Factor Models

Lecture 15: Factor Models MIT 18.S096 Dr. Kempthorne

Fall 2013

MIT 18.S096

Factor Models

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Factor Models

Linear Factor Model Macroeconomic Factor Models Fundamental Factor Models Statistical Factor Models: Factor Analysis Principal Components Analysis Statistical Factor Models: Principal Factor Method

Outline

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Factor Models Linear Factor Model Macroeconomic Factor Models Fundamental Factor Models Statistical Factor Models: Factor Analysis Principal Components Analysis Statistical Factor Models: Principal Factor Method

MIT 18.S096

Factor Models

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Factor Models

Linear Factor Model Macroeconomic Factor Models Fundamental Factor Models Statistical Factor Models: Factor Analysis Principal Components Analysis Statistical Factor Models: Principal Factor Method

Linear Factor Model Data: m assets/instruments/indexes: i = 1, 2, . . . , m n time periods: t = 1, 2, . . . , n m-variate random vector for each time period: xt = (x1,t , x2,t , . . . , xm,t )0 E.g., returns on m stocks/futures/currencies; interest-rate yields on m US Treasury instruments.

Factor Model xi,t = αi + β1,i f1,t + β2,i f2,t + · · · + βk,i fk,t + i,t = αi + β 0i f t + i,t where αi : intercept of asset i f t = (f1,t , f2,t , . . . , fK ,t )0 : common factor variables at period t (constant over i) β i = (β1,i , . . . , βK ,i )0 : factor loadings of asset i (constant over t) i,t : the specific factor of asset i at period t. MIT 18.S096

Factor Models

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Factor Models

Linear Factor Model Macroeconomic Factor Models Fundamental Factor Models Statistical Factor Models: Factor Analysis Principal Components Analysis Statistical Factor Models: Principal Factor Method

Linear Factor Model Linear Factor Model: Cross-Sectional Regressions xt = α + Bf t + t , for each t ∈ {1, 2 . . . , T }, where     0   α1 β1 1,t    α2   β 02       2,t     α =  .  (m × 1); B =  .  = βi,k (m × K ); t =  .  (m × 1)  ..   ..   ..  0 αm βm m,t α and B are the same for all t. {f t } is (K −variate) covariance stationary I (0) with E [f t ] = µf Cov [f t ] = E [(f t − µf )(f t − µf )0 ] = Ωf {t } is m-variate white noise with: E [t ] = 0m Cov [t ] = E [t 0t ] = Ψ Cov [t , t 0 ] = E [t 0t 0 ] = 0 ∀t 6= t 0 2 ) where Ψ is the (m × m) diagonal matrix with entries (σ12 , σ22 , . . . , σm σi2 = var (i,t ), the variance of the ith asset specific factor. The two processes {f t } and {t } have null cross-covariances: 0 MIT E [(f18.S096 0 − 0Models t − µf )(tFactor m) ] =

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Factor Models

Linear Factor Model Macroeconomic Factor Models Fundamental Factor Models Statistical Factor Models: Factor Analysis Principal Components Analysis Statistical Factor Models: Principal Factor Method

Linear Factor Model Summary of Parameters α: (m × 1) intercepts for m assets B: (m × K ) loadings on K common factors for m assets µf : (K × 1) mean vector of K common factors Ωf : (K × K ) covariance matrix of K common factors 2 ): m asset-specific variances Ψ = diag (σ12 , . . . , σm

Features of Linear Factor Model The m−variate stochastic process {xt } is a covariance-stationary multivariate time series with Conditional moments: E [xt | f t ] = α + Bf t Cov [xt | f t ] = Ψ Unconditional moments: E [xt ] = µx = α + Bµf Cov [xt ] = Σx = BΩf B 0 + Ψ MIT 18.S096

Factor Models

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Linear Factor Model Macroeconomic Factor Models Fundamental Factor Models Statistical Factor Models: Factor Analysis Principal Components Analysis Statistical Factor Models: Principal Factor Method

Factor Models

Linear Factor Model Linear Factor Model: Time Series Regressions xi = 1T αi + Fβ i + i ,

for each . . , m},  asseti ∈ {1, 2 . i,t xi,1  .  .   .  .   .   .    xi =  xi,t  i =  i,t     ..   ..  .   . x1,T i,T

where         

f 01  .  .  .  0 F =  ft   .  .. f 0T 





        =      

f1,1 .. . f1,t ... f1,T

f2,1 .. . f2,t ... f2,T

··· .. . ··· ... ···

fK ,1 .. . fK ,t .. . fK ,T

        

