Digesting Anomalies An Investment

1.Conceptual Framework
2.Factor Construction
3.Empirical Results
Conceptual Framework
An economic model
• Stochastic general equilibrium model
• Two dates, 0 and 1.
• ROE increases from −0.80% to 5.00% per
quarter.
3.2.1 Earnings momentum (SUE-1) and price momentum (R6-6)
Result:
• 4 significant alphas in the Fama-French model
stronger in small firms than in big firms (e.g., Bernard and Thomas 1990;
Fama and French 2008), we control for size when constructing the
investment and ROE factors. Sorting jointly with size is also standard in
• 3 in the Carhart model
• 1 in the q-factor model
More:
• The q-factor model captures momentum via the
ROE factor (loadings increase from −0.22 to 0.26).
Digesting Anomalies: An
Investment
Approach
ABSTRACT
An empirical q-factor model consisting of the market factor, a size factor,
an investment factor, and a profitability factor largely summarizes the
else equal, high expected profitability stocks should earn higher expected
returns than low expected profitability stocks earn.
2.1 Factor construction
Significant Anomalies (35)
FF 27
C 19
Q
5
Q-factor Loadings and Economic Fundamentals
3.2 Detailed results for selected anomalies
The author reported more detailed results for selected anomalies.
cross section of average stock returns. A comprehensive examination of
nearly 80 anomalies reveals that about one-half of the anomalies are
insignificant in the broad cross section. More importantly, with a few
s
Pi0=E0[M1(Pi1+Di1)] or E0[M1ri1
]=1
s
stock return ∶ ri1
≡(Pi1+Di1)/Pi0
stochastic discount factor: M1≡ρU’(C1)/U’(C0)
The free cash flow of date 0:
Di0≡Πi0Ai0−Ii0−(a/2)(Ii0/Ai0) 2 Ai0
investment and ROE independently helps orthogonalize the 2 new factors.
2X3X3 ME x （I/A）x ROE
2.2 Empirical properties
3. Empirical Results
74 primary anomalies, 6 categories:
An economic model
Firm i chooses Ii0 to maximize the cum-dividend equity value at
the beginning of date 0:
The first principle for investment is given by:
which are the key testing portfolios for Fama and French (1993, 1996).
3.2.1 Earnings momentum (SUE-1) and price momentum (R6-6)
Result:
• 5 significant alphas in the Fama-French model
• Earnings momentum
• Price momentum
which are classic anomalies to the Fama-French model.
• accrual anomaly
which the q-factor model fails to capture.
• 25 size and book-to-market portfolios
quarter.
3.2.1 Earnings momentum (SUE-1) and price momentum (R6-6)
Chan, Jegadeesh, and Lakonishok (1996) show that momentum profits are short-lived.
The authors calculated the returns of earnings and price momentum deciles for holding periods
constructing the value factor, HML, and the momentum factor, UMD. HML
is from a double 2-by-3 sort on size and book-to-market, and UMD is from a
double 2-by-3 sort on size and prior 2–12 month returns. Finally, sorting on
An economic model
Using the definition of Di0，we can derive the ex-dividend equity value from
Equation
as Pi0=E0[M1Πi1Ai1] at the optimum
Rewrite the stock return as
longer than 6 months.
• t to t +11
• t +12 to t +35
• t +36 to t +59
In addition, at the beginning of month t , we sort all stocks on their prior 6-month returns from t−7 to
exceptions, ce is at least comparable to, and
in many cases better than that of the Fama-French (1993) 3-factor model
and the Carhart (1997) 4-factor model in capturing the remaining
We measure investment-to-assets, I/A, as the annual change in total assets
(Compustat annual item AT) divided by 1-year-lagged total assets. We
measure profitability as ROE, which is income before extraordinary items
significant anomalies.
Q-factor model
NPV rule
• Investment cost=Project present
•

=


value=
• 3 in the Carhart model
• 0 in the q-factor model
More:
• The q-factor model captures momentum via the
ROE factor (loadings increase from−0.74 to 0.28).
• ROE increases from −0.71% to 3.43% per
ΠitAit , (i=0,1)
• Investment :I
Ai1=Ii0
• Investment entails quadratic adjustment costs:
(a/2)(Ii0/Ai0) 2 Ai0
An economic model
• The first principle for consumption:
t−2, skipping month t−1, and calculate value-weighted decile returns for the holding periods
(1) Momentum
(4) Profitability
(2) Value-versus-growth
(5) Intangibles
(3) Investment
(6) Trading frictions
Insignificant Anomalies(38)
average high-minus-low decile returns and t-statistics
1 =(Pi1+Di1)/Pi0=Πi1Ai1/E0[M1Πi1Ai1]=Πi1/E0[M1Πi1]
So
then implies
1 = Πi1 /(1+a(Ii0/Ai0).
Implications
• Taking the expectation on both sides of 1 = Πi1 /(1+a(Ii0/Ai0) yields:
• This equation predicts that, all else equal, high investment stocks should
earn lower expected returns than low investment stocks earn, and that, all
• A representative household and heterogeneous firms
An economic model
• expected utility of representative household :
Max[U(C0)+ρE0[U(C1)]
• The operating cash flow of firm i:
(Compustat quarterly item IBQ) divided by 1-quarter-lagged book equity。
We construct the q-factors from a triple 2-by-3-by-3 sort on size, I/A, and
ROE. Because both the investment and earnings effects in the data are