Risk Measurement: From Quantitative Measures to Management Decisions

Preface
Contents
1 Introduction
	1.1 1968–1988: Premises
		1.1.1 The Triggering Elements
			1.1.1.1 A US Default
			1.1.1.2 Herstatt Bank
	1.2 Basel I
	1.3 Basel II
		1.3.1 Basel II.5
	1.4 Basel III
	1.5 Solvency II Directives
	1.6 Successive Criticisms
		1.6.1 Basel I
		1.6.2 Basel II
		1.6.3 Basel III
		1.6.4 Solvency II
	References
2 Financial Institutions: A Regulation Review Through the Risk Measurement Prism
	2.1 Credit Risk
		2.1.1 Standardised Approach
		2.1.2 Internal Ratings-Based Approach
	2.2 Credit Value Adjustment
	2.3 Operational Risk
		2.3.1 Basic Indicator Approach
		2.3.2 Standardised Approach
		2.3.3 The Advanced Measurement Approach (AMA)
		2.3.4 Standardised Measurement Approach or New Standardized Approach
	2.4 Market Risk
		2.4.1 The Fundamental Review of the Trading Book
		2.4.2 Standardised Approach
		2.4.3 Internal Models Approach
	2.5 IFRS9 as a Risk Regulation
	2.6 The Stress-Testing Framework
	References
3 The Traditional Risk Measures
	3.1 Measures of Dispersion
		3.1.1 Distance Between Representative Values
		3.1.2 Deviation from a Central Value
			3.1.2.1 The Variance
			3.1.2.2 The Expected Absolute Deviation
			3.1.2.3 Semi-Interquartile Deviation
		3.1.3 Mean Absolute Difference
	3.2 Risk Measurement: Portfolio Theory Philosophy
		3.2.1 Modern Portfolio Theory
		3.2.2 Efficient Frontier with No Risk-Free Asset: Risk Management in Essence
		3.2.3 Risk-Free Asset and the Capital Allocation Line
		3.2.4 Risk Management Through MPT: Systematic Risk and Specific Risk
		3.2.5 Capital Asset Pricing Model
		3.2.6 Arbitrage Pricing Theory
		3.2.7 Downside Risk Measures
			3.2.7.1 Below a Reference Point Risk Measures
		3.2.8 Downside Deviation
		3.2.9 Sortino Ratio
	3.3 Quantile Risk Measure and Affiliated
		3.3.1 VaR
		3.3.2 Expected Shortfall
		3.3.3 Tail Value at Risk
		3.3.4 CoVaR
		3.3.5 Entropic Risk Measures: ρerm and EVaR
	3.4 Distortion Risk Measure
	References
4 Univariate and Multivariate Distributions
	4.1 Univariate Distributions
		4.1.1 Some Recalls on the Probabilistic Characteristic of a Risk Factor
		4.1.2 Examples of Parametric Distributions
		4.1.3 Non-parametric Modelling for a Distribution
		4.1.4 Distorted Distributions
			4.1.4.1 Shift of a Distribution
			4.1.4.2 Multimodal Distributions
	4.2 Multivariate Approach
		4.2.1 Definition of a Copula
		4.2.2 Properties
		4.2.3 Examples of Copulas
		4.2.4 Tail Dependence Concept
	Appendix: Estimation
		Distribution\'s Parameters Estimation
		Copulas\' Parameters Estimation
	References
5 Extensions for Risk Measures: Univariate and Multivariate Approaches
	5.1 New Risk Measure Computed for One Risk Factor
		5.1.1 Introduction
		5.1.2 The Spectral Risk Measure
		5.1.3 The Spectrum as a Risk Measure
		5.1.4 A Spatial Risk Measure
		5.1.5 Risk Measure Associated with a Multimodal Distribution
		5.1.6 Some Remarks
	5.2 VaR in High Dimension
		5.2.1 Archimedean and Extreme Value Copulas in High Dimension
		5.2.2 Lower- and Upper-Orthant VaR, and MultivariateQuantiles
	References
6 Linear Dynamics
	6.1 Introduction
	6.2 Stochastic Processes
		6.2.1 Definition of a Stochastic Process
		6.2.2 Stationarity Framework
		6.2.3 Examples of Particular Stochastic Processes: The Related Random Walk Models
			6.2.3.1 Strong Gaussian White Noise Model
			6.2.3.2 LogNormal Model
			6.2.3.3 The Geometric Random Walk Model
			6.2.3.4 Are the Assumptions Realistic?
	6.3 Models Used in the Classical Portfolio Theory
		6.3.1 Introduction
		6.3.2 Regression Model
			6.3.2.1 The Model
			6.3.2.2 Best Linear Prediction
	6.4 ARMA Processes
		6.4.1 Stationary Solution for ARMA(p, q) Model
		6.4.2 Second Order Properties of an ARMA (p, q) Process
			6.4.2.1 Autocorrelation Function of an ARMA(p,q) Process
			6.4.2.2 Partial Autocorrelation Function
		6.4.3 Estimation
			6.4.3.1 Yule–Walker Estimators
			6.4.3.2 Moments\' Method
			6.4.3.3 Maximum Likelihood Method
			6.4.3.4 Remarks
		6.4.4 Model\'s Selection
		6.4.5 Forecasting
	References
7 Risks and Non-Linear Dynamics
	7.1 GARCH Modelling
		7.1.1 Introduction
		7.1.2 Description of Some Heteroskedastic Processes
			7.1.2.1 ARCH(p) Model
			7.1.2.2 ARCH Regression Process
			7.1.2.3 GARCH(p,q) Process
			7.1.2.4 Heteroskedasticity in the Mean
			7.1.2.5 Asymmetric GARCH Processes
		7.1.3 Properties of Gaussian GARCH Processes
			7.1.3.1 Properties of an ARCH(p) Process
			7.1.3.2 Properties of a GARCH(p,q) Process
			7.1.3.3 IGARCH Model
		7.1.4 GARCH Models with Non-Gaussian Distributions
		7.1.5 Inference for GARCH Processes
			7.1.5.1 Parameter Estimation
			7.1.5.2 Tests
			7.1.5.3 Volatility Forecasts
	7.2 Markov Switching Modelling
		7.2.1 Model Formalisation
		7.2.2 Some Properties
			7.2.2.1 Stationarity Conditions
			7.2.2.2 Auto-Covariance Function
			7.2.2.3 Unconditional Law
		7.2.3 Simulations Studies with Gaussian Noises
			7.2.3.1 Role of Transition\'s Probabilities
			7.2.3.2 Role of Model\'s Parameters
			7.2.3.3 Summary
		7.2.4 Simulations Studies with Non-Gaussian Noises
			7.2.4.1 Noises with Symmetrical Distribution Laws
			7.2.4.2 Dependent Noises
		7.2.5 Estimation Procedures
	7.3 Dynamical VaR and Expected Shortfall Measures: Illustration
		7.3.1 VaR and ES Measures Computed Using ParametricModels
		7.3.2 The RiskMetrics Approach
		7.3.3 The GARCH Approach
		7.3.4 Markov-Switching Approach
		7.3.5 VaR and ES Computed Using the Empirical Histogram
		7.3.6 VaR and ES Computed Using Copulas
	7.4 References on Dynamical Modelling for Financial Applications
	References