The implementation of sound quantitative risk models is a vital concern for all financial institutions, and this trend has accelerated in recent years with regulatory processes such as Basel II. This book provides a comprehensive treatment of the theoretical concepts and modelling techniques of quantitative risk management and equips readers - whether financial risk analysts, actuaries, regulators, or students of quantitative finance - with practical tools to solve real-world problems. The authors cover methods for market, credit, and operational risk modelling; place standard industry approaches on a more formal footing; and describe recent developments that go beyond, and address main deficiencies of, current practice. The book's methodology draws on diverse quantitative disciplines, from mathematical finance through statistics and econometrics to actuarial mathematics. Main concepts discussed include loss distributions, risk measures, and risk aggregation and allocation principles. A main theme is the need to satisfactorily address extreme outcomes and the dependence of key risk drivers. The techniques required derive from multivariate statistical analysis, financial time series modelling, copulas, and extreme value theory. A more technical chapter addresses credit derivatives. Based on courses taught to masters students and professionals, this book is a unique and fundamental reference that is set to become a standard in the field.
"This book is a compendium of the statistical arrows that should be in any quantitative risk manager's quiver. It includes extensive discussion of dynamic volatility models, extreme value theory, copulas, and credit risk. Academics, Ph.D. students, and quantitative practitioners will find many new and useful results in this important volume."
- Robert F. Engle III, 2003 Nobel Laureate in Economic Sciences, Michael Armellino Professor in the Management of Financial Services at New York University's Stern School of Business
"This book provides a framework and a useful toolkit for analysis a wide variety of risk management problems. Common pitfalls are pointed out, and mathematical sophistication is used in pursuit of useful and usable solutions. Every financial institution has a risk management department that looks at aggregated portfolio-wide risks on longer time scales, and at risk exposure to large, or extreme, market movements. Risk managers are always on the lookout for good techniques to help them do their jobs. This very good book provides these techniques and addresses an important, and under-developed, area of practical research."
- Martin Baxter, Nomura International
"McNeil, Frey, and Embrechts present a wide-ranging yet remarkably clear and coherent introduction to the modelling of financial risk. Unlike most finance texts, where the focus is on pricing individual instruments, the primary focus in this book is the statistical behavior of portfolios of risky instruments, which is, after all, the primary concern of risk management. This ought to be a core text in every risk manager's training, and a useful reference for experienced professionals."
- Michael Gordy
"There is no book that provides the type of rigorous and detailed coverage of risk management topics that this book does. This could become the book on quantitative risk management."
- Riccardo Rebonato, Royal Bank of Scotland, author of Modern Pricing of Interest-Rate Derivatives
CHAPTER 1: Risk in Perspective
1.1 Risk
1.1.1 Risk and Randomness
1.1.2 Financial Risk
1.1.3 Measurement and Management
1.2 A Brief History of Risk Management
1.2.1 From Babylon to Wall Street
1.2.2 The Road to Regulation
1.3 The New Regulatory Framework
1.3.1 Basel II
1.3.2 Solvency 2
1.4 Why Manage Financial Risk?
1.4.1 A Societal View
1.4.2 The Shareholder's View
1.4.3 Economic Capital
1.5 Quantitative Risk Management
1.5.1 The Nature of the Challenge
1.5.2 QRM for the Future
CHAPTER 2: Basic Concepts in Risk Management
2.1 Risk Factors and Loss Distributions
2.1.1 General Definitions
