Quantitative risk management : concepts, techniques and tools

Cover
Title
Copyright
Dedication
Contents
Preface
I An Introduction to Quantitative Risk Management
	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 Regulatory Framework
			1.3.1 The Basel Framework
			1.3.2 The Solvency II Framework
			1.3.3 Criticism of Regulatory Frameworks
		1.4 Why Manage Financial Risk?
			1.4.1 A Societal View
			1.4.2 The Shareholder’s View
		1.5 Quantitative Risk Management
			1.5.1 The Q in QRM
			1.5.2 The Nature of the Challenge
			1.5.3 QRM Beyond Finance
	2 Basic Concepts in Risk Management
		2.1 Risk Management for a Financial Firm
			2.1.1 Assets, Liabilities and the Balance Sheet
			2.1.2 Risks Faced by a Financial Firm
			2.1.3 Capital
		2.2 Modelling Value and Value Change
			2.2.1 Mapping Risks
			2.2.2 Valuation Methods
			2.2.3 Loss Distributions
		2.3 Risk Measurement
			2.3.1 Approaches to Risk Measurement
			2.3.2 Value-at-Risk
			2.3.3 VaR in Risk Capital Calculations
			2.3.4 Other Risk Measures Based on Loss Distributions
			2.3.5 Coherent and Convex Risk Measures
	3 Empirical Properties of Financial Data
		3.1 Stylized Facts of Financial Return Series
			3.1.1 Volatility Clustering
			3.1.2 Non-normality and Heavy Tails
			3.1.3 Longer-Interval Return Series
		3.2 Multivariate Stylized Facts
			3.2.1 Correlation between Series
			3.2.2 Tail Dependence
II Methodology
	4 Financial Time Series
		4.1 Fundamentals of Time Series Analysis
			4.1.1 Basic Definitions
			4.1.2 ARMA Processes
			4.1.3 Analysis in the Time Domain
			4.1.4 Statistical Analysis of Time Series
			4.1.5 Prediction
		4.2 GARCH Models for Changing Volatility
			4.2.1 ARCH Processes
			4.2.2 GARCH Processes
			4.2.3 Simple Extensions of the GARCH Model
			4.2.4 Fitting GARCH Models to Data
			4.2.5 Volatility Forecasting and Risk Measure Estimation
	5 Extreme Value Theory
		5.1 Maxima
			5.1.1 Generalized Extreme Value Distribution
			5.1.2 Maximum Domains of Attraction
			5.1.3 Maxima of Strictly Stationary Time Series
			5.1.4 The Block Maxima Method
		5.2 Threshold Exceedances
			5.2.1 Generalized Pareto Distribution
			5.2.2 Modelling Excess Losses
			5.2.3 Modelling Tails and Measures of Tail Risk
			5.2.4 The Hill Method
			5.2.5 Simulation Study of EVT Quantile Estimators
			5.2.6 Conditional EVT for Financial Time Series
		5.3 Point Process Models
			5.3.1 Threshold Exceedances for Strict White Noise
			5.3.2 The POT Model
	6 Multivariate Models
		6.1 Basics of Multivariate Modelling
			6.1.1 Random Vectors and Their Distributions
			6.1.2 Standard Estimators of Covariance and Correlation
			6.1.3 The Multivariate Normal Distribution
			6.1.4 Testing Multivariate Normality
		6.2 Normal Mixture Distributions
			6.2.1 Normal Variance Mixtures
			6.2.2 Normal Mean–Variance Mixtures
			6.2.3 Generalized Hyperbolic Distributions
			6.2.4 Empirical Examples
		6.3 Spherical and Elliptical Distributions
			6.3.1 Spherical Distributions
			6.3.2 Elliptical Distributions
			6.3.3 Properties of Elliptical Distributions
			6.3.4 Estimating Dispersion and Correlation
		6.4 Dimension-Reduction Techniques
			6.4.1 Factor Models
			6.4.2 Statistical Estimation Strategies
			6.4.3 Estimating Macroeconomic Factor Models
			6.4.4 Estimating Fundamental Factor Models
			6.4.5 Principal Component Analysis
	7 Copulas and Dependence
		7.1 Copulas
			7.1.1 Basic Properties
			7.1.2 Examples of Copulas
			7.1.3 Meta Distributions
			7.1.4 Simulation of Copulas and Meta Distributions
			7.1.5 Further Properties of Copulas
		7.2 Dependence Concepts and Measures
			7.2.1 Perfect Dependence
			7.2.2 Linear Correlation
			7.2.3 Rank Correlation
			7.2.4 Coefficients of Tail Dependence
		7.3 Normal Mixture Copulas
			7.3.1 Tail Dependence
			7.3.2 Rank Correlations
			7.3.3 Skewed Normal Mixture Copulas
			7.3.4 Grouped Normal Mixture Copulas
		7.4 Archimedean Copulas
			7.4.1 Bivariate Archimedean Copulas
			7.4.2 Multivariate Archimedean Copulas
