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Factor ModelsLecture 15: Factor ModelsMIT 18.S096Dr. KempthorneFall 2013MIT 18.S096Factor Models1

Factor ModelsLinear Factor ModelMacroeconomic Factor ModelsFundamental Factor ModelsStatistical Factor Models: Factor AnalysisPrincipal Components AnalysisStatistical Factor Models: Principal Factor MethodOutline1Factor ModelsLinear Factor ModelMacroeconomic Factor ModelsFundamental Factor ModelsStatistical Factor Models: Factor AnalysisPrincipal Components AnalysisStatistical Factor Models: Principal Factor MethodMIT 18.S096Factor Models2

Factor ModelsLinear Factor ModelMacroeconomic Factor ModelsFundamental Factor ModelsStatistical Factor Models: Factor AnalysisPrincipal Components AnalysisStatistical Factor Models: Principal Factor MethodLinear Factor ModelData:m assets/instruments/indexes: i 1, 2, . . . , mn time periods: t 1, 2, . . . , nm-variate random vector for each time period:xt (x1,t , x2,t , . . . , xm,t )0E.g., returns on m stocks/futures/currencies;interest-rate yields on m US Treasury instruments.Factor Modelxi,t αi β1,i f1,t β2,i f2,t · · · βk,i fk,t i,t αi β 0i f t i,twhereαi : intercept of asset if 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.S096Factor Models3

Factor ModelsLinear Factor ModelMacroeconomic Factor ModelsFundamental Factor ModelsStatistical Factor Models: Factor AnalysisPrincipal Components AnalysisStatistical Factor Models: Principal Factor MethodLinear Factor ModelLinear Factor Model: Cross-Sectional Regressionsxt α Bf t t ,for each t {1, 2 . . . , T }, where 0 α1β1 1,t α2 β 02 2,t α . (m 1); B . βi,k (m K ); t . (m 1) . . . 0αmβm m,tα and B are the same for all t.{f t } is (K variate) covariance stationary I (0) withE [f t ] µfCov [f t ] E [(f t µf )(f t µf )0 ] Ωf{ t } is m-variate white noise with:E [ t ] 0mCov [ t ] E [ t 0t ] ΨCov [ t , t 0 ] E [ t 0t 0 ] 0 t 6 t 02 ) whereΨ is the (m m) diagonal matrix with entries (σ12 , σ22 , . . . , σmσi2 var ( i,t ), the variance of the ith asset specific factor.The two processes {f t } and { t } have null cross-covariances:0MITE [(f18.S0960 0Modelst µf )( tFactorm) ] 4

Factor ModelsLinear Factor ModelMacroeconomic Factor ModelsFundamental Factor ModelsStatistical Factor Models: Factor AnalysisPrincipal Components AnalysisStatistical Factor Models: Principal Factor MethodLinear Factor ModelSummary of Parametersα: (m 1) intercepts for m assetsB: (m K ) loadings on K common factors for m assetsµf : (K 1) mean vector of K common factorsΩf : (K K ) covariance matrix of K common factors2 ): m asset-specific variancesΨ diag (σ12 , . . . , σmFeatures of Linear Factor ModelThe m variate stochastic process {xt } is acovariance-stationary multivariate time series withConditional moments:E [xt f t ] α Bf tCov [xt f t ] ΨUnconditional moments:E [xt ] µx α BµfCov [xt ] Σx BΩf B 0 ΨMIT 18.S096Factor Models5

Linear Factor ModelMacroeconomic Factor ModelsFundamental Factor ModelsStatistical Factor Models: Factor AnalysisPrincipal Components AnalysisStatistical Factor Models: Principal Factor MethodFactor ModelsLinear Factor ModelLinear Factor Model: Time Series Regressionsxi 1T αi Fβ i i ,for each. . , m}, asset i {1, 2 . i,txi,1 . . . . . . xi xi,t i i,t . . . .x1,T i,Twhere f 01 . . . 0F ft . .f 0T f1,1.f1,t.f1,Tf2,1.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 ITLinear Factor Model: Multivariate RegressionX [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.S096Factor Models6

