Innovation, Good Idiosyncratic Volatility, and

Stock Returns1

Praveen Kumara Dongmei Lib

a C.T. Bauer College of Business, University of Houston, Houston, TX 77204. Email: pkumar@uh.edu. b Moore School of Business, University of South Carolina, Columbia, SC 29208. Email: dongmei.li@moore.sc.edu.                                                             1 We thank Gurdip Bakshi, Louis Chan, Yong Chen, Phil Dybvig, Adlai Fisher, Paolo Fulghieri, Tom George, Joao Gomes, Gustavo Grullon, Dirk Hackbarth, David Hirshleifer, Kris Jacobs, Dirk Jenter, Michael Lemmon, Neil Pearson, Jeff Pontiff, Lawrence Schmidt, Ken Singleton, Selale Tuzel, Neng Wang, Lu Zhang, Guofu Zhou, and Haoxian Zhu for helpful discussions or comments on the issues examined in this paper.

Innovation, Good Idiosyncratic Volatility, and

Stock Returns

February 26, 2017

Abstract

We theoretically and empirically examine the relation of idiosyncratic volatility of stock

returns (IVOL) to future expected returns of innovative and mature firms. Our theoreti-

cal framework shows that stochastic improvements in the value of future innovations raise

contemporaneous IVOL and predict higher future expected returns for big innovative firms.

Therefore, for such firms, there exists a "good" IVOL that is positively related to future

stock performance, and the strength of this relation increases with innovative capacity in-

vestment. However, no such predictive relation exists for mature or small innovative firms.

Our empirical analysis finds strong and robust support for these predictions. The IVOL

of big innovative firms is significantly and positively related to their future average stock

returns and alphas, and this relation strengthens with their recent asset growth. However,

we detect no such relation for mature or small innovative firms.

Keywords: Innovation; Growth Options; Idiosyncratic Volatility; Stock Returns

JEL classification codes: G34, G24, O31

1. Introduction

The negative cross-sectional relation of a stock’s idiosyncratic volatility (IVOL) to its

average future return appears now well-established (e.g., Ang et al., 2006; Chen et al., 2012)

and attracts substantial interest. The observed empirical relation seems inconsistent with

the classical asset pricing view that only systematic risk should be priced in equilibrium.1

A growing literature provides various possible explanations for the negative IVOL-return

relation.2 In particular, Babenko et al. (2016) argue that IVOL is negatively related to

expected returns because it represents conditioning information in a dynamic asset-pricing

framework. We theoretically show and empirically verify that high IVOL reflects positive

information on future expected returns for big innovative firms. That is, high IVOL predicts

higher future stock performance in a class of economically important firms.3

Take, for example, innovative firms that, at any given time, are working to develop

future technological innovations. The literature on the economics of innovation argues that

successful innovative firms invest in R&D and innovative capacity prior to the finalization

and marketing of the innovation (the “arrival”stage) to improve its economic value in the

post-arrival stage (Cohen and Levinthal, 1990; Kumar and Li, 2016). In particular, such firms

invest to enhance buyer utility from the innovation and develop capabilities and technological

strengths in the pre-arrival phase to exploit the new ideas and opportunities that often

arise following the initial commercialization of the innovation (Schumpeter, 1942; Maclaurin,

1953).4 However, because of technological uncertainty, progress in this process is uncertain,

and signals regarding the value of the prospective innovation arrive stochastically during

1Theoretical deviations from the classical paradigm typically predict a positive relation of IVOL to ex-pected returns (e.g., Merton, 1987; Barberis and Huang, 2001; Malkiel and Xu, 2002).

2See, e.g., Barinov (2013), Chen et al. (2014), Stambaugh, Yu, and Yu (2015), Babenko, Boguth, andTserlukevich (2016), Hou and Loh (2016), and Herskovic et al. (2016).

3For example, in our empirical analysis, the market capitalization of big innovative firms is about 56% ofthe sample universe.

4Specifically, innovative firms attempt to improve the commercial value of innovations by investing in R&Dex ante to maintain their first mover advantage (Lieberman and Montgomery, 1988), or realize economies ofscope and internalize gains from knowledge spillovers and ancillary innovations (Henderson and Cockburn,1996; Adner, 2012), or nurture disruptive innovations that can eventually replace market leaders (Chris-tensen, 1997).

1

the pre-arrival phase. To fix ideas, consider a hi-tech firm developing a new generation

of a telecommunication device. In the pre-arrival phase, as the firm internally overcomes

technological hurdles, it may release information on the “exciting”potential capabilities of

the new model and/or give a more precise time window on its launch. These news events

should affect the volatility of both the systematic and idiosyncratic components of the firm’s

stock returns. If high IVOL reflects information on (past or realized) successful efforts that

enhance the value of future innovations, then it is intuitively plausible that – for certain

types of innovative firms – IVOL in the pre-arrival phase is positively related to their future,

post-arrival expected returns.

We develop a theoretical framework that distinguishes the predictive information con-

tent of IVOL between mature and innovative firms. The value of mature firms is composed

entirely of assets-in-place (AIP). However, innovative firms have both AIP and uncertain

arrival of innovations (or growth options), and financial markets stochastically receive sig-

nals regarding future innovation-value enhancements in the pre-arrival phase. Our model

shows that such news events need not always amplify IVOL. Depending on the composition

of the firm’s AIP, specifically, the relative value weights of the systematic and idiosyncratic

components of the AIP, good incremental news regarding the value of prospective innova-

tions can increase or decrease IVOL. In particular, we show that good news relating to the

value of future innovations can increase the IVOL of big innovative firms in the pre-arrival

phase. Therefore, for such firms, there can be a positive relation of IVOL to future stock

performance.

Our analysis highlights the substantial difference in the relation of IVOL to future stock

performance between big innovative firms and mature/small innovative firms. Similar to

Babenko et al. (2016), for mature firms (that only have AIP), a positive non-systematic

earnings shock simultaneously increases IVOL and reduces the firm’s risk factor loading,

leading to a negative relation of IVOL to contemporaneous expected returns. Importantly,

because mature firms only have AIP and earnings shocks are random, there is no predictive

2

or conditioning information in IVOL regarding future expected returns. In contrast, the pre-

arrival value for innovative firms is composed of AIP and the potential innovation. Under

certain conditions that depend on the firm’s asset composition (described in details below),

the shocks affecting future innovation value can increase current IVOL as well as future

expected returns.

Profits from innovations (both past and future potential innovations) have a firm-specific

(or idiosyncratic) component and a systematic (or economy-wide) component in our model.

The firm-specific component arises from local or highly specialized markets that may protect

the innovation profits from economy-wide shocks, while the systematic component increases

with the product market size of the innovation.5

The empirical literature on the product life cycle (or diffusion) of innovations suggests

a dynamic evolution of the relative weights of the idiosyncratic and systematic components

of innovations’cash flows. Initially, innovations appeal to a relatively small set of adopters,

i.e., have localized markets. But successful innovations then exhibit fast market growth that

eventually slows as the technology matures – the well-known “S-shaped” diffusion curve

(Mansfield, 1968). Indeed, big innovative firms attain their larger size through a superior

ability to extract economic rents from past successful innovations (Klepper, 1996). Hence,

before future innovations are successfully developed and introduced to the market, the AIP

of big innovative firms is more likely associated with past successful innovations in their

mature stage. In contrast, the AIP of small innovative firms is more likely associated with

past innovations that are at an early stage (or commercially unsuccessful). In sum, the AIP

of big innovative firms tend to have a relatively larger systematic component than that of

small innovative firms. But, for all innovative firms (big or small), the decomposition of the

value of a future growth option (following its development and introduction to the market) is

5To illustrate, consider the case of a pharmaceutical innovation targeted to a specific disease. There maybe a “core” market for the new drug – for example, patients in advanced stages of the disease – andthis market will tend to be invariant to systematic shocks. But the wider is the market for the drug, themore likely the market includes buyers for whom taking the drug is an elective, and the higher will be thesensitivity of profits to systematic shocks.

3

similar – initially there is a relatively large idiosyncratic component and then the systematic

component grows as the new innovation gains market size over time.

We show in a partial equilibrium asset-pricing model that such dynamic patterns in

the composition of innovative firms’AIP generate a positive relation of IVOL to future

stock performance for big innovative firms. Moreover, this relation should be stronger for

big innovative firms with greater innovative capacity investment, which increases expected

enhancement in innovation value. However, the positive equilibrium predictive relation of

IVOL to future stock performance need not hold for mature or small innovative firms.

Empirical tests of our model require proxies for identifying innovative firms, which in our

framework involve uncertain innovation generation. For reasons explicated in the literature

(see Franzen, Rodgers, and Simin, 2007), firms’disclosures of R&D expenditures may not be

the most reliable method to identifying truly innovative firms. These concerns are amplified

in our study because we need empirical proxies for uncertain innovation option generation.6

Therefore, we utilize multiple innovation proxies. In addition to R&D expenditures, we use as

innovation proxies two prominent characteristics of innovation-driven firms – low leverage or

debt-to-equity ratio (DTE) and high market-to-book assets (MABA) (Cao, Simin, and Zhao,

2008) – and also whether a firm operates in growth-option-intensive industries (Grullon,

Lyandres, and Zhdanov, 2012). Moreover, guided by the model, we use total assets to proxy

for firm size.

We find strong empirical support for the predictions of our conceptual framework. For

big innovative firms, IVOL is significantly positively related to future average stock returns

6Specifically, the level of R&D expenditures may not be a good proxy for the (magnitude of) innovationgeneration uncertainty faced by a firm. Innovation or R&D projects typically involve multiple stages (see,e.g., Berk, Green, and Naik, 2004). Later stages of projects may require large R&D expenditures eventhough the innovation generation uncertainty is relatively low. For example, in drug development projects,a drug candidate typically needs to go through a pre-clinical trial and three phases of clinical trials (I, II,and III) sequentially before getting Food and Drug Administration (FDA) approval. The R&D expenserequirement for the Phase III trial, which jointly tests for a drug’s safety, effi cacy, and proper dosage istypically significantly higher than that for Phase I and II trials, which test separately for safety and effi cacy(respectively) with a relatively small number of subjects. However, the probability of getting FDA approvalfor a drug candidate in the Phase III trial is significantly higher than that for a drug candidate in Phase Ior II trials.

4

and alphas. Moreover, the strength of this positive relation increases with asset growth,

other things held fixed. However, we detect no such relation for mature firms or for small

innovative firms. We verify that the positive predictive IVOL-return relation among big

innovative firms is not driven by time-varying loadings on standard risk factors. Moreover,

for our overall sample, we replicate the significant cross-sectional negative relation of IVOL

to future average stock returns and alphas. That is, we ensure that the novel findings on the

dynamic relation of IVOL to future stock performance for big innovative firms is not due to

differences in the sample period or the calculation procedure of IVOL.

Our empirical analysis employs both independent triple sorts (based on with versus with-

out innovative activity, big versus small asset size, and high versus low IVOL) and Fama-

MacBeth (1973) cross-sectional regressions. Overall, the triple sorts show that the IVOL

return spreads (high IVOL minus low IVOL) for big innovative firms turn significantly pos-

itive in the second year and/or third year after portfolio formation, in terms of monthly

value-weighted returns and alphas from the Carhart (1997) four-factor model, the Carhart

model augmented with the liquidity factor (Pastor and Stambaugh, 2003), and the recently

developed Fama-French five-factor model (Fama and French, 2015). These IVOL return

spreads are economically substantial and statistically significant. For example, using low

DTE as the innovation proxy, we find IVOL spreads of 1.59% and 1.51% for monthly Fama-

French five-factor alphas in the second and third years after portfolio formation, respec-

tively, among big innovative firms; and the corresponding spreads for the monthly returns

are 1.25% and 1.16%, respectively. Furthermore, big innovative firms with high IVOL drive

these large return spreads, which is consistent with our model and the intuition that high

IVOL for such firms on average signals higher value of future innovations. The results from

the Fama-MacBeth regressions, controlling for many well-known return predictors (such as

size, book-to-market ratio, momentum, return reversals, ROA, asset growth, and skewness),

are also consistent with the predictions of the model.

To the best of our knowledge, this is the first study to show theoretically and empirically

5

that IVOL may contain positive predictive information on future stock performance and

growth option values for an important class of firms, namely, big innovative firms. Thus,

high IVOL can reflect good news for shareholders of such firms.7 Our analysis supports the

view that IVOL contains conditioning information on future returns (Babenko et al., 2016).

But by providing a structural model of the relation of IVOL to future innovation (or growth

option) value, we theoretically show and empirically verify the heterogeneous relation of

firms’IVOL to their future stock returns in terms of innovative activity and size.8

Our results also extend the recent literature that examines the dynamic stock return

implications of uncertain growth option generation. Kumar and Li (2016) interpret capital

investment or asset growth by innovative firms as investment in innovative capacity and

examine its dynamic implications for stock returns, investment, and profitability. We find

that big innovative firms with high IVOL also tend to have high asset growth. However,

our results are robust to controlling for asset growth. Therefore, our analysis uncovers a

novel dynamic relation of IVOL to future stock returns.9 Furthermore, as mentioned earlier,

the positive IVOL-return relation among big innovative firms intensifies with asset growth.

7Bartram, Brown, and Stulz (2012) and Segal, Shaliastovich, and Yaron (2015) also highlight the existenceof “good”volatility or uncertainty at the macro or country level, which predicts better aggregate economicoutcomes. But these papers do not link IVOL to subsequent superior stock returns at the firm level.

8Our theoretical framework and empirical results, which focus on the positive dynamic relation of IVOLto future stock returns of big innovative firms, are very distinct from the papers mentioned earlier thattheoretically find a positive cross-sectional relation of IVOL to expected returns. For example, Merton(1987) and Malkiel and Xu (2002) show that IVOL is a priced risk factor if investors have limited knowledgeof market opportunities, or cannot hold the market portfolio. And Barberis and Huang (2001) show thatthere may be a positive cross-sectional relation of mean returns and volatility if investors exhibit loss aversionwith respect to fluctuations in individual stocks that they own. In contrast, we focus on the informationcontent of the IVOL of innovative firms with respect to the value of future potential innovation(s) andhence future stock returns. We note that the channels emphasized in the previous literature – incompleteinformation or loss aversion – cannot explain the sharp contrast in the dynamic IVOL-return relation thatwe observe between big innovative firms and mature/small innovative firms.

