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<p>Preface xii</p> <p><b>1 Introduction 1</b></p> <p>1.1 Our Subject and Why It Matters 1</p> <p>1.2 Players, Roles, and Risk Measures 2</p> <p>1.3 Book Contents and Structure 4</p> <p>1.4 What’s in It for the Practitioner? 7</p> <p>1.5 Where to Start 9</p> <p><b>2 The Insurance Market and Our Case Studies 13</b></p> <p>2.1 The Insurance Market 13</p> <p>2.2 Ins Co.: A One-Period Insurer 15</p> <p>2.3 Model vs. Reality 16</p> <p>2.4 Examples and Case Studies 17</p> <p>2.5 Learning Objectives 25</p> <p><b>Part I Risk 27</b></p> <p><b>3 Risk and Risk Measures 29</b></p> <p>3.1 Risk in Everyday Life 29</p> <p>3.2 Defining Risk 30</p> <p>3.3 Taxonomies of Risk 31</p> <p>3.4 Representing Risk Outcomes 36</p> <p>3.5 The Lee Diagram and Expected Losses 40</p> <p>3.6 Risk Measures 54</p> <p>3.7 Learning Objectives 60</p> <p><b>4 Measuring Risk with Quantiles, VaR, and TVaR 63</b></p> <p>4.1 Quantiles 63</p> <p>4.2 Value at Risk 70</p> <p>4.3 Tail VaR and Related Risk Measures 85</p> <p>4.4 Differentiating Quantiles, VaR, and TVaR 102</p> <p>4.5 Learning Objectives 102</p> <p><b>5 Properties of Risk Measures and Advanced Topics 105</b></p> <p>5.1 Probability Scenarios 105</p> <p>5.2 Mathematical Properties of Risk Measures 110</p> <p>5.3 Risk Preferences 124</p> <p>5.4 The Representation Theorem for Coherent Risk Measures 130</p> <p>5.5 Delbaen’s Differentiation Theorem 137</p> <p>5.6 Learning Objectives 141</p> <p>5.A Lloyd’s Realistic Disaster Scenarios 142</p> <p>5.B Convergence Assumptions for Random Variables 143</p> <p><b>6 Risk Measures in Practice 147</b></p> <p>6.1 Selecting a Risk Measure Using the Characterization Method 147</p> <p>6.2 Risk Measures and Risk Margins 148</p> <p>6.3 Assessing Tail Risk in a Univariate Distribution 149</p> <p>6.4 The Intended Purpose: Applications of Risk Measures 150</p> <p>6.5 Compendium of Risk Measures 153</p> <p>6.6 Learning Objectives 156</p> <p><b>7 Guide to the Practice Chapters 157</b></p> <p><b>Part II Portfolio Pricing 161</b></p> <p><b>8 Classical Portfolio Pricing Theory 163</b></p> <p>8.1 Insurance Demand, Supply, and Contracts 163</p> <p>8.2 Insurer Risk Capital 168</p> <p>8.3 Accounting Valuation Standards 178</p> <p>8.4 Actuarial Premium Calculation Principles and Classical Risk Theory 182</p> <p>8.5 Investment Income in Pricing 186</p> <p>8.6 Financial Valuation and Perfect Market Models 189</p> <p>8.7 The Discounted Cash Flow Model 192</p> <p>8.8 Insurance Option Pricing Models 200</p> <p>8.9 Insurance Market Imperfections 210</p> <p>8.10 Learning Objectives 213</p> <p>8.A Short- and Long-Duration Contracts 215</p> <p>8.B The Equivalence Principle 216</p> <p><b>9 Classical Portfolio Pricing Practice 217</b></p> <p>9.1 Stand-Alone Classical PCPs 217</p> <p>9.2 Portfolio CCoC Pricing 223</p> <p>9.3 Applications of Classical Risk Theory 224</p> <p>9.4 Option Pricing Examples 227</p> <p>9.5 Learning Objectives 231</p> <p><b>10 Modern Portfolio Pricing Theory 233</b></p> <p>10.1 Classical vs. Modern Pricing and Layer Pricing 233</p> <p>10.2 Pricing with Varying Assets 235</p> <p>10.3 Pricing by Layer and the Layer Premium Density 238</p> <p>10.4 The Layer Premium Density as a Distortion Function 239</p> <p>10.5 From Distortion Functions to the Insurance Market 245</p> <p>10.6 Concave Distortion Functions 252</p> <p>10.7 Spectral Risk Measures 255</p> <p>10.8 Properties of an SRM and Its Associated Distortion Function 259</p> <p>10.9 Six Representations of Spectral Risk Measures 261</p> <p>10.10 Simulation Interpretation of Distortion Functions 263</p> <p>10.11 Learning Objectives 264</p> <p>10.A Technical Details 265</p> <p><b>11 Modern Portfolio Pricing Practice 271</b></p> <p>11.1 Applying SRMs to Discrete Random Variables 271</p> <p>11.2 Building-Block Distortions and SRMs 275</p> <p>11.3 Parametric Families of Distortions 280</p> <p>11.4 SRM Pricing 285</p> <p>11.5 Selecting a Distortion 292</p> <p>11.6 Fitting Distortions to Cat Bond Data 298</p> <p>11.7 