to Actuarial Mathematics by A. K. Gupta Bowling Green State University, Bowling Green, Ohio, U. S. A. and T. Varga National Pension Insurance Fund. Budapest, Hungary SPRINGER-SCIENCE+BUSINESS MEDIA, B. V. A C. I. P. Catalogue record for this book is available from the Library of Congress. ISBN 978-90-481-5949-9 ISBN 978-94-017-0711-4 (eBook) DOI 10. 1007/978-94-017-0711-4 Printed on acid-free paper All Rights Reserved (c) 2002 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2002 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner. To Alka, Mita, and Nisha AKG To Terezia and Julianna TV TABLE OF CONTENTS PREFACE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix CHAPTER 1. FINANCIAL MATHEMATICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. 1. Compound Interest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. 2. Present Value. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1. 3. Annuities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 CHAPTER 2. MORTALITy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 2. 1 Survival Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 2. 2. Actuarial Functions of Mortality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 2. 3. Mortality Tables. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 CHAPTER 3. LIFE INSURANCES AND ANNUITIES . . . . . . . . . . . . . . . . . . . . . 112 3. 1. Stochastic Cash Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 3. 2. Pure Endowments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 3. 3. Life Insurances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 3. 4. Endowments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 3. 5. Life Annuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 CHAPTER 4. PREMIUMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 4. 1. Net Premiums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 4. 2. Gross Premiums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215 Vll CHAPTER 5. RESERVES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 5. 1. Net Premium Reserves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 5. 2. Mortality Profit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 5. 3. Modified Reserves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 ANSWERS TO ODD-NuMBERED PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Consumer Attitudes Toward Credit Insurance provides the findings of a survey of approximately 3600 individuals who had the opportunity to purchase credit life insurance in conjunction with all types of consumer loans, except first mortgages and credit cards. The survey that forms the basis of the book was conducted in 1993 by the Credit Research Center at Purdue University's Krannert Graduate School of Management. It replicates and expands upon four previous national studies of credit insurance consumers, done between 1970 and 1985. Despite the generally positive findings of prior research with respect to consumer attitudes toward credit insurance, several open questions remain of interest to policy makers, specifically the question of whether coercion is involved in the sale of the insurance. Consumer Attitudes Toward Credit Insurance addresses these outstanding issues. It presents a profile of who is currently being served by the credit insurance market, as well as the reasons borrowers purchase the product and their experience with the offer of credit insurance at point of sale.

Introduction This book includes terms of reference and offers an augmented volume of relevant work initiated within the comprehensive concept of "Knowledge Management and Risk Governance." The latter stood for the initial title of an ad-hoc meeting held in Ascona, Switzerland, organized by the Technological Risk Management Unit of the Joint Research Centre of the European Commission (JRC) and the KOVERS Centre of Excellence in Risk and Safety Sciences of the Swiss Federal Institute of Technology, ETH Zurich. Background Risk governance, in addition to the continuous interest of researchers, has recently attracted the attention of policy-makers and the media and the concern of the public. New and emerging risks in various fields and a number of risk-related issues increased the public interest and prompted for a new framework in dealing with risks. The Conference on Science and Governance organized by the European Commission in October 2000 is one of the international forums addressing this issue. Other recent events such as the establishment of the International Risk Governance Council outline the importance of the governance concept in relation to that of risk management (see www.irgc.org). At the same time noticeable progress has been made in Information Technologies and Decision Support, passing from the process of information PREFACE xvi to the process of knowledge. In this context new tools and methods became available, whose application in risk management may be beneficial.