αi and β i = (β1,i , . . . , βK ,i ) are regression parameters. i is the T -vector of regression errors with Cov (i ) = σi2 IT Linear Factor Model: Multivariate Regression X = [x1 | · · · |xm ], E = [1 | · · · |m ], B = [β 1 | · · · |β m ], X = 1T α0 + FB + E (note that B equals the transpose of cross-sectional B) MIT 18.S096

Factor Models

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Factor Models

Linear Factor Model Macroeconomic Factor Models Fundamental Factor Models Statistical Factor Models: Factor Analysis Principal Components Analysis Statistical Factor Models: Principal Factor Method

Outline

1

Factor Models Linear Factor Model Macroeconomic Factor Models Fundamental Factor Models Statistical Factor Models: Factor Analysis Principal Components Analysis Statistical Factor Models: Principal Factor Method

MIT 18.S096

Factor Models

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Factor Models

Linear Factor Model Macroeconomic Factor Models Fundamental Factor Models Statistical Factor Models: Factor Analysis Principal Components Analysis Statistical Factor Models: Principal Factor Method

Macroeconomic Factor Models Single Factor Model of Sharpe (1970) xi,t = αi + βi RMt + i,t i = 1, . . . , m t = 1, . . . , T where RMt is the return of the market index in excess of the risk-free rate; the market risk factor. xi,t is the return of asset i in excess of the risk-free rate. K = 1 and the single factor is f1,t = RMt . Unconditional cross-sectional covariance matrix of the assets: 2 ββ 0 + Ψ where Cov (xt ) = Σx = σM 2 σM = Var (RMt ) β = (β1 , . . . , βm )0 2 Ψ = diag (σ12 , . . . , σm ) MIT 18.S096

Factor Models

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Factor Models

Linear Factor Model Macroeconomic Factor Models Fundamental Factor Models Statistical Factor Models: Factor Analysis Principal Components Analysis Statistical Factor Models: Principal Factor Method

Estimation of Sharpe’s Single Index Model Single Index Model satisfies the Generalized Gauss-Markov assumptions so the least-squares estimates (α ˆ i , βˆi ) from the time-series regression for each asset i are best linear unbiased estimates (BLUE) and the MLEs under Gaussian assumptions. x i = 1T α ˆ i + RM βˆi + ˆi Unbiased estimators of remaining parameters: σ ˆi2 = (ˆ0i ˆi )/(T − 2) PT P 2 ¯ M )2 ]/(T − 1) with R ¯ M = ( T RMt )/T σ ˆM = [ t=1 (RMt − R t=1 2 ˆ = diag (σ Ψ ˆ12 , . . . , σ ˆm )

Estimator of unconditional covariance matrix: 2 ββ \ ˆx = σ ˆ ˆ0 + Ψ ˆ Cov (xt ) = Σ ˆM MIT 18.S096

Factor Models

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Factor Models

Linear Factor Model Macroeconomic Factor Models Fundamental Factor Models Statistical Factor Models: Factor Analysis Principal Components Analysis Statistical Factor Models: Principal Factor Method

Macroeconomic Multifactor Model The common factor variables {f t } are realized values of macro econonomic variables, such as Market risk Price indices (CPI, PPI, commodities) / Inflation Industrial production (GDP) Money growth Interest rates Housing starts Unemployment See Chen, Ross, Roll (1986). “Economic Forces and the Stock Market”

Linear Factor Model as Time Series Regressions xi = 1T αi + Fβ i + i , where F = [f 1 , f 2 , . . . f T ]0 is the (T × K ) matrix of realized values of (K > 0) macroeconomic factors. Unconditional cross-sectional covariance matrix of the assets: Cov (xt ) = BΩf B 0 + Ψ where B = (β 1 , . . . , β m )0 is (m × K ) MIT 18.S096

Factor Models

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Factor Models

Linear Factor Model Macroeconomic Factor Models Fundamental Factor Models Statistical Factor Models: Factor Analysis Principal Components Analysis Statistical Factor Models: Principal Factor Method

Estimation of Multifactor Model Multifactor model satisfies the Generalized Gauss-Markov ˆ i (K × 1) assumptions so the least-squares estimates α ˆ i and β from the time-series regression for each asset i are best linear unbiased estimates (BLUE) and the MLEs under Gaussian assumptions. ˆ i + ˆi x i = 1T α ˆ i + Fβ Unbiased estimators of remaining parameters: σ ˆi2 = (ˆ0i ˆi )/[T − (k + 1)] 2 ˆ = diag (σ Ψ ˆ2, . . . , σ ˆm ) PT 1 ˆ ¯ Ωf = [ t=1 (f t − f)(f t − ¯f)0 ]/(T − 1) PT with ¯f = ( f t )/T t=1

