Pricing in general insurance

Written with both the student and the practicing actuary in mind, this pragmatic textbook and professional reference: • Complements the standard pricing methods with a description of t

w w w c r c p r e s s c o m

Based on the syllabus of the actuarial industry course on general insurance

pricing — with additional material inspired by the author’s own experience as

a practitioner and lecturer — Pricing in General Insurance presents pricing as

a formalised process that starts with collecting information about a particular

policyholder or risk and ends with a commercially informed rate The main

strength of this approach is that it imposes a reasonably linear narrative on the

material and allows the reader to see pricing as a story and go back to the big

picture at any time, putting things into context.

Written with both the student and the practicing actuary in mind, this pragmatic

textbook and professional reference:

• Complements the standard pricing methods with a description of

techniques devised for pricing specific products (e.g., non-proportional

reinsurance and property insurance)

• Discusses methods applied in personal lines when there is a large amount

of data and policyholders can be charged depending on many rating factors

• Addresses related topics such as how to measure uncertainty, incorporate

external information, model dependency, and optimize the insurance

structure

• Provides case studies, worked-out examples, exercises inspired by past

exam questions, and step-by-step methods for dealing concretely with

specific situations

Pricing in General Insurance delivers a practical introduction to all aspects of

general insurance pricing, covering data preparation, frequency analysis, severity

analysis, Monte Carlo simulation for the calculation of aggregate losses, burning

cost analysis, and more

PRICING IN

GENERAL INSURANCE

PRICING IN GENERAL INSURANCE

Pietro Parodi

© 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)

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Visit the Taylor & Francis Web site at

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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

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

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

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

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

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

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

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

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

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

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

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

23.2 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

23.3 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

23.4 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

23.5 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

Contents

23.5.3.2 Exposure 405

23.5.3.3 Frequency Analysis 405

23.5.3.4 Severity Analysis 406

23.5.3.5 Periodic Payment Orders 406

23.5.3.6 Aggregate Loss Modelling 407

23.5.3.7 Rating Factor Analysis 407

23.5.4 Product Liability 407

23.5.4.1 Integrated Occurrence Basis 408

23.5.4.2 Frequency Analysis 408

23.5.4.3 Severity Analysis 408

23.6 Questions 409

24 Catastrophe Modelling 413

24.1 Structure of a Catastrophe Model 413

24.1.1 Hazard Module 414

24.1.2 Vulnerability Module 415

24.1.2.1 Inventory Database 416

24.1.3 The Financial Module 416

24.2 Outputs of Catastrophe Model 417

24.2.1 Occurrence Exceedance Probability 417

24.2.2 Aggregate Exceedance Probability 418

24.2.3 Return Period 419

24.2.4 Average Annual Loss 419

24.2.5 Standard Deviation 419

24.3 Frequency-Severity Analysis Based on the ELT 419

24.4 Calculation of the Severity Distribution 420

24.5 Key Perils Modelled 421

24.6 What Role for Actuaries? 421

24.7 Questions 423

Section IV Advanced Topics 25 Credibility Theory 427

25.1 Gentle Start: A Noninsurance Example 427

25.1.1 Measuring the Uncertainty of the Observation-Based Estimate 429

25.1.2 Measuring the Relevance of the Benchmark Information 429

25.1.3 Calculating the Credibility Factor 430

25.1.4 Calculating the Credibility Estimate 431

25.2 What’s All This Got to Do with General Insurance? 432

25.2.1 Terminology 432

25.2.2 Client’s ‘True’ Risk Premium Rate 433

25.2.3 Client’s Estimated Risk Premium Rate, and Its Uncertainty 433

25.2.4 Market’s Risk Premium Rate 433

25.2.5 Credibility Estimate 433

25.2.6 An Example 434

25.3 Applications 435

25.3.1 Credibility of a Frequency/Severity Model 435

25.3.2 Mixing Experience and Exposure Rating 436

25.4 Specific Approaches to Credibility 43725.4.1 Classical Credibility 43825.4.1.1 Criterion for Full Credibility 43825.4.1.2 Partial Credibility 44125.4.2 Bayesian Credibility 44225.4.2.1 Normal/Normal Case 44225.4.2.2 Empirical Bayesian Approach 44325.4.2.3 Pros and Cons of the Bayesian Approach to Credibility 44325.4.2.4 Other Interesting Cases 44325.4.3 Bühlmann Credibility 44425.4.4 Bühlmann–Straub Credibility 44425.5 Appendix: Proof of the Credibility Formula for Uncertainty-Based

