Technology Adoption and the Capital Age Spread∗
Xiaoji Lin†
Berardino Palazzo‡
Fan Yang§
May 11, 2016
Abstract
We explore the asset pricing implications of an investment-based model that features a
stochastic technology frontier and costly technology adoption. Firms adopt the latest
technology embodied in new capital to reach a stochastic technology frontier, but this
decision entails an adoption cost. The model predicts that old capital firms are more
risky and hence offer a higher returns than young capital firms. This is because old
capital firms are more likely to upgrade their capital in the near future and hence are
more exposed to shocks driving the technology frontier. Our empirical analysis supports
the model’s predictions. We find an annual return spread of 7% between old and young
capital firms. The CAPM fails in explaining this return spread.
JEL Classification: E23, E44, G12
Keywords: Technology adoption, technology frontier shock, vintage capital, investment,
capital age, stock returns
∗
We thank Frederico Belo, Yen-cheng Chang, René Stulz, and Lu Zhang for their comments. All errors are
our own.
†
Department of Finance, Fisher College of Business, The Ohio State University, 2100 Neil Avenue, Columbus
OH 43210. e-mail:lin.1376@osu.edu
‡
Department of Finance, Questrom School of Business, Boston University, 595 Commonwealth Avenue,
Boston, MA 02215. e-mail:bpalazzo@bu.edu
§
Finance Department, School of Business, University of Connecticut, 2100 Hillside Road, Storrs, CT 06269.
e-mail:fan.yang@business.uconn.edu
1
1
Introduction
Over the last few decades, the nature of economic growth and productivity advancement
has transformed profoundly: technological changes taking the form of adopting the new and
more productive capital goods—especially in information and communication equipment and
software—have represented the major source of output growth in the United States and Europe
(Jorgenson, 2001). Productivity growth embodied in new capital has accelerated significantly
over the past 30 years, from 2 percent per year in the 1960s to 4.5 percent in the 1990s (Gordon,
1990; Cummins and Violante, 2002). These findings have motivated a growing literature that
investigates the link between technological progress and asset prices (e.g., Greenwood and
Jovanovic 1999, Jovanovic and Rousseau 2001, Laitner and Stolyarov (2003), Albuquerque and
Wang (2008), Papanikalaou 2011, Jermann and Quadrini 2012, Garleanu, Panageas, and Yu
(2012), Kogan and Papanikalaou (2014), Garlappi and Song (2014), among many others).
In this paper, we follow the idea of asynchronous technology adoption in Jovanovic and
Stolyarov (2000) to study the implications of firms’ heterogeneity in technical efficiency for
the cross-section of equity returns. We do so by developing an investment-based model that
features a stochastic technology frontier and costly technology adoption. In our model, the
technology frontier, which all firms have access to, follows a stochastic process driven by a
systematic shock. Facing this shock and the standard aggregate and firm-specific productivity
shocks, firms can decide to adopt the latest vintage capital, which is more efficient, or to keep
operating with the existing vintage capital which will become obsolete (i.e., less productive)
over time.
In the model, firms incur a cost when adopting the latest technology. This cost consists of
two parts: a linear variable cost and a fixed adoption cost. The adoption costs arise because
not all firms’ existing expertise (human capital or workers’ skills) can be applied to the new
technology. In particular, the fixed adoption cost delays the adoption decision of large firms
(i.e., firms with relatively new capital vintage and high productivity). On the other hand,
because of the linear costs, the further a firm’s installed capital is from the technology frontier,
the more costly the adoption of the latest capital vintage. This feature of the model mimics the
behavior of many of the value firms in real world. Rather than upgrading their capital, they
2
keep using their vintage capital until bankruptcy. The benefit of adopting the latest capital
vintage is a more productive installed capital. Thus, firms trade-off the cost of adoption and
the benefit of more efficient technology embodied in the new capital.
The key insight of the model is that firms adopting the latest technology (young capital age
firms) are less risky than non-adopting firms (old capital age firms). In the model, young capital
age firms are characterized by high investment, low book-to-market ratio, high productivity,
and high market equity. Thus, the model provides a novel explanation for the cross sectional
variations of stock returns associated with capital age, investment, book-to-market, firm-specific
productivity, and size.
The economic mechanism behind the model’s results is as follows. There are essentially two
types of firms in the economy: adopting firms and non-adopting firms. The capital installed
for adopting firms has already been upgraded with the latest technology. More importantly,
even when facing a positive productivity shock, the adopting firms will delay further investment
because of the fixed adoption cost. Hence, their continuation value is less exposed to the advance
of the technology frontier or equivalently, to the technology frontier’s shock. In contrast, nonadopting firms are more likely to upgrade their capital in the near future if they face a positive
productivity shock. Therefore, their continuation value loads more on the technology frontier
shock. This mechanism allows the model not only to generate a capital age premium but also
other important features of the cross—section of equity returns like a value premium, a size
premium, an investment rate spread and a firm–level productivity spread.
Through several comparative static exercises, we show that the existence of technology
adoption costs is important for the model to capture the cross sectional return spreads. In
particular, without fixed adoption costs (not convex costs), the model generates value, size, and
investment spreads that are much smaller relative to their benchmark values and the average
capital age in the economy drops dramatically. The model mechanism departs from the existing
literature because it does not rely on convex adjustment costs or on an irreversibility constraint
to generate sizable differences in cross–sectional returns.
By linking vintage capital to firm risk, the model sheds light on the relationship between
firms’ capital age and expected returns. More specifically, firms with newer capital (younger
capital age) are less risky. This is a novel prediction distinct from the standard investment3
based models where capital vintage is homogeneous across firms and there is no distinction
between new and old capital. Because we do not directly observe firms’ capital age in the data,
we follow Salvanes and Tveteras (2004) and use firm–level investment data to build a measure
of capital age that we use to test the main model’s prediction.
We form portfolio sorting firms on their capital age and we find evidence consistent with the
model prediction: U.S. publicly traded companies with young capital age earn lower average
returns than ones with old capital age. In particular, a spread portfolio of stocks that goes long
on young firms and short on old firms generates a significant equally–weighted (value-weighted)
spread of 7% (6%) per year.1 We also verify if the capital age premium survives when we
control for other firm–level characteristics that are correlated with investment activities and
the cross–section of equity returns like size, book–to–market, and investment rate. We find
that the capital age premium remains positive and significant both when we form double–
sorted portfolios and when we run Fama–MacBeth cross–sectional regressions using individual
stock returns.
2
The model
In this section, we describe and solve an investment–based asset pricing model that features a
stochastic technology frontier and fixed adoption costs. In each period, a firm can choose to
invest and adopt the latest technology or keep using its old capital. Firms choose the optimal
investment policy to maximize their shareholders’ value.
2.1
Production technology
Firms use their physical capital (Kt ) to produce a homogeneous good (Yt ) while facing an
aggregate productivity shock (Xt ) and a firm-specific productivity shock (Zt ). To save on
notation, we omit firm index j whenever possible. To simplify the model solution, we assume
1
Zhang (2006) and Jiang, Lee and Zhang (2005) document the similar finding regarding firm age and the
expected returns, but they attribute their findings to information uncertainty and behavior bias, respectively.
4
a constant return to scale production function:
Yt = Xt Zt Kt .
(1)
This assumption is not uncommon in the literature (e.g., Eisfeldt and Papanikolaou (2013))
and does not drive our major results.
Log aggregate productivity (xt = log Xt ) follows a random walk process with a constant
growth rate gx ,
xt+1 = xt + gx + σx et+1 ,
(2)
where σx measures the volatility and et+1 is an independent and identically distributed (i.i.d.)
standard normal variable.
Log firm-specific productivity (zt = log Zt ) follows an AR(1) process,
zt+1 = ρz zt + (1 − ρz )z̄ + σz ut+1 ,
(3)
where ρz is the persistence, σz is its conditional volatility, and z̄ is the long run mean of firmspecific productivity. ut+1 denotes an i.i.d standard normal variable. In the model, all firms
are ex ante identical. But they differ in the realization of their firm-specific productivity and
hence their physical capital. The various realized paths of firm-specific productivity generate
an artificial cross–section of firms that we use to study the cross–section of equity returns.
2.2
Costly technology adoption
We denote the stock of general and scientific technology of the entire economy with Nt .
Following Parente and Prescott (1994), Greenwood and Yorukoglu (1997), and Cooper,
Haltiwanger, and Power (1999), we assume that the technology frontier Nt grows at an i.i.d.
stochastic rate,
Nt+1 = Nt egN +σN ηt ,
(4)
where gN is the average log growth rate and σN is the volatility. ηt denotes an i.i.d standard
normal variable. The timing of ηt and the technology frontier is slightly different from the
5
productivity shocks. The technology frontier at t + 1 is determined by the shock (ηt ) at t so
that there is no uncertainty in the investment cost of adopting the technology frontier at t.
Given the productivity shocks (xt , zt ) and the level of technology, Nt , the firm chooses
between adopting the latest technology, Nt+1 , or continue using the existing vintage capital,
Kt , for another period. Hence the capital stock for the firm evolves as follows:
Kt+1

 (1 − δ) K if φ = 0
t
t
=
,
 N
if φt = 1
t+1
(5)
where δ is the rate of depreciation for capital. The choice variable in this problem is φt where
φt = 1 means that new technology is adopted in period t and the existing vintage capital is
replaced. Accordingly, investment rate is given by

 0
if φt = 0
It
it =
=
.
Kt  Nt+1 − (1 − δ) if φt = 1
Kt
(6)
The gain of technology adoption is that the new capital is more efficient than old vintage
as it reflects the current technological progress. This can be seen by comparing two series of
capital over time: {N0 , N1 , N2 , ..., Nt } and N0 , (1 − δ) N0 , (1 − δ)2 N0 , ..., (1 − δ)t N0, . The
first series represents the case where the firm adopts the latest technology every period and is
able to stay on the technology frontier in the entire history, whereas the second case represents
another case where the firm is unable to adopt the latest technology and remains operating the
old vintage capital all the time. As the technology frontier evolves over time, the capital of the
firm in the first case is on average more productive in terms of efficiency unit (units of output
to be produced) than the capital in the second case which is effectively obsolete. For example,
1
2
at t, the expected capital of the first firm is E[Nt ] = egN t+ 2 σN t N0 , which can be an order of
magnitude more efficient than the capital of the second firm, (1 − δ)t N0 , when t is large.
