The Investment CAPM
Lu Zhang∗
The Ohio State University
and NBER
December 2015†
Abstract
A new class of Capital Asset Pricing Models arises from the first principle of real investment for individual firms. Conceptually as “causal” as the consumption CAPM, yet
empirically more tractable, the investment CAPM emerges as a leading asset pricing
paradigm. Firms do a good job in aligning investment policies with costs of capital, and
this alignment drives many empirical patterns that are anomalous in the consumption
CAPM. Most important, integrating the anomalies literature in finance and accounting
with neoclassical economics, the investment CAPM succeeds in mounting an efficient
markets counterrevolution to behavioral finance in the past 15 years.
∗
Fisher College of Business, The Ohio State University, 760A Fisher Hall, 2100 Neil Avenue, Columbus OH 43210;
and NBER. Tel: (614) 292-8644, fax: (614) 292-7062, and e-mail: zhanglu@fisher.osu.edu.
†
I thank Hang Bai, Zhengyu Cao, Alex Edmans, Andrei Gonçalves, Kewei Hou, Xingzhou Li, Stephen Penman,
Michael Weisbach, and especially Chen Xue for helpful comments.
Contents
1 The Investment CAPM
1
2 The q-Factor Model
5
2.1
Why A Factor Model? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.2
Economic Intuition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.2.1
Investment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.2.2
Profitability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.2.3
Beyond Investment and Profitability . . . . . . . . . . . . . . . . . . . . . . .
9
2.3
Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.4
Factors War . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4.1
The Empire Strikes Back . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4.2
A Better Factor Model That Explains More Anomalies . . . . . . . . . . . . . 14
2.4.3
2.4.2.1
Conceptual Grounds . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.4.2.2
Empirical Grounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Which ROE Factor? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4.3.1
2.4.3.2
2.4.4
2.4.5
2.5
Lagging the Deflator: The Irrelevance of Gross Profitability . . . . . 20
The Importance of Monthly Sorts . . . . . . . . . . . . . . . . . . . 22
Understanding the Low Risk Anomaly . . . . . . . . . . . . . . . . . . . . . . 22
2.4.4.1
The Idiosyncratic Volatility Anomaly . . . . . . . . . . . . . . . . . 24
2.4.4.2
The Beta Anomaly
. . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Independent Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.5.1
2.5.2
Related Studies on the Investment Premium
. . . . . . . . . . . . . . . . . . 31
2.5.1.1
Different Forms of the Investment Premium . . . . . . . . . . . . . . 31
2.5.1.2
The Cross-sectional Variation of the Investment Premium . . . . . . 34
2.5.1.3
Applications in the Accrual Anomaly . . . . . . . . . . . . . . . . . 36
Related Studies on the Profitability Premium . . . . . . . . . . . . . . . . . . 40
2.5.2.1
Post-earnings-announcement Drift . . . . . . . . . . . . . . . . . . . 40
2.5.2.2
Fundamental Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3 The Investment CAPM: Structural Estimation and Tests
3.1
The Multiperiod Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.1.1
3.1.2
3.2
44
Expected Stock Returns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.1.1.1
Liu, Whited, and Zhang (2009) . . . . . . . . . . . . . . . . . . . . . 47
3.1.1.2
Liu and Zhang (2014) . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Equity Valuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2.1
The Neoclassical q-Theory of Investment . . . . . . . . . . . . . . . . . . . . . 53
1
3.2.2
Investment-Based Asset Pricing Tests . . . . . . . . . . . . . . . . . . . . . . 54
3.2.2.1
Aggregate Stock Market . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2.2.2
Cross-sectional Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2.2.3
New Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4 Quantitative Investment Theories
58
4.1
Real Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.2
Neoclassical Investment Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
4.3
Controversies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.4
4.3.1
Is Value Riskier Than Growth? . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4.3.2
Does the Conditional CAPM Explain the Value Premium? . . . . . . . . . . 65
4.3.3
What Explains the Failure of the CAPM? . . . . . . . . . . . . . . . . . . . . 67
4.3.3.1
Two-shock Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.3.3.2
Disasters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.4.1
Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.4.2
Beyond Value and Momentum . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.4.3
4.4.2.1
Equity Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.4.2.2
Real Estate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.4.2.3
4.4.2.4
Inventories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
Intangibles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
Debt Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5 The Big Picture
5.1
5.2
5.3
80
The Investment CAPM: A Historical Perspective . . . . . . . . . . . . . . . . . . . . 80
5.1.1
Origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.1.2
Oblivion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.1.3
Rebirth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
Complementarity with the Consumption CAPM . . . . . . . . . . . . . . . . . . . . 87
5.2.1
Marshall’s Scissors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
5.2.2
The Aggregation Critique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.2.3
Beta Measurement Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
An Efficient Markets Counterrevolution to Behavioral Finance
. . . . . . . . . . . . 92
5.3.1
A Dark Age of Finance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
5.3.2
Do Psychological Biases Explain Asset Pricing Anomalies? . . . . . . . . . . 94
5.3.2.1
Are Investors More Biased Than Managers? . . . . . . . . . . . . . 94
5.3.2.2
Are U.S. Investors More Biased Than Chinese Investors? . . . . . . 95
5.3.2.3
Limits to Limits to Arbitrage? . . . . . . . . . . . . . . . . . . . . . 97
5.3.2.4
Measurement Ahead of Theory? . . . . . . . . . . . . . . . . . . . . 97
2
5.3.3
The Joint-Hypothesis Problem Squared . . . . . . . . . . . . . . . . . . . . . 98
5.3.3.1
A Tribute to Fama and French (1993) . . . . . . . . . . . . . . . . . 99
5.3.3.2
Interpreting Factors: The Investment CAPM Perspective . . . . . . 99
5.3.3.3
Beyond the “Risk Doctrine” . . . . . . . . . . . . . . . . . . . . . . 101
6 Challenges for Future Work
102
6.1
Where Do We Stand? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.2
Where Are We Going? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
A Data
121
B Derivations
122
C Timing Alignment
123
3
1
The Investment CAPM
Consider a two-period stochastic general equilibrium model. The economy has three defining characteristics of neoclassical economics: (i) Agents have rational expectations; (ii) consumers maximize
utility, and firms maximize their market value of equity; and finally (iii) markets clear.
There are two dates, t and t + 1. The economy is populated by a representative household
and heterogeneous firms, indexed by i = 1, 2, . . . , N . The representative household maximizes its
expected utility, U (Ct ) + ρEt [U (Ct+1 )], in which ρ is the time preference coefficient, and Ct and
Ct+1 are consumption expenditures in t and t + 1, respectively. Let Pit be the ex-dividend equity,
and Dit the dividend of firm i at period t. The first principle of consumption says that:
Pit = Et [Mt+1 (Pit+1 + Dit+1 )]
⇒
S
] = 1,
Et [Mt+1 rit+1
(1)
S
in which rit+1
≡ (Pit+1 + Dit+1 )/Pit is firm i’s stock return, and Mt+1 ≡ ρU ′ (Ct+1 )/U ′ (Ct ) is the
household’s stochastic discount factor. Equation (1) can be rewritten as:
S
Et [rit+1
] − rf t = β M
it λM t ,
(2)
S
in which rf t ≡ 1/Et [Mt+1 ] is the real interest rate, β M
it ≡ −Cov(rit+1 , Mt+1 )/Var(Mt+1 ) is the
consumption beta, and λM t ≡ Var(Mt+1 )/Et [Mt+1 ] is the price of the consumption risk.
Equation (2) is the consumption-based Capital Asset Pricing Model (the consumption CAPM),
which was first derived by Rubinstein (1976), Lucas (1978), and Breeden (1979). The classic
CAPM, due to Treynor (1962), Sharpe (1964), Lintner (1965), and Mossin (1966), can be derived
as a special case of the consumption CAPM under quadratic utility or exponential utility with
normally distributed returns (Cochrane 2005, p. 152–155).
On the production side, firms produce a single commodity to be consumed or invested. Firm i
starts with the productive capital, Kit , operates in both dates, and exits at the end of date t + 1
with a liquidation value of zero. The rate of capital depreciation is set to be 100%. Firms differ
1
in capital, Kit , and profitability, Xit , both of which are known at the beginning of date t. The
operating profits are given by Πit ≡ Xit Kit . Firm i’s profitability at date t + 1, Xit+1 , is stochastic,
and is subject to aggregate shocks affecting all firms simultaneously, as well as firm-specific shocks
affecting only firm i. Let Iit be the investment for date t, then Kit+1 = Iit . Investment entails
quadratic adjustment costs, (a/2)(Iit /Kit )2 Kit , in which a > 0 is a constant parameter.
Firm i uses its operating profits at date t to pay investment and adjustment costs. If the net
cash flow, Dit ≡ Xit Kit − Iit − (a/2)(Iit /Kit )2 Kit , is positive, the firm distributes it back to the
household. A negative Dit means external equity raised by the firm from the household. At date
t + 1, firm i uses capital, Kit+1 , to obtain operating profits, Xit+1 Kit+1 , which is in turn distributed
as dividends back to the household, Dit+1 ≡ Xit+1 Kit+1 . With only two dates, firm i does not
invest in date t + 1, i.e., Iit+1 = 0, and the ex-dividend equity value, Pit+1 , is zero.
Taking the household’s stochastic discount factor, Mt+1 , as given, firm i chooses Iit to maximize
the cum-dividend equity value at the beginning of date t:
Pit + Dit
=
max
{Iit }
"
a
Xit Kit − Iit −
2
Iit
Kit
2
#
Kit + Et [Mt+1 Xit+1 Kit+1 ] .
(3)
The first principle of investment for firm i says that:
1+a
Iit
= Et [Mt+1 Xit+1 ].
Kit
(4)
Intuitively, the marginal costs of investment, consisting of the purchasing price (unity) and the
marginal adjustment costs, a(Iit /Kit ), must equal marginal q, which is the present value of the
marginal benefits of investment given by the marginal product of capital, Xit+1 .
The first principle of investment can be rewritten without the stochastic discount factor, Mt+1 ,
as in Cochrane (1991). In particular, equation (3), when combined with Dit ≡ Xit Kit − Iit −
(a/2)(Iit /Kit )2 Kit , implies that the ex-dividend equity value at the optimum is:
Pit = Et [Mt+1 Xit+1 Kit+1 ].
2
(5)
The stock return can then be rewritten as:
S
rit+1
=
Pit+1 + Dit+1
Xit+1 Kit+1
Xit+1
=
=
.
Pit
Et [Mt+1 Xit+1 Kit+1 ]
Et [Mt+1 Xit+1 ]
(6)
The investment CAPM arises from combining equations (4) and (6):
S
rit+1
=
Xit+1
.
1 + a(Iit /Kit )
(7)
Intuitively, firm i keeps investing until the date-t marginal costs of investment, 1 + a(Iit /Kit ), equal
the marginal benefits of investment at t + 1, Xit+1 , discounted to date t with the stock return,
S , as the discount rate. Equivalently, the ratio of the marginal benefits of investment at t + 1
rit+1
S .
divided by the marginal costs of investment at t equals the discount rate, rit+1
Most important, the investment CAPM, as an asset pricing theory, gives rise to cross-sectionally
varying expected returns. The investment CAPM predicts that, all else equal, high investment stocks
should earn lower expected returns than low investment stocks, and that stocks with high expected
profitability should earn higher expected returns than stocks with low expected profitability. When
expected returns vary cross-sectionally in equilibrium, stock prices will adjust in a way that connects expected returns to characteristics. In particular, stock prices will not adjust to conform to
a cross-sectionally constant discount rate, meaning that characteristics do not predict returns. A
cross-sectionally constant discount rate can only occur if all stocks are equally risky.
The intuition behind the investment CAPM is just capital budgeting. Investment predicts returns because given expected profitability, high costs of capital imply low net present values of new
capital and low investment, and low costs of capital imply high net present values of new capital
and high investment. Profitability predicts returns because high expected profitability relative to
low investment must imply high discount rates. The high discount rates are necessary to offset the
high expected profitability to induce low net present values of new capital and low investment. If
the discount rates were not high enough, firms would instead observe high net present values of new
3
capital and invest more. Conversely, low expected profitability relative to high investment must
imply low discount rates. If the discount rates were not low enough to counteract the low expected
profitability, firms would instead observe low net present values of new capital and invest less.
The consumption CAPM and the investment CAPM are the two sides of the same coin in general equilibrium, delivering identical expected returns (Lin and Zhang 2013). To see this insight,
combining the consumption CAPM in equation (2) and the investment CAPM in equation (7) yields:
S
rf t + β M
it λM t = Et [rit+1 ] =
Et [Xit+1 ]
.
1 + a(Iit /Kit )
(8)
Intuitively, the consumption CAPM, which is derived from the first principle of consumption, connects expected returns to consumption betas. The consumption CAPM predicts that consumption
betas are sufficient statistics for expected returns. Once consumption betas are controlled for, characteristics should not affect the cross section of expected returns. This prediction is the organizing
framework for the bulk of empirical finance and capital markets research in accounting.
The investment CAPM, which is derived from the first principle of investment, connects expected returns to characteristics. It predicts that characteristics are sufficient statistics for expected
returns. Once characteristics are controlled for, consumption betas should not affect expected returns. As such, the consumption CAPM, and the classic CAPM as its special case, miss what
neoclassical economics has to say about the cross section of expected returns from the production
side altogether. Derived from the two-period manager’s problem, the investment CAPM is the dual
proposition to the classic CAPM, which is in turn derived from a two-period investor’s problem.
This article serves three purposes. First, I review the investment CAPM literature within a
unified conceptual framework, identify its main strands, and clarify their interconnections. Second,
I explain the big picture. I trace the origin of the investment CAPM to at least Fisher (1930,
The Theory of Interest). I describe the rebirth of the investment CAPM as an inevitable response
to the anomalies literature and its challenge to efficient markets (Fama 1965, 1970) and rational
4
expectations (Muth 1961; Lucas 1972). Finally, I hope to make the investment CAPM literature accessible to doctoral students, economists who do not specialize in asset pricing, as well as investment
management professionals. While mostly a review, some new insights and results are also furnished.
The rest of the paper unfolds as follows. Section 2 shows how the investment CAPM is tested as
a factor model. Section 3 shows the structural estimation and tests of the multiperiod investment
CAPM. Section 4 reviews quantitative investment theories. Section 5 explains the big picture of the
investment CAPM, and clarifies its relations to the consumption CAPM and behavioral finance.
Finally, Section 6 speculates how the investment CAPM literature can evolve going forward.
2
The q-Factor Model
Hou, Xue, and Zhang (2015a) test the investment CAPM with the Black, Jensen, and Scholes
(1972) portfolio approach. In their q-factor model, the expected return of an asset in excess of
the risk-free rate, denoted E[Ri − Rf ], is described by the sensitivities of its returns to the market
factor, a size factor, an investment factor, and a profitability (return on equity, ROE) factor:
E[Ri − Rf ] = β iMKT E[MKT] + β iME E[rME ] + β iI/A E[rI/A ] + β iROE E[rROE ],
(9)
in which E[Ri − Rf ] is the expected excess return, E[MKT], E[rME ], E[rI/A ], and E[rROE ] are the
expected factor premiums, and β iMKT , β iME , β iI/A , and β iROE are the factor loadings, respectively.
Strictly speaking, the investment CAPM in equation (7) only motivates the investment and ROE
factors. Hou, Xue, and Zhang (2015a) include the size factor to make the q-factor model more palatable for evaluating mutual funds, in which size is a popular investment style. However, Hou et al.
show that while the size factor helps the q-factor model fit the average returns across the size deciles,
its incremental role in a broad set of anomaly deciles is minimal. Finally, equation (7) is primarily a
cross-sectional model, and the size, investment, and ROE factors are all zero-investment portfolios.
As such, the market factor is also included to anchor the cross-sectional average of expected returns,
while the task of fitting the cross-sectional variation of expected returns is left to the q-factors.
5
2.1
Why A Factor Model?
Why test the investment CAPM via a factor model? After all, equation (7) expresses the expected
stock return as a nonlinear function of the expected profitability and investment-to-capital, not
a linear function of the profitability and investment factors. Two popular approaches in empirical finance are the Black-Jensen-Scholes (1972) factor regressions and the Fama-MacBeth (1973)
cross-sectional regressions. Hou, Xue, and Zhang (2015a) opt to use factor regressions.
Cross-sectional regressions have several weaknesses that can be alleviated in factor regressions.
By weighting firm-month observations equally, including those for microcaps (stocks with market
capitalization below the 20th NYSE percentile), cross-sectional regressions are sensitive to outliers.
Despite accounting for nearly 60% of the total number of stocks, microcaps are on average only
about 3% of the market capitalization of the NYSE-Amex-NASDAQ universe (Fama and French
2008). Due to high transaction costs and lack of liquidity, microcaps are largely irrelevant for the
broad cross section. By judiciously choosing breakpoints and return weights, such as NYSE breakpoints and value-weighting portfolio returns, the factor approach is largely immune to these issues.
In addition, cross-sectional regressions impose a misspecified linear relation between characteristics and expected stock returns. However, the relation is nonlinear in theory. The factor approach
imposes fewer parametric restrictions. In particular, no linear relation is imposed between a sorting
variable and expected returns. However, it should be noted that the q-factor model does require
expected returns to be linear in the expected investment and profitability factor returns. Finally,
equation (7) is only a parametric example of the investment CAPM. In a multiperiod setup, additional terms would appear in the numerator of the expected return (Section 3). As such, Hou,
Xue, and Zhang (2015a) only use equation (7) to motivate the negative investment-expected return
relation and the positive profitability-expected return relation, but downplay its parametric form.
In this sense, the largely nonparametric factor approach is ideal.
6
2.2
Economic Intuition
2.2.1
Investment
Figure 1 shows that many cross-sectional patterns, such as equity issuance, accruals, valuation
ratios, and long-term reversal, are likely different manifestations of the investment effect.
Figure 1. The Investment Effect
✻
Y -axis: The discount rate
High composite issuance firms
Firms with high long-term prior returns
Low market leverage firms
Growth firms with low book-to-market
High net stock issues firms
Equity (convertible bond) issuers
High investment firms
✒
Low investment firms
Matching nonissuers
Low net stock issues firms
Value firms with high book-to-market
High market leverage firms
Firms with low long-term prior returns
Low composite issuance firms
✠
0
✲
X-axis: Investment-to-capital
The negative relation between average returns and equity issuance (Ritter 1991; Loughran and
Ritter 1995) is consistent with the investment effect. The flow of funds constraint of firms implies
that a firm’s uses of funds must equal its sources of funds. As such, all else equal, issuers must
invest more, and earn lower average returns than nonissuers (Lyandres, Sun, and Zhang 2008).
In addition, total asset growth predicts returns with a negative slope (Cooper, Gulen, and Schill
2008). However, asset growth is the most comprehensive measure of investment-to-capital, in which
capital is interpreted as all productive assets, and investment is the change in total assets. As such,
asset growth is the premier manifestation of the investment effect.
7
The value premium (Rosenberg, Reid, and Lanstein 1985) is also consistent with the investment
effect. Combining equations (4) and (5) implies that the marginal costs of investment, 1+a(Iit /Kit ),
equal market equity-to-capital, Pit /Kit+1 . Without debt, Pit /Kit+1 equals market-to-book equity.
Intuitively, value firms with low Pit /Kit+1 should invest less, and earn higher expected returns than
growth firms with high Pit /Kit+1 . More generally, firms with high valuation ratios have more growth
opportunities, invest more, and earn lower expected returns than firms with low valuation ratios.
The investment effect also manifests as long-term reversal (De Bondt and Thaler 1985). High
valuation ratios of growth firms are often associated with a stream of positive shocks to prior stock
returns, and low valuation ratios of value firms with a stream of negative shocks to prior stock
returns. As such, firms with high long-term prior returns tend to be growth firms that invest more,
and earn lower expected returns than firms with low long-term prior returns.
2.2.2
Profitability
In addition to the investment effect, equation (7) also gives rise to the profitability effect. Controlling for investment, firms with high expected profitability should earn higher expected returns than
firms with low expected profitability. For any portfolio sorts that produce wider cross-sectional
expected return spreads associated with expected profitability than those with investment, their
average returns can be interpreted with the common implied sort on expected profitability.
In particular, earnings momentum winners that have recently experienced positive shocks to
profitability tend to be more profitable, with higher expected profitability, than earnings momentum losers that have recently experienced negative shocks to profitability. The profitability effect
then implies that earnings momentum winners should earn higher expected returns than earnings
momentum losers (Chan, Jegadeesh, and Lakonishok 1996).
In addition, shocks to earnings are positively correlated with stock returns contemporaneously.
Intuitively, firms with positive earnings shocks tend to experience immediate stock price increases,
whereas firms with negative earnings shocks tend to experience immediate stock price decreases.
8
As such, the profitability effect is also consistent with price momentum, i.e., stocks that have performed well recently continue to earn higher average returns in the subsequent six months than
stocks that have performed poorly recently (Jegadeesh and Titman 1993). Finally, firms that are
more financially distressed are less profitable, and all else equal, should earn lower expected returns
than firms that are less financially distressed. As such, the profitability effect is also consistent
with the distress anomaly (Dichev 1998; Campbell, Hilscher, and Szilagyi 2008).
2.2.3
Beyond Investment and Profitability
The anomaly variables described so far are directly related to investment and profitability. Hou,
Xue, and Zhang (2015a) test the q-factor model with substantially more anomalies. As noted, the
consumption CAPM and the investment CAPM are theoretically equivalent in general equilibrium,
delivering identical expected returns. In contrast to the consumption CAPM, which says that
consumption betas are sufficient to explain the cross section of expected returns, the investment
CAPM says that a few characteristics are sufficient. Hou et al. take this prediction seriously, and
test the q-factor model with an exhaustive array of anomaly variables, including those that are not
directly related to investment and profitability. The consumption CAPM, especially the CAPM,
has played a fundamental role in empirical finance. Because of its theoretical equivalence with the
consumption CAPM, the investment CAPM perhaps holds the equal promise.
2.3
Empirical Results
Hou, Xue, and Zhang (2015a) test the q-factor model with time series regressions. If the model is
well specified, the intercepts should be economically small and statistically insignificant.
Investment-to-capital, I/A, is the annual change in total assets divided by one-year-lagged total
assets, and profitability, ROE, is quarterly earnings (income before extraordinary items) divided by
one-quarter-lagged book equity. The q-factors are from a triple 2 × 3 × 3 sort on size, I/A, and ROE
(Appendix A). From 1972 to 2012, the size factor earns an average return of 0.31% per month (t =
2.12), the investment factor 0.45% (t = 4.95), and the ROE factor 0.58% (t = 4.81). Both the invest9
ment and ROE factors survive the Fama-French (1993) three-factor model and the Carhart (1997)
four-factor model, which adds the momentum factor, UMD, to the three-factor model. In particular,
the investment factor has a three-factor alpha of 0.33% (t = 4.85) and a Carhart alpha of 0.28% (t =
3.85), and the ROE factor has a three-factor alpha of 0.77% (t = 6.94) and a Carhart alpha of 0.50%
(t = 4.75). Finally, the correlation between the two q-factors is only 0.06, which is insignificant.
Hou, Xue, and Zhang (2015a) examine a large set of nearly 80 anomaly variables that cover
all the major categories of anomalies, including momentum, value-versus-growth, investment, profitability, intangibles, and trading frictions. To form testing deciles, NYSE breakpoints and valueweighted decile returns are used to alleviate the impact of microcaps, following Fama’s (1998) recommendation. When forming annually sorted portfolios, Hou et al. follow the Fama-French (1993)
timing. At the end of June of each year t, stocks are grouped into deciles with the NYSE breakpoints
of, say, book-to-market, at the fiscal year ending in calendar year t − 1. Monthly value-weighted
returns are calculated from July of year t to June of t + 1, and the deciles are rebalanced in June.
In contrast to the Fama-French (1993) gold standard for annual portfolio sorts, the procedure
for aligning the timing of accounting variables with subsequent stock returns in monthly sorts is less
clear in the existing literature. Many published studies seem to suffer from look-ahead bias. Hou,
Xue, and Zhang (2015a) attempt to set a clear standard that is immune from look-ahead bias. For
quarterly earnings, Hou et al. use the Compustat quarterly data in the months immediately after
the earnings announcement dates (Compustat quarterly item RDQ). For all the other quarterly
items that are different from earnings, Hou et al. impose a four-month lag between the accounting
variables and subsequent stock returns. The crux is that unlike earnings, other quarterly items are
typically not available upon earnings announcement dates.1
The first major result in Hou, Xue, and Zhang (2015a) is that many anomalies (about 50% of
1
Many firms announce their earnings for a given quarter through a press release, and then file SEC reports several
weeks later. Easton and Zmijewski (1993, p. 117) document a median reporting lag of 46 days for NYSE/Amex
firms and 52 days for NASDQA firms. More recently, Chen, DeFond, and Park (2002, p. 230) report that only 37%
of quarterly earnings announcements include balance sheet information.
10
those examined) are insignificant at the 5% level in the broad cross section. Echoing Harvey, Liu,
and Zhu (2015), the evidence suggests that many claims in the anomalies literature are exaggerated. More important, using the remaining 35 significant anomalies as testing deciles, Hou et al.
run an empirical horse race between the q-factor model, the Fama-French three-factor model, and
the Carhart model. It should be emphasized that Hou et al. select testing deciles based on whether
the average high-minus-low return for a given anomaly variable is significant, not whether its FamaFrench alpha is significant. As such, the criterion does not bias against the three-factor model.
The q-factor model outperforms the Fama-French three-factor and Carhart models. Across the
35 significant high-minus-low deciles, the average magnitude of the q-model alphas is 0.2% per
month, which is lower than 0.33% in the Carhart model and 0.55% in the three-factor model. Only
five out of 35 high-minus-low deciles have significant q-model alphas at the 5% level. In contrast, 19
high-minus-low deciles have significant Carhart alphas, and 27 have significant Fama-French alphas.
The q-factor model also has the lowest mean absolute value of alphas across all 35 sets of deciles,
0.11% per month, which is lower than 0.12% in the Carhart model and 0.16% in the three-factor
model. Finally, the Gibbons, Ross, and Shanken (GRS, 1989) test on the null that the alphas are
jointly zero across a given set of deciles rejects the q-factor model at the 5% level in 20 out of 35
sets of deciles, but the Carhart model in 24, and the three-factor model in 28 sets of deciles.
The q-factor model outperforms the Carhart model and the three-factor model in all except
for the value-versus-growth category, in which the models are largely comparable. Across the nine
momentum anomalies, the average magnitude of the high-minus-low alphas is 0.19% per month
in the q-factor model, 0.29% in the Carhart model, and 0.85% in the three-factor model. Two
high-minus-low deciles have significant q-model alphas, in contrast to five Carhart alphas and all
nine Fama-French alphas. The GRS test rejects the q-factor model and the Carhart model in six
out of nine sets of deciles, but rejects the three-factor model in all nine.
Consistent with the broad implications of the investment CAPM (Section 2.2.3), most anoma-
11
lies turn out to be different manifestations of the investment and profitability effects. Price and
earnings momentum are mainly connected to the ROE factor. For the high-minus-low momentum
deciles, all the ROE factor loadings are significant, but the investment factor loadings are mostly
insignificant. The investment factor is mainly responsible for capturing the value-minus-growth
anomalies. Their investment factor loadings are all more than five standard errors from zero, but
the ROE factor loadings are mostly insignificant. The investment factor is mainly responsible for
the investment anomalies, the ROE factor for the profitability anomalies, and a combination of the
two factors accounts for the anomalies in the intangibles and trading frictions categories.
2.4
Factors War
The q-factor model poses to end the almost quarter-century reign of the Fama-French three-factor
model, along with its extension, the Carhart four-factor model, as the workhorse model for empirical
finance. Perhaps not surprisingly, the q-factor model touches off a firestorm of controversy.2
2.4.1
The Empire Strikes Back
Fama and French (2015a) incorporate their own versions of the investment and profitability factors
into their original three-factor model to form a five-factor model:3
E[Ri − Rf ] = bi E[MKT] + si E[SMB] + hi E[HML] + ri E[RMW] + ci E[CMA].
(10)
MKT, SMB, and HML are the market, size, and value factors that first appear in their three-factor
model. The two new factors resemble the q-factors. RMW (robust-minus-weak) is the difference
between the returns on diversified portfolios of stocks with robust and weak operating profitability.
2
Although first appearing in October 2012 as NBER working paper 18435, the Hou-Xue-Zhang (2015a) article
is a new incarnation of the previous work circulated as “Neoclassical factors” (as NBER working paper 13282, July
2007), “An equilibrium three-factor model” (January 2009), “Production-based factors” (April 2009), “A better
three-factor model that explains more anomalies” (June 2009), and “An alternative three-factor model” (April
2010). The frequent title changes make it evident that the June 2009 title was fought against vigorously, albeit
unsuccessfully. The insight that investment and profitability are fundamental forces in the cross section of expected
stock returns in the investment CAPM first appears in Zhang (2005a, NBER working paper 11322, May 2005).
3
Their June 2013 draft, Fama and French (2013), adds only a profitability factor, and subsequent drafts, starting
from the November 2013 draft, also add an investment factor.
12
CMA (conservative-minus-aggressive) is the difference between the returns on diversified portfolios
of low and high investment stocks. Operating profitability, OP, is revenues minus costs of goods
sold, minus selling, general, and administrative expenses, minus interest expenses, all scaled by book
equity (Novy-Marx 2013). Investment, Inv, is the change in book assets divided by one-year-lagged
book assets (same as in Hou, Xue, and Zhang 2015a). The factors are constructed from independent
2×3 sorts by interacting size with, separately, book-to-market, OP, and Inv (Appendix A).
Fama and French (2015a) motivate their five-factor model from the Miller and Modigliani (1961)
valuation theory. From the dividend discounting model, the market value of firm i’s stock, Pit , is
the present value of its expected dividends, Pit =
P∞
τ =1 E[Dit+τ ]/(1
+ ri )τ , in which Dit is divi-
dends, and ri is the firm’s long-term average expected stock return, i.e., the internal rate of return.
The clean surplus relation says that dividends equal earnings minus the change in book equity,
Dit+τ = Πit+τ − △Bit+τ , in which △Bit+τ ≡ Bit+τ − Bit+τ −1 . It follows that:
Pit
=
Bit
P∞
τ =1 E[Πit+τ
− △Bit+τ ]/(1 + ri )τ
.
Bit
(11)
Fama and French (2015a) argue that the valuation equation (11) makes three theoretical predictions. First, fixing everything except the current market value, Pit , and the expected stock return,
ri , a low Pit , or a high book-to-market equity, Bit /Pit , implies a high expected return. Second, fixing
everything except the expected profitability and the expected stock return, high expected profitability implies a high expected return. Third, fixing everything except the expected growth in book
equity and the expected return, high expected book equity growth implies a low expected return.
Fama and French (2015a) use operating profitability as the proxy for the expected profitability
and past investment as the proxy for the expected investment. Empirically, the five-factor model
outperforms the original three-factor model in pricing testing portfolios formed on size, book-tomarket, investment, and profitability. However, these portfolios are formed on the same variables
underlying the factors. Fama and French (2015b) extend their set of testing portfolios to include
13
accruals, net share issues, momentum, volatility, and the market beta, which are still a small subset
of the comprehensive universe of anomalies examined in Hou, Xue, and Zhang (2015a).
2.4.2
A Better Factor Model That Explains More Anomalies
Hou, Xue, and Zhang (2015b) compare the q-factor model with the Fama-French five-factor model.
On both economic and empirical grounds, the q-factor model is a better factor model than the
five-factor model, which appears to be a noisy version of the q-factor model.
2.4.2.1
Conceptual Grounds
Hou, Xue, and Zhang (2015b) raise four critiques on Fama and
French’s (2015a) rationale of the five-factor model based on the valuation theory.
First, Fama and French (2015a) derive the relations between book-to-market, investment, and
profitability only with the internal rate of return, but argue that the difference between the oneperiod-ahead expected return and the internal rate of return is not important (p. 2):
Most asset pricing research focuses on short-horizon returns—we use a one-month horizon in our tests. If
each stock’s short-horizon expected return is positively related to its internal rate of return—if, for example,
the expected return is the same for all horizons—the valuation equation implies that the cross-section of
expected returns is determined by the combination of current prices and expectations of future dividends.
Hou, Xue, and Zhang (2015b) estimate the internal rate of returns for RMW and CMA with
accounting-based models (Gebhardt, Lee, and Swaminathan 2001), and show that these estimates
deviate greatly from their one-month-ahead average returns. Whereas the average returns of RMW
are significantly positive, its estimates of the internal rate of returns are often significantly negative.
Second, Fama and French (2015a) argue that the value factor is a separate factor in valuation
theory, but find it to be redundant in describing average returns in the data. However, as noted, the
investment CAPM implies that the value premium is just a different manifestation of the investment
effect. The first principle of investment says that the marginal costs of investment, which rise with
investment, equal marginal q, which is in turn closely related to market-to-book equity. As such,
the value factor is redundant in the presence of the investment factor in the investment CAPM.
14
Third, Fama and French (2015a) motivate their investment factor, CMA, from the negative relation between the expected investment and the internal rate of returns in valuation theory. However,
Hou, Xue, and Zhang (2015b) reformulate the valuation equation with the one-period-ahead expected return, and show that its relation with the expected investment is more likely to be positive.
In particular, the definition of return, Pit = (Et [Dit+1 ] + Et [Pit+1 ])/(1 + Et [rit+1 ]), combined
with the clean surplus relation, implies that:
Pit =
Et [Πit+1 − △Bit+1 ] + Et [Pit+1 ]
.
1 + Et [rit+1 ]
(12)
Dividing both sides by Bit and rearranging yield:
Pit
Bit
=
Pit
Bit
=
Et
h
Πit+1
Bit
i
− Et
h
△Bit+1
Bit
i
+ Et
h
Pit+1
Bit+1
1+
△Bit+1
Bit
i
1 + Et [rit+1 ]
h
i
h
i
i
h
Pit+1
△Bit+1 Pit+1
−
1
+
E
+
E
Et ΠBit+1
t Bit+1
t
Bit
Bit+1
it
1 + Et [rit+1 ]
.
,
(13)
(14)
Fixing everything except Et [△Bit+1 /Bit ] and Et [rit+1 ], high Et [△Bit+1 /Bit ] is likely to be associated with high Et [rit+1 ], because Pit+1 /Bit+1 − 1 tends to be positive in the data. More generally,
leading equation (14) by one period at a time and recursively substituting Pit+1 /Bit+1 in the same
equation implies a positive Et [△Bit+τ /Bit ]-Et [rit+1 ] relation for all τ ≥ 1.4
Fourth, after arguing for the negative relation between the expected investment and the expected
return, Fama and French (2015a) use past investment as the proxy for the expected investment.
However, while past profitability forecasts future profitability, past investment does not forecast
future investment. In annual cross-sectional regressions of future book equity growth on current
asset growth, the average R2 starts at 5% in year one, drops quickly to zero in year four, and
remains at zero through year ten. The low persistence of micro-level investment is well known in
the lumpy investment literature (Dixit and Pindyck 1994; Doms and Dunne 1998; Whited 1998).
4
Lettau and Ludvigson (2002) show that high aggregate risk premiums forecast high long-term aggregate investment growth rates. Using cross-sectional regressions, Fama and French (2006) report a weakly positive relation
between the expected investment and the expected returns. However, Aharoni, Grundy, and Zeng (2013) report a negative relation, and attribute the difference to firm-level variables, as opposed to per share variables in Fama and French.
15
In all, Fama and French’s (2015a) conceptual foundation for the five-factor model is built on
sand. Most important, the third and fourth critiques imply that the investment factor can only be
derived from market-to-book in the valuation equation (11), augmented with the economic linkage
between investment and value, which is in turn a key insight from the investment CAPM. Also, without the redundant HML, the five-factor model collapses to (a noisy version of) the q-factor model.
2.4.2.2
Empirical Grounds
Hou, Xue, and Zhang (2015a) start their sample in January 1972,
restricted by the availability of quarterly earnings announcement dates and quarterly book equity
data. To make the sample more comparable to Fama and French (2015a), who start their sample
in July 1963, Hou et al. (2015b) extend the sample for the q-factors back to January 1967. Prior
to 1972, the most recent quarterly earnings from the fiscal quarter ending at least four months
prior to the portfolio formation are used, and the fourth quarter book equity data from Compustat
annual files are used to impute quarterly book equity with clean surplus accounting.
