Lecture Notes
Liu, Whited, and Zhang (2009, J. of Political Economy):
Investment-Based Expected Stock Returns
Lu Zhang1
1
The Ohio State University
and NBER
BUSFIN 920: Theory of Finance
The Ohio State University
Autumn 2011
Outline
What and Why
Key Results
Model
Econometric Methods
Matching Expected Stock Returns
Matching Expected Returns and Variances
Summary and Future Work
Outline
What and Why
Key Results
Model
Econometric Methods
Matching Expected Stock Returns
Matching Expected Returns and Variances
Summary and Future Work
Introduction
What
We derive and test q-theory implications for the cross-section of
expected stock returns
Introduction
Motivation: Many characteristics-return relations in capital markets research
Realized returns
z}|{
rjt+1
Abnormal returns
=
Et [rjt+1 ]
| {z }
+
z}|{
jt+1
Expected returns
Use the q-theory of investment to link expected returns to firm
characteristics
Outline
What and Why
Key Results
Model
Econometric Methods
Matching Expected Stock Returns
Matching Expected Returns and Variances
Summary and Future Work
Key Results
Average predicted vs. realized returns, ten SUE portfolios, the CAPM
Average predicted returns
0.3
0.25
0.2
Low
High
0.15
0.1
0.05
0.05
0.1
0.15
0.2
0.25
Average realized returns
0.3
Key Results
Average predicted vs. realized returns, ten SUE portfolios, the Fama-French model
Average predicted returns
0.3
0.25
0.2
Low
High
0.15
0.1
0.05
0.05
0.1
0.15
0.2
0.25
Average realized returns
0.3
Key Results
Average predicted vs. realized returns, ten SUE portfolios, the consumption-CAPM
Average predicted returns
0.3
0.25
Low
High
0.2
0.15
0.1
0.05
0.05
0.1
0.15
0.2
0.25
Average realized returns
0.3
Key Results
Average predicted vs. realized returns, ten SUE portfolios, the q-theory model
0.3
Average predicted returns
High
0.25
0.2
0.15
0.1
Low
0.05
0.05
0.1
0.15
0.2
0.25
Average realized returns
0.3
Key Results
Average predicted vs. realized returns, ten B/M portfolios, the CAPM
Average predicted returns
0.3
0.25
0.2
Low
High
0.15
0.1
0.05
0.05
0.1
0.15
0.2
0.25
Average realized returns
0.3
Key Results
Average predicted vs. realized returns, ten B/M portfolios, the Fama-French model
Average predicted returns
0.3
0.25
0.2
0.15
Low
High
0.1
0.05
0.05
0.1
0.15
0.2
0.25
Average realized returns
0.3
Key Results
Average predicted vs. realized returns, ten B/M portfolios, the consumption-CAPM
Average predicted returns
0.3
0.25
0.2
High
Low
0.15
0.1
0.05
0.05
0.1
0.15
0.2
0.25
Average realized returns
0.3
Key Results
Average predicted vs. realized returns, ten B/M portfolios, the q-theory model
Average predicted returns
0.3
0.25
High
0.2
Low
0.15
0.1
0.05
0.05
0.1
0.15
0.2
0.25
Average realized returns
0.3
Key Results
Average predicted vs. realized returns, ten CI portfolios, the CAPM
Average predicted returns
0.3
0.25
0.2
0.15
0.1
High
0.05
0.05
0.1
Low
0.15
0.2
0.25
Average realized returns
0.3
Key Results
Average predicted vs. realized returns, ten CI portfolios, the Fama-French model
Average predicted returns
0.3
0.25
0.2
0.15
0.1
0.05
0.05
High
0.1
Low
0.15
0.2
0.25
Average realized returns
0.3
Key Results
Average predicted vs. realized returns, ten CI portfolios, the consumption-CAPM
Average predicted returns
0.3
0.25
High
Low
0.2
0.15
0.1
0.05
0.05
0.1
0.15
0.2
0.25
Average realized returns
0.3
Key Results
