.3.1.3 Implementation............................................................................. 249 17.3.1.4 Output Example............................................................................ 249 17.3.2 Panjer Recursion........................................................................................... 249 17.3.2.1 Practical Issues............................................................................... 250 17.3.2.2 Implementation............................................................................. 251 17.4 Monte Carlo Simulation............................................................................................ 251 17.4.1 Practical Issues..............................................................................................254 17.4.2 Implementation.............................................................................................254 17.5 Coverage Modifications............................................................................................ 255 17.5.1 Gaussian Approximation............................................................................. 255 17.5.2 Fast Fourier Transform................................................................................. 256 17.5.2.1 Aggregate Retained Losses.......................................................... 256 17.5.2.2 Aggregate Ceded Losses.............................................................. 257 17.5.2.3 More Complex Structures............................................................ 257 17.5.3 Panjer Recursion........................................................................................... 258 17.5.4 The Monte Carlo Simulation....................................................................... 258 17.6 A Comparison of the Different Methods at a Glance........................................... 260 17.6.1 Conclusions.................................................................................................... 261 17.7 The Monte Carlo Simulation for the Individual Risk Model............................... 261 17.8 Appendix A: R Code for Producing an Aggregate Loss Distribution via the Monte Carlo Simulation...................................................................................... 261 17.9 Appendix B: R Code for Producing an Aggregate Loss Distribution via Fast Fourier Transform.............................................................................................. 263 17.10 Questions..................................................................................................................... 266 18. Identifying, Measuring and Communicating Uncertainty........................................ 269 18.1 Process Uncertainty................................................................................................... 270 18.1.1 How to Calculate Its Effect.......................................................................... 270 18.1.2 How to Communicate It............................................................................... 270 18.2 Parameter Uncertainty.............................................................................................. 270 18.2.1 How to Calculate Its Effect.......................................................................... 271 18.2.2 How to Communicate It............................................................................... 272 18.3 Model Uncertainty..................................................................................................... 273 18.3.1 How to Calculate Its Effect.......................................................................... 273 Contents xv 18.3.2 How to Communicate It............................................................................... 273 18.4 Assumption/Data Uncertainty................................................................................ 273 18.4.1 How to Calculate Its Effect.......................................................................... 274 18.4.1.1 Claims Inflation............................................................................. 274 18.4.1.2 Open Claims.................................................................................. 274 18.4.2 How to Communicate It............................................................................... 275 18.5 Approximation Errors in Calculations................................................................... 276 18.5.1 How to Calculate Its Effect.......................................................................... 276 18.5.2 How to Communicate It............................................................................... 276 18.6 Applications to the Risk Costing Process............................................................... 277 18.7 Questions..................................................................................................................... 277 19. From Costing to Pricing..................................................................................................... 279 19.1 Cost Plus in Detail..................................................................................................... 280 19.1.1 Expected Losses............................................................................................ 280 19.1.2 Allowance for Uncertainty.......................................................................... 280 19.1.3 Other Costs.................................................................................................... 283 19.1.4 Investment Income.......................................................................................284 19.1.5 Loading for Profit.......................................................................................... 285 19.1.6 Technical Premium....................................................................................... 286 19.1.7 Actual Premium Charged........................................................................... 286 19.2 Capital Considerations in Pricing............................................................................ 286 19.2.1 Overall Methodology................................................................................... 287 19.2.2 Risk Measures............................................................................................... 288 19.2.2.1 Coherent Risk Measures.............................................................. 288 19.2.2.2 Examples of Risk Measures......................................................... 289 19.2.3 Capital Allocation Methods........................................................................ 291 19.2.3.1 Proportional Spread...................................................................... 292 19.2.3.2 Game Theory................................................................................. 292 19.2.4 Worked-Out Example................................................................................... 293 19.2.4.1 Method 1: Proportional Spread................................................... 294 19.2.4.2 Method 2: Game Theory.............................................................. 294 19.2.4.3 Target Loss Ratio........................................................................... 296 19.3 Price Optimisation..................................................................................................... 296 19.3.1 Determining the Demand Function.......................................................... 298 19.3.1.1 Price Tests....................................................................................... 298 19.3.1.2 Cost of a Price Test........................................................................ 299 19.3.1.3 More General Case........................................................................300 19.3.2 Maximising the Total Expected Profits.....................................................300 19.3.3 Limitations..................................................................................................... 301 19.3.4 Other Considerations/Approaches............................................................ 