Table 1 shows that, from January 1967 to December 2014, the size, investment, and ROE factors
earn on average 0.32%, 0.43%, and 0.56% per month (t = 2.42, 5.08, and 5.24), respectively. SMB,
HML, RMW, and CMA earn on average 0.28%, 0.36%, 0.27%, and 0.34% (t = 1.92, 2.57, 2.58, and
3.63), respectively. The five-factor model cannot explain the investment and ROE q-factors, leaving
alphas of 0.12% (t = 3.35) and 0.45% (t = 5.6), respectively. However, the q-factor model explains
the HML, RMW, and CMA returns, with tiny alphas of 0.03%, 0.04%, and 0.01%, respectively
(t-statistics all below 0.5). As such, RMW and CMA are noisy versions of the q-factors.
Hou, Xue, and Zhang (2015b) show that the q-factor model outperforms the five-factor model
empirically. Across 36 significant high-minus-low anomaly deciles formed with NYSE breakpoints
and value-weighted returns from January 1967 to December 2013, the average magnitude of the
q-factor alphas is 0.2% per month, which is lower than 0.36% for the five-factor model and 0.33%
for the Carhart model. Seven out of 36 alphas are significant in the q-factor model, in contrast
to 19 in the five-factor model and 21 in the Carhart model. The mean absolute alpha across the
16
Table 1 : Properties of the New Factors, January 1967 to December 2014
Source: Hou, Xue, and Zhang (2015b), with the sample extended through December 2014. rME , rI/A , and
rROE are the size, investment, ROE factors in the q-factor model, respectively. SMB, HML, RMW, and
CMA are the size, value, profitability, and investment factors from the Fama-French (2015a) five-factor
model, respectively. The data for SMB and HML in the three-factor model, SMB, HML, RMW, and CMA
in the five-factor model, as well as UMD are from Kenneth French’s Web site. m is the average return, αC
is the Carhart alpha, αq the q-model alpha, α5 is the five-factor alpha, and b, s, h, r, and c are five-factor
loadings. The t-statistics in parentheses are adjusted for heteroscedasticity and autocorrelations.
Panel A: The Hou-Xue-Zhang (2015a, b) q-factors
rME
rI/A
rROE
rME
rI/A
rROE
m
αC
β MKT
β SMB
β HML
β UMD
R2
0.32
(2.42)
0.43
(5.08)
0.56
(5.24)
0.01
(0.25)
0.29
(4.57)
0.51
(5.58)
0.01
(1.08)
−0.06
(−4.51)
−0.04
(−1.39)
0.97
(67.08)
−0.04
(−1.88)
−0.30
(−4.31)
0.17
(7.21)
0.41
(13.36)
−0.12
(−1.79)
0.03
(1.87)
0.05
(1.93)
0.27
(6.19)
0.94
α5
b
s
h
r
c
R2
0.05
(1.39)
0.12
(3.35)
0.45
(5.60)
0.00
(0.39)
0.01
(0.73)
−0.04
(−1.45)
0.98
(68.34)
−0.05
(−2.86)
−0.11
(−2.69)
0.02
(1.14)
0.04
(1.60)
−0.24
(−3.54)
−0.01
(−0.21)
0.07
(2.77)
0.75
(13.46)
0.04
(1.19)
0.82
(26.52)
0.13
(1.34)
0.95
0.52
0.40
0.84
0.52
Panel B: The Fama-French (2015a) five factors
SMB
HML
RMW
CMA
SMB
HML
RMW
CMA
m
αC
β MKT
β SMB
β HML
β UMD
R2
0.28
(1.92)
0.36
(2.57)
0.27
(2.58)
0.34
(3.63)
−0.02
(−1.24)
−0.00
(−1.79)
0.33
(3.31)
0.19
(2.83)
0.00
(0.96)
0.00
(1.79)
−0.04
(−1.32)
−0.09
(−4.42)
1.00
(89.87)
−0.00
(−1.69)
−0.28
(−3.20)
0.03
(0.86)
0.13
(8.07)
1.00
(13282.85)
−0.00
(−0.03)
0.46
(13.52)
0.00
(0.11)
−0.00
(−0.87)
0.04
(0.81)
0.04
(1.51)
0.99
αq
β MKT
β ME
β I/A
β ROE
R2
0.05
(1.48)
0.03
(0.28)
0.04
(0.42)
0.01
(0.32)
−0.00
(−0.17)
−0.05
(−1.33)
−0.03
(−0.99)
−0.05
(−3.63)
0.94
(62.40)
0.00
(0.03)
−0.12
(−1.78)
0.04
(1.68)
−0.09
(−4.91)
1.03
(11.72)
−0.03
(−0.35)
0.94
(35.26)
−0.10
(−5.94)
−0.17
(−2.17)
0.53
(8.59)
−0.11
(−3.95)
0.96
17
1.00
0.19
0.55
0.50
0.49
0.85
36 sets of deciles is 0.11% in the q-factor model, 0.12% in the five-factor model, and 0.12% in the
Carhart model. The GRS test rejects all three models at the 5% level in 25 out of 36 sets of deciles.
The q-factor model outperforms the Carhart model, which in turn outperforms the five-factor
model in accounting for momentum. Across the eight significant winner-minus-loser deciles, the
average magnitude of the alphas is 0.2% per month in the q-factor model, which is lower than
0.29% in the Carhart model and 0.76% in the five-factor model. Two out of eight high-minus-low
alphas are significant in the q-factor model, in contrast to four in the Carhart model and all eight in
the five-factor model. The q-factor model also dominates the five-factor model in the profitability
category. None of the five high-minus-low alphas are significant in the q-factor model, in contrast
to four in the five-factor model and all five in the Carhart model.
Hou, Xue, and Zhang (2015b) show that their ROE factor is more powerful than Fama and
French’s (2015a) RMW. Across the eight winner-minus-loser momentum deciles, the ROE factor
loadings are all more than 2.5 standard errors from zero. In contrast, the RMW loadings are
substantially smaller and mostly insignificant. Intuitively, formed on the most up-to-date earnings
data, the ROE factor is more effective in capturing the profitability differences between winners
and losers. In contrast, formed on more stale earnings from last fiscal year end, RMW seems fairly
weak. The investment factor is also somewhat more powerful than CMA, likely because of jointly
sorting with profitability, which gives the q-factor model an edge in the investment category.
Finally, Hou, Xue, and Zhang (2015b) address an important criticism raised by Fama and French
(2015a). Deviating from Fama (1998), Fama and French argue that small stocks pose the most serious challenges for asset pricing models, but that value-weighting returns tends to underweight small
stocks. In response, Hou et al. also form testing deciles with all-but-micro breakpoints and equalweighted returns. The universe of stocks excluding microcaps is used to calculate breakpoints,
and the stocks within a given portfolio are equal-weighted to assign sufficient weights to small
stocks. All-but-micro breakpoints and equal-weighting indeed produce more significant anomalies
18
than NYSE breakpoints and value-weighting (50 versus 36). Across the set of 50 significant highminus-low anomalies, the mean absolute alpha is 0.24% per month in the q-factor model, 0.41% in
the five-factor model, and 0.4% in the Carhart model. Also, 16 high-minus-low q-factor alphas are
significant, in contrast to 34 five-factor alphas and 37 Carhart alphas.
2.4.3
Which ROE Factor?
Fama and French (2006) test the expected profitability and expected investment effects predicted
by valuation theory, but the tests have largely failed. Cross-sectional regressions are used to predict
profitability and asset growth one, two, and three years ahead, and the fitted values from these
first-stage regressions are used as explanatory variables in second-stage cross-sectional regressions
of returns. Contrary to the hypothesized negative relation between the expected investment and
expected returns, the average slopes on the expected investment in second-stage regressions are
insignificantly positive. Also, with a long list of predictors for future profitability in the first-stage,
including lagged fundamentals, returns, analysts forecasts, and the default probability, the expected
profitability shows only insignificant, albeit positive, average slopes in the second-stage regressions.
Fama and French (2008) again fail to discover the profitability effect. Profitability is measured
as income before extraordinary items (Compustat annual item IB), minus preferred dividends (item
DVP), if available, plus income account deferred taxes (item TXDI), if available, divided by contemporaneous book equity. From annual sorts per Fama and French (1993), their Table II reports that
profitability sorts “produce the weakest average hedge portfolio returns,” and suggests that with
controls for size and book-to-market, “hedge returns do not provide much basis for the conclusion
that there is a positive relation between average returns and profitability (p. 1663).
Novy-Marx (2013) shows that a different profitability measure, gross profits-to-assets, produces
significant average hedge returns in annual sorts. It is argued that gross profits are a cleaner
accounting measure of true economic profitability than earnings. The reason is that expensed investments, such as research and development, advertising, and employee training, reduce earnings,
19
but do not increase book equity. These expensed investments give rise to higher future economic
profits, and are better captured by gross profits. As noted, Fama and French (2015a) build their
profitability factor, RMW, on the gross profitability effect.
2.4.3.1
Lagging the Deflator: The Irrelevance of Gross Profitability Panel A of Table
2 reports factor regressions for the annually formed gross profits-to-assets deciles. Consistent with
Novy-Marx (2013), the high-minus-low decile earns an average return of 0.38% per month (t = 2.62)
and a Carhart alpha of 0.49% (t = 3.39). The Fama-French five-factor alpha is 0.19% (t = 1.46),
and the q-factor alpha is 0.18% (t = 1.24). While the five-factor model can be rejected by the GRS
test at the 5% significance level (p = 0.04), the q-factor model cannot (p = 0.11).
Panel B replicates the tests in Panel A, but scales gross profits with one-year-lagged assets, as
opposed to contemporaneous assets. The average high-minus-low return falls from 0.38% per month
(t = 2.62) to 0.16% (t = 1.04). Intuitively, the gross profits-to-assets ratio equals the gross profitsto-lagged assets ratio divided by asset growth (contemporaneous assets-to-lagged assets). As such,
Novy-Marx’s (2013) gross profitability effect in Panel A is contaminated by a hidden investment
or asset growth effect. Once this investment effect is purged from the gross profits-to-lagged assets
ratio, the gross profitability effect disappears, as shown in Panel B.
Which deflator should be used to scale profits, lagged or contemporaneous assets? Because in
Compustat, both profits and assets are measured at the end of a period, economic logic implies
that profits should be scaled by one-period-lagged assets. Intuitively, profits are generated by the
one-period-lagged assets. Contemporaneous assets at the end of the period in Compustat are accumulated via the investment process over the course of the current period. Assuming one-period timeto-build, contemporaneous assets can start to generate profits only at the beginning of next period.5
5
The economic model laid out in Section 1 formalizes this intuition. In the economic model, Kit is capital
(productive assets) at the beginning of period t, or equivalently, at the end of period t−1. This timing convention
follows one-period time-to-build (Kydland and Prescott 1982). In particular, Kit+1 becomes productive only at the
beginning of period t+1, when investment, Iit , over the course of period t is completed. In contrast, economic profits,
Πit+1 = Xit+1 Kit+1 , occur over the course of period t+1, and are known only at the end of period t+1 because the
shocks, Xit+1 , are stochastic. As such, economic profits, Πit+1 , known at the end of period t+1 (equivalently, at the
20
Table 2 : Deciles on Different Gross Profitability Measures
All deciles are based on NYSE breakpoints and value-weighted returns. m is average excess returns, and tm
is its t-statistics. αC , α5 , and αq are the alphas from the Carhart model, the Fama-French five-factor model,
and the q-factor model, and tC , t5 , and tq are their t-statistics, respectively. β MKT , β ME , β I/A , and β ROE are
the loadings on the market, size, investment, and ROE factors in the q-factor model, and tMKT , tME , tI/A , and
tROE are their t-statistics, respectively. m.a.e. (mean absolute error) is the average magnitude of alphas across
the testing deciles. The GRS p-value testing that all ten alphas are jointly zero is in brackets beneath the
m.a.e. for a given factor model. All the t-statistics are adjusted for heteroscedasticity and autocorrelations.
In Panel A, at the end of June of year t, stocks are split into deciles based on gross profits, measured as total
revenue (Compustat annual item REVT) minus cost of goods sold (item COGS), divided by total assets
(item AT), all for the fiscal year ending in calendar year t − 1. Monthly decile returns are calculated from
July of year t to June of t + 1, and the deciles are rebalanced in June of t + 1. Panel B replicates Panel
A, except that the denominator of the sorting variable is total assets for the fiscal year ending in calendar
year t − 2. In Panel C, at the beginning of each month t, stocks are split into deciles based on total revenue
(Compustat quarterly item REVTQ) minus cost of goods sold (item COGSQ), divided by one-quarter-lagged
assets (item ATQ), all from the fiscal quarter ending at least four months ago. Monthly decile returns are
calculated for month t, and the deciles are rebalanced at the beginning of month t + 1.
Low
2
3
4
5
6
7
8
9
High
H−L
m.a.e.
Panel A: Gross profits-to-current assets, annual sorts, January 1967–December 2014
m
tm
αC
tC
α5
t5
0.35
1.66
−0.15
−1.51
0.07
0.85
0.40
2.04
−0.19
−2.53
−0.16
−1.98
0.46
2.21
−0.12
−1.42
−0.10
−1.13
0.49
2.32
−0.08
−0.86
−0.14
−1.57
0.62
3.05
0.06
0.74
0.01
0.17
0.57
2.76
0.03
0.38
0.01
0.16
0.51
2.28
0.14
1.52
0.03
0.36
αq
β MKT
β ME
β I/A
β ROE
tq
tMKT
tME
tI/A
tROE
0.02
0.89
−0.02
0.07
−0.27
0.21
32.32
−0.61
1.27
−5.06
−0.16
0.95
−0.05
0.39
−0.12
−1.76
33.39
−1.00
6.37
−2.62
−0.11
1.03
−0.10
0.27
−0.05
−1.21
30.28
−2.44
4.57
−0.89
−0.10
1.03
−0.04
0.27
−0.07
−1.13
40.39
−0.91
4.73
−1.58
0.04
1.04
−0.05
0.19
−0.02
0.50
42.17
−1.54
3.65
−0.37
0.08
0.99
0.07
−0.01
−0.05
1.02
38.19
1.91
−0.18
−1.01
0.16
1.04
−0.03
−0.25
−0.10
1.65
30.83
−0.67
−3.58
−1.40
0.49
2.23
0.18
1.98
0.07
0.77
0.61
2.91
0.24
3.05
0.16
2.25
0.73
3.41
0.35
3.63
0.27
2.58
0.38
2.62
0.49
3.39
0.19
1.46
0.19
0.15
0.20
0.18
0.97
0.95
0.93
0.04
−0.05
0.00
0.01
0.03
−0.40 −0.32 −0.24 −0.31
0.01
0.21
0.28
0.55
1.65
1.82
1.90
1.24
36.71 39.93 31.71
0.95
−1.38 −0.10
0.34
0.69
−5.67 −5.99 −3.38 −3.21
0.16
5.00
5.59
7.66
0.15
[0.00]
0.10
[0.04]
0.12
[0.11]
Panel B: Gross profits-to-lagged assets, annual sorts, January 1967–December 2014
m
tm
0.46
2.30
0.45
2.36
0.49
2.37
0.54
2.55
0.63
3.28
0.61
2.91
0.50
2.42
0.49
2.21
0.60
2.89
0.61
2.69
0.16
1.04
Panel C: Gross profits-to-lagged assets, monthly sorts, January 1976–December 2014
m
tm
αC
tC
α5
t5
0.39
0.59
0.46
0.51
0.62
1.56
2.94
1.98
2.22
2.81
−0.19 −0.03 −0.24 −0.20 −0.06
−1.66 −0.38 −2.57 −2.14 −0.71
0.03
0.05 −0.19 −0.24 −0.17
0.26
0.70 −2.00 −2.51 −1.94
αq
β MKT
β ME
β I/A
β ROE
tq
tMKT
tME
tI/A
tROE
0.05
0.96
−0.02
−0.04
−0.39
0.40
30.27
−0.57
−0.63
−7.40
0.04
0.86
−0.03
0.34
−0.19
0.47
33.96
−0.99
6.34
−4.46
−0.16
0.98
0.00
0.23
−0.15
−1.54
30.44
−0.06
3.89
−3.02
−0.19
1.04
−0.07
0.29
−0.06
−2.03
43.80
−1.81
5.01
−1.28
0.71
3.21
0.11
1.35
0.00
0.00
0.65
2.75
0.12
1.24
0.01
0.08
0.58
2.58
0.12
1.53
−0.03
−0.31
−0.08
0.07
1.03
1.01
−0.07
0.02
0.18
0.02
0.00
0.00
−0.84
0.78
36.20 37.52
21
−1.50
0.39
3.10
0.32
0.03 −0.02
0.16
1.03
−0.02
−0.25
−0.09
1.27
30.25
−0.53
−3.13
−1.15
0.00
0.97
−0.04
−0.22
0.13
−0.04
37.14
−1.03
−2.54
2.28
0.72
3.22
0.23
2.63
0.14
1.55
0.91
3.87
0.37
3.76
0.33
2.96
0.51
3.40
0.56
3.81
0.30
2.14
0.14
0.25
0.20
0.96
0.95
0.00
−0.02
0.09
0.11
−0.33 −0.32 −0.28
0.21
0.27
0.66
1.47
2.12
1.41
37.04 35.08 −0.10
−0.55
2.21
2.20
−5.39 −4.92 −3.03
4.56
5.67 12.26
0.17
[0.00]
0.12
[0.02]
0.11
[0.12]
2.4.3.2
The Importance of Monthly Sorts In contrast to the small and insignificant prof-
itability premium (0.16% per month, t = 1.04) from annual sorts on gross profits-to-lagged assets,
Panel C of Table 2 shows that monthly sorts on the same variable revive the profitability premium.
Due to limited coverage for the cost of goods sold (Compustat quarterly item COGSQ) and total
assets (item ATQ), the sample starts from January 1976. The high-minus-low decile earns an average return of 0.51% per month (t = 3.4) and a Carhart alpha of 0.56% (t = 3.81). The five-factor
alpha is 0.3% (t = 2.14), but the q-factor alpha is 0.2% (t = 1.41). Also, while the five-factor model
can be rejected by the GRS test (p = 0.02), the q-factor model cannot (p = 0.12).
In untabulated results, monthly sorts on gross profits-to-current assets yield an average return
of 0.5% per month (t = 3.11) for the high-minus-low decile. Its Carhart, five-factor, and q-factor
alphas are 0.55% (t = 3.46), 0.2% (t = 1.35), and 0.1% (t = 0.70), respectively. As such, although
important for annual sorts, lagging the deflator matters little in monthly sorts.
Table 3 shows further that gross profits are largely irrelevant in monthly sorts. Instead of gross
profits-to-lagged assets, this table reports results for deciles formed monthly on ROA (earnings-tolagged assets) and, separately, on ROE (earnings-to-lagged book equity). To facilitate comparison,
the sample starts at January 1976 as in Panel C of Table 2. Panel A of Table 3 shows that monthly
ROA sorts yield an average high-minus-low return of 0.58% per month (t = 2.53), which is higher
than 0.51% from sorting on gross profits-to-lagged assets. From Panel B, the average return for the
high-minus-low decile formed monthly on ROE is even higher, 0.72% (t = 2.79). As such, despite
much maligned in annual sorts, earnings perform better than gross profits in monthly sorts.
2.4.4
Understanding the Low Risk Anomaly
The low risk anomaly refers to the empirical patterns that the firm-level volatility is negatively correlated with future returns (Ang, Hodrick, Xing, and Zhang 2006), and that the realized market beta is
negatively correlated with future risk-adjusted returns (Frazzini and Pedersen 2014). Despite its rebeginning of period t+2), should be scaled by assets, Kit+1 , at the beginning of period t+1.
22
Table 3 : The ROA and ROE Deciles, Monthly Sorts, January 1976–December 2014
All deciles are based on NYSE breakpoints and value-weighted returns. m is average excess returns, and tm
is its t-statistics. αC , α5 , and αq are the alphas from the Carhart model, the Fama-French five-factor model,
and the q-factor model, and tC , t5 , and tq are their t-statistics, respectively. β MKT , β ME , β I/A , and β ROE
are the loadings on the q-factors, and tMKT , tME , tI/A , and tROE are their t-statistics. m.a.e. is the average
magnitude of alphas across the testing deciles. The GRS p-value is in brackets beneath the m.a.e. for a given
factor model. The t-statistics are adjusted for heteroscedasticity and autocorrelations. At the beginning
of each month t, stocks are split into deciles based on quarter ROA and ROE, measured as income before
extraordinary items (Compustat quarterly item IBQ) scaled by one-quarter-lagged assets (item ATQ) or
one-quarter-lagged book equity (Appendix A), respectively. Quarterly earnings are used immediately after
the most recent quarterly earnings announcement dates (item RDQ). Monthly decile returns are calculated
for month t, and the deciles are rebalanced at the beginning of month t + 1.
Low
2
3
4
5
6
7
8
9
High
H−L m.a.e.
Panel A: ROA (quarterly earnings-to-one-quarter-lagged assets)
m
tm
αC
tC
α5
t5
0.20
0.56
−0.42
−2.70
−0.14
−1.30
0.44
1.58
−0.15
−1.38
−0.16
−1.33
0.57
2.25
−0.03
−0.35
−0.13
−1.38
0.68
3.30
0.07
0.80
0.06
0.75
αq
0.16
0.10
β MKT
1.11
1.05
β ME
0.30 −0.02
β I/A
−0.45
0.06
β ROE
−1.00 −0.59
tq
1.39
1.01
tMKT
34.07
35.10
tME
4.90 −0.30
tI/A
−5.71
0.87
tROE −12.88 −12.24
0.10
1.05
−0.08
0.28
−0.46
0.98
34.16
−1.33
3.85
−6.67
0.17
0.93
−0.07
0.18
−0.20
1.74
34.87
−1.27
2.70
−2.52
0.57
2.87
−0.04
−0.57
−0.06
−0.73
0.75
3.61
0.12
1.48
0.03
0.35
0.72
0.63
3.58
3.05
0.11
0.08
1.59
0.94
0.02 −0.11
0.27 −1.56
−0.04
0.06
0.03
0.92
0.96
0.98
0.00 −0.06 −0.05
0.21
0.22
0.14
−0.07
0.05
0.07
−0.49
0.61
0.43
53.21 40.92 45.63
−0.09 −1.54 −1.49
3.58
4.15
2.90
−2.00
1.06
1.63
−0.07
0.98
−0.08
0.05
0.18
−0.85
39.44
−1.95
0.66
3.54
0.70
3.25
0.13
1.82
0.04
0.60
0.78
3.41
0.27
3.61
0.24
3.07
0.58
2.53
0.68
3.61
0.39
2.85
−0.01
0.98
0.01
−0.14
0.26
−0.10
48.96
0.34
−2.63
7.72
0.13
0.99
−0.01
−0.48
0.38
1.60
42.04
−0.31
−9.18
8.93
−0.02
−0.13
−0.31
−0.02
1.38
−0.21
−4.21
−7.40
−0.32
18.94
0.14
[0.00]
0.10
[0.10]
0.09
[0.20]
Panel B: ROE (quarterly earnings-to-one-quarter-lagged book equity)
m
tm
αC
tC
α5
t5
0.16
0.45
−0.49
−3.09
−0.24
−2.02
0.43
0.50
0.56
1.57
2.02
2.69
−0.15 −0.04 −0.03
−1.40 −0.38 −0.30
−0.15
0.00 −0.02
−1.29
0.01 −0.17
αq
0.09
0.13
β MKT
1.13
1.05
β ME
0.32
0.01
β I/A
−0.38
0.02
β ROE
−1.02 −0.64
tq
0.75
1.41
tMKT
36.03
39.36
tME
5.19
0.16
tI/A
−5.20
0.35
tROE −14.98 −14.23
0.68
3.36
0.08
0.99
0.05
0.55
0.58
2.68
−0.03
−0.32
−0.03
−0.32
0.88
3.93
0.31
3.42
0.15
1.71
0.72
2.79
0.80
3.72
0.39
2.68
0.28
0.10
0.11
0.00
0.05 −0.05
0.03
0.03
0.97
0.90
0.93
0.95
0.96
1.02
0.97
1.02
−0.11
0.07 −0.01
0.03 −0.01 −0.05 −0.04 −0.02
0.01
0.09
0.19
0.05
0.06 −0.01 −0.29 −0.15
−0.62 −0.27 −0.14 −0.07
0.04
0.25
0.29
0.48
2.99
1.05
1.24
0.01
0.63 −0.59
0.37
0.39
30.99 37.49 36.74 46.57 46.24 53.66 50.31 53.89
−1.98
2.28 −0.41
0.88 −0.44 −1.62 −1.34 −0.63
0.08
1.65
3.21
0.96
1.03 −0.16 −4.95 −2.78
−9.41 −5.38 −2.81 −1.65
0.90
7.36
8.70 12.85
−0.05
−0.11
−0.34
0.23
1.50
−0.40
−3.31
−5.36
2.34
18.46
23
0.68
3.34
0.09
1.30
0.02
0.23
0.71
3.37
0.12
1.60
0.04
0.48
0.68
3.17
0.10
1.46
0.10
1.45
0.14
[0.00]
0.08
[0.33]
0.09
[0.06]
cent attention, the low risk anomaly does not make the list of significant anomalies in Hou, Xue, and
Zhang (2015a, b). This subsection shows that the q-factor model also explains the low risk anomaly.
2.4.4.1
The Idiosyncratic Volatility Anomaly Ang, Hodrick, Xing, and Zhang (2006) de-
fine a stock’s idiosyncratic volatility as the standard deviation of the residuals from regressing the
stock’s excess returns on the Fama-French three-factor model. At the beginning of month t, stocks
are split into deciles based on the idiosyncratic volatility estimated with daily returns from month
t − 1 (with a minimum of 15 daily returns). Monthly decile returns are calculated for month t, and
the deciles are rebalanced at the beginning of month t + 1.
The idiosyncratic volatility anomaly is sensitive to breakpoints and return weights. From Panel
A in Table 4, using NYSE breakpoints and value-weighted returns yields an average return of
−0.51% per month (t = −1.62). Panel B shows that with all-but-micro breakpoints and equalweights, the average return spread rises to −0.63%, but is only marginally significant (t = −1.95).
Only when formed with all (NYSE-Amex-NASDAQ) breakpoints and value-weights as in Ang, Hodrick, Xing, and Zhang (2006), does the high-minus-low decile earns an average return of −1.28%
(t = −3.48) (Panel A, Table 5). Equal-weights reduce the average return drastically to −0.27%
(t = −0.75) (Panel B, Table 5).6 However, the CAPM and Fama-French three-factor alphas for
the high-minus-low decile are all significant, regardless of breakpoints and return weights.
The q-factor model outperforms the five-factor model in capturing the idiosyncratic volatility
anomaly. In particular, Panel A in Table 4 shows that with NYSE breakpoints and value-weighted
returns, the high-minus-low decile earns a tiny q-model alpha of −0.08% per month (t = −0.46),
in contrast to −0.34% (t = −2.25) in the five-factor model. Both the investment and ROE factors
help capture this anomaly, with loadings of −0.97 (t = −5.98) and −0.94 (t = −6.33), respectively.
However, the q-factor model is still rejected by the GRS test. With all breakpoints and value6
The evidence is consistent with Bali and Cakici (2008), who report that (i) the data frequency in estimating
idiosyncratic volatility, (ii) return weighting schemes, (iii) breakpoints, and (iv) the data screen for size, price, and
liquidity all play critical roles in determining the existence of the idiosyncratic volatility anomaly. In particular, Bali
and Cakici conclude that no robust relation exists between idiosyncratic volatility and expected returns.
24
Table 4 : The Idiosyncratic Volatility Deciles, NYSE and All-but-micro Breakpoints, January
1967 to December 2014
m is average excess returns, and tm is its t-statistics. α, α3 , αC , α5 , and αq are the alphas from the CAPM,
the Fama-French three-factor model, the Carhart model, the Fama-French five-factor model, and the qfactor model, and tα , t3 , tC , t5 , and tq are their t-statistics, respectively. β MKT , β ME , β I/A , and β ROE are the
loadings on the market, size, investment, and ROE factors in the q-factor model, and tMKT , tME , tI/A , and
tROE are their t-statistics, respectively. m.a.e. (mean absolute error) is the average magnitude of alphas across
the testing deciles. The GRS p-value testing that all ten alphas are jointly zero is in brackets beneath the
m.a.e. for a given factor model. All the t-statistics are adjusted for heteroscedasticity and autocorrelations.
Low
2
3
4
5
6
7
8
9
High
H−L
m.a.e.
Panel A: NYSE breakpoints, value-weighted returns
m
tm
α
β
tα
α3
t3
αC
tC
α5
t5
0.50
3.17
0.15
0.69
1.84
0.11
1.52
0.05
0.77
−0.07
−0.99
0.68
3.91
0.27
0.82
3.66
0.26
3.83
0.21
2.96
0.12
1.54
0.65
3.49
0.19
0.91
2.73
0.18
2.74
0.14
1.87
0.03
0.37
0.66
0.61
0.60
3.20
2.73
2.54
0.16
0.07
0.03
1.00
1.06
1.13
2.46
0.84
0.33
0.14
0.06 −0.01
2.05
0.75 −0.07
0.12
0.09
0.01
1.70
1.18
0.08
0.02 −0.03 −0.06
0.35 −0.39 −0.73
αq
β MKT
β ME
β I/A
β ROE
tq
tMKT
tME
tI/A
tROE
−0.12
0.06
0.00
0.01
0.79
0.90
0.97
1.05
−0.13 −0.12 −0.08 −0.09
0.35
0.23
0.16
0.14
0.20
0.18
0.21
0.16
−1.43
0.75 −0.04
0.12
42.90 50.45 54.33 47.63
−3.74 −3.95 −2.67 −1.81
4.08
3.09
2.24
1.93
3.27
3.90
3.68
3.34
0.01 −0.02
1.08
1.12
0.01
0.10
0.05
0.01
0.06
0.03
0.10 −0.22
49.89 48.45
0.30
1.86
0.79
0.09
1.24
0.79
0.69
2.66
0.07
1.22
0.85
0.10
1.17
0.18
2.09
0.13
1.51
0.65
2.26
−0.02
1.31
−0.17
−0.03
−0.37
0.05
0.52
0.02
0.16
0.63
1.90
−0.10
1.45
−0.77
−0.09
−0.86
0.03
0.24
0.11
1.12
−0.01
−0.02
−0.81
1.58
−4.41
−0.84
−6.24
−0.60
−3.89
−0.41
−3.45
−0.51
−1.62
−0.96
0.89
−4.02
−0.94
−5.43
−0.65
−3.39
−0.34
−2.25
0.23
1.14
0.14
−0.31
−0.06
2.12
39.49
2.97
−3.40
−0.92
0.13
1.20
0.30
−0.21
−0.17
1.29
48.89
5.58
−2.84
−3.19
0.23
1.24
0.50
−0.50
−0.32
1.98
44.24
10.20
−6.62
−4.82
−0.21
1.26
0.75
−0.62
−0.74
−1.48
29.93
12.26
−5.69
−6.65
−0.08
0.47
0.88
−0.97
−0.94
−0.46
9.09
10.65
−5.98
−6.33
0.19
[0.00]
0.18
[0.00]
0.15
[0.00]
0.10
[0.00]
0.10
[0.00]
Panel B: All-but-micro breakpoints, equal-weighted returns
m
tm
α
β
tα
α3
t3
αC
tC
α5
t5
0.59
0.73
3.70
3.85
0.25
0.30
0.67
0.85
2.96
3.79
0.08
0.14
1.20
2.20
0.08
0.17
1.12
2.50
−0.06 −0.01
−0.89 −0.13
0.81
3.88
0.32
0.96
4.01
0.18
2.79
0.22
3.37
0.06
1.08
αq
β MKT
β ME
β I/A
β ROE
tq
tMKT
tME
tI/A
tROE
−0.06
0.00
0.72
0.89
0.15
0.20
0.42
0.37
0.10
0.10
−0.79 −0.03
33.43 39.17
2.89
3.30
6.47
6.18
2.10
2.14
0.09
0.96
0.27
0.23
0.09
1.21
40.93
4.30
3.86
1.77
0.87
3.95
0.35
1.04
4.21
0.20
3.54
0.25
4.05
0.11
2.03
0.86
3.55
0.29
1.13
3.49
0.17
3.03
0.21
3.33
0.14
2.49
0.83
3.21
0.22
1.20
2.37
0.13
2.31
0.15
2.54
0.13
2.24
0.83
2.94
0.17
1.31
1.67
0.12
2.10
0.16
2.70
0.23
3.51
0.75
2.36
0.02
1.43
0.18
0.03
0.41
0.09
1.21
0.21
3.03
0.57
1.61
−0.22
1.56
−1.42
−0.20
−2.16
−0.09
−0.97
0.10
1.19
−0.04
−0.10
−0.89
1.67
−4.24
−0.81
−5.85
−0.60
−3.90
−0.34
−2.95
−0.63
−1.95
−1.14
1.00
−4.47
−0.90
−4.92
−0.68
−3.44
−0.28
−1.96
0.13
0.17
1.01
1.05
0.35
0.45
0.19
0.04
0.07
0.00
1.95
2.43
39.43 50.91
5.15
9.67
3.22
0.85
1.44 −0.12
0.16
1.08
0.58
−0.06
−0.06
2.58
66.41
19.47
−1.73
−2.31
0.29
1.13
0.62
−0.31
−0.18
3.82
68.69
22.27
−4.79
−3.44
0.32
1.17
0.75
−0.52
−0.32
3.68
54.59
17.10
−7.22
−5.55
0.22
1.25
0.82
−0.66
−0.46
2.15
47.21
16.51
−7.19
−6.41
−0.14
1.27
0.92
−0.92
−0.79
−0.91
29.62
12.25
−7.23
−6.24
−0.08
0.54
0.78
−1.34
−0.89
−0.38
9.30
6.43
−7.46
−5.36
25
0.30
[0.00]
0.21
[0.00]
0.20
[0.00]
0.14
[0.00]
0.16
[0.00]
Table 5 : The Idiosyncratic Volatility Deciles, All (NYSE-Amex-NASDAQ) Breakpoints,
January 1967 to December 2014
m is average excess returns, and tm is its t-statistics. α, α3 , αC , α5 , and αq are the alphas from the CAPM,
the Fama-French three-factor model, the Carhart model, the Fama-French five-factor model, and the qfactor model, and tα , t3 , tC , t5 , and tq are their t-statistics, respectively. β MKT , β ME , β I/A , and β ROE are the
loadings on the market, size, investment, and ROE factors in the q-factor model, and tMKT , tME , tI/A , and
tROE are their t-statistics, respectively. m.a.e. (mean absolute error) is the average magnitude of alphas across
the testing deciles. The GRS p-value testing that all ten alphas are jointly zero is in brackets beneath the
m.a.e. for a given factor model. All the t-statistics are adjusted for heteroscedasticity and autocorrelations.
Low
2
3
4
5
6
7
8
9
High
H−L
m.a.e.