9The literature examines the link between growth options and the volatility-returns relation. Grullonet al. (2012) examine the contemporaneous relation between total volatility and stock returns. They findthat this positive relation is driven by option-intensive firms, consistent with the theory that the value of areal option should be increasing in the volatility of the underlying asset. In addition, Barinov (2013) linksthe well-known value premium with the negative IVOL-return relation by arguing that the sensitivity ofgrowth options to underlying asset value is negatively related to IVOL and aggregate volatility. In contrast,we develop the implications of uncertain innovation option generation for the dynamic relation of IVOL tofuture stock returns and find a significantly positive IVOL-return relation among big innovative firms.

6

This evidence is consistent with the intuition that innovative firms’asset growth captures

investment in innovative capacity, which increases expected value of future innovations.

We organize the remaining of the paper as follows. Section 2 presents the model and

develops empirical predictions. Sections 3 and 4 describe the data and the empirical results.

Section 5 concludes. All proofs are presented in the Appendix.

2. The Model and Empirical Predictions

In this section, we outline the theoretical model and derive the main results regarding

the relation of idiosyncratic volatility to expected stock returns. We then exposit the intu-

ition underlying the main empirical predictions from the model. All the computations and

derivations are detailed in the Appendix.

2.1. Theoretical results

There are infinite number of dates in the model t = 1, 2, .... For simplicity, all firms in

our model are completely equity-financed. At t = 1, each firm is endowed with initial assets-

in-place (AIP) that generate a stream of stochastic cash flows. There are two types of firms

in the economy: mature firms and innovative firms. Mature firms only have the initial AIP

and no further growth option generation. Innovative firms start with AIP at t = 1, but also

stochastically generate an innovation – or more generally a growth option.

It turns out that the presence of uncertain growth option generation has a substantial

impact on the relation of idiosyncratic volatility to expected returns. Therefore, we examine

the cases of mature and innovative firms separately.

2.1.1. IVOL and contemporaneous expected returns of mature firms

We denote the stochastic cash flows process from the AIP by {ΠAt }∞t=1, which can be

decomposed into the firm-specific and the systematic (economy-wide) components. Specifi-

cally,

ΠAt = Zt + κXt, (1)

7

where Zt is the firm-specific component, which is random because of stochastic changes in the

firm’s competitive environment, buyers’tastes, and costs etc. Xt reflects the sole systematic

risk in the economy, and κ is the sensitivity of the firm’s cash flows to the systematic risk;

we assume κ > 0 without loss of generality.

We model the firm-specific and systematic components of the cash flows as persistent

random walks (in logs). Specifically, the law of motion of zt ≡ ln(Zt) is:

zt+1 = zt + µZ + εt+1, (2)

where µZ > 0 is a drift parameter and εt is a transient shock that is i.i.d. N(0, σ2ε).Meanwhile,

the law of motion of xt ≡ ln(Xt) is:

xt+1 = xt + ηt+1, (3)

where ηt is a transient shock that is i.i.d. N(0, σ2η).10 In general, the transient parts of the

firm-specific and systematic cash flow components are uncorrelated; that is, we take Cov(εt+1,

ηt+1) = 0.

For tractability, we utilize an exogenous pricing kernel {θt}∞t=1 following θt+1 = θt exp(−r−σ2η2−ηt+1), where r is the risk-free rate. With this formulation, we can calculate the expected

gross factor return Et[RXt,t+1

]as follows. Let WX

t denote the value of an asset with payoffs

{Xt+i}∞i=1. Note that for any t, conditional on Xt,

Et[θt+1θt

Xt+1

]= XtEt

[exp(−r −

σ2η2− ηt+1 + ηt+1)

]= Xte

−(r+σ2η2). (4)

10The implicit assumption in (3) is that the systematic component is driftless. This is without loss ofgenerality, and made for notational convenience.

8

Then forward projection gives

WXt = Et

[ ∞∑i=1

θt+iθt

Xt+i

]= Xt

∞∑i=1

e−i(r+σ2η2) =

Xt

e(r+σ2η2) − 1

. (5)

It then follows (see Section A in the Appendix) that the expected factor return is

Et[RXt,t+1

]=Et[Xt+1 +WX

t+1

]WXt

= e(r+σ2η), (6)

so that the variance of the systematic component, σ2η, also reflects the risk premium on the

factor.

Next, we compute the ex-dividend value of the mature firm, V At , and the first and second

moments of its gross return. At any t, conditional on Xt and Zt, let V AZt and V AX

t denote

the idiosyncratic and systematic components of the firm value, respectively. In other words,

V At is equal to (V AZ

t + V AXt ). Using the procedure above, we can show that V AZ

t = ZteδZ−1

,

where δZ = r − (µZ + σ2ε2

). For V AZt to be well-defined, we assume henceforth that δZ > 0.

Similarly, we can show that V AXt = κXt

eδX−1, where δX = r + σ2η. We will denote the value

weights of the two components in V At by ωAZt ≡ V AZt

V Atand ωAXt ≡ V AXt

V At, respectively. The

factor beta of the mature firm (or AIP), βM , can be computed as ∂V At∂Xt

XtV At. Since Zt and Xt

are uncorrelated, we can show that

βM =κXt

(eδX − 1)V A

t

= ωAXt . (7)

Hence, the effect of the firm-specific shock on the factor beta of AIP is

∂βM

∂Zt=

−κXt

(eδX − 1)(eδ

Z − 1)(V At )2

< 0. (8)

Next, the one-period gross return on AIP, RAt,t+1, can be expressed as a weighted aver-

age of the components’returns, i.e., ωAXt RAXt,t+1 + ωAZt RAZ

t,t+1, where RAXt,t+1 and R

AZt,t+1 denote

9

the returns on the systematic and firm-specific components, respectively.11 Since the value

weights sum to one, it follows from (7) that

RAt,t+1 = βMRAX

t,t+1 + (1− βM)RAZt,t+1, (9)

where the first term represents the systematic component, while the second term reflects the

firm-specific component. Therefore, we can compute the idiosyncratic volatility (IV OLA) of

the one-period gross return for a mature firm as

IV OL2A = (1− βM)2V ar(RAZt,t+1), (10)

which is equivalent to V ar(RAt,t+1)− (βM)2V ar(RAX

t,t+1) since Cov(RAXt,t+1, R

AZt,t+1) = 0.12 Now,

it can be shown (see the Appendix, Section B) that V ar(RAZt,t+1) = e2r(eσ

2ε − 1). Therefore,

IV OL2A = (1− βM)2e2r(eσ2ε − 1) and

∂IV OL2A∂Zt

= 2(βM − 1)e2r(eσ2ε − 1)

∂βM

∂Zt> 0 (11)

if βM < 1.

Finally, since Et[RAt,t+1

]= Et

[RAZt,t+1

]+ ωAXt

(Et[RAXt,t+1

]− Et

[RAZt,t+1

]), we get

Et[RAt,t+1

]= er + ωAXt (er+

32σ2η − er). (12)

Thus, the effect of the firm-specific cash flow shock on expected returns of mature firms is

∂Et[RAt,t+1

]∂Zt

=∂ωAXt∂Zt

(er+32σ2η − er). (13)

Using the fact that ∂ωAXt∂Zt

< 0 (from (8)), we conclude that∂Et[RAt,t+1]

∂Zt< 0. In sum, (11) and

11See the Appendix (Section B) for the detailed expressions of these components’returns.12This is derived from the assumption that Cov(εt+1, ηt+1) = 0.

10

(13) together imply that if βM < 1, then IV OLA and Et[RAt,t+1

]are negatively related. For

expositional ease, we summarize this result in the following proposition.

Proposition 1 If βM < 1, then the idiosyncratic volatility of stock return for mature firms

is negatively related to their contemporaneous expected stock returns.

2.1.2. IVOL and future expected returns of innovative firms

At t = 1, innovative firms are endowed with the AIP and cash flows are given by Equation

(1). However, based on R&D investment prior to the start of the model, each innovative

firm can stochastically generate an innovation (or growth option). Specifically, conditional

on not having generated the innovation up to period t ≥ 1, the firm generates the innovation

with probability 0 < p < 1 by the beginning of the next period t+ 1. The innovation arrival

period is denoted by T.

For simplicity, we assume that the growth option is exercised as soon as it arrives at the

beginning of period T . Specifically, following its exercise, the profits from the innovation in

any period t ≥ T are given by

Λt = λT [Yt + φXt] , (14)

where λTYt is the firm-specific cash flows from the new business, and λTφ > 0 represents the

sensitivity of the new cash flows to the systematic risk. The law of motion of yt ≡ ln (Yt) is

yt+1 = yt + µY + υt+1, (15)

where µY > 0 is a drift parameter and υt is a transient shock that is i.i.d. N(0, σ2υ).13 The

transient shocks to the firm-specific component of the initial and new cash flows, and the

shock to the systematic component, {εt, υt, ηt}, are pairwise and jointly uncorrelated. We

interpret the firm-specific component of profits from the innovation as arising from local

13The probability distribution of the initial state YT (at the innovation arrival date) is taken to be pre-specified as yT = [µY + υT ], where υT is also distributed N(0, σ2υ).

11

or highly specialized markets that may protect profits from systematic shocks. However, if

the innovation has a large, economy-wide market, then profits will be more sensitive to the

systematic risk.

Meanwhile, λT is a state variable at the time of the innovation arrival that affects future

profits from the innovation. λT is determined prior to T through R&D and other invest-

ments made by the firm (in the pre-arrival phase) to ex-ante improve the economic value of

the innovation, as we motivated in the introduction. We model the innovation-value state

variable (λT ) as an accumulation of random (innovation) value enhancement efforts prior to

the innovation arrival. At the beginning of t = 1, the firm starts with λ1 = 1. In addition,

the firm realizes an innovation value enhancement ξ̃t ∈ {ξh, ξ`}, at the beginning of each

1 ≤ t < T. We assume that 0 < ξ` < 1 < ξh, and the probability of ξ̃t = ξh is q. For a firm

to have an incentive to undertake value enhancement, we assume that ξ̄ ≡ E[ξ̃] > 1, i.e.,

ξh ∗ q + ξl ∗ (1 − q) > 1. This assumption of ex-ante successful innovation-value enhancing

efforts in the pre-arrival phase should apply generally to innovative firms who invest to im-

prove the commercial value of innovations. If this condition does not apply, then innovation

value enhancement efforts will have negative NPV. Therefore, for 1 < t ≤ T, the innovation

value state variable (at the beginning of t) can be written as

λt =τ=t−1∏τ=1

ξ̃τ = λt−1ξ̃t−1. (16)

Note that at the beginning of T, the innovation arrives, and, therefore, there is no further

realization of ξ̃ in t > T . In other words, λt = λT for t > T. Finally, for tractability, we

assume that the innovation arrival probability (p) and the value enhancement probability

(q) are uncorrelated, and are independent of the cash flow shocks, {εt, υt, ηt}.

For our purposes, it is important to deduce the effect of the innovation-value state variable

on the firm’s expected returns after the arrival/exercise of the innovation (i.e., t > T ). In

the post-arrival/exercise regime, the ex-dividend value of the firm can be written as the sum

12

of the value of the old and new businesses, i.e., Vt = V At + V N

t , where VNt denotes the value

of the new business. Similar to V At , V

Nt can also be expressed as the sum of the value of the

firm-specific cash flows (V NYt ) and systematic cash flows (V NX

t ), i.e., V Nt = V NY

t + V NXt .

Using preceding arguments, we can show that V NYt = λTYt

eδY −1, and V NX

t = λTφXt

eδX−1, where

δY = r − (µY + σ2υ2

) > 0. Given this, it is straightforward to show that returns on the firm-

specific new cash flows, RNYt,t+1, is e

(r+υt+1−σ2υ2), and returns on the systematic new cash flows,

RNXt,t+1, is e

(r+ηt+1+σ2η). Note that, under our assumption of a single systematic risk, the return

on the systematic component of the new and old businesses is the same, i.e., RNXt,t+1 = RAX

t,t+1.

We denote the post-arrival and pre-arrival value weights of the different components of

innovative firms by ω̂jt ≡V jt>TVt>T

and ω̄jt ≡V jt<TVt<T

, respectively, where j ∈ {NY, NX, AZ, AX}.

Hence the one-period return of an innovative firm in the post-exercise regime is

Rt,t+1 = ω̂AXt RAXt,t+1 + ω̂NXt RNX

t,t+1 + ω̂AZt RAZt,t+1 + ω̂NYt RNY

t,t+1, (17)

where Et[RAZt,t+1

]= Et

[RNYt,t+1

]= er, and Et

[RAXt,t+1

]= Et

[RNXt,t+1

]= e(r+

32σ2η) (see the Appen-

dix, Section C).

Using the fact that the value weights of the different components sum to one, we can

write

Et≥T [Rt,t+1] = er + (ω̂AXt + ω̂NXt )er(e32σ2η − 1). (18)

The expected returns expression in (18) has a ready interpretation. Similar to the derivation

of the factor beta for the mature firm (cf. (7)), we can exploit the fact that Yt and Xt are

uncorrelated to show that for any t the factor beta of an innovative firm is

βI =∂Vt∂Xt

Xt

Vt= ωNXt + ωAXt . (19)

In particular, the equilibrium expected returns in (18) (for the post-arrival phase) are driven

by the factor beta βI and the risk-premium er(e32σ2η−1). It also then follows that for the post-

13

arrival phase ∂Et≥T [Rt,t+1]∂λT

∝ ∂βI

∂λT= ∂(ω̂AXt +ω̂NXt )

∂λT. We can then show that (see the Appendix,

Section D), for any t ≥ T,

∂Et≥T [Rt,t+1]

∂λT∝ ω̂NXt ω̂AZt − ω̂AXt ω̂NYt > 0, (20)

if ω̂NXtω̂NYt

> ω̂AXtω̂AZt

. That is, the post-arrival expected returns increase with λT if following the

arrival and commercial development of the innovation, the value of the systematic component

of the new business relative to its firm-specific component exceeds the corresponding ratio

for the established business. Shortly (in Section 2.2), we will give an empirical content to

this condition.

