Mathematics and Statistics for Financial Risk Management E-Book

Michael B. Miller

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Serie: Wiley Finance Editions
• Sprache: Englisch

Seitenzahl: 419

Veröffentlichungsjahr: 2013

Beschreibung

Mathematics and Statistics for Financial Risk Management is a practical guide to modern financial risk management for both practitioners and academics. Now in its second edition with more topics, more sample problems and more real world examples, this popular guide to financial risk management introduces readers to practical quantitative techniques for analyzing and managing financial risk. In a concise and easy-to-read style, each chapter introduces a different topic in mathematics or statistics. As different techniques are introduced, sample problems and application sections demonstrate how these techniques can be applied to actual risk management problems. Exercises at the end of each chapter and the accompanying solutions at the end of the book allow readers to practice the techniques they are learning and monitor their progress. A companion Web site includes interactive Excel spreadsheet examples and templates. Mathematics and Statistics for Financial Risk Management is an indispensable reference for today's financial risk professional.

Leseprobe

Contents

Cover

Series

Title Page

Copyright

Preface

What's New in the Second Edition

Acknowledgments

Chapter 1: Some Basic Math

LOGARITHMS

LOG RETURNS

COMPOUNDING

LIMITED LIABILITY

GRAPHING LOG RETURNS

CONTINUOUSLY COMPOUNDED RETURNS

COMBINATORICS

DISCOUNT FACTORS

GEOMETRIC SERIES

PROBLEMS

Chapter 2: Probabilities

DISCRETE RANDOM VARIABLES

CONTINUOUS RANDOM VARIABLES

MUTUALLY EXCLUSIVE EVENTS

INDEPENDENT EVENTS

PROBABILITY MATRICES

CONDITIONAL PROBABILITY

PROBLEMS

Chapter 3: Basic Statistics

AVERAGES

EXPECTATIONS

VARIANCE AND STANDARD DEVIATION

STANDARDIZED VARIABLES

COVARIANCE

CORRELATION

APPLICATION: PORTFOLIO VARIANCE AND HEDGING

MOMENTS

SKEWNESS

KURTOSIS

COSKEWNESS AND COKURTOSIS

BEST LINEAR UNBIASED ESTIMATOR (BLUE)

PROBLEMS

Chapter 4: Distributions

PARAMETRIC DISTRIBUTIONS

UNIFORM DISTRIBUTION

BERNOULLI DISTRIBUTION

BINOMIAL DISTRIBUTION

POISSON DISTRIBUTION

NORMAL DISTRIBUTION

LOGNORMAL DISTRIBUTION

CENTRAL LIMIT THEOREM

APPLICATION: MONTE CARLO SIMULATIONS PART I: CREATING NORMAL RANDOM VARIABLES

CHI-SQUARED DISTRIBUTION

STUDENT'S t DISTRIBUTION

F-DISTRIBUTION

TRIANGULAR DISTRIBUTION

BETA DISTRIBUTION

MIXTURE DISTRIBUTIONS

PROBLEMS

Chapter 5: Multivariate Distributions and Copulas

MULTIVARIATE DISTRIBUTIONS

COPULAS

PROBLEMS

Chapter 6: Bayesian Analysis

OVERVIEW

BAYES' THEOREM

BAYES VERSUS FREQUENTISTS

MANY-STATE PROBLEMS

CONTINUOUS DISTRIBUTIONS

BAYESIAN NETWORKS

BAYESIAN NETWORKS VERSUS CORRELATION MATRICES

PROBLEMS

Chapter 7: Hypothesis Testing and Confidence Intervals

SAMPLE MEAN REVISITED

SAMPLE VARIANCE REVISITED

CONFIDENCE INTERVALS

HYPOTHESIS TESTING

CHEBYSHEV'S INEQUALITY

APPLICATION: VaR

PROBLEMS

Chapter 8: Matrix Algebra

MATRIX NOTATION

MATRIX OPERATIONS

APPLICATION: TRANSITION MATRICES

APPLICATION: MONTE CARLO SIMULATIONS PART II: CHOLESKY DECOMPOSITION

PROBLEMS

Chapter 9: Vector Spaces

VECTORS REVISITED

ORTHOGONALITY

ROTATION

PRINCIPAL COMPONENT ANALYSIS

APPLICATION: THE DYNAMIC TERM STRUCTURE OF INTEREST RATES

APPLICATION: THE STRUCTURE OF GLOBAL EQUITY MARKETS

PROBLEMS

Chapter 10: Linear Regression Analysis

LINEAR REGRESSION (ONE REGRESSOR)

LINEAR REGRESSION (MULTIVARIATE)

