Do Growth Options Explain the
Trend in Idiosyncratic Risk?
Charles Cao
Timothy Simin
Jing Zhao*
March 2005
Abstract
While several recent studies document increasing idiosyncratic volatility over the past four
decades, an explanation for this trend remains elusive. We establish a theoretic link between the
risk of growth options that managers choose and the idiosyncratic risk of equity. Empirically
both the level and variance of corporate growth options are significantly related to idiosyncratic
volatility. Accounting for growth options eliminates or reverses the upward trend in firm
specific risk. These results are robust across different exchanges, are not due to the 'internet
bubble' period, and affirm the growth options explanation of the trend in idiosyncratic volatility
over alternative explanations.
*
All authors are in the Department of Finance at the Pennsylvania State University. Contact author: Simin
(tsimin@psu.edu) Department of Finance, The Pennsylvania State University, 609 Business Administration
Building, University Park, PA 16802, (814) 865-3457. We are grateful to Keith Crocker, Craig Dunbar, Patrick
Kelly, Bill Kracaw, Michael Long, Michelle Lowry, James Miles, Chris Muscarella, Dennis Sheehan, and seminar
participants at the Pennsylvania State University, the University of Texas at Dallas, and the University of Western
Ontario for their helpful comments and suggestions.
Do Growth Options Explain the
Trend in Idiosyncratic Risk?
Abstract
While several recent studies document increasing idiosyncratic volatility over the past four
decades, an explanation for this trend remains elusive. We establish a theoretic link between the
risk of growth options that managers choose and the idiosyncratic risk of equity. Empirically
both the level and variance of corporate growth options are significantly related to idiosyncratic
volatility. Accounting for growth options eliminates or reverses the upward trend in firm
specific risk. These results are robust across different exchanges, are not due to the 'internet
bubble' period, and affirm the growth options explanation of the trend in idiosyncratic volatility
over alternative explanations.
Following Campbell, Lettau, Malkiel and Xu (2001) who document increasing firm level
return volatility, but stable market and industry return volatilities over the last four decades; there
has been a flurry of work attempting to characterize the upward trend in idiosyncratic volatility.1
We now know that increases in idiosyncratic volatility are related to both the level and variance
of profitability [Pastor and Veronesi (2003) and Wei and Zhang (2003)]; that institutional
ownership and expected earnings growth are positively related [Malkiel and Xu (2003)]; that age
is negatively related [Pastor and Veronesi (2003)]; and that expected returns are negatively
related in the cross-section [Ang, Hodrick, Xing, and Zhang (2004)]. From a macroeconomic
perspective we know that small-firm idiosyncratic volatility moves with business cycle variables
[Brown and Ferreira (2003)]. While these characterizations help in understanding the nature and
impact of the trend, a theoretically based explanation of the trend is still lacking.
In this paper we use a classic model from the corporate finance literature posited by Galai
and Masulis (1976) to relate the increase in average idiosyncratic volatility to the level and
variance of growth options. To explicitly tie the idiosyncratic component of volatility to the
investment decisions of corporate managers we focus on the Galai and Masulis result that
managers of levered firms are motivated to select the investment projects from their menu of
growth opportunities that increase the idiosyncratic variance of the firm.2 Increasing firm level
idiosyncratic risk benefits shareholders by increasing equity value while at the same time
reducing the market risk of equity. Within the Galai and Masulis model it is straightforward to
show the connection between idiosyncratic risk at the firm level and the idiosyncratic risk of
equity. As we illustrate below, this link is conditional on the capital structure of the firm.
Using investment opportunities to explain different dimensions of return variance is not
novel. The relation between investment opportunities and total firm variance can be traced back
to other classic corporate theory besides Galai and Masulis such as Myers (1977).
More
recently, Schwert (2002) suggests growth options of large firms in NASDAQ high-tech
industries may explain more volatile earnings and hence higher total equity return volatility.
Miles (1986), Berk, Green and Naik (1999), Jacquier, Titman and Yalcin (2001), and Carlson,
Fisher, and Giammarino (2003) argue that the exercise of growth options change a firm’s
exposure to systematic risk. Our innovation is to connect growth options to non-systematic risk.
1
2
Goyal and Santa-Clara (2003) and Malkiel and Xu (1999, 2003) also document this result.
See also Jensen and Meckling (1976).
1
A natural question to ask is why investors should care about increasing firm specific risk?
We know from the canonical asset pricing equation 1 = Et [ M t +1 Rt +1 ] that the covariance of
returns with marginal utility of consumption determines risk premia. The component of returns
that is correlated with the discount factor determines asset prices, not that part that can be
diversified away. In reality, transactions costs, incomplete information, market timing strategies,
and compensation in the form of stock and/or stock options expose investors to idiosyncratic
risk, e.g. Barber and Odean (2000) report that the average investment club holds 8 stocks while
an average individual investor holds only 4 stocks in their portfolio. In addition, increases in
idiosyncratic volatility impact the covariance structure of returns causing an increase in the
number of stocks required to eliminate idiosyncratic risk [Campbell et. al. (2001)]. For investors
with consumption subject to idiosyncratic risk and for investors attempting to create diversified
portfolios, trends in idiosyncratic volatility are important.
Indirect evidence indicates that growth options may be positively related to firm specific
risks. Chan, Lakonishok and Sougiannis (2001) find a positive relation between firm return
volatility and research and development (R&D) intensity. Vassalou and Apedjinou (2003)
uncover that firms with larger corporate ‘innovations’ (the change in gross profit margins not
explained by changes in capital and labor utilized) have higher firm-specific volatility. Since
firm innovations and R&D are proxies for, or at least closely related to growth options, this
evidence suggests that idiosyncratic volatility may be associated with corporate growth options.
Li, Morck, Yang, and Yeung (2003) show that lower market model R2 is associated with greater
capital market openness. If greater capital market openness allows firms to broaden their menu
of growth opportunities, then this is additional evidence in support of tie between growth options
and the trend in idiosyncratic risk. Indeed, Fama and French (2004) argue that decreasing costs
of capital have induced smaller firms to seek financing to capture riskier growth opportunities.
Zingales (2000) argues that the nature of firms has changed in important ways, several of which
are consistent with a connection between growth options and idiosyncratic risk, e.g. large
conglomerates have broken up and vertically integrated manufacturers have resigned direct
control of suppliers. Intuitively, viewing a firm as a portfolio of investment opportunities,
divestitures increase the focus of firms.
The decline in diversification of both assets and
investment possibilities increases idiosyncratic volatility.
2
Consistent with the predictions of the model, we find previously used empirical proxies
for the growth opportunities of firms, and their time-series variances, explain a substantial
portion of the trend in idiosyncratic volatility. Univariate regressions of idiosyncratic volatility
on a time trend confirm the upward drift in firm specific risk found by Campbell et al. (2001).
When we include growth options in the regressions, the coefficient on the time trend becomes
indistinguishable from zero or significantly negative indicating that after controlling for growth
options idiosyncratic volatility has remained stable or even decreased over time. The growth
options proxies and their time-series variances are positively related to idiosyncratic risk and
explain up to 63% of the variation in idiosyncratic volatility. Furthermore, we show that shocks
to the growth option proxies are significantly related to shocks in idiosyncratic volatility.
Growth options explain more of the trend in idiosyncratic volatility then previously
posited explanations. Two firm specific attributes that have shown promise in explaining the
trend in idiosyncratic volatility are age and return-on-equity. Pastor and Veronesi (2003) relate
firm specific volatility to firm profitability measured by return-on-equity.
They find that
idiosyncratic return volatility tends to be higher for firms with more uncertainty about future
profitability, with more volatile profitability, and for firms that pay no dividends suggesting a
partial explanation for the increase in idiosyncratic volatility is due to increases in the number of
firms listed at earlier ages [Fama and French (2004)]. Using an accounting identity called the
clean surplus equation, Wei and Zhang (2003) show that declining return-on-equity and
increasing return-on-equity volatility contribute to the upward trend in idiosyncratic volatility.
Return-on-equity and its time-series variance lose their explanatory power in the presence
of growth options variables.
Removing profitability from average idiosyncratic volatility
consistently leaves a significant trend component that is explainable by any of the growth option
proxies we consider.
Conversely, removing the growth option component never leaves a
significant trend. Firm age adds little to the explanatory power of the model beyond growth
options alone. The empirical evidence provides support to the predictions of the Galai and
Masulis model that growth options are related to idiosyncratic volatility.
This paper
complements the work of Malkiel and Xu (1999, 2003), Pastor and Veronesi (2003) and Wei and
Zhang (2003) by addressing the issue from a corporate investment perspective.