An Introduction to Actuarial Studies provides a contemporary guide to actuarial technique and practice. This substantially revised and extended new edition includes: New and thoroughly updated material Many more exercises with solutions to allow the reader to establish confidence in using actuarial techniques. It covers a broad range of topics representing the basic areas of actuarial science including compound interest calculations, demographic theory and techniques, and the pricing and operation of simple life assurance contracts. Numerous worked examples illustrate the principles and techniques described in the text. The text assumes no prior knowledge of actuarial work but requires mathematical knowledge at first year university level and an ability for problem solving. It is designed both for those beginning a career in actuarial work, as well as those interested in learning about basic actuarial tools and the main areas of actuarial practice. Contents: Preface; 1. Introduction; 2. Valuation of Financial Transactions; 3. Demography; 4. Actuarial Practice; 5. Valuation of Contingent Payments; References; Index

One measure of public program response to rapidly expanding older populations is the approach to old-age pensions under social insurance, social assistance, and provident fund systems. Social insurance is clearly the preferred method of meeting the income needs of the elderly, but historical, as well as current social and economic conditions are forcing many nations to reevaluate the characteristics of viable and sustainable social insurance programs. This has led to a variety of innovations in old-age pension programs development, including revised benefit formulas, raised retirement ages, increased income testing, and expanded reliance on private occupational supplemental programs.

The essays in this new international handbook analyze the impact of the economic, social, and cultural effects of aging populations on government social insurance policies. They offer a perspective on how twenty different countries have approached income maintenance programs for the elderly. Collectively, the contributors demonstrate how governments, non-governmental entities, communities, and families respond to changes in traditional income and social service support systems. They provide not only descriptions of existing programs, but also a better understanding of the factors that gave rise to their distinct characteristics. This important new collection will be required reading for everyone involved in elderly services.

The articles in this volume were first presented at the Seventh and Eighth Conferences on Economic Issues in Workers' Compensation sponsored by the National Council on Compensation Insurance. A principal objective of the Conference series has been for workers' compensation insurance researchers to apply state-of-the-art research methodologies to policy questions of interest to the workers' compensation insurance community. This community is a rather diverse group--it includes employers, insurers, injured workers, regulators, and legislators, as well as those who service or represent these groups (e.g., physicians, rehabilitation specialists, labor unions). Despite this diversity and the variety of agendas, the Conference series continues to address many important policy questions. Readers familiar with the Conference series and the four previously published volumes should notice an evolution in terms of the topics addressed in this volume. In the earlier conferences, the topics were more often concerned with the underlying causes of the tremendous increase in workers' compensation benefit payments. In the present volume, h- ever, only four of the fourteen chapters directly concern workers' c- pensation insurance benefits, while the other ten concern the pricing of workers compensation insurance. This is not to suggest that workers' compensation cost increases have abated. In 1989, workers' compensation incurred losses exceeded $45 billion to continue the annual double-digit cost increases. Two explanations can be offered for the somewhat altered focus of this volume. First, despite the continued increase in prices, the financial results for the workers' compensation insurance line continue to be poor.

This book explores the ways in which the adoption of new paradigms, processes, and technologies can lead to greater revenue, cost efficiency and control, as well as improved business agility in the insurance industry.

This book explores the central problems underlying the insurance of aviation war and terrorism risks and associated perils. It critically analyses the reasons why conventional insurance markets are unwilling or unable to provide sustainable insurance coverage for aviation war and terrorism risks in the aftermath of catastrophic events such as the terrorist events of September 11, 2001. It also examines some of the prominent concepts proposed and/or implemented after 9/11 to determine whether and to what extent these concepts avoid identified pitfalls. Like many of life s essentials, the importance of insurance is most evident when it is not available. The sheer scale and magnitude of the insurance losses that followed 9/11 caused conventional insurance markets (which hitherto had been offering generous insurance coverage for aviation war and terrorism risks to air transport operators for little or no premium) to withdraw coverage forthwith. The ensuing absence or insufficiency of commercial insurance coverage for aviation war and terrorism risks has sparked a global search for viable and sustainable alternatives. Ten years have since elapsed, and despite numerous efforts, the fundamental problems remain unresolved. The book proceeds on the premise that the underlying issues are not entirely legal in nature; they have immense economic, psychological and policy implications that cannot be underestimated. A multidisciplinary approach is therefore used in examining the issues, drawing heavily upon analytical principles adapted from law and economics and behavioural law and economics. It is hoped that the resulting study will be beneficial not only to lawyers and those interested in aviation insurance but also to economists, air transport insurance program managers, capital market investors and governmental policymakers, both at the national and international levels.

