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DISTRIBUTIONS for ACTUARIES David Bahnemann CAS MONOGRAPH SERIES NUMBER 2 DISTRIBUTIONS FOR ACTUARIES David Bahnemann CASUALTY ACTUARIAL SOCIETY This monograph contains a brief exposition of the standard probability distributions and their applications by property/casualty actuaries. The focus is on the use of parametric distributions fitted to empirical claim data to solve standard actuarial problems, such as creation of increased limit factors, pricing of deductibles, and evaluating the effect of aggregate limits. A native of Minnesota, David Bahnemann studied mathematics and statistics at the University of Minnesota and at Stanford University. After teaching mathematics at Northwest Missouri State University for 18 years, he joined the actuarial department at the St. Paul Companies in St. Paul, Minnesota. For the next 25 years he provided actuarial support to several excess and surplus lines underwriting departments. While at the St. Paul (later known as St. Paul Travelers, and then Travelers) he was involved in large-account and program pricing. During this time he also created several computer- based pricing tools for use by both underwriters and actuaries. During retirement he divides his time between White Bear Lake and Burntside Lake in Minnesota. DISTRIBUTIONS FOR ACTUARIES David Bahnemann Casualty Actuarial Society 4350 North Fairfax Drive, Suite 250 Arlington, Virginia 22203 www.casact.org (703) 276-3100 Distributions for Actuaries By David Bahnemann Copyright 2015 by the Casualty Actuarial Society. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or trans- mitted, in any form or by any means. Electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. For information on obtaining permission for use of material in this work, please submit a written request to the Casualty Actuarial Society. Library of Congress Cataloging-in-Publication Data Bahnemann, David Distributions for Actuaries / David Bahnemann 978-0-9624762-8-0 (print edition) 978-0-9624762-9-7 (electronic edition) 1. Actuarial science. 2. Distribution (Probability theory) 3. Insurance—Mathematical models. I. Bahnemann, David. Copyright 2015, Casualty Actuarial Society To Abbie, Lisa & Greta 2015 CAS Monograph Editorial Board C. K. Stan Khury, Editor in Chief Emmanuel Bardis Craig Davis Richard Fein Jesse Groman Ali Ishaq Leslie Marlo Sholom Feldblum, consultant Glenn Meyers, consultant Katya Prell, consultant Contents Foreword ............................................................................................................... vii Preface .....................................................................................................................ix Chapter 1 Introduction ................................................................................ 1 1.1 Probability Spaces ...................................................................................1 1.2 Random Variables and Probability Distributions ....................................7 1.3 Mathematical Expectation ...................................................................20 1.4 Random Samples .................................................................................25 1.5 Fitting Distributions ............................................................................28 1.6 Problems ..............................................................................................33 Chapter 2 Claim Size ...................................................................................37 2.1 Claim-Size Random Variables ..............................................................37 2.2 Limited Moments ................................................................................40 2.3 Gamma Distributions ..........................................................................46 2.4 Lognormal Distributions......................................................................52 2.5 Pareto Distributions .............................................................................55 2.6 Estimation with Modified Data ...........................................................60 2.7 Transformations ...................................................................................64 2.8 Inflation Effects....................................................................................68 2.9 Problems ..............................................................................................71 Chapter 3 Claim Counts ..............................................................................78 3.1 An Elementary Claim Process ..............................................................78 3.2 Poisson Claim Processes .......................................................................80 3.3 Parameter Uncertainty .........................................................................85 3.4 Negative Binomial Distributions ..........................................................88 3.5 Claim Contagion .................................................................................92 3.6 Portfolio Claims ...................................................................................98 3.7 Problems ............................................................................................100 Chapter 4 Aggregate Claims ......................................................................106 4.1 A Discrete Example ............................................................................106 4.2 Aggregate Distribution Properties ......................................................107 4.3 Approximation by Matching Moments ..............................................113 4.4 Recursion ...........................................................................................119 Contents 4.5 Fourier Approximation ......................................................................124 4.6 Discontinuities ...................................................................................128 4.7 Simulation .........................................................................................129 4.8 Problems ............................................................................................137 Chapter 5 Excess Claims ............................................................................142 5.1 Excess Claim Size ...............................................................................142 5.2 Excess Severity ...................................................................................145 5.3 Layers of Coverage .............................................................................149 5.4 Excess Claim Counts .........................................................................152 5.5 Inflation Effects..................................................................................153 5.6 Aggregate Layer Claims ......................................................................156 5.7 Problems ............................................................................................158 Chapter 6 Limits and Deductibles .............................................................162 6.1 Premium Concepts ............................................................................162 6.2 Increased Limit Factors ......................................................................165 6.3 Risk Load ...........................................................................................171 6.4 Aggregate Limits ................................................................................174 6.5 Deductibles ........................................................................................175 6.6 Problems ............................................................................................182 Appendix .....................................................................................................188 A.1 Distribution Approximation ..............................................................188 A.2 Answers to Selected Problems ............................................................191 A.3 References ..........................................................................................200 vi Casualty Actuarial Society Foreword This is the second monograph in the recently introduced CAS Monograph Series. A CAS monograph is an authoritative, peer reviewed, in-depth work on an important topic within the property and casualty actuarial practice. In this monograph David Bahnemann brings together two perennially important elements of actuarial practice: a solid academic presentation of parametric distributions coupled with the application of these distributions in the actuarial paradigm. Bahnemann taught mathematics at the university level for nineteen years, thus developing an excellent appreciation for what works and what does not work in presenting and conveying technical subject matter. Following that, he worked for more than two decades in applying this knowledge to all types of real actuarial problems that actuaries face every day. Hence, we have this rare presentation of mathematics that actuaries use whenever distributions are involved. This monograph is useful for those wishing to learn the subject matter for the first time as well as for practicing actuaries who wish to have in their bookcase a “desk reference manual” for use whenever faced with a problem involving parametric distributions. This work clearly is a labor of love in which Bahnemann has brought together in a single volume his entire professional life experience
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