The positive relation between λT and post-arrival/exercise expected returns of innovative

firms is important to developing predictions on the relation between their pre-arrival IVOL

and post-arrival expected returns, as we now show. In the pre-arrival phase (i.e., t < T ), the

ex-dividend value of the new venture, V Nt , depends on the beginning-of-period state variable

(λt) and the current realization of the value enhancement (ξ̃t) in the following fashion. With

probability p, the innovation is realized at the beginning of next period, i.e., t+ 1 = T. The

firm then does not make any further value enhancement efforts in t+ 1, and, therefore, the

(ex-dividend) value of the new business at the end of T is

V Nt+1=T =

λTYT

eδY − 1

+λTφXT

eδX − 1

, (21)

where λT = λtξ̃t. But with probability (1−p), the innovation is not realized at t+1, therefore,

the firm makes additional value enhancement efforts in t + 1 and realizes ξ̃t+1. Hence, for

any t < T, conditional on Xt,

V Nt (λt, ξ̃t, Xt) = pEt

[θt+1θt

(DT + V NT )

]+ (1− p)Et

[θt+1θt

V Nt+1<T (λt+1, ξ̃t+1, Xt+1)

], (22)

where DT denotes cash flows in T and is equal to λT (YT +φXT ). Now, under the assumption

14

that YT is lognormal and yT = µY + υT , using forward projection on V Nt+1<T , we show in the

Appendix (Section E) that,

V Nt<T (λt, ξ̃t, Xt) =

pλtξ̃t(eδ

Y − 1)[1− (1− p)e−rξ̄

] +pλtξ̃tφXte

σ2η2

(eδX − 1)

[1− (1− p)e−(r+

σ2η2)ξ̄

] , (23)

where ξ̄ ≡ Et(ξt+1) is time-invariant by assumption. We denote the first and second terms

on the right-hand-side of (23) by V NYt<T (λt, ξ̃t) and V

NXt<T (λt, ξ̃t, Xt), which represent the value

of the firm-specific and systematic components of the new venture, respectively. In sum, the

total value of an innovative firm at t < T is

Vt<T (λt, ξ̃t, Xt) = V AXt + V AZ

t + V NYt (λt, ξ̃t) + V NX

t (λt, ξ̃t, Xt). (24)

As mentioned earlier, ω̄jt ≡V jt<TVt<T

, where j ∈ {NY,NX,AZ,AX}.Applying the decomposition

of the one-period returns (see (17)) to the pre-arrival phase, we can write the residual or

non-systematic returns as:

Rrest,t+1 = ω̄AZt RAZ

t,t+1 + ω̄NYt RNYt,t+1, (25)

where RNYt,t+1 = (1 − p)ξ̃t+1 +

[1− (1− p)e−rξ̄

]eδYYT . Hence, the pre-arrival IVOL (the

standard deviation of the residual returns) can be computed as

IV OL2t<T = V ar(Rrest,t+1) = (ω̄AZt )2V ar(RAZ

t,t+1) + (ω̄NYt )2V ar(RNYt,t+1). (26)

Here, V ar(RAZt,t+1) = e2r(eσ

2ε − 1) and

V ar(RNYt,t+1) = (1− (1− p)e−rξ̄)2e2r(eσ2υ − 1) + (1− p)2σ2ξ , (27)

15

where σ2ξ is the (time-independent) variance of ξ̃t.14 Using (26), we show in the Appendix

(Section F) that

∂IV OL2t<T∂λt

=

(2

λt

)[ω̄At (ω̄NYt )2V ar(RNY

t,t+1)− ω̄Nt (ω̄AZt )2V ar(RAZt,t+1)

], (28)

where ω̄At and ω̄Nt denote the value weights of the initial and new businesses, respectively. It

is immediate from (28) that IVOL is independent of λt (i.e.,∂IV OL2t<T

∂λt= 0) in the case of a

“pure”innovative firm that has all its value in growth options, that is, ω̄At = ω̄AZt = 0. But

for the general case where the component value weights are all positive, we can rearrange

terms in (28) to deduce that∂IV OL2t<T

∂λt> 0 if

(ω̄Atω̄AZt

)(ω̄NYtω̄Nt

)V ar(RNY

t,t+1) >

(ω̄AZtω̄NYt

)V ar(RAZ

t,t+1). (29)

Then, under conditions (20) and (29), IV OL2t<T is positively related to Et≥T [Rt,t+1] because

λt<T is positively related to λT , which in turn is positively related to Et≥T [Rt,t+1] . In other

words,∂Et≥T [Rt,t+1]

∂IV OL2t<T∝ ∂Et [Rt,t+1 | t ≥ T ]

∂λT

∂λT∂λt

∂IV OL2t<T∂λt

> 0. (30)

Thus, we conclude,

Proposition 2 If the pre-arrival IVOL condition (cf. (29)) and the post-arrival expected

return condition (cf. (20)) hold, then the pre-arrival IVOL (IV OL2t<T ) of innovative firms

is positively related to their future expected returns (i.e., the expected returns following the

arrival and exercise of the innovation).

To generate testable empirical predictions from Proposition 2, however, we need to pro-

vide empirical content to these two conditions (cf. (29) and (20)), to which we now turn.

2.2. Empirical predictions14That is, σ2ξ = q(ξh)2 + (1− q) ∗ (ξl)2 − ξ̄2.

16

The theoretical analysis in Section 2.1 presents two main empirical predictions. Proposi-

tion 1 implies a negative relation of IVOL to contemporaneous expected returns for mature

firms that are not actively involved in generating innovations or new growth options, other

things held fixed. Proposition 2 indicates that certain types of innovative (or non-mature)

firms – specifically, those for which the assumptions of the proposition apply – there is a

positive dynamic relation of IVOL to future expected returns. To give empirical content to

Proposition 2, we relate these assumptions to observable firm characteristics.

We turn first to the post-arrival expected return condition (cf. (20)). This condition

is likely to hold for both large and small innovative firms. In general, consumers’demand

for new products is more sensitive to the state of the economy. For example, when a new

generation of smart phones is launched, the success of these products in the market depends

highly on economy-wide factors because investors can simply retain their previous phones

in case of economic downturns. Thus, we expect ω̂NXt /ω̂NYt to be large relative to ω̂AXt /ω̂AZt

generally.

However, the innovation life cycle theory (mentioned in the introduction) suggests that

the pre-arrival IVOL condition (cf. (29)) is likely to hold only for big innovative firms. At any

t < T, big innovative firms are in the later life cycle of successful innovations (from their past

innovation generation efforts) that have large market sizes (Klepper, 1996). Therefore, the

value weight of the systematic component of their AIP (ω̄AXt ) will be large relative to the non-

systematic component (ω̄AZt ), i.e., they have a larger ω̄AXt /ω̄AZt . In contrast, small innovative

firms (at t < T ) are at the early stages of innovation life cycle, or their past innovation

efforts were unsuccessful; in either case, their small or localized innovation markets imply

a relatively greater value weight of the non-systematic component of AIP, i.e., they have

a smaller ω̄AXt /ω̄AZt . In sum, at any t < T, the ratio ω̄At /ω̄AZt

(= (ω̄AXt + ω̄AZt )/ω̄AZt

)is

cross-sectionally positively related to firm size.

Meanwhile, in the pre-arrival stage (t < T ), the future potential innovation is not fully

developed or finalized, and typically there is significant uncertainty regarding its configu-

17

ration and market potential. Hence, the ratio ω̄NYt /ω̄Nt is likely to be invariant to size.

Finally, because ω̄AZt decreases with size (as suggested by the innovation life cycle theory),

the ratio ω̄AZt /ω̄NYt decreases with size as well if V NYt (or, effectively, V N

t ) tends to increase

with size. This assumption is plausible since big innovative firms generally invest heavily in

R&D to develop major innovations, such as powerful smart phones in telecommunications

and “blockbuster”drugs in pharmaceuticals, to maintain their growth and meet investors’

expectations. In sum, the likelihood of satisfying the pre-arrival IVOL condition (cf. (29))

increases with the size of innovative firms.

Combining the discussions on the post-arrival expected return condition (cf. (20)) and

the pre-arrival IVOL condition (cf. (29)) thus yields the empirical prediction that IVOL of

big innovative firms should be positively related to their future stock returns. But there is

no such prediction for small innovative firms.

Finally, we can infer from (27), (28), and (30) that ∂Et≥T [Rt,t+1]∂IV OL2t<T

increases with E[ξ̃]. The

literature suggests a positive relation of E[ξ̃] to investment in innovative capacity that raises

the firm’s ability to generate economically successful innovations (Furman, Porter, and Stern,

2002; Kumar and Li, 2016). In particular, Kumar and Li (2016) show that asset growth (AG)

of innovative firms is a good empirical proxy for investment in innovative capacity. Therefore,

we expect that the strength of the positive relation of IVOL to future expected returns for

big innovative firms increases with AG, other things being equal.

As indicated in the introduction, the existing empirical literature has verified the predic-

tion on the negative cross-sectional relation of IVOL to average future returns. However, the

prediction regarding the heterogeneous dynamic relation of IVOL to future expected returns

is novel and has not been empirically examined to our knowledge.

3. Data and Sample Statistics

3.1. Identification of big innovative firms and data

18

Empirical tests of our model require identification of big innovative firms. For the reasons

mentioned in the introduction, there are multiple innovation proxies: firms with non-missing

R&D expenditures (that is, R&D-active firms), or low DTE, or high MABA, or operating

in growth-option-intensive industries (as defined by Grullon et al., 2012).15 As mentioned

earlier, we use total assets to proxy for firm size.

Using the R&D expenditures data imposes restrictions on our sample period. As is well-

known, the accounting treatment of R&D spending has varied over time, which necessitates

careful sample selection to ensure consistency in interpreting the R&D expenditure data.

Prior to 1976, firms had substantial discretion in determining what goes into R&D and how

they report it. The R&D reporting practice was standardized in 1975 (Financial Accounting

Standards Board Statement No. 2). Our sample, therefore, is from 1976 to 2015 and consists

of firms at the intersection of COMPUSTAT and CRSP (Center for Research in Security

Prices).

We obtain accounting data from COMPUSTAT and stock returns data from CRSP.

All domestic common shares trading on NYSE, AMEX, and NASDAQ with accounting

and returns data available are included except utilities and financial firms (i.e., those with

standard industrial classification (SIC) codes beginning with 49 and those between 6000 and

6999). Moreover, following Fama and French (1993), we exclude closed-end funds, trusts,

American Depository Receipts, Real Estate Investment Trusts, units of beneficial interest,

and firms with negative book value of equity. To mitigate backfilling bias, we require firms

to be listed on Compustat for at least two years.

3.2. Summary statistics and verification of the negative IVOL-return relation

To test the predictions of the model, we use portfolio sorts and Fama-MacBeth regressions

as detailed later. Prior to these tests, we show the patterns in the data regarding the

innovation proxies, (asset) size and IVOL. It is convenient to organize the sample in terms of

15Using R&D intensity such as the ratio of R&D expense to sales generates almost identical results asusing R&D expenses.

19

portfolios based on R&D expenditures. Table 1 reports the summary statistics of the eight

R&D-assets-IVOL portfolios formed from independent triple sorts (2×2×2). Specifically, at

the end of June of each year t from 1977 to 2015, we sort firms independently into two R&D

portfolios based on R&D expenditures in fiscal year ending in calendar year t− 1, two assets

portfolios based on NYSE median breakpoints for total assets measured in fiscal year ending

in calendar year t−1, and two IVOL portfolios based on IVOL computed at the end of June

of year t as the standard deviation of residuals from regressing a firm’s daily returns over the

past 12 months on the Fama and French (1992) three factors returns. We compute IVOL

based on daily returns over one year instead of one month (as focused in Ang et al., 2006)

because innovation efforts are usually long-term projects and it generally takes time for a

firm to generate growth options. Therefore, IVOL associated with innovation generation is

more plausibly captured over horizons longer than one month. However, as we will show

shortly, the well-known negative IVOL-return relation documented in the literature (Ang et

al., 2006) continues to hold for our sample and IVOL computation.

To form the two R&D portfolios, we assign firms with missing R&D expenditures (or

non-R&D firms) into the ‘Low’R&D portfolio, and the rest into the ‘High’R&D portfolio.

Therefore, we henceforth use “low R&D”and “non-R&D”interchangeably, and “high R&D”

and “R&D” interchangeably. The intersection of these portfolios form eight R&D-assets-

IVOL portfolios. In the first column of Table 1, “L/S/L”denotes the Low R&D, Small total

assets, and Low IVOL portfolio, and so forth. For each portfolio, we report the time-series

mean of cross-sectional average characteristics measured in the fiscal year ending in calendar

year t−1 except Size (market capitalization) and IVOL measured at the end of June in year

t. R&D, Assets, and Size are in millions. BTM denotes book-to-market equity. Following

Cao, Simin, and Zhao (2008), we compute market-to-book assets (MABA) as (Total Assets

—Total Common Equity + Price × Common Shares Outstanding)/Total Assets, and debt-

to-equity ratio (DTE) as (Debt in Current Liabilities + Total Long-Term Debt + Preferred

Stock)/(Common Shares Outstanding × Price). We also report assets growth for these

20

portfolios. All variables are winsorized at the top and bottom 1% level to mitigate the

influence of outliers.

Table 1 shows that among big R&D firms, firms with low and high IVOL spend on average

213 and 101 million on R&D expenses, respectively, which dominate the R&D expenses of

small R&D firms. Although big R&D firms have a lower R&D-to-sales ratio compared with

small R&D firms, they are the main driving force for aggregate R&D spending. These big

R&D firms are also economically important – their market capitalization is about 56% of

our sample universe. The spread (in IVOL) between the high and low IVOL portfolios is

similar across the four R&D-assets groups, which suggests that the results we show later

are not driven by the variation of the IVOL spread. High IVOL firms are much smaller

than low IVOL firms in terms of assets or market capitalization. Firms with high IVOL also

tend to have higher BTM, which is consistent with the negative relation between IVOL and

size. Controlling for size and IVOL, we also observe lower DTE and higher MABA for high

R&D firms, which is consistent with the literature and reinforces the use of low DTE and

high MABA as innovation proxies. Furthermore, big R&D firms with high IVOL have the

highest asset growth, 25% on average, which is consistent with the intuition that these firms

are likely to invest more in innovative capacity (IC) investment (see Kumar and Li, 2016).