Credibility 44525.6 Questions 447

26 Rating Factor Selection and Generalised Linear Modelling 44926.1 Why Rating Factors Are Useful 45026.1.1 Limitations 45326.1.2 Some Facts about Rating Factors in Practice 45426.2 How Do We Develop a Rating-Factor Model in Practice? 45526.3 Generalised Linear Modelling 45726.3.1 Least Squares Regression and Multivariate Linear Modelling 45826.3.2 Beyond the Linear Model 46026.3.3 Examples of Applications 46226.3.4 Modelling Continuous and Categorical Variables 46326.3.5 Model Selection 46526.3.5.1 Greedy Approach to Model Selection 46626.3.5.2 Akaike Information Criterion 46626.3.5.3 Cross-Validation 46726.3.5.4 Issues with Forward/Backward Selection 46726.3.6 Practical General Insurance Example: Rating Factor Selection for Reinsurance 46826.3.7 Another Practical General Insurance Example: Modelling Claims Frequency for Personal or Commercial Lines of Business 47026.3.8 Implementation of GLM 47126.4 Other Techniques 47326.5 Questions 474

27 Multilevel Factors and Smoothing 47727.1 Credibility Smoothing 47827.1.1 An Example: Car Model Codes 47827.1.2 Credibility Smoothing and the GLM Approach 48027.2 Spatial Smoothing 48127.2.1 One-Dimensional Analogy 48127.2.2 Distance-Based Smoothing 48227.2.2.1 How It Works 48227.2.2.2 Observations 482

Contents

27.2.3 Adjacency-Based Smoothing 48327.2.3.1 How It Works 48327.2.3.2 Advantages and Disadvantages 48327.2.4 Degree of Smoothing 48327.2.5 Spatial Smoothing and the GLM Approach 48427.3 Questions 484

28 Pricing Multiple Lines of Business and Risks 48728.1 Divide and Conquer? 48728.2 Independent Case 48928.2.1 Combining Aggregate Loss Models by the Monte Carlo Simulation 48928.2.2 Combining Aggregate Loss Models by FFT 49028.2.3 Limitations 49228.3 Measuring Dependency 49228.3.1 Rank Correlation 49328.3.2 Tail Dependence 49428.4 Modelling Dependency with Copulas 49628.4.1 An Intuitive Approach to Copulas 49628.4.1.1 Relationship with Rank Correlation 49828.4.2 Notable Examples of Copulas 49828.4.2.1 Gaussian Copula 498

28.4.2.2 Student’s t Copula 499

28.4.2.3 Other Copulas 50128.5 Aggregate Loss Distribution of Two Correlated Risks 50128.5.1 Correlating the Severities of Two Risks 50128.5.1.1 Method 1: Split Aggregate Loss Model 50228.5.1.2 Method 2: Two Severity Models 50328.5.2 Correlating the Frequencies 50428.5.3 Correlating the Total Losses of Two Risks 50528.6 Aggregate Loss Distribution of an Arbitrary Number of Correlated

Risks 50628.7 Common Shock Modelling 50928.7.1 Common Shock Model 50928.7.2 Binary Common Shocks 50928.8 Appendix: R Code for Generating an Arbitrary Number of Variables

Correlated through a t Copula 512

28.9 Questions 516

29 Insurance Structure Optimisation 51929.1 Efficient Frontier Approach 51929.1.1 Cost of Insurance 51929.1.2 Benefits of Insurance 51929.1.3 Efficient Structures 52029.1.4 Limitations of the Efficient Frontier Approach 52229.2 Minimising the Total Cost of (Insurable) Risk 52229.2.1 Risk Management Options 52329.2.2 Total Cost of Risk: Definition 523

29.2.3 Choosing the Optimal Structure 525

29.2.3.1 Selecting the Best Insurance Option amongst K Options 525

29.2.4 Total Cost of Risk Calculation: A Simple Example 52529.2.5 Relationship between Total Cost of Risk and Efficient Frontier 52729.2.6 Limitations 52729.3 Questions 528

References 531

Preface

Most people with some interest in the history of science know that in 1903, Albert Einstein, unable to find a job at the university, started working as an ‘assistant examiner’ in the patent office of Berne, where he worked efficiently enough to be able also to produce four papers, one on Brownian motion (now a standard topic in the actuarial profession exams

on derivatives) and three on relativity theory and quantum mechanics (not on the labus) According to one biographer (Isaacson 2007), an opportunity had also arisen for Einstein to work in an insurance office, but he hastily turned that down arguing that that would have meant 8 hours a day of ‘mindless drudgery’ Details on the exact job descrip-tion for the position are not available, but since that was in 1903, we can rule out the pos-sibility that Einstein was offered a position as a pricing actuary in general insurance And apart from the historical impossibility (actuaries were first involved in general insurance around 1909 in the United States, to deal with workers’ compensation insurance), the job

syl-of the pricing actuary in general insurance is way too exciting – contrary perhaps to public perception – to be described as mindless drudgery: pricing risk means understanding risk and understanding risk means understanding (to some extent) how the world works (which is

after all what Einstein was after): to give but a very simple example, pricing a portfolio of properties of a company requires some understanding of what that company does (is it producing gunpowder or hosting data centres?) and what perils (natural and man-made) these properties are exposed to in the territories the company operates in