Factor ModelsLinear Factor ModelMacroeconomic Factor ModelsFundamental Factor ModelsStatistical Factor Models: Factor AnalysisPrincipal Components AnalysisStatistical Factor Models: Principal Factor MethodOutline1Factor ModelsLinear Factor ModelMacroeconomic Factor ModelsFundamental Factor ModelsStatistical Factor Models: Factor AnalysisPrincipal Components AnalysisStatistical Factor Models: Principal Factor MethodMIT 18.S096Factor Models7

Factor ModelsLinear Factor ModelMacroeconomic Factor ModelsFundamental Factor ModelsStatistical Factor Models: Factor AnalysisPrincipal Components AnalysisStatistical Factor Models: Principal Factor MethodMacroeconomic Factor ModelsSingle Factor Model of Sharpe (1970)xi,t αi βi RMt i,t i 1, . . . , m t 1, . . . , TwhereRMt is the return of the market index in excess of therisk-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 whereCov (xt ) Σx σM2σM Var (RMt )β (β1 , . . . , βm )02Ψ diag (σ12 , . . . , σm)MIT 18.S096Factor Models8

Factor ModelsLinear Factor ModelMacroeconomic Factor ModelsFundamental Factor ModelsStatistical Factor Models: Factor AnalysisPrincipal Components AnalysisStatistical Factor Models: Principal Factor MethodEstimation of Sharpe’s Single Index ModelSingle Index Model satisfies the Generalized Gauss-Markovassumptions so the least-squares estimates (α̂i , β̂i ) from thetime-series regression for each asset i are best linear unbiasedestimates (BLUE) and the MLEs under Gaussian assumptions.xi 1T α̂i RM βˆi ˆ iUnbiased estimators of remaining parameters:σ̂i2 (ˆ 0i ˆ i )/(T 2)PTPT2σ̂M [ t 1 (RMt R̄M )2 ]/(T 1) with R̄M ( t 1 RMt )/T2Ψ̂ diag (σ̂12 , . . . , σ̂m)Estimator of unconditional covariance matrix:2 ββ̂\ˆ 0 Ψ̂Cov(xt ) Σ̂x σ̂MMIT 18.S096Factor Models9

Factor ModelsLinear Factor ModelMacroeconomic Factor ModelsFundamental Factor ModelsStatistical Factor Models: Factor AnalysisPrincipal Components AnalysisStatistical Factor Models: Principal Factor MethodMacroeconomic Multifactor ModelThe common factor variables {f t } are realized values of macroecononomic variables, such asMarket riskPrice indices (CPI, PPI, commodities) / InflationIndustrial production (GDP)Money growthInterest ratesHousing startsUnemploymentSee Chen, Ross, Roll (1986). “Economic Forces and the Stock Market”Linear Factor Model as Time Series Regressionsxi 1T αi Fβ i i , whereF [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 Ψwhere B (β 1 , . . . , β m )0 is (m K )MIT 18.S096Factor Models10

Factor ModelsLinear Factor ModelMacroeconomic Factor ModelsFundamental Factor ModelsStatistical Factor Models: Factor AnalysisPrincipal Components AnalysisStatistical Factor Models: Principal Factor MethodEstimation of Multifactor ModelMultifactor model satisfies the Generalized Gauss-Markovassumptions so the least-squares estimates α̂i and β̂ i (K 1)from the time-series regression for each asset i are best linearunbiased estimates (BLUE) and the MLEs under Gaussianassumptions.xi 1T α̂i Fβ̂ i ˆ iUnbiased estimators of remaining parameters:σ̂i2 (ˆ 0i ˆ i )/[T (k 1)]2Ψ̂ diag (σ̂12 , . . . , σ̂m)PTΩ̂f [ t 1 (f t f̄)(f t f̄)0 ]/(T 1)PTwith f̄ ( t 1 f t )/TEstimator of unconditional covariance matrix:2 BΩ\ˆ ˆf Bˆ0 ΨˆCov(xt ) Σ̂x σ̂MMIT 18.S096Factor Models11

Factor ModelsLinear Factor ModelMacroeconomic Factor ModelsFundamental Factor ModelsStatistical Factor Models: Factor AnalysisPrincipal Components AnalysisStatistical Factor Models: Principal Factor MethodOutline1Factor ModelsLinear Factor ModelMacroeconomic Factor ModelsFundamental Factor ModelsStatistical Factor Models: Factor AnalysisPrincipal Components AnalysisStatistical Factor Models: Principal Factor MethodMIT 18.S096Factor Models12

