Financial Risk Forecasting E-Book

Jon Danielsson

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

Seitenzahl: 413

Veröffentlichungsjahr: 2011

Beschreibung

Financial Risk Forecasting is a complete introduction to practical quantitative risk management, with a focus on market risk. Derived from the authors teaching notes and years spent training practitioners in risk management techniques, it brings together the three key disciplines of finance, statistics and modeling (programming), to provide a thorough grounding in risk management techniques. Written by renowned risk expert Jon Danielsson, the book begins with an introduction to financial markets and market prices, volatility clusters, fat tails and nonlinear dependence. It then goes on to present volatility forecasting with both univatiate and multivatiate methods, discussing the various methods used by industry, with a special focus on the GARCH family of models. The evaluation of the quality of forecasts is discussed in detail. Next, the main concepts in risk and models to forecast risk are discussed, especially volatility, value-at-risk and expected shortfall. The focus is both on risk in basic assets such as stocks and foreign exchange, but also calculations of risk in bonds and options, with analytical methods such as delta-normal VaR and duration-normal VaR and Monte Carlo simulation. The book then moves on to the evaluation of risk models with methods like backtesting, followed by a discussion on stress testing. The book concludes by focussing on the forecasting of risk in very large and uncommon events with extreme value theory and considering the underlying assumptions behind almost every risk model in practical use - that risk is exogenous - and what happens when those assumptions are violated. Every method presented brings together theoretical discussion and derivation of key equations and a discussion of issues in practical implementation. Each method is implemented in both MATLAB and R, two of the most commonly used mathematical programming languages for risk forecasting with which the reader can implement the models illustrated in the book. The book includes four appendices. The first introduces basic concepts in statistics and financial time series referred to throughout the book. The second and third introduce R and MATLAB, providing a discussion of the basic implementation of the software packages. And the final looks at the concept of maximum likelihood, especially issues in implementation and testing. The book is accompanied by a website - href="http://www.financialriskforecasting.com/">www.financialriskforecasting.com - which features downloadable code as used in the book.

Leseprobe

Financial Risk Forecasting

For other titles in the Wiley Finance Series please see www.wiley.com/finance

Financial Risk Forecasting

The Theory and Practice of Forecasting Market Risk, with Implementation in R and Matlab

Jón Daníelsson

This edition first published 2011

Copyright © 2011 Jón Daníelsson

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ISBN 978-0-470-66943-3 (hardback)

ISBN 978-1-119-97710-0 (ebook)

ISBN 978-1-119-97711-7 (ebook)

ISBN 978-1-119-97712-4 (ebook)

A catalogue record for this book is available from the British Library.

Project management by OPS Ltd, Gt Yarmouth, Norfolk

Preface

The focus in this book is on the study of market risk from a quantitative point of view. The emphasis is on presenting commonly used state-of-the-art quantitative techniques used in finance for the management of market risk and demonstrate their use employing the principal two mathematical programming languages, R and Matlab. All the code in the book can be downloaded from the book's website at www.financialriskforecasting.com

The book brings together three essential fields: finance, statistics and computer programming. It is assumed that the reader has a basic understanding of statistics and finance; however, no prior knowledge of computer programming is required. The book takes a hands-on approach to the issue of financial risk, with the reading material intermixed between finance, statistics and computer programs.

I have used the material in this book for some years, both for a final year undergraduate course in quantitative methods and for master level courses in risk forecasting. In most cases, the students taking this course have no prior knowledge of computer programming, but emerge after the course with the ability to independently implement the models and code in this book. All of the material in the book can be covered in about 10 weeks, or 20 lecture hours.

Most chapters demonstrate the way in which the various techniques discussed are implemented by both R and Matlab. We start by downloading a sample of stock prices, which are then used for model estimation and evaluation.

The outline of the book is as follows. Chapter 1 begins with an introduction to financial markets and market prices. The chapter gives a foretaste of what is to come, discussing market indices and stock prices, the forecasting of risk and prices, and concludes with the main features of market prices from the point of view of risk. The main focus of the chapter is introduction of the three stylized facts regarding returns on financial assets: volatility clusters, fat tails and nonlinear dependence.

Chapters 2 and 3 focus on volatility forecasting: the former on univariate volatility and the latter on multivariate volatility. The aim is to survey all the methods used for volatility forecasting, while discussing several models from the GARCH family in considerable detail. We discuss the models from a theoretical point of view and demonstrate their implementation and evaluation.

This is followed by two chapters on risk models and risk forecasting: Chapter 4 addresses the theoretical aspects of risk forecasting—in particular, volatility, value-at-risk (VaR) and expected shortfall; Chapter 5 addresses the implementation of risk models.

