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456 14. Gestion de Portefeuilles multimoments et équilibre de marché Since ρα(w1, · · · , wN) is a homogeneous function of order α, its first-order derivative with respect to wi is also a homogeneous function of order α − 1. Using this homogeneity property allows us to write ∂ρα ∂wi −1 ∂ρα λ1 (w ∂wi ∗ 1, · · · , w ∗ N) = (µ(i) − µ0), i ∈ {1, · · · , N}, (106) 1 − λ1 α−1 w ∗ 1 − 1, · · · , λ1 α−1 w ∗ N = (µ(i) − µ0), i ∈ {1, · · · , N} . (107) Denoting by { ˆw1, · · · , ˆwN} the solution of this shows that the optimal weights are ∂ρα ( ˆw1, · · · , ˆwN) = (µ(i) − µ0), i ∈ {1, · · · , N}, (108) ∂wi w ∗ i = λ1 1 α−1 ˆwi. (109) Now, performing the same calculation as in the case of independent risky assets, the efficient portfolio P can be realized by investing a weight w0 of the initial wealth in the risk-free asset and the weight (1 − w0) in the risky fund Π, whose weights are given by ˜wi = Therefore, the expected return of every efficient portfolio is ˆwi N i=1 ˆwi . (110) µ = w0 · µ0 + (1 − w0) · µΠ, (111) where µΠ denotes the expected return of the market portfolio Π, while the risk, measured by ρα is so that ρα = (1 − w0) α ρα(Π) , (112) µ = µ0 + µΠ − µ0 ρα(Π) 1/α ρα 1/α . (113) This expression is the natural generalization of the relation obtained by (Markovitz 1959) for mean-variance efficient portfolios. 32
C Composition of the market portfolio In this appendix, we derive the relationship between the composition of the market portfolio and the composition of the optimal portfolio Π obtained by the minimization of the risks measured by ρα(n). C.1 Homogeneous case We first consider a homogeneous market, peopled with agents choosing their optimal portfolio with respect to the same risk measure ρα. A given agent p invests a fraction w0(p) of his wealth W (p) in the risk-free asset and a fraction 1 − w0(p) in the optimal portfolio Π. Therefore, the total demand Di of asset i is the sum of the demand Di(p) over all agents p in asset i: Di = Di(p) , (114) p 457 = W (p) · (1 − w0(p)) · ˜wi , (115) p = ˜wi · W (p) · (1 − w0(p)) , (116) p where the ˜wi’s are given by (110). The aggregated demand D over all assets is D = Di, (117) i = ˜wi · W (p) · (1 − w0(p)), (118) i p = W (p) · (1 − w0(p)). (119) p By definition, the weight of asset i, denoted by wm i , in the market portfolio equals the ratio of its capitalization (the supply Si of asset i) over the total capitalization of the market S = Si. At the equilibrium, demand equals supply, so that w m i = Si Di = S D = ˜wi. (121) Thus, at the equilibrium, the optimal portfolio Π is the market portfolio. C.2 Heterogeneous case (120) We now consider a heterogenous market, defined such that the agents choose their optimal portfolio with respect to different risk measures. Some of them choose the usual mean-variance optimal portfolios, others prefer any mean-ρα efficient portfolio, and so on. Let us denote by Πn the mean-ρα(n) optimal portfolio made only of risky assets. Let φn be the fraction of agents who choose the mean-ρα(n) efficient portfolios. By normalization, n φn = 1. The demand Di(n) of asset i from the agents optimizing with respect to ρα(n) is Di(n) = W (p) · (1 − w0(p)) · ˜wi(n), (122) p∈Sn = ˜wi(n) W (p) · (1 − w0(p)), (123) p∈Sn 33
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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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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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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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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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228 8. Tests de copule gaussienne f
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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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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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Chapitre 12 Gestion des risques gra
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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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and finally w ∗ i = χ−1 i j
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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 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
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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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