Average predicted vs. realized returns, ten CI portfolios, the q-theory model
Average predicted returns
0.3
0.25
High
0.2
Low
0.15
0.1
0.05
0.05
0.1
0.15
0.2
0.25
Average realized returns
0.3
Key Results
Predicted vs. realized stock return volatilities, joint estimation of mean and variance, the
q-theory model
0.4
Predicted volatilities
0.35
Low
0.3
0.25
0.2
0.15
0.1
High
0.05
0
0
0.1
0.2
0.3
Realized volatilities
0.4
Key Results
Average predicted vs. realized returns, joint estimation of mean and variance, the
q-theory model
Average predicted returns
0.3
0.25
Low
High
0.2
0.15
0.1
0.05
0.05
0.1
0.15
0.2
0.25
Average realized returns
0.3
Outline
What and Why
Key Results
Model
Econometric Methods
Matching Expected Stock Returns
Matching Expected Returns and Variances
Summary and Future Work
Model
The neoclassical q-theory framework à la Cochrane (1991). Firms use capital and
costlessly adjustable inputs such as labor
Operating profits, Π(Kit , Xit ), with
∂Π(Kit , Xit )
Yit
=α
∂Kit
Kit
with Yit = Sales
Capital evolves as:
Kit+1 = Iit + (1 − δit )Kit
Convex adjustment costs:
a
Φ(Iit , Kit ) =
2
Iit
Kit
2
Kit
Model
Equity-value maximization
B
One-period debt, Bit+1 , with corporate bond return rit+1
Payout, Dit , defined as:
(1−τt )[Π(Kit , Xit )−Φ(Iit , Kit )]−Iit +Bit+1 −ritB Bit +τt δit Kit +τt (ritB −1)Bit
The cum-dividend market value of the equity:
"∞
#
X
Vit ≡
max
Et
Mt+s Dit+s
∞
{Iit+s ,Kit+s+1 ,Bit+s+1 }s=0
s=0
in which Mt+1 is the stochastic discount factor, correlated with
Xit+1
Model
The investment return
I
I
Et [Mt+1 rit+1
] = 1, in which rit+1
is the investment return:
Marginal benefit of investment at time t+1
z










I
rit+1
≡
}|
#
Yit+1
a Iit+1 2
(1 − τt+1 ) α
+
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
}
"
Expected continuation value
1 + (1 − τt )
|
{z
Iit
Kit
}
Marginal cost of investment at time t
{









Model
The WACC Proposition
Ba
Ba = (1 − τ
B
Define rit+1
t+1 )rit+1 + τt+1 , then Et Mt+1 rit+1 = 1
Define Pit ≡ Vit − Dit and the stock return
S
rit+1
≡ (Pit+1 + Dit+1 )/Pit
Under constant returns to scale, the investment return is the
weighted average of stock and after-tax bond returns:
I
Ba
S
S
Iw
rit+1
= wit rit+1
+ (1 − wit )rit+1
⇒ rit+1
= rit+1
≡
I
Ba
rit+1
− wit rit+1
1 − wit
in which wit is market leverage, wit ≡ Bit+1 /(Pit + Bit+1 )
Outline
What and Why
Key Results
Model
Econometric Methods
Matching Expected Stock Returns
Matching Expected Returns and Variances
Summary and Future Work
Econometric Methods
GMM
Do expected stock returns equal expected levered investment
returns?
h
i
S
Iw
E rit+1
− rit+1
=0
Do stock return variances equal levered investment return
variances?
h
i2 h
i2 S
S
Iw
Iw
− rit+1 − E rit+1
E rit+1 − E rit+1
=0
Econometric Methods
Testing portfolios
Three sets of testing portfolios
I
Ten Standardized Unexpected Earnings (SUE) portfolios of
Chan, Jegadeesh, and Lakonishok (1996)
I
Ten book-to-market portfolios as in Fama and French (1993)
I
Ten “abnormal” investment portfolios of Titman, Wei, and Xie
(2004)
Why portfolios?
I
Larger and more reliable expected return spreads across
portfolios than across individual stocks
I
Smoothing lumpy investment as in Thomas (2002)
middle of year t to the middle of year t+1. The bottomline is that the investment return timing
matches naturally with the stock return timing from the Fama-French portfolio approach.
