Simulative Portfolio Optimization under Distributions of Hyperbolic Type - Methods and Empirical Investigation

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1 Simulative Portfolio Optimization under Distributions of Hyperbolic Type - Methods and Empirical Investigation Inaugural-Dissertation zur Erlangung des akademischen Grades eines Doktors der Wirtschafts- und Sozialwissenschaften (Dr. rer. pol.) der Friedrich-Alexander-Universität Erlangen-Nürnberg vorgelegt von Diplom-Kaufmann Nils Bierkamp aus Nürnberg

2 Referent: Professor Dr. Ingo Klein Korreferent: Professor Jürgen Kähler, Ph.D. Promotionstermin: 25. Juli 2006

3 Berichte aus der Statistik Nils Bierkamp Simulative Portfolio Optimization under Distributions of Hyperbolic Type - Methods and Empirical Investigation Shaker Verlag Aachen 2006

4 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at Zugl.: Erlangen-Nürnberg, Univ., Diss., 2006 Copyright Shaker Verlag 2006 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publishers. Printed in Germany. ISBN-10: ISBN-13: ISSN Shaker Verlag GmbH P.O. BOX D Aachen Phone: 0049/2407/ Telefax: 0049/2407/ Internet: info@shaker.de

5 To Dr. med. Willy Bierkamp, my father, my teacher, my coach, my supporter, my advisor, my idol, my hero, my friend, and by far the greatest man I will have ever known. I will miss you every day, but I will always try to make you proud.

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7 Acknowledgements It has been a long journey from the beginning of my thesis to where I stand now. There have been many people I met on this journey. Some gave me advice. Some helped me to overcome obstacles. Some gave me wings. Some provided shelter. Some accompanied me. It is impossible to mention them all here, although I am deeply indebted to all of them. There is nothing on this earth more to be prized than true friendship. There are some individuals that took key roles for the successful completion of this thesis: above all, I want to thank Prof. Dr. Ingo Klein for his valuable advice, his reliable support and for keeping me on track. Special appreciation is also offered to Prof. Dr. Jürgen Kähler for many useful comments on my manuscript and on my English. My special gratitude also belongs to PD Dr. Matthias Fischer for his great ideas, his seemingly everlasting patience in answering short questions and the inspiration he provided. Moreover, I very much thank Florian Ermer for all the fruitful discussions, his excellent eyes and the long hours he joined me. I would also like to show appreciation to Nils Büsking for removing many misuses of the English language. I owe my warmest thanks to Melanie Schäfer. The Eskimo has fifty-two names for snow because it is important to them; there ought to be as many for love. Finally, I want to express my sincere gratitude to my family. My Mum for her permanent non-academic support, my brother Jan for his good advice and my little brother Sven for making me laugh.

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9 Contents List of Figures IV List of Tables VII Frequently Used Notation IX 1 Introduction 1 2 Portfolio Optimization: A Review Classical Approach Generalization of the Assumptions Risk Measures Distributions Optimization via Simulation Distributions, Estimation and Sampling Distributions of Hyperbolic Type Univariate Generalized Hyperbolic Family Multivariate Generalized Hyperbolic (MGH) Family Multivariate Affine Generalized Hyperbolic (MAGH) Family.. 19 I

10 3.1.4 Copula-Based Generalized Hyperbolic (CBGH) Distributions Parameter Estimation Maximum Likelihood Estimation of MGH Parameters Estimation of MGH Parameters via EM-Algorithm Estimation of MAGH Parameters Estimation of CBGH Parameters Random Number Generation Sampling from a MGH- and MAGH-Distribution Sampling from a CBGH-Distribution Empirical Study Do Portfolio Strategies vary with the Distribution? Data Set Results Conclusion Providing the Setting Importance of Short Data Sets Implementation of Efficiency Conclusion Portfolio Optimization with a MGH-Distribution Experimental Setup Optimization Small Scale DAX II

11 HSI NKY Conclusion Summary 80 A Investigation: Optimal Sample Size 83 B Data Sets: Optimization 97 B.1 DAX B.2 HSI B.3 NKY Bibliography 113 III