Fisher College of Business
Working Paper Series
Charles A. Dice Center for
Research in Financial Economics
A New Approach to Measuring Riskiness in
the Equity Market: Implications for the Risk
Premium
Turan G. Bali
McDonough School of Business, Georgetown University
Nusret Cakici
Graduate School of Business, Fordham University
Fousseni Chabi-Yo
Fisher College of Business, Ohio State University
Dice Center WP 2012-9
Fisher College of Business WP 2012-03-009
Revision: August 2013
Revision: May 2013
Original: May 2010
This paper can be downloaded without charge from:
http://ssrn.com/abstract=2055380
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Working Paper Series is located at:
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fisher.osu.edu
A New Approach to Measuring Riskiness in the Equity Market:
Implications for the Risk Premium
Turan G. Balia ∗
a McDonough School of Business, Georgetown University, Washington, D.C.20057
Nusret Cakicib †
b Graduate School of Business, Fordham University, New York, NY 10023, USA
Fousseni Chabi-Yoc ‡
c Fisher College of Business, Ohio State University, Columbus, OH 43210-1144, USA
This draft: August 2013
Abstract
We introduce a new approach to measuring riskiness in the equity market. We propose option implied
and physical measures of riskiness and investigate their performance in predicting future market returns.
The predictive regressions indicate a positive and significant relation between time-varying riskiness and
expected market returns. The significantly positive link between aggregate riskiness and market risk premium remains intact after controlling for the S&P500 index option implied volatility (VIX), aggregate
idiosyncratic volatility, and a large set of macroeconomic variables. We also provide alternative explanations for the positive relation by showing that aggregate riskiness is higher during economic downturns
characterized by high aggregate risk aversion and high expected returns.
JEL C LASSIFICATION C ODES : G11, G12, G14, G33
KEY WORDS: Time-varying riskiness, risk-neutral measures, physical measures, expected returns, equity
premium.
∗ Tel.:
+1-202-687-5388; fax: +1-202-687-4031. E-mail address: tgb27@georgetown.edu
636 6776; fax: +1-212-586-0575. E-mail address: cakici@fordham.edu
‡ Tel.:+1-614-292-8477; fax: +1-614-292-7062. E-mail address: chabi-yo 1@fisher.osu.edu
† Tel.:+1-212
1. Introduction
Aumann and Serrano (2008) introduce an economic index of riskiness of gambles based on risk aversion. According to their definition, whether or not an individual takes a gamble depends on how risky the
gamble is and how averse the individual is to risk. Hence, increases in risk should affect more risk-averse
individuals more than they do less risk-averse individuals. This suggests that appropriate definitions of increases in risk and risk aversion should be closely linked. Aumann and Serrano (2008) define the riskiness
of a gamble as a function of the risk-aversion of an individual who is indifferent between accepting and
rejecting that gamble. Their riskiness index is positively homogeneous, continuous, and subadditive; respects first- and second-order stochastic dominance; and indicates that less-averse individuals accept riskier
gambles.
According to Aumann and Serrano (2008), if a gamble g is sure to yield more than h, it cannot be
considered riskier. For risk-averse investors who prefer less risky alternatives (all else equal), riskiness and
desirability are not in conflict, i.e., a less risky gamble is not always more desirable. That depends on the
investor and on other parameters in addition to riskiness, such as the mean, maximum loss, opportunities
for gain, and so on. Indeed, the decision depends on the whole distribution. Desirability is subjective:
depending on the investor, one may prefer gamble g to gamble h, whereas another prefers h to g. Riskiness,
however, is objective: it is the same for all individuals. Given two gambles, a more risk-averse individual
may well prefer the less risky gamble, whereas a less risk-averse individual may find that the opportunities
provided by the riskier gamble outweigh the risk involved.
In asset pricing literature, there is still an ongoing debate on how to quantify risk and how investors
choose among risky assets. Indeed, Aumann and Serrano (2008, p. 811) points out “The concept of risky
investment is commonplace in financial discussions and seems to have clear conceptual content. But when
one thinks about it carefully and tries to pin it down, it is elusive. Can one measure riskiness objectively independently of the person or entity taking the risk?”
In this paper, we relate expected future returns to riskiness, based on the conceptualization in Aumann
and Serrano (2008). We show that equity investments become less desirable when riskiness in the equity
market rises, and hence investors are less willing to hold equity or they demand extra compensation in
the form of higher expected return to accept equity investments in riskier times. Therefore, we expect a
positive relation between riskiness and expected returns.
We introduce a generalized measure of physical riskiness that nests the empirical measure proposed
1
by Aumann and Serrano (2008) based on the assumption of normality. Since the distribution of market
returns is typically skewed, peaked around the mean (leptokurtic) and has fat tails, we propose a measure
of aggregate riskiness for the U.S. equity market based on the mean, standard deviation, and higher order
moments of the empirical return distribution of the S&P 500 index.
In addition to the generalized measure of physical riskiness under the objective probability measure,
we propose option implied measures of riskiness based on the risk-neutral distribution of market returns.
We provide a model-independent measure of riskiness that can be obtained from the prices of S&P 500
index options and does not rely on any particular assumptions about the return distribution. Suppose an
investor needs to find a one-month ahead expected riskiness of a stock market portfolio. Under the physical
measure, riskiness can only be obtained from the past historical data (e.g., daily returns over the past one
year) and the investor has to use this historical measure to proxy for future riskiness. However, this physical
(or historical) measure may not provide an accurate characterization of the market’s expectation of future
riskiness. Using the prices of S&P 500 index options in the calculation of riskiness solves this problem
by making future riskiness observable because index option prices incorporate the market’s expectation of
future return distribution.
After introducing the option implied and physical measures of riskiness, we investigate their performance in predicting future returns on the U.S. equity market. The intertemporal relation between risk and
return in the aggregate stock market has been one of the most extensively studied topics in financial economics. Most asset pricing models postulate a positive relation between the market portfolio’s expected
return and risk, which is often defined by the variance or standard deviation of market returns. In his
seminal paper, Merton (1973) shows that the conditional expected return on the aggregate stock market
is a linear function of its conditional variance plus a hedging demand component that captures investors’
motive to hedge against unfavorable shifts in the investment opportunity set. Despite the importance of
the risk-return tradeoff and the theoretical appeal of Merton’s result, the asset pricing literature has not yet
reached an agreement on the existence of such a positive risk-return tradeoff.
This paper examines the intertemporal relation between the newly proposed measures of riskiness and
future returns on the aggregate stock market. We generate time-varying measures of aggregate riskiness
for the U.S. equity market based on the objective and risk-neutral probability measures. The physical measures of aggregate riskiness are estimated using the empirical return distribution of the S&P 500 index.
The risk-neutral measures of aggregate riskiness are obtained from the prices of S&P 500 index options.
The predictive regressions indicate a positive and significant relation between time-varying riskiness and
2
expected market returns. This result is somewhat stronger for the option implied measures of aggregate
riskiness compared to the physical measures. The significantly positive link between riskiness and equity
premium remains intact after controlling for the S&P 500 index option implied volatility, aggregate idiosyncratic volatility of individual stocks, and a large set of macroeconomic and financial variables associated
with business cycle fluctuations.
A large number of studies also investigate the intertemporal relation between macroeconomic variables
and market returns: Expected returns are found to be related to business cycle fluctuations (e.g., Keim and
Stambaugh (1986), Campbell and Shiller (1988), Fama and French (1988, 1989), Fama (1990), Kandel and
Stambaugh (1990), and Ferson and Harvey (1991)). Earlier studies find that risk premia on stocks covary
negatively with current economic activity: investors require higher (lower) expected returns in recessions
(booms). As a supporting evidence for the countercyclical behavior of expected returns, average stock
returns are found to be higher during periods of lower economic growth and after stock market declines.
We present a theoretical framework that justifies the positive link between aggregate riskiness and equity premium. Our empirical results not only confirm the positive theoretical relation between riskiness
and market returns, but they also provide evidence that increases in riskiness and risk aversion are closely
linked, consistent with the theoretical arguments of Aumann and Serrano (2008). In addition to the theoretical framework, we provide an alternative, macroeconomic based explanation for the strong positive
relation between riskiness and market risk premium by testing whether aggregate riskiness is higher during
economic downturns characterized by lower economic activity and higher expected returns. The results
indicate a significantly positive relation between time-varying measures of riskiness and lower economic
activity defined by the Chicago Fed National Activity Index and the Aruoba, Diebold, and Scotti (2009)
business conditions index. We also find that aggregate riskiness is higher when (i) the growth rate of nominal and real GDP is lower; (ii) the unemployment rate is higher; and (iii) aggregate default risk is higher.
These results provide a macroeconomic based explanation for our empirical finding that time-varying measures of riskiness positively predict future returns on the aggregate stock market.
Another potential explanation for the positive relation between aggregate riskiness and expected market
returns can be based on a time-varying or state-dependent nature of investors’ risk aversion. During large
falls of the market and periods of poor economic growth, aggregate risk aversion increases due to short sale,
liquidity, or financing constraints that hurt especially on the downside. The increased risk aversion implies
higher expected returns next period. In addition to the story due to constraints, the consumption-based
asset pricing model of Campbell and Cochrane (1999), the time-varying risk of rare economic disasters
3
introduced by Barro (2006, 2009), and the psychological factors or behavioral biases proposed by Black
(1988) provide further theoretical support for our empirical findings.
The remainder of the paper is organized as follows. Section 2 provides the original, physical measure
of riskiness developed by Aumann and Serrano (2008). Section 3 presents a generalized measure of physical riskiness. Section 4 introduces a risk-neutral option implied measure of riskiness. Section 5 provides a
theoretical framework that justifies the positive relation between aggregate riskiness and equity premium.
Section 6 contains the data and variable definitions. Section 7 investigates the significance of an intertemporal relation between aggregate riskiness and expected market returns. Section 8 tests whether aggregate
riskiness is higher during periods of lower economic activity. Section 9 concludes the paper.
2. The Original Concept of Riskiness
Aumann and Serrano (2008) assume a von Neumann-Morgenstern utility function for money which is
strictly monotonic, strictly concave, and twice continuously differentiable, and defined over the entire real
line. A gamble g is a random variable with real values – interpreted as dollar amounts – some of which are
negative, and that has positive expectation. That is, an individual with utility function u accepts a gamble
g at wealth w if E[u(w + g)] > u(w), where E stands for “expectation”.
Aumann and Serrano’s measure of riskiness is based on a “duality” axiom between riskiness and risk
aversion and positive homogeneity of degree one:
DUALITY: Roughly duality says that less risk-averse decision makers accept riskier gambles. Define
an index Q as a positive real-valued function on gambles and assume that gamble g is riskier than gamble
h, i.e., Q(g) > Q(h). If an individual i is more risk-averse than individual j, then whenever the individual i
accepts g at some wealth w, and Q(g) > Q(h), then the individual j accepts h at w.
POSITIVE HOMOGENEITY: Positive homogeneity represents the cardinal nature of riskiness, i.e.,
Q(tg) > tQ(g) for all positive numbers t. If g is a gamble, positive homogeneity implies that 2g is at least
“twice as” risky as g, not just “more” risky.
Aumann and Serrano (2008) show that for each gamble g, there is a unique positive number R(g) with
[
(
)]
g
E exp −
=1
R (g)
(1)
The index of riskiness denoted by R (g) in eq. (1) satisfies duality and positive homogeneity. Aumann and
4
Serrano (2008) consider an agent with constant absolute risk aversion (CARA) coefficient γ, who is indifferent between accepting and rejecting g. Applying eq. (1) to the CARA utility function, u (x) = − exp(−γx),
gives R (g) = 1γ . However, this example neglects the distributional parameters of g. In Section 3, we will
take into account the empirical distribution of g when computing the physical measure of riskiness.
In this paper, we focus on the riskiness measure of Aumann and Serrano (2008) to investigate the significance of a positive link between riskiness and equity premium. In addition to Aumann and Serrano
(2008), the interested reader may wish to consult Foster and Hart (2009, 2010) and Hart (2011) for further
understanding of the recently developed measures of riskiness. There is also continued research on riskiness: Bali, Cakici, and Chabi-Yo (2011) introduce a generalized measure of riskiness that nests the original
measures proposed by Aumann and Serrano (2008) and Foster and Hart (2009). Bakshi, Chabi-Yo, and
Gao (2011) develop a theoretical model in which investors may reduce their holdings in risky assets when
the change in riskiness is higher. Kadan and Liu (2011) use the riskiness measures as performance indices
for equity portfolios.
3. A Generalized Measure of Physical Riskiness
While R(g) represents an objective measure of riskiness, the decision to reject or accept risky assets
depends on risk aversion. For an investor with relative risk aversion γ and some initial wealth W0 , it is
shown in Aumann and Serrano (2008, Eq 4.3.1, p. 817) that the investor will
accept a risky asset with payoff W0 g if W0 + min{W0 g} > γR (W0 g)
(2)
W0 g if W0 + max{W0 g} < γR (W0 g)
(3)
reject a risky asset with payoff
where R (W0 g) = W0 R (g), and R (g) is defined in equation (1). The riskiness of a risky asset depends on the
distribution of the asset’s return. To understand better how riskiness relates to characteristics of the risky
assets,
(
)
√
• Assume that g ∼ N E [g] , Var [g] , it can be shown that
[
(
)]
(
)
g
E [g] Var [g]
E exp −
= exp −
+ 2
.
R (g)
R (g) 2R (g)
5
(4)
Combining this expression with Equation (1), it follows that
R (g) =
1 Var [g]
.
2 E [g]
(5)
• Assume that g follows a skew-normal distribution with a shape parameter νg , a location parameter
µg , and a scale parameter σg : g ∼ SN (µg , σg , νg ). The skew normal distribution is defined in Azzalini
(1985). The distribution is right skewed if νg > 0 and is left skewed if νg < 0. The normal distribution
1
is recovered when νg = 0, which also implies µg = E [g] and σg = (Var [g]) 2 .
Lemma 1 Assume that g follows a skew-normal distribution: gSN (µg , σg , νg ). It can be shown that
 

[
]
)2
νg 
1 (
log (E [exp (κ1 g)]) = 2 κ1 σ2g + µg − µ2g + log 2Φ κ1 √
,
2σg
1 + ν2
(6)
g
where Φ is the cumulative distribution function of a normal N (0, 1).
1
We use Lemma 1, with κ1 = − R(g)
and show that

 
]
[(
( [
(
)])
)2
νg 
g
1
1 2
1
.
log E exp −
= 2
−
σ + µg − µ2g + log 2Φ −
.√
R (g)
2σg
R (g) g
R (g) 1 + ν2
g
(7)
( [
(
)])
g
Using the definition of the riskiness measure (see equation (1)), log E exp − R(g)
= 0, and
equation (6) reduces to
(
1
R (g)
 

)
σ2g
ν
1
g
 = 0.
− µg + log 2Φ −
.√
2R (g)
R (g) 1 + ν2
(8)
g
The riskiness measure R (g) is solution to (7). Under a skew normal distribution, equation (7) shows
clearly that the riskiness R (g) depends on µg , σg and the parameter νg that characterizes the skewness of the risky asset. Since changes in macroeconomic conditions (such as recessions and market
crashes) affect the distribution of assets in place (via νg ), one may argue that changes in macroeconomic conditions lead to changes in riskiness, and hence affects investor’s decision to accept, reject,
increase, or decrease her investment in risky assets.
Aumann and Serrano (2008) assume a Normal distribution (with riskiness given in equation (5)) to illustrate the meaning of their riskiness measure. Since the empirical distribution of stock returns is typically
skewed and thick-tailed, we use a Taylor series expansion and propose a measure of aggregate riskiness for
6
the U.S. equity market based on the mean, standard deviation, and higher order moments of the empirical
return distribution of the S&P 500 index. Under the physical measure, the Aumann and Serrano (2008)
riskiness measure RAS,P is solution to:
(
(
Et exp −
))
gt+1
RAS,P
i,t
= 1.
(9)
(
)
gt+1
The Taylor expansion series of exp − AS,P around the expected value of g produces
Ri,t
(
exp −
)
gt+1
RAS,P
i,t
(
/ exp −
Et (gt+1 )
)
RAS,P
i,t
≃ 1−
−
1
RAS,P
i,t
(gt+1 − Et (gt+1 )) +
1
1
2
(
) (gt+1 − Et (gt+1 )) (10)
2! AS,P 2
Ri,t
1
1
1
1
3
4
(
)3 (gt+1 − Et (gt+1 )) + (
) (gt+1 − Et (gt+1 ))
3! AS,P
4! AS,P 4
Ri,t
Ri,t
We apply the expectation operator to (9) and deduce
(
1 − exp
+
Et (gt+1 )
RAS,P
i,t
)
+
3
1
1
1
1
(
)2 Var [gt+1 ] − (
)3 (Var [gt+1 ]) 2 SKEW [gt+1 ]
2! AS,P
3! AS,P
Ri,t
Ri,t
(11)
1
1
2
(
) (Var [gt+1 ]) KURT [gt+1 ] = 0
4! AS,P 4
Ri,t
where Var [gt+1 ], SKEW [gt+1 ], and KURT [gt+1 ] represent the variance, skewness, and kurtosis of asset i
return. The Aumann and Serrano measure of riskiness RAS,P
is solution to (10).
i,t
To estimate the generalized measures of physical riskiness for each month from January 1996 to October 2010, we first compute the mean, variance, skewness, and kurtosis of daily returns on the S&P 500
index over the past 1, 3, 6, and 12 months and then numerically back out the physical measure of riskiness
from equation (10).