We then turn to risk analysis in options and bonds; Chapter 6 demonstrates such analytical methods as delta-normal VaR and duration-normal VaR, while Chapter 7 addresses Monte Carlo simulation methods for derivative pricing and risk forecasting.

After developing risk models their quality needs to be evaluated—this is the topic of Chapter 8. This chapter demonstrates how backtesting and a number of methodologies can be used to evaluate and compare the risk forecast methods presented earlier in the book. The chapter concludes with a comprehensive discussion of stress testing.

The risk forecast methods discussed up to this point in the book are focused on relatively common events, but in special cases it is necessary to forecast the risk of very large, yet uncommon events (e.g., the probability of events that happen, say, every 10 years or every 100 years). To do this, we need to employee extreme value theory—the topic of Chapter 9.

In Chapter 10, the last chapter in the book, we take a step back and consider the underlying assumptions behind almost every risk model in practical use and discuss what happens when these assumptions are violated. Because financial risk is fundamentally endogenous, financial risk models have the annoying habit of failing when needed the most. How and why this happens is the topic of this chapter.

There are four appendices: Appendix A introduces the basic concepts in statistics and the financial time series referred to throughout the book. We give an introduction to R and Matlab in Appendices B and C, respectively, providing a discussion of the basic implementation of the software packages. Finally, Appendix D is focused on maximum likelihood, concept, implementation and testing. A list of the most commonly used abbreviations in the book can be found on p. xvii. This is followed by a table of the notation used in the book on p. xix.