Panel A: All breakpoints, value-weighted returns
m
tm
α
β
tα
α3
t3
αC
tC
α5
t5
αq
β MKT
β ME
β I/A
β ROE
tq
tMKT
tME
tI/A
tROE
0.56
3.50
0.19
0.73
2.67
0.17
2.87
0.13
2.09
0.04
0.67
0.56
3.10
0.11
0.89
2.01
0.11
2.24
0.10
2.07
0.03
0.64
0.59
2.78
0.06
1.04
1.05
0.08
1.31
0.12
1.89
0.06
0.77
0.60
2.51
0.01
1.16
0.15
0.06
0.75
0.09
1.27
0.09
1.08
0.67
2.48
0.03
1.28
0.27
0.08
1.04
0.11
1.36
0.21
2.71
0.53
1.68
−0.18
1.39
−1.34
−0.12
−1.15
−0.02
−0.21
0.08
0.90
0.49
1.42
−0.26
1.48
−1.63
−0.18
−1.48
−0.06
−0.45
0.14
1.31
0.16
0.44
−0.64
1.59
−3.64
−0.58
−4.34
−0.40
−3.18
−0.19
−1.66
−0.15
−0.35
−0.97
1.63
−4.37
−0.98
−6.06
−0.67
−3.67
−0.44
−2.88
−0.72
−1.66
−1.50
1.55
−5.56
−1.57
−7.18
−1.26
−5.09
−0.96
−4.50
−1.28
−3.48
−1.69
0.82
−5.42
−1.74
−7.16
−1.39
−5.03
−1.01
−4.16
0.02
0.04
0.10
0.12
0.24
0.21
0.81
0.93
1.04
1.10
1.15
1.19
−0.16 −0.10 −0.05
0.13
0.27
0.40
0.22
0.08 −0.07 −0.23 −0.39 −0.51
0.14
0.09
0.02 −0.04 −0.12 −0.34
0.29
0.72
1.30
1.38
2.82
1.91
52.23 57.22 43.27 47.38 62.48 43.91
−5.93 −3.70 −0.95
3.24
8.40
7.71
3.38
1.40 −1.17 −3.83 −6.81 −6.33
2.90
2.42
0.39 −0.84 −2.69 −5.05
0.25
1.21
0.55
−0.71
−0.45
1.93
34.84
8.60
−6.91
−6.37
−0.03
1.27
0.67
−0.78
−0.59
−0.24
33.21
9.84
−6.49
−6.82
−0.17
1.24
0.84
−0.81
−0.94
−0.97
23.34
10.45
−5.71
−6.94
−0.66
1.15
0.90
−0.66
−1.15
−2.67
18.89
8.65
−3.40
−6.67
−0.68
0.34
1.06
−0.89
−1.29
−2.39
5.04
9.24
−3.65
−6.20
0.39
[0.00]
0.39
[0.00]
0.30
[0.00]
0.22
[0.00]
0.19
[0.00]
Panel B: All breakpoints, equal-weighted returns
m
tm
α
β
tα
α3
t3
αC
tC
α5
t5
αq
β MKT
β ME
β I/A
β ROE
tq
tMKT
tME
tI/A
tROE
0.65
3.71
0.33
0.64
3.30
0.14
1.75
0.15
1.86
0.01
0.09
0.05
0.64
0.32
0.36
0.03
0.59
24.31
5.19
5.78
0.72
0.79
3.72
0.33
0.90
3.61
0.15
2.40
0.19
3.25
0.04
0.65
0.93
3.85
0.41
1.02
4.07
0.21
3.77
0.28
4.76
0.14
2.57
0.97
3.61
0.40
1.12
3.42
0.22
3.58
0.30
4.81
0.19
3.01
0.98
3.35
0.37
1.21
2.75
0.21
3.33
0.33
4.88
0.26
3.55
0.93
2.89
0.27
1.31
1.81
0.12
1.81
0.27
3.50
0.24
2.96
0.93
2.61
0.23
1.38
1.28
0.08
0.93
0.28
2.70
0.30
2.91
0.71
1.87
−0.01
1.41
−0.03
−0.17
−1.60
0.09
0.64
0.11
0.81
0.64
1.52
−0.09
1.45
−0.38
−0.28
−1.79
0.06
0.30
0.13
0.68
0.38
0.82
−0.35
1.44
−1.12
−0.57
−2.49
−0.19
−0.67
−0.07
−0.28
−0.27
−0.75
−0.68
0.81
−2.29
−0.71
−2.91
−0.34
−1.15
−0.08
−0.30
0.09
0.21
0.28
0.86
0.95
1.00
0.41
0.53
0.67
0.26
0.18
0.07
0.02 −0.03 −0.10
1.20
2.77
3.36
31.89 33.03 35.62
5.65
6.81
8.77
4.14
2.78
0.94
0.55 −0.61 −2.27
0.39
1.02
0.77
−0.09
−0.25
4.01
36.34
10.97
−1.12
−4.14
0.40
1.08
0.86
−0.18
−0.38
3.44
37.59
12.62
−1.74
−4.81
0.53
1.08
0.96
−0.33
−0.56
3.59
35.49
15.67
−2.59
−5.63
0.41
1.08
1.00
−0.39
−0.73
2.05
25.97
12.60
−2.22
−5.55
0.55
1.05
1.09
−0.53
−1.00
1.92
20.56
13.94
−2.40
−5.72
0.39
1.01
1.18
−0.47
−1.24
1.05
14.16
10.91
−1.62
−5.49
0.33
0.37
0.86
−0.82
−1.28
0.88
4.86
7.17
−2.74
−5.30
26
0.28
[0.00]
0.22
[0.00]
0.21
[0.00]
0.15
[0.00]
0.33
[0.00]
weighted returns, Table 5 shows that the high-minus-low decile earns a q-model alpha of −0.68%
(t = −2.39), which is lower in magnitude than −1.01% (t = −4.16) in the five-factor model.
2.4.4.2
The Beta Anomaly As in Frazzini and Pedersen (2014), a stock’s market beta is esti-
mated as β̂ i = ρ̂σ̂ i /σ̂ m , in which σ̂ i and σ̂ m are the stock and market volatilities, and ρ̂ is their correlation. Volatilities are estimated from daily log returns over a one-year rolling window (with a minimum of 120 daily returns). The correlation is estimated with overlapping three-day log returns over
a five-year rolling window (with a minimum of 750 daily returns). At the beginning of each month t,
stocks are split into deciles based on β̂ i estimated at the end of month t − 1. Monthly decile returns
are calculated for the current month t, and the deciles are rebalanced at the beginning of month t+1.
Consistent with Frazzini and Pedersen (2014), Table 6 shows that sorting on β̂ i yields no
significant average return spread. The high-minus-low decile earns only −0.22% per month (t =
−0.65) with NYSE breakpoints and value-weighted returns. In addition, the beta sorts yield a large
spread of 1.13 in the post-ranking market beta, which leads to a significant high-minus-low alpha of
−0.79% (t = −3.25) in the CAPM. The three-factor model fails to capture the beta anomaly, with a
high-minus-low alpha of −0.57% (t = −2.51). Both the Carhart and five-factor models capture the
anomaly. In particular, the five-factor alpha for the high-minus-low decile is −0.27% (t = −1.18).
The q-factor model captures the beta anomaly, leaving a high-minus-low alpha of only 0.01%
per month (t = 0.05). The post-ranking beta spread of 0.85 is offset by the significant spreads
in the investment factor loadings, −1.11, and in the ROE factor loadings, −0.47. However, the
q-factor model is still rejected by the GRS test. The five-factor model is also rejected. The results
with all-but-micro breakpoints and equal-weighted returns are largely similar. The high-minus-low
decile has a q-model alpha of 0.16% (t = 0.58) and a five-factor alpha of −0.22% (t = −1.07).
Frazzini and Pedersen (2014) also construct a betting-against-beta factor, rBAB , which goes
long in a low-beta portfolio and short a high-beta portfolio, both scaled to a beta of unity. Frazzini
and Pedersen show that rBAB earns a significantly positive average return, which also survives the
27
Table 6 : The Beta Deciles, January 1967 to December 2014
m is average excess returns, and tm is its t-statistics. α, α3 , αC , α5 , and αq are the alphas from the CAPM,
the Fama-French three-factor model, the Carhart model, the Fama-French five-factor model, and the qfactor model, and tα , t3 , tC , t5 , and tq are their t-statistics, respectively. β MKT , β ME , β I/A , and β ROE are the
loadings on the market, size, investment, and ROE factors in the q-factor model, and tMKT , tME , tI/A , and
tROE are their t-statistics, respectively. m.a.e. (mean absolute error) is the average magnitude of alphas across
the testing deciles. The GRS p-value testing that all ten alphas are jointly zero is in brackets beneath the
m.a.e. for a given factor model. All the t-statistics are adjusted for heteroscedasticity and autocorrelations.
Low
2
3
4
5
6
7
8
9
High
H−L
0.30
0.81
−0.52
1.62
−3.09
−0.49
−3.13
−0.27
−1.62
−0.32
−1.94
−0.22
−0.65
−0.79
1.13
−3.25
−0.57
−2.51
−0.31
−1.27
−0.27
−1.18
m.a.e.
Panel A: NYSE breakpoints, value-weighted returns
m
tm
α
β
tα
α3
t3
αC
tC
α5
t5
0.52
3.41
0.27
0.49
2.35
0.08
0.72
0.04
0.34
−0.04
−0.38
0.61
4.09
0.31
0.59
3.11
0.15
1.81
0.12
1.38
0.06
0.68
0.64
3.85
0.27
0.72
3.26
0.17
2.05
0.13
1.46
0.03
0.35
0.71
3.98
0.30
0.81
3.15
0.21
2.49
0.16
1.88
−0.02
−0.27
0.54
2.80
0.09
0.89
1.08
−0.01
−0.16
−0.04
−0.45
−0.22
−2.85
0.57
2.65
0.07
0.98
0.76
0.00
−0.01
−0.01
−0.07
−0.20
−2.60
0.55
2.43
0.03
1.03
0.28
−0.04
−0.42
0.01
0.10
−0.24
−2.59
0.49
1.84
−0.12
1.19
−1.09
−0.19
−1.78
−0.14
−1.30
−0.39
−3.76
0.51
1.80
−0.15
1.29
−1.20
−0.20
−1.69
−0.10
−0.89
−0.31
−2.60
αq
β MKT
β ME
β I/A
β ROE
tq
tMKT
tME
tI/A
tROE
−0.11
0.00
0.00 −0.07
0.58
0.67
0.79
0.92
0.08
0.01
0.00 −0.05
0.54
0.46
0.37
0.44
0.14
0.12
0.14
0.25
−0.84 −0.05 −0.05 −0.71
17.91 24.70 31.59 35.31
1.41
0.32
0.09 −0.92
5.27
6.75
5.82
4.98
1.79
1.87
2.30
4.23
−0.22
0.98
−0.05
0.42
0.16
−2.38
43.28
−1.00
5.14
2.69
−0.19
1.06
−0.04
0.33
0.16
−2.06
34.45
−0.98
3.81
2.20
−0.22 −0.40 −0.19 −0.10
0.01
1.10
1.24
1.28
1.43
0.85
0.00
0.08
0.11
0.32
0.24
0.28
0.26 −0.02 −0.57 −1.11
0.17
0.22
0.05 −0.33 −0.47
−2.07 −3.38 −1.49 −0.55
0.05
40.15 37.91 32.40 28.41 11.70
0.00
1.25
1.57
4.67
2.62
3.06
2.35 −0.20 −5.23 −6.69
2.26
2.71
0.68 −2.83 −2.81
0.21
[0.01]
0.15
[0.01]
0.10
[0.22]
0.18
[0.00]
0.15
[0.02]
Panel B: All-but-micro breakpoints, equal-weighted returns
m
tm
α
β
tα
α3
t3
αC
tC
α5
t5
αq
β MKT
β ME
β I/A
β ROE
tq
tMKT
tME
tI/A
tROE
0.66
4.27
0.37
0.57
3.78
0.17
1.94
0.14
1.48
0.06
0.71
0.81
4.75
0.44
0.73
4.90
0.26
3.51
0.25
3.22
0.15
2.12
0.87
4.42
0.43
0.87
4.57
0.24
3.60
0.24
3.20
0.11
1.75
0.82
0.77
3.88
3.29
0.33
0.24
0.97
1.05
3.33
2.41
0.16
0.06
2.16
0.88
0.16
0.08
2.18
1.08
0.00 −0.07
0.02 −1.03
0.87
3.63
0.30
1.13
3.27
0.15
2.41
0.17
2.82
0.07
1.22
0.86
3.24
0.22
1.25
2.33
0.08
1.24
0.13
2.11
0.03
0.52
0.73
2.50
0.05
1.36
0.43
−0.03
−0.37
0.06
0.88
0.01
0.20
0.62
1.85
−0.16
1.54
−1.13
−0.18
−1.83
0.00
0.05
0.01
0.15
0.38
0.89
−0.56
1.85
−2.63
−0.51
−3.01
−0.18
−1.00
−0.16
−1.04
−0.29
−0.80
−0.93
1.28
−3.57
−0.67
−3.02
−0.31
−1.31
−0.22
−1.07
0.02
0.61
0.25
0.46
0.09
0.20
21.68
4.96
6.50
1.59
0.12
0.74
0.32
0.36
0.11
1.34
31.73
6.32
5.87
2.01
0.08
0.86
0.40
0.33
0.15
1.04
34.09
6.73
5.00
2.73
0.01 −0.06
0.95
1.02
0.41
0.45
0.29
0.27
0.13
0.09
0.11 −0.73
43.24 45.22
6.94
8.55
4.29
4.51
2.41
1.99
0.10
1.07
0.46
0.16
0.02
1.59
58.53
9.79
3.00
0.49
0.08
1.16
0.52
0.10
−0.04
1.22
59.10
11.06
1.87
−0.98
0.10
1.20
0.61
−0.15
−0.18
1.10
60.47
20.45
−2.47
−3.29
0.22
1.29
0.66
−0.47
−0.46
1.74
40.04
15.27
−4.59
−5.35
0.18
1.49
0.76
−0.81
−0.80
0.93
26.26
9.87
−5.99
−5.49
0.16
0.88
0.51
−1.27
−0.89
0.58
11.17
4.46
−6.90
−4.73
28
0.31
[0.00]
0.18
[0.00]
0.14
[0.00]
0.07
[0.01]
0.10
[0.01]
Table 7 : Properties of the Betting-Against-Beta Factor, January 1967 to December 2014
The U.S. data for the Frazzini-Pedersen (2014) betting-against-beta factor, rBAB , are from Andrea Frazzini’s
Web site. The mean of rBAB is 0.9% per month (t = 5.56). The data for SMB, HML, RMW, and CMA in
the Fama-French five-factor model, as well as UMD are from Kenneth French’s Web site. αC is the Carhart
alpha, and β MKT , β SMB , β HML , and β UMD are the Carhart factor loadings. αq is the q-model alpha, and
β MKT , β ME , β I/A , and β ROE are the q-factor loadings. α5 is the five-factor alpha, and b, s, h, r, and c are
five-factor loadings. The t-statistics in parentheses are adjusted for heteroscedasticity and autocorrelations.
Panel A: The Carhart model
rBAB
αC
β MKT
β SMB
β HML
β UMD
R2
0.54
(3.31)
0.07
(1.33)
0.01
(0.22)
0.53
(5.96)
0.20
(3.54)
0.22
Panel B: The Fama-French (2015a) five-factor model
rBAB
α5
b
s
h
r
c
R2
0.47
(3.05)
0.09
(1.61)
0.13
(2.20)
0.29
(2.70)
0.48
(5.36)
0.37
(2.43)
0.24
Panel C: The Hou-Xue-Zhang (2015a, b) q-factor model
rBAB
αq
β MKT
β ME
β I/A
β ROE
R2
0.31
(1.78)
0.07
(1.32)
0.14
(2.26)
0.69
(6.07)
0.38
(4.02)
0.20
controls of standard factor exposures. Table 7 reports that the betting-against-beta factor earns
on average 0.9% per month (t = 5.56) in the 1967–2014 U.S. sample. The Carhart and five-factor
models cannot explain the factor return, leaving alphas of 0.54% (t = 3.31) and 0.47% (t = 3.05),
respectively. However, the q-model alpha is only 0.31% (t = 1.78). Both the investment and ROE
factor loadings are economically large and statistically significant.
2.4.5
Independent Comparison
Several studies have independently compared the Hou-Xue-Zhang (2015a, b) q-factor model with
the Fama-French (2015a) five-factor model. Their empirical results are broadly in line with those
furnished in Hou et al. (2015b). Barillas and Shanken (2015) develop a Bayesian test procedure
that allows model comparison, i.e., the computation of model probabilities for the collection of
all possible pricing models based on subsets of the given factors. Applying this new methodology
to the head-to-head contest between the q-factor model and the five-factor model, Shanken (2015)
reports model probabilities that are overwhelmingly in favor of the q-factor model (97% versus 3%).
Cooper and Maio (2015) evaluate to what extent the new factor models are consistent with the
29
Merton (1973) intertemporal CAPM, and show that the q-factor model outperforms the five-factor
model in terms of convergence with the Merton model. In particular, Hou, Xue, and Zhang’s
(2015a, b) investment and ROE factors both forecast future economic activity, but Fama and
French’s (2015a) RMW does not, in a way that is consistent with the intertemporal CAPM.
Stambaugh and Yuan (2015) independently verify the two key results in Hou, Xue, and Zhang
(2015b). First, the q-factor model explains the Fama-French (2015a) five-factor returns in time
series regressions, but the five-factor model cannot explain the q-factor returns. Second, the qfactor model outperforms the five-factor model in explaining a wide array of anomalies in the cross
section. Stambaugh and Yuan also propose a “mispricing” factor model. The mispricing factors
are formed by aggregating a stock’s information across 11 anomalies with average rankings within
two anomaly clusters that exhibit the greatest comovement in high-minus-low returns. Empirically,
the mispricing factor model and the q-factor model are “evenly matched” in factor spanning tests,
but the mispricing model has an edge in explaining anomalies, especially in the set of 11.
The edge is perhaps not surprising, however, as the two mispricing factors are basically principal
components extracted ex post from the 11 anomalies. The choice of the 11 anomalies out of a pool
of hundreds seems arbitrary. Also, stocks with prices lower than $5 are excluded. This practice is
nonstandard other than in the context of price momentum. The breakpoints for the mispricing
factors are also nonstandard, 20-60-20, as opposed to 30-40-30 in Fama and French (1993).
Finally, the q-factor model, which is based on two economic fundamentals, shows a fair amount of
explanatory power for all the 11 anomalies, casting doubt on their mispricing interpretation.
2.5
Notes
The q-factor model is built on a rich empirical literature in finance and accounting. Most important, as the q-factor model uses the Fama-French three- and five-factor models as a straw man, it is
only natural to acknowledge our enormous intellectual debt to Fama and French. Fama and French
(1992) show that size and book-to-market combine to describe average returns in cross-sectional
30
regressions, and that the relation between the market beta and average returns is flat, even when
beta is used alone. Fama and French (1993) propose the three-factor model to replace the CAPM
as the workhorse for estimating expected stock returns. Fama and French (1996) show that, except
for momentum, the three-factor model summarizes the cross section of expected returns as of the
mid-1990s. Carhart (1997) augments the three-factor model with a momentum factor. Perhaps
in response to the challenge by Hou, Xue, and Zhang (2015a), Fama and French (2015a) upgrade
their three-factor model with two new factors that resemble the investment and ROE factors in
the q-factor model. It is evident that the intellectual designs of the q-factor model, including its
econometric tests, factor construction, formation of testing portfolios, and above all, the taste of
the economic question, are all deeply influenced by Fama and French.
2.5.1
2.5.1.1
Related Studies on the Investment Premium
Different Forms of the Investment Premium
The first group of studies documents
the investment effect in various forms, and shows how it relates to other cross-sectional patterns.
Titman, Wei, and Xie (2004) show that firms that substantially increase capital investment
subsequently earn lower average returns than firms that substantially decrease capital investment.
This effect is also stronger among firms with higher cash flows, implying higher investment discretion. Titman et al. interpret the evidence as investors underreacting to the empire building
implications of increasing investment. Empire building means that managers invest for their own
private benefits rather than the benefits for shareholders (Jensen 1986).
However, the investment CAPM, derived without empire building, is consistent with the evidence that the investment effect is stronger in firms with higher cash flows. Taking the firstS /∂(I /K ) =
order derivative of equation (7) with respect to investment-to-capital yields ∂rit+1
it
it
−aXit+1 /[1 + a(Iit /Kit )]2 . Its magnitude rises with profitability, Xit+1 , meaning that the investment effect should be stronger among firms with higher cash flows.
Titman, Wei, and Xie (2004) measure abnormal investment as the ratio of capital expenditure
31
divided by sales, scaled by the prior three-year moving average of this ratio. Dividing investment
by sales makes the ratio closer to profitability than to investment. Hou, Xue, and Zhang (2015b)
show that the high-minus-low abnormal investment decile earns on average −0.31% per month (t =
−2.12) from 1967 to 2013. The q-factor alpha is −0.17% (t = −0.96), along with an insignificant
investment factor loading of 0.13 (t = 1.01) but a significant ROE factor loading of −0.19 (t = −2.1).
Anderson and Garcia-Feijóo (2006) show that sorting on book-to-market provides a large spread
in investment growth across extreme deciles, and that firms with high investment growth earn significantly lower subsequent returns on average than firms with low investment growth. Xing (2008)
shows that a low-minus-high investment growth factor contains similar information as the value
factor, and can price the 25 size and book-to-market portfolios as well as the value factor. Both
studies interpret their evidence as consistent with the investment CAPM.
Lyandres, Sun, and Zhang (2008) show that the investment effect helps interpret the new issues
puzzle (Ritter 1991; Loughran and Ritter 1995). Lyandres et al. show that adding an investment
factor into the CAPM and the Fama-French three-factor model reduces a substantial amount of the
underperformance following initial public offerings, seasoned equity offerings, and convertible debt
offerings, as well as the composite issuance effect (Daniel and Titman 2006). Also, Lyandres et al.
document that equity issuers invest much more relative to their assets than nonissuers matched on
size and book-to-market, despite similar profitability across issuers and matching nonissuers.
Cooper, Gulen, and Schill (2008) document the strong predictive power of the annual growth
rates of total assets in the cross section. Their key insight is that the investment effect is a pervasive phenomenon, going beyond specific components of investment explored in prior studies. In
particular, their Table II shows that from 1968 to 2002, the high-minus-low asset growth decile
earns on average a whopping equal-weighted return of −1.73% per month and a value-weighted
return of −1.05%. Cooper et al. argue that “bias in the capitalization of new investments leads to
a host of potential investment policy distortions,” and interpret their evidence as suggesting that
32
“such potential distortions are present and economically meaningful (p. 1648).”
Cooper, Gulen, and Schill (2008) have clearly influenced the q-factor model and the FamaFrench five-factor model. Both form the investment factor on total asset growth. However, Cooper
et al.’s evidence on the predictive power of asset growth seems exaggerated by excessively weighting
on microcaps. Their deciles are formed with all breakpoints, rather than NYSE breakpoints, and
the portfolio returns are equal-weighted to give microcaps disproportionately large weights. Hou,
Xue, and Zhang (2015b) show that from 1967 to 2013, the high-minus-low asset growth decile
earns −0.46% per month (t = −2.86) with NYSE breakpoints and value-weighted returns and
−0.72% (t = −4.84) with all-but-micro breakpoints and equal-weighted returns. Also, the investment CAPM predicts that high investment implies low subsequent returns, and that this evidence
does not necessarily mean value-destroying investment distortions such as empire building.
Butler, Cornaggia, Grullon, and Weston (2011) propose a clever identification strategy to disentangle investment and behavioral market timing explanations for the underperformance following
security issuance. The investment CAPM says that issuers invest more, and are associated with
lower costs of capital than nonissues (Lyandres, Sun, and Zhang 2008), and that investment converts
risky growth options into less risky assets in place, reducing risk and expected returns for issuers
(Carlson, Fisher, and Giammarino 2006). The market timing explanation says that managers issue
more equity relative to debt when equity is overvalued, and repurchase more equity relative to debt
when equity is undervalued. While the investment CAPM says that only the amount of net financing forecasts returns, the market timing hypothesis predicts that the composition of net financing
(equity relative to debt) is more important. Empirically, Butler et al. report pervasive evidence that
conditional on the amount of net financing, the composition of financing does not forecast returns.
Cooper and Priestley (2011) report extensive evidence that the negative relation between investment and average returns is related to macroeconomic risk. First, low investment firms have
substantially higher loadings on the Chen, Roll, and Ross (1986) macroeconomic risk factors than
33
high investment firms, especially on the growth rate of industrial production and the term spread.
Second, the large loading spreads on these two macro risk factors, combined with their large estimated risk premiums from two-pass cross-sectional regressions, account for 60–80% of the average
return spreads across extreme investment deciles. Third, macro risk loadings fall during high investment periods, but rise during disinvestment periods. Finally, the investment factor returns
contain information about future real growth rates of industrial production, gross domestic product, aggregate corporate earnings, and aggregate investment. However, Cooper and Priestley do
not conduct similar risk analysis for the positive relation between profitability and average returns.
2.5.1.2
The Cross-sectional Variation of the Investment Premium
Li and Zhang (2010)
argue that investment frictions should strengthen the investment effect. Intuitively, when investment is frictionless (adjustment costs are zero), investment would be infinitely elastic to changes in
the discount rate. As investment becomes more frictional, investment becomes less elastic to the
discount rate changes. In other words, a given magnitude of change in investment corresponds to a
higher magnitude of change in the discount rate, meaning that the investment effect in the average
returns becomes stronger as investment becomes more frictional.
The impact of investment frictions on the investment effect is likely related to that of limits
to arbitrage (Shleifer and Vishny 1997). Shleifer and Vishny argue that trading frictions from the
investors’ side make arbitrage activities costly and incomplete, allowing mispricing to persist in
the marketplace. Attributing the investment effect to mispricing, limits to arbitrage imply that it
should be stronger among firms with stronger limits to arbitrage. Because the investment CAPM
and mispricing emphasize different types of frictions that likely coexist in the data, their effects are
unlikely to be mutually exclusive. In addition, to the extent that firms with stocks that are more
costly to trade could also face higher investment costs, investment frictions and limits to arbitrage
could both at work for the same set of firms simultaneously.
Empirically, using proxies for financial constraints (asset size, payout ratio, and credit rating) to
34
measure investment frictions, Li and Zhang (2010) report only weak evidence that the investment
effect is stronger in firms with stronger investment frictions. In contrast, idiosyncratic volatility
and dollar trading volume, which are common proxies for limits to arbitrage, dominate proxies of
financial constraints in characterizing the cross-sectional variation of the investment effect.
Lam and Wei (2011) conduct more comprehensive tests by using ten proxies of limits to arbitrage, including idiosyncratic volatility, analyst coverage, analyst forecast dispersion, cash flow
volatility, the number of institutional shareholders, stock price, bid-ask spread, institutional ownership, absolute return-to-volume, and dollar trading volume, as well as four proxies of financial
constraints, firm age, asset size, payout ratio, and credit rating. Lam and Wei show that the proxies for limits to arbitrage are highly correlated with those for investment frictions. The evidence
from equal-weighted returns is consistent with both limits to arbitrage and investment frictions
hypotheses, but the evidence from value-weighted returns is weaker. Finally, in direct comparisons,
each hypothesis is supported by a fair and similar amount of evidence.
Lipson, Mortal, and Schill (2011) refute Fama and French’s (2008) conclusion that the investment effect exists only in small firms. Lipson et al. show that Fama and French’s investment measure
excludes the part of asset growth related to equity issues, which are a major source of financing for
large firms. Also, the investment effect is stronger in firms with high idiosyncratic volatility, and is
concentrated around earnings announcement dates. Finally, analysts forecast errors are higher for
faster growing firms, consistent with the mispricing interpretation.
Titman, Wei, and Xie (2013) study the investment effect in international markets, and find that
it is stronger in countries with more developed markets than in countries with less developed markets. This evidence lends support to the investment CAPM because financial market development
aligns managers’ incentives with shareholders, and the investment effect arises from maximizing the
shareholder value. Titman et al. also show that the investment effect is not related to corporate
governance or trading costs across countries, inconsistent with the mispiricing interpretation.
35
Watanabe, Xu, Yao, and Yu (2013) conduct a more comprehensive study of the investment
effect in international equity markets. If the investment effect is due to mispricing, it should be
stronger in countries in which stocks are less efficiently priced and mispricing is more difficult to
arbitrage away. In contrast, if the investment effect is due to optimal capital budgeting, it should
be stronger in countries in which stocks are more efficiently priced.
Watanabe, Xu, Yao, and Yu (2013) use four country-level proxies for the efficiency of financial
markets, (i) Morck, Yeung, and Yu’s (2000) stock return synchronicity, which is the average R2
from firm-level market regressions within each country, and is negatively related to the amount of
firm-specific information impounded into individual stock prices; (ii) future earnings response coefficients, a measure of the information content of stock prices for future earnings; (iii) the developed
market status from the International Finance Corporation; and (iv) the importance of stock market
to the economy, which is measured by total market capitalization-to-gross domestic product, as well
as the number of publicly traded companies and initial public offerings scaled by population. The investment effect is stronger in countries with lower stock return synchronicity and higher future earnings response coefficients, in developed markets, and in economies in which stock markets are more
important. The evidence lends support to the investment CAPM, but casts doubt on mispricing.
2.5.1.3
Applications in the Accrual Anomaly
Accounting accruals (earnings minus cash
flows) allow a firm to measure its performance by recognizing economic events regardless of the
timing of cash transactions. The basic idea is to match revenues to expenses at the time of a
transaction, as opposed to a cash payment (Dechow 1994). Sloan (1996) documents that firms
with high accruals earn lower average returns than firms with low accruals. Sloan interprets the
evidence as investors overreacting to the persistence of the accrual component of earnings, when
forming earnings expectations. These naive investors are subsequently surprised, when realized
earnings of firms with high accruals fall short of, and those of firms with low accruals exceed, prior
expectations. Extending Sloan’s work, Richardson, Sloan, Soliman, and Tuna (2005) rank compo-
36
nents of accruals by their degree of reliability, and show that less reliable components lead to less
persistence in earnings and stronger accrual anomaly than more reliable components.
Several studies link accruals to investment. Fairfield, Whisenant, and Yohn (2003) decompose
the growth in net operating assets into accruals and the growth in long-term net operating assets,
and show that after controlling for current profitability, both components forecast future stock
returns with a negative slope. Fairfield et al. suggest that investors overreact to both components
equally, and that the accrual anomaly is a special case of a more general growth anomaly. Zhang
(2007) documents that accruals are positively correlated with employee growth, and that the magnitude of the accrual anomaly monotonically increases with the accruals-employee growth covariation.
Interpreting accruals as working capital investment, Wu, Zhang, and Zhang (2010) apply the
investment CAPM to the accrual anomaly. The investment CAPM helps explain Richardson, Sloan,
Soliman, and Tuna’s (2005) key finding that less reliable accruals lead to stronger accrual anomaly.
Empirically, less reliable accruals are more correlated with investment than more reliable accruals.
Conceptually, less reliable changes in noncash working capital and changes in net noncurrent
operating assets represent direct forms of investment in short-term and long-term capital,
respectively. In contrast, more reliable changes in net financial assets contain diverse components,
including marketable securities and financial liability, which are less correlated with investment.
Table 8 reports factor regressions of deciles formed on operating accruals, measured with the
balance sheet approach of Sloan (1996), scaled by average total assets. The high-minus-low decile
earns an average return of −0.29% per month (t = −2.13). Consistent with Wu, Zhang, and Zhang
(2010), a two-factor model with the market and investment factors yields a high-minus-low alpha of
−0.1% (t = −0.66). The investment factor loading is −0.53, which is more than four standard errors
from zero. However, as shown in Hou, Xue, and Zhang (2015a), the full-fledged q-factor model fails
to capture the accrual anomaly with an alpha of −0.42% (t = −2.96). The culprit is a large, positive
ROE factor loading, 0.31 (t = 4.08), which goes in the wrong way to explain the average return.7
7
Hou, Xue, and Zhang (2015a) also show that the investment factor loading for the high-minus-low accrual decile is
37
Table 8 : The Balance Sheet Operating Accruals Deciles
Operating accruals are defined as (dCA−dCASH)−(dCL−dSTD−dTP)−DP, in which dCA is the change
in current assets (Compustat annual item ACT), dCASH is the change in cash or cash equivalents (item
CHE), dCL is the change in current liabilities (item LCT), dSTD is the change in debt included in current
liabilities (item DLC), dTP is the change in income taxes payable (item TXP, zero if missing), and DP is
depreciation and amortization (item DP). At the end of June of each year t, stocks are split into deciles with
NYSE breakpoints on operating accruals for the fiscal year ending in calendar year t − 1 scaled by average
total assets (item AT) for the fiscal years ending in t − 1 and t − 2. Monthly value-weighted decile returns
are calculated from July of year t to June of t + 1, and the deciles are rebalanced in June of t + 1. m is
average excess returns, tm is its t-statistics, and m.a.e. (mean absolute error) is the average magnitude of
alphas across the testing deciles. The GRS p-value is in brackets beneath the m.a.e. for a given factor model.
All the t-statistics are adjusted for heteroscedasticity and autocorrelations.
Low
2
3
4
5
6
7
8
9
High
H−L m.a.e.
Panel A: Average returns, January 1967–December 2014
m
tm
0.61
2.40
0.55
2.61
0.56
3.06
0.63
3.42
0.62
3.09
0.60
3.19
0.62
3.19
0.42
2.00
0.44
1.93
0.32 −0.29
1.14 −2.13
Panel B: Two-factor regressions, the market and investment factors, January 1967–December 2014
α
β MKT
β I/A
tα
tMKT
tI/A
0.17
0.06
1.09
0.99
−0.26 −0.04
1.48
0.67
27.84 37.17
−2.45 −0.29
0.05
0.93
0.09
0.78
56.71
1.77
0.11
0.93
0.10
1.41
35.95
1.79
0.10
0.93
0.10
1.35
42.16
1.71
0.06
0.22 −0.03
0.05
0.07
0.91
0.92
0.96
1.05
1.15
0.16 −0.13 −0.09 −0.31 −0.79
0.88
3.54 −0.35
0.62
0.78
52.43 43.58 47.76 41.83
43.73
1.93 −2.66 −1.28 −4.32 −12.70
−0.10
0.07
−0.53
−0.66
1.46
−4.03
0.09
[0.01]
Panel C: The benchmark q-factor regressions, January 1967–December 2014
αq
β MKT
β ME
β I/A
β ROE
tq
tMKT
tME
tI/A
tROE
0.33
1.07
−0.06
−0.28
−0.22
2.65
29.98
−1.21
−3.11
−2.55
0.16
1.00
−0.12
−0.05
−0.10
1.39
38.57
−2.39
−0.45
−1.27
0.13
0.09
0.11
0.94
0.94
0.95
−0.12 −0.03 −0.08
0.08
0.10
0.10
−0.07
0.04
0.02
1.81
1.01
1.36
52.20 36.98 45.00
−3.57 −0.94 −2.14
1.62
1.76
1.78
−2.09
0.80
0.46
−0.04
0.94
−0.03
0.17
0.17
−0.48
57.17
−0.86
2.41
3.28
0.14
0.94
−0.04
−0.13
0.13
2.10
43.81
−1.21
−3.01
2.87
−0.12
0.97
0.05
−0.07
0.13
−1.57
45.38
1.72
−1.25
3.10
−0.03 −0.09
1.05
1.10
0.05
0.37
−0.30 −0.75
0.09
0.10
−0.35 −1.00
37.99
46.56
1.42
11.77
−4.48 −15.72
1.78
2.03
−0.42
0.03
0.43
−0.47
0.31
−2.96
0.86
7.60
−5.19
4.08
0.12
[0.00]
Panel D: The q-factor regressions with the cash-based ROE factor, January 1972–December 2014
α
eq
e
β
MKT
e
β
ME
e
β
I/A
e
β
ROE
e
tq
e
tMKT
e
tME
e
tI/A
e
tROE
0.14
1.11
0.03
−0.16
−0.09
1.07
26.54
0.69
−1.51
−0.92
0.12
0.12
0.10
0.15
1.02
0.94
0.94
0.92
−0.09 −0.09 −0.03 −0.10
−0.05
0.03
0.10
0.06
−0.03
0.00
0.05 −0.03
1.16
1.44
1.15
1.73
39.37 51.24 34.02 46.76
−1.95 −3.22 −1.20 −2.57
−0.44
0.66
1.97
1.14
−0.24
0.02
0.75 −0.43
−0.02
0.91
−0.06
0.12
0.21
−0.22
49.24
−1.89
2.12
2.17
38
0.13
0.93
−0.07
−0.10
0.11
1.87
38.93
−2.09
−2.45
1.93
−0.07
0.95
0.00
−0.03
0.02
−0.74
42.14
−0.13
−0.59
0.34
0.02 −0.03 −0.17
1.04
1.11
0.00
0.01
0.29
0.26
−0.26 −0.64 −0.48
−0.04 −0.14 −0.05
0.18 −0.37 −1.09
33.86
41.07
0.07
0.40
8.13
4.50
−4.20 −11.79 −4.26
−0.46 −2.52 −0.48
0.09
[0.07]
Following Ball, Gerakos, Linnainmaa, and Nikolaev (2015), Table 8 also experiments with an
alternative q-factor model with a cash-based ROE factor. The cash-based ROE is quarterly cashbased operating profits divided by one-quarter-lagged book equity (Appendix A). The sample
starts in January 1972 due to limited coverage in Compustat quarterly files in earlier years. The
construction of the cash-based q-factors is identical to that for the benchmark q-factors, except that
at the beginning of each month t, stocks are split on the cash-based ROE for the fiscal quarter ending
at least four months ago, as opposed to the ROE calculated with the latest announced earnings.
The cash-based ROE factor earns on average 0.59% per month (t = 7.5) from January 1972 to
December 2014 (untabulated). Its Carhart alpha is 0.57% (t = 7.83), and the Fama-French fivefactor alpha 0.54% (t = 6.81). The investment factor earns on average 0.33% (t = 3.28). Its Carhart
alpha is 0.17% (t = 2.09), and the five-factor alpha is close to zero. The average size factor return
is 0.28% (t = 1.9). The correlation between the benchmark and cash-based ROE factors is only
0.56. The cash-based ROE factor has an alpha of 0.42% (t = 5.1) in the q-factor model. However,
the cash-based q-factor model cannot subsume the benchmark q-factors, with alphas of 0.15% (t =
2.94), 0.15% (t = 3.55), and 0.23% (t = 2.37) for the size, investment, and ROE factors, respectively.