301 19.4 Pricing Control Cycle................................................................................................. 302 19.4.1 Product Pricing............................................................................................. 302 19.4.2 Pricing as a Cycle.......................................................................................... 302 19.4.2.1 Pricing Framework versus Pricing Decisions........................... 303 19.4.3 Is the Pricing Control Cycle Really a Cycle?.............................................304 19.4.4 Monitoring.....................................................................................................304 19.4.4.1 Monitoring ‘Production’ Costs....................................................304 19.4.4.2 Monitoring Investments...............................................................305 xvi Contents 19.4.4.3 Monitoring Rates...........................................................................305 19.4.4.4 Monitoring Sales...........................................................................306 19.4.4.5 Monitoring the Portfolio Composition.......................................306 19.4.4.6 Monitoring the Competition........................................................ 307 19.4.4.7 Features of a Good Monitoring System..................................... 307 19.5 Questions.....................................................................................................................308 Section III  Elements of Specialist Pricing 20. Experience Rating for Non-Proportional Reinsurance............................................... 311 20.1 Types of Contract....................................................................................................... 312 20.2 Inputs........................................................................................................................... 313 20.2.1 Policy Information........................................................................................ 313 20.2.2 Claims Data................................................................................................... 314 20.2.3 Exposure Data............................................................................................... 316 20.2.3.1 Examples......................................................................................... 317 20.3 Frequency Analysis................................................................................................... 317 20.3.1 Exposure and Contract Basis...................................................................... 318 20.3.2 Incorporating Rate Changes When Using Original Premium as Exposure........................................................................................................ 318 20.3.2.1 Worked-Out Example................................................................... 318 20.3.3 IBNR Analysis............................................................................................... 319 20.3.4 Selection of a Frequency Model.................................................................. 320 20.4 Severity Analysis........................................................................................................ 320 20.4.1 IBNER Analysis............................................................................................. 320 20.4.2 Selecting and Calibrating a Severity Model............................................. 321 20.5 Payment and Settlement Pattern Analysis............................................................. 321 20.5.1 Settlement Pattern......................................................................................... 321 20.5.2 Payment Pattern............................................................................................ 322 20.6 Aggregate Losses to an Excess Layer...................................................................... 322 20.6.1 The Monte Carlo Simulation....................................................................... 324 20.7 Technical Premium Calculations............................................................................. 325 20.7.1 No Reinstatement Premiums, Index Clause............................................. 325 20.7.2 Reinstatement Premiums, No Index Clause............................................. 325 20.8 Questions..................................................................................................................... 328 21. Exposure Rating for Property........................................................................................... 331 21.1 Inadequacy of Standard Experience Rating for Property Risks......................... 331 21.2 How Exposure Curves Arise....................................................................................334 21.3 Relationship with Severity Curves.......................................................................... 336 21.4 Properties of Exposure Curves................................................................................ 337 21.5 Parametrisation of Exposure Curves...................................................................... 339 21.6 Build Your Own Exposure Curve............................................................................ 341 21.7 Exposure Rating Process in Reinsurance............................................................... 341 21.7.1 Basic Process..................................................................................................342 21.7.2 How This Is Done in Practice......................................................................343 21.7.2.1 Sources of Uncertainty.................................................................345 Contents xvii 21.7.3 Making It Stochastic..................................................................................... 347 21.7.3.1 Calculation of the Aggregate Loss Distribution Using Exposure Rating............................................................................ 347 21.8 Using Exposure Rating in Direct Insurance.......................................................... 349 21.8.1 Hybrid Rating: Basic Algorithm................................................................. 349 21.8.2 Hybrid Rating: Case with No Losses......................................................... 351 21.8.2.1 Heuristic Rules for Selecting a Value of λ in the Case of No Losses....................................................................................... 352 21.8.2.2 Other Method for Selecting a Value of λ in the Case of No Losses....................................................................................... 353 21.9 Comparison with Experience Rating...................................................................... 354 21.10 Other Issues................................................................................................................ 354 21.10.1 Natural Catastrophes...................................................................................354 21.10.2 Business Interruption...................................................................................354 21.11 Questions..................................................................................................................... 355 22. Liability Rating Using Increased Limit Factor Curves................................................ 357 22.1 How ILF Curves Arise.............................................................................................. 357 22.1.1 Assumptions Underlying ILF Curves........................................................ 357 22.1.2 Definition of ILF Curves.............................................................................. 358 22.1.3 Relationship with Severity Curves............................................................. 359 22.1.3.1 Properties of ILF Curves.............................................................. 360 22.2 Applications to Excess-of-Loss Rating.................................................................... 360 22.3 Effect of Claims Inflation on ILF Curves................................................................ 