Extreme Financial Risks (eBook)

Preface: Idiosyncratic and Collective Extreme Risks 7
Contents 13
1 On the Origin of Risks and Extremes 17
1.1 The Multidimensional Nature of Risk and Dependence 17
1.2 How to Rank Risks Coherently? 20
1.2.1 Coherent Measures of Risks 20
1.2.2 Consistent Measures of Risks and Deviation Measures 23
1.2.3 Examples of Consistent Measures of Risk 26
1.3 Origin of Risk and Dependence 29
1.3.1 The CAPM View 29
1.3.2 The Arbitrage Pricing Theory (APT) and the Fama–French Factor Model 34
1.3.3 The Efficient Market Hypothesis 36
1.3.4 Emergence of Dependence Structures in the Stock Markets 40
1.3.5 Large Risks in Complex Systems 45
Appendix 46
1.A Why Do Higher Moments Allow us to Assess Larger Risks? 46
2 Marginal Distributions of Returns 49
2.1 Motivations 49
2.2 A Brief History of Return Distributions 53
2.2.1 The Gaussian Paradigm 53
2.2.2 Mechanisms for Power Laws in Finance 55
2.2.3 Empirical Search for Power Law Tails and Possible Alternatives 58
2.3 Constraints from Extreme Value Theory 59
2.3.1 Main Theoretical Results on Extreme Value Theory 61
2.3.2 Estimation of the Form Parameter and Slow Convergence to Limit Generalized Extreme Value (GEV) and Generalized Pareto (GPD) Distributions 63
2.3.3 Can Long Memory Processes Lead to Misleading Measures of Extreme Properties? 67
2.3.4 GEV and GPD Estimators of the Distributions of Returns of the Dow Jones and Nasdaq Indices 68
2.4 Fitting Distributions of Returns with Parametric Densities 70
2.4.1 Definition of Two Parametric Families 70
2.4.2 Parameter Estimation Using Maximum Likelihood and Anderson-Darling Distance 76
2.4.3 Empirical Results on the Goodness-of-Fits 78
2.4.4 Comparison of the Descriptive Power of the Di.erent Families 85
2.5 Discussion and Conclusions 92
2.5.1 Summary 92
2.5.2 Is There a Best Model of Tails? 92
2.5.3 Implications for Risk Assessment 94
Appendix 96
2.A Definition and Main Properties of Multifractal Processes 96
2.A.1 Self-similar Processes, Multiplicative Cascades and Multifractal Processes 97
2.A.2 The Multifractal Spectrum 99
2.A.3 The Multifractal Model of Asset Returns of Mandelbrot et al. 100
2.A.4 The Multifractal Random Walk (MRW) 100
2.B A Survey of the Properties of Maximum Likelihood Estimators 103
2.B.1 The Pareto Distribution 104
2.B.2 The Weibull Distribution 104
2.B.3 The Exponential Distribution 106
2.B.4 The Incomplete Gamma Distribution 106
2.C Asymptotic Variance–Covariance of Maximum Likelihood Estimators of the SE Parameters 107
2.D Testing the Pareto Model versus the Stretched-Exponential Model 109
3 Notions of Copulas 115
3.1 What is Dependence? 117
3.2 Definition and Main Properties of Copulas 119
3.3 A Few Copula Families 123
3.3.1 Elliptical Copulas 123
3.3.2 Archimedean Copulas 127
3.3.3 Extreme Value Copulas 132
3.4 Universal Bounds for Functionals of Dependent Random Variables 134
3.5 Simulation of Dependent Data with a Prescribed Copula 136
3.5.1 Simulation of Random Variables Characterized by Elliptical Copulas 136
3.5.2 Simulation of Random Variables Characterized by Smooth Copulas 138
3.6 Application of Copulas 140
3.6.1 Assessing Tail Risk 140
3.6.2 Asymptotic Expression of the Value-at-Risk 144
3.6.3 Options on a Basket of Assets 147
3.6.4 Basic Modeling of Dependent Default Risks 153
Appendix 154
3.A Simple Proof of a Theorem on Universal Bounds for Functionals of Dependent Random Variables 154
3.B Sketch of a Proof of a Large Deviation Theorem for Portfolios Made of Weibull Random Variables 156
3.C Relation Between the Objective and the Risk-Neutral Copula 159
4 Measures of Dependences 162
4.1 Linear Correlations 162
4.1.1 Correlation Between Two Random Variables 162
4.1.2 Local Correlation 166
4.1.3 Generalized Correlations Between N > 2 Random Variables
4.2 Concordance Measures 169
4.2.1 Kendall’s Tau 169
4.2.2 Measures of Similarity Between Two Copulas 173
4.2.3 Common Properties of Kendall’s Tau, Spearman’s Rho and Gini’s Gamma 176
4.3 Dependence Metric 177
4.4 Quadrant and Orthant Dependence 179
4.5 Tail Dependence 183
4.5.1 Definition 183
4.5.3 Tail Dependence for Several Usual Models 185
4.5.4 Practical Implications 192
Appendix 4.A Tail Dependence Generated by Student’s Factor Model 197
4.A.1 Proof of Lemma 4.5.1 198
4.A.2 Derivation of Equation (4.A.7) 199
4.A.3 Proof of Lemma 4.5.2 199
5 Description of Financial Dependences with Copulas 204
5.1 Estimation of Copulas 205
5.1.1 Nonparametric Estimation 205
5.1.2 Semiparametric Estimation 210
5.1.3 Parametric Estimation 215
5.1.4 Goodness-of-Fit Tests 218
5.2 Description of Financial Data in Terms of Gaussian Copulas 219
5.2.1 Test Statistics and Testing Procedure 219
5.2.2 Empirical Results 222
5.3 Limits of the Description in Terms of the Gaussian Copula 227
5.3.1 Limits of the Tests 227
5.3.2 Sensitivity of the Method 228
5.3.3 The Student Copula: An Alternative? 230
5.3.4 Accounting for Heteroscedasticity 232
5.4 Summary 234
Appendix 236
5.A Proof of the Existence of a .2-Statisticfor Testing Gaussian Copulas 236
5.B Hypothesis Testing with Pseudo Likelihood 237
6 Measuring Extreme Dependences 242
6.1 Motivations 245
6.1.1 Suggestive Historical Examples 245
6.1.2 Review of Different Perspectives 246
6.2 Conditional Correlation Coefficient 248
6.2.1 Definition 249
6.2.3 Influence of the Underlying Distribution for a Given Conditioning Set 252
6.2.4 Conditional Correlation Coefficient on Both Variables 254
6.2.5 An Example of Empirical Implementation 255
6.2.6 Summary 261
6.3 Conditional Concordance Measures 262
6.3.1 Definition 263
6.3.2 Example 264
6.3.3 Empirical Evidence 266
6.4 Extreme Co-movements 269
6.5 Synthesis and Consequences 271
Appendix 276
6.A Correlation Coe.cient for Gaussian Variables Conditioned on Both X and Y Larger Than u 276
6.A.1 Asymptotic Behavior of L(u, u .)
6.A.4 Asymptotic Behavior of the Cross Moment m11 279
6.A.5 Asymptotic Behavior of the Correlation Coefficient 280
6.B Conditional Correlation Coefficient for Student’s Variables 281
6.B.1 Proposition 281
6.B.2 Proof of the Proposition 281
6.B.3 Conditioning on Y Larger Than v 283
6.B.5 Conditioning on Y > v Versus on |Y | >
6.C Conditional Spearman’s Rho 285
7 Summary and Outlook 286
7.1 Synthesis 286
7.2 Outlook and Future Directions 289
7.2.1 Robust and Adaptive Estimation of Dependences 289
7.2.2 Outliers, Kings, Black Swans and Their Dependence 291
7.2.3 Endogeneity Versus Exogeneity 291
7.2.4 Nonstationarity and Regime Switching in Dependence 294
7.2.5 Time-Varying Lagged Dependence 295
7.2.6 Toward a Dynamical Microfoundation of Dependences 296
References 298
Index 324