Table 8 shows that the cash-based q-factor model captures the accrual anomaly. The alpha of
the high-minus-low decile is −0.17% per month (t = −1.09), and the model cannot be rejected
by the GRS test (p = 0.07). The investment factor loading is −0.48, which is close to −0.47 in
the benchmark q-factor model. More important, the cash-based ROE factor loading is only −0.05
(t = −0.48), in contrast to 0.31 (t = 4.08) for the benchmark ROE factor. Intuitively, because
earnings equal cash flows plus accruals, high accrual firms would appear more “profitable,” and load
more heavily on the earnings-based benchmark ROE factor than low accrual firms. The cash-based
ROE factor avoids this pitfall, as accruals do not enter the cash-based ROE.
close to zero, in contrast to −0.47 (t = −5.19) in Table 8. The difference arises because Hou et al. measure operating
accruals, starting from 1988, as net income (Compustat annual item NI) minus net cash flow from operations (item
OANCF) from the statement of cash flows. In untabulated results, from July 1989 to December 2014, the high-minuslow decile on this cash flows-based operating accruals earns an average return of −0.38% per month (t = −2.12). The
q-factor alpha is −0.51% (t = −2.64), the investment factor loading −0.07, and the ROE factor loading 0.28 (t = 3.06).
39
2.5.2
Related Studies on the Profitability Premium
The Hou-Xue-Zhang (2015a, b) ROE factor is built on the illustrious accounting literature on
post-earnings-announcement drift (earnings momentum) launched by Ball and Brown (1968). In
particular, the monthly sort on quarterly earnings in the ROE factor is influenced by the construction of earnings momentum (Chan, Jegadeesh, and Lakonishok 1996).
2.5.2.1
Post-earnings-announcement Drift
I do not attempt to review the literature on
post-earnings-announcement drift (see Richardson, Tuna, and Wysocki 2010 for an extensive survey). Rather, the goal is to describe how this literature informs and influences the investment
CAPM literature, and how the investment CAPM might add to the accounting literature.
Ball and Brown (1968) show that the sign and magnitude of stock returns in the post-earnings
announcement period are positively correlated with the sign and magnitude of unexpected earnings.
Foster, Olsen, and Shevlin (1984) use more sophisticated models of expected earnings, and show
that the magnitude of the post-earnings-announcement drift decreases with firm size. Bernard and
Thomas (1989) show that a disproportionately large fraction of the drift is concentrated in the
three-day period surrounding the earnings announcement dates in subsequent quarters.
The investment CAPM is consistent with the evidence that a disproportionately large fraction of
earnings momentum is concentrated in a few days surrounding the subsequent earnings announcements. Intuitively, equation (7) holds ex post in realization, state by state, period by period, an
observation first made in Cochrane (1991). In the static model, stock returns should move only when
earnings shocks hit, and all of earnings momentum should be materialized on earnings announcement dates. In the multiperiod model, investment, Iit+1 , which correlates positively but not perfectly with earnings shocks, also appears in the numerator of equation (7). As such, only a fraction
of earnings momentum should be realized on earnings announcement dates (Liu and Zhang 2014).
The investment CAPM also predicts that the magnitude of post-earnings-announcement drift
40
is larger in small firms than in big firms. In particular, equations (5) and (6) imply that:
S
rit+1
=
Kit+1
Xit+1 .
Pit
(15)
As such, the strength of the earnings-return relation decreases with the market equity, Pit . The
S ]/∂E [X
same equation also implies that the profitability effect in average returns, ∂Et [rit+1
t
it+1 ],
should be stronger in value firms with high Kit+1 /Pit than in growth firms with low Kit+1 /Pit . In
S ]/∂(K
addition, the value effect in average returns, ∂Et [rit+1
it+1 /Pit ), should be stronger in more
profitable firms than in less profitable firms (Piotroski 2000).
Bernard and Thomas (1990) present the most telling evidence suggesting that stock prices fail
to reflect the extent to which earnings deviate from a naive expectation based on a seasonally adjusted random walk, in which expected earnings are the earnings from four quarters ago. Their key
evidence is the negative correlation between unexpected earnings for quarter t and the abnormal returns around the earnings announcement for quarter t+4, in addition to the positive correlation between unexpected earnings for quarter t and the drift for quarters t+1 to t+3. The evidence is interpreted as stock prices failing to reflect fully the mean reversion of earnings changes in four quarters.
Bernard and Thomas (1990) develop clear testable hypotheses based on underreaction to earnings news. Suppose investors form naive earnings expectations based on the detrended seasonally
adjusted random walk. Let Πit+1 be the detrended earnings for stock i for quarter t + 1. The
market’s expected earnings embedded in the stock price are given by EtM [Πit+1 ] = Πit−3 , in which
EtM [·] is the market’s expectation conditional on time-t information. When Πit+1 is announced, the
market perceives the unexpected earnings as Πit+1 − EtM [Πit+1 ]. Let λi be the earnings response
coefficient. The abnormal return at the earnings announcement date for quarter t + 1 is:
S
S
rit+1
− EtM [rit+1
] = λi (Πit+1 − EtM [Πit+1 ]) = λi (Πit+1 − Πit−3 ).
41
(16)
Bernard and Thomas (1990) assume the Brown-Rozeff (1979) model for the true earnings:
Πit+1 = Πit−3 + φ(Πit − Πit−4 ) + θǫit−3 + ǫit+1 ,
(17)
in which ǫit+1 is the earnings shock at period t + 1, φ > 0, and θ is negative enough that the fourthorder autocorrelation in seasonally differenced earnings is negative. The true expected earnings is:
Et [Πit+1 ] = Πit−3 + φ(Πit − Πit−4 ) + θǫit−3 .
(18)
However, if investors use the naive earnings expectations from the seasonally adjusted random walk:
S
S
rit+1
− EtM [rit+1
] = λi (Πit+1 − EtM [Πit+1 ]) = λi (Πit+1 − Et [Πit+1 ]) + λi (Et [Πit+1 ] − EtM [Πit+1 ])
= λi ǫit+1 + λi φ(Πit − Πit−4 ) + λi θǫit−3 .
(19)
Using the time series earnings model in equation (17) then yields:
S
S
rit+1
− EtM [rit+1
] = λi ǫit+1 + λi φǫit + λi φ2 ǫit−1 + λi φ3 ǫit−2 + λi (θ + φ4 )ǫit−3 + λi ν i ,
(20)
in which ν i is a linear combination of earnings shocks prior to t − 3. As such, if the true earnings
follow the Brown-Rozeff model but investors form naive expectations from a seasonal random walk,
then the abnormal return at the earnings announcement of quarter t + 1 should have positive but
declining correlations with the earnings shocks from quarters t, t − 1, and t − 2, but a negative
correlation with the earnings shock from quarter t − 3. The evidence confirms this prediction.
Bernard and Thomas’s (1990) evidence is the “most damaging” to efficient markets (Kothari
2001, p. I-194). No rational explanation has been offered to date. However, the investment CAPM
seems to make some progress. In particular, multiplying both the numerator and the denominator
of equation (7) with Kit+1 , which is known at the beginning of period t, yields:
S
rit+1
= λit Πit+1 ,
42
(21)
in which λit ≡ 1/[(1 + a(Iit /Kit ))Kit+1 ] = 1/Pit is the earnings response coefficient.
Continue to assume that the true earnings follow the Brown-Rozeff (1979) model. Under rational expectations, equations (18) and (21) imply:
S
Et [rit+1
] = λit φ(Πit − Πit−4 ) + λit θǫit−3 + λit Πit−3
= λit φǫit + λit φ2 ǫit−1 + λit φ3 ǫit−2 + λit (θ + φ4 )ǫit−3 + λit (ν i + Πit−3 ).
(22)
The abnormal return at the earnings announcement for quarter t + 1 is:
S
S
rit+1
− Et [rit+1
] = λit ǫit+1 ,
(23)
which is unpredictable with time-t information.
However, this lack of predictability requires that the expected return in equation (22) is measured precisely. In Bernard and Thomas (1989, 1990), abnormal return is calculated as the sizeadjusted return, and size seems an imperfect control for the expected return. Suppose the last
term in equation (22), λit (ν i + Πit−3 ), is largely expected due to its long lag, but a portion of the
remainder of the expected return is mismeasured as a part of the abnormal return. The equation
then implies that the “abnormal” return at the earnings announcement of quarter t + 1 would
display positive and declining correlations with earnings shocks from prior quarters.
2.5.2.2
Fundamental Analysis In addition to post-earnings-announcement drift, the ROE
factor in Hou, Xue, and Zhang (2015a, b) is also rooted in the fundamental analysis literature in
accounting. Ou and Penman (1989) combine a large set of financial statement items into one composite metric that predicts the direction of one-year-ahead earnings changes. Lev and Thiagarajan
(1993) use conceptual arguments to select 12 fundamental signals to correlate with contemporaneous
stock returns, and to forecast future earnings growth. In contrast, Hou et al. use a single variable
(profitability) as the expected profitability proxy. This parsimony is designed to reduce estimation
errors in the proxy so as to strengthen its predictive power for future profitability and returns.
43
Abarbanell and Bushee (1997, 1998) show that several fundamental signals, including inventory
changes, changes in account receivables, gross margin, changes in selling and administrative expenses, and tax expenses-to-earnings provide information about future returns. This information
is also shown to be associated with one-year-ahead earnings news and analysts’ forecast errors.
Piotroski (2000) applies context-specific fundamental analysis to a broad portfolio of value stocks.
Nine signals are selected to capture a firm’s profitability, leverage, liquidity, source of funds, and
operating efficiency. Piotroski finds that the average return earned by a value investor can be
increased by at least 7.5% per annum through selecting financially strong value stocks.
More recently, the fundamental analysis literature and the investment CAPM literature have
started to show signs of convergence in perspectives. Penman and Zhu (2015) use a model similar
to equation (12) to connect expected returns to expected earnings and expected earnings growth.
Consistent with the equation, Penman and Zhu document that many accounting variables predict
future earnings and earnings growth, in the same direction in which these variables forecast returns.
The evidence is interpreted as “consistent with rational pricing in the sense that the returns are
those one would expect if the market were efficient in its pricing (p. 1836).”
3
The Investment CAPM: Structural Estimation and Tests
3.1
The Multiperiod Model
Building on Cochrane (1991), Liu, Whited, and Zhang (2009) derive and test the multiperiod investment CAPM. Time is discrete and the horizon infinite. Firms use capital and costlessly adjustable
inputs to produce a homogeneous output. These latter inputs are chosen each period to maximize
operating profits, which are defined as revenues minus the expenditures on these inputs. Taking
the operating profits as given, firms choose investment to maximize the market value of equity.
Let Πit = Π(Kit , Xit ) be the operating profits of firm i at time t, in which Kit is capital, and Xit
is a vector of exogenous aggregate and firm-specific shocks. Firm i has a Cobb-Douglas production
function with constant returns to scale, meaning that Π(Kit , Xit ) = Kit ∂Π(Kit , Xit )/∂Kit . The
44
marginal product of capital is parameterized as ∂Π(Kit , Xit )/∂Kit = κYit /Kit , in which κ > 0 is a
constant parameter, and Yit is sales. As such, shocks to operating profits, Xit , are reflected in sales.
Capital accumulates as Kit+1 = Iit + (1 − δ it )Kit , in which Iit is investment, and δ it is an
exogenous proportional rate of depreciation. Firms incur adjustment costs when investing. The
adjustment costs function, Φ(Iit , Kit ), is increasing and convex in Iit , decreasing in Kit , and exhibits
constant returns to scale in Iit and Kit , i.e., Φ(Iit , Kit ) = Iit ∂Φ(Iit , Kit )/∂Iit +Kit ∂Φ(Iit , Kit )/∂Kit .
A standard quadratic functional form is given by Φ(Iit , Kit ) = (a/2) (Iit /Kit )2 Kit , in which a > 0.
Firms can finance investment with one-period debt. At the beginning of time t, firm i can
issue an amount of debt, Bit+1 , which must be repaid at the beginning of period t + 1. The
B , can vary across firms and over time. Taxable corporate
gross corporate bond return on Bit , rit
profits equal operating profits less capital depreciation, adjustment costs, and interest expenses,
B − 1)B , in which adjustment costs are expensed. Let τ
Π(Kit , Xit ) − δ it Kit − Φ(Iit , Kit ) − (rit
it
t
denote the corporate tax rate at time t. The payout of firm i equals
B
B
Dit ≡ (1 − τ t )[Π(Kit , Xit ) − Φ(Iit , Kit )] − Iit + Bit+1 − rit
Bit + τ t δ it Kit + τ t (rit
− 1)Bit ,
(24)
B − 1)B is the interest tax shield.
in which τ t δ it Kit is the depreciation tax shield, and τ t (rit
it
Let Mt+1 be the stochastic discount factor from t to t+1, which is correlated with the aggregate
component of Xit+1 . We can formulate the cum-dividend market value of equity as follows:
Vit
≡
max
{Iit+s ,Kit+s+1,Bit+s+1 }∞
s=0
Et
"
∞
X
s=0
#
Mt+s Dit+s ,
(25)
subject to a transversality condition that prevents firms from borrowing an infinite amount to
distribute to shareholders, limT →∞ Et [Mt+T Bit+T +1 ] = 0.
I
I
The equity value-maximization implies that Et [Mt+1 rit+1
] = 1, in which rit+1
is the investment
45
return (see Appendix B for the detailed derivation):
I
rit+1
≡
Yit+1
+
(1 − τ t+1 ) κ K
it+1
a
2
Iit+1
Kit+1
2 i
h
+ τ t+1 δ it+1 + (1 − δ it+1 ) 1 + (1 − τ t+1 )a KIit+1
it+1
.
Iit
1 + (1 − τ t )a K
it
(26)
Ba ≡ r B
B
Ba
Let the after-tax corporate bond return be rit+1
it+1 − (rit+1 − 1)τ t+1 , then Et [Mt+1 rit+1 ] = 1.
S
Let Pit ≡ Vit − Dit be the ex-dividend equity value, rit+1
≡ (Pit+1 + Dit+1 )/Pit be the stock return,
and wit ≡ Bit+1 /(Pit + Bit+1 ) be the market leverage. Then the investment return is the weighted
average of the stock return and the after-tax corporate bond return:
I
Ba
S
= wit rit+1
+ (1 − wit ) rit+1
.
rit+1
(27)
The investment return in equation (26) is the ratio of the marginal benefits of investment at
time t + 1 to the marginal costs of investment at t. In its numerator, (1 − τ t+1 )κYit+1 /Kit+1 is the
marginal after-tax profits produced by an additional unit of capital, (1 − τ t+1 )(a/2)(Iit+1 /Kit+1 )2
is the marginal after-tax reduction in adjustment costs, τ t+1 δit+1 is the marginal depreciation tax
shield, and the last term in the numerator is the marginal continuation value of the extra unit of
capital net of depreciation. Also, the first term in brackets in the numerator divided by the denominator is analogous to a dividend yield. The second term in brackets in the numerator divided
by the denominator is analogous to a capital gain because this ratio is the growth of marginal q.
Equation (27) is exactly the weighted average cost of capital in corporate finance. Solving for
S
the stock return rit+1
from equation (27) gives the multiperiod investment CAPM :
S
Iw
rit+1
= rit+1
≡
I
Ba
rit+1
− wit rit+1
,
1 − wit
(28)
Iw is the levered investment return. Together, equations (26) and (27) imply that the
in which rit+1
weighted average cost of capital equals the ratio of the next-period marginal benefits of investment
divided by the current-period marginal costs of investment. This first principle of investment pro-
46
vides the microfoundation for the weighted average cost of capital approach to capital budgeting in
corporate finance. Intuitively, firms will keep investing until the costs of doing so, which rise with
investment, equal the present value of additional investment, which is the next-period marginal
benefits of investment discounted by the weighted average cost of capital.
3.1.1
Expected Stock Returns
The multiperiod investment CAPM implies an ex-ante restriction that the expected stock return
equals the expected levered investment return across all testing assets:
S
Iw
E rit+1
− rit+1
= 0.
(29)
The model error from the moment condition defines the investment CAPM alpha:
S
Iw
− rit+1
αiq ≡ ET rit+1
,
(30)
in which ET [·] is the sample mean of the series in brackets.
3.1.1.1
Liu, Whited, and Zhang (2009)
Liu et al. estimate the parameters a and κ via one-
stage generalized method of moments (GMM) at the portfolio level. Grouping stocks into portfolios
enlarges the average return spread across extreme portfolios, and makes it more significant by diversifying away idiosyncratic volatilities at the stock level. Accordingly, the power of asset pricing tests
increases (Black, Jensen, and Scholes 1972). The portfolio approach also has the advantage that
portfolio investment data are smooth, whereas the investment data at the firm level can be lumpy.
Iw , against average
Figure 2 shows that the scatter points of average predicted stock returns, rit+1
S , are largely aligned with the 45-degree line across the earnings surprises
realized stock returns, rit+1
(Panel A) and book-to-market deciles (Panel B). The alpha of the high-minus-low earnings surprises decile is only −0.4% per annum, and the alpha of the high-minus-low book-to-market decile
1.2%, both of which are within one standard error from zero. The mean absolute alpha is 0.7%
47
Figure 2 : Average Predicted Stock Returns versus Average Realized Stock Returns, the
Multiperiod Investment CAPM
Source: Liu, Whited and Zhang (2009). The figure plots average returns predicted from the multiperiod
investment CAPM against average realized returns across the deciles formed on earnings surprises (Panel
A) and book-to-market (Panel B). “High” is the high decile, and “Low” the low decile.
Panel A: Earnings momentum
Panel B: Book-to-market
0.3
0.3
Average predicted returns
Average predicted returns
High
0.25
0.2
0.15
0.1
0.25
High
0.2
Low
0.15
0.1
Low
0.05
0.05
0.1
0.15
0.2
0.25
Average realized returns
0.05
0.05
0.3
0.1
0.15
0.2
0.25
Average realized returns
0.3
across the earnings surprises deciles, and 2.3% across the book-to-market deciles.
However, the point estimates in Liu, Whited, and Zhang (2009) reveal a weakness. The adjustment costs parameter, a, is estimated to be 7.7 across the earnings surprises deciles, but 22.3
across the book-to-market deciles. The capital’s share parameter, κ, is 0.3 across the former, but
0.5 across the latter deciles. Ideally, if a model is well specified, the estimates of these technological
parameters should be invariant across different testing assets. Clearly, Liu et al.’s first stab at
implementing the multiperiod investment CAPM via structural estimation is far from perfect.
3.1.1.2
Liu and Zhang (2014)
Liu and Zhang apply the structural investment CAPM to
understand price momentum. Momentum deciles are rebalanced monthly, but accounting variables
in Compustat are annual. To handle this measurement difficulty, Liu and Zhang design a more
polished timing alignment procedure than Liu, Whited, and Zhang (2009). In particular, monthly
levered investment returns of a momentum decile are constructed from its annual accounting variables to match with the decile’s monthly stock returns (see Appendix C for the detailed procedure).
48
Figure 3 : Average Predicted Stock Returns versus Average Realized Stock Returns, the
Multiperiod Investment CAPM, Price Momentum
Source: Liu and Zhang (2014). Panel A is for the price momentum (prior six-month returns) deciles. “W”
is the winner decile, and “L” the loser decile. Panel B is for nine (three-by-three) portfolios formed on
book-to-market and price momentum. “L,” “M,” and “W” denote the terciles in the ascending order on
price momentum, and “1,” “2,” and “3” the terciles in the ascending order on book-to-market.
Panel B: Two-way portfolios
Panel A: Deciles
0.25
W
0.2
Average predicted returns
Average predicted returns
0.25
9
8
0.15
7
6
0.1
3
45
2
0.05
0
0
L
0.05
0.1
0.15
0.2
Average realized returns
1W
0.15
2W
1M
2M
0.1
3M
1L
2L
3L
0.05
0
0
0.25
3W
0.2
0.05
0.1
0.15
0.2
Average realized returns
0.25
Panel A of Figure 3 shows that the investment CAPM performs well across the price momentum
deciles. The scatter points of average predicted returns versus average realized returns are again
largely aligned with the 45-degree line. The winner-minus-loser decile has a small alpha of 0.4%
per annum, which is only about 2.65% of its average return of 15.09% (equal-weighted). Also, the
mean absolute alpha is 0.83%, which is small relative to the average decile return of 12.4%.
However, the point estimates from fitting the price momentum deciles are 2.52 for the adjustment costs parameter and 0.12 for the capital’s share, both of which are substantially lower than
those from fitting the book-to-market deciles reported in Liu, Whited, and Zhang (2009). Panel B of
Figure 3 crystalizes the tension between explaining value and momentum simultaneously within the
baseline model. For price momentum, the winner-minus-loser alphas across the low, median, and
high book-to-market terciles are 3.46%, −0.70%, and −6.80% per annum, respectively. More important, the high-minus-low book-to-market alphas across the low, median, and high price momentum
terciles are 11%, 10.07%, and 0.73%, respectively, which vary inversely with price momentum. As
49
Figure 4 : Event-time Evolution of Average Price Momentum Profits
Source: Liu and Zhang (2014). For 36 months after the portfolio formation, this figure plots the eventtime evolution of average stock returns (Panel A), average levered investment returns (Panel B), average
investment-to-capital growth (Panel C), and average sales-to-capital (Panel D) for the winner decile (blue
solid lines) and the loser decile (red broken lines). The average returns are in annual percent.
10
5
5
10
15
20
25
30
35
20
15
10
5
0
5
10
15
20
25
30
1.2
4
1.1
1
0.9
3.5
3
0.8
0.7
35
Sales−to−capital
15
4.5
The investment−to−capital growth
20
The stock return
Panel D: Sales-to-capital
25
The levered investment return
25
0
Panel C:
Investment-to-capital
growth
Panel B: Levered
investment returns
Panel A: Stock returns
5
10
15
20
25
30
35
2.5
5
10
15
20
25
30
35
such, the baseline investment CAPM fails to explain value and momentum simultaneously.
Armed with the point estimates from matching only average momentum profits, Liu and Zhang
(2014) also examine the reversal of momentum in long horizons as separate diagnostics on the investment CAPM’s performance. Figure 4 reports the event-time evolution during 36 months after
the portfolio formation for average stock returns as well as average levered investment returns and
their key components for the winner and loser deciles. Panel A shows that average winner-minusloser returns in the data start at 19.98% per annum in the first month in the holding period, fall
to 13.15% in month six, converge largely to zero in month ten, and turn negative afterward.
The investment CAPM goes a long way toward explaining this reversal. From Panel B, levered investment returns for the winner-minus-loser decile start at 18.21% per annum in the first
month, fall to 10.73% in month six, and further to 2.87% in month twelve. The predicted price
momentum profits converge largely to zero in month fifteen, and turn negative afterward. Panel C
shows that the expected growth plays a key role in capturing the reversal of price momentum. The
investment-to-capital growth spread starts at 39.45% in month one, weakens to 23.06% in month
50
six, converges to zero in month thirteen, and turns negative afterward. In contrast, Panel D shows
that the sales-to-capital spread is much more persistent than momentum profits.
3.1.2
Equity Valuation
Equity valuation is immensely important in theory and practice. In academia, a vast accounting
literature has built on the dividend discounting and residual income models for equity valuation
(Ohlson 1995). Widely practiced in the financial services industry, valuation is at the core of standard business school curricula with many textbook treatments (Penman 2013; Koller, Goedhart,
and Wessles 2015). Traditional valuation methods calculate the present value of future dividends,
working from the perspective of investors’ demand of risky equities.
Belo, Xue, and Zhang (2013) approach the valuation question from the perspective of managers’
supply of assets. The idea is that managers, if behaving optimally, will adjust the supply of assets to
their changes in the market value. Managers will invest until the marginal benefits of one extra unit
of assets (marginal q, which is the present value of all the future dividends generated by this extra
unit) equal the marginal costs of supplying this extra unit. With a specified capital adjustment
technology, the marginal costs of investment (marginal q) can be inferred from investment. With
constant returns to scale, the inferred marginal q provides the value for a firm’s entire capital assets.
Working within the multiperiod framework outlined in Section 3.1, Belo, Xue, and Zhang (2013)
allow the marginal adjustment costs of investment to be nonlinear:
1
Φ(Iit , Kit ) =
ν
Iit
η
Kit
ν
Kit ,
(31)
in which η > 0 is the slope, and ν > 0 the curvature parameter. The first principle of investment
implies that the market value of the firm is given by (Appendix B):
"
Pit + Bit+1 = 1 + (1 − τ t )η ν
51
Iit
Kit
ν−1 #
Kit+1 .
(32)
In addition, the investment Euler equation is given by:
1+(1−τ t )η ν
Iit
Kit
ν−1

i
h
Iit+1 ν
Yit+1
ν−1
+
η
+
δ
τ
(1 − τ t+1 ) κ K
it+1 t+1
ν
Kit+1
it+1



ν−1 = Et Mt+1 
 . (33)
Iit+1
ν
+(1 − δ it+1 ) 1 + (1 − τ t+1 )η Kit+1


Belo, Xue, and Zhang (2013) measure Tobin’s q in the data as qit = (Pit + Bit+1 )/Ait , in which
Ait is total assets, and test whether the average q in the data equals that implied in the model:
"
"
E qit − 1 + (1 − τ t )η
ν
Iit
Kit
ν−1 #
Kit+1
Ait
#
= 0.
(34)
However, this equation allows assets to be potentially misvalued, but forces managers to align investment with misvalued q. To alleviate this concern, Belo et al. estimate the valuation moment
jointly with a (scaled) investment Euler equation moment, specified as:





E








ν−1
−
1 + (1 − τ t )η ν KIitit
h
ν i
Yit+1
Iit+1
(1 − τ t+1 ) κ K
+ ν−1
ηK
+ δ it+1 τ t+1
ν
it+1
it+1
ν−1 Iit+1
ν
+(1 − δit+1 ) 1 + (1 − τ t+1 )η Kit+1
Ba +(1−w )r S
wit rit+1
it it+1











 Kit+1 


 Ait  = 0,




(35)
in which Mt+1 is specified as the inverse of the weighted average cost of capital.
Figure 5 shows that the scatter points of average predicted q versus average realized q across
the Tobin’s q deciles are largely aligned with the 45-degree line. The high-minus-low error from
the valuation moment is 0.27, which is only 6% of the high-minus-low spread in Tobin’s q, 4.5. The
high-minus-low Euler equation error is slightly larger at 0.32. The mean absolute valuation error
is 0.08, which is only 5.13% of the average q across the deciles, 1.56. The point estimates are sensible, 4.1 for the slope parameter, η, and 4.09 for the curvature parameter, ν, which is also reliably
different from two, rejecting the quadratic adjustment costs function. These estimates imply an
average adjustment costs-to-sales ratio of 3.95%, which is relatively low.
52
Figure 5 : Average Predicted q versus Average Realized q, Average Marginal Costs of
Investment versus Average Marginal Benefits of Investment, The Tobin’s q Deciles, Joint
Estimation of the Valuation Moment and the Investment Euler Equation Moment
Source: Belo, Xue, and Zhang (2013). The results are from fitting the valuation moment (34) and the
investment Euler equation moment (35) jointly via one-stage GMM. The deciles are in the ascending order.
Panel B: Investment Euler equation
Panel A: Valuation
10
Predicted
4
3
9
2
1
0
0
3.2
3.2.1
Marginal benefits of investment
5
56
34
2
1
1
7
8
2
3
Realized
4
5
4
3
9
2
7
1
0
0
5
10
8
6
45
123
1
2
3
4
Marginal costs of investment
5
Notes
The Neoclassical q-Theory of Investment
The neoclassical q-theory of investment is originally developed to explain investment behavior. This
literature is vast (see Chirinko 1993 and Caballero 1999 for surveys on aggregate investment, and
Bond and Van Reenen 2007 on microeconomic investment). Introduced by Brainard and Tobin
(1968) and Tobin (1969), the basic idea is to relate investment to average q, which is the ratio of
the market value of a firm to the replacement costs of its capital stock. Jorgenson (1963), Lucas and
Prescott (1971), Mussa (1977), and Abel (1983) provide the neoclassical formulation of q-theory,
and derive the relation between investment and marginal q. Hayashi (1982) shows that under constant returns to scale, marginal q equals average q. This proposition allows empirical studies to
replace the unobservable marginal q with the observable average q. Abel and Eberly (1994) extend
q-theory to incorporate costly reversibility and flow fixed costs of investment.
Empirically, however, q-theory’s performance has been disappointing in describing investment
53
behavior. In investment-q regressions, for instance, the slope coefficients on cash flow are typically
large and significant for firms that are financially constrained, even after controlling for q (Fazzari,
Hubbard, and Petersen 1988). In investment Euler equation tests, the investment model is rejected
in subsamples that consist of financially constrained firms (Whited 1992). The q-theory performance
deteriorates further at the micro level across firms or plants, in which fixed costs of investment and
other nonconvexities play an important role (Doms and Dunne 1998; Cooper and Haltiwanger 2006).
In contrast, q-theory is more successful in cross-sectional asset pricing. The key difference is that
asset pricing tests are conducted at the portfolio level. As noted, forming portfolios makes average
return spreads more reliable, increasing the test power. Also, because of aggregation, investment
data at the portfolio level are smooth relative to firm-level or plant-level investment data. Thomas
(2002) and Veracierto (2002), for instance, show that aggregation largely eliminates the impact of
lumpy investment on aggregate dynamics in equilibrium business cycle models. Cooper and Haltiwanger (2006) also show that the quadratic model captures most of the time series variation in aggregate investment simulated from a nonconvex adjustment costs model estimated on plant-level data.
Finally, financial constraints are not a strong predictor of cross-sectional returns in the data (Lamont, Polk, and Saá-Requejo 2001). Conceptually, financial constraints only affect expected returns
via the growth of their Lagrangian multiplier (Gomes, Yaron, and Zhang 2006). As such, financial
constraints have been largely downplayed in investment-based asset pricing tests. It remains to be
seen, however, whether financial constraints affect equity valuation, especially at the firm level.
3.2.2
3.2.2.1
Investment-Based Asset Pricing Tests
Aggregate Stock Market
Cochrane (1991) is the first to apply q-theory to study asset
prices. Cochrane derives the investment return in equation (26) in a simplified setting without debt
or taxes, and argues that the investment return should equal the return to owning capital, which is in
turn the stock return. Restoy and Rockinger (1994) prove that the equality between the investment
and stock returns holds under more general conditions, and that the equality is in effect a mathe-
54
matical restatement of that between marginal q and average q under the Hayashi (1982) conditions.
Empirically, Cochrane shows that with reasonable parameters, the mean of the aggregate investment
return matches the mean of the aggregate stock market return, meaning that the equity premium
is not a puzzle in q-theory. However, the investment return volatility is only about one half of the
stock market volatility. The correlation between investment and stock returns is as high as 40%.
Finally, investment-to-capital predicts the stock market return with a significantly negative slope.
Lettau and Ludvigson (2002) emphasize that time-varying aggregate risk premiums have implications not only for aggregate investment today, but also for future investment over long horizons.
Predictive variables for market excess returns over long horizons should also forecast long horizon
fluctuations in the growth of marginal q and investment. Evidence from long horizon regressions
of aggregate investment growth on a variety of risk premium predictors confirms this prediction.
Merz and Yashiv (2007) introduce labor into the investment model, and fit the resulting valuation equation on the aggregate U.S. data. A convex adjustment costs function with nonlinear
marginal costs is able to account for the valuation data, including the mean and volatility of the
market equity-to-output ratio. The valuation ratio predicted from the estimated model also traces
closely the valuation ratio in the data, including the stock price runup in the late 1990s. Merz
and Yashiv attribute the model’s good performance to the use of gross flows data for both investment and hiring, the joint estimation of hiring and investment including their interaction, and the
nonlinear marginal costs in the adjustment costs function.
Jermann (2010) examines the determinants of the equity premium implied by firms’ first-order
conditions. Assuming two states of the world and two capital inputs (structures and equipment),
I
Jermann rewrites Et [Mt+1 rit+1
] = 1 as a system of two equations consisting of the structures and
equipment investment returns. The system can then be solved for the two state prices, from which
the interest rate can be calculated. With reasonable parameter values, the model fits the first and
second moments of the market return and interest rate in the data, and connects these moments
55
with technologies. However, while this method derives state prices from the production side, it
seems empirically infeasible to extend it to more than two states of the world.
Working within the search model of equilibrium unemployment, Chen and Zhang (2011) argue
that with time-to-build, high aggregate risk premiums should forecast low employment growth in
the short run, but high employment growth in the long run. If there is also time-to-plan, high risk
premiums should forecast low net hiring rates in the short run, but high net hiring rates in the
long run. Evidence from predictive regressions suggests two-quarter time-to-build in the aggregate
payroll data, no time-to-plan in the aggregate hiring data, but two-quarter time-to-plan in the job
creation data for manufacturing firms. Finally, high payroll growth and high net job creation rate
in manufacturing also forecast low stock market excess returns at business cycle frequencies.
3.2.2.2
Cross-sectional Tests Cochrane (1996) conducts cross-sectional asset pricing tests
S ] = 1, in which r S
on the consumption CAPM, Et [Mt+1 rit+1
it+1 is the stock returns across the size
deciles. The stochastic discount factor, Mt+1 , is specified as a linear function of two investment
returns, which are constructed per equation (26) on the gross private domestic nonresidential and
residential investment data. Cochrane also lays out an important econometric methodology for
estimating and testing dynamic asset pricing models via GMM.
Gomes, Yaron, and Zhang (2006) study the impact of financial constraints, which are modeled as
a dividend nonnegativity constraint, on the cross section of returns. Building on Cochrane (1996),
Gomes et al. specify the stochastic discount factor as a linear function of the aggregate investment
return subject to financial constraints. With the 25 size and book-to-market portfolios as testing
assets, Gomes et al. show that financial constraints provide a common factor, but that their impact
seems to be more important in good times, when investment expenditures are exceedingly high.
Instead of Cochrane’s (1996) test framework, Whited and Wu (2006) estimate the investment
Euler equation augmented with financial constraints at the firm level via GMM. The shadow price
of external funds is specified as a linear function of several firm characteristics, as in Whited (1992).
56
Using the estimated shadow price as a financial constraints index, Whited and Wu show that there
exists a financial constraints factor, but that its average return is insignificant, albeit positive.
Balvers and Huang (2007) formulate a productivity-based asset pricing model, in which the
Solow residual is the only priced factor, and capital is the only conditioning variable. Their crosssectional tests show that value and growth stocks have similar average productivity risk. However,
growth stocks have their highest productivity risk, when the productivity risk premium is low or
even negative in good times, giving rise to lower average returns than value stocks. Intuitively,
the market value of growth firms varies more drastically with productivity shocks in booms, when
capital is abundant, as positive shocks are propagated by low investment costs.
Cochrane (1993) emphasizes that standard production technologies restrict firms from adjusting
their output across states of nature. Cochrane formulates a flexible state-contingent production
technology to overcome this limitation, and derives a stochastic discount factor as firms’ marginal
rate of transformation. Belo (2010) estimates this model by identifying the marginal rate of transformation from price and output data in the durable and nondurable goods sectors. Belo’s estimates
suggest that firms’ ability to transform output across states of nature seems high.
3.2.2.3
New Applications
Vitorino (2014) applies the investment CAPM to quantify the
impact of advertising expenditures on expected stock returns and equity valuation. Interpreting
advertising expenses as investment in brand capital, Vitorino extends the expected return test in
equation (29) and the valuation test in equation (34) to a setting with two capital inputs, physical
capital and brand capital. The model matches well the average stock returns and Tobin’s q across
portfolios formed on advertising expenses. Also, brand capital accounts for a substantial fraction,
about 23%, of firms’ market equity, and this fraction varies greatly across industries.
Cooper and Priestley (2015) apply the investment CAPM to study the expected returns and
valuation of private firms, which form a substantial part of the U.S. economy. Whereas estimating
costs of capital from the consumption CAPM requires historical stock returns data that are not
57
feasible for private firms, the investment CAPM estimates costs of capital directly from economic
fundamentals. As such, the investment CAPM uniquely befits private firms. Cooper and Priestley
examine the cross section of ten industry portfolios formed on the fraction of the sales of public
firms in a given industry to its total industry sales. The bottom two deciles, which consist of only
private firms, are identified as private industries, and the top decile as public industries.
