# How to build Fama-French model factors SMB and HML to compare sustainable index to conventional benchmark?

My goal is to analyze and compare the performance between socially responsible indices and conventional ones. I am comparing for each region (Europe, UK, World, US) a sustainable index to a conventional one (Eurostoxx Sustainability 40 Index vs Eurostoxx 50 Index, FTSE4GOOD UK vs FTSE 100, MSCI World ESG Leaders Index vs MSCI World Index, Dow Jones Sustainable Index vs Dow Jones Industrial Average). I looked at the performance of each one over the period 2016-2021 using Sharpe ratio and Jensen's alpha. None of my intercept(Jensen's alpha) for the linear regression is significant, however, there are some sustainable indices that overperform their benchmark. Therefore, I would like to explain these excess return by using the Fama-French 3 factor model + an additional factor that I would call Good Minus Bad taking into account the ESG scores. I know there is data available on Fama-French website but it is not calculated for the indices I am looking at. Therefore, I would like to know the procedure to calculate the size and value factors using my indices. The doubt I have is the following :

• a single portfolio composed of both indices combined should be used to calculate SMB and HML factors?

The regression I would like to do is the following :

E(R)_i - R_f = α_i + β_iM * [ E(R)M )- R(f )]+ β_SMB* E(R)_SMB)+ β_HML* E(R)_HML+ β_GMB * E(R)_GMB

Where : E(R〗_i) = Expected return of the asset R_f = Risk-free rate α_i = Intercept of the equation β_iM = Beta [E(R)M ) - R(f )] = Excess expected return of the market portfolio β_SMB = Beta with respect to size premium SMB = Small Minus Big, size premium β_HML = Beta with respect to value premium HML = High Minus Low, value premium β_GMB = Beta with respect to ESG premium GMB = Good Minus Bad, ESG premium

Then, the following question I have is related to the following :

• Does it make sense to build the GMB factor as follows= High ESG Score - Low ESG Score

Then, the question is with which index I am going to calculate this factor ? With a combined portfolio of both indices for each region ? or something else ?