6 Measuring Extreme Dependences (p. 227)

In this chapter, we investigate the relative information content of several measures of dependence between two random variables X and Y in various models of financial series. We consider measures of dependence especially defined for large and extreme events. These measures of dependence are of two types: (i) unconditional such as with the coefficient of tail dependence already introduced in Chap. 4 and (ii) conditional such as with the correlation coefficient conditional over a given threshold. The introduction of conditioning over values of one or both variables reaching above some threshold is a natural approach to discriminate the dependence in the tails. It explodes the concept of dependence into a multidimensional set of measures, each adapted to certain ranges spanned by the random variables. We present explicit analytical formulas as well as numerical and empirical estimations for these measures of dependence. The main overall insight is that conditional measures of dependence may be very different from the unconditional ones and can often lead to paradoxical interpretations, whose origins are explained in detail.

When the dependence properties are studied as a function of time, one can often observe that conditional measures vary with time. Such time variation has initiated a vigorous discussion in the literature on its possible economic meaning.

We review the mechanism by which conditioning provides a straightforward and general mechanism for explaining changes of correlations based on changes of volatility or of trends: for a given conditional threshold, if the volatility of one or both time series changes in some time interval, then the corresponding quantiles sampled in the conditional measure will also change; as a result, the conditional measure will not sample the same part of the tails of the distributions, effectively changing the deffnition of the conditional measure. In this explanation, the variation with time of conditional measures of dependence results solely from a change of volatility but does not respect a genuine change of dependence. In other words, a constant dependence structure together with time-varying volatility may give rise to changing conditional measures of dependence, which would be incorrectly interpreted as respecting genuine changes of dependence.

Thus, tools based upon conditional quantities should be used with caution since conditioning alone induces a change in the dependence structure which has nothing to do with a genuine change of unconditional dependence. In this respect, for its stability, the coefficient of tail dependence should be preferred to the conditional correlations. Moreover, the various measures of dependence exhibit different and sometimes opposite behaviors, showing that extreme dependence properties possess a multidimensional character that can be revealed in various ways.

As an illustration, the theoretical results and their interpretation presented below are applied to the controversial contagion problem across Latin American markets during the turmoil periods associated with the Mexican crisis in 1994 and with the Argentinean crisis that started in 2001. The analysis of several measures of dependence between the Argentinean, Brazilian, Chilean and Mexican markets shows that the above conditioning effect does not fully explain the behavior of the Latin American stock indexes, confirming the existence of a possible genuine contagion. Our analysis below suggests that the 1994 Mexican crisis has spread over to Argentina and Brazil through contagion mechanisms and to Chile only through co-movements. Concerning the recent Argentinean crisis that started in 2001, no evidence of contagion to the other Latin American countries (except perhaps in the direction of Brazil) can be found but significant co-movements are identified.