361 22.4 Derivation of ILF Curves.......................................................................................... 362 22.4.1 ISO Methodology for the Derivation of ILF Curves................................ 362 22.5 Other Issues................................................................................................................ 365 22.5.1 Index Clause.................................................................................................. 365 22.5.2 Discounting................................................................................................... 365 22.5.3 Dealing with Expenses................................................................................ 365 22.5.4 Per Claim versus per Occurrence............................................................... 365 22.6 Examples of ILF Curves............................................................................................ 365 22.6.1 ISO Curves..................................................................................................... 365 22.6.2 National Council on Compensation Insurance Curves.......................... 366 22.6.3 Riebesell Curves............................................................................................ 366 22.7 Do We Really Need ILF Curves?............................................................................. 366 22.8 Questions..................................................................................................................... 367 23. Pricing Considerations for Specific Lines of Business................................................ 369 23.1 Professional Indemnity Cover................................................................................. 369 23.1.1 Legal Framework.......................................................................................... 369 23.1.2 Policy............................................................................................................... 370 23.1.2.1 Claims-Made Basis........................................................................ 370 23.1.2.2 Retroactive Date............................................................................ 370 23.1.2.3 Reinstatements............................................................................... 370 23.1.3 Calculating Reporting Year Exposure....................................................... 370 23.1.4 Frequency Analysis...................................................................................... 373 23.1.5 Severity Analysis.......................................................................................... 374 23.1.5.1 IBNER.............................................................................................. 374 xviii 23.2 23.3 23.4 23.5 Contents 23.1.5.2 Severity Distribution.................................................................... 374 23.1.6 Systemic Effects............................................................................................. 374 23.1.7 Aggregate Loss Analysis............................................................................. 374 23.1.8 Recap on Professional Indemnity............................................................... 375 Weather Derivatives................................................................................................... 375 23.2.1 What Needs Weather Derivatives Respond To........................................ 375 23.2.2 What Weather Derivatives Are................................................................... 376 23.2.3 Market for Weather Derivatives................................................................. 377 23.2.4 Basis Risk....................................................................................................... 377 23.2.5 Valuation of Weather Derivatives............................................................... 378 23.2.5.1 Option Pricing Method................................................................ 378 23.2.5.2 Actuarial Method.......................................................................... 379 23.2.6 Actuarial Valuation Method in Detail....................................................... 379 23.2.6.1 Phase 1: Finding the Index........................................................... 379 23.2.6.2 Phase 2: Modelling the Payout....................................................384 Credit Risk.................................................................................................................. 388 23.3.1 Data Requirements....................................................................................... 389 23.3.1.1 Current Exposure Information................................................... 389 23.3.1.2 Historical Loss Information......................................................... 389 23.3.1.3 Information on the Wider Economic Context........................... 390 23.3.2 Pricing Methodology.................................................................................... 390 23.3.2.1 Probability of Default................................................................... 390 23.3.2.2 Loss Given Default........................................................................ 391 23.3.2.3 Correlation between Defaults...................................................... 392 23.3.2.4 The Monte Carlo Simulation........................................................ 393 23.3.2.5 Output............................................................................................. 394 23.3.2.6 Limitations..................................................................................... 394 Extended Warranty.................................................................................................... 395 23.4.1 Data Requirements....................................................................................... 396 23.4.2 Failure Rate Analysis................................................................................... 396 23.4.3 Frequency Analysis: From Failure Rate to Expected Claim Count....... 399 23.4.4 Severity Analysis.......................................................................................... 399 23.4.5 Aggregate Loss Model................................................................................. 399 23.4.6 Systemic Factors............................................................................................400 23.4.7 Rating Factor Analysis.................................................................................400 23.4.8 Pricing.............................................................................................................400 Miscellaneous Classes of Business.......................................................................... 401 23.5.1 Aviation Insurance....................................................................................... 401 23.5.1.1 Exposure......................................................................................... 401 23.5.1.2 Frequency Analysis....................................................................... 401 23.5.1.3 Severity Analysis........................................................................... 402 23.5.1.4 Aggregate Loss Modelling........................................................... 402 23.5.1.5 Rating Factor Analysis................................................................. 402 23.5.1.6 Pricing............................................................................................. 402 23.5.2 Business Interruption................................................................................... 403 23.5.2.1 Frequency/Severity Analysis...................................................... 403 23.5.2.2 Dependency Analysis...................................................................404 23.5.3 Commercial Motor Insurance.....................................................................404 23.5.3.1 General Modelling Considerations............................................. 405
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