These results are robust across exchanges, and are not a by-product of including the
‘internet bubble’ period. Schwert (2002) shows that the higher total volatility of NASDAQ firms
3
relative to S&P 500 firms is driven by the technology focus of firms listed on NASDAQ rather
than the size or age of the firms. Our results corroborate Schwert’s findings that high growth
option firms have higher risk. The upward trend in idiosyncratic volatility is nearly four times
larger for NASDAQ firms than for NYSE/AMEX firms. Both the level and variance of growth
options are significant in explaining the upward trend in firm specific risk for NYSE/AMEX
firms. Only the level of growth options is significant for NASDAQ firms. These results suggest
that large firms with sufficient cash flow are able to take advantage of transient investment
opportunities unlike smaller, more cash constrained firms on NASDAQ.
An implication of the Galai and Masulis model is that capital structure plays an important
role in the relationship between growth options and the idiosyncratic volatility of equity. We
show that a low total value to equity value ratio reflects more control by shareholders and hence
higher idiosyncratic volatility due to the implementation of risky projects by managers to
enhance equity values. We condition on the controlling power of shareholders in the firm by
forming portfolios sorted on the total value to equity value ratio. Growth options become less
important in accounting for firm specific risk as more of the firm is controlled by debt holders.
At the extreme, firms primarily controlled by debt holders exhibit a significant negative
relationship between idiosyncratic risk and growth options. These results indicate that leverage
increases restraints on the ability of shareholders to take advantage of risky growth opportunities
and is consistent with the ‘underinvestment’ phenomena pointed out by Myers (1977).
In the first section we clarify the link between growth options and idiosyncratic volatility
and develop testable hypotheses. Section two details the measure of idiosyncratic variance and
the explanatory variables. Section three describes the data and Section four contains the main
empirical findings. Section five examines the explanatory power of growth options relative to
alternative explanations. In section six we separate out firms listed on the NASDAQ and
examine the results excluding the ‘internet bubble’ period.
Section seven presents results
relating capital structure and idiosyncratic risk. We conclude in the final section.
1. Connecting Growth Options to Idiosyncratic Volatility
Looking to growth options as a driver of the trend in idiosyncratic volatility follows from
a moral hazard problem occurring between share holders and debt holders within a firm. This
problem clearly arises in the joint model of Galai and Masulis (1976) which combines the
4
continuous time version of the capital asset pricing model (CAPM) of Merton (1973 and 1974)
and the Black and Scholes (1973) option pricing model. In the Galai and Masulis framework the
total value of a firm’s assets, A, is the sum of the value of equity, S, and debt, D. The
instantaneous expected return of the firm is therefore
rA =
S
D
rS + rD .
A
A
Here each of the instantaneous expected return components in (1),
(1)
( rA , rS , rD )
is a linear
function of instantaneous systematic risk βi and the instantaneous market risk premium, e.g.
ri = rf + βi ( rm − rf ) , where β i =
cov ( ri ; rM )
.
σ 2 ( rM )
Under the assumptions in Galai and Masulis, the value of equity can be considered
equivalent to the value of a European call option on the value of the firm so that S is determined
by the Black-Scholes option pricing model
S = AN ( d1 ) − ke
where d1 =
ln ( A k ) + ( rf + σ 2 2 ) T
σ T
− rf T
N ( d2 ) ,
(2)
, d 2 = d1 − σ T , N(*) is the standardized normal cumulative
probability density function, k the exercise price of the option, T the time to maturity, and σ2 the
instantaneous variance of percentage changes in A. Galai and Masulis show that as the time
increment goes to zero, the dollar return of equity is proportional to the dollar return on the value
of the firm,
rS =
∆S S A ∆A S A
A
ArA = η S rA .
=
=
S
S
A
S
(3)
Substituting equation (3) into the CAPM yields a relation between the systematic risk of
equity and the systematic risk of the total value of the firm,
5
βS = S A
A
β A = ηS β A .
S
(4)
Here ηS is the elasticity of equity value with respect to firm value. The moral hazard occurs
because from (2),
∂β S
∂S
> 0 while from (4),
< 0 (if βA is stationary and > 0). Stock holders
∂σ
∂σ
have a two-fold incentive to increase the variance of the firm since doing so increases the value
of the equity while reducing the market risk of equity. It is important to note that since both of
these derivatives are taken holding the value of the firm, the debt to equity ratio, and the market
risk of the firm constant, any change in the total variance of the firm must be due to changes in
the idiosyncratic risk of firm level returns, i.e.
∂β S
∂β S 3
∂S
∂S
≡
and
≡
.
∂σ ∂σ ε A
∂σ ∂σ ε A
One way
managers influence the idiosyncratic variance of the firm is to choose investments from their
opportunity set with the most non-systematic risk.4
Similar to the relationship between the market risk of equity and the market risk of the
firm in equation (4), the idiosyncratic volatility of equity is a function of the interaction between
the idiosyncratic volatility of firm level returns and ηS, the elasticity of equity value with respect
to firm value. To see this let εi represent the idiosyncratic part of returns. Adding εi to the
CAPM equation implies that the total variance of equity can be written as σ S2 = β S2σ r2M + σ ε2S .
Substituting in equation (4) for βS and using the implication from equation (3) that σ S2 = η S2σ A2
reveals that
3
See the discussion of the Galai and Masulis model in Copeland and Weston (1988).
Galai and Masulis stress that since the investment decisions of a levered firm are controlled by stockholders, a
project requiring an initial investment of I, will be undertaken even if dA dI < 0 as long as
4
∂S ∂I = ( ∂S ∂A )( dA dI ) + ( ∂S ∂σ 2 )( dσ 2 dI ) ≥ 0 .
Stockholders are willing to take on a project that has a
negative impact on firm value as long as the benefit from increasing the idiosyncratic variance outweighs the cost.
For our purposes consider a levered firm with a choice between two mutually exclusive projects, X and Y, with equal
profitability in terms of expected net cash flow discounted for systematic risk, dA dI X = dA dIY , where σ2X < σ2Y.
Managers will always choose Y since ( ∂S ∂σ 2 )( dσ 2 dIY ) > ( ∂S ∂σ 2 )( dσ 2 dIY ) . The costs of increasing nonsystematic risk are borne by the holders of the firm’s debt.
6
σ ε2 = η S2 (σ A2 − β A2σ r2 ) = η S2σ ε2 .
S
M
A
(5)
Equation (5) shows that the idiosyncratic risk of equity is a function of the idiosyncratic variance
of firm, and through ηS, the firm’s capital structure and the slope of the function relating equity
values and total firm value.
Given the motivation of managers to seek growth opportunities that increase the
idiosyncratic variance of the firm, we model σ ε2A as a linear function of growth options, GO:
σ ε2 = η 2σ ε2 = η 2 ( β 0 + β1GO ) .
S
S
S
A
(6)
In our first set of results we use an estimate of η S in the regression
σ ε 2 = ατ + β 0η 2 + β1 ⎡⎣η 2 * GO ⎤⎦ ,
S
S
S
(7)
and test for the presence of a significant positive trend, τ. We find that the estimated η S is very
close to 1 throughout the sample. To relate our results to previously posited explanations of the
trend in idiosyncratic volatility, we also conduct tests assumingη S = 1 . This restriction has the
added advantages of reducing estimation error and simplifying the interpretation of the parameter
estimates.
Since managers disposed to increasing idiosyncratic risk can do so by selecting high risk
projects from their investment opportunity set, we expect a close association between any trend
in corporate growth options and the trend in firm specific risk. More growth opportunities
provide managers a larger menu of projects from which to choose those with higher variance.
This leads to our first hypothesis:
H1. Aggregate idiosyncratic volatility is positively related to the level of growth options.
7
We allow for the possibility that changes through time around any trend in the investment
opportunity set may also be related to idiosyncratic volatility. As the investment opportunity set
changes so does the ability of managers to impact the idiosyncratic component of firm returns. If
the menu of projects is highly variable, managers will have more opportunities to select projects
that will increase the idiosyncratic risk of the firm. Therefore we expect the increasing trend in
idiosyncratic volatility to be related to the volatility in corporate growth options:
H2. Aggregate idiosyncratic volatility is positively related to the variance of growth options.
If growth options constitute a significant portion of a firm’s value and are closely related to firm
specific shocks, ceteris paribus we expect that more growth options and higher variance of
growth options will be related to greater idiosyncratic volatility and will explain the documented
trend.
2. Idiosyncratic Volatility, Growth Options, and Controls
2.1 Idiosyncratic Volatility
We calculate aggregate idiosyncratic volatility following Campbell et al. (2001) and Wei
and Zhang (2003). Start with the capital asset pricing model of Sharpe and Lintner:
Rit = β im Rmt + ε it .
(8)
Here Rit is the excess return of firm i at time t over the Treasury bill rate, Rmt is the valueweighted market excess return, β im is the market beta of firm i, and ε it captures firm specific
shocks. To get to the measure of aggregate idiosyncratic volatility used in Campbell et al.
(2001), consider the alternative decomposition of returns for firm i into a market return and an
idiosyncratic return.
Rit = Rmt + η it .
(9)
Subtracting (9) from (8) and solving for the idiosyncratic return yields
8
η it = ε it + ( β im − 1) Rmt .
