A Neoclassical Interpretation of
Momentum
Laura Xiaolei Liu1
1
Lu Zhang2
Hong Kong University of Science and Technology
2
The Ohio State University
and NBER
The Q Group
April 9, 2014
Theme
An economic interpretation of momentum based on the neoclassical
q-theory of investment
Practical implication: Momentum might be riskier than you thought
Key Result
Price and earnings momentum
25
W
20
Average predicted returns
Average predicted returns
25
9
8
15
7
6
10
3
45
2
5
0
0
L
5
10
15
20
Average realized returns
25
20
6
15
3
L2
10
4
W
89
7
5
5
0
0
5
10
15
20
Average realized returns
25
Outline
1 The Q Model
2 Average Momentum Profits
3 Momentum Reversal
4 Long-run Risks in Momentum
5 Market States and Momentum
6 The Interaction of Momentum with Firm Characteristics
7 Risk Analysis
Outline
1 The Q Model
2 Average Momentum Profits
3 Momentum Reversal
4 Long-run Risks in Momentum
5 Market States and Momentum
6 The Interaction of Momentum with Firm Characteristics
7 Risk Analysis
The Q Model
A microfoundation for the WACC approach to capital budgeting
Marginal benefits of investment at time t+1
z

"
}|
2 #
Yit+1
a Iit+1
(1 − τt+1 ) κ
+

Kit+1 2 Kit+1


|
{z
}


Marginal product plus economy
of
scale
(net
of
taxes) 
Iit+1

 +τt+1 δit+1 + (1 − δit+1 ) 1 + (1 − τt+1 )a

Kit+1
|
{z
}
I
rit+1
≡
Expected continuation value
1 + (1 − τt )a
|
{z
Iit
Kit
}
Marginal costs of investment at time t
Ba
S
wit rit+1
+ (1 − wit )rit+1
|
{z
}
The weighted average cost of capital
I
= rit+1
{









The Q Model
Expected stock returns = expected levered investment returns?