Cooper and Priestley (2015) report three key findings. First, characteristics such as investment
and profitability play a similar role in driving the cross-sectional variation of investment returns
across both private and public firms. This evidence lends support to the investment CAPM, but
seems inconsistent with mispricing as an explanation for the role of these characteristics. The
key identification is that private firms do not have stock prices to overreact or underreact to, and
should be less affected by mispricing, if at all. Second, a four-factor model, resembling the q-factor
model, constructed on both private and public firms, performs well in describing the cross section
of investment returns, and provides the cost-of-capital estimates for private firms. Finally, private
firms have expected returns and valuation ratios that are not far from those of public firms.
4
Quantitative Investment Theories
This applied theoretical literature is pioneered by Berk, Green, and Naik (1999), who construct the
first fully specified theoretical model for the cross section of expected returns. Their key insight is
that as a consequence of optimal investment decisions, a firm’s assets in place and growth options
change in ways that give rise to cross-sectional predictability. Their influential work has inspired
a generation of (then) young theorists to enter the field of cross-sectional asset pricing.
This literature has developed along two parallel lines, real options and neoclassical investment
models. The difference is mostly technical in nature, not substantive. Projects are discrete in
real options theories, but infinitesimally small and differentiable in neoclassical investment models.
Section 4.1 reviews real options, and Section 4.2 neoclassical investment models. Section 4.3 traces
several subsequent controversies. Finally, Section 4.4 summarizes the more recent development.
58
4.1
Real Options
Berk (1995) provides a theoretical interpretation for Banz’s (1981) size effect. Consider a oneperiod economy, in which all firms have exactly the same expected end-of-period cash flows. If
firms’ cash flows have different risks, firms must differ in their market values. In particular, riskier
firms must have lower market values, and consequently, must have higher expected returns. As
such, the market value of equity predicts returns with a negative slope (see also Ball 1978).
Berk, Green, and Naik (1999) construct the first real options model for the cross section. In
their model, firms own assets in place, which currently generate cash flows, and growth options,
which allow firms to take positive net present value projects in the future. During each period, some
existing assets die off, and new projects become available to firms. A low risk project is attractive,
all else equal, and the firm that invests in it would experience a large increase in its market value.
However, the firm’s risk falls as a result of taking the low risk project, giving rise to lower expected
returns. Similarly, when a firm loses low risk assets, its market value drops, but its risk rises. Bookto-market serves as a state variable that summarizes a firm’s risk relative to its assets base, and
size describes the relative importance of assets in place and growth options. In addition, because
the composition and risk of firms’ assets are persistent in the model, expected returns are positively
correlated with lagged expected returns, giving rise to momentum profits. However, momentum in
the model is substantially more persistent or long-lived than that in the data.
Gomes, Kogan, and Zhang (2003) embed a real options model into a dynamic stochastic general
equilibrium economy. To facilitate aggregation across heterogeneous firms, Gomes et al. assume
that new potential projects are assigned randomly across firms, independent of their productivity
levels. In their model, growth options are riskier than assets in place, because growth options are
levered relative to assets in place. This leverage effect is analogous to the way that a call option
can be replicated by a levered position in the underlying asset and a short position in cash. The
size effect arises in their model, because small firms, relative to their market value, are assigned a
59
disproportionately higher fraction of risky growth options than big firms.
The value premium arises in the model because book-to-market is inversely related to the “effective duration” risk of assets in place. Value firms have less productive assets, implying higher cash
flow durations, than growth firms. A positive duration-risk relation then implies that value firms
are riskier than growth firms. However, value firms have lower cash flow durations than growth
firms in the data (Dechow, Sloan, and Soliman 2004). The counterfactual prediction in Gomes,
Kogan, and Zhang (2003) is driven by their assumption that firms have equal growth options.
Growth firms cannot invest more than value firms, and must distribute more cash flows from their
more productive assets in place as dividends in the short run.
Carlson, Fisher, and Giammarino (2004) introduce operating leverage, reversible investment,
fixed adjustment costs, and finite growth opportunities into a real options model. In their model,
growth options are also riskier than assets in place, but operating leverage offsets this effect to give
rise to the value premium. Intuitively, when the demand for a firm’s product decreases, equity value
falls relative to its capital stock, for which the book value is a proxy. Suppose that fixed operating
costs are proportional to capital, risk and expected returns rise due to higher operating leverage.
Working within a continuous time real options model, Cooper (2006) shows that the value premium arises because the excess installed capital of value firms allows them to fully benefit from
positive aggregate shocks without undertaking costly investment. In addition, negative profitability
shocks reduce the value of firms’ assets in place and expansion options, but increase the value of the
option to disinvest. This effect is more pronounced for value firms with excess capital capacity, for
which the option to disinvest is more likely to be exercised. This option provides insurance to value
firms. However, when investment is largely irreversible, the option to disinvest is less valuable.
4.2
Neoclassical Investment Models
Zhang (2005b) constructs the first quantitative investment model for the cross section of expected
returns in the neoclassical tradition of Kydland and Prescott (1982) and Long and Plosser (1983).
60
Zhang’s model uses two important features, asymmetric adjustment costs and countercyclical price
of risk, to explain the value premium. Asymmetry, meaning that firms face higher costs in cutting
than in expanding capital, gives rise to the cyclical behavior of value and growth betas.
Intuitively, in bad times, value firms are burdened with more unproductive capital, finding it
more difficult to cut their capital than growth firms. Consequently, the cash flows and returns of
value firms covary more with economic downturns. In good times, growth firms invest more, and
face higher adjustment costs to exploit favorable economic conditions. Expanding capital is less
urgent for value firms, because their previously unproductive capital has become more productive.
Because expanding capital is relatively easy, the cash flows and returns of growth firms do not
covary much more with booms. The net effect is a high risk spread between value and growth in
bad times, and a low or even negative risk spread in good times. When the price of risk is also
countercyclical, discount rates are higher in bad times, and firms’ expected continuation values are
lower than those when the price of risk is constant. Consequently, value firms want to disinvest
even more than growth firms in bad times. In all, the time-varying price of risk interacts with, and
propagates the impact of asymmetry, giving rise to a higher value premium.
The Zhang (2005b) model has been a workhorse in this theoretical literature, with many extensions (Section 4.4). A simplified model is outlined as follows. The production function is given by:
Yit = Xt Zit Kitα ,
(36)
in which 0 < α < 1 means decreasing returns to scale, Yit is output for firm i at time t, Kit is
capital, Xt is the aggregate shock, and Zit the firm-specific shock.
The log aggregate shock, xt ≡ log Xt , follows:
xt+1 = x̄(1 − ρx ) + ρx xt + σ x ǫxt+1 ,
(37)
in which x̄ is the long-run mean, ρx persistence, σ x conditional volatility, and ǫxt+1 an independently
61
and identically distributed standard normal shock. The log firm-specific shock, zit ≡ log Zit , follows:
zit+1 = ρz zit + σ z ǫzit+1 ,
(38)
in which ǫzit+1 is an independently and identically distributed standard normal shock. In addition,
ǫzit+1 and ǫzjt+1 are uncorrelated for any i 6= j, and ǫxt+1 is uncorrelated with ǫzit+1 for all i.
The stochastic discount factor, Mt+1 , is specified exogenously as:
log Mt+1 = log β + γ t (xt − xt+1 ),
(39)
γ t = γ 0 + γ 1 (xt − x̄)
(40)
in which β, γ 0 > 0, and γ 1 < 0 are constant parameters.
Let Vit ≡ V (Kit , Zit , Xt ) be the market value of firm i. The firm’s maximization problem is:
V (Kit , Zit , Xt ) = max
{Iit }
"
at
Xt Zit Kitα − f − Iit −
2
Iit
Kit
2
#
Kit + Et [Mt+1 V (Kit+1 , Zit+1 , Xt+1 )] ,
(41)
subject to the capital accumulation rule:
Kit+1 = Iit + (1 − δ)Kit ,
(42)
in which Iit is investment, and δ is the constant rate of depreciation. The first four terms in equation
(41) form current dividends, Dit , and f is fixed costs of production. Also, to capture asymmetric
adjustment costs, at depends on the sign of investment:
at = a+ 1{Iit ≥0} + a− (1 − 1{Iit ≥0} ),
(43)
in which 1{Iit ≥0} is the indicator function that equals one if Iit ≥ 0, and zero otherwise. Finally,
a− > a+ > 0, meaning that firms face higher adjustment costs when cutting than expanding capital.
62
The risk and expected return of firm i satisfy:
Et [Rit+1 ] = Rf t + β it λM t ,
(44)
in which Rf t = 1/Et [Mt+1 ] is the interest rate, Rit+1 ≡ Vit+1 /(Vit − Dit ) is the stock return,
β it ≡ −Covt [Rit+1 , Mt+1 ]/Vart [Mt+1 ] is risk, and λM t ≡ Vart [Mt+1 ]/Et [Mt+1 ] is the price of risk.
The Zhang (2005b) model is quantitatively consistent with many stylized facts related to the
value premium: (i) Value stocks earn higher average returns than growth stocks; (ii) value stocks
are riskier than growth stocks, especially in bad times when the price of risk is high; (iii) value stocks
have persistently lower profitability than growth stocks; (iv) value stocks invest less than growth
stocks; (v) the spreads in book-to-market and in earnings growth between value and growth predict
the value-minus-growth returns with positive slopes; (vi) the expected value premium is countercyclical; and (vii) the aggregate book-to-market and the cross-sectional dispersion of individual
stock returns predict aggregate costs of capital in the time series with positive slopes. However, the
unconditional market betas of value stocks are significantly higher than those of growth stocks in the
model. As a result, the CAPM alpha of the value-minus-growth decile is economically small and statistically insignificant (Lin and Zhang 2013), contradicting that in the data. Explaining this failure
of the CAPM has become an important question in the recent theoretical literature (Section 4.3.3).
4.3
Controversies
The subsequent debate has centered on three issues: (i) Is value riskier than growth? (ii) does the
conditional CAPM explain the value premium? and (iii) what explains the failure of the CAPM?
4.3.1
Is Value Riskier Than Growth?
The key prediction in Zhang (2005b) that value stocks are riskier than growth stocks, especially in
bad times when the price of risk is high, lends theoretical support to Lettau and Ludvigson (2001).
Using the log consumption-wealth ratio as a conditioning variable, Lettau and Ludvigson show that
the conditional consumption CAPM can account for the value premium. In particular, their figure
63
2 documents that value stocks have higher consumption betas than growth stocks in bad times.
In related work, Lustig and Van Nieuwerburgh (2005) show that in a model with collateralized
borrowing and lending, when the ratio of housing wealth to human wealth is low, the dispersion of
consumption growth across households is more sensitive to aggregate consumption shocks, raising
the market price of risk. A scaled version of the consumption CAPM, with the housing-to-human
wealth ratio as the conditioning variable, helps account for the value premium. Empirically, the
returns of the value stocks are more closely correlated with aggregate consumption shocks than the
returns of the growth stocks, when the housing-to-human wealth ratio is low.
However, the prediction that value is riskier than growth in bad times contradicts the earlier evidence in Lakonishok, Shleifer, and Vishny (1994). Lakonishok et al. argue that value stocks would be
fundamentally riskier than growth stocks, if value stocks underperform growth stocks in some states
of the world, and these states are on average bad states, in which investors’ marginal utility of wealth
is high, making value stocks unattractive. Lakonishok et al. compare the performance of value and
growth stocks across four states of the world, including the 25 worst months of equal-weighted stock
market returns, the remaining 88 negative index return months other than the 25 worst, the 122
positive index return months other than the 25 best, and the 25 best months. Value stocks outperform growth stocks in the worst 25 months and in the next worst 88 months in which the market
index has declined, suggesting that value stocks are not fundamentally riskier than growth stocks.
Petkova and Zhang (2005) refute the conclusion in Lakonishok, Shleifer, and Vishny (1994).
The crux is that realized market returns are a noisy measure of aggregate economic conditions.
More accurate measures are the default premium, the term spread, the short-term interest rate,
and the dividend yield, variables that are common instruments for modeling expected market risk
premiums. More important, because ex post market returns and ex ante expected risk premiums
are positively correlated, albeit weakly, what Lakonishok et al. classify as good states ex post tend
to be bad states ex ante, and vice versa, in terms of aggregate economic conditions. Classify-
64
ing states of the world instead on estimated expected market risk premiums, Petkova and Zhang
show that the value-minus-growth betas are economically large and significantly positive in bad
times (months with the 10% highest expected market risk premiums), and economically large and
significantly negative in good times (months with the 10% lowest expected market risk premiums).
4.3.2
Does the Conditional CAPM Explain the Value Premium?
Early quantitative theories mostly feature dynamic single-factor models, in which the conditional
CAPM roughly holds. Gomes, Kogan, and Zhang (2003), for instance, suggest that “the empirical
success of size and book-to-market can be consistent with a single-factor conditional CAPM model
(p. 693).” Relatedly, Lettau and Ludvigson (2001) as well as Lustig and Van Nieuwerburgh (2005)
report some empirical success in the context of the conditional consumption CAPM.
Lewellen and Nagel (2006) argue that the covariations in betas and the equity premium are too
small to explain momentum and the value premium. Conducting a new conditional CAPM test
based on direct estimates of conditional alphas and betas from short-window regressions, Lewellen
and Nagel conclude that the conditional CAPM performs as poorly as the unconditional CAPM.
Panel A of Table 9 replicates the Lewellen-Nagel (2006) short-window regressions in an extended
sample from July 1963 to December 2014. Across all test specifications, the conditional CAPM fails
to explain the value premium. In particular, when conditional alphas and betas are estimated each
quarter with daily returns, the average conditional alpha for the high-minus-low book-to-market
decile is 0.53% per month (t = 2.76). Estimating alphas and betas semiannually with daily and
weekly returns, as well as annually with monthly returns, delivers largely similar results. Also, the
average conditional betas for the high-minus-low decile are all close to zero.
However, Panel B shows that Lewellen and Nagel’s (2006) results disappear in the long sample
from July 1926 to December 2014. Across all short-window specifications, the conditional CAPM
succeeds in explaining the value premium. In particular, when conditional alphas and betas are
estimated each quarter with daily returns, the average conditional alpha for the high-minus-low
65
Table 9 : Average Conditional Alphas and Betas for the Book-to-market Deciles from
Short-window Regressions
Alphas and betas are estimated quarterly with daily returns (“Quarterly”), semiannually with daily and
weekly returns (“Semiannual 1” and “Semiannual 2,” respectively), and annually with monthly returns
(“Annual”). Average conditional alphas are in percent per month. Significant average conditional betas for
the high-minus-low decile are indicated by the superscript ⋆ . Data for the market factor, MKT, the shortterm Treasury bill rates, Rf , and the value-weighted book-to-market decile returns, Ri , are from Kenneth
French’s Web site. In “Annual,” Rit − Rf t = αi + β i0 MKTt + ǫit , the monthly beta is β i = β i0 . In weekly
regressions, Rit −Rf t = αi +β i0 MKTt +β i1 MKTt−1 +β i2 MKTt−2 +ǫit , the weekly beta is β i = β i0 +β i1 +β i2 .
In daily regressions, Rit − Rf t = αi + β i0 MKTt + β i1 MKTt−1 + β i2 [(MKTt−2 + MKTt−3 + MKTt−4 )/3] + ǫit ,
the daily beta is β i = β i0 + β i1 + β i2 . The t-statistics adjusted with heteroscedasticity and autocorrelations
are calculated with time series standard errors of alphas and betas.
Low
2
3
4
5
6
7
8
9
High
H−L
0.18
0.15
0.13
0.14
0.24
0.20
0.20
0.21
0.28
0.23
0.19
0.30
0.36
0.34
0.29
0.39
0.53
0.49
0.40
0.51
2.34
1.75
1.59
1.46
2.69
2.26
2.16
2.25
3.09
2.52
1.82
3.09
2.72
2.50
2.16
2.80
2.76
2.42
2.01
2.44
0.91
0.91
0.90
0.91
0.90
0.90
0.93
0.91
0.95
0.94
1.00
0.98
1.04
1.04
1.10
1.08
−0.03
−0.03
0.08
0.03
0.17
0.14
0.13
0.11
0.21
0.16
0.14
0.20
0.09
0.07
−0.00
0.10
0.21
0.16
0.05
0.16
0.66
0.32
−0.31
−0.03
2.55
1.99
1.65
1.39
2.55
1.87
1.43
1.95
0.75
0.56
−0.01
0.72
1.41
1.06
0.29
0.96
0.99
0.98
0.99
1.00
1.01
1.01
1.02
1.02
1.08
1.07
1.11
1.11
1.24
1.23
1.27
1.24
0.18⋆
0.19⋆
0.26⋆
0.20⋆
Panel A: July 1963–December 2014
Average conditional alphas
Quarterly
Semiannual 1
Semiannual 2
Annual
−0.17
−0.14
−0.11
−0.11
−0.01
0.00
0.03
0.02
0.03
0.03
0.02
0.05
0.04
0.02
−0.00
0.06
0.07
0.06
0.01
0.05
0.13
0.10
0.09
0.10
Average conditional alphas, t-statistics
Quarterly
Semiannual 1
Semiannual 2
Annual
−1.97
−1.57
−1.19
−1.17
−0.10
0.06
0.46
0.42
0.62
0.59
0.26
0.83
0.60
0.20
−0.05
0.66
0.94
0.69
0.07
0.57
1.89
1.44
1.22
1.26
Average conditional betas
Quarterly
Semiannual 1
Semiannual 2
Annual
1.07
1.07
1.02
1.05
1.02
1.02
1.01
1.04
0.97
0.97
1.01
1.00
0.97
0.96
0.99
0.99
0.92
0.93
0.95
0.93
0.93
0.93
0.95
0.94
Panel B: July 1926–December 2014
Average conditional alphas
Quarterly
Semiannual 1
Semiannual 2
Annual
−0.13
−0.10
−0.05
−0.07
0.03
0.03
0.07
0.08
0.04
0.05
0.05
0.07
0.04
0.01
−0.01
0.04
0.08
0.07
0.03
0.04
0.10
0.07
0.03
0.05
0.04
0.02
−0.02
−0.00
Average conditional alphas, t-statistics
Quarterly
Semiannual 1
Semiannual 2
Annual
−2.23
−1.60
−0.85
−1.07
0.65
0.77
1.36
1.61
1.06
1.09
0.94
1.33
0.74
0.22
−0.19
0.52
1.41
1.19
0.50
0.63
1.89
1.34
0.50
0.79
Average conditional betas
Quarterly
Semiannual 1
Semiannual 2
Annual
1.05
1.05
1.01
1.04
1.00
1.01
1.00
1.01
0.96
0.96
0.99
0.98
0.96
0.96
0.98
0.98
0.94
0.94
0.95
0.94
66
0.96
0.96
0.97
0.97
book-to-market decile is 0.21% per month (t = 1.41). Also, the average conditional beta for the
high-minus-low decile is 0.18 (t = 5.86). Estimating alphas and betas semiannually with daily and
weekly returns, as well as annually with monthly returns, again delivers largely similar results.
If the conditional CAPM helps explain the value premium, the expected value premium should
be countercyclical. Cohen, Polk, and Vuolteenaho (2003) show that the expected value-minusgrowth return is atypically high at times when the value-minus-growth spread in book-to-market
is wide. Chen, Petkova, and Zhang (2008) estimate the expected value premium as the expected
long-term dividend growth spread plus the expected dividend-to-price spread between value and
growth stocks. The expected value premium is largely stable around 6.1% per annum, consisting
of a dividend growth component of 4.4% and a dividend-to-price component of 1.7%. Although
countercyclical, the cyclical movement in the expected value premium is quantitatively small.
Gulen, Xing, and Zhang (2011) study the time variation of the expected value premium with
a two-state Markov switching model. Gulen et al. find that when conditional volatilities are high,
the expected excess returns of value stocks are more sensitive to aggregate economic conditions
than the expected excess returns of growth stocks. As a result, the expected value premium is time
varying, spiking upward in the high volatility state, only to decline more gradually in subsequent
periods. However, the out-of-sample predictability is close to nonexistent.
4.3.3
What Explains the Failure of the CAPM?
Table 10 reports the CAPM regressions with the book-to-market deciles. From July 1963 to December 2014, the high-minus-low decile earns on average 0.5% per month (t = 2.72). Consistent
with Fama and French (1992), the CAPM fails to explain the value premium. The high-minus-low
market beta is close to zero, and the alpha equals the average return in magnitude, 0.5% (t = 2.23).
From Panel B, in the long sample that starts from July 1926, the high-minus-low decile earns on average 0.51% (t = 2.58), which is close to that in the short sample. However, consistent with Ang and
Chen (2007), the CAPM captures the value premium in the long sample, with an insignificant alpha
67
Table 10 : Testing the CAPM with the Book-to-market Deciles
Data for the book-to-market deciles, one-month Treasury bill rates, and market excess returns are from
Kenneth French’s Web site. m is average returns, α CAPM alphas, β market betas, and tm , tα , and tβ are
their corresponding t-statistics adjusted for heteroscedasticity and autocorrelations. R2 is the goodness-of-fit.
Low
2
3
4
5
6
7
8
9
High
H−L
0.70
3.82
0.25
2.12
0.88
17.56
0.76
0.79
4.11
0.32
2.92
0.94
20.90
0.76
0.93
3.95
0.39
2.50
1.07
15.47
0.66
0.50
2.72
0.50
2.23
0.01
0.06
0.00
0.92
4.43
0.18
1.88
1.13
15.48
0.83
1.03
4.35
0.20
1.76
1.27
13.69
0.79
1.09
3.86
0.15
0.98
1.44
13.63
0.72
0.51
2.58
0.23
1.19
0.43
3.52
0.13
Panel A: July 1963–December 2014
m
tm
α
tα
β
tβ
R2
0.42
2.07
−0.12
−1.24
1.06
40.91
0.86
0.52
2.77
0.01
0.19
1.01
46.40
0.92
0.56
3.00
0.06
0.96
0.98
34.65
0.90
0.56
2.94
0.06
0.60
0.99
29.43
0.87
0.54
3.01
0.08
0.83
0.91
27.58
0.83
0.60
3.29
0.12
1.45
0.93
28.41
0.85
0.68
3.83
0.24
2.23
0.88
22.94
0.78
Panel B: July 1926–December 2014
m
tm
α
tα
β
tβ
R2
0.58
3.29
−0.08
−1.22
1.01
52.11
0.90
0.68
4.13
0.06
1.26
0.95
27.19
0.92
0.68
4.06
0.04
0.85
0.97
57.09
0.92
0.68
3.69
−0.01
−0.16
1.05
22.29
0.90
0.73
4.11
0.07
0.96
1.00
26.81
0.89
0.76
4.03
0.07
0.85
1.05
15.88
0.87
0.76
3.86
0.05
0.54
1.09
16.92
0.84
of 0.23% (t = 1.19). In particular, the high-minus-low decile has a market beta of 0.43 (t = 3.52).
As noted, the first generation quantitative models mostly rely on single-factor structure, and
consequently fail to explain the failure of the CAPM in the post-1963 sample. For example, Lin
and Zhang (2013, Table 4) report that despite a high value premium, the CAPM alpha of the
high-minus-low decile in the Zhang (2005b) model is economically small and statistically insignificant, whereas its market beta is economically large and statistically significant. Remedying this
deficiency in the early models has attracted much attention in the recent literature.
4.3.3.1
Two-shock Models
Papanikolaou (2011) constructs a general equilibrium model with
two sectors, the consumption goods sector and the investment goods sector, following Kogan (2004).
In the model, investment-specific shocks reduce the costs of producing new capital goods, thereby
lowering consumption and increasing marginal utility today, because the economy reallocates resources from producing consumption goods to investment goods. Positive investment shocks increase the value of growth opportunities relative to the value of assets in place. The value premium
arises because growth firms appreciate in value in response to, and act as a hedge against positive
68
investment shocks, which are associated with high marginal utility states. However, it seems counterintuitive to identify, for instance, the internet boom of the 1990s, during which growth stocks
outperform value stocks, with states of high marginal utility for the average investor.8
In Ai and Kiku’s (2013) real options model, growth options are long positions in assets in place
and short positions in capital goods. In good times, many firms try to exercise their growth options,
which tend to become less valuable soon afterward due to mean-reverting productivity. This effect,
combined with the limited supply of capital goods, implies that the equilibrium price of capital
goods is highly procyclical, making growth options less risky than assets in place. In addition,
aggregate consumption admits both short-run and long-run shocks as in Bansal and Yaron (2004).
Recursive utility implies that the two types of risk carry different prices of risk, giving rise to the
failure of the CAPM. It is an open question, however, whether the procyclical movements in the price
of capital goods are strong enough in the data to make growth options less risky than assets in place.
The two-sector general equilibrium model in Papanikolaou (2011) does not include a full-fledged
cross section of firms. To bring the model closer to the data, Kogan and Papanikolaou (2013) switch
to a partial equilibrium setup, following Berk, Green, and Naik (1999) and Zhang (2005b). The
Kogan-Papanikolaou model reproduces the negative relation between investment and expected returns, as well as the positive relation between profitability and expected returns in Hou, Xue, and
Zhang (2015a, b). Also, the exogenous stochastic discount factor in the Kogan-Papanikolaou model
contains both total factor productivity shocks and investment-specific shocks. As such, the CAPM
fails to explain the investment and profitability premiums in the model, as in the data.
Belo, Lin, and Bazdresch (2014) incorporate labor adjustment costs into the baseline investment model (Section 4.2). The sum of capital and labor adjustment costs is subject to stochastic
shocks that are analogous to investment-specific shocks. The exogenous stochastic discount factor
contains two shocks, including total factor productivity shocks, which carry a positive price of risk,
8
In the 1990–1999 sample, the average return of the high-minus-low book-to-market decile is −0.31% per month,
albeit insignificant due to the short sample length. Data for the decile returns are from Kenneth French’s Web site.
69
and adjustment costs shocks, which carry a negative price of risk. The key finding is that firms
with high hiring rates earn lower expected returns than firms with low hiring rates in the model, as
in the data. Intuitively, positive shocks that lower adjustment costs imply higher discount factor
(marginal utility), due to their negative price of risk. Firms with high hiring rates benefit the most
from these shocks, and their increase in value acts as a hedge against these high marginal utility
states. The CAPM fails to explain the hiring return spread, which is largely driven by shocks to
adjustment costs, whereas the market factor only summarizes shocks to total factor productivity.
Garlappi and Song (2015) argue that models with investment shocks most likely cannot explain
value or momentum. Using a cross section of 40 testing assets, consisting of the size, book-tomarket, and momentum deciles, as well as ten industry portfolios, Garlappi and Song report significantly positive risk premiums on investment shocks in the 1930–2012 sample. Reconciling their
estimates with prior studies, Garlappi and Song show that negative risk premiums can be obtained
only with the book-to-market deciles as the testing assets in the post-1963 sample. In particular,
risk premium estimates are positive with the book-to-market deciles in the pre-1963 data. Even for
the post-1963 data, risk premium estimates are positive when the testing assets are broader than
the book-to-market deciles. Theoretically, Garlappi and Song show that risk premiums for investment shocks should be positive, if investors prefer an early resolution of uncertainty, as commonly
assumed in the long run risks literature (Bansal and Yaron 2004).
Koh (2015) incorporates time-varying uncertainty (Bloom 2009) into the baseline investment
model (Section 4.2). The exogenous stochastic discount factor contains two shocks, including total
factor productivity shocks, which carry a positive price of risk, and uncertainty shocks, which carry
a negative price of risk. This two-shock model replicates the value premium and the failure of
the CAPM in explaining it. Intuitively, the value of growth options increases when uncertainty is
high. As such, growth stocks act as a hedge against positive uncertainty shocks as investors dislike
uncertainty, and earn lower expected returns than value stocks. The CAPM fails because the value
premium is driven by uncertainty shocks, but the market factor only captures productivity shocks.
70
4.3.3.2
Disasters While the two-shock models explain the failure of the CAPM in the post-
1963 sample, the models fail to explain its relative success in the long sample from 1926 onward
(Table 10). Bai, Hou, Kung, and Zhang (2015) propose an investment model with disasters to explain the CAPM performance in both samples. Their model has three key ingredients, asymmetric
adjustment costs per Zhang (2005b), recursive utility per Bansal and Yaron (2004), and disasters as
large drops in aggregate consumption and productivity growth per Rietz (1988) and Barro (2006).
Intuitively, when a disaster hits, value firms are burdened with more unproductive capital, and
find it more difficult to downsize. Consequently, value firms are more exposed to the disaster risk
than growth firms. This effect, combined with high marginal utilities of the household in disasters,
implies a high value premium. More important, the disaster risk induces strong nonlinearity in the
stochastic discount factor, of which the linear CAPM is a poor proxy. When disasters are not materialized in a finite sample, estimated market betas only measure the weak comovement between
the value-minus-growth returns and market excess returns in normal times. However, the value
premium is primarily driven by the disaster risk. As a result, the CAPM fails in normal times. In
contrast, when disasters are materialized, the estimated market betas provide an adequate account
for the strong comovement between the value-minus-growth returns and the stochastic discount
factor. As such, the CAPM explains the value premium in samples with disasters.
Bai, Hou, Kung, and Zhang (2015) also explain the beta “anomaly,” which says that the empirical relation between market betas and average returns is too flat to be consistent with the
CAPM. In simulated samples with and without disasters, sorting on pre-ranking market betas
yields small and insignificant high-minus-low average return spreads, large and significantly positive post-ranking market betas, and large and often significantly negative CAPM alphs. Intuitively,
estimated market betas are a poor proxy for true betas. Based on prior 60-month rolling windows,
pre-ranking betas are averaged over the prior five years. In contrast, true betas accurately account
for changes in aggregate and firm-specific conditions. In simulations, true betas often mean-revert
within a rolling window, giving rise to negative correlations with rolling betas.
71
4.4
Notes
Quantitative investment studies can be grouped into three categories: momentum; beyond value and
momentum, including equity issues, real estate, inventories, and intangibles; and debt dynamics.
4.4.1
Momentum
Johnson (2002) argues that the log price-to-dividend ratio is a convex function of the expected
dividend growth. As such, changes in log price-to-dividend ratio or stock returns are more sensitive
to changes in the expected growth, when the expected growth is high. If the risk related to the
expected growth is priced with a positive price of risk, expected returns should rise with expected
growth rates. All else equal, winners that have recent large positive price movements are likely to
have higher expected growth rates than losers that have recent large negative price movements.
Consequently, winners should earn higher expected returns than losers.
Sagi and Seasholes (2007) provide a real options model of momentum. Firms that have performed well recently are better poised to exploit their growth options than firms that have performed
poorly recently. These options, which are riskier than assets in place, account for a larger fraction
of the market value of winners, increasing their risk. As such, winners earn higher expected returns
than losers. This intuition is also consistent with the evidence that momentum is stronger in growth
firms than in value firms (Asness 1997), as well as the evidence that momentum is higher after positive market returns than after negative market returns (Cooper, Gutierrez, and Hameed 2004).
Liu and Zhang (2008) document that winners have temporarily higher loadings than losers on
the growth rate of industrial production. In univariate regressions, the loadings for winners and
losers are 0.63 and −0.17 in the first month, and 0.33 and 0.38 six months after portfolio formation.
In two-stage cross-sectional regressions with the size, book-to-market, and momentum deciles as
testing assets, the industrial production growth earns significantly positive risk premiums. The combined effect of loadings and risk premiums accounts for more than half of momentum profits. Also,
winners have temporarily higher average future dividend, investment, and sales growth rates than
72
losers, and the duration of the expected growth spreads matches roughly that of momentum profits.
An important weakness of this literature is that momentum and value are studied in isolation.
In the data, momentum and value are negatively correlated, yet both forecast returns with positive
slopes (Asness 1997). Explaining momentum and value together is challenging. As noted, Liu and
Zhang (2014) show that a baseline specification of the structural investment CAPM fails to fit the
expected returns across the value and momentum portfolios simultaneously (Section 3.1.1.2).
Li (2015) provides an ambitious attempt to explain value and momentum simultaneously within
the neoclassical investment framework. Li incorporates time to plan and shocks to the price of
capital goods into the baseline investment model (Section 4.2). Time to plan means that firms
make investment decisions today, but can only complete a small fraction of the investments today,
and must finish the remaining fraction in the next period. Naturally, firms that have committed
to investing more in the next period are more sensitive to shocks to the price of capital goods than
firms that have committed to investing less. The model reproduces quantitatively the coexistence of
momentum and the value premium, their negative correlation, the failure of the CAPM in explaining
momentum, the variation of momentum by market states, and the long-term reversal of momentum.
Intuitively, winners have positive recent productivity shocks, commit to invest more, and have
large negative loadings on the price shocks of capital goods. In contrast, losers have negative recent
productivity shocks, commit to invest less and even to disinvest, and have small and even positive
loadings on the price shocks. When the price of risk for the price shocks of capital goods is negative,
winners earn higher expected returns than losers. Differing from momentum, value and growth firms
have different loadings on productivity shocks, in the presence of asymmetric adjustment costs.
Because the market is mostly driven by productivity shocks, the CAPM fails to explain momentum,
but does better in explaining the value premium. Also, because time to plan is temporary, exposures
to the price shocks, and consequently momentum profits, are short-lived. In particular, after time
to plan runs its course, exposures to productivity shocks take over, giving rise to long-term reversal.
73
Finally, Li (2015) also estimates the price of risk for the price shocks of capital goods. With the
size, book-to-market, momentum, and industry portfolios as testing assets, the price of risk estimates are significantly negative in two-stage cross-sectional regressions. Because the price of capital
goods can be viewed as the inverse of investment-specific productivity, Li’s estimates are consistent
with significantly positive price of risk estimates on investment shocks in Garlappi and Song (2015).
4.4.2
4.4.2.1
Beyond Value and Momentum
Equity Issues Carlson, Fisher, and Giammarino (2006) use a real options model to
explain long-term performance following equity issues. Intuitively, equity issues are related to firm
expansion. As firms grow, investment converts risky growth options into less risky assets in place,
reducing risk and expected returns. As such, the relation between investment and expected returns
is negative, giving rise to a negative relation between equity issues and expected returns. Because
only growth options in the money are exercised, high average returns naturally precede equity issues.
Li, Livdan, and Zhang (2009) incorporate equity floatation costs into the neoclassical investment framework (Section 4.2). The model reproduces simultaneously many stylized facts: (i) The
frequency of equity issuance is procyclical; (ii) investment is negatively correlated with future returns, and this correlation is stronger in firms with higher cash flows; (iii) firms that issue equities
underperform nonissuers with similar size and book-to-market in the long run; (iv) the operating
performance of issuers substantially improves prior to equity offerings, but deteriorates afterward;
(v) firms distributing equity outperform, and the outperformance is stronger in value firms than
in growth firms; and (vi) the operating performance of firms distributing capital substantially improves prior to distribution, but deteriorates afterward. However, the model cannot fully capture
the magnitude of the outperformance following cash distributions.
Intuitively, investment and expected return are negatively correlated in the model via two
channels. First, given expected profitability, high costs of capital imply low net present values of
new capital and low investment, and low costs of capital imply high net present values of new capital
74
and high investment. This channel operates even with constant returns to scale (Section 1). Second,
with decreasing returns to scale, high investment leads to lower expected profitability, which further
reduces expected returns. Also, the flow of funds constraint means that all else equal, equity issuers
are disproportionately high investment firms, and cash distributing firms are disproportionately low
investment firms. As such, raising equity is related to high investment and low expected returns,
and distributing equity is related to low investment and high expected returns.
4.4.2.2
Real Estate Tuzel (2010) explores the link between a firm’s capital composition and
its expected returns. Firms use two capital stocks in production, real estate and other capital.
Investment in both types of capital is subject to asymmetric adjustment costs as in Zhang (2005b).
Because real estate depreciates slowly, firms with high real estate holdings face more difficulty in
bad times in downsizing, and are more sensitive to negative aggregate shocks. Empirically, firms
with high real estate holdings earn higher average returns than firms with low real estate holdings.
4.4.2.3
Inventories Belo and Lin (2012) embed inventory as a factor of production with convex
and nonconvex adjustment costs into the baseline investment model (Section 4.2). Calibrated to
match the volatilities and autocorrelations of capital and inventory investment rates, the model predicts that the inventory-to-sales ratio is persistent and countercyclical. Also, firms with low inventory growth earn higher average returns than firms with high inventory growth, consistent with data.
In independent work, Jones and Tuzel (2013) also study the relation between inventory investment and costs of capital. Empirically, Jones and Tuzel show that risk premiums, rather than real
interest rates, are strongly negatively correlated with future inventory growth at the aggregate,
industry, and firm levels. This effect is stronger for firms in industries that produce durables rather
than nondurables, exhibit more cyclical sales, require longer lead times, and are subject to more
technological innovations. A dynamic investment model with two types of capital, physical capital
and inventory, quantitatively accounts for these empirical patterns.