All firms can adopt the latest technology vintage, but it is costly to do so. We assume that
technology adoption entails an investment cost Ct given by

 0
if φt = 0
Ct =
.
 C X (f K + i K ) if φ = 1
q t
i t
t t
t
6
(7)
The investment cost per unit of investment (Cq Xt ) varies over time and it is driven by aggregate
productivity as in Jermann (1988) and Eisfeldt and Papanikolaou (2013). Without loss of
generality, we set Cq = 1. Other than the stochastic unit cost, the investment costs consists of
two parts: a fixed cost (fi Kt ) and a linear variable cost (it Kt ). Here, the fixed cost captures the
cost of learning new technology, workers training costs, and the cost of abandoning old capital.
It could also include the cost in the destruction of old organizational capital or human capital
of existing workers who are used to the old vintage capital.
The fixed investment cost in equation (7) causes asyncronous technology adoption as in
Jovanovic and Stolyarov (2000) that leads to firms’ heterogeneity in technical efficiency. In
addition, our model implies inaction when the firm chooses not to adopt the latest technology,
and an investment spike when the firm does. As a result, investment lumpiness arises in the
model, which is consistent with the firm-level investment data. This is different from the smooth
investment series implied by the q-theory of investment which is usually assumed in standard
investment-based models.
2.3
Firms’ problem
Following Berk, Green, and Naik (1999), Zhang (2005), and Eisfeldt and Papanikolaou (2013),
we directly specify the pricing kernel without explicitly modeling the consumer’s problem. The
log pricing kernel is given by
1
1
log Mt,t+1 = −r − λ2e − λ2η − λe et+1 − λη ηt+1
2
2
(8)
Both aggregate productivity shock (et+1 ) and technology frontier shock (ηt+1 ) are priced in
this economy. The drift term is designed such that the pricing kernel prices the risk-free asset
(Et [Mt,t+1 ] = e−r ). The risk-free rate (r) is assumed to be a constant so that the long–term
equity risk premium is not generated by interest rate risk. λe and λη parameterize the price of
aggregate risk and technology risk, respectively.
The firm maximizes shareholders’ value by choosing to adopt the technology frontier (φt = 1)
7
or keep using its vintage capital (φt = 0)
V (Zt , Kt , Xt , Nt ) = max Dt + Et [Mt,t+1 V (Zt+1 , Kt+1 , Xt+1 , Nt+1 )],
φt
(9)
subject to the budget constraint Dt = Yt − Ct and the capital law of motion in equation 6,
where Dt is the time t dividend.
Since both aggregate productivity and the technology frontier follow random walk processes,
the firm problem is non-stationary. We show how to obtain a detrended version of the economy
in the Appendix (Section A1).
2.4
Risk and expected stock return
In the model, risk and expected stock returns are determined endogenously along with firms’
value-maximization. Evaluating the value function at the optimum, we obtain
Vt = Dt + Et [Mt,t+1 Vt+1 ]
(10)
⇒ 1 = Et [Mt,t+1 Rt+1 ] ,
(11)
where equation (10) is the Bellman equation for the value function and equation (11) follows
from the standard formula for stock return Rt+1 = Vt+1 / [Vt − Dt ] . Note that if we define
Pt ≡ Vt − Dt as the ex-dividend market value of equity, Rt+1 reduces to the common definition
of stock return, Rt+1 ≡ (Pt+1 + Dt+1 ) /Pt .
Following Cochrane (2001 p. 19), we rewrite equation (11) as the beta-pricing form
Et [Rt+1 ] − r = βe λe + βη λη ,
(12)
where r = − log(E [Mt,t+1 ]) is the real interest rate. β = (βe , βη ) denotes the vector of the
quantities of risk and it is defined as:
β=−
Covt [Rt+1 , Mt,t+1 ]
.
Vart [Mt,t+1 ]
(13)
In the model, the prices of risks are exogenously specified in the pricing kernel. However, the
8
model is able to generate cross–sectional dispersion in risk premia thanks to the heterogeneity
in the quantities of risk (β).
3
Properties of model solutions
In this section we discuss the solution and the calibration of the model. After detrending, all
the endogenous variables are functions of three state variables: (i) the endogenous capital kt ;
(ii) the firm–level productivity zt ; and (iii) the technology shock ηt . Because the functional
forms are not available analytically, we solve for these functions numerically. Appendix A2
provides a description of the solution algorithm (value function iteration) and the numerical
implementation of the model.
The model is solved at a monthly frequency. Table 1 reports the parameter values used in
the baseline calibration. The model is calibrated using parameter values reported in previous
studies, whenever possible, or by matching a set of empirical moments. Table 2 reports the
model–generated moments together with their empirical counterparts.
To neutralize the impact of the initial condition, we simulate a panel of 5, 000 firms for
80 years with a monthly frequency to generate a stationary cross sectional distribution of
firms. Each firm is characterized by the firm–level state variables zt and kt . Then, using this
distribution of firms as initial condition, we simulate 100 panels of artificial data with sample
size of 35 years and 5, 000 firms. 35-year is chosen to match the empirical sample size. We
report the cross-sample average results as model moments. Because we do not explicitly target
the cross section of return spreads in the baseline calibration, we use these moments to evaluate
the model in Section 4.
3.1
Calibration
Stochastic processes: We set the annual average log growth of the technological frontier (12gN )
equal to 0.0138, consistent with the estimate in Greenwood, Hercowitz and Krusell (1997).2
2
We choose to calibrate the growth rate gN as that of the investment specific technological change, but the
notion of the technology frontier in the model is broader than the investment specific technological change in
Greenwood et al (1997). The quantitative implications of the model remain unchanged with different values of
the growth rate gN .
9
In the model, the aggregate productivity shock xt is essentially a profitability shock. We set
annual the average log growth of aggregate productivity (12gx ) equal to 0.012 to match the
average growth of aggregate profits and the volatility of the aggregate productivity shock to be
σx = 0.045 to match the volatility of aggregate profits. In the data, we measure aggregate profits
using data from the National Income and Product Accounts (NIPA). Given the volatility of
the aggregate productivity shock, we set the volatility of log technology frontier to σN = 0.055.
The long-run average of firm-specific productivity, z̄, is a scaling variable. We set z̄ = −3.5
so that average physical capital scaled by the technology frontier (kt ) across firms is around
0.5. To calibrate the persistence ρz and conditional volatility σz of firm-specific productivity,
we restrict these two parameters using their implications on the degree of dispersion in the
cross-sectional distribution of firms’ stock return volatilities. Thus ρz = 0.97, and σz = 0.12,
which implies an average annual volatility of individual stock returns of 30%, consistent with
Campbell at al (2001).
Firm’s technology:
The monthly capital depreciation rate (δ) is set to 0.01 as in Jermann
(1998). Without loss of generality, we normalize the average per unit cost of investment (Cq )
to 1. We set the fixed cost of technology adoption, fi = 2.8, to match the average capital age
of 6 years in the data. We also assume the presence of a fixed operating cost (fo = 0.0025) that
we calibrate to match the average capital-to-market equity ratio.
Pricing kernel: The annual real risk-free rate is chosen to match the data 12r = 0.022. We set
the price of aggregate risk to be λe = 3σx and the price of the technology risk to be λη = 6σN
by matching average stock market return and the Sharpe ratio. This implies an annual market
excess return of 7% and Sharpe ratio of 44%, values close to their empirical counterparts. These
risk prices also imply an equivalent risk aversion parameter of 3 with respect to the aggregate
productivity shock and an equivalent risk aversion parameter of 6 with respect to the technology
frontier shock. Both are within reasonable range.
10
3.2
Model solutions
Using the benchmark parametrization, we discuss how the key endogenous variables such as
the optimal technology adoption policy, investment, cum-dividend and ex -dividend firm value,
and conditional beta are determined by the underlying state variables.
3.2.1
Value functions and policy functions
Figure 1 compares the optimal investment policies for high and low productive (z) firms. Figure
2 compares the cum-dividend (v), the ex -dividend firm value (ex-div v), the book-to-market
ratio (BM), the dividend (div), the stock risk premium (E[Re ]), and the stock beta to the
technology frontier shock (βη /σN ) for high and low productive (z) firms. All variables are
detrended.
The upper panel in Figure 1 plots the optimal capital in next period (kt+1 ) as a function
of the capital in this period (kt ). The lower panel reports the corresponding investment rate
which is defined as the ratio of investment over capital installed (I/K). The vintage of the
installed capital plays a key role in shaping the technology adoption policies in the benchmark
model.
First, high productive firms (high z) optimally choose to adopt the technology frontier–
which is represented as 1 for the detrended capital–when their technology is obsolete (low level
of capital). Their investment rates are in general positive and declining in k, as shown in the
bottom panel. In contrast, low productive firms (low z) choose to keep operating with their
obsolete capital and hence do not invest. Their investment rates are zero regardless of current
capital level.
Second, high productive firms optimally decide not to adopt the newest technology when
their capital vintage is relatively recent. This is due to the fixed cost of investment in the model.
These high productive firms, which have recently updated their capital stock, optimally choose
to delay their adoption decision. These firms are characterized by high z and high capital k.
This inaction region generates lumpy investment in our model, a feature we observe in the
firm-level investment data. More importantly, this channel helps to generate a sizable number
of firms with high capital age, as in the data. In a later section, we show that the model implied
11
capital age drops significantly when removing this fixed investment cost.
Figure 2 depicts the cum-dividend (v), the ex -dividend firm value (ex-div v), the book-tomarket ratio (BM), the dividend (div), the stock risk premium (E[Re ]), and the stock beta to
the technology frontier shock (βη /σN ) as functions of detrended capital (k) for high and low
productive (z) firms. The exogenous firm-specific productivity shock (z) generates heterogenous
optimal investment decisions across firms and hence endogenizes many interesting cross sectional
patterns in firms’ characteristics. In this figure, we can observe that with the same current
capital (k), high productive firms (high z) are associated with higher cum-dividend (v) and
ex -dividend firm value (ex-div v). Thus, they have a low book-to-market ratio which is defined
as the ratio of capital over ex -dividend firm value. These are growth firms and their dividends
are negative. Negative dividend in the model can interpreted as equity issuance. These firms
choose to adopt the technology frontier and hence need to raise external fund to finance their
large investment. Most importantly, these firms are less exposed to the technology frontier
shock. If we assume that the technology frontier shock carries a positive price of risk, the
model predicts that high productive firms (high z) offer less risk premium than low productive
firms (low z). In the later section, we show that this key channel produces novel cross sectional
stock return patterns that are consistent with the empirical evidence.