4. Option Implied Measures of Riskiness
This paper contributes to the existing literature by introducing the risk-neutral option implied measures
of riskiness based on the Bakshi, Kapadia, and Madan (2003) spanning formula. They show that any
7
function of the form H(S) with E[H(S)] < ∞ can be spanned by as a collection of call and put options:
[ ] (
) [ ] ∫
H [S] = H S + S − S Hs S +
∞
S
+
HSS [K] (S − K) dK +
∫ S
0
HSS [K] (K − S)+ dK
(12)
where HS (.) and HSS (.) represent the first and second derivative of H with respect to S. We denote
gt+τ =
Si (t, τ) − Si (t)
Si (t)
(13)
the return on the risky asset i with an investment horizon τ. Si (t, τ) is the price of the individual security
at time t + τ, and Si (t) is the price of the individual security at time t. In Proposition (1), we derive a
model-independent measure of riskiness from option prices.1
Proposition 1 Let RAS
i,t be the riskiness measure of the risky asset with return-payoff gt+τ . Let C (Si (t) , K, τ)
be the price at time t of the call option with strike price K, maturity τ. Let P (Si (t) , K, τ) be the price at time
t of the put option with strike price K, maturity τ. Denote 1 + r f (t, τ) the return on the risk-free security.
RAS
i,t is the fixed point solution to (13)
r f (t, τ) 1
=
1 + r f (t, τ) RAS
i,t
∫ ∞
Si (t)
fRAS
[K]C (Si (t) , K, τ) dK +
i,t
∫ Si (t)
where
fRAS
[K] = (
i,t
−
1
)2 e
AS
Ri,t Si (t)
1
RAS
i,t
0
fRAS
[K] P (Si (t) , K, τ) dK
i,t
(14)
(K−Si (t))
Si (t)
.
(15)
Proof: Section I of the online appendix.
The advantage of the option implied measure in (13) is that it can be computed using option prices at
any time, and it is not model-dependent. We also provide in Section II of the online appendix, the option
implied measure of riskiness when the return on the underlying assets are defined in terms of log returns.
5. A Theoretical Framework for the Positive Riskiness-Return Tradeoff
In this section, we first discuss key properties of the Aumann and Serrano riskiness measure when it
is applied to returns. As shown in Auman and Serrano (2008), when it is applied to a gamble, their riski1 Aumann
and Serrano (2008) develop their physical measure of riskiness R relative to a gamble g interpreted as a random
dollar amount, whereas we introduce option implied measures of riskiness by applying R to returns on g, not to dollar amounts.
We should note that our newly proposed measure of riskiness retains the properties of R shown by Aumann and Serrano.
8
ness measure preserves desirable properties: “Unicity”, “Duality”, “Positive Homogeneity”, “Monotonicity with Respect to Stochastic Dominance”, “Continuity”, “Diluted Gambles”, and “Compound Gambles”.
To show that our riskiness measure preserves key properties as well, we need to demonstrate that
net returns ( St+1St−St ) can be viewed as payoff of a gamble. We recall that, by definition, a gamble is a
random variable that takes negative values and has positive expectation. Net returns ( St+1St−St ) take negative
values and have positive expectation because the average rate of return on the market portfolio is positive:
[
]
[ ]
f
E St+1St−St > E rt . Notice that the riskiness measure of Aumann and Serrano cannot be applied to gross
payoffs (St+1 ) because gross payoffs take positive values in all states of the world.
Second, we use a simple equilibrium model to provide theoretical and empirical evidence for the positive relation between riskiness (RtAS ) and the market return (Retm,t+1 ). Assume that zt is a time-series
predictor. Following Kirby (1998, Equation (8), page 348), the slope β in the predictive regression:
Retm,t+1 = α + βzt + εt+1 ,
can be written as
)
(
Retm,t+1 (zt − E (zt ))
.
β = −Cov mt+1 ,
Var [zt ]
(16)
where Retm,t+1 = (St+1 − St ) /St . In equilibrium, the sign of β is determined by the correlation between the
(
( ))
Stochastic Discount Factor mt+1 and Retm,t+1 RtAS − E RtAS . In our framework, zt = RtAS . The sign of
the left hand side of (15) which is the slope in our predictive regressions must be consistent with the sign
of the right hand side of (15). While any SDF can be used, we focus on a simple and well-known SDF
and investigate whether the sign of the slope in our predictive regression is consistent with asset pricing
models in equilibrium. In an environment with a representative agent with a CRRA utility who maximizes
her expected utility, the SDF takes the form
(
mt+1 =
ct+1
ct
)−γ
,
where ct+1 /ct is the consumption growth and γ is the risk aversion. To empirically investigate whether the
sign of β in the predictive regression is consistent with asset pricing models in equilibrium, one needs to
empirically check whether the sign of the slope in our predictive regressions is consistent with (16)
((
−Cov
ct+1
ct
)−γ
, Retm,t+1
9
(
)
))
RtAS − E RtAS
(
(17)
To test whether equation (16) is positive, we obtain monthly data on the U.S. aggregate consumption expenditures from the Bureau of Economic Analysis. The original data are seasonally adjusted and
cover the sample period from January 1959 to December 2010. Earlier studies estimate the risk aversion
parameter (γ) in the range of 2 to 4 (see, e.g., Lundblad (2007), Bali and Engle (2010)).
)−γ for the risk aversion parameter values of γ = 2, 3, and 4 over the
Hence, we compute mt+1 = ( CCt+1
t
sample period January 1996 - October 2010. For each measure of riskiness (1-, 3-, 6-, and 12-month), we
(
( ))
compute Retm,t+1 RtAS − E RtAS using the S&P 500 index as a proxy for the U.S. equity market.
(
( ))
Finally, we calculate the sample covariance between ( CCt+1
)−γ and Retm,t+1 RtAS − E RtAS . As pret
sented in Table I of the online appendix, the covariance estimates are negative for all values of γ, and for all
measures of riskiness, confirming the positive intertemporal relation between riskiness and market returns.
Another notable point in Table I is that the magnitude of covariances (in absolute terms) increases as the
risk aversion parameter increases from 2 to 4. Consistent with the theoretical arguments of Aumann and
Serrano (2008), our results provide supporting evidence that increases in riskiness and risk aversion are
closely linked.
6. Data and Variable Definitions
In this section, we first describe the S&P 500 index options data used to estimate the option implied
measures of riskiness. Second, we present figures and descriptive statistics of the option implied and
physical measures of riskiness. Third, we provide summary statistics of the U.S. equity market indices.
Finally, we describe the control variables used in predictive regressions.
6.1. S&P 500 index options data
The daily data on call and put option prices for the S&P 500 index, and the corresponding strikes,
maturities, and volatilities are obtained from OptionMetrics. The OptionMetrics Volatility Surface computes the interpolated implied volatility surface separately for puts and calls based on a kernel smoothing
algorithm using options with various strikes and maturities. The volatility surface data contain prices and
implied volatilities for a list of standardized options for constant maturities and deltas. A standardized option is only included if there exists enough underlying option price data on that date to accurately compute
an interpolated value. The interpolations are done each day so that no forward-looking information is used
10
in computing the volatility surface. One advantage of using the Volatility Surface is that it avoids having
to make potentially arbitrary decisions on which strikes or maturities to include when computing option
implied measures of riskiness. To be consistent with Bakshi, Kapadia, and Madan (2003) methodology,
we use out-of-the-money call and put option prices with expirations of 1, 3, 6, and 12 months to estimate
option implied measures of riskiness for the S&P 500 index. In Volatility Surface, at-the-money call (put)
options have a delta of 0.50 (-0.50). Out-of-the-money call options have delta of 0.20 to 0.50 and out-ofthe-money put options have delta of -0.20 to -0.50. We use the longest sample available from January 1996
to October 2010.
6.2. Option implied and physical measures of riskiness
Figure 1 shows the option implied measures of aggregate riskiness obtained from the S&P 500 index
options with 1, 3, 6, and 12 months to maturity. These riskiness measures represent the 1-, 3-, 6-, and 12month ahead “expected riskiness” of the U.S. equity market. A notable point in Figure 1 is that the option
implied measures of riskiness are highly correlated with each other and they all present significant timeseries variation. In particular, aggregate riskiness is extremely high during the recent financial crisis period
(2008 – 2010). Another notable point in Figure 1 is that the short-term expected riskiness of the stock
market (e.g., 1-month ahead riskiness) is higher and more volatile than the market’s long-term expected
riskiness (e.g., 12-month ahead riskiness).
Table 1, Panel A presents the descriptive statistics of the option implied measures of aggregate riskiness
for the sample period January 1996 – October 2010. The average option implied measures of riskiness
are about 2.12, 1.59, 1.44, and 1.22 for 1-, 3-, 6-, and 12-month horizons. The corresponding standard
deviations are about 3.97, 2.46, 2.13, and 1.66. These results indicate that when investors feel a great deal
of uncertainty about future market returns, they will be correspondingly uncertain about how much change
to expect in short-term returns. However, investors will not change their expectations as much about the
long-term returns. Hence, aggregate riskiness of the market will be higher and more volatile over the next
month (short-term riskiness) as compared to aggregate riskiness of the market over the next 12 months
(long-term riskiness). Similarly, when investors feel a great deal of uncertainty about financial markets
and state of the economy, their risk aversion will be higher in the short-run. In other words, economic
downturns or uncertainty about output growth, unemployment, and default risk may frighten people and
cause them to withdraw from the market in the short-run. Therefore, aggregate risk aversion is more likely
to increase as a result of unexpected changes in the short-run.
11
Figure 2 displays the physical measures of riskiness obtained from daily returns on the S&P 500 index
over the past 1, 3, 6, and 12 months. Similar to our earlier findings in Figure 1, the physical measures
also exhibit significant time-series variation. Although the level and standard deviation of the physical
measures are lower compared to the option implied measures of riskiness, the physical measures also reach
extremely high values during the recent financial crisis period (2008 – 2010). Another notable point in
Figure 2 is that the short-term riskiness of the market under the objective probability measure is also higher
and more volatile than the market’s long-term riskiness.
Panel A of Table 1 reports the descriptive statistics of the physical measures of riskiness for the period
January 1996 – October 2010. The average physical measures of riskiness obtained from the past 1, 3,
6, and 12 months of daily data are in the range of 0.36 to 0.38. The standard deviations of the physical
measures are in the range of 0.37 and 0.61. Although not as strong as in the option implied measures
of riskiness, the results in Panel A indicate that the historical (objective) measure of riskiness is higher
and more volatile in the short-run, consistent with the idea that aggregate risk aversion is more likely
to increase as a result of unexpected changes in returns and risk in the short-run. Panel A of Table 1
also presents descriptive statistics for the S&P 500 index option implied variance (VIX). Similar to the
riskiness measures, the implied volatility is also skewed to the right and has excess kurtosis. The VIX is
highly persistent as well, with the first-order serial correlation of 0.84. Another notable point in Panel A is
that the option implied measures of riskiness are more persistent than the physical measures and the VIX.
Panel B of Table 1 presents the correlations among the option implied and the physical measures of
riskiness and the VIX. As shown in Figure 1, the riskiness measures obtained from the S&P 500 index
options are highly correlated with each other; the correlations are in the range of 0.91 to 0.99. Although
the correlations among the physical measures are not as high, they are still considerably large, in the range
of 0.46 and 0.86. The physical measures obtained from the past 6 and 12 months of daily data are highly
correlated with the option implied measures of riskiness; the correlations are in the range of 0.50 to 0.78.
However, the short-term physical measures obtained from the past 1 and 3 months of daily data have
relatively low correlations with the option implied measures of riskiness; the correlations are in the range
of 0.27 to 0.55. These results also suggest that the physical riskiness measures obtained from the past 6 and
12 months of daily returns have smaller measurement errors because of the larger sample used to compute
the mean, standard deviation, and higher order moments of the return distribution.
Another notable point in Panel B is that the correlations between the option implied measures of riskiness and the VIX are in the range of 0.43 to 0.56. Although riskiness is positively associated with the
12
option implied volatility (VIX), the correlation is not as high as one would expect because riskiness takes
into account ex-ante expected measures of option implied skewness and kurtosis. Hence, riskiness provides
a broader measure of risk in the equity market.
6.3. U.S. equity market indices
The aggregate stock market portfolio is proxied by the U.S. equity market indices. Specifically, we
use the value-weighted and equal-weighted CRSP indices that contain all stocks trading at NYSE, AMEX,
and NASDAQ. In addition to these broad stock market indices, we use the S&P500, Dow Jones Industrial
Average (DJIA), and NASDAQ as well.
Table 2 presents the descriptive statistics of the monthly excess returns on these equity market indices
for the sample period January 1996 – October 2010. As expected, the average excess return on the equalweighted CRSP (EW CRSP) index is higher (0.84% per month) than the average excess return on the
value-weighted CRSP (VW CRSP) index (0.42% per month) because the equal-weighted index gives more
weight to small stocks with higher average returns. Similarly, the average excess return on the NASDAQ
containing relatively small stocks is higher than the average excess returns on the S&P500 and Dow 30
indices containing the large 500 and 30 stocks, respectively. As expected, the equal-weighted CRSP and
NASDAQ indices containing relatively more illiquid and smaller stocks are more volatile than the valueweighted CRSP, S&P500, and DJIA. Specifically, the unconditional standard deviations are 6.17% and
7.70% per month for the EW CRSP and NASDAQ, respectively. Whereas, the standard deviations are,
respectively, 4.93%, 4.72%, and 4.62% per month for VW CRSP, S&P500, and DJIA.
One common characteristic of the stock market indices in Table 2 is that their return distributions
are skewed to the left and they have excess kurtosis. The significance of departures from normality is
determined by the Jarque-Bera (JB) statistic: JB = n[(S2 /6) + (K–3)2 /24], where n denotes the number of
observations, S is skewness and K is kurtosis. The JB statistics reported in Table 2 provide strong evidence
that the monthly returns on the U.S. equity market indices are not normally distributed. Consistent with
earlier studies, the empirical distribution of market returns is negatively skewed and fat-tailed. These results
also justify our newly proposed measures of riskiness that take into account the mean, standard deviation,
and higher order moments of the physical and risk-neutral distributions.
13
6.4. Control variables
We investigate the predictive power of aggregate riskiness in forecasting future market returns after
controlling for a wide variety of volatility, macroeconomic and financial variables. Our control variables
include:
1. VIX, a ticker symbol for the Chicago Board Options Exchange (CBOE) market volatility index.
VIX is a popular measure of the implied volatility of S&P 500 index options. Often referred to as
the fear index, it represents one measure of the market’s expectation of stock market volatility over
the next 30 day period.
2. Aggregate idiosyncratic volatility (IVOL) defined as the value-weighted average of idiosyncratic
volatility of individual stocks (see, e.g., Goyal and Santa-Clara (2003), Bali, Cakici, Yan, and Zhang
(2005), and Guo and Savickas (2006)). For each stock trading at NYSE, AMEX, and NASDAQ,
we first run the CAPM regression using daily returns in a month. Then, idiosyncratic volatility
of individual stocks is defined as the standard deviation of residuals from daily return regressions.
Aggregate idiosyncratic volatility is defined as the value-weighted average idiosyncratic volatility
using market capitalization weights.
3. Realized market variance (RVAR) defined as the sum of squared 5-minute returns on the S&P 500
index in a month. The monthly RVAR data are obtained from Hao Zhou’s website:
http://sites.google.com/site/haozhouspersonalhomepage/.
4. The variance risk premium (VRP) defined as the difference between expected variance under the
risk-neutral measure and expected variance under the objective measure (Bollerslev, Tauchen, and
Zhou (2009)). The monthly VRP data are obtained from Hao Zhou’s website.
5. The default spread (DEF) defined as the difference between the monthly yields on BAA- and AAArated corporate bonds.
6. The term spread (TERM) defined as the difference between the 3-month Treasury bill and the 10year Treasury yields.
7. The relative T-bill rate (RREL) is defined as the difference between the 3-month T-bill rate and its
12-month backward moving average.2
8. The dividend-price ratio for the S&P 500 index (DIV). The monthly data are available at Robert
2 The monthly data on the 10-year Treasury yields, 3-month Tbill rate, BAA- and AAA-rated corporate bond yields are available
at the Federal Reserve website: http://www.federalreserve.gov/releases/h15/data.htm.
14
Shiller’s website: http://www.econ.yale.edu/shiller/data.htm.
9. Monthly growth rate of the U.S. industrial production (IP) obtained from the G.17 database of the
Federal Reserve Board.
