An Introduction to Actuarial Mathematics

359 Pages·2002·7.11 MB·English

by  A. K. Gupta, T. Varga (auth.)

An Introduction to Actuarial Mathematics
MATHEMATICAL MODELLING:
Theory and Applications
VOLUME 14
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An Introduction
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
© 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 ............................... 303
ApPENDIX 1. COMPOUND INTEREST TABLES ........................... 311
ApPENDIX 2. ILLUSTRATIVE MORTALITY TABLE ...................... 331
REFERENCES ................................................................................... 340
SYMBOL INDEX ............................................................................... 341
SUBJECT INDEX ............................................................................... 348
viii
PREFACE
This text has been written primarily as an introduction to the basics of
the actuarial mathematics of life insurance. The subject matter is suitable
for a one-semester or a one-year college course. Since it attempts to derive
the results in a mathematically rigorous way, the concepts and techniques of
one variable calculus and probability theory have been used throughout. A
two semester course in calculus and a one semester course in probability
theory at the undergraduate level are the usual prerequisites for the
understanding of the material.
There are five chapters in this text. Chapter 1 focuses on some
important concepts of financial mathematics. The concept of interests,
essential to the understanding of the book, is discussed here very
thoroughly. After the study of present values in general, annuities-certain
are examined.
Chapter 2 is concerned with the mortality theory. The analytical study
of mortality is followed by the introduction of mortality tables.
Chapter 3 discusses different types of life insurances in detail. First, we
examine stochastic cash flows in general. Then we study pure endowments,
whole life and term insurances, endowments, and life annuities.
Chapter 4 is devoted to premium calculations. It opens with a section on
net premiums, followed by a section discussing office premiums.
Chapter 5 deals with reserves. In addition to presenting different
reserving methods, the mortality profit is also studied here. Careful
consideration is given to the problem of negative reserves as well.
The book contains many systematically solved examples showing the
practical applications of the theory presented. Solving the problems at the
end of each section is essential for understanding the material. Answers to
odd-numbered problems are given at the end of the book.
The authors are indebted to Drs. D. K. Nagar, Jie Chen and Asoka
Ramanayake with whom discussions and whose comments on the original
manuscript, resulted in a vastly improved text. We are grateful to the
Society of Actuaries, the Institute of Actuaries and the Faculty of Actuaries,
acknowledged in the appropriate places, for permission to reproduce tables.
Thanks are also due to Ms. Anneke Pot of Kluwer Academic Publishers for
her cooperation and help. Her counsel on matters of form and style is
sincerely appreciated. Finally, the authors are thankful to Cynthia
Patterson for her efficient word processing and for her patience during the
preparation of this manuscript.
Bowling Green, Ohio A. K. Gupta
Budapest, Hungary T. Varga
August, 2001
IX
CHAPTER 1
FINANCIAL MATHEMATICS
1.1 COMPOUND INTEREST
There are many situations in every day life when we come across the
concept of interest. For example, when someone makes a deposit in a bank
account, the money will earn interest. Money can also be borrowed from a
bank, which has to be repaid later on with interest. Many car buyers who
cannot afford paying the price in cash will pay monthly installments whose
sum is usually higher than the original price of the car. This is due to the
interest which has been added to the price of the car.
In general, interest is the fee one pays for the right of using someone
else's money. In the case of a bank deposit, the bank uses the money of the
depositor, whereas in the case of a bank loan, the borrower uses the money of
the bank. The situation is slightly complicated with a car purchase. What
happens here is that the buyer of the car gets a loan from the dealer which
can be used to pay the price of the car in full. This loan has to be repaid to
the dealer in monthly installments.
Consider an interest earning bank account. Assume we make a deposit of
$X on January 1, and leave the money there until the end of the year. On
January 1 of the next year, we withdraw the money together with the
interest. We find that our money has accumulated to $Y. What is the rate
of interest for the year? Let us denote the interest rate by "i". It is worth
noting that the interest rate is often expressed as a percentage (e.g. 4%), or
as a real number (e.g. 0.04).
Let us split $Y into two parts. One is the amount of the original
deposit: $X, also called the capital content, the rest is the interest content:
$Xi. We have the following equation:
Y=X+Xi
or equivalently
Y = X(1 + i).
EXAMPLE 1.1. A sum of $340 is deposited in a bank account on January 1,
1990. The account earns interest at a rate of 3% per annum. What is the
accumulated value of the account on January 1, 1991? What are the capital
and interest contents of the accumulation?