2.1.2 Conditional and Unconditional Loss Distribution
2.1.3 Mapping of Risks:Some Examples
2.2 Risk Measurement
2.2.1 Approaches to Risk Measurement
2.2.2 Value-at-Risk
2.2.3 Further Comments on VaR
2.2.4 Other Risk Measures Based on Loss Distributions
2.3 Standard Methods for Market Risks
2.3.1 Variance -Covariance Method
2.3.2 Historical Simulation
2.3.3 Monte Carlo
2.3.4 Losses over Several Periods and Scaling
2.3.5 Backtesting
2.3.6 An Illustrative Example
CHAPTER 3: Multivariate Models
3.1 Basics of Multivariate Modelling
3.1.1 Random Vectors and Their Distributions
3.1.2 Standard Estimators of Covariance and Correlation
3.1.3 The Multivariate Normal Distribution
3.1.4 Testing Normality and Multivariate Normality
3.2 Normal Mixture Distributions
3.2.1 Normal Variance Mixtures
3.2.2 Normal Mean-Variance Mixtures
3.2.3 Generalized Hyperbolic Distributions
3.2.4 Fitting Generalized Hyperbolic Distributions to Data
3.2.5 Empirical Examples
3.3 Spherical and Elliptical Distributions
3.3.1 Spherical Distributions
3.3.2 Elliptical Distributions
3.3.3 Properties of Elliptical Distributions
3.3.4 Estimating Dispersion and Correlation
3.3.5 Testing for Elliptical Symmetry
3.4 Dimension Reduction Techniques
3.4.1 Factor Models
3.4.2 Statistical Calibration Strategies
3.4.3 Regression Analysis of Factor Models
3.4.4 Principal Component Analysis
CHAPTER 4: Financial Time Series
4.1 Empirical Analyses of Financial Time Series
4.1.1 Stylized Facts
4.1.2 Multivariate Stylized Facts
4.2 Fundamentals of Time Series Analysis
4.2.1 Basic Definitions
4.2.2 ARMA Processes
4.2.3 Analysis in the Time Domain
4.2.4 Statistical Analysis of Time Series
4.2.5 Prediction
4.3 GARCH Models for Changing Volatility
4.3.1 ARCH Processes
4.3.2 GARCH Processes
4.3.3 Simple Extensions of the GARCH Model
4.3.4 Fitting GARCH Models to Data
4.4 Volatility Models and Risk Estimation
4.4.1 Volatility Forecasting
4.4.2 Conditional Risk Measurement
4.4.3 Backtesting
4.5 Fundamentals of Multivariate Time Series
4.5.1 Basic Definitions
4.5.2 Analysis in the Time Domain
4.5.3 Multivariate ARMA Processes
4.6 Multivariate GARCH Processes
4.6.1 General Structure of Models
4.6.2 Models for Conditional Correlation
4.6.3 Models for Conditional Covariance
4.6.4 Fitting Multivariate GARCH Models
4.6.5 Dimension Reduction in MGARCH
4.6.6 MGARCH and Conditional Risk Measurement
CHAPTER 5: Copulas and Dependence
5.1 Copulas
5.1.1 Basic Properties
5.1.2 Examples of Copulas
5.1.3 Meta Distributions
5.1.4 Simulation of Copulas and Meta Distributions
5.1.5 Further Properties of Copulas
5.1.6 Perfect Dependence
5.2 Dependence Measures
5.2.1 Linear Correlation
5.2.2 Rank Correlation
5.2.3 Coefficients of Tail Dependence
5.3 Normal Mixture Copulas
5.3.1 Tail Dependence
5.3.2 Rank Correlations
5.3.3 Skewed Normal Mixture Copulas
5.3.4 Grouped Normal Mixture Copulas
5.4 Archimedean Copulas
5.4.1 Bivariate Archimedean Copulas
5.4.2 Multivariate Archimedean Copulas
5.4.3 Non-exchangeable Archimedean Copulas
5.5 Fitting Copulas to Data
5.5.1 Method-of-Moments using Rank Correlation
5.5.2 Forming a Pseudo-Sample from the Copula
5.5.3 Maximum Likelihood Estimation
CHAPTER 6: Aggregate Risk
6.1 Coherent Measures of Risk
6.1.1 The Axioms of Coherence
6.1.2 Value-at-Risk
6.1.3 Coherent Risk Measures Based on Loss Distributions
6.1.4 Coherent Risk Measures as Generalized Scenarios
6.1.5 Mean-VaR Portfolio Optimization
6.2 Bounds for Aggregate Risks
6.2.1 The General Fréchet Problem
6.2.2 The Case of VaR
6.3 Capital Allocation
6.3.1 The Allocation Problem
6.3.2 The Euler Principle and Examples
6.3.3 Economic Justification of the Euler Principle
CHAPTER 7: Extreme Value Theory
7.1 Maxima
7.1.1 Generalized Extreme Value Distribution
7.1.2 Maximum Domains of Attraction
7.1.3 Maxima of Strictly Stationary Time Series
7.1.4 The Block Maxima Method
7.2 Threshold Exceedances
7.2.1 Generalized Pareto Distribution
7.2.2 Modelling Excess Losses
7.2.3 Modelling Tails and Measures of Tail Risk
7.2.4 The Hill Method
7.2.5 Simulation Study of EVT Quantile Estimators