(10)
Letting wit denote the weight on stock i in the market portfolio at time t, the weighted average
variance across all (N) firms is then
N
∑ w Var ( R
i =1
it
it
N
N
i =1
i =1
) = Var ( Rmt ) + ∑ witVar (η it ) + 2∑ wit Cov( Rmt ,η it )
N
N
i =1
i =1
= Var ( Rmt ) + ∑ witVar (ηit ) + 2∑ wit Cov( Rmt , ε it + ( β im − 1) Rmt )
N
= Var ( Rmt ) + ∑ witVar (ηit ) .
(11)
i =1
That is, the average cross-sectional variance of returns is the sum of the market-level stock return
volatility and average firm specific volatility. The third equality holds due to equation (10), the
fact that ε it and Rmt are orthogonal, and because the weighted average beta equals one.
Using (11) we are able to estimate the idiosyncratic volatility of firm i in month t, Vit ,
using daily returns within month t as
Vit = Var (ηˆit ) = D ⎡⎣Var ( Ris − Rms ) ⎤⎦ ,
(12)
where Ris is firm i's daily excess return for each trading day s in month t, Rms is the crosssectional average of returns for all stocks available on day s in our sample weighted by the
market capitalization on day s.5 More succinctly, we measure Vit for stock i in month t as the
number of trading days, D, in month t times the sample variance of market adjusted daily returns
5
We do not extract the industry average return from individual stock return as in Campbell et al. (2001). Since the
industry component constitutes a relatively small part of the individual stock return and has remained stable over
time, its effect is negligible. As shown in Figure 1, our measure of idiosyncratic volatility is virtually identical to
that reported in Campbell et al. (2001).
9
of stock i within month t. We then take the value-weighted average of Vit across all stocks in
month t and construct the monthly time series of idiosyncratic volatility Vt for month t as
N
Vt = ∑ witVit .
(13)
i =1
Here wit is a weight based on the market capitalization of stock i at the end of month t-1.
2.2 Growth Option Proxies and Return-on-Equity
Growth opportunities are not directly observable. We use two proxies for growth options
that have been studied in previous work [e.g. Jacquier, Titman and Yalcin (2001) and Goyal,
Lehn and Racic (2002)]. These are the ratio of the market value to book value of assets (MABA)
and the ratio of the market to book value of common equity (MEBE). Using the merged
quarterly Center for Research in Stock Prices (CRSP) and COMPUSTAT industrial and research
files data items, these variables are defined as:
1. MABA = [Total Assets (data44) – Total Common Equity (data59) + Price (data14) *
Common Shares Outstanding (data61)] / Total Assets (data44)
2. MEBE = [Price (data14) * Common Shares Outstanding (data61)] / Total Common
Equity (data59)
Both the MABA and MEBE ratios proxy for corporate growth options since the market
value of assets and equity capture the market’s anticipation of future growth opportunities within
the firm while book value does not. These ratios should both be positively related to the growth
options of a firm.6 We construct time-series volatilities of the growth options and include them in
6
Other proxies for growth options are dividend yields, debt/equity, and earnings-per-share/share price, capital
expenditures/net fixed assets, and research and development expenditures/total assets. We do not use these proxies
for several reasons. In our sample negative earnings appear in approximately 27% of non-missing firm-months
making them difficult to interpret in terms of growth options. Dividend yields are zero in 63% of the sample.
Jacquier, Titman, and Yalcin (2001) note that a low or zero dividend yield may also proxy for financial distress (as
does a high debt-to-equity ratio) making it a poor proxy for growth opportunities. Quarterly capital expenditures
and R&D expenditures have only been available on COMPUSTAT since 1984 and 1989 respectively with many
missing observations. Furthermore, Adam and Goyal (2003) use a unique data set to evaluate many proxies for
growth options and determine that MABA and MEBE capture the information contained in the other proxies.
10
the analyses since it is reasonable to anticipate that not only the growth option itself, but also its
volatility is related to firm specific risk.
As noted above, return-on-equity (ROE), defined as earnings divided by book value of
common equity, has been shown to play an important role in explaining the increase in
idiosyncratic volatility through time. We include ROE and its time-series variance as control
variables in the analyses. Specifically,
ROE = Income before Extraordinary Items (data8) / Total Common Equity (data59).
2.3 The Elasticity of Equity Value to Total Firm Value
Our estimate of elasticity, ηS, is a cross-sectional value-weighted average of individual
firm ηS’s. To generate a time series of ηS for firm i, we roll the regression of the natural
logarithm of the value of equity, S, (measured as shares outstanding * share price) on the natural
logarithm of an estimate of firm value, A ,
ln Si ,t = α i + β i ln Ai ,t + ε i ,t ,
(14)
through the sample for each firm using the past 36 months of data. The β i coefficient is the
elasticity ηS,i for firm i.
We follow Vassalou and Xing (2004) and use the Black-Scholes option pricing model in
equation (2) to estimate firm value, A . Specifically, we use market capitalization at the end of
each month as the value of equity, S. The exercise price k is approximated by the book value of
bonds. We use the debt in one year (Data 44) plus half the long-term debt, (Data 9) as the book
value of bonds, where both variables are from the merged CRSP/COMPUSTAT annual file.7
The risk-free rate r is the monthly 1-year Treasury Constant Maturity Rate from the Board of
Governors of the Federal Reserve System. The time to maturity is 10 years. In the BlackScholes model, the volatility σ A2 is the annualized variance of firm value.
We adopt an iterative procedure to estimate the value of each firm. In each month we:
7
We follow Vassalou and Xing (2004) in construction of the book value of bonds using “Debt in One Year” plus
one half of the “Long-Term Debt”. They note that this method is common practice in industry.
11
(1) Estimate the volatility of equity, σ S2 , by using previous 36 months of market
capitalization, and set the initial value of σ A2 to be σ S2 ;
(2) Use the Black-Scholes model to estimate the firm value A for each month;
(3) After obtaining a monthly time-series of A, we compute its variance and use it as the
input to the Black-Scholes model for the next round iteration;
(4) Repeat steps (2) and (3) until the difference in σ A2 between two iterations is no larger
than 0.0001;
(5) Back out the value of the firm by using the final estimate of σ A2 and the Black-Scholes
model in equation (2).
We remove 80 firms (0.73 percent of the sample) that did not achieve convergence for at
least 25 percent of their sample data. The time series average of our value-weighted crosssectional ηS is 0.998 with a standard deviation of 0.023. For all 120 month periods in our sample
the average ηS ranges from 0.99 to 1.005.
3. Data
Daily stock returns are from CRSP. The accounting data are from the merged
CRSP/COMPUSTAT quarterly industrial file. The COMPUSTAT quarterly files start in 1971
and we include data through 2002.8 We include only common shares (share codes of 10 and 11)
listed on the NYSE, AMEX and NASDAQ. We exclude financial service firms (SIC codes 60006999) since their growth opportunities and capital structure differ from most firms. In each
month all the stocks included must have a non-missing return for the current month, non-missing
market capitalization at the end of the previous month, and non-negative book value of common
equity. Finally, we delete the last quarter of data for any firm delisted before our sample period
ends.
To eliminate any look-ahead bias we match the COMPUSTAT quarterly accounting
variables with the monthly return variances, calculated using daily stock returns from CRSP, by
the earnings report date in COMPUSTAT. Firms typically report earnings within three months
8
At the beginning of the sample there are many missing values. We use the sample period from 1974 to 2002 in
the subsequent analysis. Since we require 3 years of data to estimate time-series variance of growth options
variables, the sample period for the time-series analysis is from September, 1976 to December, 2002.
12
after the end of current fiscal quarter. For any firm-quarter that has non-missing accounting
variables but missing earnings report date, we assume that the firm reports its quarterly earnings
at the end of the third month after the end of the fiscal quarter. As is commonly done with these
data, we winsorize the firm-month panel data at both the upper and lower 2.5% levels to mitigate
the impacts of outliers. The original panel consists of 1,242,983 firm-month observations over
the period of 1971 through 2002.
For each month, we calculate the value-weighted averages of the variables across all
firms in the original firm-month panel and construct a value-weighted monthly time series,
where the weight is the market capitalization evaluated at the end of the prior month.9 To
calculate the time series variance of the growth options variables we require that each firm have
at least eight quarterly values within the past three years.
To confirm that our measure of idiosyncratic volatility is similar to previous measures
Figure 1 overlays our time series of Vt for the period of January 1971 to December 2002 on top
of the Campbell et. al. measure which runs from July of 1962 through 1997. The figure shows
the increase in idiosyncratic volatility and countercyclical patterns consistent with that in
Campbell et. al. (2001) and other studies. The correlation between our idiosyncratic volatility
measure and the Campbell et. al. (2001) measure is 0.98 for the overlapping time period from
January 1971 to December 1997.