10. Monthly U.S. unemployment rate (UNEMP) obtained from the Bureau of Labor Statistics.
11. Monthly consumption-wealth ratio (CAY) following Lettau and Ludvigson (2001). The original
quarterly data on CAY are available at Sydney Ludvison’s website:
http://www.econ.nyu.edu/user/ludvigsons/. A linear interpolation is used to convert quarterly data to
monthly frequency.
7. Empirical Results
In this section, we first investigate the intertemporal relation between the option implied measures of
riskiness and equity premium. Second, we examine the predictive power of physical riskiness in forecasting
market risk premium. Finally, we test whether the riskiness premium is as informative as the riskiness itself
when predicting future market returns.
7.1. Intertemporal relation between option implied riskiness and future market returns
Dynamic asset pricing models starting with Merton’s (1973) ICAPM provide a theoretical framework
that gives a positive equilibrium relation between the conditional first and second moments of excess returns on the aggregate market portfolio. However, many studies fail to identify a statistically significant
intertemporal relation between risk and return of the market portfolio. French, Schwert, and Stambaugh
(1987) find that the coefficient estimate is not significantly different from zero when they use past daily
returns to estimate the monthly conditional variance. Goyal and Santa-Clara (2003) obtain similar insignificant results using the same variance estimator. Glosten, Jagannathan, and Runkle (1993) use monthly data
and find a negative but statistically insignificant relation from the asymmetric GARCH-in-mean models.
Harrison and Zhang (1999) find a significantly positive risk-return tradeoff at one-year horizon, but they
do not find a significant relation at shorter holding periods such as one month. Using a sample of monthly
returns and implied and realized volatilities for the S&P 500 index, Bollerslev and Zhou (2006) find an
insignificant intertemporal relation between expected return and realized volatility, whereas the relation
between return and implied volatility turns out to be significantly positive.
15
Several studies even find that the intertemporal relation between risk and return is negative. Examples
include Campbell (1987), Breen, Glosten, and Jagannathan (1989), Turner, Startz, and Nelson (1989),
Nelson (1991), Glosten, Jagannathan, and Runkle (1993), Whitelaw (1994), and Harvey (2001). Using a
regime switching model, Whitelaw (2000) finds a negative unconditional relation between the mean and
variance of excess returns on the market portfolio. Using a latent vector autoregression approach, Brandt
and Kang (2004) show that although the conditional correlation between the mean and volatility of market
portfolio returns is negative, the unconditional correlation is positive due to the lead-lag correlations.
Some studies do provide evidence supporting a positive risk-return relation. Using a multivariate
GARCH-in-mean model, Bollerslev, Engle, and Wooldridge (1988) find an economically small but statistically significant risk-return tradeoff. Ghysels, Santa-Clara, and Valkanov (2005) introduce a new variance estimator that uses past daily squared returns, and find that the monthly data are consistent with a
positive relation between conditional expected excess return and conditional variance. Guo and Whitelaw
(2006) develop an asset pricing model based on Merton’s ICAPM and Campbell and Shiller (1988) loglinearization method, and find a positive relation between stock market risk and return. Using a long history
of monthly data, Lundblad (2007) estimates alternative specifications of the GARCH-in-mean model, and
finds a positive and significant risk-return tradeoff. Using a large sample of time-series and cross-sectional
data, Bali (2008) and Bali and Engle (2010) identify a positive and significant relation between expected
return and risk on equity portfolios in a multivariate GARCH framework.
In this paper, we examine the performance of aggregate riskiness in predicting future returns on the U.S.
equity market. Specifically, we estimate the time-series predictive regressions of one-month ahead excess
market returns on the option implied measures of riskiness with controlling for a large set of variables:
(
)
Retm,t+1 = α + β RtAS,Q + λXt + εm,t+1 ,
(18)
where Retm,t+1 denotes the excess market return in month t+1, RtAS,Q is the option implied Q-measure of
aggregate riskiness in month t, and Xt includes a large set of control variables: the S&P 500 index option implied volatility (VIX), aggregate idiosyncratic volatility (IVOL), default spread (DEF), term spread
(TERM), relative T-bill rate (RREL), dividend-price ratio (DIV), growth rate of industrial production (IP),
unemployment rate (UNEMP), consumption-to-wealth ratio (CAY), and the lagged excess market return
(RET).
Panel A of Table 3 presents results for the value-weighted and equal-weighted CRSP indices that con-
16
tain all stocks trading at NYSE, AMEX, and NASDAQ. For the value-weighted CRSP index, the slope
coefficient on RtAS,Q is estimated to be positive, in the range of 0.0066 to 0.0223, and highly significant
with the Newey-West (1987) t-statistics ranging from 2.28 to 3.29. Similar results are obtained for the
equal-weighted CRSP index as well: The slope coefficients on RtAS,Q are positive, in the range of 0.0067 to
0.0256, and statistically significant. As shown in Panel A, the significantly positive link between aggregate
riskiness and future market returns is robust across all measures of option implied riskiness. These results
indicate that when riskiness rises and investors choose to reject equity investments (or divest equity), then
such de-risking behavior is accompanied by higher subsequent returns.
A notable point in Panel A of Table 3 is that the S&P 500 index option implied volatility (VIX) and
aggregate idiosyncratic volatility do not predict one-month ahead returns on the market portfolio. For
the value-weighted CRSP index, there is a positive but statistically insignificant intertemporal relation between VIX and expected returns. For the equal-weighted CRSP index, the intertemporal relation between
VIX and future market returns is negative but statistically weak. There is no significant predictive relation between aggregate idiosyncratic volatility and market returns either. For both the value-weighted
and equal-weighted CRSP indices, the intertemporal relation between IVOL and future market returns is
negative but statistically weak.
These results are different from earlier studies providing evidence that the option implied variance
measured by VIX is positively and significantly related to future returns on the market portfolio. The
significantly positive link between VIX and future market returns is previously documented by Guo and
Whitelaw (2006) and Bollerslev and Zhou (2006) for different sample periods (not for the 1996-2010
period). Earlier studies do not provide conclusive results on the predictive power of average volatility of
individual stocks. Goyal and Santa-Clara (2003) find a positive and significant link between the equalweighted average stock volatility and future market returns, whereas Bali, Cakici, Yan, and Zhang (2005)
find an insignificant relation between the value-weighted average stock volatility and future market returns.
Guo and Savickas (2006) show that when the value-weighted idiosyncratic risk and aggregate stock market
volatility are jointly used to predict one-quarter ahead returns, the stock market risk-return relation is
positive, but the value-weighted idiosyncratic risk is negatively related to future market returns.
Since the significantly positive link between aggregate riskiness and future market returns remains intact after controlling for VIX and IVOL, the option implied riskiness clearly dominates the option implied
volatility and aggregate idiosyncratic volatility in terms of predicting future returns. However, since the
option implied riskiness and volatility are correlated, we investigate this issue further in the online ap-
17
pendix. In Table II of the online appendix, we examine the predictive power of riskiness without VIX in
the predictive regressions but keeping all other control variables. The results show a positive and highly
significant relation between riskiness and future market returns without VIX in the predictive regressions.
In Table III of the online appendix, we investigate the predictive power of VIX without the option implied riskiness in the predictive regressions but keeping all other control variables. We find a positive and
marginally significant relation between VIX and future market returns for the value-weighted CRSP, S&P
500, and NASDAQ indices, whereas the relation is positive but insignificant for the equal-weighted CRSP
and DJIA indices. In other words, the option implied volatility (VIX) itself is not as strong as the option
implied riskiness in our sample period 1996-2010.
Among the control variables, dividend yield (DIV), the growth rate of industrial production (IP), unemployment rate (UNEMP), and lagged return (RET) have some predictive power. Specifically, the future
returns on the value-weighted CRSP index are positively related to DIV and IP, and negatively linked with
UNEMP. The lagged excess return seems to be positively related but, its effect is weak for the valueweighted CRSP index. For the equal-weighted CRSP index, there is a significantly negative relation between future returns and UNEMP, whereas the relations between future returns and DIV and IP are positive
but weak statistically. The lagged excess return has significant predictive power for the equal-weighted
CRSP index, indicating positive serial correlation in monthly returns of small stocks. The other control
variables including the default spread, term spread, relative T-bill rate, and consumption-to-wealth ratio do
not seem to have robust, significant predictive power for one-month ahead returns.
Panel B of Table 3 provides a robustness check by presenting results for alternative stock market indices. For the S&P500, DJIA, and NASDAQ indices, the slope coefficients on the option implied measures
of riskiness are found to be positive and statistically significant. Similar to our earlier findings for the CRSP
index, the significantly positive link between aggregate riskiness and future market returns remains intact
for all measures of riskiness. Also, there is no significant relation between VIX, IVOL and market returns.
For the S&P500, NASDAQ, and DJIA indices, the intertemporal relation between VIX (IVOL) and future
market returns is positive (negative) but statistically insignificant.
7.2. Intertemporal relation between physical riskiness and future market returns
In this section, we investigate the predictive power of physical riskiness in forecasting future market
returns. We estimate the predictive regressions of one-month ahead excess market returns on the physical
18
measures of riskiness after controlling for a large number of variables:
(
)
Retm,t+1 = α + β RtAS,P + λXt + εm,t+1 ,
(19)
where Retm,t+1 denotes the excess market return in month t+1, RtAS,P is the physical, objective P-measure
of riskiness in month t, and Xt denotes a vector including the control variables: the physical measure
of market variance (RVAR), aggregate idiosyncratic volatility (IVOL), and a set of macroeconomic and
financial variables used in equation (17).3
Table 4, Panel A presents results for the value-weighted and equal-weighted CRSP indices. When
we use the past 1 month and 3 months of daily data in estimating the physical measures of riskiness,
we find no evidence for a significant link between riskiness and equity premium. However, when the
physical measures are estimated using the past 6 and 12 months of daily data, we generally find a positive
and significant link between the objective measure of riskiness and future market returns. As discussed
earlier, the physical riskiness measures obtained from the past 6 and 12 months of daily data have smaller
measurement errors because of the larger sample used to compute the mean, standard deviation, and higher
order moments of the return distribution. This drives the significant predictive power of the 6-month
and 12-month historical measures of riskiness. Panel B of Table 4 provides very similar findings for the
S&P500, DJIA, and NASDAQ indices. There is a positive and significant relation between market risk
premium and the 6-month and 12-month measures of physical riskiness, whereas the 1-month and 3-month
historical measures do not predict future market returns.
Among the control variables, only dividend yield (DIV) and the growth rate of industrial production
(IP) have a significant link with future market returns although their predictive power is sensitive to the
proxy for an equity market index. The other control variables do not have any forecasting ability for future
market returns.
We should note that the option implied measure of riskiness is limited by data availability of the S&P
500 index options (starting January 1996), but we are able to estimate much longer time-series of the
physical riskiness measure that relies on the empirical return distribution. In this section, we use daily
returns on the S&P 500 index over the past 1, 3, 6, and 12 months to estimate the physical measure of
riskiness for each month from January 1960 to October 2010. In Table IV of the online appendix, we
3 Since we test the predictive power of physical riskiness in equation (18), the risk-neutral measure of volatility (VIX) is
replaced by the physical measure of volatility (RVAR), proxied by the monthly realized variance of the S&P 500 index calculated
with high-frequency data (see Section 6.4).
19
investigate the predictive power of physical riskiness for the long sample period 1960-2010. Similar to our
findings from the options sample, 1996-2010, the results in Table IV indicate that when we use the past 1
month and 3 months of daily data in estimating the physical measures of riskiness, there is no significant
link between riskiness and equity premium. However, when the physical measures are estimated using the
past 6 and 12 months of daily data, there is a positive and significant link between the objective measure of
riskiness and future market returns. In fact, the predictive power of physical riskiness is somewhat stronger
for the long sample period 1960-2010.
7.3. Intertemporal relation between riskiness premium and future market returns
While the physical measure of riskiness under the objective probability measure (P) captures the actual
risk, the option implied measure of riskiness under the risk-neutral probability measure (Q) also incorporates the investors’ preference toward risk and the difference between the two roughly has an interpretation
as “riskiness premium”. We now test whether the riskiness premium is as informative as the riskiness
itself when forecasting future market returns. Specifically, we estimate the predictive regressions of onemonth ahead excess market returns on the spread between the option implied and the physical measures of
riskiness:
(
)
Retm,t+1 = α + β RtAS,Q − RtAS,P + λXt + εm,t+1 ,
(20)
where Retm,t+1 denotes the excess market return in month t+1, RtAS,Q − RtAS,P is the riskiness premium
in month t, and Xt denotes a vector including the control variables: the variance risk premium (VRP),
aggregate idiosyncratic volatility (IVOL), a set of macroeconomic and financial variables used in equation
(17).4
Table 5 presents results for the value-weighted and equal-weighted CRSP, S&P 500, NASDAQ, and
DJIA. After controlling for the variance risk premia and a large set of variables associated with business cycle fluctuations, the predictive regressions indicate a positive and significant relation between time-varying
measures of riskiness premium and expected market returns. This result holds for the CRSP, S&P 500 and
NASDAQ indices without any exception. For the Dow 30 index, we also find a strong relation between
expected returns and the 1-month and 3-month riskiness premium. However, the positive link between
expected returns on Dow Jones and the riskiness premium is marginally significant for 6-month and 124 Since
we test the predictive power of riskiness premium in equation (19), the risk-neutral measure of volatility (VIX) is
replaced by the variance risk premium (VRP), proxied by the difference between expected variance under the risk-neutral measure
and expected variance under the physical measure (see Section 6.4).
20
month horizons. Overall, we conclude that when predicting future market returns, the riskiness premium
is as informative as the riskiness itself.
8. Alternative Explanations for the Positive Relation between Riskiness and
Equity Premium
One possible explanation for the strong positive relation between riskiness and equity premium can be
based on a time-varying or state-dependent nature of the aggregate risk aversion. If the market declines
substantially, this could effectively raise risk aversion for investors because of constraints that bite on the
downside, e.g., short sale constraints, financing constraints due to collateral or even behavioral biases.
Consequently, the increased risk aversion leads to an increase in the next period’s expected return. In this
section, we will provide a macroeconomic based explanation of our empirical findings.
Aumann and Serrano (2008) propose a measure of riskiness based on investors’ risk tolerance. Risk
tolerance is one of the most important factors influencing investment decisions because it takes into account
investors’ ability to take risk. A conservative or risk averse investor would favor investments in which her
capital is preserved, whereas an aggressive investor can risk losing her investment to generate higher profits.
According to Aumann and Serrano (2008), aggregate riskiness is related to aggregate risk aversion of
market investors. Since aggregate risk aversion affects investors’ investment and consumption decisions, it
is natural to think that aggregate riskiness may potentially be related to macroeconomic activity. The option
implied and physical measures of aggregate riskiness introduced in the paper take into account time-series
variation in aggregate risk aversion and may potentially be linked to business cycle fluctuations.
We determine increases and decreases in real economic activity by relying on the Chicago Fed National
Activity Index (CFNAI index), which is a monthly index designed to assess production, consumption,
employment, and related inflationary pressure. The CFNAI is a weighted average of 85 existing monthly
indicators of national economic activity. It is constructed to have an average value of zero and a standard
deviation of one. Since economic activity tends toward trend growth rate over time, a positive index
reading corresponds to growth above trend and a negative index reading corresponds to growth below trend.
Since the underlying monthly macroeconomic data series are volatile, the monthly CFNAI index is also
quite volatile. The Chicago Fed generates the 3-month moving average of the CFNAI index (CFNAI MA3
index) to reduce the month-to-month volatility. In our empirical analyses, we use both the CFNAI and the
CFNAI MA3 indices.
21
In addition to the CFNAI index, we use the Aruoba-Diebold-Scotti (ADS, 2009) business conditions
index which is designed to track real business conditions at high frequency. Its underlying (seasonally
adjusted) economic indicators (weekly initial jobless claims; monthly payroll employment, industrial production, personal income less transfer payments, manufacturing and trade sales; and quarterly real GDP)
blend high- and low-frequency information and stock and flow data. The average value of the ADS index
is zero. Progressively bigger positive values indicate progressively better-than-average conditions, whereas
progressively more negative values indicate progressively worse-than-average conditions. We use the original daily data available at the Federal Reserve Bank of Philadelphia and generate the monthly ADS index
using the end-of-month daily ADS values and also by taking the averages of daily ADS values in a month.
Since both measures of the monthly ADS index generate very similar results, we report our key findings
from the end-of-month ADS index. To be consistent with the Chicago Fed’s 3-month moving average
index (CFNAI MA3 index), we also generate the 3-month moving average of the ADS index (ADS MA3).
A recession is a business cycle contraction, a general slowdown in economic activity. During recessions, many macroeconomic indicators vary in a similar way. Production, as measured by gross domestic
product (GDP), employment, investment spending, capacity utilization, household incomes, business profits, and inflation all fall, while bankruptcies and the unemployment rate rise. Hence, in addition to the
monthly CFNAI and ADS indices, we use more traditional measures of macroeconomic activity: (i) nominal and real GDP growth; (ii) unemployment rate; and (iii) default risk.
First, we obtain quarterly data on the nominal and real GDP from the Bureau of Economic Analysis.