This book is based on my experience as both a practitioner and a part-time lecturer at Cass Business School in London It was written to communicate some of the excitement

of working in this profession and to serve the fast-expanding community of actuaries involved in general insurance and especially in pricing It comes at a time when this rela-tively new discipline is coming of age and pricing techniques are slowly crystallising into

industry standards: the collective risk model is widely used to estimate the distribution of future total losses for a risk; extreme value theory is increasingly used to model large losses; generalised linear modelling is the accepted tool to model large portfolios with a significant number of rating factors – to name but a few of these techniques

This book was written by a practitioner with actuarial students specialising in general insurance* and other practitioners in mind, and its aim is therefore not foundational (such

as is, for example, the excellent Loss Models: from Data to Decisions by Klugman et al 2008)

but practical As a matter of fact, I have tried to keep constantly in mind Jerome K Jerome’s wry remark on the lack of practicality of foreign language teaching in British schools dur-ing his (Victorian) times: ‘No doubt [students] could repeat a goodly number of irregular verbs by heart; only, as a matter of fact, few foreigners care to listen to their own irregu-lar verbs recited by young Englishmen’ (Jerome 1900) This book, therefore, rarely dwells

on the more abstract points about pricing, mathematical definitions, and proofs (which might be thought of as the irregular verbs of our profession) beyond the bare minimum needed to develop an intuition of the underlying ideas and enable execution – rather, it is rich in step-by-step methods to deal concretely with specific situations and in worked-out examples and case studies

of the Actuarial Profession in the United Kingdom and India, with additional material.

Pricing as a Process

One of the main efforts of this book is to present pricing as a process because pricing activity (to an insurer) is never a disconnected bunch of techniques, but a more-or-less formalised process that starts with collecting information about a particular policyholder

or risk and ends with a commercially informed rate

The main strength of this approach is that it imposes a reasonably linear narrative on

the material and allows the student or the practitioner to see pricing as a story and go back

to the big picture at any time, putting things into context It therefore allows students to enter the pricing profession with a connection between the big and small picture clearer in their minds, and the practitioner who is switching from another area of practice (such as pensions) to hit the ground running when starting to work in general insurance

At the same time, the fact that there is roughly a chapter for each building block of the detailed risk pricing process should help the practitioner who is already involved in gen-

eral insurance pricing to use this textbook as a reference and explore each technique in depth

as needed, without the need of an exhaustive read

In practice, this is achieved as follows A few introductory chapters (Section I: Introductory Concepts) set the scene; the most important is Chapter 1, which provides an elementary but

fully worked-out example Then, Section II: The Core Pricing Process introduces a modern

generic pricing process and explores its building blocks: data preparation, frequency ysis, severity analysis, Monte Carlo simulation for the calculation of aggregate losses, and

anal-so on, with the emphasis always being on how things can be done in practice, and what the issues in the real world are Alternative, more traditional ways of pricing (such as burning cost analysis) are also explored

After learning the language of pricing, one can start learning the dialects; Section III: Elements

of Specialist Pricing goes beyond the standard process described in Section II and is devoted to pricing methods that have been devised to deal with specific types of insurance or reinsurance product, such as exposure rating for property reinsurance If the student has started becoming

a bit dogmatic about the pricing process at the end of Section II, and has started thinking that there is only one correct way of doing things, this part offers redemption: the student learns that tweaking the process is good and actually necessary in most practical cases

Finally, Section IV: Advanced Topics deals with specific methods that are applied in certain

circumstances, such as in personal lines where there is a large amount of data and holders can be charged depending on many rating factors, or when one wants to price a product that insures many different risks at the same time and the correlation between these risks are important

policy-Additional Resources

Since this book is designed to provide practical help to students and practitioners, several spreadsheets and simple examples of R code have been included to illustrate and implement some of the techniques described in this textbook Most of these spreadsheets take their name from the chapter they refer to and address IBNR calculations, frequency modelling, severity modelling, etc These and other resources (such as errata, solutions to selected problems and additional text) can be found in the book’s website, http://pricingingeneralinsurance.com

Acknowledgements

First and foremost, I would like to thank my wife, Francesca, for her support and the many

creative ways she devised for shielding me from familial invexendo, simply making this

endeavour possible; and my children, Lisa, David, Nathan and Sara, for those times they managed to breach that shield Thanks also to the rest of my family for helping in different ways, with special gratitude to my aunt for the crucial time she let me spend in her country

house in Trisobbio in the summer of 2013, retir’d from any mortal sight – except of course for

the compulsory Moon Pub evening expeditions with my friends

I am grateful to the staff and the students at Cass Business School for providing me with the motivation for starting this textbook and for the feedback on the lecture notes that became the foundation for this pursuit