Table 1 reports the descriptive statistics of monthly idiosyncratic return variance (Vt ) at
the firm-month panel level over the original period of January 1971 to December 2002. The
mean 3.7% and the median is 1.7%. The value-weighted average variance has a standard
deviation of 5.4%, is moderately skewed to the right, and has relatively fat tails. Table 1 also
presents summary statistics of the value-weighted monthly average growth options variables
MABA and MEBE, as well as ROE, and firm SIZE. MABA averages 1.8 with a median of 1.3.
The maximum (minimum) value of MABA is about 7 (0.7), which is within the usual [0.01, 100]
interval. MABA is positively skewed, fat tailed, and volatile with a standard deviation of 1.35.
Similar patterns appear for MEBE with the exception that MEBE is on average larger with a
mean (median) at 2.5 (1.6) and slightly more volatile than MABA. Our samples of MABA and
9
While not reported, we also considered the equally weighted values of all the explanatory and control variables,
even though equation (11) requires that we use value weighting for idiosyncratic volatility. We find that value
weighting is important for the explanatory ability of these data.
13
MEBE are consistent with series used in other work, i.e.. Jacquier, Titman and Yalcin (2001) and
Adam and Goyal (2003).
Return-on-equity measures profitability of a firm. In general, the firms in our sample are
profitable with positive mean ROE of 0.3%. A median firm has a ROE of about 2.4%, which is
about eight times as high as the mean ROE, implying that there exist some extremely bad
performers in our sample. This is corroborated by both the minimum value of ROE at –34% and
the negative skewness of –2.2.
Firm size, measured by the previous month-end market
capitalization, has a mean and median of $942 million and $68 million respectively.
4. Empirical Results
4.1 Growth Options and Idiosyncratic Volatility: (η S = ηS )
To evaluate the impact of growth options on the idiosyncratic variance of equity returns
we estimate a regression following the specification laid out in equation (7),
Vt = β 0 + β1τ + β 2ηS2,t −1 + β 3 ⎡⎣ηS2,t −1 * GOt −1 ⎤⎦ + β 4 ⎡⎣ηS2,t −1 * GOTSVt −1 ⎤⎦ + ε t ,
(15)
where Vt is the value-weighted idiosyncratic return variance as defined in equation (13), τ is a
time trend, η S2 is the squared elasticity of equity value with respect to firm value, and GOt is the
value-weighted average of the growth option in month t, ( GO = MABA or MEBE). We also
include a measure of the variance of growth options to test our second hypothesis. GOTSVt is
the value-weighted average of individual firm time-series variances of the growth option in
month t. All the regressions are estimated using the Generalized Method of Moments (GMM)
and the t-statistics are computed using the heteroscedasticity- and autocorrelation-consistent
standard errors of Newey-West (1987) with 12 lags.
Table 2 contains the regressions results.
Before taking into account the impact of
elasticity or growth options we check for the significance of the time trend. Consistent with
prior literature on idiosyncratic volatility, the time trend is positively and significantly related to
14
idiosyncratic volatility and explains 32% of the variation in return volatility.10 The elasticity is
negatively, but insignificantly associated with firm specific risk and the inclusion of elasticity in
the regression does not change the significance of the time trend. However, after controlling for
the product of the elasticity and either the level or time series variance of growth options, the
time trend becomes either insignificant or negatively related to firm idiosyncratic volatility. The
empirical results support our hypotheses that the upward trend in firm idiosyncratic volatility is
related to the time-series dynamics in growth options and growth options volatility, conditional
on the elasticity of equity value with respect to firm asset value.
Panel A concentrates on market-to-book assets. Consistent with our first and second
hypotheses, both the interactive terms of elasticity with respect to MABA and MABATSV are
significantly and positively associated with firm idiosyncratic volatility at the 5% significance
level and both have strong explanatory power for the dynamics of return volatility in addition to
the trend. The adjusted R-square increases from 32% with a time trend as the only explanatory
variable, to 59% and 55% when we consider either the product of η 2 and MABA or the product
of η 2 and MABATSV individually as an additional explanatory variable. When both interactive
terms are included as independent variables, both enter significantly with the expected signs and
the adjusted R-square increases from 32% to 63%.
When we use MEBE as a proxy for growth options the empirical evidence is also
consistent with the predictions of the Galai and Masulis model. Panel B shows that both the
interactive terms of elasticity η 2 with MEBE and MEBETSV are significantly and positively
related to firm idiosyncratic volatility even after controlling for the time trend. As with MABA,
once we add either the product of η 2 and MEBE or that of η 2 and MEBETSV or both to the
regression, the time trend becomes insignificant or negative. Adjusted R-squares increase to
56% for the case of the interactive term with MEBE alone and to 53% for that with MEBETSV
alone. The interactive terms of η 2 with both MEBE and MEBETSV together with the trend can
explain 60% of the idiosyncratic volatility.
The results from the analyses usingη 2 , MABA and MEBE support our hypotheses that the
products of elasticity of equity value with respect to total firm value and growth options as well
10
In panels A and B of Table 2, we report results of the univariate regression where the time trend is the only
explanatory variable. The number of firm-month observations used in Panels A and B are slightly different due to
the filters imposed on each of the variables MABA and MEBE (See Table 1).
15
as the product of elasticity and the time-series variances of growth options are positively related
to the upward trend in firm idiosyncratic volatility. In every case, once the growth options
variables are considered, the time trend becomes insignificantly different from zero or negative.
4.2 Growth Options and Idiosyncratic Volatility: (η S = 1)
While the results in the previous table clearly support the connection between growth
options and idiosyncratic volatility suggested by the Galai and Masulis model, the interactive
regression specification is not easily comparable to alternative explanations such as the
profitability hypothesis explored by Wei and Zhang (2003). Given that the estimated value of ηS
is close to one throughout the sample, we set η S2 in equation (15) equal to one and estimate the
following regression,
Vt = β 0 + β1τ + β 2GOt −1 + β 3GOTSVt −1 + ε t .
(16)
Making the assumption that η S =1 has only a marginal impact on the analysis. In Table 3
there is some variation in the estimated parameters, but no qualitative impact. Overall, the
results from the analyses using MABA and MEBE support our hypotheses that growth options
and their time-series variances are positively related to the upward trend in firm idiosyncratic
volatility. In every case once the growth options variables are considered, the time trend
becomes insignificantly different from zero or negative. Together, the level and time-series
variance of growth options account for as much as 63% of the time-series variation of aggregate
idiosyncratic volatility. The economic content of these results is that once growth options are
controlled for, either the trend in idiosyncratic volatility is nonexistent or firm specific risk has
decreased.
While Table 3 focuses on the ability of growth options to explain the trend in
idiosyncratic volatility, in Table 4 we examine whether shocks in lagged growth options
contribute to shocks in firm specific risk. To isolate the shocks in the growth options and
idiosyncratic volatility we first detrend each series by regressing them on time trend. Using the
AIC statistic, both MABA and MEBE are well described by an AR(2) model while V follows an
ARMA(3,3).
The residuals from the time-series models are the resulting shocks for each
16
variable. In both regressions shocks in the lagged growth option variables are positive and
significant at the 5 percent level providing evidence that unexpected changes in the lagged
growth options are related to unexpected changes in idiosyncratic volatility. Together with the
results in Tables 2 and 3, these results provide additional evidence that growth options are
important for explaining idiosyncratic variance.
5. Alternative Explanations
There are two plausible alternative explanations for the increase in idiosyncratic volatility
in the literature to date. In a model of investor learning, Pastor and Veronesi (2003) show that
idiosyncratic volatility is higher for younger firms with more uncertainty about future
profitability (return-on-equity) and more volatile profitability, which they suggest partially
explains increasing idiosyncratic volatility since more firms have listed at younger ages. Along
the same lines, Vuolteenaho (2002) concludes that variance of cash-flow news is more than
twice that of expected-return news and that cash-flow shocks are largely firm specific. Since
return-on-equity is actually scaled earnings (i.e.. corporate cash-flows) this implies that the
variations in return-on-equity are predominantly firm specific and will be reflected in
idiosyncratic risk. Wei and Zhang (2003) analyze the relations between idiosyncratic volatility
and return-on-equity, as well as firm age. They show that declining average corporate return-onequity and increasing return-on-equity volatility contribute to the upward trend in idiosyncratic
volatility, although they do not find firm age to be important. Given the significance of returnon-equity and the varied results on age we include both as control variables below.
5.1 Profitability vs. Growth Options: Round One
We explore the relationship between the trend in firm specific risk and growth options
controlling for return-on-equity and its time-series volatility, using the following regression:
Vt = β 0 + β1TimeTrend + β 2GOt −1 + β3GOTSVt −1 + β 4 ROEt −1 + β5 ROETSVt −1 + ε t .
(17)
17
Here ROEt is the weighted average of return-on-equity in month t and ROETSVt is the weighted
average of the three-year time-series variance of return-on-equity in month t, created following
the same procedure used to define the time-series volatility of growth options.
Panel A of Table 5 reports the regression results for MABA controlling for ROE. The
time trend is insignificant in all specifications except the first where it is significantly negative.
