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458 14. Gestion de Portefeuilles multimoments et équilibre de marché where Sn denotes the set of agents, among all the agents, who follow the optimization stragtegy with respect to ρα(n). Thus, the total demand of asset i is Di = N φn · Di(n), (124) n = N φn · ˜wi(n) W (p) · (1 − w0(p)), (125) n where N is the total number of agents. This finally yields the total demand D for all assets and for all agents D = Di, (126) i p∈Sn = N φn · ˜wi(n) W (p) · (1 − w0(p)), (127) i = N n n φn p∈Sn W (p) · (1 − w0(p)), (128) p∈Sn since i ˜wi(n) = 1, for every n. Thus, setting γn = φn p∈Sn n φn p∈Sn W (p) · (1 − w0(p)) , (129) W (p) · (1 − w0(p)) the market portfolio is the weighted sum of the mean-ρα(n) optimal portfolios Πn: w m i = Si S = Di D = γn · ˜wi(n) . (130) 34 n
D Generalized capital asset princing model Our proof of the generalized capital asset princing model is similar to the usual demontration of the CAPM. Let us consider an efficient portfolio P. It necessarily satisfies equation (105) in appendix B : 459 ∂ρα (w ∂wi ∗ 1, · · · , w ∗ N) = λ1 (µ(i) − µ0), i ∈ {1, · · · , N}. (131) Let us now choose any portfolio R made only of risky assets and let us denote by wi(R) its weights. We can thus write N i=1 wi(R) · ∂ρα (w ∂wi ∗ 1, · · · , w ∗ N) = λ1 N wi(R) · (µ(i) − µ0), (132) i=1 = λ1 (µR − µ0). (133) We can apply this last relation to the market portfolio Π, because it is only composed of risky assets (as proved in appendix B). This leads to wi(R) = w ∗ i and µR = µΠ, so that N i=1 which, by the homogeneity of the risk measures ρα, yields Substituting equation (131) into (135) allows us to obtain where w ∗ i · ∂ρα (w ∂wi ∗ 1, · · · , w ∗ N) = λ1 (µΠ − µ0), (134) α · ρα(w ∗ 1, · · · , w ∗ N) = λ1 (µΠ − µ0) . (135) µj − µ0 = β j α · (µΠ − µ0), (136) β j α = ∂ 1 ln ρα α ∂wj , (137) calculated at the point {w∗ 1 , · · · , w∗ N }. Expression (135) with (137) provides our CAPM, generalized with respect to the risk measures ρα. In the case where ρα denotes the variance, the second-order centered moment is equal to the second-order cumulant and reads Since we find C2 = w ∗ 1 · Var[X1] + 2w ∗ 1w ∗ 2 · Cov(X1, X2) + w ∗ 2 · Var[X2], (138) = Var[Π] . (139) 1 ∂C2 · 2 ∂w1 = w ∗ 1 · Var[X1] + w ∗ 2 · Cov(X1, X2) , (140) = Cov(X1, Π), (141) β = Cov(X1, XΠ) Var[XΠ] which is the standard result of the CAPM derived from the mean-variance theory. 35 , (142)
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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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Chapitre 10 La mesure du risque Dan
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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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Page 453 and 454: A Description of the data set We ha
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Page 473 and 474: w i w i 0.2 0.15 0.1 0.05 Mean−μ
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Page 477 and 478: Empirical Cumulative Distribution 1
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Page 489 and 490: Return μ 0.12 0.11 0.1 0.09 0.08 0
Page 491 and 492: Conclusions et Perspectives L’obj
Page 493 and 494: Conclusion 493 en univers gaussien
Page 495 and 496: Annexe A Evaluation de la conduite
Page 497 and 498: A.2. Eléments de contexte 497 bora
Page 499 and 500: A.4. Compétences acquises et ensei
Page 501 and 502: A.5. Conclusion 501 de la recherche
Page 503 and 504: Bibliographie ACERBI, C. (2002) :
Page 505 and 506: Bibliographie 505 BOUCHAUD, J. P. E
Page 507 and 508: 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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