Many thanks to Phillip Ellis, Eamonn McMurrough and all the colleagues at the Willis Global Solution Consulting Group, both in London and Mumbai, for providing so many opportunities of tackling new technical and commercial problems, experimenting with a large variety of different techniques and engaging in critical discussion I also thank them for their encouragement while writing this book

My deep gratitude to the Institute and Faculty of Actuaries for its invaluable support in this endeavour Special thanks to Trevor Watkins for encouraging me and overseeing the process, and to Neil Hilary, who reviewed the material, guided me through the process

of ensuring that this book be as relevant as possible to the profession and the students of the General Insurance practice area and encouraged me and advised me in countless other ways

I would like to thank Jane Weiss and Douglas Wright for their initial review of the rial and their encouragement Special thanks to Julian Lowe for his guidance and for sev-eral important structural suggestions on the first draft

mate-Many people have subsequently helped me with their feedback on various versions

of the manuscript I am especially grateful to Julian Leigh and Sophia Mealy for takingly reading the whole manuscript and for their insightful comments on both con-tent and style I am also thankful to the following people for precious suggestions on specific topics of the book: Paolo Albini (frequency analysis), Cahal Doris (catastrophe modelling), Tomasz Dudek (all the introductory chapters), Junsang Choi (catastrophe modelling), Phil Ellis, LCP (various presentational suggestions), Michael Fackler (all the reinsurance content and frequency/severity modelling), Raheal Gabrasadig (introduc-tory chapters), Chris  Gingell (energy products), Torolf Hamm (catastrophe modelling), Anish Jadav (introductory example, products), Joseph Lees (burning cost), Marc Lehmann (catastrophe modelling), Joseph  Lo (from costing to pricing, plus all the introductory chapters),  Eamonn  McMurrough (pricing control cycle), David Menezes (aggregate loss modelling, dependency modelling), Cristina Parodi (claims management), David Stebbing (aviation), Andreas Troxler (dependency modelling) and Claire Wilkinson (weather derivatives)

pains-Section I Introductory Concepts

1

Pricing Process: A Gentle Start

Pricing is a complex endeavour, which is best conceptualised as a process or even as a project: that is to say, an undertaking with a beginning and an end and a number of dif-ferent tasks to be executed in a certain order To get a fair idea of what the pricing process entails without getting bogged down into the details that inevitably need to be considered

in any realistic example, we will start looking at a very simple example of pricing, ried out in a rather traditional, non-actuarial way Despite its simplicity, this example will contain all the main ingredients of the pricing process that we will expand on in the rest

car-of the book

1.1 An Elementary Pricing Example

Suppose you are an underwriter working for a commercial insurer and you are asked to insure a company for employer’s liability Also assume that you have agreed to insure only 95% of each loss, so that the insured has an active interest in keeping the number of claims low.* We also assume that only up to £10M for each loss and in aggregate is awarded (after taking the 95% cut)

The prospective insured has given you the following table (Figure 1.1), which gives you the total losses incurred by year, from the ground up, before taking insurance into consid-eration (Although this sounds reasonable, it is actually quite unusual for an insured to be able to give you a well-structured data set with their losses More likely, insurance com-panies or claims management companies will have collected them and will have shared them with the insured.)

What price do you think the insured should pay for cover in 2015 (assume that the policy will incept on 1 January 2015 and will last one year)? It is not possible to answer this ques-tion directly at this stage For one thing, the price is a commercial decision, which depends

on many factors, the main one being: how much money do I want to make (on average) by selling this insurance?

So let us rephrase the question: assuming that my expenses (such as the expenses of dealing with policies and claims) are roughly 15% of the premium, and that I want 10% profit, how much should I charge for this policy?

A nạve method that comes to mind to reply to the question above is simply to average the total losses over the period 2005 to 2014 This yields the numbers in Figure 1.2:

legal unless the amount retained is retained by the client through a direct-writing captive domiciled in the European Union (employers’ liability policies have to be sold from the ground up and 100% of each loss has

to be insured, up to a certain limit) However, the reason for using this structure here is that it allows us to explain the stages of the rating process with an uncomplicated example.