75
4.4.2.4
Intangibles Berk, Green, and Naik (2004) develop a dynamic model of a multistage
research and development (R&D) venture. Firms learn about the potential profitability of a R&D
project throughout its life, but the idiosyncratic technical uncertainty about the project is only resolved via additional investment. As technical uncertainty is resolved, the probability of successful
and timely completion changes. This effect changes the properties of the option to suspend future
investment, which affects future cash flows of the project. Consequently, the cash flows risk has a
systematic component, even though the purely technical uncertainty is idiosyncratic in nature.
Li (2011) extends the Berk-Green-Naik (2004) model by incorporating financial constraints. A
financially constrained R&D intensive firm is more likely to suspend R&D projects, implying that
their risk increases with financial constraints. Constrained firms’ risk also increases with their R&D
intensity. Empirically, financial constraints are related to average returns, primarily among R&D
intensive firms, and R&D predicts returns only among financially constrained firms.
Lin (2012) incorporates endogenous technological progress via R&D into the baseline investment model (Section 4.2). One part of R&D is devoted to product innovation, which increases the
expected marginal benefits of physical investment. The other part of R&D is devoted to increasing
the productivity of physical investment in producing new physical capital, or equivalently, decreasing the marginal costs of physical investment. Consequently, a key prediction from the model is
that high R&D firms earn higher expected returns than low R&D firms, whereas high physical
investment firms earn lower expected returns than low physical investment firms.
Eisfeldt and Papanikolaou (2013) study the impact of organizational capital as a durable production input embodied in a firm’s key talent on the cross section. Both shareholders and key talent
have a claim on the cash flows from organizational capital, and the cash flows to shareholders are
determined in part by the sharing rule between shareholders and key talent. This sharing rule
depends on aggregate productivity shocks to organizational capital deployed in new firms, shocks
that determine the outside option of key talent. From the perspective of shareholders, investing in
76
key talent and organizational capital is risky, because unlike physical capital, shareholders do not
own all of the cash flows. Key talent can leave the firm when the outside option exceeds the inside
value. As such, firms with more organizational capital are riskier, and earn higher expected returns
than firms with less organizational capital, a prediction confirmed in the data.
Donangelo (2014) studies asset pricing implications of labor mobility, which is the flexibility of
workers to walk away from an industry in response to better opportunities in another industry. In
the model, firms compete for workers in a partially segmented labor market. The less segmented
the labor market, the greater the flexibility workers have to move in and out of an industry. Labor mobility induces a form of operating leverage that amplifies firms’ risk. Intuitively, a firm
that employs mobile workers is less flexible in adjusting wages to mitigate the impact of aggregate
shocks. Less elastic wages make the cash flows to shareholders more sensitive to aggregate shocks,
increasing risk and expected returns, a prediction that Donangelo confirms in the data.
4.4.3
Debt Dynamics
Building on the recent literature on dynamic corporate finance (Hennessy and Whited 2005, 2007;
Strebulaev and Whited 2012), several studies incorporate debt dynamics into the investment model.
Livdan, Sapriza, and Zhang (2009) examine the effect of financial constraints on risk and expected returns by embedding retained earnings, debt, costly equity, and collateral constraints into
the baseline investment model (Section 4.2). By preventing firms from financing all desired investments, collateral constraints restrict the flexibility of firms in smoothing dividends in response to
shocks. As such, more financially constrained firms are riskier and earn higher expected returns than
less financially constrained firms. Also, asset beta increases with market leverage. Intuitively, more
levered firms are burdened with more debt, which must be repaid before financing new investments.
As a result, whereas the standard leverage hypothesis in corporate finance predicts a linear relation
between market leverage and expected returns, the investment model predicts a convex relation.
Gomes and Schmid (2010) revisit the theoretical relation between financial leverage and ex77
pected stock returns by embedding taxes and defaultable debt into the baseline investment model.
Quantitatively, the model replicates the complex leverage-return relations in the data. Equity
returns are positively related to market leverage, but insensitive to book leverage. Also, market
leverage is only weakly related to equity returns after controlling for book-to-market. Intuitively,
with taxes and distress costs, leverage and investment are simultaneously determined in a way that
highly levered firms are also firms with more safe book assets and fewer risky growth options.
Garlappi and Yan (2011) study the role of potential shareholder recovery upon resolution of
financial distress in an equity valuation model. The model replicates lower returns for financially
distressed firms, stronger value premiums for firms with high default likelihood, and stronger momentum profits in low credit quality firms. Intuitively, when default probability is low, leverage
increases equity beta, but when default probability is high, potential shareholder recovery decreases
equity beta. As such, risk and expected returns are hump-shaped in default probability. Also, this
hump-shaped relation implies a similarly hump-shaped relation between the value premium and
default probability. Finally, momentum profits are stronger among distressed firms. When default
probability is low, negative realized returns raise the default probability, increasing risk and expected returns. However, when default probability is high, negative realized returns increase the
possibility of shareholder recovery, decreasing risk and expected returns.
Ozdagli (2012) incorporates tax benefits of debt, risk-free debt via collateral constraints, and
debt adjustment costs into the baseline investment model. The main driving force of the value
premium is financial leverage, which amplifies equity risk and expected returns as in the standard
leverage hypothesis in corporate finance. Ozdagli also clarifies the effect of asymmetric adjustment
costs on the value premium, by showing that the effect depends on fixed costs of operation. If
the fixed costs are sufficiently high, a higher degree of asymmetry increases the value premium.
However, when the fixed costs are low, the value premium can be negative despite the asymmetry.
Choi (2013) uses corporate bond data to examine empirically how asset risk and financial lever-
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age interact to explain the equity risk dynamics of value and growth firms. During economic
downturns, the asset risk and leverage of value firms increase, giving rise to a sharp rise in equity
betas. In contrast, the asset risk of growth firms is much less sensitive to aggregate economic conditions. The interactions of time-varying betas with the market risk premium and volatility account
for roughly 40% of the value premium in the 1982–2007 sample. Finally, asset risk and leverage are
negatively correlated in the cross section, consistent with the trade-off theory of capital structure.
Because asset risk tends to increase financial distress costs, growth firms have high asset risk and
low leverage, and value firms have low asset risk and high leverage.
Obreja (2013) incorporates taxes, equity floatation costs, bankruptcy costs, and defaultable
debt into the baseline investment model (Section 4.2). The model replicates a significantly positive
value premium jointly with a weak or even negative relation between book leverage and expected
equity returns (Penman, Richardson, and Tuna 2007). Intuitively, the equity value is depressed in
bad times because of costly reversibility and the countercyclical price of risk. However, the effect
of the price of risk is weaker for nearly default-free debt, with value hardly correlated with the
stochastic discount factor. As such, firms have incentives in bad times to levering up and using the
debt proceeds to repurchase equity. More important, the extent of this recapitalization depends on
operating leverage. If operating leverage is low, firms issue debt and retire equity, giving rise to high
book leverage, but if operating leverage is high, firms cannot add much debt without increasing
default probability. The value premium arises because value firms have either higher operating or
financial leverage than growth firms. However, the relation between book leverage and expected
returns is flat. The crux is that both high and low book leverage firms can have high risk and
expected equity returns, as long as low book leverage firms also face high operating leverage.
Finally, Kuehn and Schmid (2014) examine the impact of endogenous investment in credit risk
models. In the presence of financing and investment frictions, however, firm-level variables of asset
composition are significant determinants of credit spreads beyond leverage and asset volatility that
are the focus of traditional credit risk models with exogenous assets value. Intuitively, while asset
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volatility captures the total risk of firms’ assets, asset composition proxies for their systematic risk.
The crux is that growth options and assets in place have different exposures to aggregate business
conditions. Empirically, the size and value premiums also appear in credit spreads.
5
The Big Picture
After reviewing what the investment CAPM can do, I take a step back, and explain the big picture
as I perceive it. I provide a historical perspective of the investment CAPM in Section 5.1, and clarify
its relation with the consumption CAPM in Section 5.2, and with behavioral finance in Section 5.3.
5.1
5.1.1
The Investment CAPM: A Historical Perspective
Origin
The origin of the investment CAPM can be traced to Böhm-Bawert (1891) and Fisher (1930).
Böhm-Bawert provides three reasons why the rate of interest is positive. First, the marginal utility of income falls over time because of more resources in the future. Second, due to psychological
reasons, consumers tend to underestimate future needs. Both reasons are consumption-based determinants of the equilibrium interest rate. Böhm-Bawert’s third reason is “roundabout” production:
It is an elementary fact of experience that methods of production which take time are more productive.
That is to say, given the same quantity of productive instruments, the lengthier the productive method
employed the greater the quantity of products that can be obtained (p. 260).... Command over a sum of
present consumption goods provides us with the means of subsistence during the current economic period.
This leaves the means of production, which we may have at our disposal during this period (Labour, Uses of
Land, Capital), free for the technically more productive service of the future, and gives us the more abundant
product attainable by them in longer methods of production (p. 271).
Because production per worker rises with the length of the period of production, some counterbalancing force must exist to cause producers to choose a production period of finite length.
Böhm-Bawert (1891) argues that this counterbalancing force is the positive rate of interest. This
mechanism implies that the lower the interest rate, the longer will be the optimal period of production, and the more capital will be tied up in the production process. Conversely, the higher the
interest rate, the shorter will be the optimal period of production, and the less capital will be tied
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up in the production process. This effect is exactly the negative relation between real investment
and the discount rate (Figure 1), albeit without uncertainty.
Fisher (1930) studies the economic determinants of the real interest rate by constructing the
first general equilibrium model with both intertemporal exchange and production. His model also
shows the Fisher Separation Theorem, which justifies the maximization of the present value as the
objective of the firm, without any direct dependence on shareholder preferences. Figure 6, which
is adapted from Chart 38 in Fisher (p. 271), shows the key insights.
Figure 6. The Fisherian Equilibrium
C1 ✻
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U0
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0
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K0
In the figure, the horizontal axis labeled C0 represents consumption in date 0, and the vertical
axis C1 represents consumption in date 1. Endowed with an amount of resources, K0 , in date 0,
the agent’s problem is to choose an optimal time pattern of consumption. The agent’s preferences
are represented by indifference curves, such as U0 and U1 . There are two available ways to transfer
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resources between dates 0 and 1. The “market opportunity” represented by the two straight lines in
the figure is through borrowing and lending at the interest rate, r. The “investment opportunity”
represented by the curve K0 OK1 is through a real production technology. The curve is concave to
the origin due to diminishing returns to scale. Without the production technology, the optimum is
Q, which is the tangent point between the market line initiated at K0 and the indifference curve U0 .
With the production technology, the agent solves the optimum in two separate steps. First, the
production optimum is achieved at O, which is the tangent point between its associated market
line and the production frontier. Starting from K0 , the agent keeps moving up along the production curve K0 OK1 , until the marginal rate of transformation, △K1 /△K0 − 1, in which △K0 is
the amount of input in date 0, and △K1 is output in date 1, is equal to the interest rate, r. This
equality is achieved at the tangent point O. Prior to this point, △K1 /△K0 − 1 > r, meaning that it
is optimal to invest more. Beyond this point, △K1 /△K0 − 1 < r, meaning that it is suboptimal to
keep investing. Second, the consumption optimum is achieved at point P , through borrowing and
lending along the market line associated with O. An important implication of the Fisher Separation Theorem is that the equilibrium interest rate can be “determined” purely from the production
technology, without any direct dependence on investor preferences.
In his own words, Fisher (1930) describes the interaction and equivalence between the
consumption-based (impatience) and investment-based determinants of the interest rate as follows:
Our outer opportunities urge us to postpone present income—to shift it toward the future, because it will
expand in the process. Impatience is impatience to spend, while opportunity is opportunity to invest. The
more we invest and postpone our gratification, the lower the investment opportunity rate becomes, but the
greater the impatience rate; the more we spend and hasten our gratification, the lower the impatience rate
becomes but the higher the opportunity rate.
If the pendulum swings too far toward the investment extreme and away from the spending extreme, it is
brought back by the strengthening of impatience and the weakening of investment opportunity. Impatience is
strengthened by growing wants, and opportunity is weakened because of diminishing returns. If the pendulum
swings too far toward the spending extreme and away from the investment extreme it is brought back by the
weakening of impatience and the strengthening of opportunity for reasons opposite to those stated above.
Between these two extremes lies the equilibrium point which clears the market, and clears it at a rate of
interest registering (in a perfect market) all impatience rates and all opportunity rates (p. 177).
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Hirshleifer (1958) uses the Fisherian approach to study optimal investment decisions of firms,
with and without certain capital markets imperfections. Hirshleifer (1965, 1966, 1970) extends the
Fisherian equilibrium into a world with uncertainty, using the state-preference approach of Arrow
(1964) and Debreu (1959). Diamond (1967) and Hirshleifer (1970) also contain an early formation
of state-contingent production, a theme echoed later by Cochrane (1993) and Belo (2010).
Jorgenson (1963) formulates a model of investment demand with capital as a factor of production, and introduces the concept of the user cost of capital as the rental price of capital services.
The user cost of capital contains three components, including the interest rate, the depreciation
rate, and the rate of capital loss on investment goods, adjusted for the taxation of capital income.
Hall and Jorgenson (1967) analyze the impact of alternative tax policies on the user cost of capital.
In a justly immortal contribution, Modigliani and Miller (1958) ask the cost of capital question:
What is the “cost of capital” to a firm in a world in which funds are used to acquire assets whose yields
are uncertain; and in which capital can be obtained by many different media, ranging from pure debt instruments, representing money-fixed claims, to pure equity issues, giving holders only the right to a pro-rata
share in the uncertain venture (p. 261)?
Modigliani and Miller’s Proposition I says that in perfect capital markets, the value of a firm
equals the market value of the total cash flows produced by its assets, and is unaffected by its capital structure. Proposition II says that the cost of equity equals the cost of unlevered equity plus
a risk premium, which equals the market debt-to-equity ratio times the spread between the cost
of unlevered equity and the cost of debt. Both propositions are extensively discussed in modern
corporate finance textbooks, including Ross, Westerfield, and Jaffe (2008, p. 428–447), Berk and
DeMarzo (2009, p. 455–466), and Brealey, Myers, and Allen (2011, p. 418–434).
However, it is the relatively overlooked Proposition III in Modigliani and Miller (1958) that
contains arguably the earliest discussion of the investment CAPM:
Proposition III. If a firm in class k is acting in the best interest of the stockholders at the time of the decision,
it will exploit an investment opportunity if and only if the rate of return on the investment, say ρ∗ , is as
large as or larger than ρk . That is, the cut-off point for investment in the firm will in all cases be ρk and
will be completely unaffected by the type of security used to finance the investment. Equivalently, we may
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say that regardless of the financing used, the marginal cost of capital to a firm is equal to the average cost
of capital, which is in turn equal to the capitalization rate for an unlevered stream in the class to which the
firm belongs (p. 288, original emphasis).
Cochrane (1991) is the first in the modern era to use the investment model to study asset prices:
The logic of the production-based model is exactly analogous [to that of the consumption-based model].
It ties asset returns to marginal rates of transformation, which are inferred from data on investment (and
potentially, output and other production variables) through a production function. It is derived from the
producer’s first order conditions for optimal intertemporal investment demand. Its testable content is a restriction on the joint stochastic process of investment (and/or other production variables) and asset returns.
This restriction can also be interpreted in two ways. If we fix the return process, it is a version of the q theory
of investment. If we fix the investment process, it is a production-based asset pricing model. For example,
the production-based asset pricing model can make statements like “expected returns are high because (a
function of) investment growth is high” (p. 210, original emphasis).
Alas, when applying the investment model to the cross section, Cochrane (1996) retreats back to the
consumption CAPM, stipulates the stochastic discount factor as a linear function of the residential
and nonresidential investment returns, and performs GMM tests per Hansen and Singleton (1982).
5.1.2
Oblivion
Modern asset pricing literature is thoroughly dominated by the consumption CAPM. The term
“the investment CAPM” has not even appeared in the prior literature. The glorious history of
asset pricing is well known (Bernstein 1992; Rubinstein 2006). I only wish to understand how
classic asset pricing studies have gone down the path of ignoring the investment model altogether,
despite it receiving a symmetric treatment as the consumption model in Fisher (1930). The crux
seems to be uncertainty. Fisher’s general equilibrium model assumes certainty, and only provides
a theory of the interest rate, as opposed to a theory of risky asset prices. In hindsight, thanks to
Arrow (1964) and Debreu (1959), it is clear that asset pricing theory is basically the standard price
theory in microeconomics extended to uncertainty and over time.
However, predating the Arrow-Debreu framework, Markowitz (1952, 1959) formulates the mean
variance model of portfolio selection (see also Roy 1952). The model postulates that investors
should maximize the expected portfolio return, while minimizing portfolio variance. Treynor (1962),
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Sharpe (1964), Lintner (1965), and Mossin (1966) ask what would happen in equilibrium if investors
all follow Markowitz’s decision rule, leading to the derivation of the CAPM. Merton (1973) extends
the one-period CAPM to the intertemporal CAPM, in which state variables related to shifts in
the investment opportunity set over time are also risk factors (see also Long 1974). The classic textbooks by Fama and Miller (1972) and Fama (1976) rely almost exclusively on the mean
variance model. While acknowledging Hirshleifer’s (1965, 1966) applications of the Arrow-Debreu
approach, Fama and Miller view the mean variance model as much more operational empirically.
This operational advantage no longer exists, thanks to the methodological contributions of rational
expectations econometrics in Hansen (1982) and Hansen and Singleton (1982).
Recasting equilibrium asset pricing within the Arrow-Debreu framework, Rubinstein (1976),
Lucas (1978), and Breeden (1979) derive the consumption CAPM from the first principle of the
optimal consumption-portfolio choice problem of a representative consumer. Hansen and Singleton (1982) and Breeden, Gibbons, and Litzenberger (1989) provide early tests of the consumption
CAPM. Ludvigson (2013) offers a recent review of this literature.
Because consumers and producers are two different agents integral to the Arrow-Debreu model,
early asset pricing theorists justify their modeling choice as reducing a messy general equilibrium
problem into a tractable consumption-based partial equilibrium problem. Merton (1973) writes:
Since movements from equilibrium to equilibrium through time involve both price and quantity adjustment,
a complete analysis would require a description of both the rate of return and change in asset value dynamics.
To do so would require a specification of firm behavior in determining the supply of shares, which in turn
would require knowledge of the real asset structure (i.e., technology; whether capital is “putty” or “clay”;
etc.). In particular, the current returns on firms with large amounts (relative to current cash flow) of nonshiftable capital with low rates of depreciation will tend to be strongly affected by shifts in capitalization rates
because, in the short turn, most of the adjustment to the new equilibrium will be done by prices (p. 871).
Since the present paper examines only investor behavior to derive the demands for assets and the relative yield requirements in equilibrium, only the rate of return dynamics will be examined explicitly. Hence,
certain variable taken as exogenous in the model, would be endogenous to a full-equilibrium system (p. 871).
Similarly, Breeden (1979) also defends his abstraction from the production side:
[It] is not necessary to explicitly examine firms’ production decisions and the supply of asset shares, provided
that the assumptions made are consistent with optimal behavior of firms in a general equilibrium model. To
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be consistent with general equilibrium, prices must be recognized to be endogenously determined through
the equilibrium of supply and demand (p. 269).
Although general equilibrium asset pricing models with production have appeared sporadically
(Brock 1982; Cox, Ingersoll, and Ross 1985), the consumption CAPM has firmly dominated modern
asset pricing. The standard MBA textbook by Bodie, Kane, and Marcus (2014, Investments) uses
the CAPM as the organizing framework. Standard Ph.D. textbooks, such as Huang and Litzenberger (1988), Ingersoll (1988), Duffie (2001), Cochrane (2005), Back (2010), and Danthine and
Donaldson (2015), are all based on the consumption CAPM. In particular, Cochrane writes:
Asset pricing theory all stems from one simple concept, presented in the first page of the first chapter of
this book: price equals expected discounted payoff. The rest is elaboration, special cases, and a closet full
of tricks that make the central equations useful for one or another application (p. xiii).
All asset pricing models amount to alternative ways of connecting the stochastic discount factor to data
(p. 7, original emphasis).
In hindsight, it seems clear that Cochrane meant that consumption-based asset pricing is all about
the stochastic discount factor. Alas, his writings are sometimes misinterpreted as saying that only
the consumption CAPM is asset pricing, but the investment CAPM is not.
5.1.3
Rebirth
The consumption CAPM is an empirical failure. Starting from Ball and Brown (1968), a massive
literature on asset pricing anomalies has documented pervasive evidence that rejects the consumption CAPM, casting doubt on its premise of efficient markets. Fama (1998), Barberis and Thaler
(2003), and Richardson, Tuna, and Wysocki (2010) provide extensive reviews of the anomalies literature. A large behavioral finance literature has relied on investor sentiment to explain anomalies in
inefficient markets. The field today is divided between traditionalists, who are disturbed by anomalies, but unwilling to give up on microfoundations, and behavioralists, who view anomalies as a
resounding rejection of efficient markets in particular and neoclassical economics in general. In the
financial services industry, the anomalies literature has formed the scientific foundation for quantitative investment management. Practitioners are overwhelmingly sympathetic to the behavioral
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view, while dismissing traditionalists as theoretical purists who have lost touch with the real world.
I view anomalies not as an indictment of efficient markets, but as indications of the empirical
difficulties of the consumption CAPM. More important, I view anomalies as suggesting that the
consumption paradigm is conceptually incomplete. The genius of Breeden, Lucas, and Rubinstein
is to reduce a messy multidimensional general equilibrium asset pricing problem into a tractable
consumption-based partial equilibrium problem. Alas, this reduction, by its very construction, is
achieved by abstracting from the production side of general equilibrium. Unfortunately, anomalies
are primarily empirical relations between firm characteristics and expected returns. Because firms
are ignored, perhaps not surprisingly, the consumption CAPM fails to explain these anomalies.
Inspired by Cochrane (1991), I recognize in Zhang (2005a) that the neoclassical q-theory of
investment allows a different reduction of the general equilibrium asset pricing problem, a reduction that is symmetric and neatly complementary to the reduction in the form of the consumption
CAPM achieved by the permanent income theory of consumption. The first principle of investment
for individual firms implies that the weighted average cost of capital equals the benefits of one extra
unit of capital divided by its costs (equations 26 and 27). Intuitively, firms keep investing until the
costs of doing so rise to the level of the discounted investment benefits.
The big news for asset pricing is that the investment CAPM expresses the stock return purely
as a function of firm characteristics. The investment CAPM does not use any information on consumption or utility functions, thereby achieving a reduction symmetric to the consumption CAPM,
which does not use any information on production or investment technologies. This separation is
in the spirit of the Fisher Separation Theorem. The investment CAPM implies that the evidence
that firm characteristics forecast stock returns does not necessarily mean mispricing, as asserted in
behavioral finance. No expectational errors are ever assumed in deriving the investment CAPM.
5.2
Complementarity with the Consumption CAPM
The relation between the investment CAPM and the consumption CAPM is complementary.
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5.2.1
Marshall’s Scissors
As an asset pricing theory, the investment CAPM is as fundamental as the consumption CAPM. In
the Arrow-Debreu economy, the first principle of consumption and the first principle of investment
are two key optimality conditions. Consumption betas, expected returns, and firm characteristics
are all endogenous variables, which are in turn determined by a system of simultaneous equations
in general equilibrium. No causality runs across these endogenous variables. Neither consumption
betas nor firm characteristics are causal forces driving expected returns. As such, the investment
CAPM is no more and no less “causal” than the consumption CAPM (Lin and Zhang 2013).
Cochrane (2005) also emphasizes no causality from consumption betas to expected returns:
It is enormously tempting to slide into an interpretation that E(mx) determines p. We routinely think of
betas and factor risk prices—components of E(mx)—as determining expected returns. For example, we
routinely say things like ‘the expected return of a stock increased because the firm took on riskier projects,
thereby increasing its beta.’ But the whole consumption process, discount factor, and factor risk premia
change when the production technology changes (p. 41, original emphasis).
History tends to repeat itself. As noted, asset pricing theory is just the standard value theory
extended to uncertainty and over time. During the last quarter of the 19th century, economists debated the relative importance of demand and supply in value theory (Landreth and Colander 2002).
While David Ricardo and John Stuart Mill had stressed costs of production, William Jevons, Carl
Menger, and Léon Walras insisted that value should depend only on marginal utility. Alfred Marshall (1890) used the following analogy to resolve this controversy in his Principles of Economics:
We might as reasonably dispute whether it is the upper or under blade of a pair of scissors that cuts a piece
of paper, as whether value is governed by utility or costs of production. It is true that when one blade is
held still, and the cutting is affected by moving the other, we may say with careless brevity that the cutting
is done by the second; but the statement is not strictly accurate, and is to be excused only so long as it
claims to be merely a popular and not a strictly scientific account of what happens (Marshall 1890 [1961,
9th edition, p. 348]).
Even Fisher (1930) had to address an earlier criticism that he had neglected production:
Years after The Rate of Interest was published, I suggested the more popular term “impatience” in place
of “agio” or “time preference.” This catchword has been widely adopted, and, to my surprise, has led to a
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widespread but false impression that I had overlooked or neglected the productivity or investment opportunity side entirely. It also led many to think that, by using the new word impatience, I meant to claim a
new idea. Thus I found myself credited with being the author of “the impatience theory” which I am not,
and not credited with being the author of those parts lacking any catchword. It was this misunderstanding
which led me, after much search, to adopt the catchword “investment opportunity” as a substitute for the
inadequate term “productivity” which had come into such general use (p. viii).
5.2.2
The Aggregation Critique
Given that the investment CAPM and the consumption CAPM are parallel but complementary in
general equilibrium, why does the former seems to perform much better than the latter in the data?
The Roll critique and the Hansen-Richard critique are well known. Roll (1977) argues that the
wealth portfolio is not observable, making the CAPM tests infeasible. Hansen and Richard (1987)
argue that the conditioning information of investors is not observable, making the consumption
CAPM tests infeasible in a dynamic world. Both critiques seem less important for the investment
CAPM. Although the q-factor model includes the market factor, it is only used to anchor the time
series averages of returns, while the task of accounting for their cross-sectional variation is left to
the investment and ROE factors (Section 2). Also, the structural investment CAPM tests do not
need to specify the conditioning information of managers (Section 3).
The most damaging critique against the consumption CAPM, I believe, is aggregation. Most
consumption CAPM studies sidestep the aggregation problem by examining the optimizing behavior
of a representative consumer. However, the Sonnenschein-Mantel-Debreu theorem in general equilibrium theory states that the aggregate excess demand function is not restricted by the standard
rationality assumption on individual demands (Sonnenschein 1973; Debreu 1974; Mantel 1974).
Individuals can be fully rational, but the aggregate behavior might appear entirely different.
Kirman (1992) raises four objections to the representative consumer. First, individual maximization does not imply collective rationality, and collective maximization does not imply individual
rationality. Second, the response of the representative consumer to a parameter change might not
be the same as the aggregate response of individuals. Third, it is possible for the representative to
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exhibit preference orderings that are opposite to all the individuals’. Finally, the aggregate behavior of rational individuals might exhibit complicated dynamics, and imposing these dynamics on
one individual can lead to unnatural characteristics of the individual. In all, Kirman suggests that:
[It] is clear that the “representative” agent deserves a decent burial, as an approach to economic analysis
that is not only primitive, but fundamentally erroneous (p. 119).
In essence, aggregation in the representative consumer assumes that all 319 million Americans are
identical in preferences. In case it is still not disturbing enough, try 1.4 billion Chinese.
On the critical importance of aggregation, Deaton (1992) also writes:
I have come to believe that [representative agent models] are of limited value, and that what we have learned
from them is more methodological than substantive. Representative agents have two failings: they know too
much, and they live too long. An aggregate of individuals with finite lives, and with limited and heterogeneous information is not likely to behave like the single individual of the textbook. We are likely to learn
more about aggregate consumption by looking at microeconomic behavior, and by thinking seriously about
aggregation from the bottom up (p. ix).
The main puzzle is not why these representative agent models do not account for the evidence, but why
anyone ever thought that they might, given the absurdity of the aggregation assumptions that they require
(p. 70).
The aggregation problem cuts both ways, however. On the one hand, without confronting this
problem, imposing that asset pricing is all about the stochastic discount factor is self-defeating,
if not absurd.
On the other hand, the aggregation difficulty also means that the failure of
the consumption CAPM might have nothing to say about individual rationality. As noted, the
Sonnenschein-Mantel-Debreu theorem says that individual rationality imposes no restrictions on
aggregate consumption. A natural corollary is that the consumption CAPM is not testable!
The investment CAPM is relatively immune to the aggregation critique. The consumption
CAPM is macroeconomic in nature, and is derived from the first principle of the representative consumer, which is the aggregate stand-in for all the individuals in the economy. In contrast, the investment CAPM is microeconomic in nature, and is derived from the first principle of an individual firm.
The aggregation critique also helps dispel a popular myth that the consumption CAPM is all
encompassing, a notion that seems to originate from Mehra and Prescott (1985):
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With our structure, the process on the endowment is exogenous and there is neither capital accumulation
nor production. Modifying the technology to admit these opportunities cannot overturn our conclusion,
because expanding the set of technologies in this way does not increase the set of joint equilibrium processes
on consumption and asset prices (p. 158).
The consumption CAPM is as Marshallian as the investment CAPM, and neither is Walrasian.
The difficulty with the Mehra-Prescott model might have a lot to do with its assumption of a representative consumer. Also, the investment CAPM fits the equity premium well (Cochrane 1991).
5.2.3
Beta Measurement Errors
In addition to the aggregation critique, another relatively underappreciated problem is beta measurement errors. Fama (1991) describes the difficulties of estimating consumption betas as follows:
Consumption is measured with error, and consumption flows from durables are difficult to impute. The
model calls for instantaneous consumption, but the data are monthly, quarterly, and annual aggregates.
Finally, ... the elegance of the consumption model (all incentives to hedge uncertainty about consumption
and investment opportunities are summarized in consumption β’s) likely means that consumption β’s are
difficult to estimate because they vary through time (p. 1599).
These difficulties probably explain why the consumption CAPM often can underperform the CAPM
in empirical tests. However, the CAPM is also subject to severe measurement errors issue.
Miller and Scholes (1972) simulate random returns from the CAPM, and find that test results
on simulated data are consistent with those from the real data:
We have shown that much of the seeming conflict between these [empirical] results and the almost exactly
contrary predictions of the underlying economic theory may simply be artifacts of the testing procedures
used. The variable that measures the systematic covariance risk of a particular share is obtained from a
first-pass regression of the individual company returns on a market index. Hence it can be regarded at
best as an approximation to the perceived systematic risk, subject to the margin of error inevitable in any
sampling process, if to nothing else. The presence of such errors of approximation will inevitably weaken the
apparent association between mean returns and measured systematic risk in the critical second-pass tests.
More recently, Gomes, Kogan, and Zhang (2003) and Carlson, Fisher, and Giammarino (2004)
show how size and book-to-market can dominate empirically estimated betas in cross-sectional regressions on data simulated from economies with the conditional CAPM structure. Li, Livdan, and
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Zhang (2009) report similar results on real investment and equity issues in cross-sectional regressions. Lin and Zhang (2013) show how characteristics can easily dominate covariances in predicting
returns in Daniel and Titman’s (1997) two-way sorts on simulated data from a conditional CAPM
economy. Bai, Hou, Kung, and Zhang (2015) show how the CAPM can fail to explain the value
premium in an economy with rare disasters, and how measurement errors in estimated betas can
completely flatten the risk-return relation in an economy without any mispricing. These simulation
results do not mean that the CAPM is alive and well, but suggest that beta measurement errors
might have a lot to do with the empirical failure of the CAPM. In contrast, implemented on firm
characteristics, the investment CAPM is less plagued by beta measurement errors.
Some empiricists tend to downplay theoretical results based on simulated data. It is true that
simulated data are not real data. However, if the estimated betas from standard empirical procedures differ drastically from the true betas in a controlled laboratory, what confidence should one
have in beta estimates in the real world that is vastly more complex than our laboratories?
5.3
An Efficient Markets Counterrevolution to Behavioral Finance
Anomalies in capital markets obey standard economic principles.
5.3.1
A Dark Age of Finance
It is standard practice to equate anomalies to mispricing in behavioral finance.
Research in experimental psychology suggests that, in violation of Bayes’ rule, most people tend to “overreact” to unexpected and dramatic news events. This study of market efficiency investigates whether such
behavior affects stock prices. The empirical evidence, based on CRSP monthly return data, is consistent
with the overreaction hypothesis. Substantial weak form market inefficiencies are discovered (De Bondt and
Thaler 1985, p. 793).
Evidence presented here is consistent with a failure of stock prices to reflect fully the implications of
current earnings for future earnings.... Even more surprisingly, the signs and magnitudes of the three-day
reactions are related to the autocorrelation structure of earnings, as if stock prices fail to reflect the extent
to which each firm’s earnings series differs from a seasonal random walk (Bernard and Thomas 1990, p. 305).
The pattern that emerges is that the underperformance is concentrated among relatively young growth
companies, especially those going public in the high-volume years of the 1980s. While this pattern does not
rule out bad luck being the cause of the underperformance, it is consistent with a scenario of firms going
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public when investors are irrationally over optimistic about the future potential of certain industries which,
following Shiller (1990), I will refer to as the “fad” explanation (Ritter 1991, p. 4).
[It] is possible that the market underreacts to information about their long-term prospects of firms but
overreacts to information about their long-term prospects. This is plausible given that the nature of the
information available about a firm’s short-term prospects, such as earnings forecasts, is different from the
nature of the more ambiguous information that is used by investors to assess a firm’s longer-term prospects
(Jegadeesh and Titman 1993, p. 90).
Investor expectations of future growth appear to have been excessively tied to past growth despite the fact
that future growth rates are highly mean reverting. In particular, investors were systematically disappointed
(Lakonishok, Shleifer, and Vishny 1994, p. 1575).
The results indicate that earnings performance attributable to the accrual component of earnings exhibits lower persistence than earnings performance attributable to the cash flow component of earnings.
The results also indicate that stock prices act as if investors “fixate” on earnings, failing to distinguish fully
between the different properties of the accrual and cash flow components of earnings. Consequently, firms
with relatively high (low) levels of accruals experience negative (positive) future abnormal stock returns that
are concentrated around future earnings announcements (Sloan 1996, p. 290).
[Managers] have an incentive to put the best possible spin on both their new opportunities as well their
overall business when their investment expenditures are especially high because of their need to raise capital
as well as to justify their expenditures. If investors fail to appreciate managements’ incentives to oversell their
firms in these situations, stock returns subsequent to an increase in investment expenditures are likely to be
negative. This effect is likely to be especially important for managers who are empire builders, and invest for
their own benefits rather than the benefits of the firm’s shareholders (Titman, Wei, and Xie 2004, p. 678).
By the early 2000s, Barberis and Thaler (2003) describe the state of affairs as follows:
While the behavior of the aggregate stock market is not easy to understand from the rational point of view,
promising rational models have nonetheless been developed and can be tested against behavioral alternatives.
Empirical studies of the behavior of individual stocks have unearthed a set of facts which is altogether more
frustrating for the rational paradigm. Many of these facts are about the cross-section of average returns:
they document that one group of stocks earn higher average returns than another. These facts have come
to be known as “anomalies” because they cannot be explained by the simpliest and most intuitive model
of risk and return in the financial economist’s toolkit, the Capital Asset Pricing Model, or CAPM (p. 1087,
original emphasis).
It is my professional view that the argument for inefficient markets based on the failure of the
consumption CAPM in explaining asset pricing anomalies represents, to paraphrase Shiller (1984,
p. 459), “one of the most remarkable errors in the history of economic thought.”
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5.3.2
Do Psychological Biases Explain Asset Pricing Anomalies?
Shleifer (2000) and Barberis and Thaler (2003) describe behavioral finance as consisting of two
building blocks, psychological biases and limits to arbitrage. While psychological biases give rise
to anomalies, limits to arbitrage prevent arbitrageurs from eliminating the mispricing, at least in
the short run. However, recent empirical work has cast doubt on both building blocks.
5.3.2.1
Are Investors More Biased Than Managers?
