invest a lot despite low profitability) that, at least for small
stocks, are associated with low average returns left unex-
plained by the five-fact or model. In the Size-OP-In v sorts, the
portfolio of small stocks in the lowest OP and highest Inv
quartiles has an even lower averag e ex cess return, 0.09%per
month. In this case, the five-fact or slopes simply confirm that
the small stocks in this portfolio invest a lot despite low
profitability .
The portfolios in Tables 1 and 2 do not cleanly disen-
tangle the value, profitability , and investment effects in
av erag e returns pr edict ed by the valuation Eq. (3),butwe
shall see that they expose variation in averag e returns
sufficient to provide strong challenges in asset pricing tests.
4. Factor definitions
To examine whether the specifics of factor construction
are important in tests of asset pricing models, we use three
sets of factors to capture the patterns in average returns in
Tables 1 and 2. The three approaches are described
formally and in detail in Table 3. Here we provide a brief
summary.
The first approach augments the three factors of Fama
and French (1993) with profitability and investment fac-
tors defined like the value factor of that model. The Size
and value factors use independent sorts of stocks into two
Size groups and three B/M groups (independent 2 3
sorts). The Size breakpoint is the NYSE median market
cap, and the B/M breakpoints are the 30
th
and 70
th
percentiles of B/M for NYSE stocks. The intersections of
the sorts produce six VW portfolios. The Size factor, SMB
BM
,
is the average of the three small stock portfolio returns
minus the average of the three big stock portfolio returns.
The value factor HML is the average of the two high B/M
portfolio returns minus the average of the two low B/M
portfolio returns. Equivalently, it is the average of small
and big value factors constructed with portfolios of only
small stocks and portfolios of only big stocks.
The profitability and investment factors of the 2 3
sorts, RMW and CMA, are constructed in the same way as
HML except the second sort is either on operating profit-
ability (robust minus weak) or investment (conservative
minus aggressive). Like HML, RMW and CMA can be
interpreted as averages of profitability and investment
factors for small and big stocks.
The 2 3 sorts used to construct RMW and CMA
produce two additional Size factors, SMB
OP
and SMB
Inv
.
The Size factor SMB from the three 2 3 sorts is defined as
the average of SMB
B/M
, SMB
OP
, and SMB
Inv
. Equivalently,
SMB is the average of the returns on the nine small stock
portfolios of the three 2 3 sorts minus the average of the
returns on the nine big stock portfolios.
When we developed the three-factor model, we did not
consider alternative definitions of SMB and HML. The
choice of a 2 3 sort on Size and B/M is, however, arbitrary.
To test the sensitivity of asset pricing results to this choice,
we construct versions of SMB, HML, RMW, and CMA in the
same way as in the 2 3 sorts, but with 2 2 sorts on Size
and B/M, OP, and Inv, using NYSE medians as breakpoints
for all variables (details in Table 3).
Since HML, RMW, and CMA from the 2 3 (or 2 2)
sorts weight small and big stock portfolio returns equally,
they are roughly neutral with respect to size. Since HML is
constructed without controls for OP and Inv, however, it is
not neutral with respect to profitability and investment.
This likely means that the average HML return is a mix of
premiums related to B/M, profitability, and investment.
Similar comments apply to RMW and CMA.
To better isolate the premiums in average returns
related to Size,
B/M, OP, and Inv, the final candidate factors
Table 2
Averages of monthly percent excess returns for value-weight (VW) portfolios formed on (i) Size, B/M, and OP, (ii) Size, B/M, and Inv, and (iii) Size, OP, and
Inv; July 1963–December 2013, 606 months.
At the end of June each year t, stocks are allocated to two Size groups (Small and Big) using the NYSE median market cap as breakpoint. Stocks in each Size
group are allocated independently to four B/M groups (Low B/M to High B/M for fiscal year t1), four OP groups (Low OP to High OP for fiscal year t1), and
four Inv groups (Low Inv to High Inv for fiscal year t1) using NYSE breakpoints specific to the Size group. The table shows averages of monthly returns in
excess of the one-month Treasury bill rate on the 32 portfolios formed from each of the three sorts.
Small Big
Panel A: Portfolios formed on Size , B/M, and OP
B/M- Low 23High Low 23High
Low OP 0
.03 0.72 0.84 0.93 0.24 0.23 0.37 0.60
20.67 0.76 0.88 1.08 0.41 0.50 0.47 0.69
30.66 0.88 1.07 1.30 0.40 0.59 0.68 0.91
High OP 0.81 1.13 1.22 1.63 0.53 0.64 0.79 0.71
Panel B: Portfolios formed on Size, B/M and Inv
B/M - Low 23High Low 23High
Low Inv 0.69 0.99 1.18 1.23 0.58 0.70 0.62 0.
77
20.87 0.92 0.93 1.08 0.49 0.54 0.54 0.60
30.84 0.95 1.01 0 .97 0.49 0.54 0.56 0.72
High Inv 0.39 0.75 0.87 1.01 0.49 0.44 0.39 0.64
Panel C: Portfolios formed on Size, OP, and Inv
OP - Low 23High Low 23High
Low Inv 0.85 1.01 1.19 1.27 0.63 0.66 0.79 0.70
20.94 0.90 0.92 1.04 0.32 0.43 0.64 0.
64
30.61 0.93 0.94 1.06 0.52 0.57 0.48 0.53
High Inv 0.09 0.58 0.76 0.76 0.29 0.25 0.38 0.65
E.F. Fama, K.R. French / Journal of Financial Economics 116 (2015) 1–22 5