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300 9. Mesure de la dépendance extrême entre deux actifs financiers How to account for extreme co-movements between individual stocks and the market ∗ Y. Malevergne 1,2 and D. Sornette 1,3 1 Laboratoire de Physique de la Matière Condensée CNRS UMR 6622 Université de Nice-Sophia Antipolis, 06108 Nice Cedex 2, France 2 Institut de Science Financière et d’Assurances - Université Lyon I 43, Bd du 11 Novembre 1918, 69622 Villeurbanne Cedex, France 3 Institute of Geophysics and Planetary Physics and Department of Earth and Space Science University of California, Los Angeles, California 90095, USA email: Yannick.Malevergne@unice.fr and sornette@unice.fr fax: (33) 4 92 07 67 54 August 8, 2002 Abstract Using the framework of factor models, we study the extreme co-movements between two stocks and between a stock and the market. In this goal, we establish the general expression of the coefficient of tail dependence between the market and a stock (that is, the probability that the stock incurs a large loss, assuming that the market has also undergone a large loss) and between two stocks as a function of the parameters of the underlying factor model and of the tail parameters of the distributions of the factor and of the idiosyncratic noise of each stock. Our formula holds for arbitrary marginal distributions and in addition does not require any parameterization of the multivariate distributions of the market and stocks. The determination of the extreme parameter, which is not accessible by a direct statistical inference, is made possible by the measurement of parameters whose estimation involves a significant part of the data with sufficient statistics. Our empirical tests find a good agreement between the calibration of the tail dependence coefficient and the realized large losses over the period from 1962 to 2000. Nevertheless, a bias is detected which suggests the presence of an outlier in the form of the crash of October 1987. ∗ We acknowledge helpful discussions and exchanges with C.W.G. Granger, J.P. Laurent, V. Pisarenko, R. Valkanov and D. Zajdenweber. This work was partially supported by the James S. Mc Donnell Foundation 21st century scientist award/studying complex system. 1
9.2. Estimation du coefficient de dépendance de queue 301 Introduction The concept of extreme or “tail dependence” probes the reaction of a variable to the realization of another variable when this realization is of extreme amplitude and very low probability. The dependence, and especially the extreme dependence, between two assets or between an asset and any other exogeneous economic variable is an issue of major importance both for practioners and for academics. The determination of extreme dependences is crucial for financial and for insurance institutions involved in risk management. It is also fundamental for the establishment of a rational investment policy striving for the best diversification of the various sources of risk. In all these situations, the objective is to prevent or at least minimize the simultaneous occurrence of large losses across the different positions held in the portfolio. From an academic perspective, taking into account the extreme dependence properties provide useful yardsticks and important constraints for the construction of models, which should not underestimate or overestimate risks. From the point of view of univariate statistics, extreme values theory provides the mathematical framework for the classification and quantification of very large risks. This has been made possible by the existence of a “universal” behavior summarized by the Gnedenko-Pickands-Balkema-de Haan theorem which gives a natural limit law for peak-overthreshold values in the form of the Generalized Pareto Distribution (see Embrechts, Kluppelberg, and Mikosh (1997, pp 152-168)). Moreover, most of these univariate extreme values results are robust with respect to the time dependences observed in financial time series (see de Haan, Resnick, Rootzen and de Vries (1989) or Starica (1999) for instance). In contrast, no such result is yet available in the multivariate case. In such absence of theoretical guidelines, the alternative is therefore to impose some dependence structure in a rather ad hoc and arbitrary way. This was the stance taken for instance in Longin and Solnik (2001) in their study of the phenomenon of contagion across international equity markets. This approach, where the dependence structure is not determined from empirical facts or from an economic model, is not fully satisfying. As a remedy, we propose a new approach, which does not directly rely on multivariate extreme values theory, but rather derives the extreme dependence structure from the characteristics of a financial model of assets. Specifically, we use the general class of factor models, which is probably one of the most versatile and relevant one, and whose introduction in finance can be traced back at least to Ross (1976). The factor models are now widely used in many branches of finance, including stock return models, interest rate models (Vasicek (1977), Brennan and Schwarz (1978), Cox, Ingersoll, and Ross (1985)), credit risks models (Carey (1998), Gorby (2000), Lucas, Klaassen, Spreij and Straetmans (2001)), and so on, and are found at the core of many theories and equilibrium models. Here, we shall first focus on the characterization of the extreme dependence between stock returns and the market return and then on the extreme dependence between two stocks which share a common explaining factor. The role of the market return as a factor explaining the evolution of individual stock returns is supported both by theoretical models such as the Capital Asset Pricing Model (CAPM) (Sharpe (1964), Lintner (1965), Mossin (1966)) or the Arbitrage Pricing Theory (APT) (Ross (1976)) and by empirical studies (Fama and Beth (1973), Kandel and Staumbaugh (1987) among many others). It has even been shown in Roll (1988) that in certain dramatic circumstances, such as the October 1987 stock-market crash, the (global) market was the sole relevant factor needed to explain the stock market movements and the propagation of the crash across countries. Thus, the choice of factor models is a very natural starting point for studying extreme dependences from a general point of view. The main gain is that, without imposing any a priori ad hoc dependences other than the definition of the factor model, we shall be able to 2
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UNIVERSITE DE NICE-SOPHIA ANTIPOLIS
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4 Table des matières 2.2.2 Bulles
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6 Table des matières 7.3.2 Tests n
