# Fama French- typical time lag

I have daily prices of 400 stocks for the last 10 years. I have to create each month a portfolio of 20 stocks that minimizes variance with 2 approaches:

1) Estimate volatility with a GARCH(1,1) model each month using previous 6 months for the following 30 days. Then construct the covariance matrix with historical data and replace variance with that estimation (GARCH (1,1)). Finally, usign that I will have to get the min variance portfolio by minimizing that matrix.

2) On the other hand, I have the Fama and French daily factors. Using that I have to create another portfolio and compare the results with the previous one. Since the previous one is not using future data, I guess I have to do one of the following possibilities:

1. run a regression of returns against lagged FF factors and then use current factors to predict future returns. Where

$r(t)=f(FF_{t-1})$

and for

$E \left[r_{t+1} \right]=f(FF_t)$

1. run a regression of returns against same period FF factors and estimate future FF factors to estimate future returns.

$r(t)=f(FF_{t})$

and for

$E \left[r_{t+1} \right]=f(FF_{t+1})$

With either of those approaches, I'll get the coefficients to calculate the covariance of the returns (using the covariance of the FF plus the residuals of the regressions).

Since the Fama-French site has data with a one-month delay or more, what is the best time lag between Fama and French factors and returns to estimate betas coefficients?

Is it a common practice to estimate those factors and then applied the coefficients?

As far as I found, the typical approach is to suppose that you will have the same covariance matrix that the period where the coefficients were estimated, so actually, there's no "real" estimation of future data. It is only supposing that covariance matrix of factors will be the same as the last period. Am I right?

Thanks.

• Could you clarify this question? You want to regress a security's return on the contemporaneous return of the Fama-French factors. If estimating beta for Apple, the January 2017 return of Apple should line up with the January 2017 return of $\mathit{RMRF}_t$ etc... in your panel data. So I don't really understand what you mean by lag? – Matthew Gunn Feb 2 '18 at 16:41
• @MatthewGunn But the factors include returns.... So if I line up same periods, it means that I will have to have future factors to predict future returns. So how will I predict those factors? – GabyLP Feb 2 '18 at 17:15
• Your description is still quite imprecise and unclear to me. For your (2), is what you intend to say something like, "For each security $i$ and year $y$, run a regression of returns against same period FF factors using $n$ days of data up to the day before the end of June of year $y$ to obtain beta estimates $\beta_{i,y,f}$ where $f$ denotes the factor. Then I will use these estimates $\beta_{i,y,f}$ to form a new portfolio starting in July of year $y$. The holding period of the portfolio is one year." – Matthew Gunn Feb 2 '18 at 19:29
• In 1) you have a Full Covariance matrix supplemented by Garch. For 2) IMHO you need a Factor Model of Covariance where the factors are the FF factors (plus a residual). So you should estimate the covariance of the FF factor returns that French and Fama have computed and find the portfolio of the 400 stocks that minimizes that model of covariance. Are you familiar withthis approach to covariance modeling web.stanford.edu/~wfsharpe/mia/fac/mia_fac3.htm – noob2 Feb 2 '18 at 20:24