Econometric Methods
Timing
Figure 1: Timing Alignment between Annual Stock and Investment Returns
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
Kit+1
S
rit+1
B , r Ba
rit+1
it+1
(from July of year t
to June of t+1)
Kit+2
-
Econometric Methods
Measurement
I
Kit : gross property, plant, and equipment
I
Iit : capital expenditure minus sales of property, plant, and
equipment
I
Yit : sales
I
Bit : total long-term debt
I
Pit : market value of common equity
I
δit : the amount of depreciation divided by capital
I
B : impute bond ratings, assign corporate bond returns of a
rit+1
given rating to all firms with the same rating
I
τt : statutory tax rate of corporate income
Outline
What and Why
Key Results
Model
Econometric Methods
Matching Expected Stock Returns
Matching Expected Returns and Variances
Summary and Future Work
Expected Stock Returns
Point estimates and tests of overidentification
a
[ste]
α
[ste]
χ2
d.f.
p
m.p.e.
SUE
B/M
CI
7.68
[1.72]
0.32
[0.03]
22.34
[25.47]
0.50
[0.31]
0.97
[0.29]
0.21
[0.02]
4.37
8
0.82
0.74
5.99
8
0.65
2.32
6.52
8
0.59
1.51
Expected Stock Returns
Euler equation errors, ten SUE portfolios
Low
5
High
H−L
[tH−L ]
Panel A: Ten SUE portfolios
ei
eiFF
eiC
eiq
−1.69
−4.59
−8.07
0.26
6.56
1.96
−0.04
1.66
10.86
9.47
5.31
−0.15
12.55
14.06
13.38
−0.40
[5.53]
[5.31]
[1.35]
[−0.41]
Expected Stock Returns
Euler equation errors, ten B/M portfolios
Low
5
High
H−L
[tH−L ]
Panel B: Ten B/M portfolios
ei
eiFF
eiC
eiq
−4.91
−0.54
−5.43
−3.94
5.19
1.80
0.27
2.35
13.65
6.76
6.88
−2.73
18.56
7.30
12.31
1.21
[2.51]
[3.25]
[0.26]
[0.79]
Expected Stock Returns
Euler equation errors, ten CI portfolios
Low
5
High
H−L
[tH−L ]
Panel C: Ten CI portfolios
ei
eiFF
eiC
eiq
8.21
6.45
4.03
−0.97
5.89
1.54
0.46
2.72
1.91
0.11
−4.35
−1.45
−6.30
−6.34
−8.38
−0.49
[−3.88]
[−3.99]
[−1.35]
[−0.41]
Expected Stock Returns
Economic determinants of expected stock returns
I
rit+1
≡
Iw
rit+1
≡
Iit+1 2
Yit+1
a
+ τt+1 δit+1
(1 − τt+1 ) α K
+
2 Kit+1
it+1
h
i
+(1 − δit+1 ) 1 + (1 − τt+1 )a KIit+1
it+1
1 + (1 − τt )a KIitit
I
Ba
rit+1
− wit rit+1
1 − wit
Determinants: Yit+1 /Kit+1 , Iit+1 /Iit , δit+1 , and Iit /Kit , also wit and
B
rit+1
Expected Stock Returns
Characteristics, ten SUE portfolios
Iit /Kit
(Iit+1 /Kit+1 )/(Iit /Kit )
Yit+1 /Kit+1
δit+1
wit
B
rit+1
Low
5
High
H−L
[tH−L ]
0.12
0.89
1.52
0.08
0.30
9.44
0.11
1.00
1.50
0.08
0.28
9.76
0.12
1.06
1.83
0.08
0.21
9.38
0.00
0.17
0.31
0.00
−0.10
−0.06
[0.70]
[4.06]
[5.16]
[0.63]
[−5.83]
[−0.27]
Expected Stock Returns
Expected returns accounting, ten SUE portfolios
Iit /Kit
qit+1 /qit
Yit+1 /Kit+1
w it
Low
5
High
H−L
−2.48
−5.23
−0.78
0.13
4.45
1.76
0.39
1.89
−4.26
3.62
3.53
−1.46
−1.78
8.85
4.31
−1.58
Expected Stock Returns
Characteristics, ten B/M portfolios
Iit /Kit
(Iit+1 /Kit+1 )/(Iit /Kit )
Yit+1 /Kit+1
δit+1
wit
B
rit+1
Low
5
High
H−L
[tH−L ]
0.18
0.98
1.95
0.10
0.08
8.17
0.11
1.00
1.45
0.07
0.27
8.09
0.08
1.02
1.38
0.07
0.53
8.52
−0.10
0.04
−0.57
−0.03
0.44
0.35
[−7.95]
[0.68]
[−6.77]
[−5.01]
[12.44]
[1.05]
Expected Stock Returns