That is, we have average losses of approximately £1.30M, of which only 95% (£1.23M) will remain to the insurer to which we have to add a loading for expenses and profits, yielding

£1.23M/85%/90% ~£1.61M

Why is this not satisfactory? The reason is that we have ignored many elements that are essential for the assessment of the risk Even a cursory look at the actual losses in the last few years would make an underwriter cringe: out of the last 5 years, 4 years show total losses greater than the average (£1.30M), and 2 years show total losses greater than the premium we plan to charge Something is obviously not right So what have we forgotten?There is an extensive list of elements that we need to consider The most obvious ones are set down below We have subdivided the various steps of the process into four main

categories: risk costing (gross), which is the estimation of the future losses regardless of who pays for them; risk costing (ceded), which is the estimation of that portion of the losses that is covered by the insurer; determining the technical premium, which leads to the pre-

mium suggested by the actuary, based on the losses and other relevant costs; and finally,

commercial considerations and the actual premium, which explains why the premium gested by the actuary may be overridden

sug-Year

1,055,998 936,637 1,153,948 1,290,733 1,548,879 1,973,818 1,448,684 1,831,386 842,957

2006 2007 2008 2009 2010 2011 2012 2013 2014

Average

890,882 1,055,998 936,637 1,153,948 1,290,733 1,548,879 1,973,818 1,448,684 1,831,386 842,957

1,297,392

FIGURE 1.2

The average of losses over the period 2005 to 2014 yields £1,297,392, which we might naively take as our expected losses for 2015.

Pricing Process

1.1.1 Risk Costing (Gross)

1.1.1.1 Adjusting for ‘Incurred but Not Reported’ Claims

As a bare minimum, we will need to take into consideration that data for the year 2014 is not complete: only losses reported until 30 June 2014 have been recorded Because we only

have half a year of experience, we will need to multiply the 2014 figure by at least two so

that it covers the whole year and not only the first 6 months The actual correction should

be more than that because it takes a while until a claim is reported, and therefore the final amount for year 2014 will be larger than what we see at the end of the year

For that matter, even the 2013 figure may not be fully complete, and because in some cases liability claims take several years to be reported (asbestos-related claims – which were reported up to 30 years after the period when the worker was working with asbes-

tos – are a case in point), there might be missing claims in all of the policy years.

These missing claims are called incurred but not reported (IBNR) claims As the tialism suggests, these are losses that have already occurred but have not been reported yet We need some method to incorporate these into our analysis, perhaps by introducing appropriate multiplication factors that tell us by how much we have to multiply the total losses for each year to estimate the ultimate total claim amount

ini-Right now, we do not have all the tools necessary to include a full IBNR correction and therefore we will have to be happy with the very common, if unsatisfactory, trick of exclud-ing the most recent year(s) of experience and keep the rest of face value

Policy year Losses ()

2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 Average Average (2005–2013)

890,882 1,055,998 936,637 1,153,948 1,290,733 1,548,879 1,448,684 1,831,386 842,957 1,973,818

1,297,392 1,347,885

FIGURE 1.3

The average as before, but excluding 2014, which is considered immature.

Question: If the average delay is not too long – let us say a few months – can you think

of any work-around that will enable us to improve on our estimate without actually performing a full IBNR analysis?

Answer: Simply get rid of 2014 and assume that the others are complete This leads to

an estimate of £1.35M (see Figure 1.3)

1.1.1.2 Adjusting for Claims Inflation

Another simple correction we should make is to account for the time value of money Obviously, a claim that occurred in 2005 would have, if it occurred again under exactly

the same circumstances in 2015, a much larger payout, because of something called claims inflation Claims inflation is not the same as the retail price index inflation (RPI) or the consumer price index inflation (CPI) that we have in the United Kingdom – the types of index that you have to look at to decide whether your salary is increasing fast enough to keep up with the cost of living This is because claims inflation depends on factors that the CPI or the RPI are not necessarily capturing For example, property claims will depend on cost of repair materials, cost of repair labour and the like Liability claims will depend on wage inflation and court inflation, that is the tendency on the part of the courts to award compensations that are increasingly favourable for claimants

Let us assume that claims inflation for the table above is 5% We are not interested in discussing here whether this is a good assumption or not – at the moment of writing, it is

a commonly used assumption by underwriters for UK liability business, but things may

be changing quickly It so happens that this claims inflation is the perfect assumption to describe our data set… because we reverse-engineered things to make it so!

So how are claims revalued in practice? The ‘algorithm’ (if such a simple list of tions deserves this name) goes as follows:

1 Set the revaluation date (the date at which claims have to be revalued) to the age day at which claims will occur in the policy under consideration

This will normally be the midpoint of the policy (for instance, if a policy starts on

1 October 2015 and is an annual policy, the revaluation date will be 1 April 2016) However, if we know for a fact that claims have strong seasonality (for instance, motor claims occur more frequently in winter) the average point will be shifted However, again,

it is important to remember that claims inflation is never known for sure, and all these calculations are mired in uncertainty, and it is therefore often pointless to try to be too smart.

