distributions for actuaries

215 Pages·2015·2.57 MB·English

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
[phone]
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 in this field. The CAS is grateful
for his effort in producing this monograph as well as the gift it represents to the CAS
and its members.
C. K. “Stan” Khury
Chairperson
Monograph Editorial Board