Modèles à Facteurs
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Daniel HERLEMONT
Plan
Limites de Markowitz et du CAPM
Modèle à 1 facteur
CAPM comme modèle à 1 facteur
Modèles multi-facteurs
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Objectifs
Etudier la construction des modèles à un facteur et modèles à facteurs mutliples de la variance et des rendements espérés.
Comprendre pourquoi les modèles à facteurs peuvent être d'un intérêt théorique et pratique
Comprendre mieux le rôle de la covariance des rendements dans la diversification du risque. 3
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Limites de Markowitz et du CAPM Paramètres du modèle non observables: Espérance des rendements, variances-covariances Portefeuille de marché = tous les actifs
Modèle sensible aux erreurs (manque de robustesse) Problèmes d’estimation (à partir des historiques de cours) Rendements estimés dépendent longueur échantillon Estimation volatilité historique peu fiables pour le futur
Volume des calculs n(n -1)/2 covariances + 2n variances & rendements Un estimé avec peu d'observations aura trop peu de degré de liberte pour être précis, un estimé avec de nombreuses observations est sujet à des changements structurels à l'intérieur de l'échantillon. 4
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Simplification : modèle à 1 facteur
Expliquer l’incertitude sur les rendements par un « facteur » f (v.a. représentative de l’économie)
ri = ai + bi f+ εi avec Alors
E{εi} = 0
cov(εi,εj) = 0
cov(εi,f) = 0
σi 2 = var(ri) = bi2 var(f) + var(εi) cov(ri,f) = bi cov(f, f) = bi2 var(f) σi,j = cov(ri,rj) = bi bj var(f)
On calcule les O(n2) paramètres avec 3n+2 termes
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Réduction du risque spécifique par diversification
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Diversification
source: B. Jacquillat & B. Solnik, Marchés Financiers, 12
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2
2
OT = OM + OS
2
σ 2 = β 2σ M2 + σ ε2
Volatilité spécifique
Pythagore retrouvé T
I O
10%
20%
30%
M 40%
50%
Volatilité liée au marché OT=volatilité du titre OM=volatilité du titre liée au marché OS=volatilité spécifique du titre (diversifiable) OI = volatilité du marché β = ΟΜ/ΟΙ
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forte volatilité, fortement corrélé au marche
faible volatilité, faiblement corrélé au marche
forte volatilité, faiblement corrélé au marche
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Exemple CAC40 - Peugeot
source: B. Jacquillat & B. Solnik, Marchés Financiers, Daniel HERLEMONT
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Characteristic Line This line represents a single-factor model that has rM–rf as the factor for the variable ri–rf
r i -r f
10%
αi
12%
r M -r f
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Comments The characteristic line is in one sense more general than the CAPM, as it allows αi ≠ 0. From the CAPM viewpoint, we regard αi as the amount by which asset i is mispriced: • αi < 0 means asset i is underpriced • αi > 0 means asset i is overpriced Some financial service organizations estimate α and β for a large assortment of stocks.
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Example 8.2. Four stocks and the market. This is a reworking of Example 8.1 with a change in the factor. We use the excess market return as the factor. We assume the market consists of just the four stocks, with equal weights. Thus, the market return in any year is the average of the returns of the four stocks. Also, we are given risk-free rates for each year. Then we use the same estimating formulas as before, except that the factor is the excess return for the market. Thus, we use αi and βi instead of ai and bi respectively. Note the error variances are relatively small compared to the return variances. Extra calculations show error correlations are almost zero. This particular model works better than the first one.
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Luenberger's Table 8.2: a Factor Model with the Market Stk. 1 Stk. 2 Stk. 3 Stk. 4 Market Riskless Year 11.91 29.59 23.27 27.24 23.00 6.2 1 18.37 15.25 19.47 17.05 17.54 6.7 2 3.64 3.53 -6.58 10.2 2.70 6.4 3 24.37 17.67 15.08 20.26 19.35 5.7 4 30.42 12.74 16.24 19.84 19.81 5.9 5 -1.45 -2.56 -15.05 1.51 -4.39 5.2 6 20.11 25.46 17.8 12.24 18.90 4.9 7 9.28 6.92 18.82 16.12 12.79 5.5 8 17.63 9.73 3.05 22.93 13.34 6.1 9 15.71 25.09 16.94 3.49 15.31 5.8 10 avg. var. stdev corel w M cov. w. M beta alpha e-var.