2 Assume that all claims in each policy year equally occur at the midpoint of each policy year (or whatever point the seasonality suggests)

We will see later that this assumption can be dropped if we have a complete listing of the claims rather than the aggregate losses for each year.

3 Revalue the total claim X(t) for each year t to the amount X rev@t* (t) = X(t) × (1 + r) t *−t where t* is the year of the policy we are pricing, t is the policy year whose histori- cal claims we wish to revalue and r is the claims inflation.

Things get only slightly more complicated if, instead of a roughly constant inflation, we want to use an inflation index that is not constant, I(t) [I(t) could be, for example, the CPI or

a wage inflation index] In this case, the formula above becomes X rev@t* (t) = X(t) × I(t*)/I(t), where I(t*) is the estimated value of the index at time t*.

In Figure 1.4 we see how this changes the numbers for our simple example We have introduced a column ‘Revalued losses’ whose value is equal to that in the column ‘Losses’ multiplied by the revaluation factor (a power of 1.05) As a result, the average between 2005 and 2013 (our best bet, currently, on the losses expected in 2015) has gone up to approxi-mately £1.77M

Pricing Process

1.1.1.3 Adjusting for Exposure Changes

Something is still not right with the revalued losses above: there seems to be an upward trend in the number of aggregate losses per year, as shown clearly by the chart in Figure 1.5 For all we know, this might actually be a purely random effect: the upward trend is not huge and there is a lot of volatility around the best-fit line However, there might be a more obvious reason: the profile of the company may have changed significantly from 2005 to

2014, and the difference in the total losses may reflect this change

It is not easy to quantify the risk profile of a company, but there are certain measures that are thought to be roughly proportional to the overall risk, which in turn translates into the

total expected losses These measures are called exposure measures As an example, if your

company insures a fleet of cars, you expect that the overall risk will be proportional to the number of vehicles in the fleet or, better yet, to the vehicle-years (by which, for example, a vehicle which joins the fleet for only half a year counts as 0.5 vehicle-years) This measure seems quite good, but it is not perfectly proportional to the risk: for example, certain cars

Policy year Losses (£) Revaluation factor Revalued losses (£)

2005 2006 2007 2008 2009 2010 2011 2012 2013 2014

890,882 1,055,998 936,637 1,153,948 1,290,733 1,973,818 1,448,684 1,831,386 842,957 1,297,392 1,347,885

1.629 1.551 1.477 1.407 1.340 1.276 1.216 1.158 1.103 1.050

1,451,153 1,638,199 1,383,839 1,623,721 1,729,705 1,976,806 2,399,188 1,677,033 2,019,104 885,105 1,678,385 1,766,528

Average Average (2005–2013)

FIGURE 1.4

Calculating the average after revaluing the losses to current terms.

Revalued Losses for Each Policy Year

3,000,000 2,500,000 2,000,000 1,500,000 1,000,000 500,000

FIGURE 1.5

The total losses show an increasing trend, even after revaluation – an exposure effect?

are considered more risky than others, if only because they are more expensive and fore when they are damaged it costs more to repair them, and certain drivers are more dangerous than others One might therefore devise fancier ways to find a general measure

there-of exposure However, simple is there-often better, and there is there-often not enough evidence in the data to justify doing something too clever

In the case of employers’ liability, an exposure measure that is often used for burning cost analysis is wageroll (the total amount of salary paid to the company’s employees) Another good one is the number of employees (or, better still, employee-years, which – analogously to vehicle-years – gives appropriate weights to part-timers and employees joining/leaving during the year)

To go back to our example, we will assume that we have exposure in the form of wageroll because it gives us a chance to deal with another aspect of the exposure adjustment pro-cess (Figure 1.6)

Because wageroll is a monetary amount, the issue arises of how it should be corrected to

be brought to current day values What one normally does is to inflate past exposures with

an inflation factor (either a single inflation or an index) much in the same way as we did for

the losses Notice, however, that the inflation factor we apply to exposures does not need to be the same as that we apply to losses: in the case of wageroll, for example, it makes sense (unless we have company-specific information on their wage policy) to revalue it based on the wage inflation for that country If we assume that wage inflation is 4% (which again happens to

be exactly true for the case in point, as the numbers were shamelessly reverse-engineered

to make it so), we can produce a revalued exposure as in Figure 1.7

The last column in Figure 1.7 was obtained by adjusting the total claims by the exposure –

that is, imagining what the losses in a given year t would have been if the exposure had been that estimated for the policy renewal period t* Assuming again that t* = 2015, we can write

( )( *)

Policy year Losses () Revaluation factor Revalued losses () (wageroll, ) Exposure

2005 2006 2007 2008 2009 2010 2011 2012 2013 2014

890,882 1,055,998 936,637 1,153,948 1,190,733 1,548,879 1,973,818 1,448,684 1,831,386 842,957 1,297,392 1,347,885

