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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?

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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.

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    $\begingroup$ (+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. $\endgroup$ – Matthew Gunn Oct 4 '18 at 14:51
  • $\begingroup$ 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. $\endgroup$ – skoestlmeier Oct 4 '18 at 16:47

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