Expected returns accounting, ten B/M portfolios
Iit /Kit
qit+1 /qit
Yit+1 /Kit+1
w it
Low
5
High
H−L
−42.06
−1.92
0.16
−6.00
4.69
2.11
0.92
2.19
48.17
−4.06
−6.33
5.58
90.23
−2.14
−6.49
11.58
Expected Stock Returns
Characteristics, ten CI portfolios
Iit /Kit
(Iit+1 /Kit+1 )/(Iit /Kit )
Yit+1 /Kit+1
δit+1
wit
B
rit+1
Low
5
High
H−L
[tH−L ]
0.09
1.25
1.84
0.08
0.35
8.47
0.11
1.04
1.58
0.07
0.25
8.27
0.16
0.81
1.89
0.08
0.28
8.44
0.07
−0.44
0.05
0.00
−0.07
−0.03
[11.06]
[−7.23]
[0.38]
[−0.46]
[−2.59]
[−0.15]
Expected Stock Returns
Expected returns accounting, ten CI portfolios
Iit /Kit
qit+1 /qit
Yit+1 /Kit+1
w it
Low
5
High
H−L
2.86
0.73
0.57
1.80
3.50
2.97
−0.44
2.61
−5.67
−3.87
0.09
−0.91
−8.53
−4.60
−0.48
−2.71
Outline
What and Why
Key Results
Model
Econometric Methods
Matching Expected Stock Returns
Matching Expected Returns and Variances
Summary and Future Work
Joint Estimation
Point estimates and tests of overidentification
SUE
B/M
CI
28.88
[16.25]
0.61
[0.27]
11.48
[4.75]
0.35
[0.07]
16.23
[5.53]
0.36
[0.08]
χ2(2)
d.f.(2)
p(2)
m.p.e.(2)
5.14
8
0.74
0.03
6.18
8
0.63
0.04
6.05
8
0.64
0.02
χ2(1)
d.f.(1)
p(1)
m.p.e.(1)
5.22
8
0.73
3.45
4.38
8
0.82
2.58
4.81
8
0.78
2.22
χ2
d.f.
p
5.45
18
1.00
6.17
18
1.00
6.62
18
0.99
a
[ste]
α
[ste]
Joint Estimation
Predicted vs. realized stock return volatilities, ten B/M portfolios, the q-theory model
0.4
Predicted volatilities
0.35
0.3
High
0.25
0.2
0.15
0.1
Low
0.05
0
0
0.1
0.2
0.3
Realized volatilities
0.4
Joint Estimation
Average predicted vs. realized returns, ten B/M portfolios, the q-theory model
Average predicted returns
0.3
0.25
0.2
High
Low
0.15
0.1
0.05
0.05
0.1
0.15
0.2
0.25
Average realized returns
0.3
Joint Estimation
Predicted vs. realized stock return volatilities, ten CI portfolios, the q-theory model
0.4
High
Predicted volatilities
0.35
0.3
0.25
Low
0.2
0.15
0.1
0.05
0
0
0.1
0.2
0.3
Realized volatilities
0.4
Joint Estimation
Average predicted vs. realized returns, ten CI portfolios, the q-theory model
Average predicted returns
0.3
High
0.25
0.2
Low
0.15
0.1
0.05
0.05
0.1
0.15
0.2
0.25
Average realized returns
0.3
Outline
What and Why
Key Results
Model
Econometric Methods
Matching Expected Stock Returns
Matching Expected Returns and Variances
Summary and Future Work
Conclusion
Summary and interpretation
Summary: Derive and test the q-theory model for cross-sectional
returns
Interpretation: Portfolios of firms do a good job in aligning their
investment policies with costs of equity capital, and this alignment
drives many characteristics-return relations
Conclusion
Future Work
More realistic ingredients (decreasing returns, investment lags,
financing constraints, labor, organizational capital):
I
Balance realism and analytical tractability, empirical challenges
(data limitations)
More puzzles in cross-sectional returns (momentum, asset growth,
accruals, distress, M&As, net equity issues, governance)
An investment-based theory of corporate bond returns
Belo, Xue, and Zhang (2010): Cross-sectional Tobin’s Q
Methodology applicable in dynamic corporate finance