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8 Table des matières 12.1.1 Distri
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10 Table des matières Références
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12 m’ont souvent été d’un gra
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14 Introduction pour espérer se co
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16 Introduction les distributions e
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18 Introduction
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Chapitre 1 Faits stylisés des rent
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1.1. Rappel des faits stylisés 23
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1.1. Rappel des faits stylisés 25
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1.2. De la difficulté de représen
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1.3. Modélisation des propriétés
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Chapitre 2 Modèles phénoménologi
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2.1. Bulles rationnelles multi-dime
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2.2. Des bulles rationnelles aux kr
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Chapitre 3 Distributions exponentie
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Empirical Distributions of Log-Retu
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concerning tail fatness can be form
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To fit our two data sets, section 4
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properties, stressing the problems
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Another problem lies in the determi
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In order to obtain a process with S
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and Sornette (1999) for instance. W
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1. The Pareto distribution: 79 Fu(x
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empirical analog FN(x), estimated f
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have been chosen to converge to 1 a
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5 Comparison of the descriptive pow
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ased on the fact that the quantity
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tail (positive or negative) of the
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Equation (46) depends on c only and
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B Minimum Anderson-Darling Estimato
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C Local exponent In this Appendix,
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D Testing non-nested hypotheses wit
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where α = (c ∗ ,d ∗ ). It can
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where E0[·] denotes the expectatio
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References Andersen, J.V. and D. So
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Gouriéroux C. and A. Monfort, 1994
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Rubinstein, M., 1973, The fundament
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(a) Independent Data Stretched-Expo
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(a) Dow Jones Positive Tail Negativ
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Nasdaq Dow Jones Pos. Tail Neg. Tai
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Dow Jones positive tail Dow Jones n
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Variation coefficient Variation coe
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Mean excess function Mean excess fu
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Hill‘s estimate b u Hill‘s esti
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Wilks statistic (doubled log−like
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Tail 1−F(x) and parameter b Tail
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1.4 1.2 1 0.8 0.6 0.4 0.2 0 1 2 3 4
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Chapitre 4 Relaxation de la volatil
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Volatility Fingerprints of Large Sh
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2 Long-range memory and distinction
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Figure 2: Measuring the conditional
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the system to the accumulation of m
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Appendix C: “Conditional response
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Baillie, R.T., 1996, Long memory pr
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Chapitre 5 Approche comportementale
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5.1. Prix d’un actif et excès de
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5.2. Modèles d’opinion contre mo
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5.4. Conséquences des phénomènes
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5.5. Conclusion 157 ce qui induit u
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Chapitre 6 Comportements mimétique
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RESEARCH PAPER Q UANTITATIVE F INAN
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A Corcos et al Q UANTITATIVE F INAN
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Deuxième partie Etude des proprié
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182 7. Etude de la dépendance à l
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192 8. Tests de copule gaussienne
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194 8. Tests de copule gaussienne 1
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200 8. Tests de copule gaussienne t
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202 8. Tests de copule gaussienne T
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204 8. Tests de copule gaussienne a
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206 8. Tests de copule gaussienne c
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208 8. Tests de copule gaussienne t
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210 8. Tests de copule gaussienne (
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212 8. Tests de copule gaussienne R
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228 8. Tests de copule gaussienne f
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236 8. Tests de copule gaussienne T
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238 9. Mesure de la dépendance ext
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10.1. La théorie de l’utilité 3
