FIM702: lecture 10
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Transcript of FIM702: lecture 10
![Page 1: FIM702: lecture 10](https://reader038.fdocuments.us/reader038/viewer/2022110311/55aa1ea61a28abd17e8b48ff/html5/thumbnails/1.jpg)
Modelisation destrategies en
finance de marche
Alexander Surkov
Aspectsdynamiques
Rebalancement
Correlations
Modelisation de strategies en finance demarche
Seance 15 : Aspects dynamiques de la gestion deportefeuille
Alexander Surkov, CFA, FRM, PRM, [email protected]
Ecole de gestionUniversite de Sherbrooke
Le 26 avril 2017
![Page 2: FIM702: lecture 10](https://reader038.fdocuments.us/reader038/viewer/2022110311/55aa1ea61a28abd17e8b48ff/html5/thumbnails/2.jpg)
Modelisation destrategies en
finance de marche
Alexander Surkov
Aspectsdynamiques
Rebalancement
Correlations
Table de matiere
Aspects dynamiques de la gestion de portefeuilleStrategies de rebalancement du portefeuilleCorrelations en periode de tensions
![Page 3: FIM702: lecture 10](https://reader038.fdocuments.us/reader038/viewer/2022110311/55aa1ea61a28abd17e8b48ff/html5/thumbnails/3.jpg)
Modelisation destrategies en
finance de marche
Alexander Surkov
Aspectsdynamiques
Rebalancement
Correlations
Table de matiere
Aspects dynamiques de la gestion de portefeuilleStrategies de rebalancement du portefeuilleCorrelations en periode de tensions
![Page 4: FIM702: lecture 10](https://reader038.fdocuments.us/reader038/viewer/2022110311/55aa1ea61a28abd17e8b48ff/html5/thumbnails/4.jpg)
Modelisation destrategies en
finance de marche
Alexander Surkov
Aspectsdynamiques
Rebalancement
Correlations
Exemples de strategies de rebalancement
I Acheter et detenir (buy & hold)
I Composition constante (constant mix)
I Assurance de portefeuille (constant-proportion portfolioinsurance, CPPI)
![Page 5: FIM702: lecture 10](https://reader038.fdocuments.us/reader038/viewer/2022110311/55aa1ea61a28abd17e8b48ff/html5/thumbnails/5.jpg)
Modelisation destrategies en
finance de marche
Alexander Surkov
Aspectsdynamiques
Rebalancement
Correlations
Acheter et detenir
I Choisir la mixture, disons, � 60/40 �, ν = 0.6.
I Etant donne le montant a investir V0, disons,V0 = 100$, investir νV0 en actions, (1 − ν)V0 enobligations
Na =νV0
P(a)0
, Nb =(1 − ν)V0
P(b)0
I Detenir toujours le nombre Na d’actions et Nb
d’obligations.
![Page 6: FIM702: lecture 10](https://reader038.fdocuments.us/reader038/viewer/2022110311/55aa1ea61a28abd17e8b48ff/html5/thumbnails/6.jpg)
Modelisation destrategies en
finance de marche
Alexander Surkov
Aspectsdynamiques
Rebalancement
Correlations
Acheter et detenir
0 50 100 150 200 250 300 3500
50
100
150
200
Prix d’actions
Val
eur
du p
orte
feui
lle
![Page 7: FIM702: lecture 10](https://reader038.fdocuments.us/reader038/viewer/2022110311/55aa1ea61a28abd17e8b48ff/html5/thumbnails/7.jpg)
Modelisation destrategies en
finance de marche
Alexander Surkov
Aspectsdynamiques
Rebalancement
Correlations
Composition constante
I Choisir la mixture, disons, � 60/40 �, ν = 0.6.
I Etant donne le montant a investir V0, disons,V0 = 100$, investir νV0 en actions, (1 − ν)V0 enobligations
Na =νV0
P(a)0
, Nb =(1 − ν)V0
P(b)0
I Rebalancer le portefeuille si
I le ratio ν′ ≡ NaP(a)t /Vt depasse les limites preetablies
ν − α ≤ ν′ ≤ ν + α.I ou bien, le prix d’actions depasse les limites preetablies,
1 − α ≤ P(a)t /P
(a)0 ≤ 1 + α.