Both MABA and MABATSV are positive and significant while ROE and ROETSV never enter
significantly. Comparing these results to Panel A in Table 3, adding ROE and ROETSV in the
regressions only marginally improves the adjusted R-square (1% increase), indicating little
additional explanatory power over the growth options variables alone. Additionally, both ROE
and ROETSV, which have significant explanatory power for firm specific risk in the absence of
MABA, become insignificant when the growth options variables are included as independent
variables in the regressions.
In Panel B we find qualitatively identical results for MEBE and MEBETSV. Both growth
option related variables remain positively and significantly associated with firm idiosyncratic
volatility after adding ROE and ROETSV. Neither ROE nor ROETSV is significant and adding
them to the regressions has no impact on the adjusted R-square compared to the results in Panel
B of Table 3 where we do not control for ROE. The results in Table 5 indicate that even after
controlling for return-on-equity and the time-series volatility of ROE, growth options and growth
option volatility still play a significant role in explaining idiosyncratic volatility.
5.2 Profitability vs. Growth Options: Round Two
It is worthwhile to point out that each of the regressions in Table 5 contains a time trend
and that ROE also trends up over the sample period. This raises a concern that multicollinearity
may be inflating the standard errors in the above regressions biasing our results. To account for
this possibility we perform the following procedure to check our conclusion that the growth
option proxies play a more significant role in explaining the trend in idiosyncratic variance than
does profitability.
1) Regress idiosyncratic volatility on a constant and ROE to remove any variation in the
volatility explained by ROE.
18
2) Take the residuals from step (1) and run two regressions
a. First, regress the residuals on an intercept and the time trend and test if the
trend remains significant.
b. Second, add a growth option proxy to the regression in step (a) and test if the
trend remains significant.
3) Repeat steps (1) – (2) switching the roles of the growth option and ROE.
If the time trend is significant in part (a) of step (2), then we have evidence that ROE fails to
capture the trend. If the trend is insignificant in part (b) then we conclude that the growth option
explains the trend in idiosyncratic variance. We reverse the process in step (3) to check that we
are not generating a spurious result, i.e.. that a trend exists after first removing the growth option
component.
Table 6 contains the results of this test. The left two cells of Panel A show the results of
steps (1) and (2) when we consider MABA as a proxy for growth options. In the cell labeled Step
1 we report the regression of the idiosyncratic variance on ROE. Here, as in all the other panels,
ROE is significantly different from zero at the 5% level. In the cell labeled Step 2 we first report
the regression of the orthoganalized errors on the time trend. The trend remains a significant
explanatory variable of idiosyncratic variance in all the panels after removing the profitability
component. The second regression in the Step 2 cell adds the proxy for growth options to the
previous regression. Both MABA and MEBE are significantly positive while the trend becomes
insignificant. In the right two cells we have reversed the roles of ROE and the growth options
and in no case is the trend significant in Step (2) after removing the growth option component.
Overall, we see that removing profitability from idiosyncratic variance consistently
leaves a significant trend component that is explainable by any of the growth option proxies we
consider. Conversely, taking the growth option component out of the idiosyncratic variance
never leaves a significant trend. This evidence, together with the evidence in Table 6, provides
convincing support to the predictions of the Galai and Masulis model versus the profitability
based explanations in the literature.
19
5.3 Firm Age
Table 7 summarizes the results controlling for average firm age (AGE) calculated in the
same manner as the other explanatory variables.11 The regressions (1) and (3) are repeated from
Table 3 for comparison. Controlling for firm age, MABA and MABATSV remain positive and
MABA remains significant. While AGE is not significant it does account for some of the time
series variation of the growth option as MABATSV becomes insignificant. Including AGE in the
regression increases the adjusted R-square only marginally (1%).
The results for MEBE show that MEBE remains positive and significant after controlling
for AGE, that MEBETSV is not significant, and AGE is negative but insignificant. Again
including AGE in addition to MEBE and MEBETSV does little to increase the adjusted R-square.
We take these results to indicate that the ability of growth options to explain the upward trend in
idiosyncratic volatility is not due to growth options acting as a proxy for firm age and that
controlling for firm age does not eliminate the explanatory power of growth options. Overall age
adds little to explaining the trend and does not add much in explanatory power beyond the level
and variance of the growth options.
6. Sub-Samples Analysis
6.1 NYSE/AMEX vs. NASDAQ Firms
Schwert (2002) documents unusually high volatility of NASDAQ firms compared with
NYSE/AMEX firms over the six year period between 1995 and 2001. Comparing large and
small firm portfolios on the three national markets he concludes that the substantial increase in
total return volatility of NASDAQ stocks is due to the type of firms (hi-tech firms with more
growth options) rather than to the firm size or the immaturity of the firm. Motivated by this
finding we investigate the ability of growth options to explain the firm specific risk of NASDAQ
versus NYSE/AMEX firms. A priori, we could argue that the small, high-tech firms with large
growth options listed on NASDAQ will be more sensitive to the level and volatility of growth
options. On the other hand, the larger NYSE/AMEX firms may have more free cash flow and
more flexibility in choosing risky projects that increase the idiosyncratic variance of the firm
11
The correlation between AGE and trend is high, above 96%. To avoid problems with multicollinearity we detrend AGE. De-trending AGE only impacts the intercept and trend coefficient and their standard errors.
20
thereby increasing the value of their equity. We explore these issues by separating firms listed
on NASDAQ from those listed on the NYSE and AMEX.
Table 8 presents the summary statistics for the two samples respectively. Between 4649% of all firm-months in our full sample are from NYSE/AMEX while the rest come from
NASDAQ. Idiosyncratic volatility is on average more than two times higher for NASDAQ
firms. The mean (median) of V is 2% (0.9%) for NYSE/AMEX firms versus 5.4% (2.9%) for
NASDAQ firms. The standard deviation of V is also much higher for NASDAQ (6.4%) versus
NYSE/AMEX firms (3.4%). In both samples V is positively skewed with heavy tails indicating
some firms with large idiosyncratic volatility in both samples.
NYSE/AMEX firms on average are 3-4 times larger than NASDAQ firms in their market
capitalization. The standard deviation for NYSE/AMEX firms is about 6 times its mean while
that for NASDAQ firms is about 15 times its mean value reflecting the size variation of firms
listed on NASDAQ. In addition, SIZE is more skewed to the right with heavier tails for
NASDAQ firms. Profitability for NYSE/AMEX firms is on average positive and less variable
than is profitability of NASDAQ firms which average a negative ROE of -1.6%. Consistent with
the conjecture that NASDAQ firms have more growth options, both MABA and MEBE are on
average larger for that sample. NASDAQ firms tend to have smaller, more positively skewed,
and heavier tailed MVS, indicating more control of the firm is held by equity holders.
Table 9 provides the regression results for the analysis of the two samples respectively.
Since Adam and Goyal (2003) suggest that MABA is the most informative growth options proxy
and the results for MEBE are similar, we focus on MABA and its time-series volatility. There are
distinct differences between the average idiosyncratic variances and the ability of growth options
to explain firm specific risk on different exchanges. Regression (1) in Table 9 indicates that,
while the trend is significant for both samples, it is much more prevalent in the idiosyncratic
variance of NASDAQ firms. The coefficient on the trend is nearly four times larger for the
NASDAQ sample and the trend alone explains more than twice the variability of the firm
specific risk of that sample. Even though we cannot say that NASDAQ firms are driving the
trend in idiosyncratic volatility, it is apparent that the upward drift of idiosyncratic variance is
more pronounced for firms on NASDAQ.
While MABA is significant and drives out the trend for both exchanges, the time series
variance of MABA is important only for the NYSE/AMEX sample. Adding the variance of
21
MABA as an explanatory variable has little impact on the adjusted R-squares for the NASDAQ
samples, but doubles the explanatory power in the NYSE/AMEX sample. In regression (2) when
both the level and variance of MABA are included in the regressions, the level is significant in the
NASDAQ sample while both the level and the variance are significant in the NYSE/AMEX
sample for the one-sided test. In support of our first hypothesis, growth options are related to the
idiosyncratic variance of firms in both samples. The result that the variance of growth options is
more important for NYSE/AMEX firms is consistent with larger firms having more flexibility in
their choice and timing of investments and supports our second hypothesis.
Overall the exercise of breaking up the sample based on exchanges adds more support for
the implications of the Galai and Masulis model. Firms with more growth options exhibit a
stronger trend in their idiosyncratic volatility. Growth options are important in explaining the
significant trends in firm specific risk across exchanges while the variance of growth options
plays a significant role for larger firms who have more ability to take advantages of changes in
their investment opportunity set. Finally, firms that consist of a higher percentage of equity have
higher idiosyncratic volatility.
6.2 The ‘Internet Bubble’
Our sample extends five years beyond the sample period originally examined by
Campbell et al. (2001). It is obvious from Figure 1 that the idiosyncratic volatility between
January 1999 and December 2002 is different from the rest of the sample period. To ensure that
our results are not being driven by the ‘internet bubble’ period, we split our data into two subperiods; one running through the end of 1998 and the other starting in January of 1999 and
running through the end of 2002.
