Bernhard Pfaff. Financial Risk Modelling and Portfolio Optimization with R. Statistics in Practice

Transcription

1 Bernhard Pfaff Financial Risk Modelling and Portfolio Optimization with R Statistics in Practice

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3 Financial Risk Modelling and Portfolio Optimization with R

4 Statistics in Practice Series Advisory Editors Marian Scott University of Glasgow, UK Stephen Senn CRP-Santé, Luxembourg Wolfgang Jank University of Maryland, USA Founding Editor Vic Barnett Nottingham Trent University, UK Statistics in Practice is an important international series of texts which provide detailed coverage of statistical concepts, methods and worked case studies in specific fields of investigation and study. With sound motivation and many worked practical examples, the books show in down-to-earth terms how to select and use an appropriate range of statistical techniques in a particular practical field within each title s special topic area. The books provide statistical support for professionals and research workers across a range of employment fields and research environments. Subject areas covered include medicine and pharmaceutics; industry, finance and commerce; public services; the earth and environmental sciences, and so on. The books also provide support to students studying statistical courses applied to the above areas. The demand for graduates to be equipped for the work environment has led to such courses becoming increasingly prevalent at universities and colleges. It is our aim to present judiciously chosen and well-written workbooks to meet everyday practical needs. Feedback of views from readers will be most valuable to monitor the success of this aim. A complete list of titles in this series appears at the end of the volume.

5 Financial Risk Modelling and Portfolio Optimization with R Bernhard Pfaff Invesco Global Strategies, Germany A John Wiley & Sons, Ltd., Publication

6 This edition first published 2013 C 2013 John Wiley & Sons, Ltd Registered Office John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 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, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought. Library of Congress Cataloging-in-Publication Data Pfaff, Bernhard. Financial risk modelling and portfolio optimization with R / Bernhard Pfaff. p. cm. Includes bibliographical references and index. ISBN (cloth) 1. Financial risk Mathematical models. 2. Portfolio management. 3. R (Computer program language) I. Title. HG106.P dc A catalogue record for this book is available from the British Library. ISBN: Set in 10/12 pt Times Roman by Aptara Inc., New Delhi, India

7 Contents Preface List of abbreviations xi xiii Part I MOTIVATION 1 1 Introduction 3 Reference 5 2 A brief course in R Origin and development Getting help Working with R Classes, methods and functions The accompanying package FRAPO 20 References 25 3 Financial market data Stylized facts on financial market returns Stylized facts for univariate series Stylized facts for multivariate series Implications for risk models 32 References 33 4 Measuring risks Introduction Synopsis of risk measures Portfolio risk concepts 39 References 41 5 Modern portfolio theory Introduction 43

8 vi CONTENTS 5.2 Markowitz portfolios Empirical mean variance portfolios 47 References 49 Part II RISK MODELLING 51 6 Suitable distributions for returns Preliminaries The generalized hyperbolic distribution The generalized lambda distribution Synopsis of R packages for the GHD The package fbasics The package GeneralizedHyperbolic The package ghyp The package QRM The package SkewHyperbolic The package VarianceGamma Synopsis of R packages for GLD The package Davies The package fbasics The package gld The package lmomco Applications of the GHD to risk modelling Fitting stock returns to the GHD Risk assessment with the GHD Stylized facts revisited Applications of the GLD to risk modelling and data analysis VaR for a single stock Shape triangle for FTSE 100 constituents 79 References 82 7 Extreme value theory Preliminaries Extreme value methods and models The block maxima approach rth largest order models The peaks-over-threshold approach Synopsis of R packages The package evd The package evdbayes The package evir 91

9 CONTENTS The package fextremes The packages ismev and extremes The package POT The package QRM The package Renext Empirical applications of EVT Section outline Block maxima model for Siemens r block maxima model for BMW POT method for Boeing 105 References 110 vii 8 Modelling volatility Preliminaries The class of ARCH models Synopsis of R packages The package bayesgarch The package ccgarch The package fgarch The package gogarch The packages rugarch and rmgarch The package tseries Empirical application of volatility models 123 References Modelling dependence Overview Correlation, dependence and distributions Copulae Motivation Correlations and dependence revisited Classification of copulae Synopsis of R packages The package BLCOP The packages copula and nacopula The package fcopulae The package gumbel The package QRM Empirical applications of copulae GARCH copula model Mixed copula approaches 149 References 151