N ′a =
νVt
P(a)t
, N ′b =
(1 − ν)Vt
P(b)t
![Page 8: FIM702: lecture 10](https://reader038.fdocuments.us/reader038/viewer/2022110311/55aa1ea61a28abd17e8b48ff/html5/thumbnails/8.jpg)
Modelisation destrategies en
finance de marche
Alexander Surkov
Aspectsdynamiques
Rebalancement
Correlations
Strategie concave : composition constante
0 50 100 150 200 250 300 3500
50
100
150
200
Prix d’actions
Val
eur
du p
orte
feui
lle
![Page 9: FIM702: lecture 10](https://reader038.fdocuments.us/reader038/viewer/2022110311/55aa1ea61a28abd17e8b48ff/html5/thumbnails/9.jpg)
Modelisation destrategies en
finance de marche
Alexander Surkov
Aspectsdynamiques
Rebalancement
Correlations
Assurance de portefeuille (CPPI)
I Choisir le multiplicateur m et la borne inferieure F
NaP(a)0 = m(V0 − F ), NbP
(b)0 = V0 − NaP
(a)0
Disons, m = 2, V0 = 100$, F = 70$, ce qui donne
NaP(a)0 /V0 = 0.6.
I Rebalancer le portefeuille si le prix d’actions depasse les
limites preetablies, 1 − α ≤ P(a)t /P
(a)0 ≤ 1 + α.
N ′a =
m(V0 − F )
P(a)t
, N ′b =
Vt − N ′aP
(a)t
P(b)t
![Page 10: FIM702: lecture 10](https://reader038.fdocuments.us/reader038/viewer/2022110311/55aa1ea61a28abd17e8b48ff/html5/thumbnails/10.jpg)
Modelisation destrategies en
finance de marche
Alexander Surkov
Aspectsdynamiques
Rebalancement
Correlations
Strategie convexe : CPPI
0 50 100 150 200 250 300 3500
50
100
150
200
250
300
350
Prix d’actions
Val
eur
du p
orte
feui
lle
![Page 11: FIM702: lecture 10](https://reader038.fdocuments.us/reader038/viewer/2022110311/55aa1ea61a28abd17e8b48ff/html5/thumbnails/11.jpg)
Modelisation destrategies en
finance de marche
Alexander Surkov
Aspectsdynamiques
Rebalancement
Correlations
Les strategies en Matlab : parametres
Nsim = 500; T = 252;
mu = 0.15 / T; sigma = 0.3 / sqrt( T );
Rf = 0.02 / T;
V0 = 100; nu = 0.6;
Pa0 = 100; Pa = ones(2, Nsim) * Pa0;
Pb0 = 100; Pb = Pb0;
Na0 = nu * V0 / Pa0;
Nb0 = ( 1 - nu ) * V0 / Pb0;
Na = ones(1,Nsim) * Na0; Nb = ones(1,Nsim) * Nb0;
alpha =0.1; m = 2; F = 70;
![Page 12: FIM702: lecture 10](https://reader038.fdocuments.us/reader038/viewer/2022110311/55aa1ea61a28abd17e8b48ff/html5/thumbnails/12.jpg)
Modelisation destrategies en
finance de marche
Alexander Surkov
Aspectsdynamiques
Rebalancement
Correlations
Les strategies en Matlab : simulation
for t = 2:(T+1)
Pa(1,:) = Pa(1,:).*exp(normrnd(mu,sigma,1,Nsim));
Pb = Pb * exp( Rf );
V1 = Na0 .* Pa(1, :) + Nb0 * Pb;
V2 = Na .* Pa(1, :) + Nb * Pb;
r = Pa(1, :) ./ Pa(2, :);
idx1 = r > 1 + alpha; idx2 = r < 1 - alpha;
idx = or(idx1, idx2);
if any(idx)
Na(idx) = nu * V2(idx) ./ Pa(1, idx);
% Na(idx) = m * ( V2(idx) - F ) ./ Pa(1, idx);
Nb(idx) = (V2(idx) - Na(idx).*Pa(1, idx))/Pb;
Pa(2, idx) = Pa(1, idx);
end
end
![Page 13: FIM702: lecture 10](https://reader038.fdocuments.us/reader038/viewer/2022110311/55aa1ea61a28abd17e8b48ff/html5/thumbnails/13.jpg)
Modelisation destrategies en
finance de marche
Alexander Surkov
Aspectsdynamiques
Rebalancement
Correlations
Table de matiere
Aspects dynamiques de la gestion de portefeuilleStrategies de rebalancement du portefeuilleCorrelations en periode de tensions
![Page 14: FIM702: lecture 10](https://reader038.fdocuments.us/reader038/viewer/2022110311/55aa1ea61a28abd17e8b48ff/html5/thumbnails/14.jpg)
Modelisation destrategies en
finance de marche
Alexander Surkov
Aspectsdynamiques
Rebalancement
Correlations
Correlations en periode de tensions
I Des observations montrent que les correlationss’augmentent en periode de tensions (voir, par exemple,l’article � Quantifying the Behavior of StockCorrelations Under Market Stress �).
