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.

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,..,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=.

3 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.)

4 , 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 .

5 Factor AnalysisPrincipal Components AnalysisStatistical Factor Models : 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.)

6 , 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.

7 , 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,.., 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 .

8 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,..) (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 .

9 Principal Factor MethodBarra Industry Factor ModelSuppose themassets (i= 1,2,..,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.}

10 (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,..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.