		7.5 Fitting Copulas to Data
			7.5.1 Method-of-Moments Using Rank Correlation
			7.5.2 Forming a Pseudo-sample from the Copula
			7.5.3 Maximum Likelihood Estimation
	8 Aggregate Risk
		8.1 Coherent and Convex Risk Measures
			8.1.1 Risk Measures and Acceptance Sets
			8.1.2 Dual Representation of Convex Measures of Risk
			8.1.3 Examples of Dual Representations
		8.2 Law-Invariant Coherent Risk Measures
			8.2.1 Distortion Risk Measures
			8.2.2 The Expectile Risk Measure
		8.3 Risk Measures for Linear Portfolios
			8.3.1 Coherent Risk Measures as Stress Tests
			8.3.2 Elliptically Distributed Risk Factors
			8.3.3 Other Risk Factor Distributions
		8.4 Risk Aggregation
			8.4.1 Aggregation Based on Loss Distributions
			8.4.2 Aggregation Based on Stressing Risk Factors
			8.4.3 Modular versus Fully Integrated Aggregation Approaches
			8.4.4 Risk Aggregation and Fréchet Problems
		8.5 Capital Allocation
			8.5.1 The Allocation Problem
			8.5.2 The Euler Principle and Examples
			8.5.3 Economic Properties of the Euler Principle
III Applications
	9 Market Risk
		9.1 Risk Factors and Mapping
			9.1.1 The Loss Operator
			9.1.2 Delta and Delta–Gamma Approximations
			9.1.3 Mapping Bond Portfolios
			9.1.4 Factor Models for Bond Portfolios
		9.2 Market Risk Measurement
			9.2.1 Conditional and Unconditional Loss Distributions
			9.2.2 Variance–Covariance Method
			9.2.3 Historical Simulation
			9.2.4 Dynamic Historical Simulation
			9.2.5 Monte Carlo
			9.2.6 Estimating Risk Measures
			9.2.7 Losses over Several Periods and Scaling
		9.3 Backtesting
			9.3.1 Violation-Based Tests for VaR
			9.3.2 Violation-Based Tests for Expected Shortfall
			9.3.3 Elicitability and Comparison of Risk Measure Estimates
			9.3.4 Empirical Comparison of Methods Using Backtesting Concepts
			9.3.5 Backtesting the Predictive Distribution
	10 Credit Risk
		10.1 Credit-Risky Instruments
			10.1.1 Loans
			10.1.2 Bonds
			10.1.3 Derivative Contracts Subject to Counterparty Risk
			10.1.4 Credit Default Swaps and Related Credit Derivatives
			10.1.5 PD, LGD and EAD
		10.2 Measuring Credit Quality
			10.2.1 Credit Rating Migration
			10.2.2 Rating Transitions as a Markov Chain
		10.3 Structural Models of Default
			10.3.1 The Merton Model
			10.3.2 Pricing in Merton’s Model
			10.3.3 Structural Models in Practice: EDF and DD
			10.3.4 Credit-Migration Models Revisited
		10.4 Bond and CDS Pricing in Hazard Rate Models
			10.4.1 Hazard Rate Models
			10.4.2 Risk-Neutral Pricing Revisited
			10.4.3 Bond Pricing
			10.4.4 CDS Pricing
			10.4.5 P versus Q: Empirical Results
		10.5 Pricing with Stochastic Hazard Rates
			10.5.1 Doubly Stochastic Random Times
			10.5.2 Pricing Formulas
			10.5.3 Applications
		10.6 Affine Models
			10.6.1 Basic Results
			10.6.2 The CIR Square-Root Diffusion
			10.6.3 Extensions
	11 Portfolio Credit Risk Management
		11.1 Threshold Models
			11.1.1 Notation for One-Period Portfolio Models
			11.1.2 Threshold Models and Copulas
			11.1.3 Gaussian Threshold Models
			11.1.4 Models Based on Alternative Copulas
			11.1.5 Model Risk Issues
		11.2 Mixture Models
			11.2.1 Bernoulli Mixture Models
			11.2.2 One-Factor Bernoulli Mixture Models
			11.2.3 Recovery Risk in Mixture Models
			11.2.4 Threshold Models as Mixture Models
			11.2.5 Poisson Mixture Models and CreditRisk^+
		11.3 Asymptotics for Large Portfolios
			11.3.1 Exchangeable Models
			11.3.2 General Results
			11.3.3 The Basel IRB Formula
		11.4 Monte Carlo Methods
			11.4.1 Basics of Importance Sampling
			11.4.2 Application to Bernoulli Mixture Models
		11.5 Statistical Inference in Portfolio Credit Models
			11.5.1 Factor Modelling in Industry Threshold Models
			11.5.2 Estimation of Bernoulli Mixture Models
			11.5.3 Mixture Models as GLMMs
			11.5.4 A One-Factor Model with Rating Effect
	12 Portfolio Credit Derivatives
		12.1 Credit Portfolio Products
			12.1.1 Collateralized Debt Obligations