7.2.6 Conditional EVT for Financial Time Series
7.3 Tails of Specific Models
7.3.1 Domain of Attraction of Fréchet Distribution
7.3.2 Domain of Attraction of Gumbel Distribution
7.3.3 Mixture Models
7.4 Point Process Models
7.4.1 Threshold Exceedances for Strict White Noise
7.4.2 The POT Model
7.4.3 Self-Exciting Processes
7.4.4 A Self-Exciting POT Model
7.5 Multivariate Maxima
7.5.1 Multivariate Extreme Value Copulas
7.5.2 Copulas for Multivariate Minima
7.5.3 Copula Domains of Attraction
7.5.4 Modelling Multivariate Block Maxima
7.6 Multivariate Threshold Exceedances
7.6.1 Threshold Models Using EV Copulas
7.6.2 Fitting a Multivariate Tail Model
7.6.3 Threshold Copulas and Their Limits
CHAPTER 8: Credit Risk Management
8.1 Introduction to Credit Risk Modelling
8.1.1 Credit Risk Models
8.1.2 The Nature of the Challenge
8.2 Structural Models of Default
8.2.1 The Merton Model
8.2.2 Pricing in Merton's Model
8.2.3 The KMV Model
8.2.4 Models Based on Credit Migration
8.2.5 Multivariate Firm-Value Models
8.3 Threshold Models
8.3.1 Notation for One-Period Portfolio Models
8.3.2 Threshold Models and Copulas
8.3.3 Industry Examples
8.3.4 Models Based on Alternative Copulas
8.3.5 Model Risk Issues
8.4 The Mixture Model Approach
8.4.1 One-Factor Bernoulli Mixture Models
8.4.2 CreditRisk +
8.4.3 Asymptotics for Large Portfolios
8.4.4 Threshold Models as Mixture Models
8.4.5 Model-Theoretic Aspects of Basel II
8.4.6 Model Risk Issues
8.5 Monte Carlo Methods
8.5.1 Basics of Importance Sampling
8.5.2 Application to Bernoulli-Mixture Models
8.6 Statistical Inference for Mixture Models
8.6.1 Motivation
8.6.2 Exchangeable Bernoulli-Mixture Models
8.6.3 Mixture Models as GLMMs
8.6.4 One-Factor Model with Rating Effect
CHAPTER 9: Dynamic Credit Risk Models
9.1 Credit Derivatives
9.1.1 Overview
9.1.2 Single-Name Credit Derivatives
9.1.3 Portfolio Credit Derivatives
9.2 Mathematical Tools
9.2.1 Random Times and Hazard Rates
9.2.2 Modelling Additional Information
9.2.3 Doubly Stochastic Random Times
9.3 Financial and Actuarial Pricing of Credit Risk
9.3.1 Physical and Risk-Neutral Probability Measure
9.3.2 Risk-Neutral Pricing and Market Completeness
9.3.3 Martingale Modelling
9.3.4 The Actuarial Approach to Credit Risk Pricing
9.4 Pricing with Doubly Stochastic Default Times
9.4.1 Recovery Payments of Corporate Bonds
9.4.2 The Model
9.4.3 Pricing Formulas
9.4.4 Applications
9.5 Affine Models
9.5.1 Basic Results
9.5.2 The CIR Square-Root Diffusion 423
9.5.3 Extensions
9.6 Conditionally Independent Defaults
9.6.1 Reduced-Form Models for Portfolio Credit Risk
9.6.2 Conditionally Independent Default Times
9.6.3 Examples and Applications
9.7 Copula Models
9.7.1 Definition and General Properties
9.7.2 Factor Copula Models
9.8 Default Contagion in Reduced-Form Models
9.8.1 Default Contagion and Default Dependence
9.8.2 Information-Based Default Contagion
9.8.3 Interacting Intensities
CHAPTER 10: Operational Risk and Insurance Analytics
10.1 Operational Risk in Perspective
10.1.1 A New Risk Class
10.1.2 The Elementary Approaches
10.1.3 Advanced Measurement Approaches
10.1.4 Operational Loss Data
10.2 Elements of Insurance Analytics
10.2.1 The Case for Actuarial Methodology
10.2.2 The Total Loss Amount
10.2.3 Approximations and Panjer Recursion
10.2.4 Poisson Mixtures
10.2.5 Tails of Aggregate Loss Distributions
10.2.6 The Homogeneous Poisson Process
10.2.7 Processes Related to the Poisson Process
Appendix 494
A.1 Miscellaneous Definitions and Results
A.1.1 Type of Distribution
A.1.2 Generalized Inverses and Quantiles
A.1.3 Karamata's Theorem
About Alexander J. McNeil, Rudiger Frey and Paul Embrechts
Alexander J. McNeil is Professor of Mathematics at the Swiss Federal Institute of Technology (ETH) in Zurich. Rudiger Frey is Professor of Financial Mathematics at the University of Leipzig. Paul Embrechts, Professor of Insurance Mathematics at the Swiss Federal Institute of Technology (ETH) in Zurich, is the coauthor of "Modelling Extremal Events for Insurance and Finance".
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