Many psychological biases have been
proposed. Two prominent studies are Barberis, Shleifer, and Vishny (1998) and Daniel, Hirshleifer,
and Subrahmanyam (1998). In Barberis et al., conservatism means that individuals’ beliefs are
excessively slow to change, and representative heuristics means that after a consistent history of
earnings news over several years, investors erroneously believe that the past history is representative of future earnings news. Conservatism is used to explain earnings momentum, as investors
disregard the full information content of earnings announcements, and assign disproportionately
high weights to their priors. Representative heuristics is used to explain the value premium. After
a consistent history of, say, positive earnings news, investors become overly optimistic about future
earnings news, and overreact, pushing the stock price to unduly high levels.
In Daniel, Hirshleifer, and Subrahmanyam (1998), overconfidence means that investors overestimate the precision of, and tend to overreact to, their private information signals. Self-attribution
means that investors too strongly attribute public signals that confirm the validity of their actions
to high ability, and those that disconfirm their actions to bad luck or even sabotage. As a result,
starting with unbiased beliefs about their ability, these investors tend to view public signals as on
average confirming their prior private signals, at least in the short run. Overconfidence and selfattribution combine to produce continuous overreaction in the short run, giving rise to momentum,
and the correction of the mispricing in the long run, giving rise to the value premium.
While intuitive, recent empirical work has cast doubt on psychological biases as driving forces
of anomalies. In particular, the body of evidence documented in Liu, Whited, and Zhang (2009),
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Liu and Zhang (2014), and Hou, Xue, and Zhang (2015a, b), some of which has been independently
verified by Fama and French (2015a, b), suggests that managers of individual firms seem to do a
good job in aligning investment policies with their costs of capital, and this assignment drives many
cross-sectional patterns that are anomalous in the consumption CAPM.
If investors are psychologically biased, why would the managers of publicly traded companies be
less biased? After all, investors and managers are from the same population. The recent intermediary asset pricing literature suggests that marginal investors tend to be sophisticated institutional
investors in the financial services industry. Why would the managers of financial firms be more
biased than the managers of nonfinancial firms? Outside institutional investors, why would individual investors exhibit a variety of biases while at home making portfolio choice decisions, but
switch them off readily while at work making real investment decisions?
A more plausible explanation for the simultaneous failure of the consumption CAPM and the
relative success of the investment CAPM, I believe, is aggregation (Section 5.2.2). The investment
CAPM goes straight to the individual behavior, but the consumption CAPM works with the construct of a representative investor, which, according to the Sonnenschein-Mantel-Debreu theorem,
does not really exist, and even if it does, is almost entirely detached from individual rationality.
5.3.2.2
Are U.S. Investors More Biased Than Chinese Investors?
As noted, Titman,
Wei, and Xie (2013) and Watanabe, Xu, Yao, and Yu (2013) document that the investment effect
is stronger in developed countries than in developing countries. In particular, Titman et al.’s Table
3 reports that the equal-weighted size-adjusted return of the high-minus-low investment quintile
is −0.35% per month (t = −6.04) averaged across developed countries, in contrast to −0.17%
(t = −1.06) averaged across developing countries.
Similar evidence has been reported for price momentum. Griffin, Ji, and Martin (2003, Table
1) report that momentum profits are on average 0.51% per month (t = 3.09) in developed markets,
in contrast to 0.27% (t = 1.21) in emerging markets. Table 11, adapted from Chui, Titman, and
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Table 11 : Momentum Profits by Country
Source: Chui, Titman, and Wei (2010, Table III). The definition of developed markets and emerging markets
is from Hou, Karolyi, and Kho (2011, Table 1). WML is momentum profits in percent per month, and t is
the corresponding t-statistics. See Chui et al. for the construction of momentum profits and data source.
Panel A: Developed markets
Australia
Austria
Belgium
Canada
Denmark
Finland
France
Germany
Hong Kong
Ireland
Italy
Japan
Netherlands
New Zealand
Norway
Singapore
Spain
Sweden
Switzerland
United Kingdom
United States
Average
Panel B: Emerging markets
WML
t
1.08
0.63
0.89
1.35
0.96
0.98
0.94
0.99
0.77
0.88
0.90
−0.04
0.83
1.58
1.05
0.14
0.63
0.71
0.82
1.13
0.79
0.86
4.76
2.70
5.50
6.29
4.29
2.62
4.68
4.41
3.18
3.06
4.47
−0.18
4.40
5.01
3.77
0.47
2.24
2.27
4.39
7.08
3.44
Argentina
Bangladesh
Brazil
Chile
China
Greece
India
Indonesia
Israel
Korea
Malaysia
Mexico
Pakistan
Philippines
Poland
Portugal
South Africa
Taiwan
Thailand
Turkey
Average
WML
t
0.08
1.68
0.46
0.99
0.26
0.59
1.14
0.14
0.32
−0.34
0.10
0.69
0.46
0.37
1.76
0.31
0.94
−0.20
0.48
−0.41
0.12
2.75
0.96
3.60
0.92
1.49
2.91
0.30
1.19
−0.81
0.26
2.00
1.05
0.68
3.33
0.93
3.29
−0.48
1.10
−0.96
0.49
Wei (2010), shows that momentum profits are on average 0.86% in developed markets, in contrast
to 0.49% in emerging markets. Out of 21 developed markets, 19 show significant momentum at the
5% level, whereas only six out of 20 emerging markets show significant momentum. In particular,
momentum profits are 0.79% (t = 3.44) in the U.S., but 0.26% (t = 0.92) in China. Momentum is a
stronghold of behavioral finance. Unless investors in developed markets such as U.S. can be argued
as more psychologically biased than investors in emerging markets such as China, the evidence in
Table 11 seems an emphatic rejection of behavioral explanations.
The evidence is more consistent with the investment CAPM. The crux is that whereas behavioral
finance rides on dysfunctional, inefficient markets, the investment CAPM relies on well functioning,
efficient markets for its mechanisms to work. As such, behavioral finance predicts that anomalies
should be stronger in emerging markets, in which investors are less sophisticated, but weaker in de-
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veloped markets, in which investors are more sophisticated. In contrast, the investment CAPM predicts that anomalies should be stronger in developed markets, such as the U.S., in which the incentives of managers and shareholders are more aligned, and managers are more likely to maximize the
market equity in making investment decisions. However, anomalies should be weaker in emerging
markets, such as China, in which managers are often appointed by the government, and more likely
to pursue social objectives such as maximizing employment, rather than the market value of equity.
5.3.2.3
Limits to Limits to Arbitrage? As noted, Shleifer and Vishny (1997) argue that
trading frictions make arbitrage activities costly and incomplete, allowing mispricing to persist. Because limits to arbitrage are more severe in emerging markets than in developed markets, anomalies
should be stronger in emerging markets, contradicting Table 11 and the related evidence.
5.3.2.4
Measurement Ahead of Theory?
Koopmans (1947) emphasizes an artful combina-
tion of theory, econometrics, and measurement in scientific work:
[In] research in economic dynamics the Kepler stage and the Newton stage of inquiry need to be more intimately combined and to be pursued simultaneously. Fuller utilization of the concepts and hypotheses of
economic theory ... as a part of the processes of observation and measurement promises to be a shorter road,
perhaps even the only possible road, to the understanding of cyclical fluctuations (p. 162, original emphasis).
The historical rise of behavioral finance has been fueled by the discoveries of anomalies. As
opportunities of new anomalies inevitably deplete over time, behavioral finance seems to be in a
stage of stagnation. Despite enormous empirical contributions in the anomalies literature, what is
missing, it seems, is a coherent behavioral asset pricing framework. Even if psychological biases
can be reliably documented at the individual level, it is not clear to what extent the biases apply to
the aggregate level. In addition to the consumption CAPM, the aggregation critique also applies to
prominent behavioral models such as Barberis, Shleifer, and Vishny (1998), Daniel, Hirshleifer, and
Subrahmanyam (1998), and Barberis, Huang, and Santos (2001), all of which work with a representative investor. Also, for the most part, behavioral models have not been subject to the rigorous
structural econometric tests imposed on the consumption CAPM and the investment CAPM. While
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these tests reveal weaknesses such as unreasonable parameter estimates and parameter instability,
not going through these tests might have obscured similar problems in behavioral models.
5.3.3
The Joint-Hypothesis Problem Squared
Fama (1970, 1991, 1998) and Fama and French (1993, 1996) defend efficient markets. With the
investment CAPM, I drastically modify, but ultimately strengthen their arguments. Despite a few
technical quarrels, the opposing sides of Factors War are both led by ardent defenders of efficient
markets. In addition to lacking theoretical progress, behavioral finance is also conspicuously missing
from the enterprise of constructing the next workhorse model of empirical finance.
Fama’s (1970) definition of efficient markets needs no modification almost half a century later:
The assumptions that the conditions of market equilibrium can be stated in terms of expected returns and
that equilibrium expected returns are formed on the basis of (and thus “fully reflect”) the information set Φt
have a major empirical implication—they rule out the possibility of trading systems based only on information
in Φt that have expected profits or returns in excess of equilibrium expected profits or returns. Thus, let
xjt+1 = pjt+1 − E[pjt+1 |Φt ].
(45)
E[xjt+1 |Φt ] = 0
(46)
Then
which, by definition, says that the sequence {xjt } is a “fair game” with respect to the information sequence
{Φt }. Or, equivalently, let
zjt+1 = rjt+1 − E[rjt+1 |Φt ],
(47)
then
E[zjt+1 |Φt ] = 0,
(48)
so that the sequence {zjt } is also a “fair game” with respect to the information sequence {Φt } (p. 384–385,
original emphasis).
Twenty years after Fama (1970), with the anomalies evidence all the rage, Fama (1991) brings
the joint-hypothesis problem to the front and center of asset pricing research:
Ambiguity about information and trading costs is not, however, the main obstacle to inferences about market
efficiency. The joint-hypothesis problem is more serious. Thus, market efficiency per se is not testable. It
must be tested jointly with some model of equilibrium, an asset-pricing model. This point ... says that we can
only test whether information is properly reflected in prices in the context of a pricing model that defines the
meaning of “properly.” As a result, when we find anomalous evidence on the behavior of returns, the way it
should be split between market inefficiency or a bad model of market equilibrium is ambiguous (p. 1575–1576).
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Fama (1998) reiterates the joint-hypothesis problem, and shows that some CAPM anomalies disappear in the Fama-French three-factor model. However, as documented extensively in, for example,
Hou, Xue, and Zhang (2015a, b), the three-factor model leaves many anomalies unexplained. The
q-factor model is a direct attempt at alleviating the joint-hypothesis problem. Empirically, it represents a major step forward from the three-factor model (Section 2). Perhaps more important, theoretically, the q-factor model brings neoclassical economics to bear on the largely empirical literature.
5.3.3.1
A Tribute to Fama and French (1993)
The Fama-French three-factor model has
served its historical purpose, rather admirably. After the CAPM is firmly rejected and abandoned
in Fama and French (1992), the three-factor model fills the vacuum left by the CAPM as the
workhorse model, and takes on the burden of defending efficient markets. The mainstream asset
pricing theorists are of little help, as the consumption CAPM often underperforms the CAPM.
Alas, the largely empirical nature of the three-factor model leaves it vulnerable to the data mining critique. The conjecture of the book-to-market factor as a relative distress factor in Fama and
French (1996) is also rejected by the distress anomaly, which shows that more distressed firms earn
lower average returns than less distressed firms (Dichev 1998). More generally, the interpretation
of characteristics-based factors as sources of risk in the intertemporal CAPM or arbitrage pricing
theory has fallen on deaf ears. Both theories are silent about the identities of state variables, and
neither make the conceptual leap of linking expected returns to firm characteristics.
5.3.3.2
Interpreting Factors: The Investment CAPM Perspective
Fama and French
(1996) interpret their three-factor model as a risk factor model:
[If] assets are priced rationally, variables that are related to average returns, such as size and book-to-market
equity, must proxy for sensitivity to common (shared and thus undiversifiable) risk factors in returns. The
time-series regressions give direct evidence on this issue. In particular, the slopes and R2 values show whether
mimicking portfolios for risk factors related to size and [book-to-market] capture shared variation in stock
and bond returns not explained by other factors (p. 4–5).
[The] empirical successes of [the three-factor model] suggest that it is an equilibrium pricing model, a threefactor version of Merton’s (1973) intertemporal CAPM (ICAPM) or Ross’s (1976) arbitrage pricing theory
99
(APT). In this view, SMB and HML mimic combinations of two underlying risk factors or state variables of
special hedging concern to investors (p. 57).
This interpretation has been controversial. Daniel and Titman (1997) show that expected returns do not correlate positively with factor loadings after controlling for characteristics:
In equilibrium asset pricing models the covariance structure of returns determines expected returns. Yet we
find that variables that reliably predict the future covariance structure do not predict future returns. Our
results indicate that high book-to-market stocks and stocks with low capitalizations have high average returns whether or not they have the return patterns (i.e., covariances) of other small and high book-to-market
stocks (p. 4).
This evidence has often been interpreted as suggesting mispricing in behavioral finance:
One general feature of the rational approach is that it is loadings or betas, and not firm characteristics, that
determine average returns.... [Daniel and Titman’s (1997)] results appear quite damaging to the rational
approach (Barberis and Thaler 2003, p. 1091).
[It] is important to empirically distinguish (1) the covariance between stock returns and a given attribute
from (2) the returns attributable to the characteristic. Finding evidence in support of (1) is consistent with
a risk based explanation for the return relation, whereas finding (2) would suggest mispricing (Richardson,
Tuna, and Wysocki 2010, p. 430).
From the investment CAPM perspective, I interpret characteristics-based factor models as linear approximations of the investment CAPM in equation (7). As noted, although the equation
expresses the expected stock return as a nonlinear function of characteristics, the q-factor model
is robust to the specific functional forms in equation (7). Also, stock returns data are available at
high frequencies, and are less subject to measurement errors than characteristics.
As a more fundamental departure from Fama and French (1993, 1996), Hou, Xue, and Zhang
(2015a, b) interpret the q-factor model as just a parsimonious description of the cross section, not
necessarily a risk factor model, and the q-factor loadings as just regression slopes, not necessarily
measures of some inexplicable sources of risk. Fama and French go for the risk interpretation,
probably thinking that it is the only way to defend efficient markets, but this step is not necessary
either. The investment CAPM predicts all kinds of relations between characteristics and expected
returns, without over- or underreaction. The evidence of characteristics dominating covariances
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in Daniel and Titman (1997) can be largely attributed to measurement errors in covariances (Lin
and Zhang 2013). And the failure of the consumption CAPM can be largely attributed to the
aggregation problem and measurement errors in consumption data.
Time series and cross-sectional regressions are just two different ways of summarizing empirical
correlations, and are largely equivalent in economic terms. If a characteristic shows up significant
in cross-sectional regressions, its factor mimicking portfolio is likely to show “explanatory” power
in factor regressions. If a factor earns a significant average return in the time series, the slope of
its underlying characteristic is likely to be significant in cross-sectional regressions. Factor loadings
are no more primitive than characteristics, and vice versa, in “explaining” expected returns.
5.3.3.3
Beyond the “Risk Doctrine”
More generally, the investment CAPM broadens effi-
cient markets beyond the “risk doctrine.” An early statement of the doctrine is in Fama (1970):
Most of the available work is based only on the assumption that the conditions of market equilibrium can
(somehow) be stated in terms of expected returns. In general terms, like the two parameter model such theories would posit that conditional on some relevant information set, the equilibrium expected return on a security is a function of its “risk.” And different theories would differ primarily in how “risk” is defined (p. 384).
The colossal anomalies literature puts efficient markets in peril because of the failure of traditional
risk models. The investment CAPM shows that the peril is more apparent than real.
The risk doctrine asks what risks “explain” asset pricing anomalies. This question has been at
the center of modern asset pricing research. The investment CAPM questions the question (Lin and
Zhang 2013). As shown in Section 1, the consumption CAPM and the investment CAPM are the
two sides of the same coin in general equilibrium, delivering identical expected returns. However,
the risk doctrine only describes the consumption CAPM, which predicts that consumption risks
are sufficient statistics for expected returns, and after risks are controlled for, characteristics should
not matter. It is clear that the risk doctrine is conceptually incomplete. It does not apply to the
investment CAPM, which predicts that characteristics are sufficient statistics for expected returns,
and after characteristics are controlled for, risks should not matter.
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In general equilibrium, risks, expected returns, and characteristics are all endogenously determined simultaneously. Neither risks nor characteristics “determine” expected returns. The
investment CAPM is as fundamental as the consumption CAPM in economy theory. The concept
that risks determine expected returns is a relic and an illusion from the CAPM, because its inventors happen to take stock returns and their covariances as exogenous. Risks are no more exogenous
than investment and profitability in theory, and should not be put on a pedestal in applied work.
The risk doctrine is also impractical. Without addressing the aggregation critique, the consumption CAPM is not testable. Perhaps, it is time to heed the advice from Kuhn (1962):
[The] really pressing problems, e.g., a cure for cancer and the design of a lasting peace, are often not puzzles
at all, largely because they may not have any solution. Consider the jigsaw puzzle whose pieces are selected at
random from each of two different puzzle boxes. Since that problem is likely to defy (though it might not) even
the most ingenious of men, it cannot serve as a test of skill. In solution in any usual sense, it is not a puzzle
at all. Though intrinsic value is no criterion for a puzzle, the assured existence of a solution is (p. 36–37).
6
Challenges for Future Work
In this article, for the first time, I have attempted to articulate the broad, and hopefully internally
consistent, perspective of the investment CAPM for asset pricing. This perspective is quite a bit
different from the perspective of the consumption CAPM as well as that of behavioral finance.
6.1
Where Do We Stand?
I have reviewed three workhorse models in this rapidly expanding literature, the q-factor model
for estimating expected stock returns, the multiperiod investment CAPM for structural estimation
and tests, and quantitative investment models. The q-factor model of Hou, Xue, and Zhang (2015a,
b) achieves the important goal of dimension reduction in the colossal anomalies literature, in the
spirit of what Fama and French (1996) did twenty years ago. The structural investment CAPM of
Liu, Whited, and Zhang (2009) takes the first principle of real investment to cross-sectional returns
data, in the spirit of what Hansen and Singleton (1982) did with the first principle of consumption.
Finally, the quantitative investment model of Zhang (2005) provides a detailed neoclassical
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description of cross-sectional returns in connection with the real economy, in the spirit of what
Kydland and Prescott (1982) and Long and Plossor (1983) did for modern business cycle research.
I have also attempted to put the investment CAPM into the historical perspective. I have traced
the origin of the investment CAPM to at least Fisher (1930), who provided a symmetric treatment
of the impatience and investment opportunity theories of the equilibrium interest rate. Despite the
important writings in Hirshleifer (1958, 1970), Modigliani and Miller (1958), and Cochrane (1991),
modern asset pricing research is thoroughly dominated by the consumption CAPM. A long list
of foundational contributions, including Markowitz (1952), Sharpe (1964), Lintner (1965), Merton
(1973), Rubinstein (1976), Lucas (1978), and Breeden (1979), has put investors at the center of
inquiry, while abstracting from managers altogether. This abstraction is unjustifiable, ultimately,
because asset prices are equilibrated by both demand and supply. Fixing the supply of assets in
one’s model to focus on the demand does not mean that the supply is not important in real life. On
the contrary, the abstraction is disastrous, empirically, because the aggregation critique means that
the consumption CAPM is not testable. The investment CAPM, which is derived from the first
principle of investment for individual firms, is largely immune to the aggregation problem. In the
data, anomalies in the consumption CAPM turn out to be regularities in the investment CAPM.
6.2
Where Are We Going?
The investment CAPM literature is young, with many exciting new directions for future research.
Are the q-factor premiums reliable? While the ubiquitous data mining concern is hard to address
completely, an effective antidote is to study global financial data. Do the investment and ROE factor
premiums exist in countries outside the U.S.? Does the q-factor model absorb the explanatory power
of the Carhart (1997) four-factor model in global data as in the U.S. data? Fama and French (2015c)
provide a recent attempt in this direction, albeit with a noisy version of the q-factor model.
The investment CAPM has provided at least interpretations, if not economic explanations,
to most anomalies that have been attributed to behavioral finance. It has become important
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to disentangle the investment-based from behavioral explanations. The key identification is that
behavioral finance relies on dysfunctional, inefficient markets for its mechanisms to work, but the
investment CAPM relies on well functioning, efficient markets. As noted, in international data, both
the investment and momentum effects are stronger in developed markets than in emerging markets
(Chui, Titman, and Wei 2010; Titman, Wei, and Xie 2013). Do similar cross-country patterns also
hold for other important cross-sectional patterns, such as the value and profitability premiums?
Mutual fund performance is a vast and important literature (see Fischer and Wermers 2012
for an extensive review). Can the q-factor model help improve the measurement of mutual fund
performance? The Carhart (1997) four-factor model is the current workhorse model in this area.
In view of the superior performance of the q-factor model over the Carhart model documented
extensively in Hou, Xue, and Zhang (2015a, b), it seems natural to apply the q-factor model in this
area. Xue (2014) takes an important first step in this direction.
A significant portion of the return spreads from anomaly-based strategies occurs over a few
days surrounding earnings announcement dates (see Jegadeesh and Titman 1993 for momentum
and La Porta, Lakonishok, Shleifer, and Vishy 1997 for the value premium). Because the consumption CAPM has nothing to say about ex-post returns, this evidence has often been cited as
indicating investors’ expectational errors. However, the investment CAPM applies ex-ante as well
as ex-post, and consequently, does predict that a portion of anomalies should occur around earnings
announcements. Is there any way to disentangle these two competing explanations empirically?
Asness, Moskowitz, and Pedersen (2013) show that value and momentum are pervasive internationally across different asset classes, including individual stocks, country equity index futures,
government bonds, currencies, and commodity futures. To what extent can the q-factor model be
applied to these alternative assets? While it seems straightforward to apply the q-factor model
to international stocks, country equity indices, corporate bonds, and even real estate assets, other
asset classes such as currencies, government bonds, and commodities require additional theorizing.
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In the context of real estate investment trusts, Bond and Xue (2015) measure investment as property acquisition and profitability as property income and price appreciation, and show that the two
economic fundamentals have substantial predictive power for real estate returns.
While the investment CAPM is immune to the aggregation critique in theory, its current structural estimation, which is done at the portfolio level, is not. To be free of any aggregation bias,
the structural estimation should be done at the firm level. This step requires additional methodological innovations on handling unbalanced panel data. Can the firm-level estimation address the
parameter instability issue across different testing portfolios? Or equivalently, can the firm-level estimation account for value and momentum simultaneously in a single structural investment CAPM
specification? What about the investment and profitability premiums simultaneously?
How can the supply approach to valuation in Belo, Xue, and Zhang (2013) be applied to individual firms? Their valuation framework can be extended to incorporate additional productive inputs
such as labor and intangible assets. Their econometric framework can also be used to quantify the
valuation impact of corporate finance frictions, such as the agency conflicts between shareholders
and managers. More generally, what is the impact of time-varying and cross-sectionally varying
expected returns on corporate investment, financing, and payout policies, and vice versa? The burgeoning dynamic corporate finance literature has mostly worked with risk neutrality. Integrating
this promising literature with the investment CAPM literature seems fruitful.
The investment CAPM provides a neoclassical foundation for Graham and Dodd’s (1934) security analysis. Ou and Penman (1989) describe the traditional view as follows:
Firms’ (‘fundamental’) values are indicated by information in financial statements. Stock prices deviate
at times from these values and only slowly gravitate towards the fundamental values. Thus, analysis of
published financial statements can discover values that are not reflected in stock prices. Rather than taking
prices as value benchmarks, ‘intrinsic values’ discovered from financial statements serve as benchmarks with
which prices are compared to identify overpriced and underpriced stocks. Because deviant prices ultimately
gravitate to the fundamentals, investment strategies which produce ‘abnormal returns’ can be discovered by
the comparison of prices to these fundamental values (p. 296).
When discussing “abnormal returns,” the traditional view seems to use a cross-sectionally constant
105
discount rate as the null in efficient markets. The investment CAPM changes the big picture, by
showing cross-sectionally varying expected returns from the first principle of real investment. Much
remains to be done to integrate the investment CAPM with capital markets research in accounting.
Can the investment and profitability premiums be explained simultaneously within quantitative
investment models? What about the comovement behind the q-factors, as well as the sources of
the cross-sectional heterogeneity in investment and profitability? Hou, Xue, and Zhang (2015a, b)
document many correlations of investment and profitability with other firm characteristics behind
the explanatory power of the q-factor model. For instance, firms with high idiosyncratic volatility
are less profitable, but invest more than firms with low idiosyncratic volatility. More distressed
firms are less profitable than, but invest similarly as, less distressed firms. Can quantitative investment models explain the idiosyncratic volatility and distress anomalies, as well as the failure of
the CAPM but the success of the q-factor model in the data? Finally, the quantitative investment
literature has so far tackled the failure of the CAPM, but what about the failure of the consumption
CAPM and the failure of the Fama-French three-factor model? In all, the best is yet to come.
References
Abarbanell, Jeffery S., and Brian J. Bushee, 1997, Fundamental analysis, future earnings, and
stock prices, Journal of Accounting Research 35, 1–24.
Abarbanell, Jeffery S., and Brian J. Bushee, 1998, Abnormal returns to a fundamental analysis
strategy, The Accounting Review 73, 19–45.
Abel, Andrew B., 1983, Optimal investment under uncertainty, American Economic Review 73,
228–233.
Abel, Andrew B., and Janice C. Eberly, 1994, A unified model of investment under uncertainty,
American Economic Review 84, 1369–1384.
Aharoni, Gil, Bruce Grundy, and Qi Zeng, 2013, Stock returns and the Miller Modigliani valuation
formula: Revisiting the Fama French analysis, Journal of Financial Economics 110, 347–357.
Ai, Hengjie, and Dana Kiku, 2013, Growth to value: Option exercise and the cross section of
equity returns, Journal of Financial Economics 107, 325–349.
Anderson, Christopher W., and Luis Garcia-Feijóo, 2006, Empirical evidence on capital
investment, growth options, and security returns, Journal of Finance 61, 171–194.
106
Ang, Andrew, and Joseph Chen, 2007, CAPM over the long run: 1926–2001, Journal of Empirical
Finance 14, 1–40.
Ang, Andrew, Robert J. Hodrick, Yuhang Xing, and Xiaoyan Zhang, 2006, The cross-section of
volatility and expected returns, Journal of Finance 61, 259–299.
Arrow, Kenneth J., 1964, The role of securities in the optimal allocation of risk-bearing, Review
of Economic Studies 31, 91–96.
Asness, Clifford S., 1997, The interaction of value and momentum strategies, Financial Analysts
Journal, March/April, 29–36.
Asness, Clifford S., Tobias J. Moskowitz, and Lasse Heje Pedersen, 2013, Value and momentum
everywhere, Journal of Finance 68, 929–985.
Back, Kerry E., 2010, Asset Pricing and Portfolio Choice Theory, Oxford University Press.
Bai, Hang, Kewei Hou, Howard Kung, and Lu Zhang, 2015, The CAPM strikes back? An
investment model with disasters, working paper, The Ohio State University.
Bali, Turan G., and Nusret Cakici, 2008, Idiosyncratic volatility and the cross section of expected
returns, Journal of Financial and Quantitative Analysis 43, 29–58.
Ball, Ray, 1978, Anomalies in relations between securities’ yields and yield-surrogates, Journal of
Financial Economics 6, 103–126.
Ball, Ray, and Philip Brown, 1968, An empirical evaluation of accounting income numbers, Journal
of Accounting Research 6, 159-178.
Ball, Ray, Joseph Gerakos, Juhani T. Linnainmaa, and Valeri Nikolaev, 2015, Accruals, cash
flows, and operating profitability in the cross section of stock returns, forthcoming, Journal
of Financial Economics.
Balvers, Ronald J., and Dayong Huang, 2007, Productivity-based asset pricing: Theory and
evidence, Journal of Financial Economics 86, 405–445.
Bansal, Ravi, and Amir Yaron, 2004, Risks for the long run: A potential resolution of asset pricing
puzzles, Journal of Finance 59, 1481–1509.
Banz, Rolf W., 1981, The relationship between return and market value of common stocks, Journal
of Financial Economics 9, 3–18.
Barberis, Nicholas, Ming Huang, and Tano Santos, 2001, Prospect theory and asset prices,
Quarterly Journal of Economics 116, 1–53.
Barberis, Nicholas, Andrei Shleifer, and Robert Vishny, 1998, A model of investor sentiment,
Journal of Financial Economics 49, 307–343.
Barberis, Nicholas, and Richard Thaler, 2003, A survey of behavioral finance, in George M.
Constantinides, Milton Harris, and René M. Stulz eds., Handbook of the Economics of
Finance, p. 1053–1123, Elsevier, North Holland.
107
Barillas, Francisco, and Jay Shanken, 2015, Comparing asset pricing models, working paper,
Emory University.
Belo, Frederico, 2010, Production-based measures of risk for asset pricing, Journal of Monetary
Economics 57, 146–163.
Belo, Frederico, and Xiaoji Lin, 2011, The inventory growth spread, Review of Financial Studies
25, 278–313.
Belo, Frederico, Xiaoji Lin, and Santiago Bazdresch, 2014, Labor hiring, investment, and stock
return predictability in the cross section, Journal of Political Economy 122, 129–177.
Belo, Frederico, Chen Xue, and Lu Zhang, 2013, A supply approach to valuation, Review of
Financial Studies 26, 3029–3067.
Berk, Jonathan B., 1995, A critique of size-related anomalies, Review of Financial Studies 8,
275–286.
Berk, Jonathan B., and Peter M. DeMarzo, 2009, Corporate Finance, 2nd ed., Prentice Hall.
Berk, Jonathan B., Richard C. Green, and Vasant Naik, 1999, Optimal investment, growth options,
and security returns, Journal of Finance 54, 1153–1607.
Berk, Jonathan B., Richard C. Green, and Vasant Naik, 2004, Valuation and return dynamics of
new ventures, Review of Financial Studies 17, 1–35.
Bernard, Victor L., and Jacob K. Thomas, 1989, Post-earnings-announcement drift: Delayed price
response or risk premium? Journal of Accounting Research Supplement 27, 1-48.
Bernard, Victor L., and Jacob K. Thomas, 1990, Evidence that stock prices do not fully reflect the
implications of current earnings for future earnings, Journal of Accounting and Economics
13, 305–340.
Bernstein, Peter L., 1992, Capital Ideas: The Improbably Origins of Modern Wall Street, New
Jersey: John Wiley & Sons.
Black, Fischer, Michael C. Jensen, and Myron Scholes, 1972, The Capital Asset Pricing Model:
Some empirical tests, in Michael C. Jensen, ed., Studies in the Theory of Capital Markets
79–121, New York: Praeger.
Bloom, Nicholas, 2009, The impact of uncertainty shocks, Econometrica 77, 623–685.
Bodie, Zvi, Alex Kane, and Alan J. Marcus, 2014, Investments, McGraw-Hill Education.
Böhm-Bawerk, Eugen V., 1891, The Positive Theory of Capital, London: Macmillan. Translated
by William Smart.
Bond, Shaun, and Chen Xue, 2015, The cross section of expected real estate returns: Insights from
investment-based asset pricing, forthcoming, Journal of Real Estate Finance and Economics.
Bond, Stephen, and John Van Reenen, 2007, Microeconometric models of investment and
employment, in James J. Heckman and Edward E. Leamer, eds., Handbook of Econometrics
4417–4498, Amsterdam: North-Holland.
108
Brainard, William C., and James Tobin, 1968, Pitfalls in financial model building, American
Economic Review 58, 99–122.
Brealey, Richard A., Stewart C. Myers, and Franklin Allen, 2011, Principles of Corporate Finance,
10th ed., McGraw-Hill Irwin.
Breeden, Douglas T., 1979, An intertemporal asset pricing model with stochastic consumption
and investment opportunities, Journal of Financial Economics 7, 265–296.
Breeden, Douglas T., Michael R. Gibbons, and Robert H. Litzenberger, 1989, Empirical tests of
the consumption-oriented CAPM, Journal of Finance 44, 231–262.
Brock, William A., 1982, Asset prices in a production economy, in: John L. McCall, ed., Economics
of Information and Uncertainty, Illinois: University of Chicago Press.
Brown, Lawrence D., and Michael S. Rozeff, 1979, Univariate time-series models of quarterly
accounting earnings per share: A proposal model, Journal of Accounting Research 17, 179–
189.
Butler, Alexander W., Jess Cornaggia, Gustavo Grullon, and James P. Weston, 2011, Corporate
financing decisions, managerial market timing, and real investment, Journal of Financial
Economics 101, 666–683.
Caballero, Ricardo J., 1999, Aggregate investment, in John B. Taylor and Michael Woodford, eds.,
Handbook of Macroeconomics 813–862, Amsterdam: North-Holland.
Campbell, John Y., Jens Hilscher, and Jan Szilagyi, 2008, In search of distress risk, Journal of
Finance 63, 2899–2939.
Carhart, Mark M. 1997, On persistence in mutual fund performance, Journal of Finance 52,
57–82.
Carlson, Murray, Adlai Fisher, and Ron Giammarino, 2004, Corporate investment and asset price
dynamics: Implications for the cross section of returns, Journal of Finance 59, 2577–2603.
Carlson, Murray, Adlai Fisher, and Ron Giammarino, 2006, Corporate investment and asset price
dynamics: Implications for SEO event studies and long-run performance, Journal of Finance
61, 1009–1034.
Chan, Louis K. C., Narasimhan Jegadeesh, and Josef Lakonishok, 1996, Momentum strategies,
Journal of Finance 51, 1681–1713.
Chen, Long, Ralitsa Petkova, and Lu Zhang, 2008, The expected value premium, Journal of
Financial Economics 87, 269–280.
Chen, Long, and Lu Zhang, 2011, Do time-varying risk premiums explain labor market
performance? Journal of Financial Economics 99, 385–399.
Chen, Shuping, Mark L. DeFond, and Chul W. Park, 2002, Voluntary disclosure of balance sheet
information in quarterly earnings announcements, Journal of Accounting and Economics 33,
229–251.
109
Chen, Nai-Fu, Richard Roll, and Stephen A. Ross, 1986, Economic forces and the stock market,
Journal of Business 59, 383–403.
Chirinko, Robert S., 1993, Business fixed investment spending: Modeling strategies, empirical
results, and policy implications, Journal of Economic Literature 31, 1875–1911.
Choi, Jaewon, 2013, What drives the value premium? The role of asset risk and leverage, Review
of Financial Studies 26, 2845–2875.
Chui, Andy C. W., Sheridan Titman, and K. C. John Wei, 2010, Individualism and momentum
around the world, Journal of Finance 65, 361–392.
Cochrane, John H., 1991, Production-based asset pricing and the link between stock returns and
economic fluctuations, Journal of Finance 46, 209–237.
Cochrane, John H., 1993, Rethinking production under uncertainty, working paper, University of
Chicago.
Cochrane, John H., 1996, A cross-sectional test of an investment-based asset pricing model,
Journal of Political Economy 104, 572–621.
Cochrane, John H., 2005, Asset Pricing, New Jersey: Princeton University Press.
Cohen, Randolph B., Christopher Polk, and Tuomo Vuolteenaho, 2003, The value spread, Journal
of Finance 58, 609–641.
Cooper, Ilan, 2006, Asset pricing implications of nonconvex adjustment costs and irreversibility
of investment, Journal of Finance 61, 139–170.
Cooper, Ilan, and Paulo Maio, 2015, Equity risk factors and the intertemporal CAPM, working
paper, Norwegian Business School.
Cooper, Ilan, and Richard Priestley, 2011, Real investment and risk dynamics, Journal of Financial
Economics 101, 182–205.
Cooper, Ilan, and Richard Priestley, 2015, The expected returns and valuations of private and
public firms, forthcoming, Journal of Financial Economics.
Cooper, Michael J., Huseyin Gulen, and Michael J. Schill, 2008, Asset growth and the cross-section
of stock returns, Journal of Finance 63, 1609–1652.
Cooper, Michael J., Roberto C. Gutierrez Jr., and Allaudeen Hameed, 2004, Market states and
momentum, Journal of Finance 59, 1345–1365.
Cooper, Russell W., and John C. Haltiwanger, 2006, On the nature of capital adjustment costs,
Review of Economic Studies 73, 611–633.
Cox, John C., Jonathan E. Ingersoll, Jr., and Stephen A. Ross, 1985, An intertemporal general
equilibrium model of asset prices, Econometrica 53, 363–384.
Daniel, Kent D., David Hirshleifer, and Avanidhar Subrahmanyam, 1998, Investor psychology and
security market under- and overreaction, Journal of Finance 53, 1839–1885.
110
Daniel, Kent D., and Sheridan Titman, 1997, Evidence on the characteristics of cross sectional
variation in stock returns, Journal of Finance 52, 1–33.