I
Ba 
 S

rit+1
(a, κ) − wit rit+1

 = 0,
E rit+1 −

1
−
w
it

|
{z
}
Iw
rit+1
with the model error, αiq , as the sample average of the difference
Construct a χ2 test per Hansen (1982) based on these alphas
The Q Model
Measurement, 1963–2012
Kit : Net property, plant, and equipment (PPE)
Iit : Capital expenditure minus sales of PPE
Yit : Sales
Bit : Long-term debt plus short-term debt
Pit : The market value of common equity
δit : The amount of depreciation divided by capital
B : Imputed bond ratings, assigning corporate bond returns
rit+1
of a given rating to all firms with the same rating
τt : Statutory tax rate of corporate income
and firm i’s investment for the current fiscal year, respectively. δ it+1 and Yit+1 are the depreciate
rate and sales from the next fiscal year, respectively. Kit is firm i’s capital observed at the end of
the last fiscal year (or at the beginning of the current fiscal year).
The Q Model
Panelalignment,
A: Firms with
December
yearendfiscal yearend
Timing
firms
with fiscal
December
I
rit+1
-
(from July of year t
to June of t + 1)
τ t , Iit
(from January of year t
to December of t)
-
τ t+1 , δ it+1
Yit+1 , Iit+1
-
(from January of year t + 1
to December of t + 1)
December/January
December/January
December/January
t
June/July
t+1
June/July
t+2
Kit 6
The holding period,
February–July of year t,
for the first sub-portfolio of
a momentum portfolio
in July of year t
6 6
Kit+1
Kit+2
6
The holding period, July–December of year t,
for the sixth sub-portfolio of a momentum
portfolio in July of year t
Outline
1 The Q Model
2 Average Momentum Profits
3 Momentum Reversal
4 Long-run Risks in Momentum
5 Market States and Momentum
6 The Interaction of Momentum with Firm Characteristics
7 Risk Analysis
Average Momentum Profits
Point estimates
a
[se]
κ
[se]
Price
momentum
Earnings
momentum
2.52
[0.94]
0.12
[0.02]
5.41
[2.51]
0.17
[0.03]
Average Momentum Profits
Deciles, alphas and overall model performance
L
5
W
W−L
mae
[p-val]
0.83
[0.04]
0.63
[0.09]
Price momentum
r̄ S
αq
[t]
4.04
−1.61
−0.39
12.36
1.32
0.44
19.13
−1.21
−0.29
15.09
0.40
0.12
Earnings momentum
r̄ S
αq
[t]
10.48
−0.39
−0.09
15.48
1.05
0.25
18.95
−1.31
−0.37
8.47
−0.92
−0.36
Average Momentum Profits
Comparative statics
I
The investment return, rit+1
z
S
rit+1
=
|
}|
{
2 Yit+1
I
(1 − τt+1 ) κ Kit+1 + 2a Kit+1
+ τt+1 δit+1
it+1
h
i
+(1 − δit+1 ) 1 + (1 − τt+1 )a KIit+1
it+1
Ba
−wit rit+1
Iit
1 + (1 − τt )a Kit
1 − wit
{z
Iw
The levered investment return, rit+1
Components of expected stock returns:
Iit /Kit , Yit+1 /Kit+1 , (Iit+1 /Kit+1 )/(Iit /Kit ), and wit
}
Average Momentum Profits
Expected return components
Loser
5
Winner
W−L
[t]
Price momentum
Iit /Kit
(Iit+1 /Kit+1 )/(Iit /Kit )
Yit+1 /Kit+1
wit
0.22
0.83
3.16
0.34
0.19
0.99
3.00
0.25
0.25
1.15
4.10
0.22
0.04
0.32
0.94
−0.12
[3.6]
[15.4]
[5.6]
[−7.2]
Earnings momentum
Iit /Kit
(Iit+1 /Kit+1 )/(Iit /Kit )
Yit+1 /Kit+1
wit
0.19
0.95
3.01
0.29
0.19
1.00
3.06
0.28
0.20
1.05
3.53
0.20
0.01
0.10
0.52
−0.09
[2.2]
[5.0]
[3.7]
[−7.5]
Average Momentum Profits
Comparative statics
Loser
5
Winner
W−L
Price momentum
Iit /Kit
qit+1 /qit
Yit+1 /Kit+1
w it
−2.58
−7.26
−2.59
−1.39
3.77
1.00
−0.56
1.22
−7.23
2.66
4.13
−1.48
−4.65
9.92
6.73
−0.09
Earnings momentum
Iit /Kit
qit+1 /qit
Yit+1 /Kit+1
w it
0.62
−3.20
−1.65
−0.57
2.89
0.88
0.24
1.20
−4.54
0.88
1.71
−2.52
−5.16
4.07
3.36
−1.95
Outline
1 The Q Model
2 Average Momentum Profits
3 Momentum Reversal
4 Long-run Risks in Momentum
5 Market States and Momentum
6 The Interaction of Momentum with Firm Characteristics
7 Risk Analysis
Reversal
Price momentum,
The stock return
20
15
10
5
0
Iw
and rit+1
25
The levered investment return
25
S
rit+1
5
10
15
20
25
30
35
20
15
10
5
0
5
10
15
20
25
30
35
Reversal
Price momentum, (Iit+1 /Kit+1 )/(Iit /Kit ) and Yit+1 /Kit+1
1.2
4
Sales−to−capital
The investment−to−capital growth
4.5
1.1
1
0.9
3.5
3
0.8
0.7
5
10
15
20
25
30
35
2.5
5
10
15
20
25
30
35
Reversal
Earnings momentum,
The stock return
20
15
10
5
0
Iw
and rit+1
25
The levered investment return
25
S
rit+1
5
10
15
20
25
30
35
20
15
10
5
0
5
10
15
20
25
30
35
Reversal
Earnings momentum, (Iit+1 /Kit+1 )/(Iit /Kit ) and Yit+1 /Kit+1
1.2
4
Sales−to−capital
The investment−to−capital growth
4.5
1.1
1
0.9
3.5
3
0.8
0.7
5
10
15
20
25
30
35
2.5
5
10
15
20
25
30
35
Outline
1 The Q Model
2 Average Momentum Profits
3 Momentum Reversal
4 Long-run Risks in Momentum
5 Market States and Momentum
6 The Interaction of Momentum with Firm Characteristics
7 Risk Analysis
Long-run Risks
Long-run risks in the stock return
Bansal, Dittmar, and Lundblad (2005) show that aggregate