Before testing the dynamic IVOL—future return relation for innovative firms, we first

confirm the well-known negative IVOL—return relation documented in the literature for our

sample and the annual IVOL measure. In Table 2, we sort firms into deciles based on

idiosyncratic volatility measured at the end of June of year t. We also form a high-minus-

low (10 − 1) IVOL portfolio. We then compute monthly value-weighted portfolios returns

over the next 12 months. The table reports the average portfolio returns, excess returns

(Exret), intercepts (alphas in percentage) and factor loadings from regressions of time-series

portfolio excess returns on the Carhart (1997) four factors returns. The portfolio excess

returns are the difference between monthly portfolio returns and the one month Treasury

bill rate. Market, SMB, HML, and UMD refer to the loadings on the market, size, value,

21

and momentum factors, respectively. The heteroscedasticity-robust t-statistics are reported

in parentheses.

Consistent with the literature, Table 2 shows that firms in the highest IVOL decile

significantly underperform the lowest IVOL decile in both returns and alphas. For example,

the monthly value-weighted alpha of the high-minus-low IVOL portfolio is —1.09%, which is

significant at the 1% level.

4. Empirical tests of the theoretical predictions

4.1. Main tests

Our model predicts that IVOL of big innovative firms should be positively related to their

future stock returns. Moreover, this positive relation should be stronger for big innovative

firms with high asset growth (AG). However, the positive predictive relation of IVOL to

future stock performance need not hold for non-innovative firms, or for small innovative

firms. We note that these empirical predictions impose restrictions on innovation activity,

asset size, and IVOL.

Therefore, we first test these predictions through independent triple sorts of firms in

terms of high versus low innovation activity, small versus big asset size, and high versus

low IVOL. We use non-missing R&D expenditures, low leverage (DTE), high MABA, and

growth-option-intensive (GOI) industries, respectively, as proxies of high innovation activ-

ity.16 Conversely, missing R&D expenditures, high DTE, low MABA, and operating in

non-GOI industries indicate low innovation activity.

We have described the formation of these portfolios for the R&D-expenditure innovation

proxy in Table 1. The construction of portfolios for the other innovation proxies is similar.

At the end of June of each year t from 1977 to 2015, we sort firms independently into two

portfolios based on levels of DTE, MABA, or operating in GOI industries in the fiscal year

16In particular, in Panel D of Table 3, following Grullon et al. (2012), we include Fama and French(1997) industries 27 (precious metals), 28 (mining), 30 (oil and natural gas), 22 (electrical equipment), 32(telecommunications), 35 (computers), 36 (electronic equipment), 37 (measuring and control equipment), 12(medical equipment), and 13 (pharmaceutical products) in the growth-option-intensive industries.

22

ending in calendar year t− 1; two assets portfolios based on NYSE median breakpoints for

total assets measured in fiscal year ending in calendar year t − 1; and two IVOL portfolios

based on IVOL computed at the end of June of year t as the standard deviation of residuals

from regressing a firm’s daily returns over the past 12 months on the Fama and French

(1992) three factors returns. The intersection of these portfolios forms eight innovation-

assets-IVOL portfolios for each innovation proxy. In addition, we also form a high-minus-low

IVOL portfolio within each innovation-assets portfolio.

Turning to the timing conventions, in our model, the relation of IVOL to future stock

returns of innovative firms refers to ∂Et≥T [Rt,t+1]∂IV OL2t<T

, where T is the innovation arrival time. In our

large sample of firms (all firms at the intersection CRSP and COMPUSTAT) over a forty-

year period (1976-2015), one can not observe the start and completion of R&D projects at

the level of individual firms. Hence, we appeal to the empirical R&D literature to calibrate

T through the average duration of innovation related projects. Using surveys of innovative

firms in the U.K., Whittard et al. (2009) report an average duration of 2 years (with average

durations of 2.3 years in high-technology firms and 1.5 years in less technologically intensive

R&D-active firms). In a similar vein, Jannuzzi (2005) reports an average duration of 21—24

months for R&D projects in the energy sector of Brazil. Hence we examine the relation

of IVOL to future stock returns over the three post-formation years, where on average the

post-arrival returns should be apparent in years 2 and 3.

Panels A through D in Table 3 report the average value-weighted (VW) portfolio returns

and alphas (in percentage) from different factor models over the next three years for the four

proxies of innovation activity: R&D expenditures, DTE, MABA, and location in growth-

option-intensive (GOI) industries. The 4f alphas are from regressions of time series VW

portfolio excess returns on the Carhart (1997) four factors returns, whereas 5f alphas are

from regressions of portfolio excess returns on the Carhart factors plus the liquidity factor as

in Pastor and Stambaugh (2003).17 We also report alphas from the Fama-French five-factor

17Year 1 is from July of year t to June of year t+ 1. Year 2 is from July of year t+ 1 to June of year t+ 2.

23

model (Fama and French, 2015) later and find that the results are even stronger.

The results in Table 3 support the predictions of our conceptual framework. In Panel A,

consistent with the literature, we find significantly negative IVOL return spreads in Year 1

for small non-R&D firms and small R&D firms. For example, the monthly 5f alphas of the

high-minus-low IVOL portfolios are —0.67% (t = —4.37) and —0.48% (t = —2.86) for small non-

R&D and small R&D firms, respectively. For large non-R&D firms, the IVOL return spreads

are negative but insignificant. In contrast, for big R&D firms, the IVOL return spreads are

positive (although insignificant) in Year 1, and significantly positive in Year 2. Specifically,

the monthly returns, 4f alpha, and 5f alpha of the high-minus-low IVOL portfolio among

big R&D firms are 0.88% (t = 2.47), 0.67% (t = 2.07), and 0.70% (t = 2.13), respectively,

in Year 2. Furthermore, big R&D firms with high IVOL mainly drive these large return

spreads, which is consistent with our model, and with the intuition that high IVOL for big

innovative firms on average signals higher value of innovation-based growth options.

In Panel B, the relation between IVOL and future returns is even stronger using low

DTE as the innovation proxy. The monthly value-weighted returns, 4f alpha, and 5f alpha

of the high-minus-low IVOL portfolio among big firms with low leverage are 1.25% (t =

2.66), 1.20% (t = 2.62), and 1.20% (t = 2.59), respectively, in Year 2. Moreover, this pattern

exists in Year 3 as well. The monthly returns, 4f alpha, and 5f alpha of the high-minus-low

IVOL portfolio among big firms with low leverage in Year 3 are 1.16% (t = 2.79), 1.29% (t =

3.17), and 1.21% (t = 2.92), respectively. And in Panels C and D, we also find a significant

positive relation of IVOL to future stock returns in year 2 (MABA) and year 3 (location in

GOI industries).

With respect to firms’location in GOI industries, the results are stronger (untabulated)

when we focus only on the hi-tech industries.18 One possible reason is that in hi-tech indus-

Year 3 is from July of year t+ 2 to June of year t+ 3. Year 4 is from July of year t+ 3 to June of year t+ 4.Year 5 is from July of year t+ 4 to June of year t+ 5. The heteroscedasticity-robust t-statistics are reportedin parentheses.18Hi-tech industries include the Fama and French (1997) industries 22 (electrical equipment), 32 (telecom-

munications), 35 (computers), 36 (electronic equipment), and 37 (measuring and control equipment).

24

tries, innovations usually lead to local monopoly for some time, that is, if a firm comes up

with new product or technological breakthrough, then it can have market or pricing power

in its segment till the next innovation. Hence, IVOL in the pre-arrival phase is much more

likely to be informative of the economic value of future innovations. However, in the oil

and gas or precious metals industries, which are also classified as growth-option-intensive

industries, commodity prices are out of the control of even the largest firms and are much

more volatile owing to other types of uncertainty (such as natural disasters). Hence, stock

return IVOL in these types of industries is subject to many factors that are not in our model.

It is possible that the dynamic positive relation of IVOL to future stock performance of

big innovative firms is due to time-varying increasing factor loadings (for well-known risk

factors) of these firms. To examine this possibility, in Table 4 we report the factor loadings

from the Carhart model regressions mentioned above. For brevity, we report these only for

R&D-expenditures and low DTE as innovation proxies (but the results are similar for the

other proxies).

In Panel A of Table 4, we find that controlling for R&D and size, high IVOL firms load

significantly higher on the market and size factors compared with low IVOL firms in each

of the three succeeding years. The loadings on the other factors over different years are

not robustly related to the level of IVOL, however. These patterns are consistent with the

results from single sort reported in Table 2. And we find very similar patterns in Panel B,

where we use low DTE as the innovation proxy. In sum, the positive predictive IVOL-return

relation among big innovative firms observed in Table 3 is not driven by their time-varying

loadings on standard risk factors. This is consistent with Da, Guo, and Jagannathan (2012)

and Grullon, Lyandres, and Zhdanov (2012), who suggest that a risk factor model is likely

to generate “abnormal”returns (alphas) if it does not take into account real options, which

are particularly relevant for innovative firms.

In Table 5, we conduct time-series factor regressions as in Table 3, but with the recently

developed Fama-French (2015) five-factor model, which includes the investment and prof-

25

itability factors in addition to the traditional market, size, and value factors.19 For brevity, we

only report the monthly value-weighted alphas (in percentage) and their heteroscedasticity-

robust t-statistics in parentheses over the three post-sorting years for the eight innovation-

assets-IVOL portfolios and the IVOL hedge portfolios formed among the innovation-assets

groups. In Panel A, we use R&D-active firms to proxy for innovative firms. In Panels B,

C, and D, we use low DTE firms, high MABA firms, or firms operating in those growth-

options-intensive industries to proxy for innovative firms, respectively.

The results are even sharper. For example, in Panel A, the monthly IVOL alpha spread

among big R&D-active firms in Year 2 is 1.01% (t = 3.05). In Panel B, the monthly IVOL

alpha spreads in Years 2 and 3 among big, low DTE firms are 1.59% (t = 3.43) and 1.51%

(t = 3.73), respectively. In Panel C, the monthly IVOL alpha spreads in Years 2 and 3

among the big, high MABA firms are 1.14% (t = 2.50) and 0.83% (t = 2.14), respectively.

Similarly, in Panel D, the monthly IVOL alpha spread in Year 3 among big firms operating

in those growth-options-intensive industries is 0.88% (t = 2.32).

Next, to ensure that our results are not driven by other well-known return predictors,

we conduct monthly Fama-MacBeth (1973) cross-sectional regressions, which allow us to

control for them. Specifically, for each month j between July of year t+ i and June of year

t+ i+1 (i = 0, 1, 2, 3, 4), we regress firms’monthly stock returns, Rj, on a set of independent

variables as the following:

Rj = αj +β1jIV OLt +β2jIV OLt ∗ Innov_Bigt +β3jInnov_Bigt + g(Controls) + εj, (31)

where IV OL is a dummy variable that equals 1 if a firm’s idiosyncratic volatility (computed

at the end of each June of year t as defined earlier) is above median and 0 otherwise.

19Motivated by the q-theory of investment, Hou, Xue, and Zhang (2015) develop a four-factor model thatincludes the investment and profitability factors in addition to the market and size factors. Although thetwo models (Fama and French, 2015; Hou et al., 2015) differ in their motivation, the construction of the twonew factors is quite close. Therefore, for simplicity, we only report the results from the Fama and French(2015) five-factor model.

26

Innov_Big is a dummy variable that equals 1 if a firm is in the big innovative portfolio as

defined in Table 3, and 0 otherwise.

We use a dummy for IV OL to facilitate comparison with the portfolio sorts and to

mitigate multicollinearity between IV OL and Innov_Big. However, the results are similar

if we use raw value of IVOL. We control for size, book-to-market equity, momentum, and

short-term return reversals effects in the regressions. (The results are similar if we do not

control for return reversals as discussed below.) Specifically, ln(BTM) denotes the natural

logarithm of the ratio of book equity in the fiscal year ending in calendar year t−1 to market

equity at the end of calendar year t− 1. ln(Size) is the natural logarithm of market equity

at the end of June of year t. Momentum is the cumulative return over the prior 11 months

with a one-month gap. Reversal is the lagged monthly return. To mitigate the effects of

outliers, all independent variables (except dummies) are winsorized at the top and bottom

1%. Weighted least square regressions are used with lagged market capitalization as the

weight to mitigate effects of smaller firms and to be consistent with the VW returns in the

portfolio sorts.

The IVOL effect among big innovative firms is captured by (β1j + β2j), while that for

the other firms is reflected in β1j. As mentioned earlier, our model predicts a significantly

positive IVOL effect among big innovative firms (that is, (β1j + β2j) > 0).

Panels A through D of Table 6 report the time-series average slopes (in percentage)

and their t-statistics (in parentheses) from the regressions above for each of the innovation

proxies. The results support the predictions of the model and are consistent with the sorting

results in Table 3. Specifically, in Panel A, where R&D is the proxy of innovation, the

slope on IVOL is significantly negative in Year 1 (−0.41%, t = −2.57) and Year 2 (−0.34%,

t = −2.33), which implies that there exists a significantly negative relation between IVOL

and next two years’returns among firms other than big innovative firms. The slope on the

interaction term is significantly positive in Years 1 (0.53%, t = 2.04) and 2 (0.91%, t = 3.01).

More importantly, they are larger than the corresponding slopes on IVOL, which implies a

27

positive IVOL relation for big innovative firms. Furthermore, in Year 2 the magnitude of the

slope on the interaction term, 0.91%, almost triples that on IVOL, —0.34%. This evidence

is consistent with the prediction of our model that IVOL is positively related to future

stock returns among big innovative firms. The slopes on size, momentum, and reversals are

consistent with the literature. The slope on ln(BTM) is positive but insignificant.20

In Panel B where we use low leverage as the innovation proxy, the slope on IV OL is

significantly negative in Years 1 and 2, while the slope on the interaction term, IV OL ∗

Innov_Big is significantly positive in the first three years. Furthermore, the magnitude of

the slope on IV OL ∗ Innov_Big is in general much larger than that of the slope on IVOL.