3.2.2
Risk and expected return
After detrending the model, a stock return can be written as
Rt+1 =
Vt+1
vt+1 gN +σN ηt +∆xt+1
=
e
.
Vt − Dt
vt − dt
Since both vt and dt are only functions of state variables (zt , ηt , kt ), the first term
(14)
vt+1
vt −dt
does
not depend on the aggregate productivity shock xt . From Equation (14), the betas to aggregate
productivity shock ∆xt+1 across all the stocks equal to 1. By design, the aggregate productivity
shock does not drive the cross sectional stock return but only drives the market return. We do
this to emphasize the role played by the technology frontier shock in shaping the cross section
of equity returns. This feature also allows the model to generate a failure of the the standard
capital asset pricing model in capturing the cross sectional risk premia linked to the capital age
12
spread and the book–to–market spread.
Figure 3 reports the betas to the aggregate productivity shock and the technology frontier
shock across ten value weighted and ten equal weighted portfolio sorted on capital age and
other firm’s characteristics in the benchmark model. The betas are estimated using the model
simulated shocks and portfolio returns with a two factor model by time series regressions,
e
Rj,t+1
= αj + βj,x ∆xt+1 + βj,η σN ηt+1 .
(15)
The beta to the aggregate productivity shock (βx ) equals to 1 across all the portfolios. As a
consequence, cross–sectional differences in equity returns are not driven by different exposures
to aggregate risk. More interestingly, the model implied beta to the technology frontier (βη /σN )
increases with capital age for both the value weighted and equal weighted portfolios. Stocks
with older vintage capital load more on the technology frontier shock than stocks with younger
vintage capital. With a constant positive price for the technology frontier risk, the model
predicts that old capital stocks offer higher expected returns than young capital stocks.
4
Cross sectional stock returns
An important firm characteristic, which makes this model different from the standard
investment-based model (e.g. Zhang (2005) and Papanikolaou (2011)), is capital age. The
technology frontier represents the latest technology in capital and thus defines capital age zero.
Firms which are close from the technology frontier own relatively new capital. On the other
hand, firms which are far from the technology frontier own capital with high age. Our model
predicts a positive capital age risk premium. In this section, we perform asset pricing tests
using model–generated data to quantitatively explore this positive relation between capital age
and equity returns in the cross—section.
4.1
Capital Age sorted portfolios
In the model, we measure the capital age of a firm as the number of quarters since the firm’s
last adoption decision. Once a firm adopts the technology frontier, we reset its capital age to
13
zero by assuming that it reinstalls all of its capital using the latest vintage.
We use artificial data to create ten portfolios sorted on capital age that we rebalance at
a quarterly frequency. Table 3 reports the values of some key characteristics across the ten
capital age sorted portfolios. The model predicts that old capital firms (high age) are value
firms (high book-to-market ratio), low investment firms (low IK), low productivity firms (low
z), and small firms (low capital k).
Table 4 reports the returns and the asset pricing test results. Panel A and Panel B show
that the average return of old capital firms (column ‘O’) is higher than the average return of
young capital firms (column ‘Y’). The implied return differential (column ‘OMY’) is about 6%
per annum for value weighted portfolios and 13% per annum for equal weighted portfolios.
We also decompose the model predicted cross sectional risk premium into the price of
risk and the quantity of risk using standard asset pricing tests. This decomposition helps us
understand the channel through which the cross sectional risk premium is generated in the
model.
First, we test the standard capital asset pricing model (CAPM) using the ten value weighted
and ten equally weighted portfolios sorted on capital age as test assets. The market return is
defined as the average return across all stocks weighted by their market equity. The market
factor (Mktt+1 ) is the difference between the market return and the constant risk-free rate. We
test the CAPM using the time-series regression,
e
Rj,t+1
= αj + βj,M Mktt+1 + j,t+1 ,
(16)
e
where Rj,t+1
denotes the portfolio excess return, βj,M measures the quantity of the market risk,
and αj denotes the abnormal return. The results reported in Table 4 show that the market
risk does not explain the cross sectional risk premium. In the model, all the portfolios share
the same quantity of market risk (βj,M = 1). The market beta of the OMY portfolio is almost
zero. Consistently, the annual abnormal return of the value weighted OMY portfolio is about
6% and the annual abnormal return of the equal weighted OMY portfolio is about 13%. The
CAPM fails in explaining these portfolios sorted on capital age.
Second, we investigate a two factor model in which the excess market return is the first factor
14
and the technology frontier the second. We decompose the cross sectional risk premium into
the price of risk and quantity of risk using this two factor model. Through this decomposition,
we find that firms with old capital are more exposed to the technology frontier shock than
firms with young capital. With the assumption of a positive price for the technology risk,
the heterogenous risk exposures across firms explain the cross sectional risk premium. More
specifically, we estimate the quantity of risks using the time-series regression,
e
Rj,t+1
= αj + βj,M Mktt+1 + βj,N σN ηj,t+1 + j,t+1 ,
(17)
e
denotes the portfolio excess return, βj,M is the market beta, βj,N measures the
where Rj,t+1
quantity of risk for the technology frontier shock (σN ηj,t+1 ). The lower panel of Table 4 reports
the estimation results. Again, the market beta is flat across all the portfolios. In contrast,
the beta of the technology frontier shock (σN ηj,t+1 ) differs across portfolios. In particular,
old capital firms load more on the technology frontier shock (TFS) than young capital firms.
The beta of the OMY portfolio is 0.41 for the value weighted portfolio and 0.65 for the equal
weighted portfolio. Even though old and young capital firms have the same exposure to market
risk, they differ in their exposure to the technology frontier risk. This is the channel that
generates heterogeneity in cross–sectional equity returns in our model.
In panel C, we report the estimated risk premia for the two linear asset pricing models. The
two factor model produces the lowest mean absolute errors across the ten capital age sorted
portfolios. Not surprisingly, the estimated values for the price of risk are close to their calibrated
values.
4.2
Other portfolios
In this section, we explore the cross sectional risk premia predicted by the model other than the
capital age spread. Following the same empirical procedure in sorting stocks on their capital age,
we form ten value weighted and ten equally weighted portfolios sorted on their book-to-market
ratio (BM), investment rate (IK), market equity (ME), and firm-specific productivity (z),
seperately. The portfolios are rebalanced at quarterly frequency and the reported returns are
annualized. Table 5 reports the average excess returns, volatilities, and statistical significance
15
(t-statistics) of these portfolios. We define the the book-to-market ratio as the ratio of physical
capital over ex-dividend stock value. The quarterly investment rate is computed as the sum of
monthly investment rates over a quarter divided by the beginning of the quarter capital stock.
The model generates a value premium, an investment rate spread, a size premium, and
a productivity spread. More specifically, the model predicts a 9.7% (11.4%) value weighted
(equally weighted) annual risk premium for value firms (high BM) and 4.8% (7.6%) value
weighted (equally weighted) annual risk premium for growth firms (low BM). Thus, the model
can generate a positive value weighted (equally weighted) value premium of 4.9% (3.8%) per
year. The difference in exposure to the technology frontier risk between value and growth
portfolios is much smaller than the one between old capital age and young capital age portfolios.
This difference is reflected in the lower spreads in returns. Investment sorted portfolios have a
even smaller difference in TFS beta and a corresponding lower return spread. The average value
weighted (equally weighted) return differential between high investment and low investment
portfolios generated by the model is around -1.5% (-2.0%) per year. This result is qualitative
consistent with the findings in Xing (2009).
The differences in exposure to the technology frontier risk are more pronounced across size
and productivity sorted portfolios. Big firms and high productive firms on average have lower
return than small firms and low productive firms, respectively. In particular, the value weighted
(equally weighted) small-minus-big spread is 10% (16.7%) per annum and the value weighted
(equally weighted) low-minus-high productivity spread is 5.1% (8%) per annum in the model.
These predictions of cross sectional risk premia are also consistent with the empirical estimates
(e.g. Fama and French (1992) and Imrohoroglu and Tuzel (2013)).
To conclude, we explore how capital age and the idiosyncratic productivity shock jointly
shape the cross–section of equity returns. In the canonical one-factor neoclassical investment
model the differences in equity returns across firms are driven by their different idiosyncratic
productivities. The reason being that assets in place are less risky than growth options and
low idiosyncratic productivity firms are riskier because they have a smaller fraction of their
value tied to assets in place. Table 6 reports portfolios double sorted on capital age and on
idiosyncratic productivity. The results show that capital age is an important source of returns
variability that is not subsumed by idiosyncratic productivity. In our model, the capital age
16
spread is the highest among low idiosyncratic productivity firms and it is decreasing across
idiosyncratic productivity categories.
5
Inspecting the mechanism
This section investigates the economic mechanism in the model that generates heterogeneity
in the cross sectional returns. We solve the model with several alternative specifications and
compare the key moments with the benchmark model. Table 7 reports the results.
The role of a positive λη :
As a first experiment, we set the price of the technology frontier
shock (λη ) to be zero. In this specification, even though the technology frontier shock exists and
is systematic, it does not affect the marginal utility of investors and thus it is not a priced risk.
Stock returns load on this shock. However, the risk exposure to this shock is not associated
with any risk premium. This can be observed from Specification 2 in Table 7. The beta to
the technology frontier shock (TF-shock) after controlling for the market factor for old capital
stocks is 0.17. In contrast, the beta to the technology frontier shock for young capital stocks
is -0.25. The old-minus-young spread portfolio’s beta to the technology frontier is 0.42, a value
almost identical to the one in the benchmark case.3 However, because λη = 0, the capital age
spread drops to almost 0. It is also interesting to note that not only the capital age spread drops
to 0, all other cross sectional risk premia reduce to almost 0 as well. This is not surprising
because the technology frontier shock is the only shock to which stocks have heterogeneous
exposure in the model. Therefore, we find that a positive λη is necessary for the model to
generate cross sectional risk premia.