Daniel, Kent D., and Sheridan Titman, 2006, Market reactions to tangible and intangible
information, Journal of Finance 61, 1605–1643.
Danthine, Jean-Pierre, and John B. Donaldson, 2015, Intermediate Financial Theory, 3rd edition,
Academic Press.
Deaton, Angus, 1992, Understanding Consumption, Oxford: Clarendon Press.
De Bondt, Werner F. M., and Richard Thaler, 1985, Does the stock market overreact? Journal of
Finance 40, 793–805.
Debreu, Gerard, 1959, Theory of Value: An Axiomatic Analysis of Economic Equilibrium, New
Haven and London: Yale University Press.
Debreu, Gerard, 1974, Excess-demand functions, Journal of Mathematical Economics 1, 15–21.
Dechow, Patricia M., 1994, Accounting earnings and cash flows as measures of firm performance:
The role of accounting accruals, Journal of Accounting and Economics 18, 3–42.
Dechow, Patricia M., Richard G. Sloan, and Mark T. Soliman, 2004, Implied equity duration: A
new measure of equity risk, Review of Accounting Studies 9, 197–228.
Diamond, Peter A., 1967, The role of a stock market in a general equilibrium model with
technological uncertainty, American Economic Review 57, 759–776.
Dixit, Avinash K., and Robert S. Pindyck, 1994, Investment Under Uncertainty, New Jersey:
Princeton University Press.
Doms, Mark, and Timothy Dunne, 1998, Capital adjustment patterns in manufacturing plants,
Review of Economic Dynamics 1, 409-429.
Donangelo, Andrés, 2014, Labor mobility: Implications for asset pricing, Journal of Finance 69,
1321–1346.
Duffie, Darrell, 2001, Dynamic Asset Pricing Theory, 3rd edition, New Jersey: Princeton
University Press.
Easton, Peter D., and Mark E. Zmijewski, 1993, SEC form 10K/10Q reports and annual reports
to shareholders: Reporting lags and squared market model prediction errors, Journal of
Accounting Research 31, 113–129.
Eisfeldt, Andrea L., and Dimitris Papanikolaou, 2013, Organizational capital and the cross-section
of expected returns, Journal of Finance 68, 1365–1406.
Fairfield, Patricia M., Scott Whisenant, and Terry Lombardi Yohn, 2003, Accrued earnings and
growth: Implications for future profitability and market mispricing, The Accounting Review
78, 353–371.
Fama, Eugene F., 1965, The behavior of stock-market prices, Journal of Business 38, 34–105.
111
Fama, Eugene F., 1970, Efficient capital markets: A review of theory and empirical work, Journal
of Finance 25, 383–417.
Fama, Eugene F., 1976, Foundations of Finance, New York: Basic Books, Publishers.
Fama, Eugene F., 1991, Efficient capital markets: II, Journal of Finance 46, 1575–1617.
Fama, Eugene F., 1998, Market effciency, long-term returns, and behavioral finance, Journal of
Financial Economics 49, 283–306.
Fama, Eugene F., and Kenneth R. French, 1992, The cross-section of expected stock returns,
Journal of Finance 47, 427–465.
Fama, Eugene F., and Kenneth R. French, 1993, Common risk factors in the returns on stocks
and bonds, Journal of Financial Economics 33, 3–56.
Fama, Eugene F., and Kenneth R. French, 1996, Multifactor explanation of asset pricing
anomalies, Journal of Finance 51, 55–84.
Fama, Eugene F., and Kenneth R. French, 2006, Profitability, investment, and average returns,
Journal of Financial Economics 82, 491–518.
Fama, Eugene F., and Kenneth R. French, 2008, Dissecting anomalies, Journal of Finance 63,
1653–1678.
Fama, Eugene F., and Kenneth R. French, 2013, A four-factor model for the size, value, and
profitability patterns in stock returns, Fama-Miller working paper, University of Chicago.
Fama, Eugene F., and Kenneth R. French, 2015a, A five-factor asset pricing model, Journal of
Financial Economics 116, 1–22.
Fama, Eugene F., and Kenneth R. French, 2015b, Dissecting anomalies with a five-factor model,
forthcoming, Review of Financial Studies.
Fama, Eugene F., and Kenneth R. French, 2015c, International tests of a five-factor asset pricing
model, working paper, University of Chicago.
Fama, Eugene F., and James D. MacBeth, 1973, Risk, return, and equilibrium: Empirical tests,
Journal of Political Economy 81, 607–636.
Fama, Eugene F., and Merton H. Miller, 1972, The Theory of Finance, Illinois: Dryden Press.
Fazzari, Steven M., R. Glenn Hubbard, and Bruce C. Petersen, 1988, Financing constraints and
corporate investment, Brookings Papers on Economic Activities 1, 141–195.
Fischer, Bernd R., and Russ Wermers, 2012, Performance Evaluation and Attribution of Security
Portfolios, Academic Press.
Fisher, Irving, 1930, The Theory of Interest: As Determined by Impatience to Spend Income and
Opportunity to Invest It, 1st ed., New York: Macmillan.
Foster, George, Chris Olsen, and Terry Shevlin, 1984, Earnings releases, anomalies, and the
behavior of security returns, The Accounting Review 59, 574–603.
112
Frazzini, Andrea, and Lasse Heje Pedersen, 2014, Betting against beta, Journal of Financial
Economics 111, 1–25.
Garlappi, Lorenzo, and Zhongzhi Song, 2015, Can investment shocks explain value and
momentum? working paper, University of British Columbia.
Garlappi, Lorenzo, and Hong Yan, 2011, Financial distress and the cross-section of equity returns,
Journal of Finance 66, 789–822.
Gebhardt, William R., Charles M. C. Lee, and Bhaskaram Swaminathan, 2001, Toward an implied
cost of capital, Journal of Accounting Research 39, 135–176.
Gibbons, Michael R., Stephen A. Ross, and Jay Shanken, 1989, A test of the efficiency of a given
portfolio, Econometrica 57, 1121–1152.
Gomes, Joao F., and Lukas Schmid, 2010, Levered returns, Journal of Finance 65, 467–494.
Gomes, Joao F., Leonid Kogan, and Lu Zhang, 2003, Equilibrium cross section of returns, Journal
of Political Economy 111, 693–732.
Gomes, Joao F., Amir Yaron, and Lu Zhang, 2006, Asset pricing implications of firms’ financing
constraints, Review of Financial Studies 19, 1321–1356.
Graham, Benjamin, and David Dodd, 1934, Security Analysis, New York: Whittlesey House,
McGraw-Hill.
Griffin, John M., Xiuqing Ji, and J. Spencer Martin, 2003, Momentum investing and business
cycle risk: Evidence from pole to pole, Journal of Finance 58, 2515–2547.
Gulen, Huseyin, Yuhang Xing, and Lu Zhang, 2011, Value versus growth: Time-varying expected
stock returns, Financial Management 40, 381–407.
Hall, Robert E., and Dale W. Jorgenson, 1967, Tax policy and investment behavior, American
Economic Review 57, 391–414.
Hansen, Lars Peter, 1982, Large sample properties of generalized methods of moments estimators,
Econometrica 40, 1029–1054.
Hansen, Lars Peter, and Scott F. Richard, 1987, The role of conditioning information in deducing
testable restrictions implied by dynamic asset pricing models, Econometrica 55, 587–614.
Hansen, Lars Peter, and Kenneth J. Singleton, 1982, Generalized instrumental variables estimation
of nonlinear rational expectations models, Econometrica 50, 1269–1288.
Harvey, Campbell R., Yan Liu, and Heqing Zhu, 2015, . . . and the cross-section of expected
returns, forthcoming, Review of Financial Studies.
Hayashi, Fumio, 1982, Tobin’s marginal q and average q:
Econometrica 50, 213–224.
A neoclassical interpretation,
Hennessy, Christopher A., and Toni M. Whited, 2005, Debt dynamics, Journal of Finance 60,
1129–1165.
113
Hennessy, Christopher A., and Toni M. Whited, 2007, How costly is external financing? Evidence
from a structural estimation, Journal of Finance 62, 1705–1745.
Hirshleifer, Jack, 1958, On the theory of optimal investment decision, Journal of Political Economy
66, 329–352.
Hirshleifer, Jack, 1965, Investment decision under uncertainty: Choice-theoretic approaches,
Quarterly Journal of Economics 79, 509–536.
Hirshleifer, Jack, 1966, Investment decision under uncertainty: Applications of the state-preference
approach, Quarterly Journal of Economics 80, 252–277.
Hirshleifer, Jack, 1970, Investment, Interest, and Capital, New Jersey: Prentice-Hall.
Hou, Kewei, Chen Xue, and Lu Zhang, 2015a, Digesting anomalies: An investment approach,
Review of Financial Studies 28, 650–705.
Hou, Kewei, Chen Xue, and Lu Zhang, 2015b, A comparison of new factor models, The Ohio
State University and University of Cincinnati.
Hou, Kewei, G. Andrew Karolyi, and Bong-Chan Kho, 2011, What factors drive global stock
returns? Review of Financial Studies 24, 2527-2574.
Huang, Chi-fu, and Robert H. Litzenberger, 1988, Foundations for Financial Economics, NorthHolland.
Ingersoll, Jonathan E., Jr., 1987, Theory of Financial Decision Making, Rowman & Littlefield
Publishers.
Jegadeesh, Narasimhan and Sheridan Titman, 1993, Returns to buying winners and selling losers:
Implications for stock market efficiency, Journal of Finance 48, 65–91.
Jensen, Michael C., 1986, Agency costs of free cash flow, corporate finance, and takeovers,
American Economic Review 76, 323–329.
Jermann, Urban J., 2010, The equity premium implied by production, Journal of Financial
Economics 98, 279–296.
Johnson, Timothy C., 2002, Rational momentum effects, Journal of Finance 57, 585–608.
Jones, Christopher S., and Selale Tuzel, 2013, Inventory investment and the cost of capital, Journal
of Financial Economics 107, 557–579.
Jorgenson, Dale W., 1963, Capital theory and investment behavior, American Economic Review
53, 247–259.
Kirman, Alan P., 1992, Whom or what does the representative individual represent? Journal of
Economic Perspectives 6, 117–136.
Kogan, Leonid, 2004, Asset prices and real investment, Journal of Financial Economics 73, 411–
431.
Kogan, Leonid, and Dimitris Papanikolaou, 2013, Firm characteristics and stock returns: The role
of investment-specific shocks, Review of Financial Studies 26, 2718–2759.
114
Koh, Woo Hwa, 2015, The impact of uncertainty shocks on the cross-section of returns, working
paper, Korea Advanced Institute of Science and Technology.
Koller, Tim, Marc Goedhart, and David Wessels, 2015, Valuation: Measuring and Managing the
Value of Companies, 6th ed., New Jersey: John Wiley and Sons.
Koopmans, Tjalling C., 1947, Measurement without theory, Review of Economics and Statistics
29, 161–172.
Kothari, S. P., 2001, Capital markets research in accounting, Journal of Accounting and Economics
31, 105–231.
Kuhn, Thomas S., 1962, The Structure of Scientific Revolutions, Chicago: The University of
Chicago Press.
Kuehn, Lars-Alexander, and Lukas Schmid, 2014, Investment-based corporate bond pricing,
Journal of Finance 69, 2741–2776.
Kydland, Finn E., and Edward C. Prescott, 1982, Time to build and aggregate fluctuations,
Econometrica 50, 1345–1370.
Lakonishok, Josef, Andrei Shleifer, and Robert W. Vishny, 1994, Contrarian investment,
extrapolation, and risk, Journal of Finance 49, 1541–1578.
Lam, F. Y. Eric C., and K. C. John Wei, 2011, Limits-to-arbitrage, investment frictions, and the
asset growth anomaly, Journal of Financial Economics 102, 127–149.
Lamont, Owen, Christopher Polk and Jesús Saá-Requejo, 2001, Financial constraints and stock
returns, Review of Financial Studies 14, 529–554.
Landreth, Harry, and David C. Colander, 2002, History of Economic Thought, 4th edition,
Houghton Mifflin Company.
La Porta, Rafael, Josef Lakonishok, Andrei Shleifer, and Robert Vishny, 1997, Good news for
value stocks: Further evidence on market efficiency, Journal of Finance 52, 859–874.
Lettau, Martin, and Sydney Ludvigson, 2001, Resurrecting the (C)CAPM: A cross-sectional test
when risk premia are time-varying, Journal of Political Economy 109, 1238–1287.
Lettau, Martin, and Sydney Ludvigson, 2002, Time-varying risk premia and the cost of capital:
An alternative implication of the Q theory of investment, Journal of Monetary Economics
49, 31–66.
Lev, Baruch, and S. Ramu Thiagarajan, 1993, Fundamental information analysis, Journal of
Accounting Research 31, 190–215.
Lewellen, Jonathan, and Stefan Nagel, 2006, The conditional CAPM does not explain asset-pricing
anomalies, Journal of Financial Economics 82, 289–314.
Li, Dongmei, 2011, Financial constraints, R&D investment, and stock returns, Review of Financial
Studies 24, 2974–3007.
115
Li, Dongmei, and Lu Zhang, 2010, Does q-theory with investment frictions explain anomalies in
the cross section of returns? Journal of Financial Economics 98, 297–314.
Li, Erica X. N., Dmitry Livdan, and Lu Zhang, 2009, Anomalies, Review of Financial Studies 22,
4301–4334.
Li, Jun, 2015, Explaining momentum and value simultaneously, working paper, University of
Texas at Dallas.
Lin, Xiaoji, 2012, Endogenous technological progress and the cross-section of stock returns,
Journal of Financial Economics 103, 411–427.
Lin, Xiaoji, and Lu Zhang, 2013, The investment manifesto, Journal of Monetary Economics 60,
351–366.
Lintner, John, 1965, The valuation of risk assets and the selection of risky investments in stock
portfolios and capital budgets, Review of Economics and Statistics 47, 13–37.
Lipson, Marc L., Sandra Mortal, and Michael J. Schill, 2011, On the scope and drivers of the asset
growth effect, Journal of Financial and Quantitative Analysis 46, 1651–1682.
Liu, Laura Xiaolei, Toni M. Whited, and Lu Zhang, 2009, Investment-based expected stock
returns, Journal of Political Economy 117, 1105–1139.
Liu, Laura Xiaolei, and Lu Zhang, 2008, Momentum profits, factor pricing, and macroeconomic
risk, Review of Financial Studies 21, 2417–2448.
Liu, Laura Xiaolei, and Lu Zhang, 2014, A neoclassical interpretation of momentum, Journal of
Monetary Economics 67, 109–128.
Livdan, Dmitry, Horacio Sapriza, and Lu Zhang, 2009, Financially constrained stock returns,
Journal of Finance 64, 1827–1862.
Long, John B., Jr., 1974, Stock prices, inflation, and the term structure of interest rates, Journal
of Financial Economics 1, 131–170.
Long, John B., Jr., and Charles I. Plosser, 1983, Real business cycles, Journal of Political Economy
91, 39-69.
Loughran, Tim, and Jay R. Ritter, 1995, The new issues puzzle, Journal of Finance 50, 23–51.
Lucas, Robert E., Jr., 1972, Expectations and the neutrality of money, Journal of Economic
Theory 4, 103–124.
Lucas, Robert E., Jr., 1978, Asset prices in an exchange economy, Econometrica 46, 1429–1445.
Lucas, Robert E., Jr., and Edward C. Prescott, 1971, Investment under uncertainty, Econometrica
39, 659–681.
Ludvigson, Sydney C., 2013, Advances in consumption-based asset pricing: Empirical tests, in
George M. Constantinides, Milton Harris, and René M. Stulz eds., Handbook of the Economics
of Finance volume 2B, 799–906.
116
Lustig, Hanno, and Stijn G. Van Nieuwerburgh, 2005, Housing collateral, consumption insurance,
and risk premia: An empirical perspective, Journal of Finance 60, 1167–1219.
Lyandres, Evgeny, Le Sun, and Lu Zhang, 2008, The new issues puzzle: Testing the investmentbased explanation, Review of Financial Studies 21, 2825–2855.
Mantel, Rolf R., 1974, On the characterization of aggregate excess-demand, Journal of Economic
Theory 7, 348–353.
Markowitz, Harry M., 1952, Portfolio selection, Journal of Finance 7, 77–91.
Markowitz, Harry M., 1959, Portfolio Selection: Efficient Diversification of Investments, New
York: John Wiley & Sons.
Marshall, Alfred, 1961, Principles of Economics, 9th edition, London: Macmillan, first published
in 1890.
Mehra, Rajnish, and Edward C. Prescott, 1985, The equity premium: A puzzle, Journal of
Monetary Economics 15, 145–161.
Merton, Robert C., 1973, An intertemporal capital asset pricing model, Econometrica 41, 867–887.
Merz, Monika, and Eran Yashiv, 2007, Labor and the market value of the firm, American Economic
Review 97, 1419–1431.
Miller, Merton H., and Franco Modigliani, 1961, Dividend policy, growth, and the valuation of
shares, Journal of Business 34, 411–433.
Miller, Merton H., and Myron S. Scholes, 1972, Rate of return in relation to risk: A re-examination
of some recent findings. In Michael C. Jensen, ed. Studies in the Theory of Capital Markets.
New York: Praeger.
Modigliani, Franco, and Merton H. Miller, 1958, The cost of capital, corporation finance, and the
theory of investment, American Economic Review 48, 261–297.
Morck, Randall, Bernard Yeung, and Wayne Yu, 2000, The information content of stock markets:
Why do emerging markets have synchronous stock price movements? Journal of Financial
Economics 58, 215–260.
Mossin, Jan, 1966, Equilibrium in a capital asset market, Econometrica 34, 768–783.
Mussa, Michael L., 1977, External and internal adjustment costs and the theory of aggregate and
firm investment, Economica 44, 163–178.
Muth, John F., 1961, Rational expectations and the theory of price movements, Econometrica 29,
315–335.
Novy-Marx, Robert, 2013, The other side of value: The gross profitability premium, Journal of
Financial Economics 108, 1–28.
Obreja, Iulian, 2013, Book-to-market equity, financial leverage, and the cross-section of stock
returns, Review of Financial Studies 26, 1146–1189.
117
Ohlson, James A., 1995, Earnings, book values, and dividends in equity valuation, Contemporary
Accounting Research 11, 661–687.
Ou, Jane A., and Stephen H. Penman, 1989, Financial statement analysis and the prediction of
stock returns, Journal of Accounting and Economics 11, 295–329.
Ozdagli, Ali K., 2012, Financial leverage, corporate investment, and stock returns, Review of
Financial Studies 25, 1033–1069.
Papanikolaou, Dimitris, 2011, Investment shocks and asset prices, Journal of Political Economy
119, 639–685.
Penman, Stephen H., 2013, Financial Statement Analysis and Security Returns, 5th ed., New
York: McGraw-Hill Irwin.
Penman, Stephen H., and Julie Lei Zhu, 2015, Accounting anomalies, risk, and return, The
Accounting Review 89, 1835–1866.
Penman, Stephen H., Scott A. Richardson, and Irem Tuna, 2007, The book-to-market effect in
stock returns: Accounting for leverage, Journal of Accounting Research 45, 427–468.
Petkova, Ralitsa, and Lu Zhang, 2005, Is value riskier than growth?
Economics 78, 187–202.
Journal of Financial
Piotroski, Joseph D., 2000, Value investing: The use of historical financial statement information
to separate winners from losers, Journal of Accounting Research 38, Supplement: Studies on
accounting information and the economics of the firm, 1–41.
Restoy, Fernando, and G. Michael Rockinger, 1994, On stock market returns and returns on
investment, Journal of Finance 49, 543–556.
Richardson, Scott A., Richard G. Sloan, Mark T. Soliman, and Irem Tuna, 2005, Accrual
reliability, earnings persistence and stock prices, Journal of Accounting and Economics 39,
437–485.
Richardson, Scott A., Irem Tuna, and Peter Wysocki, 2010, Accounting anomalies and
fundamental analysis: A review of recent research advances, Journal of Accounting and
Economics 50, 410–454.
Ritter, Jay R., 1991, The long-run performance of initial public offerings, Journal of Finance 46,
3–27.
Roll, Richard, 1977, A critique of the asset pricing theory’s tests: Part I, Journal of Financial
Economics 4, 129–176.
Rosenberg, Barr, Kenneth Reid, and Ronald Lanstein, 1985, Persuasive evidence of market
inefficiency, Journal of Portfolio Management 11, 9–16.
Ross, Stephen A., 1976, The arbitrage theory of capital asset pricing, Journal of Economic Theory
13, 341–360.
Ross, Stephen A., Randolph W. Westerfield, and Jeffrey Jaffe, 2008, Corporate Finance, 8th ed.,
McGraw-Hill Irwin.
118
Roy, Andrew D., 1952, Safety first and the holding of assets, Econometrica 20, 431–449.
Rubinstein, Mark, 1976, The valuation of uncertain income streams and the pricing of options,
Bell Journal of Economics and Management Science 7, 407–425.
Rubinstein, Mark, 2006, A History of the Theory of Investments, New Jersey: John Wiley & Sons.
Sagi, Jacob S., and Mark S. Seasholes, 2007, Firm-specific attributes and the cross-section of
momentum, Journal of Financial Economics 84, 389–434.
Shanken, Jay, 2015, Discussion on “HXZ: A comparison of new factor models,” ASU Sonoran
Winter Finance Conference.
Sharpe, William F., 1964, Capital asset prices: A theory of market equilibrium under conditions
of risk, Journal of Finance 19, 425–442.
Shiller, Robert J., 1984, Stock prices and social dynamics, Brookings Papers on Economic Activity
2, 457–498.
Shiller, Robert J., 1990, Speculative prices and popular models, Journal of Economic Perspectives
4, 55–65.
Shleifer, Andrei, 2000, Inefficient Markets: An Introduction to Behavioral Finance, Oxford
University Press.
Shleifer, Andrei, and Robert W. Vishny, 1997, The limits of arbitrage, Journal of Finance 52,
35–55.
Sloan, Richard G., 1996, Do stock prices fully reflect information in accruals and cash flows about
future earnings? The Accounting Review 71, 289–315.
Sonnenschein, Hugo, 1973, Do Walras’ identity and continuity characterize the class of community
excess-demand function? Journal of Economic Theory 6, 345–354.
Stambaugh, Robert F., and Yu Yuan, 2015, Mispricing factors, working paper, University of
Pennsylvania.
Strebulaev, Ilya A., and Toni M. Whited, 2012, Dynamic models and structural estimation in
corporate finance, Foundations and Trends in Finance 6, 1–163.
Titman, Sheridan, K. C. John Wei, and Feixue Xie, 2004, Capital investments and stock returns,
Journal of Financial and Quantitative Analysis 39, 677–700.
Titman, Sheridan, K. C. John Wei, and Feixue Xie, 2013, Market development and the asset
growth effect: International evidence, Journal of Financial and Quantitative Analysis 48,
1405–1432.
Tobin, James, 1969, A general equilibrium approach to monetary theory, Journal of Money, Credit,
and Banking 1, 15–29.
Treynor, Jack L., 1962, Toward a theory of market value of risky assets, published in 1999 in
Robert A. Korajczyk, ed. Asset Pricing and Portfolio Performance, 15–22, London: Risk
Publications.
119
Tuzel, Selale, 2010, Corporate real estate holdings and the cross-section of stock returns, Review
of Financial Studies 23, 2268–2302.
Veracierto, Marcelo L., 2002, Plant-level irreversible investment and equilibrium business cycles,
American Economic Review 92, 181–197.
Vitorino, Maria Ana, 2014, Understanding the effect of advertising on stock returns and firm
value: Theory and evidence from a structural model, Management Science 60, 227–245.
Watanabe, Akiko, Yan Xu, Tong Yao, and Tong Yu, 2013, The asset growth effect: Insights from
international equity markets, Journal of Financial Economics 108, 529–563.
Whited, Toni M., 1992, Debt, liquidity constraints, and corporate investment: Evidence from
panel data, Journal of Finance 47, 1425–1460.
Whited, Toni M., 1998, Why do investment Euler equations fail?
Economic Statistics 16, 479–488.
Journal of Business and
Whited, Toni M., and Guojun Wu, 2006, Financial constraints risk, Review of Financial Studies
19, 531–559.
Wu, Jin G., X. Frank Zhang, and Lu Zhang, 2010, The q-theory approach to understanding the
accrual anomaly, Journal of Accounting Research 48, 177–223.
Xing, Yuhang, 2008, Interpreting the value effect through the Q-theory:
investigation, Review of Financial Studies 21, 1767–1795.
An empirical
Xue, Chen, 2014, An investment-based investigation of mutual fund performance, working paper,
University of Cincinnati.
Zhang, Lu, 2005a, Anomalies, NBER working paper 11322.
Zhang, Lu, 2005b, The value premium, Journal of Finance 60, 67–103.
Zhang, X. Frank, 2007, Accruals, investment, and the accrual anomaly, The Accounting Review
82, 1333–1363.
120
A
Data
To construct the q-factors, at the end of June of each year t, the median New York Stock Exchange
(NYSE) market equity is used to split all stocks into two groups, small and big. Independently, at
the end of June of year t, all stocks are sorted into three I/A groups with the NYSE breakpoints
for the low 30%, middle 40%, and high 30% of the ranked values of I/A for the fiscal year ending
in calendar year t − 1. Also, independently, at the beginning of each month, all stocks are sorted
into three ROE groups with the NYSE breakpoints for the low 30%, middle 40%, and high 30%
of the ranked values of ROE. Earnings (Compustat quarterly item IBQ) are used in the months
immediately after the most recent quarterly earnings announcement dates (item RDQ). For a firm
to enter the factor construction, the end of its fiscal quarter corresponding to the most recently
announced quarterly earnings must be within six months prior to the portfolio formation.
Taking the intersections of the two size, three I/A, and three ROE groups yields 18 portfolios,
and monthly value-weighted portfolio returns are calculated. The size factor, rME , is the difference
(small-minus-big) between the simple average of the returns for the nine small size portfolios and
the simple average of the returns for the nine big size portfolios. The investment factor, rI/A , is the
difference (low-minus-high investment) between the simple average of the returns for the six low
I/A portfolios and the simple average of the returns for the six high I/A portfolios. The ROE factor,
rROE , is the difference (high-minus-low profitability) between the simple average of the returns for
the six high ROE portfolios and the simple average of the returns for the six low ROE portfolios.
Quarterly book equity is shareholders’ equity, plus balance sheet deferred taxes and investment
tax credit (item TXDITCQ) if available, minus the book value of preferred stock. Depending on
availability, stockholders’ equity (item SEQQ), or common equity (item CEQQ) plus the carrying
value of preferred stock (item PSTKQ), or total assets (item ATQ) minus total liabilities (item
LTQ) in that order is used as shareholders’ equity. Redemption value (item PSTKRQ) if available,
or carrying value is used for the book value of preferred stock.
Fama and French (2015a) use independent (2×3) sorts by interacting size, separately, with bookto-market (B/M), OP, and Inv. The size breakpoint is the NYSE median, and the B/M, OP, and Inv
breakpoints are their respective 30th and 70th percentiles for NYSE stocks. HML is the average of
the two high B/M portfolio returns minus the average of the two low B/M portfolio returns. RMW
is the average of the two high OP portfolio returns minus the average of the two low OP portfolio
returns. CMA is the average of the two low Inv portfolio returns minus the average of the two high
Inv portfolio returns. Finally, SMB is the average of the returns on the nine small portfolios from
the three separate 2×3 sorts minus the average of the returns on the nine big portfolios.
Cash-based operating profits are revenues (Compustat quarterly item REVTQ) minus cost of
goods sold (item COGSQ) minus selling, general, and administrative expenses (item XSGAQ) plus
research and development expenditure (item XRDQ, zero if missing) minus change in accounts receivable (item RECTQ) minus change in inventory (item INVTQ) plus change in deferred revenue
(item DRCQ plus item DRLTQ) plus change in trade accounts payable (item APQ). All balance
sheet changes are quarterly changes, and are set to zero if missing.
121
B
Derivations
Let qit be the Lagrangian multiplier associated with Kit+1 = Iit + (1 − δ it )Kit . The optimality
conditions with respect to Iit , Kit+1 , and Bit+1 from maximizing equation (25) are, respectively,
∂Φ(Iit , Kit )
qit = 1 + (1 − τ t )
(B1)
∂Iit
∂Π(Kit+1 , Xit+1 ) ∂Φ(Iit+1 , Kit+1 )
qit = Et Mt+1 (1 − τ t+1 )
−
+ τ t+1 δ it+1 + (1 − δ it+1 )qit+1
∂Kit+1
∂Kit+1
(B2)
B
B
(B3)
1 = Et Mt+1 rit+1 − (rit+1 − 1)τ t+1 .
Equation (B1) equates the marginal purchase and adjustment costs of investing to the marginal
benefit, qit . Equation (B2) is the investment Euler condition, which describes the evolution of qit .
Dividing both sides of equation (B2) by qit and substituting equation (B1) yields
I
I
Et [Mt+1 rit+1
] = 1, in which rit+1
is the investment return given by:
h
i
h
i
∂Φ(iit+1 ,Kit+1 )
∂Φ(Iit+1 ,Kit+1 )
it+1 ,Xit+1 )
(1 − τ t+1 ) ∂Π(K∂K
+
τ
δ
+
(1
−
δ
)
1
+
(1
−
τ
)
−
t+1
it+1
it+1
t+1
∂Kit+1
∂Iit+1
it+1
I
rit+1
≡
.
∂Φ(Iit ,Kit )
1 + (1 − τ t ) ∂Iit
(B4)
Substituting ∂Π(Kit+1 , Xit+1 )/∂Kit+1 = κYit+1 /Kit+1 and Φ(Iit , Kit ) = (a/2)(Iit /Kit )2 Kit into
equation (B4) yields the investment return equation (26).
B ] = 1 + E [M
B
From equation (B3), Et [Mt+1 rit+1
t
t+1 (rit+1 − 1)τ t+1 ]. Due to the tax benefit
B ], is higher than one. The difof debt, the unit price of the pre-tax bond return, Et [Mt+1 rit+1
ference is the present value of the tax benefit. Because the after-tax corporate bond return is
Ba ≡ r B
B
Ba
rit+1
it+1 − (rit+1 − 1)τ t+1 , equation (B3) then implies that Et [Mt+1 rit+1 ] = 1.
To see equation (27), start with Pit + Dit = Vit and expand Vit using equations (24) and (25):
B
Bit ] − τ t Bit − Iit + Bit+1 + τ t δ it Kit =
Pit + (1 − τ t )[Π(Kit , Xit ) − Φ(Iit , Kit ) − rit
∂Φ(Iit , Kit )
∂Φ(Iit , Kit )
B
Iit −
Kit − rit Bit − τ t Bit − Iit + Bit+1 + τ t δit Kit
(1 − τ t ) Π(Kit , Xit ) −
∂Iit
∂Kit
∂Φ(Iit+1 , Kit+1 )
Iit+1
− qit (Kit+1 − (1 − δ it )Kit − Iit ) + Et [Mt+1 ((1 − τ t ) Π(Kit+1 , Xit+1 ) −
∂Iit+1
∂Φ(Iit+1 , Kit+1 )
B
−
Kit+1 − rit+1 Bit+1 − τ t+1 Bit+1 − Iit+1 + Bit+2
∂Kit+1
+ τ t+1 δ it+1 Kit+1 − qit+1 (Kit+2 − (1 − δit+1 )Kit+1 − Iit+1 ) + . . . )]
(B5)
Recursively substituting equations (B1), (B2), and (B3) yields:
B
Pit + (1 − τ t )[Π(Kit , Xit ) − Φ(Iit , Kit ) − rit
Bit ] − τ t Bit − Iit + Bit+1 + τ t δ it Kit =
∂Φ(Iit , Kit )
B
Kit − rit Bit − τ t Bit + qit (1 − δ it )Kit + τ t δ it Kit (B6)
(1 − τ t ) Π(Kit , Xit ) −
∂Kit
122
Simplifying further and using the linear homogeneity of Φ(Iit , Kit ) yields:
Pit + Bit+1 = (1 − τ t )
∂Φ(Iit , Kit )
Iit + Iit + qit (1 − δ it )Kit = qit Kit+1
∂Iit
(B7)
Finally, equation (27) results from:


B B
(1 − τ t+1 )rit+1
it+1 + τ t+1 Bit+1 + Pit+1
B B
 +(1 − τ t+1 )[Π(Kit+1 , Xit+1 ) − Φ(Iit+1 , Kit+1 ) − rit+1
it+1 ] 
−τ t+1 Bit+1 − Iit+1 + Bit+2 + τ t+1 δit+1 Kit+1
Ba
S
wit rit+1
+ (1 − wit )rit+1
=
Pit + Bit+1
1
qit+1 (Iit+1 + (1 − δ it+1 )Kit+1 ) + (1 − τ t+1 )[Π(Kit+1 , Xit+1 )
=
−Φ(Iit+1 , Kit+1 )] − Iit+1 + τ t+1 δ it+1 Kit+1
qit Kit+1
h
i
∂Φ(Iit+1 ,Kit+1 )
it+1 ,Xit+1 )
qit+1 (1 − δ it+1 ) + (1 − τ t+1 ) ∂Π(K∂K
+ τ t+1 δ it+1
−
∂Kit+1
it+1
I
=
= rit+1
. (B8)
qit
C
Timing Alignment
Consider the loser decile from the six-six momentum strategy (stocks are split into deciles at the
beginning of each month based on prior six-month returns, skipping the current month, and the
decile returns are calculated for the subsequent six months). In any given month, there are six
sub-deciles for the loser decile because of the six-month holding period. For instance, for the loser
decile in July of year t, the first sub-decile is formed at the end of January of year t based on the
prior six-month return from July to December of year t − 1. Skipping the month of January of year
t, this sub-decile’s holding period is from February to July of year t.
The Liu-Zhang (2014) timing alignment procedure contains three steps. First, each month, the
timing of firm-level characteristics at the sub-decile level is determined. The general principle is
to take firm-level characteristics from the fiscal yearend that is closest to the month in question
to measure economic variables dated time t in the model, and to take characteristics from the
subsequent fiscal yearend to measure variables dated t + 1 in the model.
Figure A1 illustrates the general principle for firms with December fiscal yearend. Firms with
fiscal year ending in other months are handled analogously. In Compustat stock variables are measured at the end of the fiscal year, and flow variables are over the course of the fiscal year. As
such, the investment return constructed from annual accounting variables goes roughly from the
midpoint of the current fiscal year to the midpoint of the next fiscal year. For firms with December
fiscal yearend, this midpoint time interval is from July of year t to June of year t + 1.
Take, for example, the first sub-decile of the loser decile in July of year t. As noted, this subdecile’s holding period is from February of year t to July of year t. For firms in this sub-decile, the
first five months (February to June) lie to the left of the applicable time interval. For these five
months, accounting variables at the fiscal yearend of calendar year t are used to measure economic
variables dated t + 1 in the model, and accounting variables at the fiscal yearend of t − 1 are used to
measure economic variables dated t in the model. However, for the last month in the holding period
(July), because the month is within the midpoint time interval, accounting variables at the fiscal
yearend of t + 1 are used to measure economic variables dated t + 1 in the model, and accounting
variables at the fiscal yearend of t are used to measure economic variables dated t in the model.
123
Second, the components of the levered investment return are constructed at the sub-decile level.
For each month, characteristics for a given sub-decile are calculated by aggregating firm characteristics over the firms in the sub-decile. For example, the sub-decile investment-to-capital for month
t, Iit /Kit , is the sum of investment for all the firms within the sub-portfolio in month t divided by
the sum of capital for the same set of firms in month t. Because the portfolio composition changes
monthly, the sub-decile characteristics also change monthly.
Figure A1: The Liu-Zhang (2014) Timing Alignment
✛
✲
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)
Dec/Jan
✛
Dec/Jan
✻
✻
✲
(from January of year t + 1
to December of t + 1)
June/July of t
Kit ✻
τ t+1 , δ it+1
Yit+1 , Iit+1
Dec/Jan
June/July of t+1
Kit+1
Kit+2
✻
The holding period,
February–July of year t,
for the first sub-decile of
a momentum decile
in July of year t
The holding period, July–December of year t,
for the sixth sub-decile of a momentum
decile in July of year t
Third, the levered investment returns for a given decile are calculated. Continue to consider
the loser decile. After obtaining its sub-decile characteristics, in each month the cross-sectional
average characteristics is computed over the six sub-deciles to obtain the characteristics for the
loser decile for each month. These characteristics are then used to construct the investment returns
per equation (26) and levered investment returns per equation (28). The investment returns are in
annual terms but vary monthly because the sub-decile characteristics change monthly.
124