Specifically, the slopes on IV OL ∗HRD_Big in Years 1 to 3 are 0.83% (t = 2.39), 1.24%

(t = 3.36), and 1.09% (t = 3.12), respectively. In contrast, the slopes on IV OL in Years 1

to 3 are —0.36% (t = −2.27), —0.32% (t = —2.22), and —0.21% (t = —1.55), respectively. The

slope on the interaction term exhibits similar pattern in Panel B, where we use MABA as the

proxy; here the slope on IV OL∗Innov_Big is significant in years 2 and 3, while in Panel D,

where operating in GOI industries is the innovation proxy, the slope on IV OL ∗ Innov_Big

becomes statistically significant in Year 3.

All these patterns are robust if we do not control for return reversals (untabulated).

Furthermore, in untabulated analysis, we control for asset growth, ROA (return on assets),

and skewness. The results are qualitatively unchanged.

4.2. The effects of asset growth

As mentioned earlier, we expect an even stronger positive IVOL-return relation for big

innovative firms with higher AG. This is because, as shown in Kumar and Li (2016), AG of

innovative firms captures investment in innovative capacity, which should increase expected

value from innovative efforts.

Specifically, in Table 7, we conduct monthly Fama-MacBeth regressions as in Table 6

with additional controls for asset growth (a dummy variable that equals 1 for firms with

20This may be because we use weighted least square regressions instead of ordinary least square regressions.

28

above-median asset growth and 0 otherwise) and its interactions with the IV OL and the

Innov_Big dummy variables. We also control for ROA since it is also a well-known return

predictor. Our focus is the average slope on the triple interaction term, IV OL∗Innov_Big∗

AG. It is significantly positive in Year 2 and/or Year 3, which is consistent with the prediction

of the model. In fact, its magnitude is substantial and dominates the sum of the slopes on

the other terms that contain IVOL – IV OL, IV OL ∗ AG, and IV OL ∗ Innov_Big. For

example, in Panel A, in which we proxy innovative firms by R&D-active firms, the slope

on IV OL ∗ Innov_Big ∗ AG is 1.14% (t = 2.23) and 1.51% (t = 2.72) in Years 2 and

3, respectively. In contrast, the corresponding slopes on IV OL ∗ Innov_Big are tiny and

insignificant, 0.16% (t = 0.46) and −0.34 (t = −0.80), respectively. The same pattern

exhibits for the other proxies of innovative firms as shown in Panels B, C, and D.

This evidence suggests that the significantly positive slope on IV OL ∗ Innov_Big re-

ported in Table 6 are driven by big innovative firms with high innovative capacity investment.

This finding is consistent with our theoretical framework.

5. Conclusions

We study the relation of IVOL of innovative firms to future stock returns. Innovative

firms actively make efforts to generate growth options, and their IVOL presumably contains

conditioning information on future growth options. Our conceptual framework shows that

the IVOL of big innovative firms is positively related to their future expected returns because

of their asset composition and intentional efforts to generate ‘blockbuster’innovations; that

is, high IVOL predicts higher future stock performance for a class of economically important

firms. Furthermore, this relation should be stronger for big innovative firms with higher

innovative capacity investment. However, this relation need not hold for mature firms, or

for small innovative firms. These predictions utilize the insights and stylized facts developed

by the literature on the economics of technological change and innovation life cycle.

Empirical analyses provide strong support for these predictions. We utilize various prox-

29

ies for innovative firms and use both independent sorts and Fama-MacBeth (1973) cross-

sectional regressions, and our results remain robust. We find that IVOL of big innovative

firms is significantly positively related to future average stock returns and alphas [relative to

the Carhart (1997) four-factor model, the Carhart model augmented by the liquidity factor

of Pastor and Stambaugh (2003), and the Fama-French (2015) five-factor model]. However,

we find no such dynamic relation for mature or small innovative firms. The positive predic-

tive IVOL-return relation among big innovative firms is not driven by time-varying loadings

on standard risk factors and is not due to differences in the sample period or the IVOL

calculation procedure compared with the literature. This relation is also intensified among

big innovative firms with higher innovative capacity investment.

While the negative cross-sectional relation of IVOL to future average returns receives

much attention, our study highlights the heterogeneous dynamic patterns between IVOL

and future stock returns that exist in the data. In particular, for big innovative firms, high

IVOL can contain good conditioning information on future economic value of growth options

and, therefore, on future stock performance.

30

Appendix for

“Innovation, Good Idiosyncratic Volatility, and Stock Returns”

A. Expected Factor Returns

Note that the expected factor return is Et[RXt,t+1

]=

Et[Xt+1+WXt+1]

WXt

. Hence, using (5) in

the text,

Et[RXt,t+1

]=

Et[Xt+1

(1 + 1

e(r+σ2η2 )−1

)]Xt/(e

(r+σ2η2) − 1)

=XtEt

[exp(ηt+1)

]e(r+

σ2η2)

Xt

= e(r+σ2η2+σ2η2) = e(r+σ

2η). (A.1)

B. Variance of AIP Returns

We have

RAt,t+1 =

Zt+1 + V AZt+1 + κXt+1 + V AX

t+1

V At

=Zt exp(µZ + εt+1)

eδZ

(eδZ−1)+ κXt exp(ηt+1)

eδX

(eδX−1)

V At

(A.2)

= ωAZt exp(µZ + εt+1 + δZ) + ωAXt exp(ηt+1 + δX). (A.3)

Hence, RAZt,t+1 = exp(µZ + εt+1 + δZ) = exp(r+ εt+1− σ2ε

2), RAX

t,t+1 = exp(r+ ηt+1 + σ2η). Using

these facts,

V ar(RAZt,t+1) = Et[(RAZ

t,t+1)2]−

(Et[RAZ

t,t+1])2

= e2r(eσ2ε − 1). (A.4)

C. Post-Arrival Returns of the New Business

31

For t ≥ T,

RNt,t+1 =

λTYt+1 + V NYt+1 + λTφXt+1 + V NX

t+1

V Nt

=λTYt exp(µY + υt+1)

eδY

(eδY −1)+ λTφXt exp(ηt+1)

eδX

(eδX−1)

V Nt

= wNYt exp(µY + υt+1 + δY ) + wNXt exp(ηt+1 + δX), (A.5)

where wNYt ≡ V NYt

V Ntand wNXt ≡ V NXt

V Ntdenote the value weights relative to the new business

V Nt . Note that R

Nt,t+1 can also be expressed as a weighted average of the individual compo-

nents’returns with wNYt and wNXt as the value weights, i.e., RNt,t+1 = wNYt RNY

t,t+1+wNXt RNXt,t+1,

where RNYt,t+1 and R

NXt,t+1 denote the returns on the idiosyncratic and systematic components of

the new cash flows, respectively. Therefore, RNYt,t+1 = exp(µY +υt+1+δ

Y ) = exp(r+υt+1− σ2υ2

),

and RNXt,t+1 = exp(ηt+1 + δX) = exp(r + ηt+1 + σ2η) = RAX

t,t+1.

D. Effect of λT on Post-Arrival Expected Returns

We have∂Et [Rt,t+1]

∂λT∝ ∂(ω̂AXt + ω̂NXt )

∂λT=Vt

φXt

eδX−1− (κ+λTφ)Xt

eδX−1∂Vt∂λT

V 2t

(A.6)

Note that

Vt =(κ+ λTφ)Xt

eδX − 1

+λTYt

eδY − 1

+Zt

eδZ − 1

. (A.7)

Therefore, we get

∂Vt∂λT

=φXt

eδX − 1

+Yt

eδY − 1

=ω̂NXt VtλT

+ω̂NYt VtλT

=VtλT

(ω̂NXt + ω̂NYt ). (A.8)

32

Thus,∂Vt∂λT

Vt= ω̂NXt +ω̂NYt

λT, and we have

VtφXt

eδX−1− (κ+λTφ)Xt

eδX−1∂Vt∂λT

V 2t

=ω̂NXtλT− (ω̂AXt + ω̂NXt )

∂Vt∂λT

Vt

=ω̂NXtλT− (ω̂AXt + ω̂NXt )

(ω̂NXt + ω̂NYt )

λT

=1

λT[ω̂NXt ω̂AZt − ω̂AXt ω̂NYt ]. (A.9)

E. Pre-Arrival IVOL of Innovative Firms

For t < T , we have

V Nt = pEt

[θt+1θt

(DT + V NT )

]+ (1− p)Et

[θt+1θt

V Nt+1<T

], (A.10)

where DT = λTYT + λTφXT . Now, under the assumption that YT is lognormal and yT =

µY + υT ,

Et[θt+1θt

(DT + V NT )

]=

λtξ̃teδY − 1

+λtξ̃tφXte

σ2η2

eδX − 1

. (A.11)

Now note that for t+ 1 < T,

V Nt+1<T = pEt+1

[θt+2θt+1

(DT + V NT )

]+ (1− p)Et+1

[θt+2θt+1

V Nt+2<T

]. (A.12)

Furthermore, we have

Et+1[θt+2θt+1

(DT + V NT )

]=λt+1ξ̃t+1eδY − 1

+λt+1ξ̃t+1φXt+1e

σ2η2

eδX − 1

. (A.13)

33

Therefore,

Et[θt+1θtEt+1

[θt+2θt+1

(DT + V NT )

]]= Et

θt+1θt

λt+1ξ̃t+1eδY − 1

+λt+1ξ̃t+1φXt+1e

σ2η2

eδX − 1

= e−rλtξ̃tξ̄

[1

eδY − 1

+φXt

eδX − 1

], (A.14)

where ξ̄ denotes Et(ξ̃t+1) = ξh ∗ q + ξl ∗ (1 − q) for any t < T and is time-invariant by

assumption. Therefore,

Et[θt+1θt

V Nt+1<T

]= pEt

[θt+1θt

(Et+1[θt+2θt+1

(DT + V NT )

])

]+(1− p)Et

[θt+1θt

(Et+1[θt+2θt+1

V Nt+2<T

])

]= pe−rλtξ̃tξ̄

[1

eδY − 1

+φXt

eδX − 1

]+(1− p)Et

[θt+1θt

(Et+1[θt+2θt+1

V Nt+2<T

])

], (A.15)

and (A.10) becomes

V Nt = pEt

[θt+1θt

(DT + V NT )

]+ (1− p)Et

[θt+1θt

V Nt+1<T

]

= p(λtξ̃t

eδY − 1

+λtξ̃tφXte

σ2η2

eδX − 1

) + (1− p)pe−rλtξ̃tξ̄[

1

eδY − 1

+φXt

eδX − 1

]+(1− p)2Et

[θt+1θt

(Et+1[θt+2θt+1

V Nt+2<T

])

]. (A.16)

Note that

V Nt+2<T = pEt+2

[θt+3θt+2

(DT + V NT )

]+ (1− p)Et+2

[θt+3θt+2

V Nt+3<T

]. (A.17)

Similarly, we can show that

Et+2[θt+3θt+2

(DT + V NT )

]=λt+2ξ̃t+2eδY − 1

+λt+2ξ̃t+2φXt+2e

σ2η2

eδX − 1

, (A.18)

34

and that

Et+1[θt+2θt+1

(Et+2[θt+3θt+2

(DT + V NT )

])

]= e−rλt+1ξ̃t+1ξ̄

[1

eδY − 1

+φXt+1

eδX − 1

]. (A.19)

Hence we have

Et[θt+1θtEt+1

[θt+2θt+1

(Et+2[θt+3θt+2

(DT + V NT )

])

]]= e−2rλtξ̃tξ̄

2

[1

eδY − 1

+φXt

eδX − 1

e−σ2η2

].

(A.20)

Therefore, the value of the firm at t < T is

V Nt<T = pEt

[θt+1θt

(DT + V NT )

]+ (1− p)Et

[θt+1θt

V Nt+1<T

]=

pλtξ̃teδY − 1

[1 + (1− p)e−rξ̄ + (1− p)2e−2rξ̄2 + ...

]+pλtξ̃tφXte

σ2η2

eδX − 1

[1 + (1− p)e−r−

σ2η2 ξ̄ + (1− p)2pe−2r−σ2η ξ̄2 + ...

]

=pλtξ̃teδY − 1

1

1− (1− p)e−rξ̄+pλtξ̃tφXte

σ2η2

eδX − 1

1

1− (1− p)e−r−σ2η2 ξ̄, (A.21)

assuming (1− p)e−rξ̄ < 1 and (1− p)e−r−σ2η2 ξ̄ < 1. In sum, for t < T,

V Nt<T (λt, Xt) =

pλtξt(eδ

Y − 1)(1− (1− p)e−rξ̄)+

pλtξtφXteσ2η2

(eδX − 1)(1− (1− p)e−r−

σ2η2 ξ̄)

. (A.22)

We can rewrite (A.22) as V Nt = V NY

t + V NXt , where

V NYt (λt, ξ̃t) =

pλtξt(eδ

Y − 1)(1− (1− p)e−rξ̄)(A.23)

V NXt (λt, ξ̃t, Xt) =

pλtξtφXteσ2η2

(eδX − 1)(1− (1− p)e−r−

σ2η2 ξ̄)

. (A.24)

F. Effect of λt on Pre-Arrival IVOL of Innovative Firms

35

∂IV OL2t<T∂λt

= 2ω̄AZt∂ω̄AZt∂λt

V ar(RAZt,t+1) + 2ω̄NYt

∂ω̄NYt∂λt

V ar(RNYt,t+1). (A.25)

Note that

∂ω̄AZt∂λt

= −VAZt

V 2t

∂Vt∂λt

= − ω̄AZt

Vt

∂Vt∂λt

(A.26)

∂Vt∂λt

=∂V NX

t

∂λt+∂V NY

t

∂λt= Vt

[ω̄NYtλt

+ω̄NXtλt

]> 0 (A.27)

∂ω̄NYt∂λt

=ω̄NYtλt

[1− ∂Vt

∂λt

λtVt

]=ω̄NYtλt

(1− ω̄NYt − ω̄NXt ). (A.28)

Therefore,

∂IV OL2t<T∂λt

= −2(ω̄AZt )2V ar(RAZt,t+1)

[ω̄NYt + ω̄NXt

λt

]+ 2(ω̄NYt )2V ar(RNY

t,t+1)

[1− ω̄NYt − ω̄NXt

λt

]=

(2

λt

)[ω̄At (ω̄NYt )2V ar(RNY

t,t+1)− ω̄Nt (ω̄AZt )2V ar(RAZt,t+1)

], (A.29)

where ω̄At ≡ 1 − ω̄NYt − ω̄NXt and ω̄Nt ≡ ω̄NYt + ω̄NXt denote the value weights of the initial

and new businesses, respectively.