1.629 1.551 1.477 1.407 1.340 1.276 1.216 1.158 1.103 1.050

1,451,153 1,638,199 1,383,839 1,623,721 1,729,705 1,976,806 2,399,188 1,677,033 2,019,104 885,105 1,678,385 1,766,528

67,556 77,285 91,336 98,789 114,596 123,289 145,317 155,574 161,797 170,192 165,000 Average

Average (2005–2013)

FIGURE 1.6

A simple look at this table shows the reason behind the year-on-year increase in total losses: the exposure has steadily gone up – or is the increase in exposure simply a consequence of wage inflation? Note that the £165,000 figure in the bottom row is not an average exposure but next year’s estimated exposure.

1.1.1.4 Adjusting for Other Risk-Profile Changes

As we mentioned above, even if the exposure remains the same, the risk may change because, for example, the proportion of employees in manual work changes or because, for example, new technologies/risk-control mechanisms are introduced that make working

safer Ideally, one would like to quantify all these changes by creating a ‘risk index’ R(t)

that can be associated with every year of experience, so that the total losses can then be

modified for each year by multiplying by R(t*)/R(t), much in the same way as we did for

claims inflation and exposure adjustments

Underwriters will be looking for such indices (more or less informally), but their exact quantification will be difficult: in the case of commercial insurance and reinsurance, they are part of the negotiations between brokers and underwriters

In the case of personal lines insurance, the sheer amount of data makes it more feasible

to assess the risk posed by each policyholder, and this takes us into the realm of rating factor analysis, which will be addressed in a subsequent unit Going back to our initial example, let us assume for simplicity that there have been no significant changes in the risk profile

1.1.1.5 Adjusting for ‘Incurred but Not Enough Reserved’ Claims

Although we have so far treated the ‘total reported losses’ amount as certain, the total

losses amount for each year t is really made up of two parts: paid losses (amounts that

Policy year Losses () Revaluation factor Revalued losses () (wageroll, ) Exposure

Revalued exposure (wageroll, )

Revalued losses + exposure correction ()

1.629 1.551 1.477 1.407 1.340 1.276 1.216 1.158 1.103 1.050

1,451,153 1,638,199 1,383,839 1,623,721 1,729,705 1,976,806 2,399,188 1,677,033 2,019,104 885,105

67,556 77,285 91,336 98,789 114,596 123,289 145,317 155,574 161,797 170,192 165,000

100,000 110,000 125,000 130,000 145,000 150,000 170,000 175,000 175,000 177,000 165,000

2,394,403 2,457,298 2,060,877 1,968,285 2,174,487 2,328,624 1,581,202 1,903,726

1,952,067 2,077,286 825,098 1,826,668

1,678,385 1,766,528

Average

Average (2005–2013)

FIGURE 1.7

Even after taking wage inflation into account, exposure can be seen to go up Our recalculated average over 2005

to 2013 once the change in exposure is taken into account is now £2.08M.

have been paid for year t until today or whose payment has been approved) and ing losses (amounts that have been reserved for claims in year t, based on our estimate

outstand-of how big the claim will become after, for example, a court decision has been made on the level of compensation) By their very nature of being estimates, outstanding losses are still subject to changes, and they end up being bigger or smaller than expected If we reserve skilfully (that is, with a good technical knowledge of the compensation process) and neutrally (that is, without deliberately underestimating or exaggerating our claims estimates), the prediction errors that we make on many different claims will, by the law of large numbers, average out and the overall estimate will be near enough the final actual outcome – at least if the number of claims is indeed large However, it is often the case that insurers make systematic upward or downward errors on estimates, whether deliberately

or not: the error amount is called the IBNER (incurred but not enough reserved, sometimes spelled out – inexplicably – as incurred but not enough reported) claims amount If we know that the total loss amount is subject to IBNER, we will want to correct our expecta-tion on the total loss amount accordingly

To allow for IBNER, we either need an oracle that tells what percentage we have to take out from or add to every reserved claim, or we need to form our own judgment as to whether there is IBNER or not This can be done by using historical information on the way that losses were reserved and were then settled This should be done ideally with histori-cal reserving information on individual losses or at least with total claims triangulations.Because, in our case, we have been given no information of this kind, we will need to skip these considerations – but it is still important to keep in mind that our estimates are more uncertain because of IBNER

1.1.1.6 Changes for Cover/Legislation

Adjustments need to be made to past claims experience if significant changes of cover are made, for example, if certain types of claims were previously covered and now are not However, one should not automatically adjust the historical data to eliminate large and unusual movements if this means excluding ‘uncomfortable’ past claims

