Lecture 15 Factor Models - MIT OpenCourseWare

1 Factor ModelsFactor ModelsMIT KempthorneFall 2013 MIT 15: Factor Models1 Factor ModelsLinear Factor ModelMacroeconomic Factor ModelsFundamental Factor ModelsStatistical Factor Models : Factor AnalysisPrincipal Components AnalysisStatistical Factor Models : Principal Factor MethodOutline1 Factor ModelsLinear Factor ModelMacroeconomic Factor ModelsFundamental Factor ModelsStatistical Factor Models : Factor AnalysisPrincipal Components AnalysisStatistical Factor Models : Principal Factor MethodMIT Models2 Factor ModelsLinear Factor ModelMacroeconomic Factor ModelsFundamental Factor ModelsStatistical Factor Models : Factor AnalysisPrincipal Components AnalysisStatistical Factor Models : Principal Factor MethodLinear Factor ModelData:massets/instruments/indexes:i= 1,2,..,mntime periods:t= 1,2,..,nm-variate random vector for each time period:xt= (x1,t,x2,t,..,xm,t) ,returns onmstocks/futures/currencies;interest-ra te yields onmUSTreasury Modelxi,t= i+ 1,if1,t+ 2,if2,t+ + k,ifk,t+ i,t= i+ ift+ i,twhere i: intercept of assetift= (f1,t,f2,t.)

2 ,fK,t) :common factorvariables at periodt(constant overi) i= ( 1,i,.., K,i) : Factor loadings of asseti(constant overt) i,t: thespecific Factor ofassetiat 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= +Bft+ t,for each t { 1, ,T}, where 1 2 =.(m1);B=. , 11t 2 . m .. m =[[ i,k ]] 2,t (m K); t= .1).. m, (m t andBare the same for allt.{ft}is (K variate) covariancestationaryI(0) withE[ft]= fCov[ft]=E[(ft f)(ft f) ] = f{ t}ism-variate white noise with:E[ t]=0mCov[ t]=E[ t t] = Cov[ t, t ]=E[ t ] =0t t6=t is the (m m) diagonal matrixwith entries ( 2, 2,.., 2) where12m 2=var( ii,t), the variance of theithasset specific two processes{ft}and{ t}have null cross-covariances:E[(ft f)( t 0m) ] =MIT Models4 Factor ModelsLinear Factor ModelMacroeconomic Factor ModelsFundamental Factor ModelsStatistical Factor Models : Factor AnalysisPrincipal Components AnalysisStatistical Factor Models : Principal Factor MethodLinear Factor ModelSummary of Parameters : (m 1) intercepts formassetsB: (m K) loadings onKcommon factors formassets f: (K 1) mean vector ofKcommon factors f: (K K) covariance matrix ofKcommon factors =diag( 2.

3 , 2m):masset-specific variances1 Features of Linear Factor ModelThem variate stochastic process{xt}is acovariance-stationary multivariatetime series withConditional moments:E[xt|ft]= +BftCov[xt|ft]= Unconditional moments:E[xt]= x= +B fCov[xt]= x=B fB + MIT Models5 Factor ModelsLinear Factor ModelMacroeconomic Factor ModelsFundamental Factor ModelsStatistical Factor Models : Factor AnalysisPrincipal Components AnalysisStatistical Factor Models : Principal Factor MethodLinear Factor ModelLinear Factor model : Time Series Regressionsxi=1T i+F i+ i,for each asset i {1,2..,m}, wherex f x= i,1f .. ff ixi,ti=. i,t 11,.. K, 12, .. i,t . = F ..x1, i,T f = f1,tf2,t fK,t .. f1,Tf2,T fTK,T iand i= ( 1,i,.., K,i) are regression parameters. iis theT-vector of regression errors withCov( i) = 2iITLinear Factor model : Multivariate RegressionX= [x1| |xm],E= [ 1| | m],B= [ 1| | m],X=1T +FB+E(note thatBequals the transpose of cross-sectionalB)MIT Models6 Factor ModelsLinear Factor ModelMacroeconomic Factor ModelsFundamental Factor ModelsStatistical Factor Models : Factor AnalysisPrincipal Components AnalysisStatistical Factor Models : Principal Factor MethodOutline1 Factor ModelsLinear Factor ModelMacroeconomic Factor ModelsFundamental Factor ModelsStatistical Factor Models : Factor AnalysisPrincipal Components AnalysisStatistical Factor Models : Principal Factor MethodMIT Models7 Factor ModelsLinear Factor ModelMacroeconomic Factor ModelsFundamental Factor ModelsStatistical Factor Models : Factor AnalysisPrincipal Components AnalysisStatistical Factor Models .

4 Principal Factor MethodMacroeconomic Factor ModelsSingle Factor model of Sharpe (1970)xi,t= i+ iRMt+ i,ti= 1,..,m t= 1,..,TwhereRMtis the return of the market index in excess of therisk-free rate; themarket risk ,tis the return of assetiin excessof the risk-free 1 and the single Factor isf1,t= cross-sectional covariance matrix of the assets:Cov(xt) = x= 2M + where 2M=Var(RMt) = ( 1,.., m) =diag( 21,.., 2m)MIT 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 ( i, i) from thetime-series regression for each assetiare best linear unbiasedestimates (BLUE) and the MLEs under Gaussian i=1T i+RM i+ iUnbiased estimators of remaining parameters: 2i= ( i i)/(T 2) 2M= [ Tt=1(RMt RM)2]/(T 1) with RM= ( Tt=1 RMt)/T =diag( 21.)