consumption risks in cash flows help explain momentum profits:
!
8
1X
gc,t−k + ui,t
gi,t = γi
8
k=1
gi,t : demeaned log real dividend growth of momentum decile i
gc,t : demeaned log real growth rate of aggregate consumption
γi : cash flow exposure to long-run consumption growth
Long-run Risks
Long-run risks in the investment return
Define the cash flow in the investment return as:
"
#
Yit+1
a Iit+1 2
?
Dit+1 = (1 − τt+1 ) κ
+
+ τt+1 δit+1
Kit+1 2 Kit+1
? : demeaned log real growth of D ?
Let gi,t
it+1 for momentum decile i:
8
?
gi,t
= γi?
1X
gc,t−k
8
!
+ ui,t
k=1
γi? : cash flow exposure to long-run consumption growth
Long-run Risks
Evidence
Price momentum
L
5
W
γi
[se]
−3.09
0.18
14.94
[4.41]
[1.27]
[9.04]
γi?
Earnings momentum
[se]
4.21 [2.09]
5.52 [1.20]
15.95 [2.87]
W−L 19.28 [11.66] 11.74 [2.78]
γi
[se]
−1.27 [2.51]
1.21 [2.20]
3.70 [1.94]
γi?
[se]
6.77 [2.06]
5.82 [1.49]
9.02 [1.64]
4.97 [3.43] 2.26 [1.67]
Outline
1 The Q Model
2 Average Momentum Profits
3 Momentum Reversal
4 Long-run Risks in Momentum
5 Market States and Momentum
6 The Interaction of Momentum with Firm Characteristics
7 Risk Analysis
Market States and Momentum
Cooper, Gutierrez, and Hameed (2004), UP (DOWN) defined as
the market returns nonnegative (negative) over the prior year
State
DOWN
UP
DOWN
UP
Price momentum
Profits
[t]
2.21
9.89
9.19
6.87
[0.62] r S
[5.04] r S
[4.50] r Iw
[4.64] r Iw
Earnings momentum
Profits
[t]
1.31
5.04
4.50
4.64
[0.40]
[6.73]
[2.46]
[6.51]
rS
rS
r Iw
r Iw
Outline
1 The Q Model
2 Average Momentum Profits
3 Momentum Reversal
4 Long-run Risks in Momentum
5 Market States and Momentum
6 The Interaction of Momentum with Firm Characteristics
7 Risk Analysis
Interaction
GMM tests, price momentum
a
[se]
κ
[se]
p-val
mae
Size
Age
Trading
volume
Credit
ratings
Stock return
volatility
Book-tomarket
2.33
0.70
0.09
0.01
0.00
3.66
2.37
0.95
0.12
0.01
0.00
1.29
2.76
0.93
0.12
0.01
0.00
1.67
1.97
0.83
0.12
0.01
0.00
1.68
3.17
0.82
0.12
0.02
0.00
1.92
3.44
0.89
0.13
0.01
0.00
3.10
Interaction
GMM tests, earnings momentum
a
[se]
κ
[se]
p-val
mae
Size
Age
Trading
volume
Credit
ratings
Stock return
volatility
Book-tomarket
2.74
0.60
0.09
0.01
0.00
4.37
2.75
1.55
0.12
0.02
0.27
1.08
2.56
1.32
0.12
0.02
0.00
2.30
1.14
0.72
0.11
0.01
0.00
1.35
2.74
0.76
0.12
0.02
0.00
1.95
7.20
2.36
0.16
0.02
0.01
2.88
Interaction
Size and price (earnings) momentum
25
25
1W
Average predicted returns
Average predicted returns
1W
20
1M
2W
15
1L
2M
10
3W
2L
5
0
0
3M
3L
5
10
15
20
Average realized returns
25
1M
20
15
2W
2M
1L
10
3W
2L
5
0
0
3M
3L
5
10
15
20
Average realized returns
25
Interaction
Book-to-market and price (earnings) momentum
25
3W
20
1W
15
Average predicted returns
Average predicted returns
25
2W
1M
2M
10
3M
1L
2L
3L
5
0
0
5
10
15
20
Average realized returns
25
20
3W
1W
2W
15
1M
2M 3M
1L
10
2L
3L
5
0
0
5
10
15
20
Average realized returns
25
Outline
1 The Q Model
2 Average Momentum Profits
3 Momentum Reversal
4 Long-run Risks in Momentum
5 Market States and Momentum
6 The Interaction of Momentum with Firm Characteristics
7 Risk Analysis
Risk Analysis
The consumption-investment model
Testing the consumption CAPM and the Q model jointly:
S
f
E [Mt+1 (rit+1
− rt+1
)] = 0,
f
E [Mt+1 (rt+1
/it+1 )] = 1,
f
Iw
E [Mt+1 (rit+1
− rt+1
)] = 0,
S
Iw
− rit+1
] = 0.
E [rit+1
Mt+1 = ρ (Ct+1 /Ct )−γ . Annual data
Risk Analysis
R : CCAPM with r , CCAPM with r Iw , Q-moment
S
25
20
20
15
10
5
0
−5
−10
−10 −5
2
8
67
3 45
9
W
15
10
5
L 2 345 7 8 9 W
6
0
−5
L
0 5 10 15 20
Average real stock returns
25
25
−10
−10 −5 0 5 10 15 20 25
Average real levered investment returns
Average levered investment returns
25
Average real predicted rIw
Average real predicted stock returns
6
20
W
15
8
10
2
5
3
9
67
45
L
0
−5
−10
−10 −5
0 5 10 15 20
Average realized stock returns
25
Risk Analysis
SUE: CCAPM with r , CCAPM with r Iw , Q-moment
25
20
20
15
9
L2 4 56 78W
3
10
5
0
−5
−10
−10 −5
25
15
10
5
0
L4 7
23 5 6
8
9W
−5
0 5 10 15 20
Average real stock returns
25
−10
−10 −5 0 5 10 15 20 25
Average real levered investment returns
Average levered investment returns
25
Average real predicted rIw
Average real predicted stock returns
S
20
15
10
3
L2
4
6
5
W
89
7
5
0
−5
−10
−10 −5
0 5 10 15 20
Average realized stock returns
25
Conclusion
Summary and future work
The Q model broadly consistent with many aspects of momentum:
Managers align investment properly with costs of capital
Momentum per se might not imply investor irrationality
Several directions in the future:
Value and momentum jointly, industry-specific parameters
Value and momentum in currencies, International Q model
Recent developments in CCAPM in the joint model