Estimator of unconditional covariance matrix: 2 BΩ \ ˆx = σ ˆ ˆf B ˆ0 + Ψ ˆ Cov (xt ) = Σ ˆM MIT 18.S096

Factor Models

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Factor Models

Linear Factor Model Macroeconomic Factor Models Fundamental Factor Models Statistical Factor Models: Factor Analysis Principal Components Analysis Statistical Factor Models: Principal Factor Method

Outline

1

Factor Models Linear Factor Model Macroeconomic Factor Models Fundamental Factor Models Statistical Factor Models: Factor Analysis Principal Components Analysis Statistical Factor Models: Principal Factor Method

MIT 18.S096

Factor Models

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Factor Models

Linear Factor Model Macroeconomic Factor Models Fundamental Factor Models Statistical Factor Models: Factor Analysis Principal Components Analysis Statistical Factor Models: Principal Factor Method

Fundamental Factor Models The common-factor variables {f t } are determined using fundamental, asset-specific attributes such as Sector/industry membership. Firm size (market capitalization) Dividend yield Style (growth/value as measured by price-to-book, earnings-to-price, ...) Etc. BARRA Approach (Barr Rosenberg) Treat observable asset-specific attributes as factor betas Factor realizations {f t } are unobservable, but are estimated. MIT 18.S096

Factor Models

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Factor Models

Linear Factor Model Macroeconomic Factor Models Fundamental Factor Models Statistical Factor Models: Factor Analysis Principal Components Analysis Statistical Factor Models: Principal Factor Method

Fama-French Approach (Eugene Fama and Kenneth French) For every time period t, apply cross-sectional sorts to define factor realizations For a given asset attribute, sort the assets at period t by that attribute and define quintile portfolios based on splitting the assets into 5 equal-weighted portfolios. Form the hedge portfolio which is long the top quintile assets and short the bottom quintile assets.

Define the common factor realizations for period t as the period-t returns for the K hedge portfolios corresponding to the K fundamental asset attributes. Estimate the factor loadings on assets using time series regressions, separately for each asset i. MIT 18.S096

Factor Models

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Factor Models

Linear Factor Model Macroeconomic Factor Models Fundamental Factor Models Statistical Factor Models: Factor Analysis Principal Components Analysis Statistical Factor Models: Principal Factor Method

Barra Industry Factor Model Suppose the m assets (i = 1, 2, . . . , m) separate into K industry groups (k = 1, . . . , K ) For each asset i , define the factor loadings (k = 1, . . . K ) 1 if asset i is in industry group k βi,k = 0 otherwise These loadings are time invariant. For time period t, denote the realization of the K factors as f t = (f1t , . . . , fKt )0 These K − vector realizations are unobserved. The Industry Factor Model is Xi,t = βi,1 f1t + · · · + βi,K fKt + it , ∀i, t where = σ2 , ∀i var ( ) it

cov (it , fkt ) cov (fk 0 t , fkt )

i

= =

0, [Ωf ]k 0 ,k , MIT 18.S096

∀i, k, t ∀k 0 , k, t Factor Models

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Factor Models

Linear Factor Model Macroeconomic Factor Models Fundamental Factor Models Statistical Factor Models: Factor Analysis Principal Components Analysis Statistical Factor Models: Principal Factor Method

Barra Industry Factor Model Estimation of the Factor Realizations For each time period t consider the cross-sectional regression for the factor model: xt = Bf t + t (α = 0 so it does not appear) with     0  x1,t  x2,t  xt =  .  .. xm,t

β1 1,t  2,t   β 02        (m × 1); B =  .  = βi,k (m × K ); t =  ..  (m × 1)  ..   .  β 0m m,t 0 where E [t ] = 0m , E [t t ] = Ψ, and Cov (f t ) = Ωf . Compute ˆf t by least-squares regression of xt on B with regression parameter f t . B is (m × K ) matrix of indicator variables (same for all t) B 0 B = diag (m1 , . . . mK ), P where mk is the count of assets i in industry k, and K k=1 mk = m. 0 0 ˆf t = (B B)−1 B xt (vector of industry averages!) ˆ t = xt − Bˆf t    

MIT 18.S096

Factor Models

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Factor Models

Linear Factor Model Macroeconomic Factor Models Fundamental Factor Models Statistical Factor Models: Factor Analysis Principal Components Analysis Statistical Factor Models: Principal Factor Method