The role of adoption costs:
The presence of fixed technology adoption cost is key to match
both firms’ investment dynamics and cross sectional risk premia. In Specification 3, we set
fi = 0, i.e., there is no fixed investment cost. In this case, the average capital age drops from
24 quarters in the benchmark case to 11 quarters. This is because the inaction region becomes
3
The choice of the price of risk can also affect stock’s risk exposure (betas) through the channel of driving
the optimal investment rate and hence stock valuation.
17
smaller and large (i.e., relatively young) firms with a positive productivity shock are more
willing to adopt the latest capital vintage. The reduction of the inaction region also causes a
sharp reduction in the capital age spread that is now less than 50% smaller than the benchmark
value. Given that size and capital age in our model are highly correlated, the removal of the
fixed technology adoption cost also causes a sharp decrease in the size premium.
The role of operating costs: The removal of the fixed operating cost does not cause dramatic
deviations from the benchmark economy.
Since this variable does not affect the optimal
investment policy, the mean capital age remains the same as in the benchmark economy. On
the other hand, the level of the fixed cost has a large impact on the model’s ability to generate
a value premium. In the model without a fixed operating cost, we are only able to generate an
annualized value premium of 2.82%, a value much smaller than the one observed in the data.
The role of technology risk:
In the last specification, we explore the role of technology risk.
When this risk is negligible (i.e., σN = 0.001) and the price of risk stays the same (λη = 6), there
is virtually no heterogeneity in cross-sectional equity returns. In addition, the mean capital age
becomes much higher than its value in the benchmark economy.
6
Empirical Evidence
In this section, we provide evidence on the relation between capial age and the cross–section
of equity returns. We first describe the data and the methodology we use to build a firm–
level measure of capital age. Then, we perform a battery of asset pricing tests and show that
capital age carries a positive risk premium (age premium), as predicted by the model. The
age premium survives when we control for other firm–level characteristics that are correlated
with investment activities and the cross–section of equity returns: size, book–to–market, and
investment rate.
18
6.1
Data
Stock prices and quantities come from CRSP; accounting data come from COMPUSTAT
Quarterly. We restrict our sample to companies listed in the three major stock exchanges
(AMEX, NYSE, and NASDAQ). We exclude companies non incorporated in the USA, and we
also exclude financials (SIC codes from 6000 up to 6999), utilities (SIC codes from 4900 up to
4999), and R&D–intensive sectors4 (SIC codes 737, 384, 382, 367, 366, 357, and 283) from our
sample. In addition, we only consider ordinary common shares in CRSP. The sample period is
1981q1–2013q4. All the quantities are winsorized at the top and bottom 1% to attenuate the
impact of outliers.
We measure capital age following the methodology in Salvanes and Tveteras (2004). We
start by defining an initial measure of firm–level capital stock (Ki,0 ) for firm i using gross
property plant and equipment (item ppegtq) and an initial measure of firm–level capital age.
The latter quantity is calculated using the average industry capital age from the BEA5 .
Then we recursively build a measure of firm–level capital stock using
N
Ki,t+1 = Ki,t + Ii,t
,
(18)
N
where Ii,t
is net investment of firm i between period t and t + 1. Net investment is defined
as the difference in net property plant and equipment (item ppentq) between two consecutive
N
, where δj is the depreciation of
quarters. We define gross investment as Ii,t = δj Ki,t + Ii,t
industry j calculated using depreciation data from the BEA. All the quantities are expressed
in 2009 constant dollars using the seasonal adjusted implicit price deflator for non–residential
fixed investment.
Once we have a time–series of capital stock and gross investment observations at the firm
level, we define the capital age of firm i at time t as the following weighted average:
AGEi,t
P
t−j−1
(1 − δj )t Ki,0 (AGEi,0 + t) + t−1
Ii,j (t − j)
j=0 (1 − δj )
=
.
Ki,t
4
(19)
We define R&D–intensive sectors following the definition of Brown, Petersen, and Fazzari (2009).
We use Table 3.10E in the BEA dataset, which reports the historical-cost average age of private equipment.
Industries are defined using the North American Industry Classification System (NAICS)
5
19
The above expression implies that at time t = 0 a firm has a capital age equal to AGEi,0 ; at time
t = 1 the capital age becomes ((1−δj )Ki,0 (AGEi,0 +1)+Ii,0 )/Ki,1 , where Ki,1 = (1−δj )Ki,0 +Ii,0 ;
at time t = 2 the capital age becomes ((1 − δj )2 Ki,0 (AGEi,0 + 2) + (1 − δj )Ii,0 2 + Ii,1 )/Ki,2 ,
where Ki,2 = (1 − δj )2 Ki,0 + (1 − δj )Ii,0 + Ii,1 and so on. For each firm, we eliminate the first
eigth quarters to minimize the impact of the initial choice for the capital age value.
The other key quantities in our analysis are the (gross) investment rate, profitability, equity
issuance activity, market capitalization (i.e., size), the book–to–market ratio, and firm–level
N
productivity. We measure the investment rate as gross investment (δj Ki,t + Ii,t
) divided by
the beginning of the period capital stock (Ki,t ). Profitability is income before extraordinary
items (item ibq) divided by the previous quarter book value of equity. The latter quantity is
constructed following Hou, Xue, and Zhang (2015) and it is equal to shareholders’ equity (item
seqq) plus deferred taxes and investment tax credit (item txditcq, if available) minus the book
value of preferred stock (item pstkrq). If shareholders’ equity is not available, we use common
equity (item ceqq) plus the carrying value of the preferred stock (item pstkq). If common equity
is not available, we measure shareholders’ equity as the difference between total assets (item atq)
and total liabilities (item ltq). The book–to–market ratio is the book value of equity divided
by the market capitalization (item prccq times item cshoq) at the end of the fiscal quarter.
We measure the equity issuance activity using the sale of common and preferred stocks (item
sstky) net of the purchase of common and preferred stocks (item prstkcy) and cash dividends
(item dvy) scaled by the beginning of the period book value of assets (item atq)6 . We measure
firm–level productivity using the annual estimates provided by Imrohoroglu and Tuzel (2014).
We normalize the latter quantity to have a zero unconditional mean. Market capitalization is
calculated using data from CRSP and it is equal to the number of shares outstanding (item
shrout) multiplied by the share price (item prc). When size is reported in levels, we express it
in 2009 dollar using the personal consumption expenditure price deflator.
6
In the Compustat quarterly database, sale of common and preferred stocks, purchase of common and
preferred stocks, and cash dividends are reported as year-to-date proceeds. As an example, equity issues (sstky)
in quarter 2 is the sum of the equity issues from quarters 1 and 2. The quarter 3 equity issue is the sum of equity
issues from quarters 1, 2, and 3. To get the actual equity issues in quarter 3, we subtract the quarter 2 equity
issues from the quarter 3 equity issues. The same procedure is used to derive quarterly equity repurchases and
quarterly cash dividends distributions.
20
6.2
Summary Statistics
Table 8 reports the summary statistics for our measure of capital age and other variables used
in the empirical analysis. The average (median) capital age in the sample is 23 (23) quarters
with a volatility of 9 quarters. Consistent with the model, capital age is negatively correlated
with the investment rate and firm–level productivity and positively correlated with the book–
to–market (Table 9). Differently from the model, the data show a positive relation between
capital age and firms size.
6.3
Portfolio Analysis
In this section, we replicate the portfolio analysis performed with the artificial data. We form
portfolios at a quarterly frequency at the beginning of January, April, July, and October of
each year starting in 1981 and ending in 2014. Accounting–based characteristics are evaluated
at least six months prior to portfolio formation, while market capitalization is calculated at the
end of the last month preceding portfolio formation.
We start analyzing portfolios sorted on capital age. Then, we form portfolios double sorting
on capital age and other firm–level characteristics that have been successful in explaining the
cross–section of equity returns: size, book–to–market, and investment rate.
6.3.1
One–way sorts
Table 10 reports the time–series average of the cross–sectional mean for capital age, book–
to–market, investment rate, market size (log), profitability, equity issuance, and firm–level
productivity across ten portfolios sorted on capital age.
Young firms have an average capital age of 9 quarters and a half, while the Old firms’ value
is four time as large. The sorting on capital age is also able to generate a sizable spread in the
investment rate, the issuance activity, and the firm–level productivity. The differences across
the top and bottom decile for these variables are more than half of their unconditional standard
deviation.
Table 11 reports excess equity returns across the ten portfolios together with the
corresponding risk–adjusted returns derived using the CAPM. Consistent with the model, firms
21
with older capital in place earn an excess return over firms with younger capital in place. The
annualized equally–weighted age premium is 7.44% and is statistically significant at the 1%
level. The corresponding risk–adjusted age premium is larger, significant at the 1% level, and
equal to 9.72% per year. The age premium becomes smaller when returns are value weighted,
however it remains statistically different from zero. In this case, the annualized value is 5.52%
for raw returns and 8.04% for risk–adjusted returns.
6.3.2
Two–way sorts
Tables 12, 13, and 14 report the results for portfolios double–sorted on capital age and size,
book–to–market, and investment rate, respectively. We perform this analysis to verify if the
age premium survives when we control for firm–level characteristics that are correlated with
investment activities and the cross–section of equity returns.
Table 12 shows that size does not subsume the age premium, which remains always positive
and significant. Panel A shows that the age premium is not monotonic across size categories
and it is larger for medium size firms. This features survives when we use value–weighted
returns (Panel C) and when we calculate risk–adjusted returns using the CAPM (Panel B and
Panel D).
When we condition on book–to–market, the age premium remains positive but is not
significant in some instances when we use value–weighted returns (Table 13 ). Also in this
case, the age premium is not monotonic across book–to–market categories. Differently from
age and size sorted portfolios, the age premium is smaller for medium book–to–market firms.
To conclude, we report the equity returns across age and investment rate sorted portfolios.
Since our capital age measure relies on investment rates, we might find not significant age premia
across investment rate categories. This is not the case. The age premium is significant most
of the time for equally weighted portfolios both when we use raw returns (Panel A) and when
we use risk–adjusted returns (Panel B). The results are weaker when we use value–weighted
portfolios. In this case, the age premium remains positive but it is statistically different from
zero only for low investment rate firms.
22
6.4
Fama–MacBeth regressions
In Table 15 we report the results of Fama–MacBeth cross–sectional regressions using individual
stock returns. The reported coefficient is the average slope from month–by–month regressions
and the corresponding t-statistic is the average slope divided by its time-series standard error.