5 , 2m)Estimator of unconditional covariance matrix:Cov (xt) = x= 2 M + MIT 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{ft}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 i+F i+ i,whereF= [f1,f2,..fT] is the (T K) matrix of realized values of(K>0) macroeconomic cross-sectional covariance matrix of the assets:Cov(xt) =B fB + whereB= ( 1,.., m) is (m K)MIT 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 iand i(K 1)from the time-series regression for each assetiare best linearunbiased estimates (BLUE) and the MLEs under i=1T i+Fi+ iUnbiased estimators of remaining parameters: 2i= ( i i)/[T (k+ 1)] =diag( 21.

6 , 2m) f= [ Tt=1(ft f)( ft f) ]/(T 1)with fT= (t=1ft)/TEstimator of unconditional covariance matrix:Cov (xt) = 2 x= B fB M+ MIT Models11 Factor ModelsLinear Factor ModelMacroeconomic Factor ModelsFundamental Factor ModelsStatistical Factor Models : Factor AnalysisPrincipal Components AnalysisStatistical Factor Models : Principal Factor MethodOutline1 Factor ModelsLinear Factor ModelMacroeconomic Factor ModelsFundamental Factor ModelsStatistical Factor Models : Factor AnalysisPrincipal Components AnalysisStatistical Factor Models : Principal Factor MethodMIT 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{ft}are determined usingfundamental, asset-specific attributes such asSector/industry size (market capitalization)Dividend yieldStyle (growth/value as measured by price-to-book,earnings-to-price.

7 (Barr Rosenberg)Treat observable asset-specific attributes asfactor betasFactor realizations{ft}are unobservable, butare 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 periodt,apply cross-sectionalsorts to define Factor realizationsFor a given asset attribute, sort the assets atperiodtby that attribute and define quintileportfolios based on splitting the assets into 5equal-weighted the hedge portfolio which is long the topquintile assets and short the bottom the common Factor realizations for periodtas the period-treturns for theKhedgeportfolios corresponding to theKfundamentalasset the Factor loadings on assets using timeseries regressions, separately for each Models14 Factor ModelsLinear Factor ModelMacroeconomic Factor ModelsFundamental Factor ModelsStatistical Factor Models : Factor AnalysisPrincipal Components AnalysisStatistical Factor Models : Principal Factor MethodBarra Industry Factor ModelSuppose themassets (i= 1,2.)

8 ,m) separate intoKindustry groups (k= 1,..,K)For each asseti{, define the Factor loadings (k= 1,..K)1if assetiis in industry groupk i,k=0otherwiseThese loadings are time periodt, denote the realization of theKfactors asft= (f1t,..,fKt) TheseK vectorrealizations are Industry Factor model isXi,t= i,1f1t+ + i,KfKt+ it, i,twherevar( 2it)= ,i icov( it,fkt)=0, i,k,tcov(fk t,fkt)=[ f]k ,k, k ,k,tMIT 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 periodtconsider the cross-sectional regression forthe Factor model :xt=Bft+ t( = 0 so it does not appear)withxt= x1,t 1 1,t x2,t 2 [[ ]] 2,t . (m 1);B= . = i,k(m 1). K);t=.. (m..m .x,t m m,t whereE[ t] =0m,E[ t t] = ,andCov(ft) = f . Compute ftby least-squares regression ofxtonBwith regression (m K) matrix of indicator variables (same for allt)B B=diag(m1.}

9 MK),wheremkis the count ofassetsiin industryk,and Kk=1mk=m. ft= (B B) 1B xt(vector of industry averages!) t=xt B ftMIT 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 Matrix f=1T 1 Tt=1( ft f)( ft f) f=1fT T t=1tEstimation of Residual Covariance Matrix =diag( 21,.., 2m)where 2i=1T 1 Tt=1[ i,t i]2 i=1TT t=1 i,tEstimation of Industry FactorModel Covariance Matrix =B fB+ MIT Models17 Factor ModelsLinear Factor ModelMacroeconomic Factor ModelsFundamental Factor ModelsStatistical Factor Models : Factor AnalysisPrincipal Components AnalysisStatistical Factor Models : Principal Factor MethodBarra Industry Factor ModelFurther DetailsInefficiency of least squares estimates due toheteroscedasticity in.

10 Resolution: apply Generalized Least Squares (GLS) estimating in the cross-sectional Factor realizations can be rescaled to representfactormimicking portfoliosThe Barra Industry Factor model can be expressed as aseemingly unrelated regression (SUR) modelMIT Models18 Factor ModelsLinear Factor ModelMacroeconomic Factor ModelsFundamental Factor ModelsStatistical Factor Models : Factor AnalysisPrincipal Components AnalysisStatistical Factor Models : Principal Factor MethodOutline1 Factor ModelsLinear Factor ModelMacroeconomic Factor ModelsFundamental Factor ModelsStatistical Factor Models : Factor AnalysisPrincipal Components AnalysisStatistical Factor Models : Principal Factor MethodMIT Models19 Factor ModelsLinear Factor ModelMacroeconomic Factor ModelsFundamental Factor ModelsStatistical Factor Models : Factor AnalysisPrincipal Components AnalysisStatistical Factor Models : Principal Factor MethodStatistical Factor ModelsThe common- Factor variables{ft}are hidden (latent) and theirstructure is deduced from analysis of the observed returns/data{xt}.