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40

Table 1. Summary Statistics

At the end of June of each year t from 1977 to 2015, we sort firms independently into two R&D

portfolios based on R&D expenditures in fiscal year ending in calendar year t – 1, two assets

portfolios based on NYSE median breakpoints for total assets measured in fiscal year ending in

calendar year t – 1, and two idiosyncratic volatility (IVOL) portfolios based on IVOL, which is

computed at the end of June of year t as the standard deviation of residuals from regressing a firm’s

daily returns over the past 12 months on the Fama and French (1992) three factors. To form the

two R&D portfolios, we assign firms with missing R&D expenditures into the Low R&D portfolio,

and the rest into the High R&D portfolio. The intersection of these portfolios form eight R&D-

assets-IVOL portfolios. In the first column, “L/S/L” denotes the Low R&D, Small total assets, and

Low IVOL portfolio, and so forth. For each portfolio, we report the time-series mean of cross-

sectional average characteristics measured in the fiscal year ending in calendar year t – 1 except

Size (market capitalization) and IVOL, measured at the end of June in year t. R&D, Assets, and

Size are in millions. BTM denotes book-to-market equity. Following Cao, Simin, and Zhao (2008),

we compute market-to-book assets (MABA) as (Total Assets – Total Common Equity + Price ×

Common Shares Outstanding)/Total Assets, and debt-to-equity ratio (DTE) as (Debt in Current

Liabilities + Total Long-Term Debt + Preferred Stock) / (Common Shares Outstanding × Price).

AG denotes the total asset growth. All variables are winsorized at the top and bottom 1% level.

Financial and utility firms are excluded.

Rank R&D/Sales R&D Assets IVOL Size BTM DTE MABA AG

L/S/L 363 0.02 445 0.84 0.53 1.61 0.16

L/S/H 159 0.05 139 1.05 0.89 1.64 0.17

L/B/L 5629 0.02 5138 0.82 0.76 1.49 0.16

L/B/H 3407 0.04 1503 1.36 2.40 1.22 0.21

H/S/L 0.14 13.62 330 0.02 565 0.70 0.32 1.96 0.16

H/S/H 0.91 8.28 107 0.05 197 0.76 0.42 2.45 0.17

H/B/L 0.04 213.28 6771 0.02 8087 0.66 0.47 1.75 0.13

H/B/H 0.09 100.68 2889 0.04 2334 1.20 1.61 1.51 0.25

41

Table 2. Idiosyncratic Volatility and Stock Returns

At the end of June of each year t, we sort firms into deciles based on idiosyncratic volatility

measured as the standard deviation of residuals from regressing a firm’s daily returns over the past

12 months on the Fama and French (1992) three factors. We also form a high-minus-low (10-1)

IVOL portfolio. We then compute monthly value-weighted portfolios returns for the next 12

months. The table reports the average portfolio returns, excess returns (Exret), intercepts (alphas

in percentage) and factor loadings from regressions of time series portfolio excess returns on the

Carhart (1997) four factors returns. The portfolio excess returns are the difference between

monthly portfolio returns and the one month Treasury bill rate. Market, SMB, HML, and UMD

refer to the loadings on the market, size, value, and momentum factors, respectively. The

heteroscedasticity-robust t-statistics are reported in parentheses. Financial and utility firms are

excluded. The sample period for stock returns is from July of 1977 to December 2015.

IVOL Return Exret Alpha Market SMB HML UMDLow (1) 1.03 0.64 0.13 0.87 -0.26 0.02 0.03

(5.71) (3.54) (2.75) (61.21) (-11.75) (0.92) (1.76)Middle (5) 1.16 0.77 0.02 1.29 0.64 -0.43 -0.09

(3.33) (2.20) (0.17) (37.16) (7.80) (-7.38) (-2.55)High (10) 0.36 -0.04 -0.96 1.33 1.51 -0.30 -0.17

(0.75) (-0.07) (-3.80) (20.25) (14.87) (-2.53) (-2.49)High-Low -0.68 -0.68 -1.09 0.46 1.77 -0.32 -0.20

(-1.67) (-1.67) (-3.95) (6.41) (15.75) (-2.39) (-2.56)

42

Table 3. Innovation, Idiosyncratic Volatility, and Stock Returns

(Portfolio Sorts—Returns and Alphas)

In Panel A, at the end of June of each year t from 1977 to 2015, we sort firms independently into

two R&D portfolios based on R&D expenditures in fiscal year ending in calendar year t – 1, two

assets portfolios based on NYSE median breakpoints for total assets measured in fiscal year ending

in calendar year t – 1, and two idiosyncratic volatility (IVOL) portfolios based on IVOL as defined

in Table 1. To form the two R&D portfolios, we assign firms with missing R&D expenditures into

the Low R&D portfolio, and the rest into the High R&D portfolio. The intersection of these

portfolios form eight R&D-assets-IVOL portfolios. We also form a high-minus-low IVOL

portfolio within each R&D-assets portfolio. The table reports the average value-weighted (VW)

portfolio returns and alphas (in percentage) from different factor models over the next three years

and their heteroscedasticity-robust t-statistics in parentheses. 4f alphas are from regressions of time

series VW portfolio excess returns on the Carhart (1997) four factors returns. 5f alphas are from

regressions of portfolio excess returns on the Carhart factors plus the liquidity factor as in Pastor

and Stambaugh (2003). Year 1 is from July of year t to June of year t + 1. Year 2 is from July of

year t + 1 to June of year t + 2. Year 3 is from July of year t + 2 to June of year t + 3. Financial

and utility firms are excluded. The sample period for stock returns is from July of 1977 to

December 2015. In Panels B, C, and D, we use low leverage (DTE), high MABA, and growth-

option-intensive industries, respectively, as alternative proxies of innovation. DTE and MABA are

defined in Table 1. In Panel D, High GO refers to firms operating among those growth-option-

intensive industries as defined as in Grullon, Lyandres, and Zhdanov (2012), which include Fama

and French (1997) industries 27 (precious metals), 28 (mining), 30 (oil and natural gas), 22

(electrical equipment), 32 (telecommunications), 35 (computers), 36 (electronic equipment), 37

(measuring and control equipment), 12 (medical equipment), and 13 (pharmaceutical products).

43

Panel A. R&D as innovation proxy

Return 4f Alpha 5f Alpha Return 4f Alpha 5f Alpha Return 4f Alpha 5f AlphaLow/Small Low (L) 1.11 -0.05 -0.05 1.09 -0.05 -0.07 1.10 -0.01 -0.04

(4.68) (-0.57) (-0.64) (4.40) (-0.60) (-0.81) (4.44) (-0.16) (-0.43)High (H) 0.61 -0.67 -0.73 0.66 -0.59 -0.62 0.81 -0.41 -0.45

(1.87) (-5.35) (-5.77) (2.08) (-4.93) (-5.22) (2.51) (-3.25) (-3.62)H-L -0.50 -0.63 -0.67 -0.43 -0.54 -0.55 -0.29 -0.39 -0.41

(-2.86) (-4.16) (-4.37) (-2.82) (-3.97) (-4.05) (-1.98) (-2.91) (-3.05)Low/Big Low (L) 0.99 -0.04 -0.07 1.01 -0.05 -0.08 1.03 -0.01 -0.05

(4.63) (-0.60) (-1.03) (4.68) (-0.61) (-1.02) (4.77) (-0.19) (-0.60)High (H) 0.71 -0.31 -0.41 1.09 -0.24 -0.24 1.48 0.24 0.25

(1.62) (-0.92) (-1.30) (2.54) (-0.78) (-0.75) (3.32) (0.65) (0.65)H-L -0.29 -0.26 -0.34 0.07 -0.20 -0.17 0.45 0.26 0.30

(-0.87) (-0.80) (-1.07) (0.22) (-0.65) (-0.53) (1.21) (0.70) (0.77)High/Small Low (L) 1.29 0.20 0.22 1.16 0.12 0.12 1.19 0.21 0.20

(4.55) (2.57) (2.71) (4.03) (1.49) (1.45) (4.13) (2.53) (2.45)High (H) 0.90 -0.30 -0.26 0.90 -0.19 -0.15 1.04 -0.02 0.03

(2.09) (-1.90) (-1.64) (2.16) (-1.33) (-1.05) (2.51) (-0.13) (0.20)H-L -0.39 -0.51 -0.48 -0.26 -0.30 -0.26 -0.16 -0.22 -0.18

(-1.96) (-3.00) (-2.86) (-1.38) (-2.11) (-1.85) (-0.84) (-1.55) (-1.21)High/Big Low (L) 1.01 0.11 0.11 1.04 0.11 0.11 1.03 0.10 0.11

(4.97) (2.72) (2.80) (5.02) (2.54) (2.65) (4.94) (2.33) (2.51)High (H) 1.38 0.28 0.30 1.92 0.78 0.82 1.61 0.58 0.57

(3.11) (1.02) (1.07) (4.10) (2.42) (2.48) (3.25) (1.80) (1.63)H-L 0.36 0.17 0.19 0.88 0.67 0.70 0.58 0.48 0.46

(1.12) (0.63) (0.67) (2.47) (2.07) (2.13) (1.51) (1.46) (1.30)

R&D/Assets Ranks

IVOL Rank

Year 1 Year 2 Year 3

44

Panel B. Low DTE as innovation proxy

Return 4f Alpha 5f Alpha Return 4f Alpha 5f Alpha Return 4f Alpha 5f AlphaHigh/Small Low (L) 1.26 0.05 0.05 1.24 0.02 0.01 1.26 0.09 0.07

(4.78) (0.63) (0.55) (4.51) (0.20) (0.09) (4.49) (0.99) (0.75)High (H) 1.06 -0.29 -0.31 1.02 -0.32 -0.32 1.01 -0.19 -0.22

(2.80) (-2.57) (-2.69) (2.67) (-2.94) (-2.84) (2.81) (-1.82) (-2.08)H-L -0.20 -0.35 -0.36 -0.22 -0.33 -0.32 -0.25 -0.28 -0.29

(-1.06) (-2.34) (-2.35) (-1.23) (-2.47) (-2.33) (-1.68) (-2.29) (-2.31)High/Big Low (L) 1.02 0.02 0.01 1.05 0.05 0.04 1.04 0.05 0.05

(4.79) (0.44) (0.27) (4.78) (0.98) (0.74) (4.69) (0.91) (0.78)High (H) 0.97 -0.11 -0.22 1.08 -0.28 -0.27 0.81 -0.40 -0.45

(2.24) (-0.41) (-0.83) (2.51) (-1.10) (-1.06) (1.88) (-1.41) (-1.56)H-L -0.05 -0.14 -0.23 0.03 -0.33 -0.31 -0.23 -0.45 -0.50

(-0.17) (-0.50) (-0.87) (0.10) (-1.31) (-1.21) (-0.76) (-1.59) (-1.70)Low/Small Low (L) 1.20 0.16 0.17 1.08 0.06 0.06 1.10 0.13 0.12

(4.21) (2.12) (2.22) (3.71) (0.81) (0.70) (3.80) (1.63) (1.43)High (H) 0.72 -0.42 -0.38 0.75 -0.33 -0.31 0.85 -0.17 -0.13

(1.62) (-2.49) (-2.24) (1.76) (-2.18) (-2.10) (1.99) (-1.17) (-0.92)H-L -0.48 -0.58 -0.55 -0.32 -0.39 -0.37 -0.24 -0.30 -0.25

(-2.24) (-3.25) (-3.08) (-1.62) (-2.52) (-2.39) (-1.18) (-1.95) (-1.63)Low/Big Low (L) 0.97 0.16 0.16 0.94 0.11 0.11 1.02 0.15 0.15

(4.31) (2.98) (2.91) (4.27) (1.84) (1.82) (4.50) (2.49) (2.50)High (H) 1.50 0.53 0.46 2.20 1.30 1.30 2.18 1.43 1.36

(2.95) (1.50) (1.28) (3.95) (2.87) (2.84) (4.16) (3.55) (3.28)H-L 0.53 0.37 0.30 1.25 1.20 1.20 1.16 1.29 1.21

(1.34) (1.06) (0.84) (2.66) (2.62) (2.59) (2.79) (3.17) (2.92)

DTE/Assets Ranks

IVOL Rank

Year 1 Year 2 Year 3

45

Panel C. High MABA as innovation proxy

Return 4f Alpha 5f Alpha Return 4f Alpha 5f Alpha Return 4f Alpha 5f AlphaLow/Small Low (L) 1.28 0.04 0.03 1.21 0.00 -0.02 1.14 -0.07 -0.11

(5.17) (0.57) (0.38) (4.71) (0.00) (-0.22) (4.32) (-0.83) (-1.23)High (H) 1.21 -0.11 -0.13 1.12 -0.14 -0.15 1.15 -0.06 -0.08

(3.40) (-0.98) (-1.20) (3.26) (-1.38) (-1.50) (3.40) (-0.58) (-0.73)H-L -0.07 -0.15 -0.16 -0.09 -0.14 -0.14 0.01 0.01 0.03

(-0.40) (-1.12) (-1.20) (-0.59) (-1.15) (-1.12) (0.09) (0.09) (0.21)Low/Big Low (L) 1.11 -0.02 -0.04 1.20 0.02 -0.01 1.13 -0.03 -0.08

(4.98) (-0.24) (-0.53) (5.24) (0.22) (-0.13) (4.78) (-0.40) (-0.93)High (H) 0.94 -0.27 -0.32 1.04 -0.40 -0.34 1.04 -0.15 -0.16

(2.29) (-1.10) (-1.31) (2.51) (-1.56) (-1.34) (2.49) (-0.51) (-0.54)H-L -0.17 -0.25 -0.28 -0.15 -0.41 -0.33 -0.09 -0.12 -0.08

(-0.59) (-1.07) (-1.19) (-0.53) (-1.57) (-1.27) (-0.29) (-0.41) (-0.29)High/Small Low (L) 1.20 0.12 0.14 1.10 0.07 0.06 1.16 0.18 0.17