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10.2. Les mesures de risque cohére
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10.3. Les mesures de fluctuations 3
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10.4. Conclusion 361 et de manière
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Chapitre 11 Portefeuilles optimaux
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11.2. Prise en compte des grands ri
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11.3. Equilibre de marché 367 s’
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11.5. Annexe 369 Collective Origin
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11.5. Annexe 371 point λ = 1. Thus
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Chapitre 12 Gestion des risques gra
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12.1. Comprendre et gérer les risq
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12.2. Minimiser l’impact des gran
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Chapitre 13 Gestion de portefeuille
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VaR-Efficient Portfolios for a Clas
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dependence: from independence to co
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given a series of return {rt}t foll
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eads cV (u1, · · · , un) = 1
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THEOREM 4 (TAIL EQUIVALENCE FOR A S
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Independent Assets Comonotonic Asse
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3.2 Typical recurrence time of larg
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and finally w ∗ i = χ−1 i j
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where the ˆwi’s are solution of
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Choosing x > y > Aɛ, and applying
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Expanding fi(xi) around x ∗ i yie
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for all h ∈ AC. Thus, integrating
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B Asymptotic distribution of the su
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This yields × h+ Sc−2 − 2 dh
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References Acerbi, A. and D. Tasche
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Malevergne, Y. and D. Sornette, 200
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ln(T/T 0 ) T Figure 2: Logarithm
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Chapitre 14 Gestion de Portefeuille
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Multi-Moments Method for Portfolio
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choosen risk measure. Section 5 pre
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The variance of the return X of an
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This property is verified for all c
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µn=α of order n = 2, 4, 6 and 8.
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these two assets or portfolios. The
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Due to the homogeneity property of
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invested in the risky fund depends
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C. Then, if g1(X1), · · · , gn(X
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Differentiating with respect to x1,
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7.2 Transformation of the modified
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In the sequel, we will first evalua
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8.3 Non-symmetric assets In the cas
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y the variance will then increase).
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A Description of the data set We ha
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thus and finally 1 λ1 n−1 = w
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C Composition of the market portfol
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D Generalized capital asset princin
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This brings us back to the problem
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F.2 General case We now consider a
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Harvey, C.R. and A. Siddique, 2000,
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µ µ2 1/2 µ4 1/4 µ6 1/6 µ8 1/8
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Mean (10 −3 ) Variance (10 −3 )
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μ (daily return) x 10−3 2.5 2 1.
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w i w i 0.2 0.15 0.1 0.05 Mean−μ
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Figure 6: Schematic representation
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Empirical Cumulative Distribution 1
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Z 2 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0
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Z 2 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0
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10 0 10 −1 10 0 10 −1 10 −1 1
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10 0 10 −1 10 0 10 −1 10 −1 1
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κ 22 20 18 16 14 12 10 8 6 4 2 Exc
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Return μ 0.12 0.11 0.1 0.09 0.08 0
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Conclusions et Perspectives L’obj
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Conclusion 493 en univers gaussien
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Annexe A Evaluation de la conduite
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A.2. Eléments de contexte 497 bora
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A.4. Compétences acquises et ensei
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A.5. Conclusion 501 de la recherche
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Bibliographie ACERBI, C. (2002) :
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Bibliographie 505 BOUCHAUD, J. P. E
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Bibliographie 507 DE FINETTI, B. (1
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Bibliographie 509 GRANGER, C. W. ET
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Bibliographie 511 LILLO, F. ET R. N
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Bibliographie 513 PATTON, A. (2001)
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Bibliographie 515 SUSMEL, R. (1996)
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