Stephen J. Mildenhall - Pricing Insurance Risk

Stephen J. Mildenhall and John A. Major

This edition first published 2022

© 2022 John Wiley & Sons, Inc.

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions.

The right of Stephen J. Mildenhall and John A. Major to be identified as the authors of this work has been asserted in accordance with law.

Registered Office

John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USA

Editorial Office

111 River Street, Hoboken, NJ 07030, USA

For details of our global editorial offices, customer services, and more information about Wiley products visit us at www.wiley.com.

Wiley also publishes its books in a variety of electronic formats and by print-on-demand. Some content that appears in standard print versions of this book may not be available in other formats.

Limit of Liability/Disclaimer ofWarranty

While the publisher and authors have used their best efforts in preparing this work, they make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives, written sales materials or promotional statements for this work. The fact that an organization, website, or product is referred to in this work as a citation and/or potential source of further information does not mean that the publisher and authors endorse the information or services the organization, website, or product may provide or recommendations it may make. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for your situation. You should consult with a specialist where appropriate. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages.

Library of Congress Cataloging-in-Publication Data Names: Mildenhall, Stephen J., author. | Major, John A., author. Title: Pricing insurance risk : theory and practice / Stephen J Mildenhall, John A Major. Description: Hoboken, NJ : John Wiley and Sons, 2022. | Includes bibliographical references and index. Identifiers: LCCN 2021062792 (print) | LCCN 2021062793 (ebook) | ISBN 9781119755678 (hardback) | ISBN 9781119755692 (pdf) | ISBN 9781119756521 (epub) | ISBN 9781119756538 (ebook) Subjects: LCSH: Risk (Insurance) | Risk management. Classification: LCC HG8054.5 .M55 2022 (print) | LCC HG8054.5 (ebook) | DDC 368–dc23/eng/20220111 LC record available at https://lccn.loc.gov/2021062792LC ebook record available at https://lccn.loc.gov/2021062793

Cover Design: Wiley

Cover Image: © zef art/Shutterstock

Set in 9.5/12.5pt STIXTwoText by Integra Software Services Pvt. Ltd, Pondicherry, India

Dedicated to our wives, Helen and Diane

1 Cover

2 Title page Pricing Insurance Risk Theory and Practice Stephen J. Mildenhall and John A. Major

3 Copyright

4 Dedication

5 Preface

6 1 Introduction 1.1 Our Subject and Why It Matters1.2 Players, Roles, and Risk Measures1.3 Book Contents and Structure1.4 What’s in It for the Practitioner?1.5 Where to Start

7 2 The Insurance Market and Our Case Studies 2.1 The Insurance Market2.2 Ins Co.: A One-Period Insurer2.3 Model vs. Reality2.4 Examples and Case Studies2.5 Learning Objectives

8 Part I Risk 3 Risk and Risk Measures 3.1 Risk in Everyday Life3.2 Defining Risk3.3 Taxonomies of Risk3.4 Representing Risk Outcomes3.5 The Lee Diagram and Expected Losses3.6 Risk Measures3.7 Learning Objectives 4 Measuring Risk with Quantiles, VaR, and TVaR 4.1 Quantiles4.2 Value at Risk4.3 Tail VaR and Related Risk Measures4.4 Differentiating Quantiles, VaR, and TVaR4.5 Learning Objectives 5 Properties of Risk Measures and Advanced Topics 5.1 Probability Scenarios5.2 Mathematical Properties of Risk Measures5.3 Risk Preferences5.4 The Representation Theorem for Coherent Risk Measures5.5 Delbaen’s Differentiation Theorem5.6 Learning Objectives5.A Lloyd’s Realistic Disaster Scenarios5.B Convergence Assumptions for Random Variables 6 Risk Measures in Practice 6.1 Selecting a Risk Measure Using the Characterization Method6.2 Risk Measures and Risk Margins6.3 Assessing Tail Risk in a Univariate Distribution6.4 The Intended Purpose: Applications of Risk Measures6.5 Compendium of Risk Measures6.6 Learning Objectives 7 Guide to the Practice Chapters

9 Part II Portfolio Pricing 8 Classical Portfolio Pricing Theory 8.1 Insurance Demand, Supply, and Contracts8.2 Insurer Risk Capital8.3 Accounting Valuation Standards8.4 Actuarial Premium Calculation Principles and Classical Risk Theory8.5 Investment Income in Pricing8.6 Financial Valuation and Perfect Market Models8.7 The Discounted Cash Flow Model8.8 Insurance Option Pricing Models8.9 Insurance Market Imperfections8.10 Learning Objectives8.A Short- and Long-Duration Contracts8.B The Equivalence Principle 9 Classical Portfolio Pricing Practice 9.1 Stand-Alone Classical PCPs9.2 Portfolio CCoC Pricing9.3 Applications of Classical Risk Theory9.4 Option Pricing Examples9.5 Learning Objectives 10 Modern Portfolio Pricing Theory 10.1 Classical vs. Modern Pricing and Layer Pricing10.2 Pricing with Varying Assets10.3 Pricing by Layer and the Layer Premium Density10.4 The Layer Premium Density as a Distortion Function10.5 From Distortion Functions to the Insurance Market10.6 Concave Distortion Functions10.7 Spectral Risk Measures10.8 Properties of an SRM and Its Associated Distortion Function10.9 Six Representations of Spectral Risk Measures10.10 Simulation Interpretation of Distortion Functions10.11 Learning Objectives10.A Technical Details 11 Modern Portfolio Pricing Practice 11.1 Applying SRMs to Discrete Random Variables11.2 Building-Block Distortions and SRMs11.3 Parametric Families of Distortions11.4 SRM Pricing11.5 Selecting a Distortion11.6 Fitting Distortions to Cat Bond Data11.7 Resolving an Apparent Pricing Paradox11.8 Learning Objectives

10 Part III Price Allocation 12 Classical Price Allocation Theory 12.1 The Allocation of Portfolio Constant CoC Pricing12.2 Allocation of Non-Additive Functionals12.3 Loss Payments in Default12.4 The Historical Development of Insurance Pricing Models12.5 Learning Objectives 13 Classical Price Allocation Practice 13.1 Allocated CCoC Pricing13.2 Allocation of Classical PCP Pricing13.3 Learning Objectives 14 Modern Price Allocation Theory 14.1 The Natural Allocation of a Coherent Risk Measure14.2 Computing the Natural Allocations14.3 A Closer Look at Unit Funding14.4 An Axiomatic Approach to Allocation14.5 Axiomatic Characterizations of Allocations14.6 Learning Objectives 15 Modern Price Allocation Practice 15.1 Applying the Natural Allocations to Discrete Random Variables15.2 Unit Funding Analysis15.3 Bodoff’s Percentile Layer of Capital Method15.4 Case Study Exhibits15.5 Learning Objectives