			12.1.2 Credit Indices and Index Derivatives
			12.1.3 Basic Pricing Relationships for Index Swaps and CDOs
		12.2 Copula Models
			12.2.1 Definition and Properties
			12.2.2 Examples
		12.3 Pricing of Index Derivatives in Factor Copula Models
			12.3.1 Analytics
			12.3.2 Correlation Skews
			12.3.3 The Implied Copula Approach
	13 Operational Risk and Insurance Analytics
		13.1 Operational Risk in Perspective
			13.1.1 An Important Risk Class
			13.1.2 The Elementary Approaches
			13.1.3 Advanced Measurement Approaches
			13.1.4 Operational Loss Data
		13.2 Elements of Insurance Analytics
			13.2.1 The Case for Actuarial Methodology
			13.2.2 The Total Loss Amount
			13.2.3 Approximations and Panjer Recursion
			13.2.4 Poisson Mixtures
			13.2.5 Tails of Aggregate Loss Distributions
			13.2.6 The Homogeneous Poisson Process
			13.2.7 Processes Related to the Poisson Process
IV Special Topics
	14 Multivariate Time Series
		14.1 Fundamentals of Multivariate Time Series
			14.1.1 Basic Definitions
			14.1.2 Analysis in the Time Domain
			14.1.3 Multivariate ARMA Processes
		14.2 Multivariate GARCH Processes
			14.2.1 General Structure of Models
			14.2.2 Models for Conditional Correlation
			14.2.3 Models for Conditional Covariance
			14.2.4 Fitting Multivariate GARCH Models
			14.2.5 Dimension Reduction in MGARCH
			14.2.6 MGARCH and Conditional Risk Measurement
	15 Advanced Topics in Multivariate Modelling
		15.1 Normal Mixture and Elliptical Distributions
			15.1.1 Estimation of Generalized Hyperbolic Distributions
			15.1.2 Testing for Elliptical Symmetry
		15.2 Advanced Archimedean Copula Models
			15.2.1 Characterization of Archimedean Copulas
			15.2.2 Non-exchangeable Archimedean Copulas
	16 Advanced Topics in Extreme Value Theory
		16.1 Tails of Specific Models
			16.1.1 Domain of Attraction of the Fréchet Distribution
			16.1.2 Domain of Attraction of the Gumbel Distribution
			16.1.3 Mixture Models
		16.2 Self-exciting Models for Extremes
			16.2.1 Self-exciting Processes
			16.2.2 A Self-exciting POT Model
		16.3 Multivariate Maxima
			16.3.1 Multivariate Extreme Value Copulas
			16.3.2 Copulas for Multivariate Minima
			16.3.3 Copula Domains of Attraction
			16.3.4 Modelling Multivariate Block Maxima
		16.4 Multivariate Threshold Exceedances
			16.4.1 Threshold Models Using EV Copulas
			16.4.2 Fitting a Multivariate Tail Model
			16.4.3 Threshold Copulas and Their Limits
	17 Dynamic Portfolio Credit Risk Models and Counterparty Risk
		17.1 Dynamic Portfolio Credit Risk Models
			17.1.1 Why Dynamic Models of Portfolio Credit Risk?
			17.1.2 Classes of Reduced-Form Models of Portfolio Credit Risk
		17.2 Counterparty Credit Risk Management
			17.2.1 Uncollateralized Value Adjustments for a CDS
			17.2.2 Collateralized Value Adjustments for a CDS
		17.3 Conditionally Independent Default Times
			17.3.1 Definition and Mathematical Properties
			17.3.2 Examples and Applications
			17.3.3 Credit Value Adjustments
		17.4 Credit Risk Models with Incomplete Information
			17.4.1 Credit Risk and Incomplete Information
			17.4.2 Pure Default Information
			17.4.3 Additional Information
			17.4.4 Collateralized Credit Value Adjustments and Contagion Effects
Appendix
	A.1 Miscellaneous Definitions and Results
		A.1.1 Type of Distribution
		A.1.2 Generalized Inverses and Quantiles
		A.1.3 Distributional Transform
		A.1.4 Karamata’s Theorem
		A.1.5 Supporting and Separating Hyperplane Theorems
	A.2 Probability Distributions
		A.2.1 Beta
		A.2.2 Exponential
		A.2.3 F
		A.2.4 Gamma
		A.2.5 Generalized Inverse Gaussian
		A.2.6 Inverse Gamma
		A.2.7 Negative Binomial
		A.2.8 Pareto
		A.2.9 Stable
	A.3 Likelihood Inference
		A.3.1 Maximum Likelihood Estimators
		A.3.2 Asymptotic Results: Scalar Parameter
		A.3.3 Asymptotic Results: Vector of Parameters
		A.3.4 Wald Test and Confidence Intervals
		A.3.5 Likelihood Ratio Test and Confidence Intervals
		A.3.6 Akaike Information Criterion
References
Index