There is no discernable trend in the latter period either visually or statistically. The
regression of idiosyncratic volatility on a time trend for that period produces an insignificant
coefficient on the trend and an adjusted R2 of 0.03. Table 10 contains the results of regressing
idiosyncratic volatility on MABA and MEBE for the earlier period: 1976-1998. The first
regression indicates that the trend is significantly positive and explains about 16% of the
variation in idiosyncratic volatility. Individually both growth option proxies subsume the trend
and both proxies are significantly positive and in both cases the variables explain 22% of the
22
variation in idiosyncratic volatility. These results show that the predictions of the Galai and
Masulis model are not a byproduct of including the ‘internet bubble’ in our sample.12
7. Controlling for the Total Value to Equity Value Ratio
While an important assumption in the Galai and Masulis model rules out debt covenants,
the ability of managers to influence the level of σ εV should be conditional on capital structure in
the sense that it reflects the amount of shareholder control over which projects are undertaken.
The intuition, via Jensen and Meckling (1976), is straightforward.
Firms with a higher
proportion of value in equity are controlled more by shareholders than creditors.
Since
shareholders are motivated by both lower market risk and higher share value to invest in riskier
projects with more growth options (at the expense of debt holders), we expect a more
pronounced positive relationship between growth options and the idiosyncratic risk of equity for
firms dominated by equity financing.
Firms with more value held by debt holders have
management being restrained by the cost of capital provided by debt holders. We expect to see a
relatively weaker relation between growth options and the idiosyncratic risk of equity in such
firms as the moral hazard problem is at least partially mitigated.13
We define a measure of capital structure as the ratio of market value of total assets to
market value of common equity, MVS.
To calculate MVS, we use the merged
CRSP/COMPUSTAT quarterly industrial and research file items to define quarterly market value
of assets and we define the monthly market value of common equity as the product of month-end
share price and shares outstanding from the CRSP monthly stock file (Mkt. Cap.).
MVS = [Total Assets (data44) – Total Common Equity (data59) + Price (data14) *
Common Shares Outstanding (data61)] / Mkt. Cap.
On average, MVS is 2.20 with a lower median value at 1.6. The standard deviation of MVS is
about 1.54.
12
The market crash in October, 1987 is a large outlier in the earlier sample. Removing that data point leaves the
results we present qualitatively unchanged.
13
The “under investment” phenomenon outlined in Myers (1977) and Myers and Maljuf (1984) is also a possible
explanation.
23
A simple method of controlling for changes in capital structure is to condition on MVS by
sorting firms into MVS portfolios and re-estimating the regressions of idiosyncratic volatility on
growth options proxied by MABA and MEBE. To do this we partition, in each month, all firms
into five value-weighted portfolios based on MVS. Portfolio 1 contains firms in the lowest MVS
quintile. Table 11 documents the results of regressing idiosyncratic volatility on MABA within
each of the five portfolios sorted by MVS in ascending order. We report the results for MABA
only since the results for MEBE are qualitatively similar.
For each of the five data sets regression (1) shows that the trend alone is positive,
significant at standard levels, and explains between 25% and 35% of the variation in
idiosyncratic volatility. Regression (2) in each panel adds the level and time-series variance of
the growth option proxy to the specification. Moving from Portfolio 1 to Portfolio 5 we see that
both the significance of MABA and its ability to explain the trend drops off as the level of
ownership by debt holders increases. For firms with the highest MVS (Portfolio 5), the inclusion
of MABA has almost no impact on the point estimate of the time trend. The significant trend in
this portfolio is unsurprising given that a high level of debt is commonly considered a precursor
to distress.
These regressions show that as the controlling power of shareholders in the firm increases
relative to debt holders, growth options become more important in accounting for firm specific
risk due to the decreasing restraints on the ability of shareholders to take advantage of corporate
growth opportunities and assume risky investments. In the case where debt holders are likely to
have the most influence on managers, more choice in terms of growth options significantly
reduces the idiosyncratic risk of firms. The significant negative coefficient on the level of MABA
for the highest MVS portfolio is consistent with the intuition that for firms with high levels of
monitoring by bondholders, more choice in the menu of growth options leads to managers
picking safer projects rather than riskier projects. The implication of relaxing the assumption
ruling out debt covenants in the Galai and Masulis model is that the positive and significant
association between growth option variables and firm specific risk will decline moving from the
portfolio of firms mainly controlled by the shareholders, Portfolio 1, to the portfolio of firms
whose total values are mainly in the hands of the debt holders, Portfolio 5.
24
8. Conclusions
This paper provides a theoretically based explanation of why individual stocks have
become more volatile by linking the trend in equity return volatility to the investment decisions
of corporate managers. We use the classic corporate finance model of Galai and Masulis (1976)
to argue that a significant portion of the trend in average idiosyncratic volatility can be explained
by changes in the level and variance of growth options conditional on the capital structure of
firms. We maintain that a moral hazard problem present in levered firms motivates managers
acting on behalf of shareholders to select investment projects that increase the idiosyncratic
variance of the firm at the expense of debt holders. The level of debt to equity reflects the
relative control of managers to select risky projects and impacts their ability to increase equity
values by increasing diversifiable firm level risk.
The empirical evidence supports the hypotheses that both the level and variance of
aggregate growth options are positively related to the upward trend in idiosyncratic volatility.
After controlling for growth options the trend in idiosyncratic volatility disappears or becomes
negative. The evidence also supports other classic corporate theory showing the importance of
capital structure for our growth option based explanation. In particular, we show that firms
primarily controlled by shareholders exhibit a positive relationship between idiosyncratic
volatility and both the level and variance of growth opportunities, while firms controlled
primarily by debt holders demonstrate decreasing idiosyncratic volatility as the level of their
investment opportunity set increases. Our results are robust across different exchanges as well as
sub-samples and the data clearly support growth options over alternative explanations of the
trend in idiosyncratic volatility.
Showing that growth options play an important role in explaining changes in firm
specific risk begs the question of why the level and variance of growth options have changed.
The most obvious source of increased growth options comes from the trend towards more open
capital and goods markets. Globalization of the world’s economies provides managers with
more opportunities for growth while at the same time increasing competition. Zingales (2000)
points out that increased worldwide competition increases the demand for process innovation and
improvements in quality. Our results are consistent with globalization of markets providing
more opportunities for growth and more competition which forces manages to seek out riskier
projects to increase shareholder value.
While Irvine and Pontiff (2004) show increasing
25
competition can raise the variability of cash flows thereby increasing idiosyncratic volatility, the
variability of cash flows in part comes from riskier growth options. Beck, Demirgüç-Kunt and
Maksimovic (2004) suggest that the legal and institutional structures which are more favorable to
investment contribute to the availability of growth opportunities in the economy. As more so
called ‘third world’ economies mature into stable marketplaces, the menu of growth options for
many firms continues to increase, suggesting that our results may hold internationally as well.
The strong results linking idiosyncratic risk and growth options following from the Galai
and Masulis model have implications for the development of incentive packages for corporate
managers and for the understanding of risks faced by debt holders.
From an investors
perspective our results suggest consideration of growth options is important for understanding
time variation in the covariance structure of returns.
Alternatively, the results suggest an
important role for firm specific information in the information set of conditional asset pricing
models.
26
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29
Value-Weighted Average Idiosyncratic Variance of Returns
0.045
0.040
0.035
0.030
0.025
0.020
0.015
0.010
0.005
0.000
196207 196609 197011 197501 197903 198305 198707 199109 199511 200001
Figure 1: Value-weighted Average Idiosyncratic Variance of Returns
This figure plots the value-weighted monthly time series of aggregate idiosyncratic variance.
The dotted line is the Campbell et al. (2001) data and the solid line is the data in this paper.
30
Table 1: Panel Data Summary Statistics: 1971:1 – 2002:12
Summary statistics of monthly average idiosyncratic return variance, return-on-equity, total to equity value ratio,
and growth options at the firm-month panel level from January 1971 to December 2002. V denotes the monthly
average idiosyncratic return variance; SIZE the previous month end market capitalization in million dollars; MABA
market value to book value of assets; MEBE market value to book value of common equity; ROE return-on-equity
and MVS the ratio of market value of assets to market value of common equity. All variables are winsorized at the
2.5% and 97.5% levels.