Then, we use a linear interpolation assuming a constant month-to-month GDP growth in a quarter and
generate the monthly series of the nominal and real GDP growth. We acquire the monthly data on the U.S.
unemployment rate (UNEMP) from the Bureau of Labor Statistics. Finally, we collect the monthly data on
the BAA- and AAA-rated corporate bond yields from the Federal Reserve Board, and define the aggregate
default risk (DEF) as the difference between the monthly yields on BAA- and AAA-rated corporate bonds.
To provide a potential explanation for the positive relation between aggregate riskiness and expected
market returns, we now test whether aggregate riskiness is higher during economic downturns characterized
by lower economic activity and higher expected returns.
Table 6 reports the correlation matrix for the option implied and physical measures of aggregate riskiness, the CFNAI, CFNAI MA3, ADS, and ADS MA3 economic activity indices, the nominal and real
GDP growth, the unemployment rate, and the aggregate default risk for the period January 1996 – October
2010. Table 6 clearly indicates a significantly positive link between time-varying measures of riskiness
22
and lower economic activity defined by the CFNAI and ADS indices. Specifically, the correlations are
all negative and significant at the 1% level. The correlations reported in Table 6 also show that aggregate
riskiness is higher when (i) the growth rate of nominal and real GDP is lower; (ii) the unemployment rate
is higher; and (iii) aggregate default risk is higher.
We now test the significance of a contemporaneous relation between aggregate riskiness and economic
downturns using time-series regressions. Specifically, we estimate the regressions of economic activity
indices on the riskiness measures:
CFNAIt = α + βRtAS + εt
(21)
ADSt = α + βRtAS + εt ,
(22)
where RtAS denotes the options implied and the physical measures of riskiness. As shown in Panels A and
B of Table 7, there is a negative and significant relation between aggregate riskiness and the CFNAI, ADS
indices, implying a significantly positive link between riskiness and lower economic activity.5 As a further
robustness check, we run the above regressions using the nominal GDP growth, unemployment rate, and
default risk as a proxy for the economic and financial downturns. The highly significant t-statistics on the
slope coefficients and the large R2 values in Panels C, D, and E of Table 7 provide evidence that aggregate
riskiness is higher when output growth is lower, unemployment rate is higher, and default risk is higher.
Since aggregate risk aversion is higher during economic downturns, investors demand extra compensation in the form of higher expected return to take higher perceived risk during bad states of the economy.
Overall, these results provide a macroeconomic based explanation for our empirical finding that timevarying measures of riskiness positively predict future returns on the aggregate stock market. In Sections
III, IV, and V of the online appendix, we provide three alternative explanations based on the consumptionbased asset pricing models, the time-varying risk of rare economic disasters, and the psychological factors.
Consistent with these alternative explanations, in Section VI of the online appendix, we show that the
intertemporal relation between riskiness and future market returns is stronger during economic recessions.
9. Conclusion
Aumann and Serrano (2008) develop an objective measure of riskiness that looks for the critical utility
regardless of wealth. Their riskiness measure is introduced based on the physical return distribution of
5 At an earlier stage of the study, we also use the CFNAI MA3 and ADS MA3 indices in our regressions and the results turn
out to be very similar to those reported in Panels A and B of Table 7. They are available upon request.
23
risky assets. For illustrative purposes only, Aumann and Serrano (2008) present an empirical counterpart
of their riskiness measure based on the normal distribution. This paper introduces a generalized measure of
physical riskiness that nests the original, empirical riskiness measure of Aumann and Serrano (2008). Since
the distribution of market returns is typically skewed, leptokurtic, and has fat tails, we provide a measure of
aggregate riskiness for the equity market based on the mean, standard deviation, and higher order moments
of the empirical return distribution. In addition to the generalized measure of physical riskiness under the
objective probability measure, this paper develops a new measure of riskiness based on the risk-neutral
return distribution of underlying assets. Riskiness of an underlying financial security (e.g., equity market
index) is derived from the prices of derivative securities written on the underlying asset (i.e., prices of
call and put options on the S&P 500 index). The newly proposed forward-looking measures of riskiness
condense option implied risk-neutral probability distribution to a scalar and satisfy the monotonicity and
duality conditions.
We generate time-varying measures of aggregate riskiness for the U.S. equity market based on the objective and risk-neutral probability measures. The physical (objective) measures of aggregate riskiness are
estimated using daily returns on the S&P 500 index over the past 1, 3, 6, and 12 months. The option implied (risk-neutral) measures of aggregate riskiness are obtained from the prices of S&P 500 index options
with 1, 3, 6, and 12 months to maturity. After introducing the option implied and historical measures of
riskiness, we investigate their performance in predicting future returns on the U.S. equity market for the
period 1996-2010. The predictive regressions indicate a positive and significant relation between timevarying riskiness and expected market returns. The significantly positive link between riskiness and equity
premium remains intact after controlling for the S&P500 index option implied volatility (VIX), aggregate
idiosyncratic volatility, and a large set of macroeconomic and financial variables. These results indicate that
equity investments become less attractive when riskiness in the equity market rises, and hence investors are
less interested in holding equity or they demand extra compensation in the form of higher expected return
to accept equity investments in riskier times.
We present a theoretical framework that justifies the positive link between aggregate riskiness and equity premium. We also provide a macroeconomic based explanation for the positive relation between aggregate riskiness and expected market returns by showing that aggregate riskiness is higher during economic
downturns characterized by lower economic activity and higher expected returns. The results indicate a
significantly positive relation between riskiness and lower economic activity defined by the Chicago Fed
National Activity Index and the Aruoba, Diebold, and Scotti (2009) business conditions index. We also
24
find that aggregate riskiness is higher when GDP growth is lower, unemployment rate is higher, and default risk is higher. Consistent with the original definition of Aumann and Serrano (2008), higher riskiness
corresponds to periods with higher risk aversion and the increased risk aversion implies higher expected
returns next period.
We explore alternative explanations for the positive link between riskiness and equity premium based
on a time-varying or state-dependent nature of investors’ risk aversion. The consumption-based asset pricing model of Campbell and Cochrane (1999), the time-varying risk of rare economic disasters introduced
by Barro (2006, 2009), and the psychological factors proposed by Black (1988) provide evidence that
risk-aversion is higher during recessionary periods and large falls of the market, and more risk-averse individuals are more reluctant to invest on riskier assets, so they demand extra compensation in the form of
higher expected return to induce them to take higher perceived riskiness during economic downturns. Since
increases in riskiness and risk-aversion are closely linked, these studies provide an alternative explanation
for our empirical finding that time-varying measures of riskiness positively predict future returns on the
aggregate stock market.
25
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28
Table 1: Aggregate Riskiness Measures and the VIX
This table shows the descriptive statistics of the option implied and physical measures of aggregate riskiness
and the S&P 500 index option implied variance (VIX). The option implied measures of aggregate riskiness
are obtained from the S&P 500 index options with 1, 3, 6, and 12 months to maturity. The generalized
physical measures of aggregate riskiness are obtained from the daily returns on the S&P 500 index over
the past 1, 3, 6, and 12 months. Panel A reports the mean, standard deviation, maximum, minimum,
skewness, kurtosis, and AR(1) statistics of the aggregate riskiness measures and the VIX. Panel B presents
the correlation matrix for the option implied and physical measures of aggregate riskiness and the VIX.
The sample period is from January 1996 to October 2010.
Panel A. Descriptive Statistics
Implied Measures of Riskiness
Physical Measures of Riskiness
VIX
1-month 3-month 6-month 12-month 1-month 3-month 6-month 12-month
Mean
2.1208
1.5868
1.4417
1.2244
0.3796
0.3743
0.3711
0.3644
0.0047
Std. Dev.
3.9728
2.4622
2.1311
1.6644
0.6143
0.5283
0.4535
0.3702
0.0040
Maximum 20.3960 10.8540 9.5582
8.0583
5.5901
4.0648
2.7804
1.7749
0.0299
Minimum 0.0948
0.1089
0.1207
0.1551
0.0389
0.0447
0.0469
0.0616
0.00091
Skewness
2.62
2.14
2.13
2.16
5.53
4.72
3.64
2.52
3.08
Kurtosis
9.37
6.41
6.37
6.83
40.84
29.35
17.51
9.26
16.17
AR(1)
0.947
0.973
0.989
0.997
0.748
0.898
0.957
0.980
0.838
Panel B. Correlation Matrix
AS,Q
AS,Q
AS,Q
AS,P
AS,P
AS,P
AS,P
RAS,Q
1-month R3-month R6-month R12-month R1-month R3-month R6-month R12-month
RAS,Q
1-month
RAS,Q
3-month
RAS,Q
6-month
RAS,Q
12-month
1
VIX
0.93
0.91
0.91
0.38
0.55
0.72
0.78
0.56
1
0.99
0.97
0.29
0.40
0.52
0.70
0.45
1
0.99
0.27
0.39
0.50
0.68
0.43
1
0.30
0.43
0.54
0.66
0.46
1
0.84
0.66
0.46
0.88
1
0.86
0.62
0.83
1
0.81
0.77
1
0.61
RAS,P
1-month
RAS,P
3-month
RAS,P
6-month
RAS,P
12-month
VIX
1
29
Table 2: Descriptive Statistics of the U.S. Equity Market Indices
This table presents the descriptive statistics of the monthly excess returns on the U.S. equity market indices:
the value-weighted NYSE/AMEX/NASDAQ (VW CRSP), the equal-weighted NYSE/AMEX/NASDAQ
(EW CRSP), S&P500, NASDAQ, and Dow Jones Industrial Average (DJIA). The excess market return is
defined as the monthly return on the aggregate market portfolio in excess of the risk-free rate. The riskfree interest rate is measured by the one-month T-bill rate. The table reports the mean, standard deviation,
maximum, minimum, skewness, kurtosis, and the Jarque-Bera (JB) statistics. JB = n[(S2 /6) + (K–3)2 /24] is
a formal test statistic for testing whether the returns are normally distributed, where n denotes the number
of observations, S is skewness and K is kurtosis. The JB statistic is distributed as the Chi-square with two
degrees of freedom. The last row presents the p-values for the JB statistics in square brackets. The sample
period is from January 1996 to October 2010.
VW CRSP EW CRSP S&P500 NASDAQ
DJIA
Mean
0.0042
0.0084
0.0022
0.0053
0.0028
Std. Dev.
0.0493
0.0617
0.0472
0.0770
0.0462
Maximum
0.1104
0.2196
0.0938
0.2154
0.1047
Minimum
-0.1854
-0.2068
-0.1702
-0.2341
-0.1556
Skewness
-0.7448
-0.2680
-0.6341
-0.3581
-0.5566
Kurtosis
3.9184
4.5616
3.6803
3.5287
3.7991
JB
22.71
[0.00%]
20.22
[0.00%]
15.36
5.88
13.93
[0.05%] [5.30%] [0.09%]
30
Table 3: Option Implied Measures of Riskiness and Future Market Returns
This table presents results from the predictive regressions of one-month ahead excess market returns on the
option implied measures of riskiness obtained from the S&P 500 index options with 1, 3, 6, and 12 months
to maturity. Panel A presents results for the value-weighted and equal-weighted NYSE/AMEX/NASDAQ
(CRSP) index. Panel B presents results for the S&P 500, Dow Jones Industrial Average (DJIA), and NASDAQ. The control variables include the S&P 500 index option implied volatility (VIX), aggregate idiosyncratic volatility (IVOL), default spread (DEF), term spread (TERM), relative T-bill rate (RREL), dividendprice ratio (DIV), growth rate of industrial production (IP), unemployment rate (UNEMP), consumptionto-wealth ratio (CAY), and the lagged excess market return (RET). The Newey-West (1987) t-statistics are
reported in parentheses. The last row presents the R2 values. The sample period is January 1996 – October
2010.
Panel A. NYSE/AMEX/NASDAQ (CRSP) Index
Value-Weighted CRSP Index
Equal-Weighted CRSP Index
1-month 3-month 6-month 12-month 1-month 3-month 6-month 12-month
intercept 0.1930 0.2742 0.2914
(3.19) (3.19) (3.59)
0.2771
(4.10)
0.1452 0.2513 0.2774
(1.76) (2.49) (3.14)
0.2585
(3.15)
0.0256
(3.22)
RtAS,Q
0.0066 0.0163 0.0195
(2.35) (2.28) (2.77)
0.0223
(3.29)
0.0067 0.0184 0.0227
(1.77) (2.26) (3.11)
VIX
2.2232 0.9576 1.0090
(1.16) (0.42) (0.46)
0.9934
(0.45)
-0.0480 -1.4622 -1.4153 -1.3692
(-0.02) (-0.55) (-0.56) (-0.55)
IVOL -0.9784 -1.1620 -1.0882 -0.8011
(-1.35) (-1.63) (-1.55) (-1.17)
-0.0289 -0.2333 -0.1559 0.1759
(-0.02) (-0.20) (-0.14) (0.15)
DEF
-1.3485 0.1973 0.0077 -0.7235
(-0.63) (0.09) (0.00) (-0.36)
-3.4524 -1.8057 -1.9519 -2.5812
(-2.18) (-1.05) (-1.16) (-1.59)
TERM 0.4729 0.5955 0.6438
(1.25) (1.39) (1.52)
0.5915
(1.52)
0.6187 0.8477 0.9384
(1.16) (1.58) (1.81)
0.8701
(1.71)
RREL
0.2785 0.5873 0.6320
(0.65) (1.43) (1.49)
0.5757
(1.32)
0.3020 0.6224 0.6753
(0.60) (1.25) (1.30)
0.6084
(1.13)
DIV
0.0507 0.0686 0.0762
(1.92) (2.62) (2.91)
0.0803
(3.07)
0.0419 0.0642 0.0739
(1.13) (1.73) (2.03)
0.0782
(2.13)
IP
1.4753 1.3695 1.4162
(1.86) (1.81) (1.88)
1.5034
(1.94)
1.4847 1.3727 1.4269
(1.51) (1.48) (1.55)
1.5281
(1.62)
UNEMP -0.0163 -0.0293 -0.0307 -0.0265
(-2.44) (-2.52) (-2.88) (-3.40)
-0.0150 -0.0317 -0.0344 -0.0292
(-1.72) (-2.48) (-3.27) (-3.40)
CAY
-0.0435 0.1284 0.1255
(-0.22) (0.61) (0.60)
0.0618
(0.31)
0.0831 0.2537 0.2487
(0.28) (0.82) (0.81)
0.1718
(0.57)
RET
0.2080 0.1833 0.1630
(2.14) (1.94) (1.67)
0.1605
(1.68)
0.2738 0.2667 0.2543
(2.95) (2.91) (2.80)
0.2550
(2.88)
R2
15.96% 17.46% 17.73% 17.54%
31
12.93% 14.67% 15.20% 14.94%
32
1.5195
(0.52)
0.0363
(3.80)
0.5753
(1.38)
0.3004 0.5854 0.6276
(0.74) (1.49) (1.54)
0.0505 0.0679 0.0752
(2.06) (2.72) (2.94)
1.3661 1.2669 1.3100
(1.86) (1.81) (1.88)
RREL
DIV
IP
15.71% 17.29% 17.55% 17.40%
R2
0.1146
(1.16)
0.1642 0.1383 0.1180
(1.65) (1.42) (1.18)
RET
0.0864
(0.48)
-0.0102 0.1488 0.1455
(-0.06) (0.80) (0.79)
CAY
UNEMP -0.0142 -0.0266 -0.0280 -0.0241
(-2.16) (-2.29) (-2.55) (-3.05)
1.3916
(1.95)
0.0793
(3.11)
0.4665
(1.21)
TERM 0.3406 0.4683 0.5133
(0.92) (1.09) (1.20)
-3.6530 -2.1741 -2.3267 -2.9199
(-2.44) (-1.33) (-1.44) (-1.87)
1.3803
(1.43)
0.1114
(2.47)
0.7318
(1.03)
1.0921
(1.43)
0.0887
(1.36)
0.1961
(0.56)
9.70% 11.81% 12.81% 12.65%
0.1163 0.1053 0.0909
(1.74) (1.59) (1.37)
0.0839 0.3085 0.3005
(0.23) (0.87) (0.86)
-0.0216 -0.0445 -0.0503 -0.0437
(-2.04) (-2.79) (-3.39) (-3.60)
1.3152 1.1666 1.2377
(1.27) (1.20) (1.30)
0.0568 0.0877 0.1042
(1.21) (1.92) (2.32)
0.3195 0.7408 0.8226
(0.45) (1.08) (1.17)
0.6584 0.9891 1.1687
(0.86) (1.26) (1.48)
-4.3898 -2.4244 -2.7850 -3.8240
(-1.56) (-0.90) (-1.05) (-1.46)
3.4895 1.5581 1.5463
(1.24) (0.51) (0.54)
0.0087 0.0246 0.0317
(2.27) (2.69) (3.65)
DEF
0.9854
(0.47)
0.0209
(3.21)
0.4260
(3.77)
-2.0049 -2.2826 -2.1904 -1.7242
(-1.26) (-1.45) (-1.39) (-1.10)
2.1920 0.9677 1.0155
(1.19) (0.44) (0.48)
0.0061 0.0152 0.0182
(2.29) (2.19) (2.60)
0.2487 0.3960 0.4474
(2.29) (3.03) (3.61)
NASDAQ index
1-month 3-month 6-month 12-month
IVOL -0.9035 -1.0616 -0.9822 -0.7112
(-1.43) (-1.70) (-1.61) (-1.20)
VIX
RtAS,Q
0.2665
(3.83)
S&P 500 index
1-month 3-month 6-month 12-month
0.5169
(0.28)
0.0146
(2.53)
0.1800
(2.43)
1.3854
(2.31)
0.0568
(1.96)
0.2243
(0.52)
0.0041
(0.01)
0.0519
(0.54)
0.0915
(0.46)
11.07% 12.24% 12.23% 12.21%
0.0841 0.0652 0.0541
(0.87) (0.69) (0.56)
0.0326 0.1387 0.1325
(0.17) (0.69) (0.67)
-0.0063 -0.0162 -0.0166 -0.0141
(-0.94) (-1.44) (-1.52) (-1.79)
1.3693 1.2987 1.3289
(2.21) (2.19) (2.25)
0.0358 0.0493 0.0536
(1.32) (1.80) (1.89)
0.0539 0.2364 0.2602
(0.13) (0.58) (0.61)
-0.1189 0.0146 0.0299
(-0.29) (0.03) (0.07)
-2.5942 -1.7006 -1.8057 -2.2277
(-1.64) (-1.06) (-1.13) (-1.40)
-0.5471 -0.6508 -0.5909 -0.4018
(-0.77) (-0.92) (-0.84) (-0.57)
1.4060 0.4396 0.5533
(0.82) (0.23) (0.30)
0.0038 0.0109 0.0126
(1.61) (1.74) (1.94)
0.1160 0.1801 0.1871
(1.82) (2.07) (2.16)
DJIA index
1-month 3-month 6-month 12-month
Panel B. Alternative Stock Market Indices: S&P 500, NASDAQ, and DJIA
intercept 0.1839 0.2626 0.2792
(3.12) (3.02) (3.30)
Table 3 (continued)
Table 4: Physical Measures of Riskiness and Future Market Returns
This table presents results from the predictive regressions of one-month ahead excess market returns on
the generalized physical measures of riskiness obtained from daily returns on the S&P 500 index over
the past 1, 3, 6, and 12 months. Panel A presents results for the value-weighted and equal-weighted
NYSE/AMEX/NASDAQ (CRSP) index. Panel B presents results for the S&P 500, Dow Jones Industrial
Average (DJIA), and NASDAQ. The control variables include the physical measure of market variance
(RVAR), aggregate idiosyncratic volatility (IVOL), default spread (DEF), term spread (TERM), relative Tbill rate (RREL), dividend-price ratio (DIV), growth rate of industrial production (IP), unemployment rate
(UNEMP), consumption-to-wealth ratio (CAY), and the lagged excess market return (RET). The NeweyWest (1987) t-statistics are reported in parentheses. The last row presents the R2 values. The sample period
is January 1996 – October 2010.
