Pricing Insurance Risk

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 xii1 Introduction 11.1 Our Subject and Why It Matters 11.2 Players, Roles, and Risk Measures 21.3 Book Contents and Structure 41.4 What’s in It for the Practitioner? 71.5 Where to Start 92 The Insurance Market and Our Case Studies 132.1 The Insurance Market 132.2 Ins Co.: A One-Period Insurer 152.3 Model vs. Reality 162.4 Examples and Case Studies 172.5 Learning Objectives 25Part I Risk 273 Risk and Risk Measures 293.1 Risk in Everyday Life 293.2 Defining Risk 303.3 Taxonomies of Risk 313.4 Representing Risk Outcomes 363.5 The Lee Diagram and Expected Losses 403.6 Risk Measures 543.7 Learning Objectives 604 Measuring Risk with Quantiles, VaR, and TVaR 634.1 Quantiles 634.2 Value at Risk 704.3 Tail VaR and Related Risk Measures 854.4 Differentiating Quantiles, VaR, and TVaR 1024.5 Learning Objectives 1025 Properties of Risk Measures and Advanced Topics 1055.1 Probability Scenarios 1055.2 Mathematical Properties of Risk Measures 1105.3 Risk Preferences 1245.4 The Representation Theorem for Coherent Risk Measures 1305.5 Delbaen’s Differentiation Theorem 1375.6 Learning Objectives 1415.A Lloyd’s Realistic Disaster Scenarios 1425.B Convergence Assumptions for Random Variables 1436 Risk Measures in Practice 1476.1 Selecting a Risk Measure Using the Characterization Method 1476.2 Risk Measures and Risk Margins 1486.3 Assessing Tail Risk in a Univariate Distribution 1496.4 The Intended Purpose: Applications of Risk Measures 1506.5 Compendium of Risk Measures 1536.6 Learning Objectives 1567 Guide to the Practice Chapters 157Part II Portfolio Pricing 1618 Classical Portfolio Pricing Theory 1638.1 Insurance Demand, Supply, and Contracts 1638.2 Insurer Risk Capital 1688.3 Accounting Valuation Standards 1788.4 Actuarial Premium Calculation Principles and Classical Risk Theory 1828.5 Investment Income in Pricing 1868.6 Financial Valuation and Perfect Market Models 1898.7 The Discounted Cash Flow Model 1928.8 Insurance Option Pricing Models 2008.9 Insurance Market Imperfections 2108.10 Learning Objectives 2138.A Short- and Long-Duration Contracts 2158.B The Equivalence Principle 2169 Classical Portfolio Pricing Practice 2179.1 Stand-Alone Classical PCPs 2179.2 Portfolio CCoC Pricing 2239.3 Applications of Classical Risk Theory 2249.4 Option Pricing Examples 2279.5 Learning Objectives 23110 Modern Portfolio Pricing Theory 23310.1 Classical vs. Modern Pricing and Layer Pricing 23310.2 Pricing with Varying Assets 23510.3 Pricing by Layer and the Layer Premium Density 23810.4 The Layer Premium Density as a Distortion Function 23910.5 From Distortion Functions to the Insurance Market 24510.6 Concave Distortion Functions 25210.7 Spectral Risk Measures 25510.8 Properties of an SRM and Its Associated Distortion Function 25910.9 Six Representations of Spectral Risk Measures 26110.10 Simulation Interpretation of Distortion Functions 26310.11 Learning Objectives 26410.A Technical Details 26511 Modern Portfolio Pricing Practice 27111.1 Applying SRMs to Discrete Random Variables 27111.2 Building-Block Distortions and SRMs 27511.3 Parametric Families of Distortions 28011.4 SRM Pricing 28511.5 Selecting a Distortion 29211.6 Fitting Distortions to Cat Bond Data 29811.7 Resolving an Apparent Pricing Paradox 30411.8 Learning Objectives 306Part III Price Allocation 30712 Classical Price Allocation Theory 30912.1 The Allocation of Portfolio Constant CoC Pricing 30912.2 Allocation of Non-Additive Functionals 31212.3 Loss Payments in Default 32412.4 The Historical Development of Insurance Pricing Models 32612.5 Learning Objectives 33713 Classical Price Allocation Practice 33913.1 Allocated CCoC Pricing 33913.2 Allocation of Classical PCP Pricing 34713.3 Learning Objectives 34814 Modern Price Allocation Theory 34914.1 The Natural Allocation of a Coherent Risk Measure 34914.2 Computing the Natural Allocations 36514.3 A Closer Look at Unit Funding 36914.4 An Axiomatic Approach to Allocation 38514.5 Axiomatic Characterizations of Allocations 39214.6 Learning Objectives 39415 Modern Price Allocation Practice 39715.1 Applying the Natural Allocations to Discrete Random Variables 39715.2 Unit Funding Analysis 40415.3 Bodoff’s Percentile Layer of Capital Method 41315.4 Case Study Exhibits 42115.5 Learning Objectives 439Part IV Advanced Topics 44116 Asset Risk 44316.1 Background 44316.2 Adding Asset Risk to Ins Co. 44416.3 Learning Objectives 44717 Reserves 44917.1 Time Periods and Notation 44917.2 Liability for Ultimate Losses 45017.3 The Solvency II Risk Margin 46117.4 Learning Objectives 46818 Going Concern Franchise Value 46918.1 Optimal Dividends 46918.2 The Firm Life Annuity 47218.3 Learning Objectives 47619 Reinsurance Optimization 47719.1 Background 47719.2 Evaluating Ceded Reinsurance 47719.3 Learning Objectives 48120 Portfolio Optimization 48320.1 Strategic Framework 48320.2 Market Regulation 48420.3 Dynamic Capital Allocation and Marginal Cost 48520.4 Marginal Cost and Marginal Revenue 48720.5 Performance Management and Regulatory Rigidities 48820.6 Practical Implications 49020.7 Learning Objectives 491A Background Material 493A.1 Interest Rate, Discount Rate, and Discount Factor 493A.2 Actuarial vs. Accounting Sign Conventions 493A.3 Probability Theory 494A.4 Additional Mathematical Terminology 500B Notation 503References 507Index 523