(4.43) (1.78) (1.91) (3.99) (0.94) (0.84) (4.22) (2.41) (2.26)High (H) 0.70 -0.48 -0.46 0.72 -0.36 -0.33 0.86 -0.18 -0.15

(1.68) (-3.07) (-2.95) (1.77) (-2.53) (-2.40) (2.15) (-1.38) (-1.16)H-L -0.50 -0.60 -0.59 -0.38 -0.42 -0.39 -0.30 -0.36 -0.32

(-2.44) (-3.57) (-3.52) (-2.02) (-2.84) (-2.66) (-1.59) (-2.51) (-2.24)High/Big Low (L) 0.98 0.10 0.10 1.02 0.11 0.12 1.02 0.12 0.13

(4.79) (2.70) (2.74) (4.93) (2.76) (2.75) (4.98) (2.75) (2.91)High (H) 1.26 0.24 0.14 1.79 0.83 0.86 1.62 0.73 0.62

(2.57) (0.67) (0.39) (3.47) (1.99) (2.01) (3.25) (1.96) (1.63)H-L 0.28 0.13 0.03 0.77 0.72 0.74 0.63 0.61 0.49

(0.73) (0.38) (0.09) (1.79) (1.68) (1.70) (1.58) (1.61) (1.27)

MABA/Assets Ranks

IVOL Rank

Year 1 Year 2 Year 3

46

Panel D. Growth-option-intensive industries as innovation proxy

Return 4f Alpha 5f Alpha Return 4f Alpha 5f Alpha Return 4f Alpha 5f AlphaLow/Small Low (L) 1.18 0.06 0.07 1.13 0.01 0.00 1.13 0.09 0.09

(4.78) (0.86) (0.87) (4.44) (0.17) (0.05) (4.39) (1.09) (1.02)High (H) 0.89 -0.32 -0.32 0.87 -0.24 -0.24 1.05 -0.03 -0.04

(2.26) (-2.25) (-2.16) (2.29) (-1.88) (-1.79) (2.74) (-0.25) (-0.28)H-L -0.29 -0.39 -0.39 -0.26 -0.26 -0.24 -0.08 -0.13 -0.12

(-1.33) (-2.39) (-2.32) (-1.30) (-1.77) (-1.64) (-0.40) (-0.82) (-0.81)Low/Big Low (L) 1.04 0.06 0.07 1.07 0.06 0.07 1.08 0.08 0.09

(4.87) (0.99) (1.08) (4.93) (0.99) (1.11) (4.97) (1.29) (1.48)High (H) 1.03 -0.16 -0.23 1.42 0.10 0.13 1.22 0.02 -0.05

(2.45) (-0.62) (-0.91) (3.24) (0.35) (0.46) (3.08) (0.07) (-0.18)H-L -0.01 -0.22 -0.29 0.34 0.04 0.07 0.13 -0.06 -0.14

(-0.03) (-0.85) (-1.15) (1.09) (0.14) (0.23) (0.49) (-0.24) (-0.53)High/Small Low (L) 1.32 0.21 0.23 1.24 0.21 0.22 1.14 0.14 0.12

(4.26) (2.07) (2.26) (3.87) (1.81) (1.83) (3.57) (1.21) (1.06)High (H) 0.81 -0.43 -0.41 0.82 -0.31 -0.29 0.84 -0.24 -0.21

(1.87) (-2.36) (-2.29) (1.95) (-1.86) (-1.75) (2.05) (-1.54) (-1.32)H-L -0.51 -0.64 -0.64 -0.42 -0.52 -0.50 -0.30 -0.38 -0.33

(-2.65) (-3.56) (-3.61) (-2.38) (-3.23) (-3.16) (-1.73) (-2.63) (-2.23)High/Big Low (L) 0.98 0.10 0.08 0.99 0.09 0.07 0.97 0.08 0.06

(4.76) (1.38) (1.16) (4.73) (1.24) (1.01) (4.64) (1.01) (0.80)High (H) 0.98 0.03 -0.01 1.15 0.10 0.09 1.94 0.87 0.84

(2.05) (0.09) (-0.03) (2.31) (0.27) (0.24) (3.77) (2.26) (2.08)H-L 0.00 -0.06 -0.09 0.17 0.01 0.02 0.97 0.80 0.78

(0.00) (-0.20) (-0.28) (0.42) (0.03) (0.06) (2.34) (2.12) (1.98)

GO/Assets Ranks

IVOL Rank

Year 1 Year 2 Year 3

47

Table 4. Innovation, Idiosyncratic Volatility, and Stock Returns (Portfolio Sorts—Factor Loadings)

We report the factor loadings from the Carhart (1997) four-factor regressions described in Table 3. All factors are defined in Table 2.

Panel A. R&D as innovation proxy

Mkt SMB HML UMD Mkt SMB HML UMD Mkt SMB HML UMDLow/Small Low (L) 0.99 0.47 0.22 0.02 1.01 0.46 0.18 0.00 1.01 0.43 0.17 0.00

(40.20) (6.07) (4.68) (0.53) (39.10) (5.62) (4.19) (-0.12) (37.39) (5.61) (3.77) (-0.15)High (H) 1.17 0.97 -0.01 -0.02 1.15 0.86 0.03 0.00 1.17 0.82 0.04 -0.02

(32.29) (13.51) (-0.08) (-0.42) (34.85) (14.10) (0.64) (0.14) (35.02) (11.35) (0.64) (-0.60)H-L 0.18 0.50 -0.22 -0.03 0.14 0.40 -0.15 0.01 0.16 0.39 -0.13 -0.02

(4.55) (9.31) (-2.76) (-0.62) (4.47) (6.66) (-2.73) (0.20) (5.14) (7.92) (-2.45) (-0.55)Low/Big Low (L) 1.01 -0.09 0.11 0.02 1.00 -0.11 0.13 0.02 0.99 -0.12 0.14 0.03

(52.77) (-2.36) (2.82) (0.86) (48.68) (-2.62) (3.37) (1.09) (46.73) (-2.87) (3.53) (1.18)High (H) 1.34 0.63 -0.22 -0.39 1.30 0.63 0.16 -0.02 1.23 0.28 0.27 -0.04

(17.61) (5.00) (-1.70) (-2.96) (17.39) (4.53) (1.17) (-0.26) (14.46) (2.10) (1.52) (-0.50)H-L 0.33 0.72 -0.33 -0.41 0.30 0.74 0.03 -0.04 0.24 0.40 0.13 -0.07

(4.44) (6.08) (-2.50) (-3.07) (4.22) (6.07) (0.24) (-0.61) (2.80) (2.98) (0.78) (-0.86)High/Small Low (L) 1.06 0.63 -0.30 0.00 1.08 0.56 -0.29 -0.05 1.07 0.55 -0.31 -0.08

(53.23) (12.75) (-7.97) (-0.04) (49.41) (15.00) (-6.50) (-2.23) (51.73) (11.99) (-8.57) (-3.47)High (H) 1.37 1.15 -0.70 -0.12 1.31 1.07 -0.75 -0.12 1.28 1.03 -0.74 -0.10

(32.48) (16.64) (-8.52) (-1.82) (34.75) (16.74) (-11.40) (-2.34) (32.30) (14.58) (-10.38) (-1.95)H-L 0.32 0.52 -0.40 -0.12 0.23 0.50 -0.46 -0.07 0.21 0.49 -0.43 -0.02

(7.82) (7.83) (-4.74) (-1.71) (6.83) (9.03) (-7.29) (-1.36) (5.65) (7.22) (-5.94) (-0.30)High/Big Low (L) 0.96 -0.16 -0.13 -0.01 0.96 -0.16 -0.12 -0.01 0.95 -0.16 -0.11 -0.01

(90.51) (-9.83) (-7.49) (-1.34) (82.25) (-9.62) (-6.21) (-1.25) (78.11) (-9.48) (-5.59) (-0.94)High (H) 1.41 0.73 -0.29 -0.34 1.42 0.58 -0.34 -0.21 1.49 0.52 -0.29 -0.41

(19.96) (7.32) (-2.60) (-4.40) (18.77) (4.00) (-2.59) (-2.56) (18.50) (2.34) (-1.79) (-5.29)H-L 0.45 0.89 -0.16 -0.32 0.46 0.74 -0.23 -0.19 0.53 0.69 -0.19 -0.40

(6.26) (8.94) (-1.43) (-4.18) (5.97) (5.20) (-1.66) (-2.43) (6.42) (3.06) (-1.14) (-5.04)

R&D/Assets Ranks

IVOL Rank

Year 1 Year 2 Year 3

48

Panel B. Low DTE as innovation proxy

Mkt SMB HML UMD Mkt SMB HML UMD Mkt SMB HML UMDHigh/Small Low (L) 1.03 0.61 0.31 -0.01 1.05 0.64 0.30 0.01 1.07 0.60 0.22 0.01

(36.03) (7.78) (6.89) (-0.26) (38.17) (8.46) (6.56) (0.34) (40.22) (9.06) (5.06) (0.22)High (H) 1.26 1.18 0.01 -0.07 1.28 1.05 -0.10 0.00 1.21 0.94 -0.12 -0.02

(39.74) (21.91) (0.25) (-1.98) (43.40) (17.40) (-1.97) (-0.01) (40.04) (11.96) (-2.69) (-0.76)H-L 0.24 0.57 -0.30 -0.06 0.24 0.40 -0.41 -0.01 0.14 0.35 -0.34 -0.03

(5.80) (7.68) (-4.07) (-1.50) (6.23) (6.42) (-6.61) (-0.28) (4.51) (7.36) (-6.81) (-0.70)High/Big Low (L) 0.98 -0.09 0.21 0.00 1.00 -0.13 0.15 0.00 0.99 -0.12 0.14 0.00

(59.99) (-3.11) (6.28) (-0.20) (57.99) (-3.66) (4.93) (0.21) (56.73) (-4.30) (4.61) (0.02)High (H) 1.40 0.73 -0.08 -0.40 1.48 0.53 0.14 -0.09 1.38 0.55 0.25 -0.22

(21.69) (6.39) (-0.69) (-3.55) (20.82) (3.92) (1.26) (-1.56) (15.96) (2.74) (1.65) (-2.45)H-L 0.42 0.82 -0.29 -0.40 0.48 0.66 -0.01 -0.09 0.39 0.68 0.11 -0.22

(6.33) (7.44) (-2.42) (-3.56) (7.08) (5.52) (-0.09) (-1.67) (4.42) (3.44) (0.73) (-2.55)Low/Small Low (L) 1.03 0.57 -0.30 0.01 1.05 0.49 -0.28 -0.04 1.04 0.47 -0.29 -0.06

(54.01) (12.19) (-8.27) (0.46) (45.23) (12.44) (-6.16) (-1.84) (50.76) (10.69) (-8.52) (-2.45)High (H) 1.33 1.07 -0.77 -0.09 1.27 0.98 -0.77 -0.11 1.27 0.96 -0.77 -0.12

(30.65) (15.01) (-9.44) (-1.33) (31.17) (13.63) (-10.95) (-2.00) (30.87) (12.79) (-10.63) (-2.38)H-L 0.31 0.50 -0.47 -0.10 0.22 0.49 -0.48 -0.07 0.24 0.49 -0.48 -0.06

(7.29) (7.57) (-5.78) (-1.38) (6.14) (7.78) (-7.28) (-1.26) (5.84) (6.85) (-6.45) (-1.08)Low/Big Low (L) 0.96 -0.21 -0.29 -0.01 0.94 -0.17 -0.25 -0.01 0.94 -0.18 -0.22 0.01

(77.30) (-12.09) (-14.00) (-0.79) (66.99) (-8.23) (-9.70) (-0.55) (70.38) (-7.94) (-9.11) (0.69)High (H) 1.37 0.44 -0.72 -0.19 1.21 0.39 -0.74 -0.16 1.30 0.11 -0.78 -0.40

(16.46) (3.35) (-5.06) (-2.30) (10.97) (2.15) (-4.49) (-1.63) (14.10) (0.68) (-4.89) (-3.91)H-L 0.41 0.65 -0.43 -0.18 0.27 0.56 -0.49 -0.15 0.35 0.28 -0.56 -0.40

(4.85) (4.90) (-2.98) (-2.19) (2.39) (3.12) (-2.92) (-1.56) (3.82) (1.72) (-3.47) (-3.88)

DTE/Assets Ranks

IVOL Rank

Year 1 Year 2 Year 3

49

Table 5. Innovation, Idiosyncratic Volatility, and Stock Returns

(Portfolio Sorts— Alphas from Fama-French 5f Model)

As in Table 3, we form eight innovation-assets-IVOL portfolios from independent triple sorts (2

by 2 by 2). In Panels A, B, C, and D, we use high R&D, low leverage (DTE), high MABA, and

growth-option-intensive industries, respectively, as proxies of innovation. DTE and MABA are

defined in Table 1. Growth-option-intensive industries are defined in Table 3. We report the

monthly value-weighted alphas (in percentage) estimated from the Fama-French five-factor model

(Fama and French, 2015) and their heteroscedasticity-robust t-statistics in parentheses over the

five post-sorting years. The five factors are the market, size, value, investment, and operating

profitability factors.