Value-Weighted CRSP Index
Equal-Weighted CRSP Index
1-month 3-month 6-month 12-month 1-month 3-month 6-month 12-month
intercept 0.0849 0.0894 0.1418
(1.42) (1.51) (2.53)
RtAS,P
0.0105 0.0033 0.0605
(0.71) (0.14) (3.59)
0.1281
(1.85)
0.0428 0.0513 0.1036
(0.58) (0.70) (1.41)
0.0838
(1.04)
0.0381
(1.78)
0.0337 0.0051 0.0628
(1.56) (0.19) (2.52)
0.0356
(1.33)
RVAR -3.1085 -1.9990 -1.8567 -1.3070
(-1.46) (-1.24) (-2.05) (-1.13)
-7.2901 -3.5089 -3.2781 -2.8042
(-2.43) (-1.63) (-2.81) (-1.92)
IVOL
0.3827 0.4258 -0.2066 -0.3496
(0.45) (0.48) (-0.23) (-0.39)
1.2359 1.3763 0.7200
(0.94) (1.00) (0.51)
DEF
0.4657 0.2995 -4.6990 -1.2897
(0.30) (0.13) (-2.62) (-0.71)
2.0292 1.9460 -3.1114 0.6571
(1.00) (0.72) (-1.16) (0.28)
0.6495
(0.44)
TERM -0.1217 -0.0907 0.2048 -0.0198
(-0.29) (-0.22) (0.52) (-0.04)
0.1394 0.2178 0.5202
(0.27) (0.41) (0.97)
0.2781
(0.51)
RREL
0.3284 0.3420 0.1582
(0.77) (0.76) (0.36)
0.3955
(0.95)
0.3579 0.4203 0.2331
(0.68) (0.76) (0.45)
0.4765
(0.96)
DIV
0.0518 0.0537 0.0582
(1.68) (1.77) (2.04)
0.0545
(1.73)
0.0440 0.0485 0.0525
(1.15) (1.24) (1.42)
0.0480
(1.20)
IP
1.8298 1.7839 1.8453
(1.76) (1.75) (2.08)
1.6804
(1.76)
1.9656 1.7896 1.8463
(1.50) (1.43) (1.69)
1.6831
(1.45)
UNEMP -0.0008 -0.0010 -0.0033 -0.0053
(-0.21) (-0.26) (-0.78) (-0.94)
-0.0009 -0.0014 -0.0037 -0.0053
(-0.18) (-0.29) (-0.74) (-0.91)
CAY
0.1159 0.1082 -0.2043 0.0732
(0.58) (0.48) (-0.92) (0.36)
0.1056 0.1154 -0.1997 0.0971
(0.36) (0.35) (-0.60) (0.34)
RET
0.0868 0.0831 0.0722
(1.08) (1.03) (0.85)
0.2064 0.1912 0.1833
(2.51) (2.49) (2.32)
R2
0.0646
(0.84)
11.84% 11.77% 15.72% 13.75%
33
0.1734
(2.34)
13.21% 12.62% 15.33% 13.70%
34
-0.0780 -0.0690 -5.0091 -1.9779
(-0.05) (-0.03) (-2.92) (-1.16)
DEF
0.0542 0.0551 0.0599
(1.83) (1.91) (2.21)
1.6813 1.6457 1.7181
(1.76) (1.75) (2.10)
DIV
IP
0.0445 0.0433 0.0350
(0.53) (0.52) (0.40)
11.79% 11.75% 15.64% 14.17%
RET
R2
0.0229
(0.29)
0.1382 0.1419 -0.1700 0.0877
(0.76) (0.69) (-0.82) (0.47)
CAY
UNEMP -0.0001 -0.0003 -0.0024 -0.0047
(-0.04) (-0.07) (-0.59) (-0.86)
1.5515
(1.77)
0.0567
(1.90)
1.7054
(1.40)
0.0667
(1.24)
0.3946
(0.55)
0.0898
(0.11)
6.75%
6.41%
9.26%
0.0614 0.0485 0.0297
(0.97) (0.81) (0.48)
7.41%
0.0382
(0.64)
0.2399 0.1569 -0.1804 0.2163
(0.67) (0.39) (-0.47) (0.60)
-0.0016 -0.0027 -0.0057 -0.0082
(-0.27) (-0.42) (-0.85) (-0.95)
2.0952 1.9387 1.9273
(1.53) (1.54) (1.71)
0.0592 0.0694 0.0720
(1.14) (1.30) (1.42)
0.2356 0.2665 0.0669
(0.32) (0.35) (0.09)
0.3481 0.3610 0.1773
(0.86) (0.85) (0.42)
RREL
0.4078
(1.02)
-0.1088 0.0673 0.4164
(-0.15) (0.09) (0.58)
TERM -0.1937 -0.1807 0.1106 -0.0940
(-0.47) (-0.45) (0.29) (-0.21)
0.7575 -0.6223 -6.3916 -1.3494
(0.30) (-0.18) (-2.04) (-0.47)
-0.0268 0.1466 -0.7559 -0.8656
(-0.01) (0.08) (-0.38) (-0.43)
0.0504
(1.65)
0.4716 0.5000 -0.0981 -0.3108
(0.63) (0.66) (-0.13) (-0.41)
0.0469 0.0239 0.0866
(1.80) (0.78) (3.24)
IVOL
0.0401
(1.97)
0.1700
(1.46)
-8.0673 -3.5350 -2.4119 -1.5830
(-2.58) (-1.94) (-1.97) (-0.92)
0.0064 0.0004 0.0573
(0.44) (0.02) (3.63)
0.1063 0.1329 0.1934
(1.05) (1.24) (1.88)
NASDAQ index
1-month 3-month 6-month 12-month
RVAR -2.4813 -1.7249 -1.7082 -1.1424
(-1.25) (-1.16) (-1.97) (-1.07)
RtAS,P
0.1314
(1.99)
S&P 500 index
1-month 3-month 6-month 12-month
0.0350
(2.08)
0.0975
(1.62)
1.4868
(2.11)
0.0432
(1.43)
9.85% 10.16% 11.12% 11.76%
-0.0031 -0.0003 0.0048 -0.0144
(-0.04) (0.00) (0.06) (-0.18)
0.1213 0.1852 -0.0661 0.0641
(0.58) (0.85) (-0.28) (0.32)
0.0021 0.0024 0.0010 -0.0017
(0.56) (0.61) (0.25) (-0.35)
1.5281 1.5025 1.6152
(2.02) (1.96) (2.34)
0.0428 0.0396 0.0447
(1.41) (1.32) (1.50)
0.0794 0.1057 -0.0431 0.0993
(0.20) (0.26) (-0.10) (0.25)
-0.4168 -0.4778 -0.2580 -0.3515
(-1.03) (-1.17) (-0.64) (-0.83)
-0.1904 0.6806 -3.0581 -1.9355
(-0.14) (0.36) (-1.62) (-1.36)
0.5710 0.5538 0.1985 -0.1688
(0.65) (0.65) (0.22) (-0.18)
-0.6876 -0.8617 -1.4472 -0.9686
(-0.32) (-0.62) (-1.69) (-1.04)
-0.0066 -0.0128 0.0325
(-0.44) (-0.72) (2.09)
0.0609 0.0503 0.0893
(1.08) (0.89) (1.56)
DJIA index
1-month 3-month 6-month 12-month
Panel B. Alternative Stock Market Indices: S&P 500, NASDAQ, and DJIA
intercept 0.0884 0.0901 0.1419
(1.53) (1.58) (2.62)
Table 4 (continued)
Table 5: Riskiness Premium and Future Market Returns
This table presents results from the predictive regressions of one-month ahead excess market returns on the
riskiness premium defined as the difference between the option implied and the physical measures of riskiness. Panel A presents results for the value-weighted and equal-weighted NYSE/AMEX/NASDAQ (CRSP)
index. Panel B presents results for the S&P 500, Dow Jones Industrial Average (DJIA), and NASDAQ.
The control variables include the variance risk premium (VRP), aggregate idiosyncratic volatility (IVOL),
default spread (DEF), term spread (TERM), relative T-bill rate (RREL), dividend-price ratio (DIV), growth
rate of industrial production (IP), unemployment rate (UNEMP), consumption-to-wealth ratio (CAY), and
the lagged excess market return (RET). The Newey-West (1987) t-statistics are reported in parentheses.
The last row presents the R2 values. The sample period is January 1996 – October 2010.
Value-Weighted CRSP Index
Equal-Weighted CRSP Index
1-month 3-month 6-month 12-month 1-month 3-month 6-month 12-month
intercept
0.2065 0.3335 0.2750
(3.38) (3.98) (3.20)
0.2051
(2.83)
0.1533 0.2980 0.2281
(1.85) (2.67) (2.29)
0.1597
(1.78)
RtAS,Q − RtAS,P 0.0072 0.0218 0.0199
(2.70) (3.25) (2.75)
0.0170
(2.75)
0.0070 0.0227 0.0197
(2.22) (2.87) (2.60)
0.0168
(2.77)
0.3046
(0.10)
-2.9509 -3.7067 -2.4970 -2.5466
(-0.99) (-1.25) (-0.76) (-0.77)
VRP
0.1990 -0.7434 0.3578
(0.07) (-0.26) (0.12)
IVOL
-0.1683 -0.6513 -0.7076 -0.2828
(-0.21) (-0.94) (-1.06) (-0.44)
0.6263 0.1534 0.1314
(0.51) (0.14) (0.12)
0.5521
(0.49)
DEF
-2.5307 -0.1520 0.1604 -0.8724
(-1.81) (-0.11) (0.12) (-0.59)
-1.2018 1.0635 1.3466
(-0.60) (0.58) (0.76)
0.3212
(0.17)
TERM
0.5534 0.8233 0.5416
(1.43) (1.95) (1.16)
0.3293
(0.76)
0.6952 1.0434 0.7110
(1.29) (1.87) (1.20)
0.5029
(0.88)
RREL
0.1939 0.6500 0.6727
(0.47) (1.58) (1.54)
0.4902
(1.10)
0.2189 0.6803 0.6881
(0.42) (1.28) (1.25)
0.5083
(0.90)
DIV
0.0626 0.0859 0.0763
(2.38) (3.14) (2.84)
0.0687
(2.45)
0.0501 0.0774 0.0664
(1.34) (1.91) (1.74)
0.0590
(1.47)
IP
1.6108 1.3579 1.4290
(1.86) (1.73) (1.71)
1.5834
(1.84)
1.6304 1.3697 1.4552
(1.62) (1.48) (1.49)
1.6077
(1.61)
UNEMP
-0.0177 -0.0391 -0.0304 -0.0178
(-2.69) (-3.71) (-2.63) (-2.20)
-0.0152 -0.0390 -0.0284 -0.0160
(-1.83) (-2.98) (-2.28) (-1.85)
CAY
0.0049 0.2527 0.2667
(0.02) (1.32) (1.40)
0.1441
(0.74)
0.0973 0.3311 0.3348
(0.33) (1.17) (1.19)
0.2133
(0.74)
RET
0.1467 0.1410 0.1539
(1.40) (1.39) (1.45)
0.1554
(1.47)
0.2016 0.2159 0.2315
(2.10) (2.31) (2.38)
0.2331
(2.33)
R2
15.37% 19.30% 15.99% 14.50%
35
14.12% 17.22% 14.52% 13.60%
36
0.4298 0.6998 0.4203
(1.13) (1.68) (0.91)
0.2241 0.6527 0.6707
(0.57) (1.66) (1.60)
0.0622 0.0849 0.0751
(2.55) (3.35) (2.97)
1.4838 1.2438 1.3132
(1.85) (1.72) (1.69)
TERM
RREL
DIV
IP
15.17% 19.23% 15.86% 14.26%
R2
0.1168
(1.07)
0.1174 0.1074 0.1158
(1.10) (1.04) (1.06)
RET
0.1629
(0.94)
0.0296 0.2626 0.2769
(0.17) (1.58) (1.65)
-0.0159 -0.0365 -0.0277 -0.0156
(-2.45) (-3.53) (-2.33) (-1.90)
1.4583
(1.82)
0.0673
(2.58)
0.4975
(1.18)
CAY
UNEMP
-2.7996 -0.6120 -0.3103 -1.2520
(-2.12) (-0.48) (-0.24) (-0.87)
DEF
0.2086
(0.49)
-0.1404 -0.5911 -0.6457 -0.2498
(-0.21) (-1.00) (-1.13) (-0.46)
IVOL
0.6005
(0.21)
0.6007 -0.3544 0.6445
(0.23) (-0.14) (0.22)
0.0155
(2.52)
RtAS,Q − RtAS,P 0.0067 0.0206 0.0186
(2.63) (3.18) (2.52)
VRP
0.1949
(2.72)
S&P 500 index
1-month 3-month 6-month 12-month
1.1918
(0.33)
0.0285
(2.91)
0.3152
(2.54)
1.4732
(1.38)
0.0930
(1.90)
0.6193
(0.85)
0.7006
(0.89)
8.88% 12.49% 11.40%
0.0892 0.0885 0.1016
(1.21) (1.21) (1.36)
0.1375 0.4735 0.5189
(0.38) (1.42) (1.60)
9.78%
0.0996
(1.33)
0.3168
(0.92)
-0.0246 -0.0564 -0.0515 -0.0307
(-2.29) (-3.34) (-3.02) (-2.35)
1.5198 1.1620 1.2162
(1.37) (1.13) (1.17)
0.0745 0.1095 0.1054
(1.53) (2.28) (2.22)
0.1849 0.8171 0.9217
(0.27) (1.17) (1.28)
0.8224 1.2676 1.0500
(1.04) (1.59) (1.26)
-2.9867 0.1901 0.6299 -1.1041
(-1.13) (0.08) (0.28) (-0.45)
-0.8407 -1.5123 -1.6838 -0.9781
(-0.52) (-1.00) (-1.15) (-0.65)
1.2048 -0.0791 1.3426
(0.35) (-0.02) (0.38)
0.0097 0.0309 0.0332
(2.71) (3.24) (3.46)
0.2717 0.4624 0.4301
(2.38) (3.31) (3.16)
NASDAQ index
1-month 3-month 6-month 12-month
0.5278
(0.22)
0.0095
(1.65)
0.1202
(1.59)
1.4176
(2.15)
0.0456
(1.56)
0.1733
(0.40)
0.0544
(0.54)
0.1418
(0.76)
11.11% 14.24% 11.63% 10.43%
0.0631 0.0533 0.0544
(0.64) (0.55) (0.54)
0.0479 0.2132 0.2197
(0.25) (1.17) (1.20)
-0.0081 -0.0255 -0.0167 -0.0069
(-1.23) (-2.57) (-1.36) (-0.82)
1.4354 1.2494 1.3144
(2.17) (2.09) (2.03)
0.0438 0.0633 0.0533
(1.61) (2.31) (1.88)
-0.0036 0.3070 0.3024
(-0.01) (0.72) (0.68)
-0.0305 0.2421 -0.0253 -0.2139
(-0.07) (0.57) (-0.05) (-0.49)
-2.1370 -0.6919 -0.4682 -1.0525
(-1.53) (-0.54) (-0.37) (-0.75)
-0.0716 -0.4019 -0.4173 -0.1429
(-0.09) (-0.56) (-0.60) (-0.21)
0.5959 -0.1975 0.5327
(0.28) (-0.10) (0.23)
0.0045 0.0159 0.0130
(1.96) (2.79) (1.79)
0.1301 0.2367 0.1782
(2.05) (2.97) (1.96)
DJIA index
1-month 3-month 6-month 12-month
Panel B. Alternative Stock Market Indices: S&P 500, NASDAQ, and DJIA
0.1986 0.3220 0.2635
(3.35) (3.94) (3.01)
intercept
Table 5 (continued)
Table 6: Correlation between Aggregate Riskiness and Economic/Financial Downturns
This table presents the correlation matrix for the option implied and the physical measures of aggregate
riskiness, the CFNAI, CFNAI MA3, ADS, and ADS MA3 economic activity indices, the growth rate of
nominal and real GDP, unemployment rate, and default risk for the sample period January 1996 – October
2010. The option implied measures of aggregate riskiness are computed from the S&P500 index options
with 1, 3, 6, and 12 months to maturity. The generalized physical measures of aggregate riskiness are
obtained from daily returns on the S&P500 index over the past 1, 3, 6, and 12 months. The CFNAI and the
CFNAI MA3 economic activity indices are obtained from the Federal Reserve Bank of Chicago. The daily
data on the Aruoba, Diebold, and Scotti (ADS) business conditions index are obtained from the Federal
Reserve Bank of Philadelphia . Quarterly data on the nominal and real GDP are obtained from the Bureau
of Economic Analysis. Monthly data on the U.S. unemployment rate (UNEMP) are obtained from the
Bureau of Labor Statistics. The default spread (DEF) is defined as the difference between the monthly
yields on BAA- and AAA-rated corporate bond yields. The monthly data on the BAA- and AAA-rated
corporate bond are available at the Federal Reserve website.
