Thursday, November 25, 2010

Multi-Factor Models – Applying the Lessons Learned from the Numbers

In Finance 1 last year, we were introduced to the idea of multi-factor models (MFM) originally explained by Fama and French as an alternative to the traditional Capital Asset Pricing Model (CAPM) for assessing systematic risk. Additional factors include small versus big (SML) and value versus growth (HML).

In our Business Analysis and Valuation class, we discussed a merger case in which a large company acquired a smaller company. We talked about what would be the best way to approximate beta. The method I used (which was the best method I could conceive, I’d be happy to hear criticism or suggestions otherwise) was to weight the betas by market cap and take an average.

However, there was some discussion about the fact that one entity was much smaller than the other. While we were having a conversation of what that would actually mean, I would suggest a mathematical method for expressing the quantitative effect of size, using FF’s MFM.

  1. 1. Express both company’s re as a three factor MFM
    Re1 – RFR = beta1 (Rm – RFR) + betas1 (SMB) + betav1 (HML)
    Re2 – RFR = beta2 (Rm – RFR) + betas2 (SMB) + betav2 (HML)
  2. Take the larger company’s size beta and apply it to the smaller entity
    betasnew = betas2
  3. Recalculate re for both entities
    Re1new – RFR = beta1 (Rm – RFR) + betasnew (SMB) + betav1 (HML)
    Re2new – RFR = beta2 (Rm – RFR) + betasnew (SMB) + betav2 (HML)
  4. Take a weighted average (by market cap) as the expected return of the combined entity

By taking the larger company’s size beta for both, what you are saying is that you expect the smaller company to have the size “characteristics” of the larger entity. I might even be more appropriate to add the size factors (Would that be appropriate? As the MFM is a linear regression, is it appropriate to add these factors?) and use that for new return on equity for each entity as it relates to the combined entity.

betasnew = betas1 + betas2

While there are some significant assumptions which are required for this to work, it is the best solution I can conjure based on information given. I would really appreciate any additional ideas for creating a more robust model.


Lepto said...


New fan - really like the concept, I'm doing something similar at

I think you've hit on a very interesting forecasting problem here. 2 Comments:

1. Why not use the weighted average approach on the function. For instance the parent may have several different business lines (I'm thinking of the classic Disney example Damodaran uses in his Applied Corporate Finance text book.) With beta's for each business function you control for idiosyncratic risk in the operation, but you let the firm size be absorbed in the single beta. Then when you add the target, you would allocate their revenue to the appropriate function, re-weight, and find the new average beta.

Kudos on your use of the 3 factor model (why not use the 4 factor model that incorporates momentum?), but I would caution against replacing a coefficient value from one model to another.

With time series data there is often autocorrolation between variables. This is another way of saying the SMB parameter is influenced by the HML and RM parameters. Basically the SMB interacts with the other variables in the minimization process. Because of this the SMB_big will bring along HML_big and RM_big adjustment characteristics when you slap it into the small regression model. There are a lot of different ways of testing this, I'd suggest cracking open "basic econometrics" by gujarati. The bottom line is that you can't take a parameter from one model into another without having a biased model.

Now that's the econometrician answer, as always even if it's not supposed to work, but if it works 'well enough' then power to you!

Joshua Wong said...

Great comments, Lepto. Much appreciated.

Actually, I did do the weighted average approach (we didn't actually have all factors, only the CAPM beta) and I'm glad that you like that answer as it was the best one I could come up with.

I think you've made some fantastic points regarding the multi factor model. Often we forget where these models come from (and the math behind them) and need to understand that this formula comes from a statistical regression and factors can't simply be replacing coefficients as you mention.