Factor ModelsLinear Factor ModelMacroeconomic Factor ModelsFundamental Factor ModelsStatistical Factor Models: Factor AnalysisPrincipal Components AnalysisStatistical Factor Models: Principal Factor MethodFundamental Factor ModelsThe common-factor variables {f t } are determined usingfundamental, asset-specific attributes such asSector/industry membership.Firm size (market capitalization)Dividend yieldStyle (growth/value as measured by price-to-book,earnings-to-price, .)Etc.BARRA Approach (Barr Rosenberg)Treat observable asset-specific attributes asfactor betasFactor realizations {f t } are unobservable, butare estimated.MIT 18.S096Factor Models13

Factor ModelsLinear Factor ModelMacroeconomic Factor ModelsFundamental Factor ModelsStatistical Factor Models: Factor AnalysisPrincipal Components AnalysisStatistical Factor Models: Principal Factor MethodFama-French Approach (Eugene Fama and Kenneth French)For every time period t, apply cross-sectionalsorts to define factor realizationsFor a given asset attribute, sort the assets atperiod t by that attribute and define quintileportfolios based on splitting the assets into 5equal-weighted portfolios.Form the hedge portfolio which is long the topquintile assets and short the bottom quintileassets.Define the common factor realizations for periodt as the period-t returns for the K hedgeportfolios corresponding to the K fundamentalasset attributes.Estimate the factor loadings on assets using timeseries regressions, separately for each asset i.MIT 18.S096Factor Models14

Factor ModelsLinear Factor ModelMacroeconomic Factor ModelsFundamental Factor ModelsStatistical Factor Models: Factor AnalysisPrincipal Components AnalysisStatistical Factor Models: Principal Factor MethodBarra Industry Factor ModelSuppose the m assets (i 1, 2, . . . , m) separate into Kindustry groups (k 1, . . . , K )For each asset i , define the factor loadings (k 1, . . . K )1 if asset i is in industry group kβi,k 0 otherwiseThese loadings are time invariant.For time period t, denote the realization of the K factors asf t (f1t , . . . , fKt )0These K vector realizations are unobserved.The Industry Factor Model isXi,t βi,1 f1t · · · βi,K fKt it , i, twhere σ2 , ivar ( )itcov ( it , fkt )cov (fk 0 t , fkt )i 0,[Ωf ]k 0 ,k ,MIT 18.S096 i, k, t k 0 , k, tFactor Models15

Factor ModelsLinear Factor ModelMacroeconomic Factor ModelsFundamental Factor ModelsStatistical Factor Models: Factor AnalysisPrincipal Components AnalysisStatistical Factor Models: Principal Factor MethodBarra Industry Factor ModelEstimation of the Factor RealizationsFor each time period t consider the cross-sectional regression forthe factor model:xt Bf t t(α 0 so it does not appear)with 0 x1,t x2,t xt . .xm,tβ1 1,t 2,t β 02 (m 1); B . βi,k (m K ); t . (m 1) . . β 0m m,t0where E [ t ] 0m , E [ t t ] Ψ, and Cov (f t ) Ωf .Compute f̂ 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 ),Pwhere mk is the count of assets i in industry k, and Kk 1 mk m.00f̂ t (B B) 1 B xt (vector of industry averages!)ˆ t xt B f̂ t MIT 18.S096Factor Models16

Factor ModelsLinear Factor ModelMacroeconomic Factor ModelsFundamental Factor ModelsStatistical Factor Models: Factor AnalysisPrincipal Components AnalysisStatistical Factor Models: Principal Factor MethodBarra Industry Factor ModelEstimation of Factor Covariance MatrixP 0Ω̂f T 1 1 Tt 1 (f̂ t f̂)(f̂ t f̂)Pˆf̂ T1 Tt 1 f tEstimation of Residual Covariance Matrix Ψ̂2)Ψ̂ diag (σ̂12 , . . . , σ̂mwherePσ̂i2 T 1 1 T i,t ˆi ]2t 1 [ˆP ˆi T1 Tˆi,tt 1 Estimation of Industry Factor Model Covariance MatrixΣ̂ B 0 Ω̂f B Ψ̂MIT 18.S096Factor Models17