# Fama-French Data from daily to monthly returns

Ken French on his website publishes daily, monthly and yearly returns for the Fama-French 3 Factors model which are excess market (Rm-Rf), small-minus-big (SMB) and high-minus-low (HML) returns.

I don't understand how he converts daily to monthly returns. For example for the last month the daily returns are

           Mkt-RF     SMB     HML      RF
20150501    1.01   -0.33   -0.60   0.000
20150504    0.32    0.06    0.16   0.000
20150505   -1.19   -0.10    0.34   0.000
20150506   -0.31    0.62   -0.20   0.000
20150507    0.39    0.03   -0.43   0.000
20150508    1.21   -0.54   -0.21   0.000
20150511   -0.39    0.67   -0.11   0.000
20150512   -0.27    0.00    0.11   0.000
20150513    0.01    0.02   -0.06   0.000
20150514    1.01   -0.10   -0.36   0.000
20150515    0.05   -0.26   -0.01   0.000
20150518    0.44    0.72   -0.09   0.000
20150519   -0.09   -0.08    0.03   0.000
20150520   -0.05    0.21   -0.09   0.000
20150521    0.23   -0.31    0.09   0.000
20150522   -0.22   -0.11   -0.14   0.000
20150526   -1.01   -0.04   -0.02   0.000
20150527    0.93    0.33   -0.39   0.000
20150528   -0.11    0.11    0.07   0.000
20150529   -0.58    0.02    0.05   0.000


And the monthly returns are

        Mkt-RF     SMB     HML      RF
201505    1.36    0.92   -1.89    0.00


For example to convert the daily Mkt-RF return to a monthly returns I use the following formula

$$\text{ret}_\text{monthly} = \left(\prod_{i\in\text{day}} \left(\frac{\text{Mkt-RF}_i}{100} + 1\right) - 1 \right)*100$$

which is

$$\text{ret}_\text{monthly} = \left[\left( \left(\frac{1.01}{100} + 1\right)\times \left(\frac{0.32}{100} + 1\right)\times\cdots\times \left(\frac{(-0.58}{100} + 1\right) \right) - 1\right]\times100$$

So I find the following monthly returns

               CUSTOM CALCULATIONS
Mkt-RF     SMB     HML      RF
201505    1.35    0.91   -1.85    0.00


I don't understand why I get these differences. What am I doing wrong?

• Couldn't this be due to rounding error as there are only two decimal places? – Gary Upper Jul 10 '15 at 2:16
• I think it might be due to regression differences vs rounding errors. I think what happens is that they regress daily data using daily factors & also regress monthly data with monthly factors. @conighion – Rime Jul 10 '15 at 11:21
• @Rime that is correct as well. – pyCthon Jul 10 '15 at 11:25
• @Rime They don't do any regression to get Fama-French factors. – John Apr 17 '17 at 16:10

You are doing it right. The differences are rounding issues and can be safely ignored for any practical purpose.

You're compounding correctly but the discrepancy is not just because of rounding. SMB and HML are formed as averages of 6 and 4 different portfolios, respectively. As French's website explains, this results from cutting all stocks into 2x3 SizexBook portfolios. French compounds each of these portfolios to the proper horizon (eg monthly) and then averages these portfolios to get SMB and HML. This is not the same as directly compounding SMB and HML from daily data.

This is because compounding SMB and HML daily data assumes daily rebalancing to equal weights of the portfolios that constitute them. French does not assume this rebalancing for longer horizons but instead, holds the constituent portfolios to the proper horizon before SMB and HML are formed at the end of the horizon. This applies to the weekly, monthly and annual factors he posts.

• I'm not sure if I follow the point in your second paragraph. I think you mean the weight of the portfolios that constitute SMB. So like the 1/3 long and 1/3 short is rebalancing every day. Rather than the weight of the stock portfolios that are the underlying? – John Apr 17 '17 at 16:16
• Yes. You just need to track the 6 underlying portfolios. No need to reform them at the stock level. The portfolio data is available on Ken French's website too so you don't need access to another data source. – Cyurmt Apr 18 '17 at 18:22