Statistics
V
SIZE
MABA
MEBE
ROE
MVS
Firm-months
1,242,983
1,242,983
1,176,690
1,242,983
1,238,385
1,176,690
Mean
0.037
942.6
1.786
2.516
0.003
2.174
Median
0.017
67.9
1.280
1.619
0.024
1.603
Std. Dev
0.054
7217.1
1.354
2.608
0.089
1.536
Skewness
2.607
30.5
2.373
2.425
-2.211
2.360
Kurtosis
9.804
1397.7
8.600
9.076
8.762
8.720
31
Table 2: Time Series Regressions of Monthly Idiosyncratic Variance on a Time Trend,
Elasticity of Equity to Firm Value, and the Product of Elasticity and Growth Options:
1976:9 – 2002:12
Time series regressions of idiosyncratic return variance on a time trend and the explanatory variables
suggested by the Galai and Masulis model. The dependent variable is the monthly average idiosyncratic
return variance, V. The independent variables include a time trend, the elasticity of equity value with respect
to firm value, η S2 , as well as the products of the elasticity and the level (time-series variance) of market-tobook value of assets, MABA (MABATSV), market-to-book value of common equity, MEBE (MEBETSV):
i.e.. η S2 *MABA, η S2 *MABATSV, η S2 *MEBE , and η S2 *MEBETSV. We use the generalized method of
moments (GMM) to estimate the model. The t-statistics (in parentheses) use heteroscedasticity and
autocorrelation consistent standard errors based on the method of Newey and West (1987) with 12 lags.
1
2
3
4
5
6
7
Intercept
0.002
(1.914)
0.038
(1.440)
-0.005*
(-2.326)
0.005*
(6.195)
-0.049
(-1.573)
-0.031
(-1.284)
-0.058
(-1.816)
Panel A: Market Value of Assets to Book Value of Assets (MABA)
η S2
η S2 *MABA
η S2 *MABATSV
Trend
3.906*
(3.726)
4.040*
-0.036
(3.615)
(-1.311)
-2.648*
0.008*
(-2.060)
(4.342)
-1.329
0.032*
(-1.202)
(3.128)
-3.050*
0.045
0.009*
(-2.318)
(1.449)
(4.465)
-1.620
0.036
0.033*
(-1.411)
(1.476)
(3.168)
-3.834*
0.057
0.006*
0.017*
(-3.155)
(1.795)
(3.349)
(2.649)
Panel B: Market Value of Equity to Book Value of Equity (MEBE)
η S2
η S2 *MEBE
η S2 *MEBETSV
Intercept
Trend
1
0.002
3.906*
(1.914)
(3.726)
2
0.038
4.040*
-0.036
(1.440)
(3.615)
(-1.311)
3
0.001
-2.948*
0.004*
(0.409)
(-2.014)
(3.882)
4
0.005*
-1.536
0.007*
(6.216)
(-1.200)
(2.867)
5
-0.043
-3.371*
0.044
0.004*
(-1.504)
(-2.195)
(1.536)
(3.955)
6
-0.034
-1.892
0.039
0.007*
(-1.375)
(-1.410)
(1.576)
(2.901)
7
-0.061
-4.689*
0.064
0.003*
0.004*
(-1.900)
(-3.160)
(1.992)
(3.276)
(2.860)
* indicates significance at 5% level using a two-sided t-test.
Adj. R2
0.32
0.32
0.59
0.55
0.59
0.55
0.63
Adj. R2
0.32
0.32
0.56
0.52
0.57
0.53
0.61
32
Table 3: Time Series Regressions of Monthly Idiosyncratic Variance on a
Time Trend and Growth Options: 1976:9 – 2002:12
Time series regressions of idiosyncratic return variance on a time trend, total firm value to equity value, growth
options and the time-series variances of growth options. The dependent variable is the monthly average idiosyncratic
return variance, V. The independent variables include a time trend, and the level (time-series variance) of market-tobook value of assets, MABA (MABATSV), market-to-book value of common equity, MEBE (MEBETSV) as well as
the market value of assets to market value of common equity, MVS. We use the generalized method of moments
(GMM) to estimate the model. The t-statistics (in parentheses) use heteroscedasticity and autocorrelation consistent
standard errors based on the method of Newey and West (1987) with 12 lags.
1
2
3
4
Panel A: Market Value of Assets to Book Value of Assets (MABA)
Intercept
Trend
MABA
MABATSV
0.002
3.906 *
(1.91)
(3.73)
-0.005 *
-2.494
0.008 *
(-2.29)
(-1.93)
(4.26)
0.005 *
-1.310
0.032 *
(6.15)
(-1.19)
(3.12)
-0.001
-3.152 *
0.006 *
0.016 *
(-0.64)
(-2.49)
(3.04)
(2.43)
Panel B: Market Value of Equity to Book Value of Equity (MEBE)
Intercept
Trend
MEBE
MEBETSV
1
0.002 *
3.869 *
(1.98)
(3.68)
2
0.001
-2.927 *
0.004 *
(0.43)
(-2.04)
(3.90)
3
0.005 *
-1.463
0.007 *
(6.27)
(-1.18)
(2.88)
4
0.003 *
-3.883 *
0.003 *
0.004 *
(2.02)
(-2.61)
(3.08)
(2.59)
* indicates significance at 5% level using a two-sided t-test.
Adj. R2
0.32
0.58
0.54
0.61
Adj. R2
0.31
0.56
0.52
0.60
33
Table 4: Regression of Shocks to Idiosyncratic Volatility on Lagged Shocks to
Growth Options
Regressions of shocks to detrended idiosyncratic volatility on shocks to detrended
growth options. Based on AIC criterion, both MABA and MEBE are AR (2) process
and V is ARMA (3, 3) process. The residuals from the respective time-series models
are the resulting shocks for each variable.
Regression of shocks to V on Lagged shocks to MABA
( Shocks to V )t = α + β ( Shocks to MABA )t −1 + ε t
α
β
0.000
(0.08)
0.008*
(2.84)
Adj. R2
0.03
Regression of shocks to V on Lagged shocks to MEBE
( Shocks to V )t = α + β ( Shocks to MEBE )t −1 + ε t
α
β
0.000
0.004*
(0.05)
(2.49)
* indicates significance at 5% level using a two-sided t-test.
Adj. R2
0.02
34
Table 5: Regressions of Monthly Idiosyncratic Return Variance on a Time Trend,
Return-on-Equity and Growth Options Variables: 1976:9 – 2002:12
Time series regressions of idiosyncratic return variance on time trend, return-on-equity, growth options, and the timeseries variances of return-on-equity and growth options. The dependent variable is monthly average idiosyncratic return
variance (V). ROE denotes return-on-equity and ROETSV indicates the time-series variance of ROE. The other
independent variables include a time trend, and the level (time-series variance) of market-to-book value of assets, MABA
(MABATSV), market-to-book value of common equity, MEBE (MEBETSV). We use the generalized method of moments
(GMM) to estimate the model. The t-statistics (in parentheses) use heteroscedasticity and autocorrelation consistent
standard errors based on the method of Newey and West (1987) with 12 lags.
Reg.
1
2
3
Intercept
-0.003
(-0.96)
0.005 *
(5.46)
-0.005
(-0.90)
Panel A: Market to Book Value of Assets (MABA)
Trend
ROE
ROETSV
MABA
-2.554 *
-0.043
0.008 *
(-1.98)
(-0.65)
(4.46)
-3.392
4.409
(-1.77)
(1.22)
-1.807
0.065
-3.736
0.006 *
(-0.90)
(0.81)
(-0.84)
(2.77)
Panel B: Market to Book Value of Equity (MEBE)
Intercept
Trend
ROE
ROETSV
MEME
1
0.003
-3.101 *
-0.072
0.004 *
(1.13)
(-2.18)
(-1.12)
(4.21)
2
0.005 *
-4.082 *
5.498
(5.28)
(-1.98)
(1.54)
3
0.001
-2.599
0.040
-3.944
0.003 *
(0.13)
(-1.21)
(0.51)
(-0.80)
(2.73)
* indicates significance at 5% level using a two-sided t-test.
MABATSV
Adj. R2
0.58
0.028 *
(2.80)
0.017 *
(2.73)
0.56
0.62
MEBETSV
Adj. R2
0.56
0.006 *
(2.70)
0.004 *
(2.81)
0.54
0.60
35
Table 6: Two-Step Sensitivity Test
Regression results of the two-step sensitivity tests. In the left two cells of each panel we do the following: In step 1 we
regress idiosyncratic return variance on an intercept and return-on-equity (ROE). In step 2 we (a) regress the residual
from step 1 regression on an intercept and a time trend, then (b) add a growth option proxy to the regression in (a). In the
right two cells we reverse the roles of ROE and the growth options and repeat each regression. The definitions of the
variables are provided in Tables 2 and 4.