The problem of solvency is, in fact, as old as insurance. The history of the industry knows many ways to meet the risks involved with underwriting, such as spreading the risk portfolio (Cato, Senior already applied it), risk selection, reserve funds, reinsurance, etc. Whilst these measures too often proved ineffective, the establish ment of legislative control and public supervision ensued. However, not until the last few decades has the solvency issue become an ob ject of intensive studies, very much thanks to the progress of related empirical and theoretical knowledge, and in the under standing of the concerned complicated processes. The research activities have grown extensively in many countries in recent years. The more the studies advance the more new relevant aspects are detected and a great variety of alternative proposals have come up for discussion. Therefore, it has become necessary to attempt a survey of the whole problem area in order to be able to place the quite numerous pieces of knowledge in their proper context, and also, among other things, to avoid the pitfalls of handling isolated problems omitting vital tie-ins to the environment. Many of the rele vant problems and subproblems are still lacking adequate and well tested solutions. Therefore, a survey of the whole problem area can also hopefully serve as guidance for future research efforts."

Provides a comprehensive and accessible introduction to general insurance pricing, based on the author’s many years of experience as both a teacher and practitioner. Suitable for students taking a course in general insurance pricing, notably if they are studying to become an actuary through the UK Institute of Actuaries exams. No other title quite like this on the market that is perfect for teaching/study, and is also an excellent guide for practitioners.

The increasingly risky environment in which companies operate is characterized by a rising number of risk components, factors, sources, and drivers. The identification, evaluation, and management of these risks require the capability to coordinate various skills within a company and in upstream and downstream relationships. This handbook provides an integrated approach to the assessment, transfer, and communication of critical risks and highlights emerging methodologies that can help to protect businesses from adverse events and their effects. It explains how different risk management perspectives should be combined, and in particular how the corporate governance vision should be integrated with the perspectives of operations management, financial management, and business continuity management. In this sense the handbook provides concrete directions on how to develop a risk management team and culture, taking into account business challenges and employing appropriate managerial tools.

This text provides a handbook for anyone involved in the current London Market. It takes the reader through the full remit of reinsurance practice from the development of reinsurance, methods and types of reinsurance, reinsurance markets and placement of risk, to the legal contract and wordings, the London Market slip, claims, proportional treaty and run-off. Full appendices are included giving examples of slips, cover wordings and key clauses.

Using institutional theory to explain innovation and merging academic and critical analysis with practical recommendations, this book provides a full and rich account of how new products are brought to market; considering both the successes and failures in equal measure. This book takes the meeting point of two seemingly incongruous schools of theoretical thought to enlighten the debate surrounding product innovation. In doing so it: illustrates how institutional forces come to shape the interest, priorities and behaviour of organizational members in the development and implementation process of incremental product innovation investigates the failed innovative attempts of established organizations demonstrates the importance of organizational and intra-organizational forces for innovative success. The insight it offers into the organization of product innovation processes in the financial services sector and the guidelines it sets up for their improvement makes Innovation and Institutions essential reading for those working in or studying the banking, finance and insurance sector who have an interest in innovation studies.

The mathematical theory of non-life insurance developed much later than the theory of life insurance. The problems that occur in the former field are far more intricate for several reasons: 1. In the field oflife insurance, the company usually has to pay a claim on the policy only once: the insured dies or the policy matures only once. It is with only a few particular types of policy (for instance, sickness insurance, when the insured starts working again after a period of sickness) that a valid claim can be made on a number of different occasions. On the other hand, the general rule in non-life insurance is that the policyholder is liable to be the victim of several losses (in automobile insurance, of course, but also in burglary and fire insurance, householders' comprehensive insurance, and so on). 2. In the field of life insurance, the amount to be paid by the company excluding any bonuses-is determined at the inception of the policy. For the various types of life insurance contracts, the sum payable on death or at maturity of the policy is known in advance. In the field of non-life insurance, the amount of a loss is a random variable: the cost of an automobile crash, the partial or totalloss of a building as a result of fire, the number and nature of injuries, and so forth."

