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98 3. Distributions exponentielles étirées contre distributions régulièrement variables so that the binding function can be expressed as b † (c,d) = c · e −( u d ) c −1 u c Γ 0, , when c > 0 (89) d In the case when c goes to zero and c and d are related by equation (47), we can still calculate b † . Indeed, we can first show that u c ∼ d 1 1 c T T ln xi −1 = u 1 c · ˆb. (90) Now, using the asymptoptic relation we conclude that ∑ i=1 e x · Γ(0,x) ∼ x −1 , as x → ∞, (91) b † (ĉ, ˆ d) = ˆb, as ĉ → 0. (92) This result is in fact natural because the PD model can be seen formally as the limit of the SE model for c → 0 under the condition as previously exposed in section 4.1 u c c · → β, d as c → 0 , (93) Now, following Mizon and Richard (1986), the model (SE) with pdf f1 is said to encompass the model (PD) with pdf f2 if the best representative of (PD) -with parameter b ∗ - is also the distribution nearest to the best representative of (SE) -with parameters (c ∗ ,d ∗ ). Thus, (SE) is said to encompass (PD) if and only if b ∗ = b † (c ∗ ,d ∗ ). The reverse situation can be considered in order to study the encompassing of the model (SE) by the model (PD). Such situation occurs if and only if c ∗ D.1.3 Wald encompassing test d ∗ c † (b∗ ) = d † (b∗ ) . (94) We first test the encompassing of (PD) into (SE), namely the null hypothesis H0 = {b ∗ = b † (c ∗ ,d ∗ )}. Under this null hypothesis, it can be shown (Gouriéroux and Monfort 1994) that the random variable √ T ˆb − b † (ĉ, ˆ d) is asymptotically normally distributed with zero mean and variance V given by V = K −1 C12]K −1 22 [C22 −C21C −1 11 + K −1 −1 22 [C21C11 − ˜C21K −1 11 22 + ]C11[C −1 11 C12 − K −1 11 ˜C12]K −1 22 where the expression of the coefficients involved in V will be given below. Thus, under H0, the random variable ξT = T ˆb − b † (ĉ, d) ˆ ˆV −1 ˆb − b † (ĉ, d) ˆ , where ˆV −1 is a consistent estimator of V −1 , follows asymptotically a χ2-distribution with one degrees of freedom. The matrix K11 is given by K11(i, j) = −E0 ∂ 2 ln f1(x|c ∗ ,d ∗ ) ∂αi∂α j 34 (95) , i, j = 1,2, (96)
where α = (c ∗ ,d ∗ ). It can be consistently estimated by The coefficient K22 is given by ˆK11(1,1) = 1 ĉ u u − ln2 + ĉ2 dˆ dˆ 1 T T ∑ ln i=1 2 ĉ xi xi , (97) dˆ dˆ ˆK11(1,2) = ˆK11(2,1) = c ĉ u u ln − d dˆ dˆ 1 T ĉ xi xi T ∑ ln , (98) ˆ i=1 d dˆ 2 ˆK11(2,2) ĉ = . (99) dˆ K22 = −E0 which is consistently estimated by ˆK22 = ˆb −2 . ∂ 2 ln f (x|b ∗ ) ∂b ∗2 99 = 1 , (100) b∗2 Now, we have to calculate the two components of the vector ˜C12 = ˜C t 21 . Its first component is ˜C12(1) ∂ln f1(x|c = E1 ∗ ,d ∗ ) ∂c∗ · ∂ln f2(x|b∗ ) ∂b∗ , (101) 1 u = E1 + ln c∗ d∗ u d∗ c∗ + ln x x − ln d∗ d∗ x d∗ c∗ 1 u · + ln b∗ d∗ − ln x d∗ ,(102) 1 u = + ln c∗ d∗ u d∗ c∗ 1 u 1 u · + ln − + ln b∗ d c∗ d∗ u d∗ c∗ · E1 ln x d∗ 1 u + + ln b∗ d∗ · E1 ln x d∗ − E1 ln x d∗ x d∗ c∗ 2 x − E1 ln d∗ 2 x + E1 ln d∗ x d∗ c∗ (103) . Some simple calculations show that E1 E1 ln x d · x c d 2 x ln d · x c d = 1 u + ln c d · u c + E1 ln d x , (104) d 2 u = ln d · u c + d 2 c E1 ln x 2 x + E1 ln , (105) d d which allows us to show that the first and third terms cancel out, and it remains ˜C12(1) = 1 E1 ln c∗ x d∗ − ln u d∗ u d∗ c∗ E1 ln x d∗ − ln u d∗ . (106) The second component is ˜C12(2) = ∂ln f1(x|c E1 ∗ ,d ∗ ) ∂d∗ · ∂ln f2(x|b∗ ) ∂b∗ = , E1 − (107) c∗ d∗ u 1 + d∗ c∗ x − d∗ c∗ 1 u · + ln b∗ d∗ − ln x d∗ = , − (108) c∗ d∗ u 1 + d∗ c∗ 1 u + ln b∗ d∗ u − 1 + d∗ c∗ E1 ln x d∗ − 1 u + ln b∗ d∗ x E1 d∗ c∗ + E1 ln x d∗ x d∗ c∗ . (109) 35
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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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2.2. Des bulles rationnelles aux kr
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Page 85 and 86: 5 Comparison of the descriptive pow
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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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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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