Panel A: Market Value to Book Value of Assets (MABA)
First on ROE then on MABA
First on MABA then on ROE
Step 2:
Step 2 :
Step 1:
Step 1:
Vt = β 0 + β1ROEt −1 + ε t
Vt = β0 + β1MABAt −1 + ε t ε t = α 0 + α1Trend + α 2 ROEt −1 + ηt
ε t = α0 + α1Trend + α 2 MABAt −1 + ηt
Intercept
-0.010
(-1.40)
ROE
0.404 *
(2.40)
Intercept
Trend
-0.004 *
(-3.91)
-0.009 *
(-3.83)
2.625 *
(2.60)
-2.047
(-1.38)
MABA
Intercept
-0.004
(-1.77)
MABA
0.006 *
(5.14)
0.006 *
(2.78)
Intercept
Trend
ROE
0.001
(1.06)
0.000
(-0.07)
-0.589
(-0.84)
-0.683
(-1.24)
0.030
(0.36)
Panel B: Market Value to Book Value of Common Equity (MEBE)
Step 1:
First on ROE then on MEBE
Step 2 :
Vt = β 0 + β1ROEt −1 + ε t
Intercept
-0.010
(-1.38)
ROE
0.402 *
(2.38)
ε t = α 0 + α1Trend + α 2 MEBEt −1 + ηt
Intercept
Trend
MEBE
-0.004 *
2.606 *
(-3.88)
(2.57)
-0.005 *
-2.034
0.003 *
(-3.88)
(-1.19)
(2.36)
* indicates significance at 5% level using a two-sided t-test.
Step 1:
First on MEBE then on ROE
Step 2:
Vt = β0 + β1MEBEt −1 + εt
Intercept
0.000
(0.12)
MEBE
0.002 *
(4.78)
ε t = α 0 + α1Trend + α 2 ROEt −1 + ηt
Intercept
Trend
ROE
0.001
(1.08)
0.000
(-0.03)
-0.594
(-0.80)
-0.676
(-1.18)
0.026
(0.30)
36
Table 7: Time Series Regression of Idiosyncratic Volatility on Firm Age
Time series regressions of idiosyncratic return variance on a time trend, de-trended firm age, growth options and
the time-series variances of growth options. The dependent variable is monthly average idiosyncratic return
variance (V). The independent variables include a time trend, the cross-sectional average of firm ages (AGE) detrended, and the level (time-series variance) of market-to-book value of assets, MABA (MABATSV), of market-tobook value of common equity, MEBE (MEBETSV). We use the generalized method of moments (GMM) to
estimate the model. The t-statistics (in parentheses) use heteroscedasticity and autocorrelation consistent standard
errors based on the method of Newey and West (1987) with 12 lags.
Intercept
1
Trend
MABA
0.006*
(3.04)
0.005*
(2.76)
MABATSV
0.016*
(2.43)
0.009
(1.27)
-0.001
-3.152*
(-0.64)
(-2.49)
2
-0.001
-1.276
(-0.58)
(-1.06)
3
0.003
-3.883*
(2.02)
(-2.61)
4
0.002
-1.042
(1.51)
(-0.63)
* indicates significance at 5% level using a two-sided t-test.
MEBE
MEBETSV
AGE
Adj. R2
0.61
-0.001
(-1.61)
0.003*
(3.08)
0.002*
(2.62)
0.004*
(2.59)
0.002
(0.97)
0.62
0.60
-0.001
(-1.84)
0.61
37
Table 8: Summary Statistics of Idiosyncratic Return Variance, Growth Options, Return-onEquity and Total to Equity Value: NYSE/AMEX vs. NASDAQ Firms
Summary statistics of idiosyncratic return variance, growth options, return-on-equity and total to equity value for
firms listed on NYSE/AMEX vs. firms on NASDAQ at the firm-month panel level from January 1971 to
December 2002. We split the full sample into two sub-samples: NYSE/AMEX and NASDAQ firms, according to
the exchange on which a firm is listed each month. Both samples are winsorized at the 2.5% and 97.5% levels for
all variables. The definitions of the variables are provided in Table 1.
Statistics
V
SIZE
ROE
MABA
MEBE
MVS
No. of Firm-months
% No. Obs.
Mean
Median
Std.Dev
Skewness
Kurtosis
605,911
49
0.020
0.009
0.034
4.333
25.78
NYSE/AMEX Firms
605,911
604,654
49
49
1564.1
0.023
148.1
0.030
8737.8
0.060
21.4
-3.009
737.7
18.06
540,804
46
1.430
1.157
0.880
3.388
17.89
605,911
49
1.950
1.368
1.946
3.181
15.65
540,804
46
2.472
1.931
1.590
2.019
7.046
No. of Firm-months
%No. Obs.
Mean
Median
Std.Dev
Skewness
Kurtosis
637,072
51
0.054
0.029
0.064
1.935
6.102
NASDAQ Firms
637,072
633,731
51
51
351.5
-0.016
39.5
0.015
5319.4
0.106
57.5
-1.654
4115.7
5.611
635,886
54
2.089
1.470
1.591
1.822
5.660
637,072
51
3.055
1.968
3.012
1.943
6.303
635,886
54
1.920
1.371
1.441
2.867
11.61
38
Table 9: Time Series Regressions of Idiosyncratic Return Variance on a Time Trend and Growth Options:
NYSE/AMEX vs. NASDAQ Firms
Time series regressions of idiosyncratic return variance on a time trend, market-to-book assets (MABA), and the time-series variance of MABA (MABATSV))
for NYSE/AMEX vs. NASDAQ firms. The dependent variable is monthly average idiosyncratic return variance (V). We use the generalized method of
moments (GMM) to estimate the model. The t-statistics (in parentheses) use heteroscedasticity and autocorrelation consistent standard errors based on the
method of Newey and West (1987) with 12 lags.
Intercept
Trend
NYSE/AMEX Firms
MABA
MABATSV
0.003*
2.396*
(3.29)
(3.02)
-0.004
-2.228
0.007*
2
(-1.73)
(-1.91)
(3.52)
0.004*
-1.176
0.034*
3
(6.14)
(-1.93)
(4.15)
0.000
-2.278*
0.004
0.022*
4
(0.11)
(-2.16)
(1.71)
(3.14)
* indicates significance at 5% level using a two-sided t-test.
1
Adj. R2
Intercept
Trend
0.21
0.004*
(2.57)
-0.001
(-0.30)
0.004*
(4.42)
-0.000
(-0.09)
8.404*
(6.32)
1.143
(0.57)
5.606*
(2.40)
-2.024
(-0.52)
0.44
0.46
0.49
NASDAQ Firms
MABA
MABATSV
Adj. R2
0.47
0.006*
(2.65)
0.006*
(2.78)
0.54
0.008
(0.85)
0.009
(1.18)
0.48
0.55
39
Table 10: Sub-Sample Analysis: 1976:09-1998:12
Time series regressions of idiosyncratic return variance on a time trend, growth options over sample period:
1976:09-1998:12. The dependent variable is monthly average idiosyncratic return variance (V). The
independent variables include a time trend, the level of market-to-book value of assets (MABA) and market-tobook value of common equity (MEBE). We use the generalized method of moments (GMM) to estimate the
model. The t-statistics (in parentheses) use heteroscedasticity and autocorrelation consistent standard errors
based on the method of Newey and West (1987) with 12 lags.
Intercept
Trend
0.004
(8.05)*
0.002
(1.00)
0.003
(4.59)*
1.529*
(4.00)
-0.250
(-0.25)
-0.506
(-0.46)
Sub-Sample Period: 1976:09 -1998:12
MABA
MEBE
Adj. R2
0.16
0.003*
(2.00)
0.22
0.001*
(2.03)
0.22
* indicates significance at 5% level using a two-sided t-test.
40
Table 11: Portfolio Regressions of Idiosyncratic Volatility on Growth Options
This table provides the results of regressing idiosyncratic volatility on growth options within portfolios sorted by
the market value of assets to market value of common equity, MVS. The dependent variable is monthly average
idiosyncratic return variance (V). The independent variables include a time trend and the level (time-series
variance) of market-to-book value of assets, MABA (MABATSV). We use the generalized method of moments
(GMM) to estimate the model. The t-statistics (in parentheses) use heteroscedasticity and autocorrelation
consistent standard errors based on the method of Newey and West (1987) with 12 lags.
Reg.
Intercept
Trend
Portfolio 1
1
0.002
5.355*
(Low MVS)
(1.32)
(3.87)
2
-0.001
-5.462
(-0.24)
(-1.49)
Portfolio 2
1
0.003*
3.041*
(3.46)
(3.61)
2
-0.001
-3.701
(-0.34)
(-1.42)
Portfolio 3
1
0.003*
2.874*
(2.95)
(3.69)
2
-0.001
-2.000
(-0.17)
(-1.07)
Portfolio 4
1
0.002*
2.933*
(2.67)
(3.98)
2
0.009*
3.420*
(2.20)
(2.00)
Portfolio 5
1
0.003*
4.204*
(High MVS)
(2.20)
(3.48)
2
0.033*
4.200*
(3.05)
(2.37)
* indicates significance at 5% level using a two-sided t-test.
MABA
MABATSV
Adj. R2
0.35
0.003*
(1.98)
0.018*
(2.15)
0.45
0.28
0.006*
(2.11)
0.015
(1.44)
0.42
0.25
0.006
(1.39)
0.043*
(2.95)
0.34
0.31
-0.007
(-1.42)
0.034
(1.51)
0.38
0.26
-0.031*
(-2.49)
0.170*
(5.63)
0.60
41