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386 12. Gestion des risques grands et extrêmes returns on the index returns, which shows that there is significantly less linear correlation between P 1 and the index (correlation coefficient of 0.52 for both the equally weighted and the minimum variance P 1 ) compared with P 2 and the index (correlation coefficient of 0.73 for the equally weighted P 2 and of 0.70 for the minimum variance P 2 ). Theoretically, it is possible to construct two random variables with small correlation coefficient and large λ and vice versa. Recall that the correlation coefficient and the tail-dependence coefficient are two opposite end-members of dependence measures. The correlation coefficient quantifies the dependence between relatively small moves while the tail-dependence coefficient measures the dependence during extreme events. The finding that P 1 comes with both the smallest correlation and the smallest tail-dependence coefficients suggests that they are not independent properties of assets. This intuition is in fact explained and encompassed by the factor model since the larger β is, the larger the correlation coefficient and the larger the tail dependence. Diversifying away extreme shocks may provide a useful diversification tool for less extreme dependences, thus improving the potential usefulness of a strategy of portfolio management based on tail dependence proposed here. As a final remark, the almost identical values of the coefficients of tail dependence for negative and positive tails show that assets that are the most likely to suffer from the large losses of the market factor are also those that are the most likely to take advantage of its large gains. This has the following consequence: minimising the large concomitant losses between the stocks and the market means renouncing the potential concomitant large gains. This point is well exemplified by our two portfolios (see figure 3): P 2 obviously underwent severe negative co-movements but it also enjoyed large gains with the large positive movements of the index. In contrast, P 1 is almost completely decoupled from the large negative movements of the market but is also insensitive to the large positive movements of the index. Thus, a good dynamic strategy seems to be: invest in P 1 during bearish or trend-less market phases and prefer P 2 in a bullish market. ■ Yannick Malevergne is a PhD student at the University of Nice-Sophia Antipolis and at the ISFA Actuarial School – University of Lyon. Didier Bingham N, C Goldie and J Teugel, 1987 Regular variation Cambridge University Press, Cambridge Boyer B, M Gibson and M Lauretan, 1997 Pitfalls in tests for changes in correlations International Finance Discussion Paper 597, Board of the Governors of the Federal Reserve System Coles S, J Heffernan and J Tawn, 1999 Dependence measures for extreme value analysis Extremes 2, pages 339–365 Embrechts P, A McNeil and D Straumann, 1999 Correlation: pitfalls and alternatives Risk May, pages 69–71 Embrechts P, A McNeil and D Straumann, 2002 Correlation and dependence in risk management: properties and pitfalls In Risk Management: Value at Risk and Beyond, edited by M Dempster, pages 176–223, Cambridge University Press, Cambridge Hull J and A White, 1987 The option pricing on assets with stochastic volatilities Journal of Finance 42, pages 281–300 Hult H and F Lindskog, 2002 Multivariate extremes, aggregation and dependence in elliptical distributions Forthcoming in Advances in Applied Probability 34(3) Ledford A and J Tawn, 1998 Concomitant tail behaviour for extremes Advances in Applied Probability 30, pages 197–215 REFERENCES Appendix: empirical estimation of the coefficient of tail dependence We show how to estimate the coefficient of tail dependence between an asset X and the market factor Y related by the relation (6) where ε is an idiosyncratic noise uncorrelated with X. Given a sample of N realisations {X 1 , X 2 , ... , X N } and {Y 1 , Y 2 , ... , Y N } of X and Y, we first estimate the coefficient β using the ordinary least square estimator. Let β ^ denote its estimate. Then, using Hill’s estimator, we obtain the tail index ν^ of the factor Y: ⎡ k 1 ˆ = ⎢ ∑ logY −logY ⎣⎢ k j= 1 νk j, N k, N where Y 1, N ≥ Y 2, N ≥ ... ≥ Y N, N are the order statistics of the N realisations of Y. The constant l is non-parametrically estimated with the formula: FXu X l = lim ≃ u→1 −1 F u Y for k = o(N), which means that k must remain very small with respect to N but large enough to ensure an accurate determination of l. Figure 2 presents l ^ as a function of k/N. Finally, using equation (7), the estimated λ ^ is: ˆ max , ˆ + 1 λ = νˆ { 1 βˆ } l Sornette is a CNRS research director at the University of Nice-Sophia Antipolis and professor of geophysics at the University of California at Los Angeles. They acknowledge helpful discussions with Jean-Paul Laurent. This work was partially supported by the James S McDonnell Foundation twenty-first century scientist award/studying complex system. e-mail: Yannick.Malevergne@unice.fr and sornette@unice.fr Malevergne Y and D Sornette, 2002a Investigating extreme dependences: concepts and tools Working paper, available at http://papers.ssrn.com/sol3/papers.cfm?abstract_id = 303465 Malevergne Y and D Sornette, 2002b Tail dependence of factor models Working paper, available at http://papers.ssrn.com/sol3/papers.cfm?abstract_id = 301266 Mandelbrot B, 1997 Fractals and scaling in finance: discontinuity, concentration Springer-Verlag, New York Nelsen R, 1998 An introduction to copulas Lectures Notes in Statistics 139, Springer Verlag, New York Ross S, 1976 The arbitrage theory of capital asset pricing Journal of Economic Theory 17, pages 254–286 Rubinstein M, 1973 The fundamental theorem of parameter-preference security valuation Journal of Financial and Quantitative Analysis 8, pages 61–69 Sharpe W, 1964 Capital asset pricing: a theory of market equilibrium under conditions of risk Journal of Finance 19, pages 425–442 Taylor S, 1994 Modeling stochastic volatility Mathematical Finance 4, pages 183–204 −1 Y ( ) ( ) kN , kN , ⎤ ⎥ ⎦⎥ WWW.RISK.NET ● NOVEMBER 2002 RISK 133
Chapitre 13 Gestion de portefeuille sous contraintes de capital économique : l’exemple de la Value-at-Risk et de l’Expected-Shortfall A l’aide d’une famille de distributions de Weibull modifiées, englobant à la fois des distributions super et sous-exponentielles (dont l’interêt a été souligné au chapitre 3), nous paramétrons les distributions de rendements des actifs financiers. L’hypothèse de copule gaussienne est utilisée pour modéliser la dépendance entre les actifs. Cela nous permet d’obtenir les expressions analytiques des queues de la distribution P (S) du rendement S d’un portefeuille composé de tels actifs. Nous montrons que les queues de la distribution P (S) demeurent asymptotiquement des distributions de Weibull modifiées avec un facteur d’échelle χ fonction du poids des actifs dans le portefeuille et dont l’expression différe selon le comportement super ou sous-exponentiel des actifs. Nos traitons alors en détails le problème de la minimisation du risque pour de tels portefeuilles, les mesures de risque considérées étant la Value-at- Risk et l’Expected-Shortfall, que nous montrons être asymptotiquement équivalentes dans le cadre de la représentaion adoptée. 387
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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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228 8. Tests de copule gaussienne f
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238 9. Mesure de la dépendance ext
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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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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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