APPLICATION: FACTOR ANALYSIS

APPLICATION: STRESS TESTING

PROBLEMS

Chapter 11: Time Series Models

RANDOM WALKS

DRIFT-DIFFUSION MODEL

AUTOREGRESSION

VARIANCE AND AUTOCORRELATION

STATIONARITY

MOVING AVERAGE

CONTINUOUS MODELS

APPLICATION: GARCH

APPLICATION: JUMP-DIFFUSION MODEL

APPLICATION: INTEREST RATE MODELS

PROBLEMS

Chapter 12: Decay Factors

MEAN

VARIANCE

WEIGHTED LEAST SQUARES

OTHER POSSIBILITIES

APPLICATION: HYBRID VaR

PROBLEMS

Appendix A: Binary Numbers

Appendix B: Taylor Expansions

Appendix C: Vector Spaces

Appendix D: Greek Alphabet

Appendix E: Common Abbreviations

Appendix F: Copulas

Answers

References

About the Author

About the Companion Website

Index

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Cover Design: Wiley Cover Image, top: © Epoxy / Jupiter Images Cover Image, bottom: © iStockphoto.com / Georgijevic

Copyright © 2014 by Michael B. Miller. All rights reserved.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada.

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Library of Congress Cataloging-in-Publication Data:

Miller, Michael B. (Michael Bernard), 1973– Mathematics and statistics for financial risk management / Michael B. Miller. — 2nd Edition. pages cm. — (Wiley finance) Includes bibliographical references and index. ISBN 978-1-118-75029-2 (hardback); ISBN 978-1-118-757555-0 (ebk); ISBN 978-1-118-75764-2 (ebk) 1. Risk management—Mathematical models. 2. Risk management—Statistical methods. I. Title. HD61.M537 2013 332.01′5195—dc23

2013027322

Preface

The recent financial crisis and its impact on the broader economy underscores the importance of financial risk management in today's world. At the same time, financial products and investment strategies are becoming increasingly complex. It is more important than ever that risk managers possess a sound understanding of mathematics and statistics.

Mathematics and Statistics for Financial Risk Management is a guide to modern financial risk management for both practitioners and academics. Risk management has made great strides in recent years. Many of the mathematical and statistical tools used in risk management today were originally adapted from other fields. As the field has matured, risk managers have refined these tools and developed their own vocabulary for characterizing risk. As the field continues to mature, these tools and vocabulary are becoming increasingly standardized. By focusing on the application of mathematics and statistics to actual risk management problems, this book helps bridge the gap between mathematics and statistics in theory and risk management in practice.

Each chapter in this book introduces a different topic in mathematics or statistics. As different techniques are introduced, sample problems and application sections demonstrate how these techniques can be applied to actual risk management problems. Exercises at the end of each chapter, and the accompanying solutions at the end of the book, allow readers to practice the techniques learned and to monitor their progress.

This book assumes that readers have a solid grasp of algebra and at least a basic understanding of calculus. Even though most chapters start out at a very basic level, the pace is necessarily fast. For those who are already familiar with the topic, the beginning of each chapter serves as a quick review and as an introduction to selected vocabulary terms and conventions. Readers who are new to these topics may find they need to spend more time in the initial sections.

Risk management in practice often requires building models using spreadsheets or other financial software. Many of the topics in this book are accompanied by an icon, as shown here.

These icons indicate that Excel examples can be found at John Wiley & Sons' companion website for Mathematics and Statistics for Financial Risk Management, Second edition at www.wiley.com/go/millerfinance2e.

You can also visit the author's website, www.risk256.com, for the latest financial risk management articles, code samples, and more. To provide feedback, contact the author at [email protected].

What's New in the Second Edition

The biggest change to the second edition is the addition of two new chapters. The first new chapter, Chapter 5: Multivariate Distributions, explores important ­concepts for measuring the risk of portfolios, including joint distributions and ­copulas. The other new chapter, Chapter 6: Bayesian Analysis, expands on what was a short ­section in the first edition. The breadth and depth of this new chapter more accurately reflect the importance of Bayesian statistics in risk management today. Finally, the second edition includes many new problems, corrections, and small improvements to topics covered in the first edition. These included expanded ­sections on value at risk model validation, and generalized auto-regressive ­conditional ­heteroscedasticity (GARCH).

Acknowledgments

This book would not have been possible without the help of many individuals. I would like to thank Jeffrey Garnett, Steve Lerit, Riyad Maznavi, Hyunsuk Moon, Elliot Noma, Eldar Radovici, and Barry Schachter for taking the time to read early drafts. The book is certainly better for their comments and feedback.

I would also like to thank everybody at John Wiley & Sons for their help in bringing this book together.

Finally, and most importantly, I would like to thank my wife, Amy, who not only read over early drafts and talked me through a number of decisions, but also put up with countless nights and weekends of typing and editing. For this and much, much more, thank you.