Implied Measures of Riskiness
Physical Measures of Riskiness
AS,Q
AS,Q
AS,Q
R1-month
RAS,Q
3-month R6-month R12-month
AS,P
AS,P
AS,P
RAS,P
1-month R3-month R6-month R12-month
CFNAI
-0.54
-0.40
-0.39
-0.41
-0.54
-0.67
-0.73
-0.62
CFNAI MA3
-0.63
-0.48
-0.46
-0.49
-0.57
-0.69
-0.81
-0.75
ADS
-0.49
-0.33
-0.32
-0.36
-0.57
-0.68
-0.75
-0.60
ADS MA3
-0.55
-0.39
-0.38
-0.40
-0.59
-0.69
-0.80
-0.70
nominal -0.65
GROW T HGDP
-0.50
-0.48
-0.51
-0.59
-0.73
-0.81
-0.74
real
GROW T HGDP
-0.53
-0.36
-0.35
-0.38
-0.58
-0.68
-0.73
-0.60
UNEMP
0.83
0.92
0.92
0.90
0.16
0.24
0.36
0.58
DEF
0.74
0.58
0.57
0.60
0.64
0.80
0.88
0.77
37
Table 7: Aggregate Riskiness and Economic Downturns
This table presents results from the contemporaneous regressions of economic indicators on the option
implied and the physical measures of riskiness. The option implied measures of aggregate riskiness are
computed from the S&P500 index options with 1, 3, 6, and 12 months to maturity. The generalized
physical measures of aggregate riskiness are obtained from daily returns on the S&P500 index over the
past 1, 3, 6, and 12 months. Economic downturns are proxied by lower CFNAI, CFNAI MA3, ADS, and
ADS MA3 indices; lower nominal and real GDP growth; higher unemployment rate; and higher default
risk. The Newey-West (1987) t-statistics are reported in parentheses. The last column presents the R2
values. The sample period is January 1996 – October 2010.
Panel A. Contemporaneous relation between CFNAI index and Aggregate Riskiness
Implied Measures of Riskiness
Physical Measures of Riskiness
Intercept 1-month 3-month 6-month 12-month 1-month 3-month 6-month 12-month R2
0.0393 -0.1227
29.33%
(0.39) (-3.42)
0.0102
-0.1456
15.88%
(0.10)
(-2.05)
0.0146
-0.1633
14.97%
(0.13)
(-2.01)
0.0534
-0.2240
17.16%
(0.45)
(-2.01)
0.0773
-0.7854
28.75%
(0.77)
(-4.28)
0.2046
-1.1367
44.55%
(2.22)
(-6.70)
0.3150
-1.4438
52.95%
(4.12)
(-16.65)
0.3307
-1.5093 38.56%
(2.49)
(-3.48)
Panel B. Contemporaneous relation between ADS index and Aggregate Riskiness
Implied Measures of Riskiness
Physical Measures of Riskiness
Intercept 1-month 3-month 6-month 12-month 1-month 3-month 6-month 12-month R2
-0.0253 -0.1056
24.38%
(-0.14) (-2.74)
-0.0683
-0.1140
10.92%
(-0.32)
(-2.02)
-0.0652
-0.1276
10.26%
(-0.58)
(-1.86)
-0.0267
-0.1817
12.68%
(-0.22)
(-1.97)
0.0517
-0.7927
32.87%
(0.54)
(-5.28)
0.1587
-1.0899
45.96%
(1.72)
(-7.33)
0.2698
-1.3984
55.75%
(3.32)
(-16.69)
0.2554
-1.3808 36.22%
(1.82)
(-2.99)
38
Table 7 (continued)
Panel C. Contemporaneous relation between GDP Growth and Aggregate Riskiness
Implied Measures of Riskiness
Physical Measures of Riskiness
Intercept 1-month 3-month 6-month 12-month 1-month 3-month 6-month 12-month
5.6403 -0.4779
(19.33) (-3.61)
5.5672
-0.5930
(17.42)
(-2.24)
5.5837
-0.6639
(16.93)
(-2.18)
5.7146
-0.8886
(15.39)
(-2.12)
5.6922
-2.8072
(17.37)
(-4.46)
6.1352
-4.0303
(21.70)
(-7.24)
6.5666
-5.2272
(27.40)
(-17.81)
6.7671
-5.8570
(17.20)
(-4.64)
R2
42.18%
24.92%
23.42%
25.59%
34.79%
53.05%
65.74%
55.02%
Panel D. Contemporaneous relation between Unemployment Rate and Aggregate Riskiness
Implied Measures of Riskiness
Physical Measures of Riskiness
Intercept 1-month 3-month 6-month 12-month 1-month 3-month 6-month 12-month
4.8472 0.3372
(38.69) (4.75)
4.6088
0.6009
(46.52)
(18.11)
4.5578
0.6968
(44.86)
(17.11)
4.4969
0.870
(38.76)
(9.96)
5.4044
0.4161
(19.98)
(2.32)
5.2938
0.7176
(19.71)
(2.82)
5.0880
1.2783
(19.69)
(4.41)
4.6347
2.5386
(20.93)
(7.84)
39
R2
69.45%
84.71%
85.32%
81.17%
2.53%
5.56%
13.00%
34.17%
Table 7 (continued)
Panel E. Contemporaneous relation between Default Spread and Aggregate Riskiness
Implied Measures of Riskiness
Physical Measures of Riskiness
Intercept 1-month 3-month 6-month 12-month 1-month 3-month 6-month 12-month R2
0.0081 0.0009
54.08%
(20.90) (3.97)
0.0082
0.0012
33.26%
(17.92)
(2.37)
0.0082
0.0013
31.94%
(17.16)
(2.35)
0.0079
0.0018
35.76%
(14.17)
(2.37)
0.0081
0.0051
40.39%
(16.63)
(4.58)
0.0073
0.0074
64.00%
(17.14)
(7.49)
0.0065
0.0095
77.13%
(17.04)
(17.77)
0.0063
0.0103 59.54%
(8.50)
(3.86)
40
Option Implied Measures of Aggregate Riskiness
25
20
15
10
5
0
Jan 96
Aug−97
Apr−99
Dec−00
1−month
Aug−02
Apr−04
3−month
Dec−05
6−month
Aug−07
Apr−09
Dec−10
12−month
Figure 1. Option Implied Measures of Aggregate Riskiness
This figure presents the option implied measures of aggregate riskiness obtained from the S&P 500 index
options with 1, 3, 6, and 12 months to maturity.
41
Physical Measures of Aggregate Riskiness
6
5
4
3
2
1
0
Jan 96
Aug−97
Apr−99
Dec−00
1−month
Aug−02
Apr−04
3−month
Dec−05
6−month
Aug−07
Apr−09
Dec−10
12−month
Figure 2. Physical Measures of Aggregate Riskiness
This figure presents the physical measures of aggregate riskiness obtained from daily returns on the S&P
500 index over the past 1, 3, 6, and 12 months.
42
A New Approach to Measuring Riskiness in the Equity Market:
Implications for the Risk Premium
Online Appendix
Section I provides a derivation of the option implied measure of riskiness based on simple returns. Section II presents the corresponding solutions for log returns. Sections III, IV, and V provide three alternative explanations for our empirical findings based on the consumption-based asset pricing models, the
time-varying risk of rare economic disasters, and the psychological factors. Section VI investigates the
intertemporal relation between riskiness and future market returns during economic recessions.
I. Recovering the Aumann and Serrano (2008) riskiness measure from option prices: The case of simple returns
We use simple returns, and derive the Aumann and Serrano (2008) riskiness measures from option
prices. Theorem A in Aumann and Serrano (2008) shows that for each gamble gt+τ , there is a unique
positive number Rt [gt+τ ] such that
)
(
g
− R t+τ
gt+τ ]
[
t
− 1 = 0.
Et e
(1)
UNDER THE RISK NEUTRAL MEASURE, (1) can be expressed as
(
)
g
− t+τ
Et∗ e Rt [gt+τ ] − 1 = 0
(2)
where
gt+τ =
Si (t, τ) − Si (t)
Si (t)
(3)
represents the return on the risky asset i with an investment horizon τ. Notice that, under the risk neutral
measure
Et∗ (gt+τ ) = r f (t, τ) .
(4)
where r f (t, τ) represents the risk-free rate for the time period [t,t + τ]. Since
Et∗
)
(
g
− R t+τ
gt+τ ]
[
t
− 1 is finite,
e
we can use the Bakshi and Madan (2000) spanning formula:
[ ] (
) [ ] ∫
H [S] = H S + S − S Hs S +
S
∞
HSS [K] (S − K)+ dK +
∫ S
0
HSS [K] (K − S)+ dK.
(5)
We use the return’s definition (3) and apply the Bakshi and Madan (2000) formula (5) to
gt+τ
t [gt+τ ]
−R
H [S (t, τ)] = e
− 1.
(6)
with S = Si (t). We obtain
e
gt+τ
t [gt+τ ]
−R
(
)
1
− 1 = (Si (t, τ) − Si (t)) −
Si (t) Rt [gt+τ ]
)
(
∫ ∞
(K−S (t))
− S (t)R ig
1
[
]
e i t t+τ (Si (t, τ) − K)+ dK
+
2
2
Si (t) Si (t) Rt [gt+τ ]
(
)
∫ Si (t)
(K−S (t))
− S (t)R ig
1
e i t [ t+τ ] (K − Si (t, τ))+ dK.
+
2 (t) R2 [g
]
S
0
t t+τ
i
1
(7)
Now, we apply the expectation operator to (7) and get
1
r f (t, τ)
Rt [gt+τ ]
)
(K−S (t))
− S (t)R ig
1
[
]
=
e i t t+τ Et∗ (Si (t, τ) − K)+ dK
2
2
Si (t) Si (t) Rt [gt+τ ]
(
)
∫ Si (t)
(K−S (t))
− S (t)R ig
1
+
e i t [ t+τ ] Et∗ (K − Si (t, τ))+ dK.
Si2 (t) Rt2 [gt+τ ]
0
∫ ∞
(
(8)
Notice that the prices of the call and put options are:
1
E ∗ (Si (t, τ) − K)+ = C (Si (t) , K, τ) ,
(1 + r f (t, τ)) t
1
E ∗ (K − Si (t, τ))+ = P (Si (t) , K, τ) .
(1 + r f (t, τ)) t
Hence (8) can be written as
r f (t, τ)
1
(1 + r f (t, τ)) Rt [gt+τ ]
(
)
(K−S (t))
− S (t)R ig
1
=
e i t [ t+τ ] C (Si (t) , K, τ) dK
2
2
Si (t) Si (t) Rt [gt+τ ]
(
)
∫ Si (t)
(K−S (t))
− S (t)R ig
1
+
e i t [ t+τ ] P (Si (t) , K, τ) dK.
Si2 (t) Rt2 [gt+τ ]
0
∫ ∞
(9)
The riskiness measure Rt [gt+τ ] is, therefore, solution to (9).
II. Recovering the Aumann and Serrano (2008) riskiness measure from option prices: The case of log returns
We use log returns, and derive the riskiness measure of Aumann and Serrano (2008) from option prices.
We denote the log return: gt+τ = log SSi (t,τ)
.
i (t)
UNDER THE RISK NEUTRAL MEASURE, the riskiness measure in Theorem A of Aumann and
Serrano (2008) is
(
)
Et∗ e−gt+τ /Rt [gt+τ ] − 1 = 0.
(10)
We apply Bakshi and Madan (2000) formula (5) to
H [Si (t, τ)] = e−gt+τ /Rt [gt+τ ] − 1
2
(11)
with S = Si (t). We obtain
e
−gt+τ /Rt [gt+τ ]
(
)
1
− 1 = (Si (t, τ) − Si (t)) −
(12)
Rt [gt+τ ] Si (t)
)
)
((
∫ ∞
− R g1
log S K(t)
1
1
i
t [ t+τ ]
+
+
e
(Si (t, τ) − K)+ dK
Rt [gt+τ ] K 2 [Rt [gt+τ ] K]2
Si (t)
((
)
)
∫ Si (t)
− R g1
log S K(t)
1
1
[
]
i
+
(K − Si (t, τ))+ dK.
+
e t t+τ
Rt [gt+τ ] K 2 [Rt [gt+τ ] K]2
0
Now, we apply the expectation operator to (12) and get
1
r f (t, τ)
Rt [gt+τ ]
∫ ∞
=
)
((
1
1
−R
1
t [gt+τ ]
log S K(t)
)
i
+
e
Et∗ (Si (t, τ) − K)+ dK (13)
2
2
R
[g
]
K
Si (t)
t t+τ
[Rt [gt+τ ] K]
((
)
)
∫ Si (t)
− R g1
log S K(t)
1
1
i
+
+
e t [ t+τ ]
Et∗ (K − Si (t, τ))+ dK.
2
2
Rt [gt+τ ] K
0
[Rt [gt+τ ] K]
Hence, (13) can be written as
r f (t, τ)
1
(1 + r f (t, τ)) Rt [gt+τ ]
∫ ∞
=
((
)
1
1
−R
1
t [gt+τ ]
log S K(t)
)
i
+
e
C (Si (t) , K, τ) dK(14)
Rt [gt+τ ] K 2 [Rt [gt+τ ] K]2
((
)
)
∫ Si (t)
log S K(t)
− R g1
1
1
i
+
+
e t [ t+τ ]
P (Si (t) , K, τ) dK.
Rt [gt+τ ] K 2 [Rt [gt+τ ] K]2
0
Si (t)
Therefore, Rt [gt+τ ] is the fixed-point solution to (14).
III. Time-Varying Risk Aversion
By adding a slow-moving habit or time-varying subsistence level to the standard power utility function, Campbell and Cochrane (1999) introduce a consumption-based asset pricing model that explains the
predictability of excess market returns. According to their model in which investors have time-varying risk
aversion, as consumption declines toward the habit in a business cycle through, the curvature of the utility
function rises (i.e., investors become more risk averse), so risky asset prices fall and expected returns rise.