15.00 90.26 9.50 0.81 65.09 0.90 1.94 31.52
14.34 107.23 10.35 0.84 73.62 1.02 0.34 32.07
10.90 162.20 12.74 0.93 100.79 1.40 -6.11 21.36
15.09 68.25 8.26 0.70 48.99 0.68 3.82 34.98
13.83 72.12 8.49 1 72.12
5.84
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Beta
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Le CAPM comme modèle à facteur Modèle à facteur
Le CAPM (ou MEDAF) impose des restrictions sur alpha, peut être vu comme un cas particulier du modèle à facteur
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Le CAPM comme modèle à facteur
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Security Market Line
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=SLOPE(E187:E246;$D187:$D246)
tendance des betas à revenir vers la moyenne c'est à dire 1 Blume 1975
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Modèles multi-facteurs
Objectifs: + de réalisme, erreurs εi + petites utilisation de 3 à 15 facteurs Sélection des facteurs: Facteurs externes (macroéconomiques): PIB, CPI (indice des prix), taux de chômage...
Facteurs extraits: analyse en composantes principales de Q Caractéristiques de l’entreprise PER, dividendes, prévisions de bénéfices, credit rating…
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Multifactor Models (brief overview) Multifactor models are more realistic and can give better results – less error - than single factor models. They also require more computational effort. Suppose there are two factors, f1 and f2. The return model for asset i can then be written as ri = ai + b1i f1 + b2i f2 + ei ai: the intercept b1i, b2i: factor loadings ei: the error
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Error Assumptions • expected value of each error is 0 • error is uncorrelated with the two factors • error is uncorrelated with the other assets Note. The two factors may be correlated with each other.
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Some results E{ri} = ai + b1i E{f1} + b2i E{f2} var{ri} = b1i2 var {f1) + 2 b1i b2i cov{f1,f2} + b2i2 var {f2} + var {ei} cov{ri,rj} = b1i b1j var {f1} + (b1i b2j + b2i b1j) cov{f1,f2} + b2i b2j var {f2} , for i ≠ j We can compute the b1i’s and b2i’s by solving the following 2 by 2 linear system (system obtained by forming cov{ri,f1} and cov{ri,f2} ): b1i var {f1} + b2i cov{f1,f2} = cov{ri,f1} b1i cov{f1,f2} + b2i var {f2} = cov{ri,f2} 30
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Facteurs Macro Economiques - Roll-Ross 1994
Confidence Factor différence des taux (spread) entre obligation corporate et obligations d'état
Time Horizon Factor spread entre les taux d'obligations d'état à 20 ans et taux à 3 mois
Inflation Factor Business Cycle Factor
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Facteurs/attributs sociétés - Exemple BARRA - Grinold
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Modèle à facteurs - exercice
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Modèle à facteurs - exercice
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Modèle à facteurs - exercice
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Modèle à facteurs - exercice
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Modèle à facteurs - exercice
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Modèle à facteurs - exercice
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Modèle à facteurs - exercice
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Modèle à facteurs - exercice
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Modèle à facteurs - exercice
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Modèle à facteurs - exercice
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Modèle à facteurs - exercice
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Modèle à facteurs - exercice
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Modèle à facteurs - exercice
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Mean Blur Example 8.3. Monthly Rates of Return an Estimation of Mean Years 4 5
mo. 1 2 3 4 5 6 7 8 9 10 11 12
1
2
3
-8.65 8.61 5.50 2.04 7.51 -2.50 2.28 1.85 5.86 1.37 3.17 9.23
2.61 -2.38 -3.28 7.45 7.96 -9.37 -7.27 -5.30 5.69 5.24 2.94 1.94