Jón Daníelsson

Contents

1 Financial markets, prices and risk

1.1 Prices, returns and stock indices

1.1.1 Stock indices

1.1.2 Prices and returns

1.2 S&P 500 returns

1.2.1 S&P 500 statistics

1.2.2 S&P 500 statistics in R and Matlab

1.3 The stylized facts of financial returns

1.4 Volatility

1.4.1 Volatility clusters

1.4.2 Volatility clusters and the ACF

1.5 Nonnormality and fat tails

1.6 Identification of fat tails

1.6.1 Statistical tests for fat tails

1.6.2 Graphical methods for fat tail analysis

1.6.3 Implications of fat tails in finance

1.7 Nonlinear dependence

1.7.1 Sample evidence of nonlinear dependence

1.7.2 Exceedance correlations

1.8 Copulas

1.8.1 The Gaussian copula

1.8.2 The theory of copulas

1.8.3 An application of copulas

1.8.4 Some challenges in using copulas

1.9 Summary

2 Univariate volatility modeling

2.1 Modeling Volatility

2.2 Simple volatility models

2.2.1 Moving average models

2.2.2 EWMA model

2.3 GARCH and conditional volatility

2.3.1 ARCH

2.3.2 GARCH

2.3.3 The “memory” of a GARCH model

2.3.4 Normal GARCH

2.3.5 Student-t GARCH

2.3.6 (G)ARCH in mean

2.4 Maximum likelihood estimation of volatility models

2.4.1 The ARCH(1) likelihood function

2.4.2 The GARCH(1,1) likelihood function

2.4.3 On the importance of σ1

2.4.4 Issues in estimation

2.5 Diagnosing volatility models

2.5.1 Likelihood ratio tests and parameter significance

2.5.2 Analysis of model residuals

2.5.3 Statistical goodness-of-fit measures

2.6 Application of ARCH and GARCH

2.6.1 Estimation results

2.6.2 Likelihood ratio tests

2.6.3 Residual analysis

2.6.4 Graphical analysis

2.6.5 Implementation

2.7 Other GARCH-type models

2.7.1 Leverage effects and asymmetry

2.7.2 Power models

2.7.3 APARCH

2.7.4 Application of APARCH models

2.7.5 Estimation of APARCH

2.8 Alternative volatility models

2.8.1 Implied volatility

2.8.2 Realized volatility

2.8.3 Stochastic volatility

2.9 Summary

3 Multivariate volatility models

3.1 Multivariate volatility forecasting

3.1.1 Application

3.2 EWMA

3.3 Orthogonal GARCH

3.3.1 Orthogonalizing covariance

3.3.2 Implementation

3.3.3 Large-scale implementations

3.4 CCC and DCC models

3.4.1 Constant conditional correlations (CCC)

3.4.2 Dynamic conditional correlations (DCC)

3.4.3 Implementation

3.5 Estimation comparison

3.6 Multivariate extensions of GARCH

3.6.1 Numerical problems

3.6.2 The BEKK model

3.7 Summary

4 Risk measures

4.1 Defining and measuring risk

4.2 Volatility

4.3 Value-at-risk

4.3.1 Is VaR a negative or positive number?

4.3.2 The three steps in VaR calculations

4.3.3 Interpreting and analyzing VaR

4.3.4 VaR and normality

4.3.5 Sign of VaR

4.4 Issues in applying VaR

4.4.1 VaR is only a quantile

4.4.2 Coherence

4.4.3 Does VaR really violate subadditivity?

4.4.4 Manipulating VaR

4.5 Expected shortfall

4.6 Holding periods, scaling and the square root of time

4.6.1 Length of holding periods

4.6.2 Square-root-of-time scaling

4.7 Summary

5 Implementing risk forecasts

5.1 Application

5.2 Historical simulation

5.2.1 Expected shortfall estimation

5.2.2 Importance of window size

5.3 Risk measures and parametric methods

5.3.1 Deriving VaR

5.3.2 VaR when returns are normally distributed

5.3.3 VaR under the Student-t distribution

5.3.4 Expected shortfall under normality

5.4 What about expected returns?

5.5 VaR with time-dependent volatility

5.5.1 Moving average

5.5.2 EWMA

5.5.3 GARCH normal

5.5.4 Other GARCH models

5.6 Summary

6 Analytical value-at-risk for options and bonds

6.1 Bonds

6.1.1 Duration-normal VaR

6.1.2 Accuracy of duration-normal VaR

6.1.3 Convexity and VaR

6.2 Options

6.2.1 Implementation

6.2.2 Delta-normal VaR

6.2.3 Delta and gamma

6.3 Summary

7 Simulation methods for VaR for options and bonds

7.1 Pseudo random number generators

7.1.1 Linear congruental generators

7.1.2 Nonuniform RNGs and transformation methods

7.2 Simulation pricing

7.2.1 Bonds

7.2.2 Options

7.3 Simulation of VaR for one asset

7.3.1 Monte Carlo VaR with one basic asset

7.3.2 VaR of an option on a basic asset

7.3.3 Options and a stock

7.4 Simulation of portfolio VaR

7.4.1 Simulation of portfolio VaR for basic assets

7.4.2 Portfolio VaR for options

7.4.3 Richer versions

7.5 Issues in simulation estimation

7.5.1 The quality of the RNG

7.5.2 Number of simulations

7.6 Summary

8 Backtesting and stress testing

8.1 Backtesting

8.1.1 Market risk regulations

8.1.2 Estimation window length

8.1.3 Testing window length

8.1.4 Violation ratios

8.2 Backtesting the S&P 500

8.2.1 Analysis

8.3 Significance of backtests

8.3.1 Bernoulli coverage test

8.3.2 Testing the independence of violations

8.3.3 Testing VaR for the S&P 500

8.3.4 Joint test

8.3.5 Loss-function-based backtests

8.4 Expected shortfall backtesting

8.5 Problems with backtesting

8.6 Stress testing

8.6.1 Scenario analysis

8.6.2 Issues in scenario analysis

8.6.3 Scenario analysis and risk models

8.7 Summary

9 Extreme value theory

9.1 Extreme value theory

9.1.1 Types of tails

9.1.2 Generalized extreme value distribution

9.2 Asset returns and fat tails

9.3 Applying EVT

9.3.1 Generalized Pareto distribution

9.3.2 Hill method

9.3.3 Finding the threshold

9.3.4 Application to the S&P 500 index

9.4 Aggregation and convolution

9.5 Time dependence

9.5.1 Extremal index

9.5.2 Dependence in ARCH

9.5.3 When does dependence matter?

9.6 Summary

10 Endogenous risk

10.1 The Millennium Bridge

10.2 Implications for financial risk management

10.2.1 The 2007–2010 crisis

10.3 Endogenous market prices

10.4 Dual role of prices

10.4.1 Dynamic trading strategies

10.4.2 Delta hedging

10.4.3 Simulation of feedback

10.4.4 Endogenous risk and the 1987 crash

10.5 Summary

APPENDICES

A Financial time series

A.1 Random variables and probability density functions

A.1.1 Distributions and densities

A.1.2 Quantiles

A.1.3 The normal distribution

A.1.4 Joint distributions

A.1.5 Multivariate normal distribution

A.1.6 Conditional distribution

A.1.7 Independence

A.2 Expectations and variance

A.2.1 Properties of expectation and variance

A.2.2 Covariance and independence

A.3 Higher order moments

A.3.1 Skewness and kurtosis

A.4 Examples of distributions

A.4.1 Chi-squared (χ2)