The key assumption in their model is that investors’ utility functions depend on the past history of aggregate
consumption, so they capture a “Catching up with the Joneses” motive. Investors are more risk averse in
recessions, when their consumption is low relative to past aggregate consumption. They are less risk averse
in booms, when their consumption is high, and so gambling feels less threatening. These countercyclical
3
movements in risk aversion make investors want to be compensated more for holding risky assets (such as
stocks) in recessions. Thus, the consumption-based asset pricing model of Campbell and Cochrane (1999)
generates expected returns that are high in recessions.
Combining the theoretical results of Campbell and Cochrane (1999) with the riskiness definition of
Aumann and Serrano (2008), we conclude that increases in risk and risk aversion are closely linked. Since
aggregate risk-aversion is higher during recessions and individuals with higher risk-aversion are more
reluctant to invest on riskier assets, they expect higher return to induce them to take higher perceived
riskiness during recessionary periods.
The countercyclical variation in the aggregate risk aversion and the market risk premium shown by
Campbell and Cochrane (1999) is also identified by Chan and Kogan (2002) in a continuous time, infinitehorizon exchange economy populated by heterogeneous agents whose individual risk aversion is constant
over time but varies across the population. Chan and Kogan (2002) show that the aggregate risk premium
in such an economy exhibits countercyclical variation due to endogenous changes in the cross-sectional
distribution of wealth. Relatively risk-tolerant investors hold a higher proportion of their wealth in stocks.
Therefore, lower economic growth reduces the fraction of aggregate wealth controlled by such investors
and hence their contribution to the aggregate risk aversion. Thus, to induce them to hold the entire stock
market in the aggregate, the equilibrium compensation for risk must rise.
Cecchetti, Lam, and Mark (1990) introduce an equilibrium model of asset pricing in which asset prices
are proportional to the endowment. An interesting feature of their model is that the factor of proportionality
depends on the relative risk aversion coefficient and whether the economy is currently in a high-growth or
low-growth state. According to the parameter estimates of Cecchetti, Lam, and Mark (1990), when the
economy is currently known to be in a low-growth state (or recession), the economy is more likely to move
to a high-growth state into the future. This implies that agents anticipate high future levels of the endowment. This has two effects on asset prices that work in opposite directions in their model. First, there
is an intertemporal relative price effect in which the higher expected future endowment implies a lower
relative price of future goods. This induces agents to want to increase savings and to increase their demand
for assets. The increased asset demand arising from this intertemporal relative price effect works to raise
current asset prices. Working in the opposite direction is a substitution effect arising from agents’ attempts
to smooth their consumption. When the expected future endowment is high, the consumption smoothing
motive leads agents to increase current consumption in an anticipation of higher future investment income.
To finance higher current consumption, they attempt to sell off part of their asset holdings, which in equi-
4
librium results in falling asset prices. When the agents become more risk averse, then the intertemporal
consumption smoothing effect is more likely to dominate the intertemporal relative price effect, indicating
lower asset prices or higher expected returns.
More recent papers have studied the performance of the Campbell-Cochrane model in other asset markets. Wachter (2006) shows that a quantitative implementation of a model with time-varying risk aversion
can simultaneously explain the predictability of stock returns and long-term government bonds. Wachter
(2006) provides a unified explanation of pricing for stocks and bonds. Wachter (2006) also finds that the
real rate is countercyclical, so long-term real bonds are assets with low payoffs in recessions. As a consequence, investors demand positive average compensation for holding these bonds, generating an upward
sloping real yield curve. Chen, Dufresne, and Goldstein (2009) apply the Campbell-Cochrane model to
corporate bond markets. A challenge in these markets is that yields on BAA-rated corporate bonds are
much higher than those on AAA-rated bonds, despite the fact that the default probabilities of BAA bonds
are only slightly higher than those of AAA bonds. A model with time-varying risk aversion can account
for high BAA-AAA spreads, because investors are sensitive to the timing of defaults: defaults of BAA
bonds are more likely to happen in recessions, when risk aversion is high. Therefore, investors want to be
compensated with high yields for a small average amount of exposure to default.
IV. Time-Varying Risk of Rare Economic Disasters
Rietz (1988) introduce the idea that rare disasters in consumption make investors worry more about
holding stocks and hence may explain a large equity premium. Disasters are rare, so their frequency, size,
and duration are difficult to measure. Although economic disasters provide new interpretations of average
risk premiums, they do not provide any mechanism for stock return volatility. To generate volatility and
predictability of stock returns, the probability of a disaster has to vary over time, so that consumption
growth is heteroskedastic. Recent literature combines disasters with such time-varying risk.
Barro (2006) calibrates disaster probabilities from the twentieth century global history (World War I,
the Great Depression, and World War II), and explains high equity premiums and volatile stock returns.
His empirical analyses indicate a disaster probability of 1.5-2% per year with a distribution of declines
in per capita GDP ranging between 15% and 64%. Characterizing economic disasters or sharp economic
contractions by the time-varying probability of disaster, the size of contractions, the probability of default,
and the extent of default, Barro (2006) provides evidence for large equity premium during highly volatile
5
periods.
In a follow-up paper, Barro (2009) introduces a model with Epstein-Zin-Weil preferences and rare
economic disasters, and explains large equity premium if the coefficient of relative risk aversion is in the
range of three and four. Bali and Engle (2010) investigate the intertemporal capital asset pricing model
(ICAPM) of Merton (1973) using the dynamic conditional correlation (DCC) model of Engle (2002).
The risk-aversion coefficient within the ICAPM-DCC model is estimated to be between two and four and
highly significant. The risk-aversion estimates in Bali and Engle (2010) provide supporting evidence for
the theoretical findings of Barro (2009).
Overall, we conclude that asset pricing models with time-varying risk of rare economic disasters provide further evidence that during sharp contractions or extremely large falls of the market, aggregate riskaversion becomes higher and individuals with higher risk-aversion demand higher expected return from
risky financial securities to induce them to take higher perceived risk.
V. Behavioral Biases
Black (1988) indicates that the level of the market is affected by the public’s confidence in the market
and the breadth of its participation. The market will be higher when participation is broad instead of
narrow. When more people are willing to share in the risk of the market, each one bears less risk. This
means that the expected return on the market can be lower and the market level higher. He calls this element
“liquidity”. When there is broad public participation in the stock market, the level of the market will be
high, and a change in one group’s desired holdings will not cause a big change in price. Such a market will
be less volatile than one with narrow participation, all else equal.
Black (1988) argues that people may avoid trading because they have little confidence in the market.
They may feel that the market is too volatile; that it may close unexpectedly just when they want to trade;
that it may be so congested at high volume times that trading will be hard; or that traders with extra
information on the market behavior have an unfair advantage. Feelings that have no apparent factual basis
can affect liquidity too. An increase in volatility, decline in output growth, or extremely large falls of the
market can scare people off, even when it is due to a change in tastes or technology. Since the causes of
volatility or the causes of sharp declines in the economy and financial markets are not observable, even
economists may decide that an increase in asset prices is capricious, and they may urge investors to be
cautious.
6
Black (1988) emphasizes that whatever the original reasons for recessions or financial market downturns, it frightens people. The sharp decline and the high volatility may cause people to withdraw from the
market. The market will be low when participation is narrow. In bad times, when stocks are trading at low
prices, investors could be well aware that prices are likely to go up, but they may worry about taking on
the extra risk associated with holding more stocks. Investors may also be facing more risk in times when
expected returns are high. During the financial crisis, we observe significant declines in stock prices and
still households do not want to buy more stocks. A plausible explanation is that they are worried about
losing their jobs and prefer holding cash. Overall, these behavioral biases indicate that investors require
higher expected return to induce them to invest on riskier financial products during recessionary periods.
Campbell and Cochrane (1999) also provide supporting evidence for the aforementioned psychological
factors. They indicate that variation across assets in expected returns is driven by variation across assets
in covariances with recessions far more than by variation across assets in covariances with consumption
growth. Therefore, investors fear stocks because they do badly in occasional serious recessions, not because stock returns are correlated with declines in wealth or consumption. Campbell and Cochrane (1999)
indicate that during recessions participation will be narrow because investors will reduce their demand for
risky assets. The reduced asset demand will decrease current asset prices and increase expected future
returns.
VI. Intertemporal relation between option implied riskiness and future market returns during recessions
In this section, we estimate the intertemporal relation between option implied measures of riskiness
and future market returns during economic recessions. Specifically, we estimate the following predictive
regression of one-month ahead excess market returns on the riskiness measures with recession dummy:
(
)
(
)
Retm,t+1 = α0 + α1 Dt + β1 RtAS,Q + β2 RtAS,Q Dt + λXt + εm,t+1 ,
(15)
where Retm,t+1 denotes the excess market return in month t+1, RtAS,Q is the option implied Q-measure of
aggregate riskiness in month t, and Xt includes a large set of control variables: the S&P 500 index option implied volatility (VIX), aggregate idiosyncratic volatility (IVOL), default spread (DEF), term spread
(TERM), relative T-bill rate (RREL), dividend-price ratio (DIV), growth rate of industrial production (IP),
7
unemployment rate (UNEMP), consumption-to-wealth ratio (CAY), and the lagged excess market return
(RET).
The Federal Reserve Bank of Chicago denotes the CFNAI index value of -0.7 as a turning point indicating economic recession. Hence, we generate a recession dummy based on the CFNAI index. Dt is the
recession dummy in equation (15) and takes the value of one when the CFNAI index is below -0.7 and
zero otherwise. If the intertemporal relation between riskiness and future market returns is stronger during
economic recessions, we expect the slope coefficient attached to the interaction dummy, RtAS,Q Dt , to be
positive; β2 is expected to be positive and statistically significant in equation (15).
Table V of this online appendix presents results from the predictive regressions of one-month ahead
excess market returns on the option implied measures of riskiness during recessions vs. normal/boom
periods. The results, reported for the CRSP and S&P 500 indices, show that the slope coefficient β2
is positive and highly significant for all measures of riskiness without any exception, implying that the
predictive power of aggregate riskiness is indeed stronger during economic downturns.
8
References
Aumann, R. J., Serrano, R., 2008. An economic index of riskiness. Journal of Political Economy 116,
810–836.
Bakshi, G., Madan, D., 2000. Spanning and derivative-security valuation. Journal of Financial Economics
55, 205–238.
Bali, T. G., Engle, R. F., 2010. The intertemporal capital asset pricing model with dynamic conditional
correlations. Journal of Monetary Economics 57, 377–390.
Barro, R. J., 2006. Rare disasters and asset markets in the twentieth century. Quarterly Journal of Economics 121, 823–866.
Barro, R. J., 2009. Rare disasters, asset prices, and welfare costs. American Economic Review 99, 243–264.
Black, F., 1988. An equilibrium model of the crash. In: S. Fischer (ed.) NBER Macroeconomics Annual
MIT Press, Cambridge, Massachusetts.
Campbell, J., Cochrane, J., 1999. Force of habit: A consumption-based explanation of aggregate stock
market behavior. Journal of Political Economy 107, 205–251.
Cecchetti, S. G., Lam, P. S., Mark, N. C., 1990. Mean reversion in equilibrium asset prices. American
Economic Review 80, 398–418.
Chan, Y. L., Kogan, L., 2002. Catching up with the joneses: Heterogeneous preferences and the dynamics
of asset prices. Journal of Political Economy 110, 1255–1285.
Chen, L., Dufresne, P. C., Goldstein, R. S., 2009. On the relation between the credit spread puzzle and the
equity premium puzzle. Review of Financial Studies 22, 3367–3409.
Engle, R. F., 2002. Dynamic conditional correlation: A simple class of multivariate generalized autoregressive conditional heteroskedasticity models. Journal of Business and Economic Statistics 20, 339–350.
Merton, R. C., 1973. An intertemporal asset pricing model. Econometrica 41, 867–887.
Rietz, T. A., 1988. The equity risk premium: A solution. Journal of Monetary Economics 22, 117–131.
Wachter, J. A., 2006. A consumption-based model of the term structure of interest rates. Journal of Financial Economics 79, 365–399.
9
Table I: Empirical Test for the Positive Relation between Riskiness and Market Returns
( AS ))
( AS
−γ and Ret
This table presents the covariance between ( CCt+1
)
R
−
E
Rt
for the risk aversion
m,t+1
t
t
parameter values of γ = 2, 3, and 4, and for each measure of riskiness (1-, 3-, 6-, and 12-month). The
sample period is from January 1996 to October 2010.
γ=2
γ=3
γ=4
RtAS,Q 1-month
-0.000107
-0.000162
-0.000219
RtAS,Q 3-month
-0.000073
-0.000109
-0.000147
RtAS,Q 6-month
-0.000054
-0.000081
-0.000109
RtAS,Q 12-month
-0.000044
-0.000066
-0.000089
10
Table II: Option Implied Measures of Riskiness and Future Market Returns
This table presents results from the predictive regressions of one-month ahead excess market returns on the
option implied measures of riskiness obtained from the S&P 500 index options with 1, 3, 6, and 12 months
to maturity. Panel A presents results for the value-weighted and equal-weighted NYSE/AMEX/NASDAQ
(CRSP) index. Panel B presents results for the S&P 500, Dow Jones Industrial Average (DJIA), and NASDAQ. The control variables include the aggregate idiosyncratic volatility (IVOL), default spread (DEF),
term spread (TERM), relative T-bill rate (RREL), dividend-price ratio (DIV), growth rate of industrial
production (IP), unemployment rate (UNEMP), consumption-to-wealth ratio (CAY), and the lagged excess
market return (RET). The Newey-West (1987) t-statistics are reported in parentheses. The last row presents
the R2 values. The sample period is January 1996 - October 2010.