Barra Industry Factor Model Estimation of Factor Covariance Matrix PT ¯ˆ ˆ ¯ˆ 0 ˆf = 1 ˆ Ω t=1 (f t − f)(f t − f) T −1 P ˆ f¯ˆ = T1 T t=1 f t ˆ Estimation of Residual Covariance Matrix Ψ 2) ˆ = diag (σ Ψ ˆ12 , . . . , σ ˆm

where

P σ ˆi2 = T 1−1 T i,t − ¯ˆi ]2 t=1 [ˆ P ¯ˆi = T1 T ˆi,t t=1  Estimation of Industry Factor Model Covariance Matrix ˆ = B 0Ω ˆf B + Ψ ˆ Σ MIT 18.S096

Factor Models

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Factor Models

Linear Factor Model Macroeconomic Factor Models Fundamental Factor Models Statistical Factor Models: Factor Analysis Principal Components Analysis Statistical Factor Models: Principal Factor Method

Barra Industry Factor Model Further Details Inefficiency of least squares estimates due to heteroscedasticity in Ψ. Resolution: apply Generalized Least Squares (GLS) estimating Ψ in the cross-sectional regressions. The factor realizations can be rescaled to represent factor mimicking portfolios The Barra Industry Factor Model can be expressed as a seemingly unrelated regression (SUR) model

MIT 18.S096

Factor Models

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Factor Models

Linear Factor Model Macroeconomic Factor Models Fundamental Factor Models Statistical Factor Models: Factor Analysis Principal Components Analysis Statistical Factor Models: Principal Factor Method

Outline

1

Factor Models Linear Factor Model Macroeconomic Factor Models Fundamental Factor Models Statistical Factor Models: Factor Analysis Principal Components Analysis Statistical Factor Models: Principal Factor Method

MIT 18.S096

Factor Models

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Factor Models

Linear Factor Model Macroeconomic Factor Models Fundamental Factor Models Statistical Factor Models: Factor Analysis Principal Components Analysis Statistical Factor Models: Principal Factor Method

Statistical Factor Models The common-factor variables {f t } are hidden (latent) and their structure is deduced from analysis of the observed returns/data {xt }. The primary methods for extraction of factor structure are: Factor Analysis Principal Components Analysis Both methods model the Σ, the covariance matrix of ˆ, {xt , t = 1, . . . , T } by focusing on the sample covariance matrix Σ computed as follows: X = [x1 : · · · xT ] (m × T ) X∗ = X · (IT − T1 1T 10T ) (‘de-meaned’ by row) ˆ x = 1 X∗ (X∗ )0 Σ T MIT 18.S096

Factor Models

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Factor Models

Linear Factor Model Macroeconomic Factor Models Fundamental Factor Models Statistical Factor Models: Factor Analysis Principal Components Analysis Statistical Factor Models: Principal Factor Method

Factor Analysis Model Linear Factor Model as Cross-Sectional Regression xt = α + Bf t + t , for each t ∈ {1, 2 . . . , T } ( m equations expressed in vector/matrix form) where α and B are the same for all t. {f t } is (K −variate) covariance stationary I (0) with E [f t ] = µf , Cov [f t ] = Ωf {t } is m-variate white noise with E [t ] = 0m and Cov [t ] = Ψ = diag (σi2 )

Invariance to Linear Tranforms of f t For any (K × K ) invertible matrix H define f ∗t = Hft and B ∗ = BH −1 Then the linear factor model holds replacing f t and B xt = α + B ∗ f ∗t + t = α + BH −1 Hf t + t = α + Bf t + t and replacing µf and Ωf with Ω∗f = Cov (f ∗t ) = Cov (Hf t ) = HCov (f t )H 0 = HΩf H 0 µ∗f = Hµf MIT 18.S096

Factor Models

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Factor Models

Linear Factor Model Macroeconomic Factor Models Fundamental Factor Models Statistical Factor Models: Factor Analysis Principal Components Analysis Statistical Factor Models: Principal Factor Method

Factor Analysis Model Standard Formulation of Factor Analysis Model Orthonormal factors: Ωf = IK 1 This is achieved by choosing H = ΓΛ− 2 , where Ωf = ΓΛΓ0 is the spectral/eigen decomposition with orthogonal (K × K ) matrix Γ and diagonal matrix Λ = diag (λ1 , . . . , λK ), where λ1 ≥ λ2 ≥ · · · ≥ λK > 0. Zero-mean factors: µf = 0K This is achieved by adjusting α to incorporate the mean contribution from the factors: α∗ = α + Bµf Under these assumptions the unconditional covariance matrix is Cov (xt ) = Σx = BB 0 + Ψ MIT 18.S096