10 viii CONTENTS Part III PORTFOLIO OPTIMIZATION APPROACHES Robust portfolio optimization Overview Robust statistics Motivation Selected robust estimators Robust optimization Motivation Uncertainty sets and problem formulation Synopsis of R packages The package covrobust The package fportfolio The package MASS The package robustbase The package robust The package rrcov The package Rsocp Empirical applications Portfolio simulation: Robust versus classical statistics Portfolio back-test: Robust versus classical statistics Portfolio back-test: Robust optimization 182 References Diversification reconsidered Introduction Most diversified portfolio Risk contribution constrained portfolios Optimal tail-dependent portfolios Synopsis of R packages The packages DEoptim and RcppDE The package FRAPO The package PortfolioAnalytics Empirical applications Comparison of approaches Optimal tail-dependent portfolio against benchmark Limiting contributions to expected shortfall 211 References Risk-optimal portfolios Overview Mean VaR portfolios Optimal CVaR portfolios Optimal draw-down portfolios 227

11 CONTENTS 12.5 Synopsis of R packages The package fportfolio The package FRAPO Packages for linear programming The package PerformanceAnalytics Empirical applications Minimum-CVaR versus minimum-variance portfolios Draw-down constrained portfolios Back-test comparison for stock portfolio 247 References Tactical asset allocation Overview Survey of selected time series models Univariate time series models Multivariate time series models Black Litterman approach Copula opinion and entropy pooling Introduction The COP model The EP model Synopsis of R packages The package BLCOP The package dse The package farma The package forecast The package MSBVAR The package PairTrading The packages urca and vars Empirical applications Black Litterman portfolio optimization Copula opinion pooling Protection strategies 299 References 310 Appendix A Package overview 314 A.1 Packages in alphabetical order 314 A.2 Packages ordered by topic 317 References 320 Appendix B Time series data 324 B.1 Date-time classes 324 B.2 The ts class in the base package stats 327 B.3 Irregular-spaced time series 328 ix

12 x CONTENTS B.4 The package timeseries 330 B.5 The package zoo 332 B.6 The packages tframe and xts 334 References 337 Appendix C Back-testing and reporting of portfolio strategies 338 C.1 R packages for back-testing 338 C.2 R facilities for reporting 339 C.3 Interfacing databases 339 References 340 Appendix D Technicalities 342 Index 343

13 Preface The project for this book began in mid At that time, financial markets were in distress and far from operating smoothly. The impact of the US real estate crisis could still be felt and the sovereign debt crisis in some European countries was beginning to emerge. Major central banks implemented measures to avoid a collapse of the inter-bank market by providing liquidity. Given the massive financial book and real losses sustained by investors, it was also a time when quantitatively managed funds were in jeopardy and investors questioned the suitability of quantitative methods for protecting their wealth from the severe losses they had made in the past. Two years later not much has changed, though the debate on whether quantitative techniques per se are limited has ceased. Hence, the modelling of financial risks and the adequate allocation of wealth is still as important as it always has been, and these topics have gained in importance driven by experiences since the financial crisis started in the latter part of the previous decade. The content of the book is aimed at these two topics by acquainting and familiarizing the reader with market risk models and portfolio optimization techniques that have been proposed in the literature. These more recently proposed methods are elucidated by code examples written in the R language, a freely available software environment for statistical computing. This book certainly could not have been written without the public provision of such a superb piece of software as R and the numerous package authors who have greatly enriched this software environment. I therefore wish to express my sincere appreciation and thanks to the R Core Team members and all the contributors and maintainers of the packages cited and utilized in this book. By the same token, I would like to apologize to those authors whose packages I have not mentioned. This can only be ascribed to my ignorance of their existence. Second, I would like to thank John Wiley & Sons, Ltd. for the opportunity to write on this topic, in particular Ilaria Meliconi who initiated this book project in the first place and Heather Kay and Richard Davies for their careful editorial work. A special thank belongs to Richard Leigh for his meticulous and mindful copy-editing. Needless to say, any errors and omissions are entirely my responsibility. Finally, I owe a debt of profound gratitude

14 xii PREFACE to my beloved wife, Antonia, who while bearing the burden of many hours of solitude during the writing of this book remained a constant source of support. This book includes an accompanying website. Please visit go/financial_risk Bernhard Pfaff Kronberg im Taunus

15 List of abbreviations 2OLS 3OLS ACF ADF AMPL ANSI AP APARCH API ARCH AvDD BL BP CDaR CLI CLT COM COP CPPI CRAN CVaR DBMS DD DE DR EDA EGARCH EM EMA EP ERS ES EVT FIML Two-stage ordinary least-squares Three-stage ordinary least-squares Autocorrelation function Augmented Dickey Fuller A modelling language for mathematical programming American National Standards Institute Active premium Asymmetric power ARCH Application programming interface Autoregressive conditional heteroscedasticity Average draw-down Black Litterman Break point Conditional draw-down at risk Command line interface Central limit theorem Component object model Copula opinion pooling Constant proportion portfolio insurance Comprehensive R Archive Network Conditional value at risk Data Base Management System Draw-down Differential evolution Diversification ratio Exploratory data analysis Exponential GARCH Expectation maximization Exponentially weighted mean Entropy pooling Elliott Rothenberg Stock Expected shortfall Extreme value theory Full-information maximum likelihood