Resolving an Apparent Pricing Paradox 304</p> <p>11.8 Learning Objectives 306</p> <p><b>Part III Price Allocation 307</b></p> <p><b>12 Classical Price Allocation Theory 309</b></p> <p>12.1 The Allocation of Portfolio Constant CoC Pricing 309</p> <p>12.2 Allocation of Non-Additive Functionals 312</p> <p>12.3 Loss Payments in Default 324</p> <p>12.4 The Historical Development of Insurance Pricing Models 326</p> <p>12.5 Learning Objectives 337</p> <p><b>13 Classical Price Allocation Practice 339</b></p> <p>13.1 Allocated CCoC Pricing 339</p> <p>13.2 Allocation of Classical PCP Pricing 347</p> <p>13.3 Learning Objectives 348</p> <p><b>14 Modern Price Allocation Theory 349</b></p> <p>14.1 The Natural Allocation of a Coherent Risk Measure 349</p> <p>14.2 Computing the Natural Allocations 365</p> <p>14.3 A Closer Look at Unit Funding 369</p> <p>14.4 An Axiomatic Approach to Allocation 385</p> <p>14.5 Axiomatic Characterizations of Allocations 392</p> <p>14.6 Learning Objectives 394</p> <p><b>15 Modern Price Allocation Practice 397</b></p> <p>15.1 Applying the Natural Allocations to Discrete Random Variables 397</p> <p>15.2 Unit Funding Analysis 404</p> <p>15.3 Bodoff’s Percentile Layer of Capital Method 413</p> <p>15.4 Case Study Exhibits 421</p> <p>15.5 Learning Objectives 439</p> <p><b>Part IV Advanced Topics 441</b></p> <p><b>16 Asset Risk 443</b></p> <p>16.1 Background 443</p> <p>16.2 Adding Asset Risk to Ins Co. 444</p> <p>16.3 Learning Objectives 447</p> <p><b>17 Reserves 449</b></p> <p>17.1 Time Periods and Notation 449</p> <p>17.2 Liability for Ultimate Losses 450</p> <p>17.3 The Solvency II Risk Margin 461</p> <p>17.4 Learning Objectives 468</p> <p><b>18 Going Concern Franchise Value 469</b></p> <p>18.1 Optimal Dividends 469</p> <p>18.2 The Firm Life Annuity 472</p> <p>18.3 Learning Objectives 476</p> <p><b>19 Reinsurance Optimization 477</b></p> <p>19.1 Background 477</p> <p>19.2 Evaluating Ceded Reinsurance 477</p> <p>19.3 Learning Objectives 481</p> <p><b>20 Portfolio Optimization 483</b></p> <p>20.1 Strategic Framework 483</p> <p>20.2 Market Regulation 484</p> <p>20.3 Dynamic Capital Allocation and Marginal Cost 485</p> <p>20.4 Marginal Cost and Marginal Revenue 487</p> <p>20.5 Performance Management and Regulatory Rigidities 488</p> <p>20.6 Practical Implications 490</p> <p>20.7 Learning Objectives 491</p> <p><b>A Background Material 493</b></p> <p>A.1 Interest Rate, Discount Rate, and Discount Factor 493</p> <p>A.2 Actuarial vs. Accounting Sign Conventions 493</p> <p>A.3 Probability Theory 494</p> <p>A.4 Additional Mathematical Terminology 500</p> <p><b>B Notation 503</b></p> <p>References 507</p> <p>Index 523</p>

<p><b>Stephen J. Mildenhall</b> 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. </p> <p><b>John A. Major</b> 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.

<p><b>A comprehensive framework for measuring, valuing, and managing risk </b></p> <p><i>Pricing Insurance Risk: Theory and Practice</i> 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. <p>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. <p>Readers will discover: <ul><li>Definitions, classifications, and specifications of risk</li> <li>An in-depth treatment of classical risk measures and premium calculation principles</li> <li>Properties of risk measures and their visualization</li> <li>A logical framework for spectral and coherent risk measures</li> <li>How risk measures for capital and pricing are distinct but interact</li> <li>Why the cost of capital, not capital itself, should be allocated</li> <li>The natural allocation method and how it unifies marginal and risk-adjusted probability approaches</li> <li>Applications to reserve risk, reinsurance, asset risk, franchise value, and portfolio optimization</li></ul> <p>Perfect for actuaries working in the non-life or general insurance and reinsurance sectors, <i>Pricing Insurance Risk: Theory and Practice</i> 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.

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