For estimation of the parameters of the factor model, you can use a cross sectionnal regression model by gathring all the individual securities model in a single ?big? factor model. If you assume, that you have 3 factors, then the model at time t for each asset is given by

4 Practical aspects For estimation of the parameters of the factor model1 you can use a cross sectionnal regression model by gathring ali the individual securities model in a single \"big" factor model. If you assume. that you have 3 factors, then the model at time t for each asset- is given by T5 = 1:- + fflrm - Tn] + bf'rsua: + iii-frame: + ct.- + Eu {4} with \"I"; = 11-: Ty\" and moreover the E11 are independent of the factors and satisfyr ofwheni=j, andt=s U otherwise. cov[Ei:,Ejs} = { 4.1 Time Series Model for a given Security If we consider T observations of the exoess return of Security 3: stacked in a column Til vector R,- = r\" , we have the time series regression model for Security 1': Tier Rf=1fi+Fi+Eifri=1525... {5} where '33 I i = bf is the [3 by 1} vector of Betas a fi TML Til TSMEH T'HMLl DE: 5 = 5 E istheisthe(Tx3]matrixof 1"} \"Farr T'rr Tamar 'T'HMLr observations of the factors. I the residual term E; is a [T by 1} vector satisfying Efsisi} 2 33h. The previous model is wellsuited for a regression to estimate the coefcients of the model using data for the securities and the factors. 4.2 Cross Sectional Mode! Alternatively, we can use a cross sectional formulation that can be useful for risk analysis including the derivation of the covariance matrix of the returns. If we Tit dene R: = Tm , the Tvector of all Securities excess returns at time t, then we TE: can write Ri=o+Bft+stfort=1,2,...T1 {'5} where 2 3? b: a :3 = 5 3 isabematrix, "5 e e" [an n n 11 TM: 7'11 a ft 2 1'3 MB: is the Trector of factors returns at time t. rHML-t I lE[E:E:If} = D = diog[o1...,crz} 1'! The cross sectional model implies that if H; is the covariance of the factors, then eeem} = nerE' + o (7} which impiies that mlRl = rri + 0i (3} and \"(Rita Rjtl = lain 133' (9} 4.3 Min Variance with 15% Target Return As a consequence of the previous section, the benchmark problem designed as nun-variance with a 15% annual return target is to he formulated as follows: minn MTELLJ' + Mm prQm mp} urE '1 pTw = 15% n (19} where p.- = r; + ee er) + seem + seem + are and s = BfB' + o. The inputs Amp, and Q are dened as in equation [1}