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Elliptically Contoured Models in Statistics and Portfolio Theory
fully revises the first detailed introduction to the theory of
matrix variate elliptically contoured distributions. There are two
additional chapters, and all the original chapters of this classic
text have been updated. Resources in this book will be valuable for
researchers, practitioners, and graduate students in statistics and
related fields of finance and engineering. Those interested in
multivariate statistical analysis and its application to portfolio
theory will find this text immediately useful. In multivariate
statistical analysis, elliptical distributions have recently
provided an alternative to the normal model. Elliptical
distributions have also increased their popularity in finance
because of the ability to model heavy tails usually observed in
real data. Most of the work, however, is spread out in journals
throughout the world and is not easily accessible to the
investigators. A noteworthy function of this book is the collection
of the most important results on the theory of matrix variate
elliptically contoured distributions that were previously only
available in the journal-based literature. The content is organized
in a unified manner that can serve an a valuable introduction to
the subject. "
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 .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
In multivariate statistical analysis, elliptical distributions have
recently provided an alternative to the normal model. Most of the
work, however, is spread out in journals throughout the world and
is not easily accessible to the investigators. Fang, Kotz, and Ng
presented a systematic study of multivariate elliptical
distributions, however, they did not discuss the matrix variate
case. Recently Fang and Zhang have summarized the results of
generalized multivariate analysis which include vector as well as
the matrix variate distributions. On the other hand, Fang and
Anderson collected research papers on matrix variate elliptical
distributions, many of them published for the first time in
English. They published very rich material on the topic, but the
results are given in paper form which does not provide a unified
treatment of the theory. Therefore, it seemed appropriate to
collect the most important results on the theory of matrix variate
elliptically contoured distributions available in the literature
and organize them in a unified manner that can serve as an
introduction to the subject. The book will be useful for
researchers, teachers, and graduate students in statistics and
related fields whose interests involve multivariate statistical
analysis. Parts of this book were presented by Arjun K Gupta as a
one semester course at Bowling Green State University. Some new
results have also been included which generalize the results in
Fang and Zhang. Knowledge of matrix algebra and statistics at the
level of Anderson is assumed. However, Chapter 1 summarizes some
results of matrix algebra.
Elliptically Contoured Models in Statistics and Portfolio Theory
fully revises the first detailed introduction to the theory of
matrix variate elliptically contoured distributions. There are two
additional chapters, and all the original chapters of this classic
text have been updated. Resources in this book will be valuable for
researchers, practitioners, and graduate students in statistics and
related fields of finance and engineering. Those interested in
multivariate statistical analysis and its application to portfolio
theory will find this text immediately useful. In multivariate
statistical analysis, elliptical distributions have recently
provided an alternative to the normal model. Elliptical
distributions have also increased their popularity in finance
because of the ability to model heavy tails usually observed in
real data. Most of the work, however, is spread out in journals
throughout the world and is not easily accessible to the
investigators. A noteworthy function of this book is the collection
of the most important results on the theory of matrix variate
elliptically contoured distributions that were previously only
available in the journal-based literature. The content is organized
in a unified manner that can serve an a valuable introduction to
the subject.
In multivariate statistical analysis, elliptical distributions have
recently provided an alternative to the normal model. Most of the
work, however, is spread out in journals throughout the world and
is not easily accessible to the investigators. Fang, Kotz, and Ng
presented a systematic study of multivariate elliptical
distributions, however, they did not discuss the matrix variate
case. Recently Fang and Zhang have summarized the results of
generalized multivariate analysis which include vector as well as
the matrix variate distributions. On the other hand, Fang and
Anderson collected research papers on matrix variate elliptical
distributions, many of them published for the first time in
English. They published very rich material on the topic, but the
results are given in paper form which does not provide a unified
treatment of the theory. Therefore, it seemed appropriate to
collect the most important results on the theory of matrix variate
elliptically contoured distributions available in the literature
and organize them in a unified manner that can serve as an
introduction to the subject. The book will be useful for
researchers, teachers, and graduate students in statistics and
related fields whose interests involve multivariate statistical
analysis. Parts of this book were presented by Arjun K Gupta as a
one semester course at Bowling Green State University. Some new
results have also been included which generalize the results in
Fang and Zhang. Knowledge of matrix algebra and statistics at the
level of Anderson is assumed. However, Chapter 1 summarizes some
results of matrix algebra.
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 .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
seit Jahrzehnten betreut die Technische Versuchs- und Forschungs-
anstalt (TVFA) der TU-Wien stahl- und maschinenbauliche Aspekte
oesterreichischer Wasserkraftanlagen. Zwischen der TVFA und dem
Institut fur Wasserkraftmaschinen und Pumpen besteht eine lang-
jahrige, fruchtbare Zusammenarbeit auf diesem Gebiet. Das Insti-
tut fur Werkstoffkunde und Materialprufung (vormals Institut fur
Mechanische Technologie I und Baustofflehre) betreut wiederum
Grundlagen und Verfahren des Arbeitsgebiets. Die im Schuljahr
1900/1901 unter L. Tetmajer gegrundete Techni- sche Versuchsanstalt
hat in ihrer wechselvollen Geschichte her- vorragende Fachleute als
Vorstande und als Mitarbeiter gehabt. A. Leon und A. Slattenschek
begannen mit Arbeiten auf dem Wasser- kraftanlagensektor; E. Uhlir
hat dann in jahrzehntelanger uner- mudlicher Arbeit die
Dienstleistungen der TVFA fur die Wasser- kraftanlagen oesterreichs
ausgebaut und zum anerkannt hohen Stand dieser Anlagen wesentlich
beigetragen. Diese Herren hatten in den letzten Jahren u.a. sehr
eng mit F. Schulz seitens der Wasser- kraftmaschinen beim Bau und
der Projektierung von Wasserkraft- anlagen zusammengearbeitet. Es
ist den veranstaltenden Instituten eine Ehre und deren Mit-
arbeitern eine besondere Freude, neben einem Ruckblick auf die
achtzigjahrige Geschichte der TVFA und auf das Wirken von E.Uhlir
neue Arbeiten des Fachgebietes vorstellen-Zu koennen.
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