The reported R–squared is the time–series average of the cross–sectional R-squared.
All
the control variables are divided by their unconditional standard deviation to facilitate the
comparison across regressions. We include the previous month return in all regressions to
control for persistence in equity returns.
Regression (1) shows that a one standard deviation increase in capital age leads to a
significant increase of 0.165% in the average monthly equity return. Table 10 shows that
the difference in average capital age between Old and Young firms is equivalent to 3.5 standard
deviations. The coefficient in Column 1 implies a difference in expected returns of 0.58% per
month, which is equivalent to an annualized different of 6.96%, a value close to the equally–
weighted age premium reported in Table 11.
In regression (2) to (7), we include one characteristic at the time. The coefficient on capital
age remains positive and significant in all cases. In regression (8), we include all the regressors
and also in this case the coefficient on capital age is positive and significant, albeit its magnitude
is 40% smaller than the value in regression (1) . In this case, a one standard deviation increase
in capital age leads to a significant increase of 0.36% in the average monthly equity return,
which translates in an annualized return of 4.32%.
7
Conclusion
We study the relationship among technology adoption, technology frontier shocks, and expected
stock returns in a dynamic vintage capital model. Our results show that technology adoption
combined with a systematic technology frontier shock is an important determinant of the cross
section of returns. In our setting, technology adopting firms are less risky than non-adopting
firms because they are more likely to adopt the technology frontier in the near future in the
face of a positive productivity shock. In the model, adopting firms are high investment, growth,
young capital age, high productive, and high market equity firms. We find the empirical results
23
to be consistent with the model predictions. In particular, we find an annual return spread of
7% between old and young capital firms. The CAPM fails in explaining this return spread.
The model can explain many other cross sectional risk premiums as well.
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26
A1
Detrending the model
Because both aggregate productivity and the technology frontier follow random walk processes
and hence non-stationary, we need to detrend the model before we can apply the value function
iteration method to solve the model. Define detrended variables
vt =
Vt
x
e t Nt
Kt
Nt
Dt
= xt .
e Nt
(20)
kt =
(21)
dt
(22)
After some algebra, the firm problem is equivalent to
vt = max dt + eA+σN ηt Et e−λη ηt+1 vt+1 ,
(23)
dt = ezt kt − fo kt − Cq (fi + it ) kt 1{φt =1}
(24)
1
1
A = −r + gx + gN + σx2 − σx λe − λ2η .
2
2
(25)
φt
where
1{φt =1} is an indicator function which equals 1 if the firm adopts the technology frontier (φt = 1)
and 0 otherwise.
After detrending, the firm’s investment rate is given by
it =



0
(26)


 egN +σN ηt + δ − 1,
kt
and capital dynamics is given by
kt+1 = (1 + it − δ)kt e−gN −σN ηt .
27
(27)
After detrending, there are three state variables in this economy (zt , ηt , kt ). The firm value
and policy functions such as optimal investment rate are functions of these three state variables.
A2
Numerical algorithm
We use the value function iteration procedure to solve the detrended firm’s maximization
problem as in Section A1. The value function and the optimal adoption decisions (φt = 0
or 1) are solved on a grid in a discrete state space. We specify one grid of 100 points for
capital, with a upper bound k̄ = 1. The upper bound represents the technology frontier in the
detrended model. The grid for capital are constructed recursively, following McGrattan (1999),
that is, ki = ki−1 + ck1 exp(ck2 (i − 2)), where i = 1,...,100 is the index of grids points and ck1
and ck2 are two constants chosen to provide the desired number of grid points and the upper
bound k̄, given a pre-specified lower bound k. The advantage of this recursive construction is
¯
that more grid points are assigned around k, where the value function has most of its curvature.
¯
The technology frontier shock ηt is an i.i.d. standard normal shock. We discretize ηt into 5
grid points using Gauss-Hermite quadrature. The firm-specific productivity (zt ) has continuous
support in the theoretical model, but it has to be transformed into discrete state space for the
numerical implementation. Because the firm-specific productivity process zt is highly persistent,
we use the method described in Rouwenhorst (1995) for a quadrature of the Gaussian shocks.
We use 5 grid points for the zt . In all cases, the results are robust to finer grids as well.
Once the discrete state space is available, the conditional expectation can be carried out simply
as a matrix multiplication. Cubic spline interpolation is used extensively to obtain optimal
investment and hiring that do not lie directly on the grid points. Finally, we use a simple
discrete global search routine in maximizing the firm’s problem.
28
Table 1: Parameter values under benchmark calibration
This table presents the calibrated parameter values of the benchmark model.
Parameter
Value
Description
Stochastic process
12gN
0.0138
Average log growth of the technology frontier
σN
0.055
Volatility of log technology frontier
12gx
0.012
Average log growth of aggregate productivity
σx
0.045
Volatility of log aggregate productivity
ρz
0.97
Persistence of firm-specific productivity
z̄
-3.5
Long run average of firm-specific productivity
σz
0.12
Conditional volatility of firm-specific productivity
0.01
Rate of capital depreciation
Technology
δ
Cq
1
fi
2.8
fo
0.0025
Average per unit cost of investment
Fixed cost of technology adoption
Fixed operating cost
Pricing kernel
12r
0.022
The real risk-free rate
λe /σx
3
Risk price of aggregate productivity shock
λη /σN
6
Risk price of the technology frontier shock
29
Table 2: Unconditional moments under the benchmark calibration
This table presents the selected moments in the data and implied by the model under the benchmark calibration.
The reported statistics in the model are averages from 100 samples of simulated data, each with 3,600 firms and
30 years of monthly observations. Benchmark refers to the benchmark model. We report the cross-simulation
averaged annual moments. The data moments are estimated from a sample from 1981 to 2014.
Moments
Data
Model
Real risk-free rate
0.022
0.022
Volatility of real risk-free rate
0.029
0
Market premium
0.057
0.068
0.35
0.44
0.067
0.049
0.62
0.51
0.3
0.27
Std. dev. of aggregate profits
0.14
0.17
Mean capital age (in quarters)
23.19
24.13
Market Sharpe ratio
Value premium
Average capital-to-market-equity ratio
Average individual stock volatility
30
Table 3: Characteristics of capital age sorted portfolios
This table reports the characteristics of ten value weighted (VW) and ten equal weighted (EW) portfolios
sorted on capital age in the model. Column‘O’ reports the portfolio consists of firms with the oldest capital
and Column ‘Y’ reports the portfolio consists of firms with the youngest capital. Column ‘OMY’ reports
the difference between Portfolio ‘O’ and Portfolio ‘Y’. The reported statistics in the model are averages from
100 samples of simulated data, each with 3,600 firms and 30 years of monthly observations. Capital age in the
model is the number of quarters for firms since the last technology adoption. To be consistent with the empirical
analysis, the portfolio is rebalanced at quarterly frequency. All returns are annualized.
Y
2
3
4
5
6
7
8
9
O OMY
Age
3.13 7.49 11.48 15.51 19.5 23.46 27.44 31.82 37.66 50.76
BM
0.47 0.49
0.5
IK
0.76 0.05
0
ME
1.75 1.54
z
k
D/P (%)
0.51 0.52
47.63
0.52
0.53
0.53
0.53
0.49
0.02
0
0
0
0
0
0
-0.76
1.37
1.23 1.11
1
0.89
0.79
0.68
0.54
-1.21
0.25 0.17
0.12
0.08 0.05
0.02
-0.02
-0.07
-0.18
-0.43
-0.68
0.79 0.73
0.66
0.6 0.55
0.5
0.45
0.4
0.34
0.24
-0.55
1.5 1.43
1.37
1.29 1.18
1.06
0.84
0.48
-0.2
-1.56
-3.06
0
31
Table 4: Capital age sorted portfolios: returns and asset pricing tests
This table reports the excess returns and the asset pricing test results of ten value weighted (VW) and ten
equal weighted (EW) portfolios sorted on capital age in the model. Panel A and Panel B report raw and
risk-adjusted excess returns for value weighted and equally weighted portfolios, respectively. Column‘O’ reports
the portfolio consisting of firms with the oldest capital and Column ‘Y’ reports the portfolio consisting of firms
with the youngest capital. Column ‘OMY’ reports the return difference of Portfolio ‘O’ and Portfolio ‘Y’. The
test results of two linear asset pricing models are reported in Panel C. The first linear model is the standard
capital asset pricing model (CAPM) which uses the value weighted market excess return as a single factor.
The other linear model is a two factor model (2F) in which the value weighted market excess return and the
technology frontier shock are the two factors. The reported statistics in the model are averages from 100 samples
of simulated data, each with 3,600 firms and 30 years of monthly observations. Capital age in the model is the
number of quarters for firms since the last technology adoption. To be consistent with the empirical analysis,
the portfolio is rebalanced at quarterly frequency. All returns are annualized.
