Why Daily Fama-French Factors May Not Compound to Monthly Returns

# Fama-French Data from daily to monthly returns


# 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?

## Answer by Cyurmt (score 5)

https://quant.stackexchange.com/a/33067

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.

## Answer by phdstudent (score 4)

https://quant.stackexchange.com/a/18813

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