I Cependant, la croissance de correlations peut etrecontribuee par le biais lie a l’observation des correlationsen periode d’une forte volatilite.
I Simulation 1 : 5000 simulation de T = 62 pairesd’observations aleatoires correlees, la correlation estimeevs. les volatilites estimees.
I Simulation 2 : 500 simulations de N = 2520 pairesd’observations aleatoires correlees, une fenetre roulantede T = 62, la correlation estimee correspondante a laperiode des volatilites maximales.
![Page 15: FIM702: lecture 10](https://reader038.fdocuments.us/reader038/viewer/2022110311/55aa1ea61a28abd17e8b48ff/html5/thumbnails/15.jpg)
Modelisation destrategies en
finance de marche
Alexander Surkov
Aspectsdynamiques
Rebalancement
Correlations
Simulation 1 : ρ = 0.1
0.7 0.8 0.9 1 1.1 1.2 1.3−0.4
−0.2
0
0.2
0.4
√
σ1σ2
ρ12
![Page 16: FIM702: lecture 10](https://reader038.fdocuments.us/reader038/viewer/2022110311/55aa1ea61a28abd17e8b48ff/html5/thumbnails/16.jpg)
Modelisation destrategies en
finance de marche
Alexander Surkov
Aspectsdynamiques
Rebalancement
Correlations
Simulation 1 : ρ = 0.5
0.7 0.8 0.9 1 1.1 1.2 1.30
0.2
0.4
0.6
0.8
√
σ1σ2
ρ12
![Page 17: FIM702: lecture 10](https://reader038.fdocuments.us/reader038/viewer/2022110311/55aa1ea61a28abd17e8b48ff/html5/thumbnails/17.jpg)
Modelisation destrategies en
finance de marche
Alexander Surkov
Aspectsdynamiques
Rebalancement
Correlations
Simulation 1 : ρ = 0.9
0.7 0.8 0.9 1 1.1 1.2 1.30.75
0.8
0.85
0.9
0.95
1
√
σ1σ2
ρ12
![Page 18: FIM702: lecture 10](https://reader038.fdocuments.us/reader038/viewer/2022110311/55aa1ea61a28abd17e8b48ff/html5/thumbnails/18.jpg)
Modelisation destrategies en
finance de marche
Alexander Surkov
Aspectsdynamiques
Rebalancement
Correlations
Simulation 1 en Matlab
T = 62; Nsim = 5000;
s = [1 1]; c = 0.5;
sigma = diag(s) * [1 c; c 1] * diag(s);
vols = NaN(Nsim,1);
cors = NaN(Nsim,1);
for j = 1:Nsim
x = mvnrnd( zeros(2,1), sigma, T );
covs = cov( x );
vols(j) = sqrt( prod( diag( covs ) ) );
cors(j) = covs(1,2) / vols(j);
end
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Modelisation destrategies en
finance de marche
Alexander Surkov
Aspectsdynamiques
Rebalancement
Correlations
Simulation 2
0.4 0.5 0.6 0.7 0.80
20
40
60
80
100
120
ρ12
N. d
’obs
.
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Modelisation destrategies en
finance de marche
Alexander Surkov
Aspectsdynamiques
Rebalancement
Correlations
Simulation 2 : periode aleatoire
0.2 0.4 0.6 0.80
20
40
60
80
100
120
ρ12
N. d
’obs
.
![Page 21: FIM702: lecture 10](https://reader038.fdocuments.us/reader038/viewer/2022110311/55aa1ea61a28abd17e8b48ff/html5/thumbnails/21.jpg)
Modelisation destrategies en
finance de marche
Alexander Surkov
Aspectsdynamiques
Rebalancement
Correlations
Simulation 2 en Matlab
Nsim = 500; T = 62; N = 2520;
cors = NaN(Nsim,1);
for i = 1:Nsim
volm = 0;
x = mvnrnd(zeros(2,1), sigma, N);
for j = 1:(N-T+1)
covs = cov( x(j:(T+j-1),:) );
vol = sqrt( prod( diag(covs) ) );
if vol > volm
volm = vol; corm = covs(1,2) / vol;
end
end
cors(i) = corm;
end