E[re ]
[t]
SR
MKT
[t]
TFS
[t]
Alpha
[t]
Alpha2F
[t]
E[re ]
[t]
SR
MKT
[t]
TFS
[t]
Alpha
[t]
Alpha2F
[t]
Y
2
3
4.09
1.36
0.25
5.26
1.76
0.33
6.06
2.05
0.38
1.00
116.79
-0.11
-11.51
1.00
144.64
-0.10
-12.48
1.00
162.53
-0.09
-11.13
-3.38
-3.90
-1.58
-2.53
-2.21
-2.93
-0.50
-1.13
-1.42
-2.14
0.09
0.35
5.07
1.69
0.31
6.12
2.05
0.38
6.88
2.32
0.43
1.00
125.90
-0.08
-8.61
1.00
148.99
-0.08
-8.99
1.00
156.25
-0.06
-7.34
-4.72
-4.60
-1.30
-2.08
-3.68
-3.99
-0.37
-0.71
-2.94
-3.52
0.10
0.42
4
5
6
7
Panel A: Value Weighted
Portfolio returns and Sharpe ratios
6.80
7.42
8.09
8.72
2.30
2.49
2.69
2.84
0.42
0.46
0.50
0.53
1.00
155.76
-0.06
-7.46
Risk factor betas
1.00
1.00
1.01
142.24
126.33
115.67
-0.03
0.02
0.07
-2.94
1.41
5.33
8
9
O
OMY
9.43
3.01
0.56
9.98
3.09
0.58
10.03
2.98
0.56
5.94
3.56
0.67
1.01
103.33
0.13
9.03
1.01
91.58
0.21
12.88
1.00
80.44
0.30
16.93
0.00
0.33
0.41
22.45
1.68
2.59
0.49
1.12
2.17
2.44
0.07
0.18
2.20
1.81
-0.96
-1.74
5.58
3.12
0.62
1.26
12.29
3.76
0.70
14.57
4.20
0.78
18.47
4.79
0.89
13.39
5.61
1.01
1.01
81.78
0.21
9.55
1.01
78.98
0.34
15.41
1.01
82.07
0.57
30.17
0.01
0.88
0.65
29.35
2.01
2.81
0.25
0.45
4.12
3.64
-0.02
-0.10
7.81
4.01
-0.35
-0.65
12.54
4.54
0.95
1.94
Pricing errors: CAPM and 2 factor model
-0.70
-0.13
0.47
1.03
-1.20
-0.29
0.98
2.02
0.48
0.62
0.69
0.60
1.40
1.52
1.51
1.28
Panel B: Equally Weighted
Portfolio returns and Sharpe ratios
7.71
8.50
9.46
10.65
2.59
2.83
3.09
3.38
0.48
0.52
0.57
0.63
1.00
144.92
-0.03
-3.68
Risk factor betas
1.00
1.00
1.01
122.99
104.42
91.30
0.00
0.05
0.12
-0.01
3.21
6.14
Pricing errors: CAPM and 2 factor model
-2.14
-1.43
-0.57
0.50
-2.93
-2.27
-1.11
0.92
0.42
0.48
0.46
0.33
1.27
1.22
1.00
0.65
32
Table 4: Capital age sorted portfolios: returns and asset pricing tests (Cont.)
Panel C: Cross Sectional Tests
Value Weighted
CAPM
2F
CAPM
Equally Weighted
2F
CAPM
2F
CAPM
2F
1st stage
2nd stage
1st stage
2nd stage
bM
2.95
2.41
3.04
2.58
3.86
2.56
3.55
2.65
[t]
2.60
2.07
2.78
2.29
3.38
2.15
3.25
2.31
bT
3.48
3.98
5.11
5.54
[t]
3.03
3.86
5.17
6.06
MAE
1.58
0.66
1.74
1.10
33
3.01
0.52
3.07
1.04
Table 5: Other standard portfolios
This table reports the excess returns of ten value weighted (VW) and ten equal weighted (EW) portfolios sorted
on book-to-market ratio (BM), investment rate (IK), market equity (ME), and firm-specific productivity (z).
Panel A and Panel B report raw and risk-adjusted excess returns for value weighted and equally weighted
portfolios, respectively. Column‘H’ reports the portfolio consisting of firms in the highest sorting category and
Column ‘L’ reports the portfolio consisting of firms in the lowest sorting category. Column ‘H-L’ reports the
return difference of Portfolio ‘H’ and Portfolio ‘L’. The reported statistics in the model are averages from 100
samples of simulated data, each with 3,600 firms and 30 years of monthly observations. To be consistent with
the empirical analysis, the portfolio is rebalanced at quarterly frequency. All returns are annualized.
9.73 4.90
3.18 5.59
0.59 1.09
10 IK
10 ME
H H-L
L
H
H-L
Panel A: Value Weighted
Portfolio returns and Sharpe ratios
7.19
5.74 -1.46
13.89
3.87 -10.02
2.41
1.93 -3.00
3.87
1.28 -4.69
0.45
0.36 -0.56
0.73
0.24 -0.88
1.00
1.00 0.00
139.85 119.69 0.31
-0.05
0.05 0.10
-5.12
4.08 5.02
Risk factor betas
1.00
1.00 0.00
1.01
1.00 -0.01
273.62 166.30 -0.38
97.46 110.61 -0.74
0.01
-0.04 -0.05
0.42
-0.13 -0.55
2.07
-5.89 -5.43
30.00 -12.49 -29.96
L
E[re ]
[t]
SR
MKT
[t]
TFS
[t]
Alpha
[t]
Alpha2F
[t]
E[re ]
[t]
SR
MKT
[t]
TFS
[t]
Alpha
[t]
Alpha2F
[t]
4.82
1.61
0.30
10 BM
H
H-L
L
10 z
H
H-L
10.81
3.40
0.63
5.72
1.93
0.36
-5.09
-4.73
-0.89
1.01
1.00 -0.01
107.42 189.18 -0.84
0.16
-0.06 -0.22
10.23
-8.25 -10.75
and 2 factor model
5.45
-4.00 -9.45
3.49
-3.91 -3.98
0.17
-1.26 -1.44
0.59
-1.82 -2.58
3.26
4.02
0.26
1.41
-1.56
-3.35
0.06
0.25
-4.82
-4.02
-0.21
-0.84
11.37 3.76
3.63 3.93
0.68 0.79
Panel B: Equally Weighted
Portfolio returns and Sharpe ratios
10.20
8.24 -1.96
20.57
3.92 -16.65
3.32
2.73 -3.37
5.14
1.30 -6.00
0.62
0.51 -0.63
0.96
0.24 -1.09
14.82
4.34
0.81
6.83
2.31
0.43
-7.98
-5.40
-1.01
1.00
1.00 0.00
112.13 102.15 0.22
0.06
0.12 0.06
3.75
7.39 2.46
Risk factor betas
1.00
1.00 0.00
1.01
1.00 -0.02
171.93 158.09 -0.48
82.37 111.76 -1.03
0.11
0.04 -0.07
0.61
-0.13 -0.74
13.65
4.55 -5.54
32.44 -12.64 -29.63
-2.68
-4.37
-0.09
-0.13
7.61
2.49
0.46
-2.27
-3.39
-1.12
-1.88
2.09
4.00
0.56
1.61
1.37
2.77
0.91
2.18
4.77
4.58
0.65
1.19
3.64
3.49
2.03
2.32
Pricing errors: CAPM
0.16
-1.23 -1.39
0.76
-2.90 -2.68
0.02
-0.09 -0.11
0.08
-1.38 -0.43
L
Pricing errors: CAPM
0.21
-1.62 -1.83
0.96
-3.18 -2.99
0.02
-0.10 -0.13
0.09
-1.66 -0.51
34
and 2 factor model
9.79
-5.85 -15.64
4.48
-4.66 -4.81
0.13
-1.23 -1.36
0.45
-1.69 -2.36
1.01
1.00 -0.01
79.64 199.28 -0.80
0.30
-0.03 -0.34
13.86
-4.59 -13.19
4.52
4.27
0.17
0.73
-2.95
-4.24
0.00
0.03
-7.47
-4.44
-0.17
-0.64
Table 6: Capital Age and Idiosyncratic Productivity Sorted Portfolios
This table reports the excess returns of 25 value weighted (VW) and 25 equally weighted (EW) portfolios sorted
on capital age and idiosyncratic productivity. Panel A and Panel B report quantities for value weighted and
equally weighted portfolios, respectively.
Panel A: Value Weighted
Low z
2
3
4
High z
HML
Old
4.61
4.38
4.33
4.31
4.30
-0.30
2
5.99
5.93
5.90
5.91
5.97
-0.02
3
7.24
7.26
7.28
7.25
7.07
-0.16
4
9.69
9.17
8.92
8.35
7.23
-2.47
Young
11.82
9.15
7.94
7.17
5.37
-6.45
OMY
7.21
4.77
3.61
2.87
1.06
Panel B: Equally Weighted
Low z
2
3
4
High z
HML
Old
5.96
5.54
5.41
5.26
5.01
-0.95
2
7.14
6.91
6.78
6.69
6.57
-0.57
3
8.41
8.34
8.28
8.25
8.13
-0.29
4
11.30
11.14
10.94
10.60
9.59
-1.71
Young
18.01
15.12
12.85
12.04
8.61
-9.41
OMY
12.06
9.58
7.44
6.77
3.60
35
36
Market
Mean
7.70
5. σN = 0.001
8.88
4. fo = 0
3.77
3. fi = 1
6.16
2. λη = 0
6.83
1. Benchmark
15.73
16.13
16.17
21.01
15.88
31.72
24.13
11.10
39.01
24.13
Risk Prem. Volatility Capital Age
Market
O
4.20
5.27
7.74
7.87
5.81 11.95
1.94
5.73
4.09 10.03
Y
E[re ]
0.14
6.15
2.26
-0.46
5.94
OMY
O
-0.27 0.11
-0.12 0.29
-0.07 0.08
-0.25 0.17
-0.11 0.30
Y
0.38
0.41
0.15
0.42
0.41
OMY
TFS Beta
IK
ME
z
-4.11
-0.66
-2.98
-0.68
0.07
0.01
0.09
-0.45
2.82 -1.52 -10.16 -5.78
4.56 -1.39
1.14 -0.29
4.90 -1.46 -10.02 -5.09
BM
Spread E[re ]
This table reports key moments generated by several alternative specifications. Model 1 is the benchmark economy. Model 2 sets the price of the
technology frontier shock (λη ) to zero. Model 3 sets fi = 0, i.e., there is no fixed investment cost. Model 4 sets fo = 0, i.e., there is no fixed operating
cost. Model 5 sets the volatility of the technology frontier to a lower value equal to 0.001. The reported statistics in the model are averages from 100
samples of simulated data, each with 3,600 firms and 30 years of monthly observations. To be consistent with the empirical analysis, the portfolio is
rebalanced at quarterly frequency. All returns are annualized.