50

Year 1 Year 2 Year 3 Year 1 Year 2 Year 3Low/Small Low (L) -0.21 -0.20 -0.16 -0.09 -0.10 -0.01

(-3.00) (-2.66) (-2.10) (-1.23) (-1.44) (-0.10)High (H) -0.50 -0.49 -0.35 -0.12 -0.12 -0.09

(-4.03) (-4.01) (-2.61) (-1.06) (-0.95) (-0.87)H-L -0.29 -0.28 -0.19 -0.03 -0.01 -0.09

(-2.22) (-2.30) (-1.42) (-0.22) (-0.10) (-0.75)Low/Big Low (L) -0.23 -0.24 -0.23 -0.10 -0.07 -0.09

(-3.59) (-3.44) (-3.34) (-1.93) (-1.36) (-1.64)High (H) -0.33 -0.32 0.04 -0.14 -0.18 -0.56

(-0.98) (-0.98) (0.10) (-0.51) (-0.68) (-1.89)H-L -0.10 -0.09 0.27 -0.04 -0.10 -0.47

(-0.31) (-0.27) (0.68) (-0.15) (-0.41) (-1.59)High/Small Low (L) 0.27 0.18 0.27 0.22 0.12 0.19

(3.08) (2.13) (2.97) (2.70) (1.44) (2.06)High (H) 0.08 0.11 0.30 -0.01 0.00 0.17

(0.55) (0.89) (2.14) (-0.07) (-0.01) (1.26)H-L -0.19 -0.07 0.03 -0.23 -0.12 -0.02

(-1.16) (-0.48) (0.17) (-1.41) (-0.86) (-0.10)High/Big Low (L) 0.06 0.05 0.04 0.10 0.05 0.10

(1.61) (1.25) (0.98) (1.98) (0.82) (1.54)High (H) 0.46 1.06 0.55 0.78 1.64 1.61

(1.66) (3.22) (1.73) (2.16) (3.56) (3.97)H-L 0.40 1.01 0.51 0.67 1.59 1.51

(1.45) (3.05) (1.57) (1.90) (3.43) (3.73)

A. R&D as innovation proxy B. Low DTE as innovation proxyInnovation/ Assets Ranks

IVOL Rank

Fama-French 5f Alpha Fama-French 5f Alpha

51

Year 1 Year 2 Year 3 Year 1 Year 2 Year 3Low/Small Low (L) -0.09 -0.11 -0.17 -0.05 -0.08 0.01

(-1.28) (-1.41) (-2.13) (-0.71) (-1.04) (0.15)High (H) 0.03 -0.02 0.04 0.09 0.07 0.25

(0.30) (-0.23) (0.38) (0.60) (0.50) (1.59)H-L 0.12 0.09 0.21 0.14 0.15 0.24

(0.99) (0.76) (1.95) (0.88) (1.01) (1.38)Low/Big Low (L) -0.15 -0.09 -0.11 -0.09 -0.09 -0.07

(-2.11) (-1.23) (-1.39) (-1.53) (-1.43) (-1.16)High (H) -0.30 -0.32 -0.20 0.02 0.22 -0.08

(-1.14) (-1.22) (-0.69) (0.09) (0.61) (-0.29)H-L -0.15 -0.23 -0.09 0.12 0.31 -0.01

(-0.61) (-0.85) (-0.31) (0.45) (0.88) (-0.02)High/Small Low (L) 0.16 0.10 0.21 0.37 0.40 0.27

(2.16) (1.40) (2.59) (3.56) (3.60) (2.26)High (H) -0.06 -0.03 0.14 -0.03 -0.02 0.09

(-0.44) (-0.27) (1.15) (-0.21) (-0.14) (0.66)H-L -0.22 -0.14 -0.06 -0.41 -0.42 -0.18

(-1.40) (-1.01) (-0.42) (-2.69) (-2.91) (-1.34)High/Big Low (L) 0.02 0.03 0.03 0.09 0.07 0.06

(0.45) (0.77) (0.78) (1.34) (0.98) (0.73)High (H) 0.46 1.17 0.86 0.08 0.44 0.94

(1.31) (2.60) (2.23) (0.24) (1.08) (2.42)H-L 0.44 1.14 0.83 -0.01 0.37 0.88

(1.26) (2.50) (2.14) (-0.03) (0.92) (2.32)

Fama-French 5f Alpha Fama-French 5f Alpha

C. High MABA as innovation proxy

D. Growth-option-intensive industries as innovation proxy

IVOL Rank

Innovation/ Assets Ranks

52

Table 6. Innovation, Idiosyncratic Volatility, and Stock Returns

(Fama-MacBeth Regressions)

Each month between July of year t + i and June of year t + i + 1 (i = 0, 1, 2), we regress firms’

monthly stock returns on a set of independent variables. Weighted least square regressions are

used with lagged market capitalization as the weight. The table reports the time-series average

slopes (in percentage) and their t-statistics. Year 1 is from July of year t to June of year t + 1. Year

2 is from July of year t + 1 to June of year t + 2. Year 3 is from July of year t + 2 to June of year t

+ 3. IVOL is a dummy variable that equals 1 if a firm’s idiosyncratic volatility (computed at the

end of each June as defined in Table 1) is above median and 0 otherwise. Innov_Big is a dummy

variable that equals 1 if a firm is in the innovative and big assets portfolio as defined in Table 3,

and 0 otherwise. ln(BTM) denotes the natural logarithm of book-to-market equity (BTM), the ratio

of book equity in the fiscal year ending in calendar year t – 1 to market equity at the end of calendar

year t – 1. ln(Size) is the natural logarithm of market equity at the end of June of year t. Momentum

is the cumulative return over the prior 11 months with a one-month gap. Reversal is the lagged

monthly return. All independent variables (except dummies) are winsorized at the top and bottom

1%. In Panels A, B, C, and D, we use high R&D, low leverage (DTE), high MABA, and growth-

option-intensive industries, respectively, as alternative proxies of innovation. Specifically, in Panel

A, Innov_Big is a dummy variable that equals 1 if a firm is in the high R&D and big assets portfolio

as defined in Table 3, and 0 otherwise. In Panel B (C), Innov_Big is a dummy variable that equals

1 if a firm has above-median total assets and below-median DTE (above-median MABA), and 0

otherwise. In Panel D, Innov_Big is a dummy variable that equals 1 if the firm has above-median

assets and operate in one of those growth-option-intensive industries as defined as in Grullon,

Lyandres, and Zhdanov (2012), which include Fama and French (1997) industries 27 (precious

metals), 28 (mining), 30 (oil and natural gas), 22 (electrical equipment), 32 (telecommunications),

35 (computers), 36 (electronic equipment), 37 (measuring and control equipment), 12 (medical

equipment), and 13 (pharmaceutical products). DTE and MABA are defined in Table 1. The table

reports the time-series average slopes (in percentage) and their t-statistics. Financial and utility

firms are excluded. The sample period for stock returns is from July of 1977 to December 2015.

53

Panel A. R&D as innovation proxyYear IVOL IVOL*Innov_Big Innov_Big ln(BTM) ln(Size) Momentum Reversal Intercept1 -0.41 0.53 0.11 0.03 -0.08 0.50 -2.37 1.55

(-2.57) (2.04) (1.58) (0.44) (-1.95) (2.49) (-4.05) (3.43)2 -0.34 0.91 0.12 0.04 -0.05 0.47 -2.39 1.34

(-2.33) (3.01) (1.72) (0.55) (-1.13) (2.21) (-3.88) (2.81)3 -0.20 0.69 0.04 0.00 -0.03 0.46 -2.28 1.18

(-1.39) (1.90) (0.52) (0.00) (-0.66) (2.09) (-3.57) (2.45)Panel B. Low DTE as innovation proxyYear IVOL IVOL*Innov_Big Innov_Big ln(BTM) ln(Size) Momentum Reversal Intercept1 -0.36 0.83 0.04 0.05 -0.06 0.52 -2.38 1.45

(-2.27) (2.39) (0.55) (0.64) (-1.52) (2.55) (-4.06) (3.24)2 -0.32 1.24 0.08 0.07 -0.04 0.49 -2.39 1.27

(-2.22) (3.36) (1.09) (0.91) (-0.83) (2.27) (-3.89) (2.72)3 -0.21 1.09 0.06 0.01 -0.03 0.47 -2.27 1.14

(-1.55) (3.12) (0.74) (0.19) (-0.58) (2.12) (-3.56) (2.44)Panel C. High MABA as innovation proxyYear IVOL IVOL*Innov_Big Innov_Big ln(BTM) ln(Size) Momentum Reversal Intercept1 -0.35 0.21 0.02 0.03 -0.06 0.50 -2.36 1.47

(-2.31) (0.58) (0.32) (0.48) (-1.63) (2.50) (-4.06) (3.36)2 -0.34 0.93 0.00 0.05 -0.03 0.48 -2.35 1.26

(-2.43) (2.25) (-0.03) (0.61) (-0.75) (2.22) (-3.84) (2.77)3 -0.21 0.74 -0.06 -0.01 -0.01 0.46 -2.30 1.07

(-1.59) (1.95) (-0.78) (-0.14) (-0.34) (2.09) (-3.61) (2.38)Panel D. Growth-option-intensive industries as innovation proxyYear IVOL IVOL*Innov_Big Innov_Big ln(BTM) ln(Size) Momentum Reversal Intercept1 -0.32 0.26 -0.02 0.04 -0.06 0.51 -2.53 1.40

(-2.05) (0.90) (-0.18) (0.62) (-1.44) (2.56) (-4.42) (3.21)2 -0.26 0.39 -0.05 0.05 -0.02 0.47 -2.54 1.17

(-1.75) (1.10) (-0.43) (0.74) (-0.53) (2.23) (-4.24) (2.57)3 -0.23 1.22 -0.06 0.00 -0.02 0.48 -2.42 1.10

(-1.74) (3.16) (-0.61) (0.07) (-0.39) (2.21) (-3.89) (2.40)

54

Table 7. Effects of Asset Growth (Fama-MacBeth Regressions)

We conduct monthly Fama-MacBeth regressions as in Table 6. In addition to those independent variables included in Table 6, we control

for asset growth (AG), return on assets (ROA), and additional interaction terms, IVOL*Innov_Big*AG, IVOL*AG, and Innov_Big*AG.

AG is a dummy variable that equals 1 for firms with above-median asset growth, and 0 otherwise. IVOL*Innov_Big*AG is a triple

interaction term consisting of three dummy variables. ROA is net income divided by total assets. All the other variables are defined as

in Table 6. The table reports the time-series average slopes (in percentage) and their t-statistics.

A. R&D as innovation proxy

Year IVOLIVOL*Innov

_BigIVOL*Innov

_Big*AGIVOL*

AGInnov_

BigAG

Innov_Big*AG

ln(BTM) Reversal ROA ln(Size) Momentum Intercept

1 -0.09 0.20 0.40 -0.27 0.12 -0.06 0.02 0.06 -2.66 1.05 -0.09 0.46 1.59(-0.51) (0.60) (0.61) (-1.76) (1.45) (-0.76) (0.17) (0.78) (-4.57) (2.47) (-2.08) (2.28) (3.56)

2 -0.11 0.16 1.14 -0.21 0.04 -0.05 0.15 0.08 -2.66 0.81 -0.05 0.40 1.40(-0.69) (0.46) (2.23) (-1.38) (0.52) (-0.55) (1.50) (1.09) (-4.33) (1.71) (-1.20) (1.86) (2.97)

3 -0.10 -0.34 1.51 -0.05 0.03 0.02 0.08 0.02 -2.62 0.22 -0.03 0.44 1.20(-0.57) (-0.80) (2.72) (-0.30) (0.32) (0.30) (0.77) (0.28) (-4.17) (0.40) (-0.76) (1.98) (2.55)

B. Low DTE as innovation proxy

Year IVOLIVOL*Innov

_BigIVOL*Innov

_Big*AGIVOL*

AGInnov_

BigAG

Innov_Big*AG

ln(BTM) Reversal ROA ln(Size) Momentum Intercept

1 -0.04 -0.22 1.23 -0.31 0.02 -0.04 0.02 0.06 -2.67 0.94 -0.07 0.47 1.49(-0.22) (-0.43) (2.07) (-2.27) (0.23) (-0.56) (0.16) (0.74) (-4.60) (2.19) (-1.70) (2.31) (3.37)

2 -0.10 -0.05 1.69 -0.20 -0.01 -0.02 0.14 0.09 -2.67 0.64 -0.03 0.41 1.29(-0.66) (-0.12) (3.07) (-1.38) (-0.06) (-0.27) (1.03) (1.13) (-4.37) (1.36) (-0.81) (1.92) (2.78)

3 -0.13 0.45 0.63 -0.03 0.02 0.06 0.04 0.02 -2.59 0.03 -0.02 0.46 1.12(-0.79) (1.02) (1.11) (-0.21) (0.23) (0.66) (0.27) (0.23) (-4.16) (0.06) (-0.53) (2.09) (2.36)

55

C. High MABA as innovation proxy

Year IVOLIVOL*Innov

_BigIVOL*Innov

_Big*AGIVOL*

AGInnov_

BigAG

Innov_Big*AG

ln(BTM) Reversal ROA ln(Size) Momentum Intercept

1 -0.08 -0.33 0.59 -0.19 -0.05 -0.15 0.13 0.05 -2.66 1.04 -0.07 0.45 1.57(-0.47) (-0.77) (0.97) (-1.30) (-0.50) (-1.88) (1.19) (0.70) (-4.60) (2.47) (-1.77) (2.24) (3.64)

2 -0.08 -0.13 1.60 -0.26 0.02 0.05 -0.01 0.08 -2.67 0.77 -0.04 0.41 1.27(-0.52) (-0.39) (2.94) (-1.81) (0.21) (0.62) (-0.06) (1.04) (-4.35) (1.61) (-0.84) (1.90) (2.80)

3 -0.12 0.01 0.76 -0.02 -0.11 0.03 0.10 0.00 -2.59 0.20 -0.02 0.45 1.08(-0.70) (0.04) (1.54) (-0.11) (-1.05) (0.35) (0.84) (0.00) (-4.13) (0.38) (-0.37) (2.04) (2.42)

D. Growth-option-intensive industries as innovation proxy

Year IVOLIVOL*Innov

_BigIVOL*Innov

_Big*AGIVOL*

AGInnov_

BigAG

Innov_Big*AG

ln(BTM) Reversal ROA ln(Size) Momentum Intercept

1 -0.08 0.41 0.04 -0.19 0.01 -0.03 -0.04 0.07 -2.85 1.13 -0.06 0.47 1.41(-0.45) (1.03) (0.07) (-1.28) (0.07) (-0.38) (-0.34) (0.90) (-5.04) (2.67) (-1.53) (2.38) (3.25)

2 -0.11 0.28 0.16 -0.12 -0.07 0.04 0.06 0.08 -2.85 0.73 -0.03 0.41 1.20(-0.69) (0.72) (0.31) (-0.78) (-0.59) (0.51) (0.51) (1.02) (-4.77) (1.55) (-0.72) (1.94) (2.64)

3 -0.10 -0.22 1.92 -0.13 -0.02 0.09 -0.03 0.02 -2.69 0.19 -0.02 0.46 1.09(-0.57) (-0.53) (3.07) (-0.79) (-0.19) (1.12) (-0.25) (0.25) (-4.39) (0.37) (-0.48) (2.13) (2.42)