# How are Fama French Factor Returns for the last 3 and 12 months calculated?

In the Fama/French data library the monthly research factors for the Fama-French-3-Factor-Model and the Fama-French-5-Factor-Model are presented.

I don't see how they are calculating the factor returns in the "Last 3 Months" and "Last 12 Months" columns.

Date      Mkt.RF  SMB   HML   RMW   CMA   RF
Sep 2016   0.25  1.75 -1.49 -1.92 -0.04 0.02
Oct 2016  -2.02 -3.97  4.16  1.24  0.20 0.02
Nov 2016   4.86  6.81  8.27 -0.50  3.67 0.01
Dec 2016   1.82  0.39  3.61  0.98 -0.25 0.03
Jan 2017   1.94 -1.28 -2.68  0.13 -0.94 0.04
Feb 2017   3.57 -2.14 -1.79  0.62 -1.74 0.04
Mar 2017   0.17  0.81 -3.17  0.83 -1.00 0.03
Apr 2017   1.09  0.51 -1.91  2.13 -1.55 0.05
May 2017   1.06 -3.07 -3.75  1.32 -1.84 0.06
Jun 2017   0.78  2.46  1.32 -2.13 -0.06 0.06
Jul 2017   1.87 -1.59 -0.28 -0.58 -0.14 0.07
Aug 2017   0.17 -1.87 -2.26  0.44 -2.44 0.07
Sep 2017   2.51  4.81  3.04 -1.51  1.62 0.09


The table above shows the most recent data from their website. I tried to calculate the arithmetic mean return over the last 3 months and also the geometric mean return but the numbers don't match up. These are the numbers I fail to come up with:

Fama/French 5 Research Factors (2x3)
Sep 2017   Last 3 Months   Last 12 Months
Rm-Rf     2.51        4.61           19.24
SMB       4.81        1.20            1.11
HML       3.04        0.38            4.86
RMW      -1.51       -1.69            3.59
CMA       1.62       -1.08           -5.58


It is clear that the first column consists of the last row data from the above dataset. But how to calculate the second and third column values?

It is correct to use the geometric return. Calculating the factors for the last 3 months following

$$r_{\mathrm{3month}} = \left( \prod_{1\le i \le3} \left(\frac{\mathrm{factor}_i}{100} + 1\right) - 1 \right) \cdot 100$$

where $$\mathrm{factor}_i$$ is the appropriate monthly return of a certain risk factor, is correct.

This approach gives the following returns for the last 3 months ($$FF$$ are the original ones and $$Own$$ the calculated ones with the above formula):

          Last 3 Months FF   Last 3 Months - Own
Rm-Rf         4.61                  4.60
SMB           1.20                  1.21
HML           0.38                  0.43
RMW          -1.69                 -1.65
CMA          -1.08                 -1.00


Why are there differences in the calculation?

You may look at these answers:

• Rounding issues (which can be ignored for practical purpose).

• Lack of rebalancing the factor mimicking portfolios when applying the geometric return.

• (+1) Quick LaTex comment though: if you're going to use words in equations, the proper way is to use roman letters: \mathrm{factor}_t. Notice the improper kerning between $f$ and $a$ in in $factor_t$? In contrast, the properly spaced, roman letter $\mathrm{factor}_t$ is much better. Personally, I would just use f_t which renders as $f_t$. Using words as variables is unwieldy. Oct 4 '18 at 14:51
• Thanks @MatthewGunn for your very helpful comment (+1)! I just didn't know about \mathrm{} and totally agree about its nicer formatting (and appreciate that you let me know how i can improve my posts here). I just edited my answer above and will forbear to use whole words as variables in future posts. Oct 4 '18 at 16:47