Table 7: Alternative model specifications
Table 8: Summary Statistics
This table reports the mean, standard deviation, 25% percentile, 50% percentile, 75% percentile, and number of
available observations for the variables used in the empirical analysis. Capital Age is a measure of tangible capital
N
)
age constructed following Salvanes and Tveteras (2004). Investment Rate is gross investment (δj Ki,t + Ii,t
N
divided by the beginning of the period capital stock (Ki,t ), where Ii,t is defined as the difference between net
property plant and equipment (item ppentq) and δj is the depreciation of industry j calculated using depreciation
data from the BEA. Return on Equity is income before extraordinary items (item ibq) divided by the previous
quarter book value of equity. The latter quantity is constructed following Hou, Xue, and Zhang (2015) and
it is equal to shareholders’ equity (item seqq) plus deferred taxes and investment tax credit (item txditcq, if
available) minus the book value of preferred stock (item pstkrq). If shareholders’ equity is not available, we use
common equity (item ceqq) plus the carrying value of the preferred stock (item pstkq). If common equity is not
available, we measure shareholders’ equity as the difference between total assets (item atq) and total liabilities
(item ltq). Equity Issuance is the sale of common and preferred stocks (item sstky) net of the purchase of
common and preferred stocks (item prstkcy) and cash dividends (item dvy) scaled by the beginning of the
period book value of assets (item atq). Size is calculated using data from CRSP and it is equal to the number
of shares outstanding (item shrout) multiplied by the share price (item prc). When size is reported in levels,
we express it in 2009 dollar using the personal consumption expenditure price deflator. Book–to–Market is
the book value of equity divided by the market capitalization (item prccq times item cshoq) at the end of the
fiscal quarter. Productivity is the annual measure of firm–level productivity provided by Imrohoroglu and Tuzel
(2014). We restrict our sample to companies listed in the three major stock exchanges (AMEX, NYSE, and
NASDAQ). We exclude companies non incorporated in the USA, and we also exclude financials (SIC codes from
6000 up to 6999), utilities (SIC codes from 4900 up to 4999), and R&D–intensive sectors (SIC codes 737, 384,
382, 367, 366, 357, and 283) from our sample. We also exclude observations with less than 8 quarters of reported
values for capital age to minimize the impact of the initial choice for the capital age value. The sample period
is 1977q1–2013q4. All the data are winsorized at the top and bottom 1% to attenuate the impact of outliers.
mean
sd
p25
p50
p75
obs
Capital Age
23.196
8.790
16.778
22.771
29.068
175,363
Investment Rate
0.041
0.074
0.012
0.027
0.050
175,363
Return on Equity
0.020
0.068
0.007
0.026
0.044
174,560
Equity Issuance
-0.002
0.026
-0.006
-0.001
0.000
158,220
Log Size (2009 dollars)
2.357
6.479
0.082
0.344
1.434
175,311
Book–to–Market
0.756
0.562
0.381
0.610
0.954
174,634
Productivity
0.000
0.349
-0.184
-0.001
0.183
35,228
37
Table 9: Correlation Matrix
This table reports the pairwise correlations of the variables described in Table 1. The 1%, 5%, and 10%
significance levels are denoted with ∗∗∗ , ∗∗ , and ∗ , respectively.
Capital Investment
Age
Capital Age
ROE
Rate
Equity
Size
Book
Issuance (2009 $) Market
1.00
Investment Rate
-0.26∗∗∗
1.00
Return on Equity
0.07∗∗∗
0.09∗∗∗
1.00
Equity Issuance
-0.15∗∗∗
0.12∗∗∗
-0.16∗∗∗
1.00
Log Size (2009 dollars)
0.09∗∗∗
-0.01∗∗
0.13∗∗∗
-0.11∗∗∗
1.00
Book-to-Market
0.10∗∗∗
-0.13∗∗∗
-0.20∗∗∗
0.02∗∗∗
-0.18∗∗∗
Productivity
-0.13∗∗∗
0.14∗∗∗
0.33∗∗∗
-0.09∗∗∗
0.30∗∗∗
38
1.00
-0.30∗∗∗ 1.00
Table 10: Capital Age–Sorted Portfolios, Characteristics
Portfolios are rebalanced at a quarterly frequency starting in January 1981 and ending in October 2014. We use
quarterly accounting data that precede portfolio formation by at least six months. This table reports the time
series average of cross–sectional mean values of firms characteristics across ten capital age–sorted portfolios.
The row OM Y reports the difference between the largest (Old) and smallest (Young) capital age portfolios, the
associated robust t-statistic is reported in parentheses.
Portfolio
Age
BM
IK
Size
ROE
Issue
TFP
Young
9.504
0.683
0.089
1.413
0.003
0.008
0.061
2
14.162
0.689
0.058
1.641
0.014
0.003
0.037
3
17.025
0.722
0.049
1.812
0.019
-0.000
0.007
4
19.422
0.754
0.041
2.011
0.020
-0.001
-0.015
5
21.701
0.758
0.038
2.404
0.022
-0.003
0.002
6
23.997
0.767
0.034
2.286
0.023
-0.004
-0.014
7
26.449
0.767
0.029
2.678
0.025
-0.005
-0.009
8
29.168
0.788
0.026
2.848
0.025
-0.006
-0.033
9
32.551
0.842
0.022
2.919
0.023
-0.006
-0.058
Old
39.165
0.913
0.017
2.964
0.021
-0.006
-0.146
OMY
29.661
0.230
-0.071
1.551
0.017
-0.013
-0.207
(150.444)
(12.863)
(-37.815)
(12.915)
(12.063)
(-31.539)
(-16.275)
39
Table 11: Capital Age–Sorted Portfolios, Equity Returns
Portfolios are rebalanced at a quarterly frequency starting in January 1981 and ending in October 2014. This
table reports the time series average of monthly excess return across ten age–sorted portfolios. Column REW
EW
reports equally–weighted excess returns. Column αCAP
M reports the risk–adjusted equally–weighted excess
VW
VW
returns produced by the CAPM. Column R
reports value–weighted excess returns. Column αCAP
M reports
the risk–adjusted value–weighted excess returns produced by the CAPM. The row OM Y reports the difference
between the largest (Old) and smallest (Young) capital age portfolios, the associated robust t-statistic is reported
in parentheses. The reported returns are annualized versions of their monthly counterpart.
Equally Weighted
Value Weighted
REW
EW
αCAP
M
RV W
EW
αCAP
M
Young
7.32
-1.80
5.52
-3.36
2
10.44
2.16
6.84
-1.44
3
12.96
5.04
9.48
1.44
4
11.64
3.96
7.92
0.12
5
12.24
4.68
8.40
1.20
6
12.84
5.52
8.16
1.32
7
12.96
5.76
7.44
1.20
8
14.04
6.96
8.64
2.40
9
14.88
8.04
10.08
3.96
Old
14.76
8.04
11.04
4.68
OMY
7.44
9.72
5.52
8.04
3.71
5.39
2.06
3.55
40
Table 12: Capital Age and Size Sorted Portfolios
Portfolios are rebalanced at a quarterly frequency starting in January 1981 and ending in October 2014. Size
is the market capitalization at the end of the month preceding portfolio formation. Age is a measure of tangible
capital age constructed following Salvanes and Tveteras (2004) that precedes portfolio formation by at least
six months. Y oung is the bottom capital age quintile, Age3 is the third capital age quintile, Old is the top
capital age quintile, and OM Y is the difference between the top and bottom capital age quintiles. Small is the
bottom size quintile, Size3 is the third size quintile, Large is the top size quintile, and SM B is the difference
between the bottom and top size quintiles. Panel A (Panel C) reports the average realized equally–weighted
(value–weighted) equity returns in excess of the riskfree rate and the corresponding tstatistics. Panel B (Panel
D) the corresponding risk–adjusted equally–weighted (value–weighted) excess returns produced by the CAPM
and the corresponding tstatistics. GRS and p(GRS) are the GibbonsRossShanken test statistics (Gibbons et al.
[1989]) and the corresponding pvalue respectively; m.a.e. is the mean absolute error of the risk adjusted excess
equity returns. The tstatistics are evaluated following Newey and West [1987] and using 12 lags. The reported
returns are annualized versions of their monthly counterpart.
Panel A: Equally Weighted Raw Returns
Returns
t-statistics
Y oung
Age3
Old
OM Y
Y oung
Age3
Old
OM Y
Small
14.90
18.33
20.60
5.70
3.19
4.40
5.33
2.67
BM3
6.81
13.22
13.31
6.50
1.77
4.18
4.08
3.24
Large
6.34
8.28
11.24
4.90
1.80
3.10
3.94
2.50
SM B
8.56
10.06
9.37
2.64
3.61
3.81
Panel B: Equally Weighted CAPM α
α
t-statistics
Y oung
Age3
Old
OM Y
Y oung
Age3
Old
OM Y
Small
7.43
11.78
14.51
7.08
1.97
3.60
5.13
3.23
BM3
-2.57
5.71
5.97
8.54
-1.02
2.15
2.34
4.65
Large
-2.31
0.79
4.47
6.78
-1.35
0.42
2.31
3.82
SM B
9.74
10.98
10.04
0.03
3.03
3.84
4.00
0.12
GRS= 4.48
p(GRS)= 0.00
41
m.a.e.=5.52
Table 11: Capital Age and Size Sorted Portfolios (Cont.)
Panel C: Value Weighted Raw Returns
Returns
t-statistics
Y oung
Age3
Old
OM Y
Y oung
Age3
Old
OM Y
Small
12.36
14.85
17.73
5.36
2.69
3.56
4.66
2.38
BM3
6.95
12.95
13.10
6.15
1.84
4.11
3.96
3.09
Large
5.94
7.80
10.01
4.07
1.63
2.98
3.86
1.80
SM B
6.43
7.05
7.72
1.66
2.04
2.95
Panel D: Value Weighted CAPM α
α
t-statistics
Y oung
Age3
Old
OM Y
Y oung
Age3
Old
OM Y
Small
4.52
8.16
11.53
7.01
1.21
2.53
4.25
2.97
BM3
-2.36
5.50
5.80
8.16
-0.96
2.05
2.28
4.39
Large
-2.47
0.99
4.01
6.48
-1.82
0.56
2.53
3.38
SM B
6.99
7.17
7.52
0.04
1.82
2.07
2.89
-0.17
GRS= 3.52
p(GRS)= 0.00
42
m.a.e.=4.75
Table 13: Capital Age and Book–to–Market Sorted Portfolios
Portfolios are rebalanced at a quarterly frequency starting in January 1981 and ending in October 2014. BM
is the book–to–market value that precedes portfolio formation by at least six months. Age is a measure of
tangible capital age constructed following Salvanes and Tveteras (2004) that precedes portfolio formation by at
least six months. Y oung is the bottom capital age quintile, Age3 is the third capital age quintile, Old is the top
capital age quintile, and OM Y is the difference between the top and bottom capital age quintiles. Low is the
bottom book–to–market quintile, BM3 is the third book–to–market quintile, High is the top book–to–market
quintile, and HM L is the difference between the top and bottom book–to–market quintiles. Panel A (Panel
C) reports the average realized equally–weighted (value–weighted) equity returns in excess of the riskfree rate
and the corresponding tstatistics. Panel B (Panel D) the corresponding risk–adjusted equally–weighted (value–
weighted) excess returns produced by the CAPM and the corresponding tstatistics. GRS and p(GRS) are the
GibbonsRossShanken test statistics (Gibbons et al. [1989]) and the corresponding pvalue respectively; m.a.e. is
the mean absolute error of the risk adjusted excess equity returns. The tstatistics are evaluated following Newey
and West [1987] and using 12 lags. The reported returns are annualized versions of their monthly counterpart.