Also, adjustment may have to be made for changes in legislation As an example, the recently introduced Jackson reforms for liability cases prescribe fixed legal fees for claims below a certain amount, in an attempt to reduce litigation costs: if the liability policy you’re trying to price includes legal costs, you will need to look back at all past claims and see what the retrospective effect of this reform would have been to predict your liabilities for next year

1.1.1.7 Other Corrections

Adjustments are also made to correct for unusual experience (say, a very large claim that could have happened everywhere in the market), for different weather conditions, for cur-rency effects and for anything else that we think is appropriate These corrections are common practice among underwriters, but one should always bear in mind that these corrections, although they purport to increase the reliability of our estimates, also add various layers of uncertainty and of arbitrariness to our estimates

As actuaries, we are obliged to make it clear what assumptions our estimates are based

on and to communicate the uncertainty with which we make certain corrections

Bringing all considerations together and going back to our initial example, we have seen that after taking into account claims inflation, exposure correction and IBNR (if only by

Pricing Process

excluding the last year), and disregarding other corrections such as for IBNER and risk profile changes, we estimate for 2015 gross losses (that is, from the ground up, without tak-ing any retention into account) of

Expected gross losses ~ £2.08M

However, for the premium to be calculated, we are only interested in the portion of the losses that are ceded (transferred) to the insurer – which is why, at this stage, an analysis

of the ceded losses is required This is normally not an easy task without the appropriate mathematical techniques, but in our illustrative example, the insurance structure is so simple that we can calculate it with simple arithmetic

1.1.2 Risk Costing (Ceded)

As we have stated at the beginning, we only insure 95% of each loss, and there is a limit of

£10M on each loss and in aggregate for each year

The latter piece of information does not change our calculations in our rather crude method, so we will just ignore it As to the 95% quota share arrangement, this can be taken into account by simply multiplying the expected losses (£2.08M) by 95%, obtaining

Expected ceded losses = 95% of £2,077,286M ~ £1.97M

1.1.3 Determining the Technical Premium

To move from the expected ceded losses to the technical premium, that is, the premium that should be charged for the risk from a purely technical point of view based on the com-pany’s objective, we need to consider the expenses incurred by the insurer for running the business and specifically underwriting the risk and handling the claims (Section 1.1.3.1); consider the income that derives from being able to invest the premiums while waiting for claims to occur and then being settled (Section 1.1.3.2); consider the profit that the firm needs/wants to make (Section 1.1.3.3)

1.1.3.1 Loading for Costs

We have assumed that 15% of the premium will serve to cover the expenses As a quence, we can write the expected losses loaded for expenses as

conse-Expected ceded losses + Expenses = £1,973,422/85% ~ £2.32M

Other costs that we have ignored in this simple example will, in general, include mission charges to be paid to brokers and reinsurance costs

com-1.1.3.2 Discounting for Investment Income

If we assume, for illustration purposes, that the average delay between the receipt of the premium and the combined payout of claims and expenses for employers’ liability is 3

years and that the insurer is able to invest the premium with a return of 3% per annum, this will have an effect on the premium that we will be able to charge We can then write

Expected ceded losses + Expenses − Income = £2,321,673/1.033 ~ £2.12M

The idea being that if we charge £2.12M for premium today, we will have £2.32M to pay for claims and expenses in 3 years’ time

1.1.3.3 Loading for Capital/Profit

Finally, we need to take into account that the company will need to make a 10% profit out

of writing insurance Hence, the technical premium is

Expected losses + Expenses − Income + Profit = £2,124,659/0.9 ~ £2.36M

Note that the 10% profit implicitly takes into consideration the loading for capital, that is, the fact that the company will need capital for underwriting a risk to remain solvent and abide by the regulator’s requirements and that the investors will require a certain return

on the capital that they have put into the firm The capital loading necessary for each

policy will be provided by the capital modelling function of the firm (this will be expanded

upon in Section 19.2)

The figure obtained after loading for costs and profit and discounting for investment income is the technical premium and can more simply be obtained from the expected ceded losses applying all loadings in one go:

If the technical premium is charged, we expect a loss ratio (the ratio between the claims

incurred before expenses and the premium) of

1.1.4 Commercial Considerations and the Actual Premium

The premium that the underwriter is going to charge might be quite different from the technical premium calculated above because of commercial considerations: it could be less because of the desire to retain old business/acquire new business, or it could be more

to signal that the insurer is not happy with a particular risk or with a particular level of retention by the insured

On the other hand, this flexibility might be limited by the need for the underwriter to achieve a certain return on capital for the portfolio for which he is responsible and other company guidelines

Market considerations may also lead to decline the underwriting of the risk, if the market conditions (what we will later on call the ‘underwriting cycle’) are such that it is only possible

to underwrite it at a loss or in any case at a profitability level that is unacceptable to the insurer