Stephen J. Mildenhall has extensive general insurance experience, having worked in primary and reinsurance pricing, broking, and education since 1992. He is a Fellow of the Casualty Actuarial Society, an Associate of the Society of Actuaries, and holds a PhD degree in mathematics from the University of Chicago.

John A. Major has served as a research leader and data scientist in diverse insurance contexts, contributing to the state of the art in areas such as claim fraud detection, insurance-linked securities, terrorism risk, and catastrophe modeling. Since 2004, much of his attention has focused on the shareholder value of risk transformation. His publications in over a dozen books and journals have been cited in hundreds of scholarly articles. He is an Associate of the Society of Actuaries and holds a Master's degree in mathematics from Harvard University.

Preface xii

1 Introduction 1

1.1 Our Subject and Why It Matters 1

1.2 Players, Roles, and Risk Measures 2

1.3 Book Contents and Structure 4

1.4 What’s in It for the Practitioner? 7

1.5 Where to Start 9

2 The Insurance Market and Our Case Studies 13

2.1 The Insurance Market 13

2.2 Ins Co.: A One-Period Insurer 15

2.3 Model vs. Reality 16

2.4 Examples and Case Studies 17

2.5 Learning Objectives 25

Part I Risk 27

3 Risk and Risk Measures 29

3.1 Risk in Everyday Life 29

3.2 Defining Risk 30

3.3 Taxonomies of Risk 31

3.4 Representing Risk Outcomes 36

3.5 The Lee Diagram and Expected Losses 40

3.6 Risk Measures 54

3.7 Learning Objectives 60

4 Measuring Risk with Quantiles, VaR, and TVaR 63

4.1 Quantiles 63

4.2 Value at Risk 70

4.3 Tail VaR and Related Risk Measures 85

4.4 Differentiating Quantiles, VaR, and TVaR 102

4.5 Learning Objectives 102

5 Properties of Risk Measures and Advanced Topics 105

5.1 Probability Scenarios 105

5.2 Mathematical Properties of Risk Measures 110

5.3 Risk Preferences 124

5.4 The Representation Theorem for Coherent Risk Measures 130

5.5 Delbaen’s Differentiation Theorem 137

5.6 Learning Objectives 141

5.A Lloyd’s Realistic Disaster Scenarios 142

5.B Convergence Assumptions for Random Variables 143

6 Risk Measures in Practice 147

6.1 Selecting a Risk Measure Using the Characterization Method 147

6.2 Risk Measures and Risk Margins 148

6.3 Assessing Tail Risk in a Univariate Distribution 149

6.4 The Intended Purpose: Applications of Risk Measures 150

6.5 Compendium of Risk Measures 153

6.6 Learning Objectives 156

7 Guide to the Practice Chapters 157

Part II Portfolio Pricing 161

8 Classical Portfolio Pricing Theory 163

8.1 Insurance Demand, Supply, and Contracts 163

8.2 Insurer Risk Capital 168

8.3 Accounting Valuation Standards 178

8.4 Actuarial Premium Calculation Principles and Classical Risk Theory 182

8.5 Investment Income in Pricing 186

8.6 Financial Valuation and Perfect Market Models 189

8.7 The Discounted Cash Flow Model 192

8.8 Insurance Option Pricing Models 200

8.9 Insurance Market Imperfections 210

8.10 Learning Objectives 213

8.A Short- and Long-Duration Contracts 215

8.B The Equivalence Principle 216

9 Classical Portfolio Pricing Practice 217

9.1 Stand-Alone Classical PCPs 217

9.2 Portfolio CCoC Pricing 223

9.3 Applications of Classical Risk Theory 224

9.4 Option Pricing Examples 227

9.5 Learning Objectives 231

10 Modern Portfolio Pricing Theory 233

10.1 Classical vs. Modern Pricing and Layer Pricing 233

10.2 Pricing with Varying Assets 235

10.3 Pricing by Layer and the Layer Premium Density 238

10.4 The Layer Premium Density as a Distortion Function 239

10.5 From Distortion Functions to the Insurance Market 245

10.6 Concave Distortion Functions 252

10.7 Spectral Risk Measures 255

10.8 Properties of an SRM and Its Associated Distortion Function 259

10.9 Six Representations of Spectral Risk Measures 261

10.10 Simulation Interpretation of Distortion Functions 263

10.11 Learning Objectives 264

10.A Technical Details 265

11 Modern Portfolio Pricing Practice 271

11.1 Applying SRMs to Discrete Random Variables 271

11.2 Building-Block Distortions and SRMs 275

11.3 Parametric Families of Distortions 280

11.4 SRM Pricing 285

11.5 Selecting a Distortion 292

11.6 Fitting Distortions to Cat Bond Data 298

11.7 Resolving an Apparent Pricing Paradox 304

11.8 Learning Objectives 306

Part III Price Allocation 307

12 Classical Price Allocation Theory 309

12.1 The Allocation of Portfolio Constant CoC Pricing 309

12.2 Allocation of Non-Additive Functionals 312

12.3 Loss Payments in Default 324

12.4 The Historical Development of Insurance Pricing Models 326

12.5 Learning Objectives 337

13 Classical Price Allocation Practice 339

13.1 Allocated CCoC Pricing 339

13.2 Allocation of Classical PCP Pricing 347

13.3 Learning Objectives 348

14 Modern Price Allocation Theory 349

14.1 The Natural Allocation of a Coherent Risk Measure 349

14.2 Computing the Natural Allocations 365

14.3 A Closer Look at Unit Funding 369

14.4 An Axiomatic Approach to Allocation 385

14.5 Axiomatic Characterizations of Allocations 392

14.6 Learning Objectives 394

15 Modern Price Allocation Practice 397

15.1 Applying the Natural Allocations to Discrete Random Variables 397

15.2 Unit Funding Analysis 404

15.3 Bodoff’s Percentile Layer of Capital Method 413

15.4 Case Study Exhibits 421

15.5 Learning Objectives 439

Part IV Advanced Topics 441

16 Asset Risk 443

16.1 Background 443

16.2 Adding Asset Risk to Ins Co. 444

16.3 Learning Objectives 447

17 Reserves 449

17.1 Time Periods and Notation 449

17.2 Liability for Ultimate Losses 450

17.3 The Solvency II Risk Margin 461

17.4 Learning Objectives 468

18 Going Concern Franchise Value 469

18.1 Optimal Dividends 469

18.2 The Firm Life Annuity 472

18.3 Learning Objectives 476

19 Reinsurance Optimization 477

19.1 Background 477

19.2 Evaluating Ceded Reinsurance 477

19.3 Learning Objectives 481

20 Portfolio Optimization 483

20.1 Strategic Framework 483

20.2 Market Regulation 484

20.3 Dynamic Capital Allocation and Marginal Cost 485

20.4 Marginal Cost and Marginal Revenue 487

20.5 Performance Management and Regulatory Rigidities 488

20.6 Practical Implications 490

20.7 Learning Objectives 491

A Background Material 493

A.1 Interest Rate, Discount Rate, and Discount Factor 493

A.2 Actuarial vs. Accounting Sign Conventions 493

A.3 Probability Theory 494

A.4 Additional Mathematical Terminology 500

B Notation 503

References 507

Index 523

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