Panel A: Equally Weighted Raw Returns
Returns
t-statistics
Y oung
Age3
Old
OM Y
Y oung
Age3
Old
OM Y
Low
3.75
8.12
10.83
7.08
0.96
2.70
3.74
3.11
BM3
10.90
12.47
14.31
3.41
3.04
4.11
4.55
1.61
High
15.72
17.13
21.04
5.32
3.24
4.01
5.15
2.59
HM L
11.97
9.01
10.20
4.41
3.14
4.18
Panel B: Equally Weighted CAPM α
α
t-statistics
Y oung
Age3
Old
OM Y
Y oung
Age3
Old
OM Y
Low
-5.63
0.50
4.05
9.68
-2.74
0.27
1.98
4.79
BM3
2.79
4.81
7.19
4.39
0.93
1.91
3.21
2.03
High
7.46
10.12
14.27
6.80
2.02
3.00
4.70
3.39
HM L
13.10
9.61
10.22
-0.24
4.82
3.33
4.19
-1.19
GRS= 3.89
p(GRS)= 0.00
43
m.a.e.=5.50
Table 12: Capital Age and Book–to–Market Sorted Portfolios (Cont.)
Panel C: Value Weighted Raw Returns
Returns
t-statistics
Y oung
Age3
Old
OM Y
Y oung
Age3
Old
OM Y
Low
4.71
7.03
9.64
4.93
1.13
2.36
4.00
1.70
BM3
7.79
8.22
10.43
2.63
2.18
3.01
3.29
1.26
High
8.48
14.10
14.32
5.84
1.68
3.04
4.28
1.48
HM L
3.77
7.08
4.68
0.86
1.93
1.80
Panel D: Value Weighted CAPM α
α
t-statistics
Y oung
Age3
Old
OM Y
Y oung
Age3
Old
OM Y
Low
-4.30
0.19
3.82
8.12
-2.43
0.09
2.22
3.55
BM3
-0.09
1.27
3.61
3.69
-0.04
0.55
1.75
1.80
High
-0.00
5.62
7.01
7.01
-0.00
1.82
2.61
1.82
HM L
4.30
5.43
3.19
-0.09
0.99
1.53
1.21
-0.24
GRS= 1.26
p(GRS)= 0.18
44
m.a.e.=2.69
Table 14: Capital Age and Investment Rate Sorted Portfolios
Portfolios are rebalanced at a quarterly frequency starting in January 1981 and ending in October 2014. IK
is the investment rate that precedes portfolio formation by at least six months. Age is a measure of tangible
capital age constructed following Salvanes and Tveteras (2004) that precedes portfolio formation by at least
six months. Y oung is the bottom capital age quintile, Age3 is the third capital age quintile, Old is the top
capital age quintile, and OM Y is the difference between the top and bottom capital age quintiles. Low is the
bottom investment rate quintile, IK3 is the third investment rate quintile, High is the top investment rate
quintile, and HM L is the difference between the top and bottom investment rate quintiles. Panel A (Panel
C) reports the average realized equally–weighted (value–weighted) equity returns in excess of the riskfree rate
and the corresponding tstatistics. Panel B (Panel D) the corresponding risk–adjusted equally–weighted (value–
weighted) excess returns produced by the CAPM and the corresponding tstatistics. GRS and p(GRS) are the
GibbonsRossShanken test statistics (Gibbons et al. [1989]) and the corresponding pvalue respectively; m.a.e. is
the mean absolute error of the risk adjusted excess equity returns. The tstatistics are evaluated following Newey
and West [1987] and using 12 lags. The reported returns are annualized versions of their monthly counterpart.
Panel A: Equally Weighted Raw Returns
Returns
t-statistics
Y oung
Age3
Old
OM Y
Y oung
Age3
Old
OM Y
Low
10.95
16.00
18.01
7.06
2.52
3.89
4.97
3.27
IK3
9.59
12.55
12.32
2.73
2.42
4.25
3.92
1.19
High
6.46
10.77
13.67
7.21
1.74
3.23
3.49
2.51
HM L
-4.49
-5.23
-4.34
-2.03
-2.27
-1.54
Panel B: Equally Weighted CAPM α
α
t-statistics
Y oung
Age3
Old
OM Y
Y oung
Age3
Old
OM Y
Low
2.13
8.46
10.85
8.71
0.66
2.61
3.99
4.17
IK3
1.78
5.28
5.53
3.75
0.56
2.53
2.52
1.66
High
-2.47
2.96
6.57
9.04
-1.08
1.24
2.24
3.16
HM L
-4.60
-5.50
-4.28
0.03
-2.08
-2.38
-1.51
0.11
GRS= 4.2362
p(GRS)= 0.0000
45
m.a.e.=5.2191
Table 13: Capital Age and Investment Rate Sorted Portfolios Sorted Portfolios (Cont.)
Panel C: Value Weighted Raw Returns
Returns
t-statistics
Y oung
Age3
Old
OM Y
Y oung
Age3
Old
OM Y
Low
5.43
12.98
11.13
5.70
1.37
4.19
3.61
2.12
IK3
8.75
10.02
10.22
1.48
2.02
3.72
3.74
0.50
High
4.69
7.73
8.31
3.62
1.13
2.34
1.87
0.87
HM L
-0.73
-5.25
-2.81
-0.27
-2.13
-0.83
Panel D: Value Weighted CAPM α
α
t-statistics
Y oung
Age3
Old
OM Y
Y oung
Age3
Old
OM Y
Low
-3.05
6.38
4.45
7.50
-1.34
2.74
2.01
3.00
IK3
0.93
3.12
4.17
3.24
0.34
1.37
2.37
1.16
High
-4.34
0.07
0.73
5.07
-2.18
0.03
0.20
1.25
HM L
-1.30
-6.30
-3.72
-0.20
-0.49
-2.60
-1.11
-0.57
GRS= 1.7782
p(GRS)= 0.0130
46
m.a.e.=2.5268
47
420
0.014
R-squared
(0.041)
(0.040)
0.137***
(0.062)
-0.469***
(6)
(0.044)
0.174***
(0.061)
-0.501***
(7)
(0.039)
0.102***
(0.061)
-0.505***
(8)
0.025
420
0.021
420
0.016
420
0.020
420
0.016
372
0.020
0.040
372
(0.034)
(0.048)
411
0.012
(0.035)
(0.034)
-0.087*
-0.058
(0.052)
(0.055)
-0.057
0.250***
(0.031)
(0.032)
0.208***
-0.134***
(0.052)
(0.052)
-0.128***
0.177***
0.264***
(0.065)
(0.042)
0.166***
(0.058)
-0.554***
(5)
(0.067)
(0.042)
0.139***
(0.058)
-0.521***
(4)
-0.217***
(0.041)
(0.042)
0.128***
(0.058)
-0.543***
(3)
-0.280***
0.174***
(0.058)
(0.059)
0.165***
-0.536***
(2)
-0.519***
Observations
TFP
Issuance
Roe
IK
Book-to-Market
Size
Capital Age
Lagged Return
(1)
This table reports the results of FamaMacbeth regressions of individual stock excess returns on their lagged value and firms’ characteristics. The
reported coefficient is the average slope from month–by–month regressions and the corresponding t-statistic is the average slope divided by its timeseries standard error. The reported R–squared is the time–series average of the crosssectional R-squared. The sample period is from 1980m1 to
2014m12. The 1%, 5%, and 10% significance levels are denoted with ∗∗∗ , ∗∗ , and ∗ , respectively.
Table 15: Fama–MacBeth Regressions
Figure 1: Investment policy functions
k t+1
1
0.5
low z
high z
0
0
0.2
0.4
0.6
0.8
1
0.6
0.8
1
k
i
1500
1000
500
0
0
0.2
0.4
k
This figure compares the optimal investment policy for high and low productive (z) firms. We fix the detrended
aggregate productivity and the detrended technology frontier at their long run means. The upper panel reports
the optimal capital in next period as a function of the capital in this period for both high and low productive
firms. The lower panel reports the optimal investment rate (I/K) as a function of the capital for both high and
low productive firms.
48
Figure 2: Value and other policy functions
low z
v
4
high z
ex-div v
4
2
2
0
0
0
0.5
1
0
0.5
k
k
BM
1
1
D/P
0.5
0
0.5
-0.5
0
-1
0
0.5
1
0
k
k
β /σ
e
E[R ] (%)
3
0.5
η
2
2
1
1
0
0
1
N
-1
0
0.5
1
0
k
0.5
1
k
This figure compares cum-dividend (v), ex -dividend firm value (ex-div v), book-to-market ratio (BM), dividend
(div), stock risk premium (E[Re ]), and stock beta to the technology frontier shock (βη /σN ) as functions of the
detrended capital (k) for high and low productive (z) firms. We fix the detrended aggregate productivity and
the detrended technology frontier at their long run means.
49
Figure 3: Model predicted capital age portfolio betas to the aggregate productivity shock and
the technology frontier shock
2
β x (AGE)
0.5
1
0
0
-0.5
1 2 3 4 5 6 7 8 9 10
2
1 2 3 4 5 6 7 8 9 10
β x (BM)
0.2
1
0
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
β x (IK)
0.1
1
0
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
β x (ME)
0.5
1
β η /σ N (ME)
0
0
-0.5
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
β (z)
2
β η /σ N (IK)
0.05
0
2
β η /σ N (BM)
0.1
0
2
β η /σ N (AGE)
β /σ (z)
x
0.4
1
η
N
0.2
0
0
1 2 3 4 5 6 7 8 9 10
1 2 3 4 5 6 7 8 9 10
This figure reports the model predicted betas to the aggregate productivity shock (∆xt+1 ) and the technology
frontier shock (σN ηt+1 ) across ten value weighted and ten equal weighted portfolios sorted on capital age and
other characteristics.
50