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 2012
Outline
1 What/Why
2 Key Results
3 Model
4 Econometric Methods
5 Matching Expected Stock Returns
6 Matching Expected Returns and Variances
7 Summary and Future Work
Outline
1 What/Why
2 Key Results
3 Model
4 Econometric Methods
5 Matching Expected Stock Returns
6 Matching Expected Returns and Variances
7 Summary and Future Work
Introduction
What
We derive and test q-theory implications for the cross-section of
expected stock returns
Introduction
Why: Many characteristics-return relations in asset pricing
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
1 What/Why
2 Key Results
3 Model
4 Econometric Methods
5 Matching Expected Stock Returns
6 Matching Expected Returns and Variances
7 Summary and Future Work
Key Results
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
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
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
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
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
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
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
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
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
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
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
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
1 What/Why
2 Key Results
3 Model
4 Econometric Methods
5 Matching Expected Stock Returns
6 Matching Expected Returns and Variances
7 Summary and Future Work
Model
The neoclassical q-theory of investment. Firms use capital and
costlessly adjustable inputs such as labor
Operating profits, Π(Kit , Xit ), with
Yit
∂Π(Kit , Xit )
=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 pricing kernel, 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

"
}|
2 #
Yit+1
a Iit+1
(1 − τt+1 ) α
+

K
2
Kit+1

it+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 )
|
{z
Iit
Kit
}
Marginal cost of investment at time t
{









Model
The WACC Proposition
Ba = (1 − τ
B
Ba
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
1 What/Why
2 Key Results
3 Model
4 Econometric Methods
5 Matching Expected Stock Returns
6 Matching Expected Returns and Variances
7 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
− E rit+1
− E rit+1
E rit+1
− rit+1
=0
Econometric Methods
Testing portfolios
Three sets of testing portfolios
Ten Standardized Unexpected Earnings (SUE) portfolios of
Chan, Jegadeesh, and Lakonishok (1996)
Ten book-to-market portfolios as in Fama and French (1993)
Ten “abnormal” investment portfolios of Titman, Wei, and Xie
(2004)
Why portfolios?
Larger and more reliable expected return spreads across
portfolios than across individual stocks
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
Kit : gross property, plant, and equipment
Iit : capital expenditure minus sales of property, plant, and
equipment
Yit : sales
Bit : total long-term debt
Pit : market value of common equity
δit : the amount of depreciation divided by capital
B : impute bond ratings, assign corporate bond returns of a
rit+1
given rating to all firms with the same rating
τt : statutory tax rate of corporate income
Outline
1 What/Why
2 Key Results
3 Model
4 Econometric Methods
5 Matching Expected Stock Returns
6 Matching Expected Returns and Variances
7 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
(1 − τt+1 )
I
rit+1
≡
Iw
rit+1
≡
Yit+1
αK
it+1
a
2
Iit+1
Kit+1
2 +
+ τt+1 δit+1
i
h
+(1 − δit+1 ) 1 + (1 − τt+1 )a KIit+1
it+1
Iit
1 + (1 − τt )a Kit
I
Ba
rit+1
− wit rit+1
1 − wit
B
Yit+1 /Kit+1 , Iit+1 /Iit , δit+1 , and Iit /Kit ; wit and 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
1 What/Why
2 Key Results
3 Model
4 Econometric Methods
5 Matching Expected Stock Returns
6 Matching Expected Returns and Variances
7 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
1 What/Why
2 Key Results
3 Model
4 Econometric Methods
5 Matching Expected Stock Returns
6 Matching Expected Returns and Variances
7 Summary and Future Work
Conclusion
Summary and interpretation
Derive and test the q-theory model for cross-sectional returns
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, such as decreasing returns, investment
lags, financing constraints, labor, organizational capital:
Balancing realism and analytical tractability
More puzzles in cross-sectional returns (momentum, asset growth,
accruals, distress, M&As, net equity issues, governance, etc)
Methodology applicable in dynamic corporate finance
An investment-based theory of corporate bond returns
Belo, Xue, and Zhang (2011): Fit cross-sectional equity valuation
ratios; firm-level estimation, allow industry specific parameters