Factor Models

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Factor Models

Linear Factor Model Macroeconomic Factor Models Fundamental Factor Models Statistical Factor Models: Factor Analysis Principal Components Analysis Statistical Factor Models: Principal Factor Method

Factor Analysis Model Maximum Likelihood Estimation For the model xt = α + Bf t + t α and B are vector/matrix constants. All random variables are Normal/Gaussian:

xt i.i.d. Nm (α, Σx ) f t i.i.d. NK (0K IK ) t i.i.d. Nm (0m , Ψ) Cov (xt ) = Σx = BB 0 + Ψ

Model Likelihood L(α, Σx ) = p (x1 , . . . , xT | α, Σ) QT = [p(xt | α, Σ)]
Qt=1 T −m/2 |Σ|− 21 exp − 1 (x − α)0 Σ−1 (x − α) ] = t t x t=1 [(2π) h i P2 T 0 Σ−1 (x − α) = (2π)−Tm/2 |Σ|− 2 exp − 21 T (x − α) t t x t=1 MIT 18.S096

Factor Models

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Factor Models

Linear Factor Model Macroeconomic Factor Models Fundamental Factor Models Statistical Factor Models: Factor Analysis Principal Components Analysis Statistical Factor Models: Principal Factor Method

Factor Analysis Model Log Likelihood of the Factor Model l(α, Σx ) = log L(α, Σx ) (2π) − K2 log (|Σ|) = − TK 2 log P 0 −1 − 21 T t=1 (xt − α) Σx (xt − α) Maximum Likelihood Estimates (MLEs) The MLEs of α, B, Ψ are the values which Maximize l(α, Σx ) Subject to: Σx = BB 0 + Ψ The MLEs are computed numerically applying the Expectation-Maximization (EM) algorithm* * Optional Reading: Dempster, Laird, and Rubin (1977), Rubin and Thayer (1983). MIT 18.S096

Factor Models

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Factor Models

Linear Factor Model Macroeconomic Factor Models Fundamental Factor Models Statistical Factor Models: Factor Analysis Principal Components Analysis Statistical Factor Models: Principal Factor Method

Factor Analysis Model ML Specification of the Factor Model ˆ and Ψ ˆ. Apply EM algorithm to compute α ˆ and B Estimate factor realizations {f t } Apply the cross-sectional regression models for each time period t: ˆ f t + ˆt xt − α ˆ =B Solving for ˆf as the regression parameter estimates of the regression of observed xt on the estimated factor loadings matrix. Taking account of the heteroscedasticity in , apply GLS estimates: ˆ −1 (xt − α ˆf t = [B ˆ 0Ψ ˆ −1 B] ˆ −1 [B ˆ 0Ψ ˆ )]

(Optional) Consider coordinate rotations of orthonormal factors as alternate interpretations of model. MIT 18.S096

Factor Models

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Factor Models

Linear Factor Model Macroeconomic Factor Models Fundamental Factor Models Statistical Factor Models: Factor Analysis Principal Components Analysis Statistical Factor Models: Principal Factor Method

Factor Analysis Model

Further Details of ML Specification Estimated factor realizations can be rescaled to represent factor mimicking portfolios Likelihood Ratio test can be applied to test for the number of factors. ˜ − l(α ˆ , Ψ)] ˆ Test Statistic: LR(K ) = 2[l(α ˜ , Σ) ˆ, B where H0 : K factors are sufficient to model Σ and ˜ are the MLEs with no factor-model restrictions. α ˜ and Σ

MIT 18.S096

Factor Models

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Factor Models

Linear Factor Model Macroeconomic Factor Models Fundamental Factor Models Statistical Factor Models: Factor Analysis Principal Components Analysis Statistical Factor Models: Principal Factor Method

Outline

1

Factor Models Linear Factor Model Macroeconomic Factor Models Fundamental Factor Models Statistical Factor Models: Factor Analysis Principal Components Analysis Statistical Factor Models: Principal Factor Method

MIT 18.S096

Factor Models

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Factor Models

Linear Factor Model Macroeconomic Factor Models Fundamental Factor Models Statistical Factor Models: Factor Analysis Principal Components Analysis Statistical Factor Models: Principal Factor Method

Principal Components Analysis (PCA) An m−va  riaterandom variable: x1  ..  x =  . , with E [x] = α ∈