A comprehensive framework for measuring, valuing, and managing risk
Pricing Insurance Risk: Theory and Practice delivers an accessible and authoritative account of how to determine the premium for a portfolio of non-hedgeable insurance risks and how to allocate it fairly to each portfolio component.
The authors synthesize hundreds of academic research papers, bringing to light little-appreciated answers to fundamental questions about the relationships between insurance risk, capital, and premium. They lean on their industry experience throughout to connect the theory to real-world practice, such as assessing the performance of business units, evaluating risk transfer options, and optimizing portfolio mix.
Readers will discover: - Definitions, classifications, and specifications of risk - An in-depth treatment of classical risk measures and premium calculation principles - Properties of risk measures and their visualization - A logical framework for spectral and coherent risk measures - How risk measures for capital and pricing are distinct but interact - Why the cost of capital, not capital itself, should be allocated - The natural allocation method and how it unifies marginal and risk-adjusted probability approaches - Applications to reserve risk, reinsurance, asset risk, franchise value, and portfolio optimization
Perfect for actuaries working in the non-life or general insurance and reinsurance sectors, Pricing Insurance Risk: Theory and Practice is also an indispensable resource for banking and finance professionals, as well as risk management professionals seeking insight into measuring the value of their efforts to mitigate, transfer, or bear nonsystematic risk.
Table of Contents
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