6.39 -1.22 1.12 3.69 .28 3.61 -1.45 6.83 2.32 -3.79 -.52 2.77
r^ σ
3.02 0.52 1.67 0.01 1.76 2.06 1.37 0.17 5.01 5.88 3.21 3.81 2.98 3.24 4.66 3.55
-4.52 2.30 -3.96 -.84 .35 6.96 4.23 .21 .14 -6.48 -1.11 2.86
6
1.28 4.49 .14 7.58 -2.63 5.02 3.15 -.51 -.47 -.19 7.04 1.18 3.68 1.61 2.74 2.62 -2.08 -2.32 1.73 -3.08 6.18 5.42 .38 2.93
7
8
-1.44 -4.34 1.24 8.92 -.46 8.28 5.33 -1.01 3.77 4.18 -2.27 4.91
3.30 3.75 3.95 -3.13 -0.31 -0.89 -6.39 -0.60 -0.76 1.92 -3.97 5.18
Overall
1.32 4.12
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Mean Blur The Table 8.3 reports monthly returns simulated from iid normal random variables with E{r} = 1 % and Var{r} = 4.33%. Note • how much the r^ values vary from year to year • the overall E{r} estimate is 33% high • the standard deviation estimates vary less from year to year • the overall standard deviation estimate is not bad. Refer to the histogram of monthly returns, Figure 8.4. "It is impossible to determine an accurate estimate of the true mean from the samples." 47
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Histogram of monthly returns
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Mean Blur
"This is the historical blur problem for the measurement of E{r}. It is basically impossible to measure E{r} to within workable accuracy using historical data. Furthermore, the problem cannot be improved much by changing the period length. If longer periods are used, each sample is more reliable, but fewer independent samples are obtained in any year. Conversely, if smaller periods are used, more samples are available, but each is worse in terms of the ratio of standard deviation to mean value. The problem of mean blur is a fundamental difficulty."
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Blur for Factor Models "The blur phenomenon applies to the parameters of a factor model, but mainly to the determination of a. In fact the presence of a-blur can be deduced from the mean-blur phenomenon, but we omit the details. The inherently poor accuracy of α estimates is reflected in the so-called Beta Book, published by Merrill Lynch (example in Table 8.4) .... the reported standard deviation for α is typically larger than the value of α itself. The related error in estimating β is somewhat better."
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Tilting Away from Equilibrium Mean-variance theory suggests that the efficient fund of risky assets would be the market portfolio. Many investors are not satisfied with this conclusion and consider that a superior solution can be computed by solving Markowitz problem directly. Historical data may not be enough to solve the Markowitz problem. Compromise solution: combine CAPM with an additional information
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Equilibrium Means Rates of return implied by CAPM E{rie} = rf + βi (E{rM} – rf) βI can be estimated from data, and E{rM} can be estimated using consensus (expert) opinions CAPM rates of return may differ from true rate E{ri} = E{rie} + ei , where ei has zero mean. Historical rates of return also differ from true rates E{ri} = E{rih} + ei
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Example: Double use of data (Ex. 8.2)
Average rates of return implied by CAPM and historical rates are not equal. Both estimates have errors, but they can be combined to form new estimates, called tilt.
avg. var. cov. w. M beta CAPM tilt
Stk. 1 15.00 90.26 65.09 0.90 13.05 13.82
Stk. 2 14.34 107.23 73.62 1.02 14.00 14.14
Stk. 3 10.90 162.20 100.79 1.40 17.01 14.17
Stk. 4 15.09 68.25 48.99 0.68 11.27 12.57
Market 13.83 72.12 72.12
Riskless 5.84
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For example, for stock 1 rate of return implied by CAPM E{r1e} = rf + βi (E{rM} – rf) = 5.84 + .9(13.83-5.84) = 13.05 To form a new, combined, estimates we calculate the variance for each estimate (errors in the CAPM model are ignored except error in E{rM} ) σih = σi /100.5 , σie = βi σM / 100.5 Tilts E{ri} =[E{rie}/(σ σie)2+ E{rih}/(σ σih)2 ]/ [1/(σ σie)2+ 1/(σ σih)2] =13.82 54
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