This book addresses an experiment in funding money damage claims in England from 2000 to 2013. The model - recoverable conditional fees - was unique and has remained so. It covers the development, amendment and effective abolition of the model, as well as the process of policy development and the motivation and objectives of the policy makers.

th This book is published to commemorate the 50 Anniversary of the S.S. Huebner Foundation for Insurance Education. Administered at the Wharton School of the University of Pennsylvania, the Huebner Foundation was established in 1941 to strengthen insurance education at the collegiate level by increasing the number of professors specializing in insurance and enriching the literature in the field. The financial support of leading life insurance companies has enabled the Foundation to provide post-graduate education for prospective insurance teachers and scholars. Through its fellowship program, the Foundation supports students in the Ph.D. program in Risk and Insurance at the Wharton School. The success of the Foundation is measured by the accomplishments of its alumni. Former Huebner Fellows play leading roles in every major area of insurance education. Fellows teach insurance to tens of thousands of undergraduate and MBA students each year and have written hundreds of books and thousands of articles on insurance. Fellows hold leadership positions at the American College, the Life Office Management Association, and the Certified Employee Benefit Specialist Program. The Foundation was created in honor of Dr. Solomon S. Huebner, a pioneer in insurance education. Dr. Huebner taught the first organized course on the economics of insurance ever offered at the collegiate level in 1904. An internationally recognized author and teacher, Dr. Huebner had a profound impact on both insurance education and the insurance industry. He served on the faculty of the Wharton School for more than nearly fifty years.

This book explores the profound transformation that has taken place in European insurance legislation since January 2016. Expert contributions discuss the changes that have taken place in the supervision of insurance and reinsurance undertakings through an economic risk-based approach. They outline the European insurance market before going on to show how Solvency II and Insurance Distribution Directive (IDD) are expected to generate significant benefits and have a positive impact on all parties involved in the insurance industry, the supervisory authorities and the insured. They also show how Solvency II is likely to benefit the economy as a whole, promoting more efficient allocation of capital and risk in a financial stability framework. This volume will be of interest to academics and researchers in the field of insurance regulation.

This series explores the central and unique role of organizational ethics in creating and sustaining a flourishing, pluralistic, free enterprise economy. It examines how profit seeking and not-for-profit organizations can be conceived and designed to satisfy legitimate human needs in an ethical and meaningful way. The authors submit rigorous research studies from a wide variety of academic perspectives including: business management, philosophy, sociology, psychology, religion, accounting, finance, and marketing. It focuses on ethical issues in the insurance industry and includes a variety of disciplines with authors from over 30 countries. The papers were selected from the best presentations at the Twelfth Annual International Conference Promoting Business Ethics, held Oct. 2005 in Manhattan.

Two different applications have been considered, automobile claims from Massachusetts and health expenses from the Netherlands. We have fit 11 different distributions to these data. The distributions are conveniently nested within a single four parameter distribution, the generalized beta of the second type. This relationship facilitates analysis and comparisons. In both cases the GB2 provided the best fit and the Burr 3 is the best three parameter model. In the case of automobile claims, the flexibility of the GB2 provides a statistically siE;nificant improvement in fit over all other models. In the case of Dutch health expenses the improvement of the GB2 relative to several alternatives was not statistically significant. * The author appreciates the research assistance of Mark Bean, Young Yong Kim and Steve White. The data used were provided by Richard Derrig of The Massachusetts Automobile Rating and Accident Prevention Bureau and by Bob Van der Laan and The Silver Cross Foundation for the medical insurance claim data. 2~ REFERENCES Arnold, B. C. 1983. Pareto Distributions. Bartonsville: International Cooperative Publishing House. Cummins, J. D. and L. R. Freifelder. 1978. A comparative analysis of alternative maximum probable yearly aggregate loss estimators. Journal of Risk and Insurance 45:27-52. *Cummins, J. D., G. Dionne, and L. Maistre. 1987. Application of the GB2 family of distributions in collective risk theory. University of Pennsylvania: Mimeographed manuscript. Hogg, R. V. and S. A. Klugman. 1983. On the estimation of long tailed skewed distributions with actuarial applications.