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.

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,..,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 =.

3 (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.

4 (K K) covariance matrix ofKcommon factors =diag( 2,.., 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 .

5 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 .

6 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 : 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.)

7 , 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.

8 2i= ( i i)/(T 2) 2M= [ Tt=1(RMt RM)2]/(T 1) with RM= ( Tt=1 RMt)/T =diag( 21,.., 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).

9 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.

10 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 .