Value-Weighted CRSP Index
Equal-Weighted CRSP Index
1-month 3-month 6-month 12-month 1-month 3-month 6-month 12-month
intercept 0.2028 0.2860 0.3042
(3.35) (3.59) (3.98)
RtAS,Q
0.0076 0.0175 0.0209
(2.99) (3.07) (3.75)
0.2892
(4.39)
0.1451 0.2373 0.2642
(1.76) (2.42) (2.94)
0.2465
(2.98)
0.0239
(4.61)
0.0067 0.0167 0.0209
(2.01) (2.55) (3.48)
0.0236
(4.00)
IVOL -0.5235 -1.0026 -0.9109 -0.6072
(-0.62) (-1.19) (-1.10) (-0.77)
-0.0382 -0.4687 -0.3984 -0.0843
(-0.03) (-0.36) (-0.32) (-0.07)
DEF
-1.3618 -0.2905 -0.4699 -1.1217
(-0.67) (-0.15) (-0.25) (-0.61)
-2.9326 -1.5311 -1.6647 -2.3500
(-1.97) (-0.94) (-1.07) (-1.55)
TERM 0.5096 0.6305 0.6804
(1.34) (1.51) (1.64)
0.6260
(1.62)
0.6178 0.7959 0.8905
(1.15) (1.44) (1.63)
0.8263
(1.55)
RREL
0.1735 0.5659 0.6117
(0.42) (1.37) (1.42)
0.5522
(1.25)
0.3044 0.6584 0.7082
(0.63) (1.36) (1.40)
0.6450
(1.23)
DIV
0.0558 0.0722 0.0803
(2.05) (2.61) (2.89)
0.0848
(3.04)
0.0418 0.0603 0.0697
(1.10) (1.54) (1.79)
0.0737
(1.87)
IP
1.5930 1.4071 1.4602
(2.00) (1.89) (1.98)
1.5527
(2.08)
1.4822 1.3134 1.3626
(1.57) (1.48) (1.55)
1.4579
(1.64)
UNEMP -0.0176 -0.0310 -0.0325 -0.0280
(-2.76) (-3.14) (-3.52) (-4.05)
-0.0150 -0.0293 -0.0322 -0.0274
(-1.77) (-2.55) (-3.22) (-3.40)
CAY
0.0184 0.1624 0.1614
(0.10) (0.92) (0.92)
0.0923
(0.52)
0.0816 0.1978 0.1945
(0.30) (0.75) (0.74)
0.1256
(0.47)
RET
0.1621 0.1642 0.1412
(1.95) (2.00) (1.70)
0.1390
(1.64)
0.2747 0.2905 0.2789
(3.26) (3.34) (3.29)
0.2787
(3.26)
R2
15.17% 17.33% 17.59% 17.40%
11
12.93% 14.48% 15.01% 14.77%
12
0.0390
(4.48)
0.5520
(1.30)
0.1968 0.5638 0.6071
(0.49) (1.42) (1.46)
0.0562 0.0718 0.0798
(2.24) (2.80) (3.04)
1.4830 1.3050 1.3545
(2.01) (1.89) (1.98)
RREL
DIV
IP
14.86% 17.15% 17.39% 17.24%
R2
0.0929
(1.08)
0.1188 0.1189 0.0958
(1.38) (1.43) (1.14)
RET
0.1155
(0.72)
0.0483 0.1821 0.1805
(0.28) (1.16) (1.15)
CAY
UNEMP -0.0155 -0.0284 -0.0298 -0.0257
(-2.47) (-2,85) (-3.12) (-3.65)
1.4407
(2.09)
0.0841
(3.19)
0.5024
(1.31)
TERM 0.3788 0.5051 0.5517
(1.01) (1.22) (1.33)
-3.1749 -1.9136 -2.0571 -2.7102
(-2.25) (-1.21) (-1.34) (-1.82)
1.4713
(1.56)
0.1194
(2.46)
0.6878
(0.94)
1.1665
(1.50)
0.0751
(1.20)
0.2354
(0.71)
8.80% 11.65% 12.64% 12.50%
0.0872 0.0928 0.0772
(1.37) (1.48) (1.24)
0.1597 0.3573 0.3485
(0.44) (1.10) (1.07)
-0.0248 -0.0479 -0.0535 -0.0464
(-2.32) (-3.16) (-3.59) (-3.74)
1.5329 1.2399 1.3203
(1.48) (1.27) (1.40)
0.0672 0.0946 0.1117
(1.36) (1.94) (2.30)
0.1265 0.6995 0.7837
(0.18) (1.00) (1.09)
0.7805 1.0708 1.2472
(0.99) (1.35) (1.55)
-3.6866 -1.9912 -2.3512 -3.4810
(-1.33) (-0.76) (-0.93) (-1.38)
0.0105 0.0268 0.0340
(2.94) (3.38) (4.25)
0.4474
(3.73)
DEF
0.0226
(4.42)
0.2724 0.4190 0.4700
(2.36) (3.14) (3.59)
-1.2199 -1.9962 -1.8845 -1.3918
(-0.75) (-1.24) (-1.18) (-0.88)
0.0070 0.0165 0.0197
(2.90) (2.92) (3.46)
0.2792
(4.21)
IVOL -0.4253 -0.8884 -0.7902 -0.5057
(-0.59) (-1.20) (-1.09) (-0.74)
RtAS,Q
intercept 0.1945 0.2752 0.2927
(3.34) (3.47) (3.75)
NASDAQ index
1-month 3-month 6-month 12-month
0.0154
(3.10)
0.1863
(2.54)
1.4103
(2.44)
0.0593
(1.99)
0.2086
(0.47)
0.0213
(0.05)
0.0409
(0.50)
0.1062
(0.58)
10.70% 12.21% 12.18% 12.17%
0.0556 0.0566 0.0424
(0.66) (0.71) (0.53)
0.0691 0.1534 0.1509
(0.36) (0.87) (0.85)
-0.0070 -0.0170 -0.0175 -0.0149
(-1.07) (-1.65) (-1.74) (-1.97)
1.4418 1.3153 1.3523
(2.35) (2.25) (2.34)
0.0394 0.0511 0.0560
(1.43) (1.81) (1.93)
–0.0199 0.2240 0.2454
(-0.05) (0.54) (0.57)
-0.0991 0.0305 0.0491
(-0.25) (0.07) (0.11)
-2.2838 -1.5842 -1.6603 -2.1179
(-1.55) (-1.05) (-1.12) (-1.41)
-0.2269 -0.5681 -0.4803 -0.2887
(-0.29) (-0.70) (-0.60) (-0.37)
0.0044 0.0114 0.0133
(1.95) (2.14) (2.37)
0.1220 0.1856 0.1941
(1.93) (2.25) (2.33)
DJIA index
1-month 3-month 6-month 12-month
Panel B. Alternative Stock Market Indices: S&P 500, NASDAQ, and DJIA
S&P 500 index
1-month 3-month 6-month 12-month
Table II (continued)
Table III: VIX and Future Market Returns
This table presents results from the predictive regressions of one-month ahead excess market returns on
the S&P 500 index option implied volatility (VIX). Equity market index is proxied by the value-weighted
and equal-weighted CRSP, S&P 500, NASDAQ, and DJIA indices. The control variables include the aggregate idiosyncratic volatility (IVOL), default spread (DEF), term spread (TERM), relative T-bill rate
(RREL), dividend-price ratio (DIV), growth rate of industrial production (IP), unemployment rate (UNEMP), consumption-to-wealth ratio (CAY), and the lagged excess market return (RET). The Newey-West
(1987) t-statistics are reported in parentheses. The last row presents the R2 values. The sample period is
January 1996 - October 2010.
VW CRSP EW CRSP S&P500 NASDAQ DJIA
intercept
0.0806
(1.28)
0.0321
(0.39)
0.0811
(1.38)
0.1029
(0.97)
0.0503
(0.90)
VIX
3.4449
(1.83)
1.2405
(0.54)
3.2763
(1.78)
5.2115
(1.74)
2.0387
(1.18)
IVOL
-1.0181
(-1.44)
-0.0695
(-0.06)
-0.9340 -2.0548 -0.5589
(-1.49) (-1.30) (-0.84)
DEF
-1.2964
(-0.77)
0.8074
(0.37)
-1.6770 -1.6239 -1.3288
(-1.06) (-0.58) (-0.89)
TERM
-0.1436
(-0.33)
0.0019
(0.00)
-0.2257 -0.1379 -0.4835
(-0.54) (-0.18) (-1.20)
RREL
0.4963
(1.16)
0.5211
(1.04)
0.5001
(1.23)
0.6035
(0.85)
0.1773
(0.45)
DIV
0.0343
(1.11)
0.0254
(0.61)
0.0355
(1.25)
0.0358
(0.70)
0.0262
(0.93)
IP
1.4387
(1.66)
1.4487
(1.37)
1.3318
(1.66)
1.2689
(1.14)
1.3447
(2.04)
UNEMP
-0.0011
(-0.23)
0.0003
(0.06)
-0.0002 -0.0018 0.0027
(-0.05) (-0.25) (0.63)
CAY
0.0519
(0.24)
0.1752
(0.55)
0.0792
(0.40)
0.1987
(0.52)
0.0928
(0.47)
RET
0.1997
(2.19)
0.2720
(2.93)
0.1523
(1.63)
0.1185
(1.79)
0.0700
(0.75)
R2
12.88%
10.93%
12.92%
7.54%
9.90%
13
14
-0.0045 -0.0049 -0.0055 -0.0056
(-1.82) (-1.91) (-2.14) (-2.16)
0.0349 0.0385 0.0479
(1.44) (1.57) (1.90)
RREL -0.0028 -0.0031 -0.0034 -0.0036
(-1.39) (-1.48) (-1.66) (-1.71)
0.4181
(1.14)
0.0003
(0.18)
0.0153 0.0175 0.0239
(0.77) (0.87) (1.16)
0.3604 0.4018 0.4343
(0.97) (1.10) (1.19)
DIV
IP
UNEMP 0.0005 0.0011 0.0012
(0.25) (0.55) (1.75)
3.88%
R2
4.74%
0.0391 0.0330 0.0295
(0.95) (0.79) (0.71)
RET
4.13%
0.2106 0.2023 0.1959
(1.95) (1.84) (1.75)
CAY
4.56%
0.0298
(0.73)
0.2293
(2.10)
7.25%
7.56%
8.40%
0.2048 0.1991 0.1882
(4.58) (4.43) (4.28)
0.0542 0.0404 0.0286
(0.41) (0.30) (0.21)
0.0009 0.0016 0.0019
(0.39) (0.66) (0.79)
0.2745 0.3262 0.3798
(0.59) (0.70) (0.82)
-0.0645 -0.1134 -0.1583 -0.1620
(-0.25) (-0.41) (-0.57) (-0.58)
TERM -0.0721 -0.1151 -0.1430 -0.1553
(-0.33) (-0.51) (-0.64) (-0.68)
0.0259
(1.20)
0.3670 -0.1156 -0.7243 -0.2424
(0.42) (-0.12) (-0.70) (-0.25)
0.3850
(0.46)
0.8892 0.5083 0.1101
(1.19) (0.60) (0.12)
DEF
7.94%
0.1926
(4.41)
0.0741
(0.54)
0.0007
(0.29)
0.3456
(0.75)
0.0486
(1.88)
0.1919
(0.27)
0.4110 0.3058 0.2063
(0.59) (0.44) (0.29)
0.0256
(2.27)
IVOL -0.2372 -0.3440 -0.4120 -0.4402
(-0.45) (-0.62) (-0.75) (-0.80)
0.3827 0.0159 0.0303
(0.46) (1.56) (2.86)
-3.9798 -1.1460 -1.1911 -0.6970
(-0.46) (-1.11) (-1.46) (-0.90)
0.0206
(2.41)
RVAR -5.0884 -1.5462 -1.4898 -1.1959
(-0.66) (-2.05) (-2.84) (-2.33)
0.4845 0.0123 0.0211
(0.65) (1.04) (2.21)
-0.0178 -0.0181 -0.0181 -0.0156
(-1.17) (-1.17) (-1.18) (-1.02)
intercept -0.0035 -0.0033 -0.0032 -0.0013
(-0.29) (-0.27) (-0.26) (-0.11)
RtAS,P
Equal-Weighted CRSP Index
1-month 3-month 6-month 12-month
Value-Weighted CRSP Index
1-month 3-month 6-month 12-month
0.0203
(2.46)
0.4242
(0.52)
0.2665
(2.49)
0.0006
(0.35)
0.4485
(1.30)
0.0144
(0.69)
3.68%
3.92%
4.44%
4.43%
-0.0069 -0.0117 -0.0136 -0.0151
(-0.16) (-0.27) (-0.32) (-0.36)
0.2488 0.2407 0.2351
(2.34) (2.22) (2.13)
0.0008 0.0013 0.0015
(0.42) (0.68) (0.76)
0.3925 0.4296 0.4580
(1.12) (1.26) (1.34)
0.0037 0.0059 0.0116
(0.19) (0.30) (0.58)
-0.0028 -0.0031 -0.0034 -0.0036
(-1.48) (-1.56) (-1.73) (-1.82)
-0.1122 -0.1497 -0.1740 -0.1917
(-0.53) (-0.68) (-0.80) (-0.87)
0.9206 0.5737 0.2205
(1.24) (0.69) (0.25)
-0.2126 -0.3009 -0.3603 -0.3987
(-0.42) (-0.56) (-0.68) (-0.75)
-3.9842 -1.7136 -1.6467 -1.4139
(-0.53) (-2.32) (-3.57) (-3.14)
0.3752 0.0112 0.0189
(0.51) (0.89) (1.96)
-0.0033 -0.0033 -0.0033 -0.0014
(-0.28) (-0.28) (-0.27) (-0.12)
S&P 500 index
1-month 3-month 6-month 12-month
This table presents results from the predictive regressions of one-month ahead excess market returns on the generalized physical measures of riskiness
obtained from daily returns on the S&P 500 index over the past 1, 3, 6, and 12 months. The results are reported for the value-weighted and equalweighted CRSP and the S&P 500 indices. The control variables include the physical measure of market variance (RVAR), aggregate idiosyncratic
volatility (IVOL), default spread (DEF), term spread (TERM), relative T-bill rate (RREL), dividend-price ratio (DIV), growth rate of industrial
production (IP), unemployment rate (UNEMP), consumption-to-wealth ratio (CAY), and the lagged excess market return (RET). The Newey-West
(1987) t-statistics are reported in parentheses. The last row presents the R2 values. The sample period is January 1960 - October 2010.
Table IV: Physical Measures of Riskiness and Future Market Returns for the Long Sample Period
15
This table presents results from the predictive regressions of one-month ahead excess market returns on the option implied measures of riskiness during
recessions vs. normal/boom periods. The recession dummy (Dum) in predictive regressions takes the value of one when the CFNAI index is below
-0.7 and zero otherwise. The results are reported for the value-weighted and equal-weighted CRSP and the S&P 500 indices. The control variables
include the S&P 500 index option implied volatility (VIX), aggregate idiosyncratic volatility (IVOL), default spread (DEF), term spread (TERM),
relative T-bill rate (RREL), dividend-price ratio (DIV), growth rate of industrial production (IP), unemployment rate (UNEMP), consumption-towealth ratio (CAY), and the lagged excess market return (RET). The Newey-West (1987) t-statistics are reported in parentheses. The last row presents
the R2 values. The sample period is January 1996 - October 2010.
Table V: Option Implied Measures of Riskiness and Future Market Returns During Recessions
16
0.2914
(4.16)
0.3698 0.5507 0.6632
(0.90) (1.30) (1.65)
0.3461 -0.1308 -0.1749 -0.3170
(0.65) (-0.21) (-0.26) (-0.46)
0.0330 0.0650 0.0728
(1.26) (2.41) (2.74)
1.7190 0.9814 1.0222
(2.10) (1.10) (1.17)
-0.0100 -0.0260 -0.0291 -0.0256
(-1.43) (-2.17) (-2.94) (-3.41)
-0.0439 0.0234 -0.0017 -0.0888
(-0.20) (0.10) (-0.01) (-0.39)
DEF
TERM
RREL
DIV
IP
UNEMP
19.58% 19.73% 20.97% 21.27%
R2
0.1306
(1.38)
0.1956 0.1650 0.1395
(2.03) (1.77) (1.46)
RET
1.1195
(1.25)
0.0764
(2.79)
0.6491
(1.64)
-3.3052 -3.3749 -4.0925 -5.3084
(-2.02) (-1.76) (-2.16) (-3.05)
DEF
-1.1352 -0.9749 -0.9133 -0.6065
(-1.54) (-1.31) (-1.22) (-0.80)
IVOL
0.4497
(0.22)
2.7942 0.7688 0.6485
(1.38) (0.34) (0.30)
0.0199
(3.28)
RtAS,Q ×Dum 0.0089 0.0088 0.0134
(3.39) (2.09) (2.78)
VIX
0.0208
(3.43)
0.0000 -0.0279 -0.0321 -0.0370
(0.00) (-1.65) (-1.84) (-2.01)
0.1356 0.2682 0.2984
(2.25) (2.93) (3.74)
Value-Weighted CRSP Index
1-month 3-month 6-month 12-month
-0.0033 0.0138 0.0176
(-0.81) (1.77) (2.53)
RtAS,Q
Dum
intercept
Table V (continued)
0.2727
(3.24)
0.0211
(2.56)
0.0236
(3.51)
0.4242
(0.35)
0.9211
(1.67)
1.0088
(0.99)
0.0746
(2.05)
0.2395
2.96
15.87% 16.36% 17.52% 17.83%
0.2698 0.2614 0.2450
(2.92)
3.01
(2.89)
0.1193 0.1300 0.1014 -0.0037
(0.36) (0.39) (0.31) (-0.01)
-0.0106 -0.0285 -0.0325 -0.0278
(-1.22) (-2.25) (-3.28) (-3.30)
1.9671 0.8422 0.9047
(2.08) (0.83) (0.91)
0.0282 0.0632 0.0718
(0.77) (1.77) (2.05)
0.6091 -0.2301 -0.2652 -0.4515
(1.01) (-0.29) (-0.33) (-0.53)
0.6360 0.8005 0.9484
(1.17) (1.45) (1.78)
-1.1527 -1.4336 -2.2154 -3.6408
(-0.51) (-0.55) (-0.85) (-1.45)
-0.2318 0.0165 0.0753
(-0.20) (0.01) (0.06)
0.4450 -1.5223 -1.6280 -1.7884
(0.19) (-0.57) (-0.66) (-0.75)
0.0082 0.0086 0.0136
(2.69) (1.66) (2.16)
-0.0023 0.0159 0.0206
(-0.50) (1.91) (2.95)
0.0092 -0.0339 -0.0380 -0.0442
(0.74) (-1.76) (-1.94) (-2.17)
0.0986 0.2498 0.2853
(1.21) (2.45) (3.27)
Equal-Weighted CRSP Index
1-month 3-month 6-month 12-month
0.2812
(3.88)
0.4677
(0.23)
0.0189
(3.30)
0.0195
(3.25)
0.5206
(1.35)
1.0085
(1.16)
0.0763
(2.83)
0.0842
(0.84)
19.78% 19.62% 20.85% 21.12%
0.1494 0.1179 0.0928
(1.53) (1.22) (0.93)
-0.0210 0.0470 0.0218 -0.0591
(-0.10) (0.22) (0.10) (-0.28)
-0.0072 -0.0235 -0.0264 -0.0233
(-1.07) (-1.94) (-2.58) (-3.04)
1.5552 0.8799 0.9153
(-1.98) (1.02) (1.08)
0.0312 0.0649 0.0726
(1.28) (2.48) (2.77)
0.3081 -0.1181 -0.1626 -0.2900
(0.60) (-0.18) (-0.24) (-0.41)
0.2033 0.4235 0.5307
(0.51) (1.00) (1.33)
-3.5241 -3.7029 -4.4098 -5.5470
(-2.28) (-2.05) (-2.46) (-3.38)
-1.0463 -0.8657 -0.7948 -0.5033
(-1.65) (-1.33) (-1.21) (-0.76)
2.8105 0.7723 0.6596
(1.43) (0.36) (0.32)
0.0093 0.0084 0.0129
(3.71) (2.10) (2.79)
-0.0044 0.0129 0.0165
(-1.15) (1.68) (2.34)
-0.0024 -0.0274 -0.0316 -0.0360
(-0.23) (-1.57) (-1.75) (-1.89)
0.1219 0.2576 0.2868
(2.10) (2.78) (3.44)
S&P 500 index
1-month 3-month 6-month 12-month