A.4.2 Student-t

A.4.3 Bernoulli and binomial distributions

A.5 Basic time series concepts

A.5.1 Autocovariances and autocorrelations

A.5.2 Stationarity

A.5.3 White noise

A.6 Simple time series models

A.6.1 The moving average model

A.6.2 The autoregressive model

A.6.3 ARMA model

A.6.4 Random walk

A.7 Statistical hypothesis testing

A.7.1 Central limit theorem

A.7.2 p-values

A.7.3 Type 1 and type 2 errors and the power of the test

A.7.4 Testing for normality

A.7.5 Graphical methods: QQ plots

A.7.6 Testing for autocorrelation

A.7.7 Engle LM test for volatility clusters

B An introduction to R

B.1 Inputting data

B.2 Simple operations

B.2.1 Matrix computation

B.3 Distributions

B.3.1 Normality tests

B.4 Time series

B.5 Writing Functions in R

B.5.1 Loops and repeats

B.6 Maximum likelihood estimation

B.7 Graphics

C An introduction to Matlab

C.1 Inputting data

C.2 Simple operations

C.2.1 Matrix algebra

C.3 Distributions

C.3.1 Normality tests

C.4 Time series

C.5 Basic programming and M-files

C.5.1 Loops

C.6 Maximum likelihood

C.7 Graphics

D Maximum likelihood

D.1 Likelihood functions

D.1.1 Normal likelihood functions

D.2 Optimizers

D.3 Issues in ML estimation

D.4 Information matrix

D.5 Properties of maximum likelihood estimators

D.6 Optimal testing procedures

D.6.1 Likelihood ratio test

D.6.2 Lagrange multiplier test

D.6.3 Wald test (W)

Bibliography

Index

Acknowledgments

This book is based on my years of teaching risk forecasting, both at undergraduate and master level, at the London School of Economics (LSE) and other universities, and in various executive education courses. I am very grateful to all the students and practitioners who took my courses for all the feedback I have received over the years.

I was fortunate to be able to employ an exemplary student, Jacqueline Li, to work with me on developing the lecture material. Jacqueline's assistance was invaluable; she made significant contributions to the book. Her ability to master all the statistical and computational aspects of the book was impressive, as was the apparent ease with which she mastered the technicalities. She survived the process and has emerged as a very good friend.

A brilliant mathematician and another very good friend, Maite Naranjo at the Centre de Recerca Matemàtica, Bellaterra in Barcelona, agreed to read the mathematics and saved me from several embarrassing mistakes.

Two colleagues at the LSE, Stéphane Guibaud and Jean-Pierre Zigrand, read parts of the book and verified some of the mathematical derivations.

My PhD student, Ilknur Zer, who used an earlier version of this book while a masters student at LSE and who currently teaches a course based on this book, kindly agreed to review the new version of the book and came up with very good suggestions on both content and presentation.

Kyle T. Moore and Pengfei Sun, both at Erasmus University, agreed to read the book, with a special focus on extreme value theory. They corrected many mistakes and made good suggestions on better presentation of the material.

I am very grateful to all of them for their assistance; without their contribution this book would not have seen the light of day.

Jón Daníelsson

Abbreviations

ACF   Autocorrelation functionAR   AutoregressiveARCH   Autoregressive conditional heteroskedasticityARMA   Autoregressive moving averageCCC   Constant conditional correlationsCDF   Cumulative distribution functionCLT   Central limit theoremDCC   Dynamic conditional correlationsDJIA   Dow Jones Industrial AverageES   Expected shortfallEVT   Extreme value theoryEWMA   Exponentially weighted moving averageGARCH   Generalized autoregressive conditional heteroskedasticityGEV   Generalized extreme valueGPD   Generalized Pareto distributionHS   Historical simulationIID   Identically and independently distributedJB test   Jarque–Bera testKS test   Kolmogorov–Smirnov testLB test   Ljung–Box testLCG   Linear congruental generatorLM   Lagrange multiplierLR   Likelihood ratioMA   Moving averageMC   Monte CarloML   Maximum likelihoodMLE   Maximum likelihood estimationMVGARCH   Multivariate GARCHNaN   Not a numberNLD   Nonlinear dependenceOGARCH   Orthogonal GARCHP/L   Profit and lossPC   Principal componentPCA   Principal components analysisPDF   Probability density functionPOT   Peaks over thresholdsQML   Quasi-maximum likelihoodQQ plot   Quantile–quantile plotRN   Random numberRNG   Random number generatorRV   Random variableSV   Stochastic volatilityVaR   Value-at-riskVR   Violation ratio