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Finance and Economics Discussion Series: 2011-45 Screen Reader version ^♣

Risk, Uncertainty, and Expected Returns¹

Turan G. Bali² and Hao Zhou³

First Draft: March 2011
This Version: September 2011

Keywords: Risk, uncertainty, expected returns, ICAPM, time-series and cross-sectional stock returns, variance risk premium, consumption-based asset pricing model.

Abstract:

A consumption-based asset pricing model with risk and uncertainty implies that the time-varying exposures of equity portfolios to the market and uncertainty factors carry positive risk premiums. The empirical results from the size, book-to-market, and industry portfolios as well as individual stocks indicate that the conditional covariances of equity portfolios (individual stocks) with market and uncertainty predict the time-series and cross-sectional variation in stock returns. We find that equity portfolios that are highly correlated with economic uncertainty proxied by the variance risk premium (VRP) carry a significant premium relative to portfolios that are uncorrelated or lowly correlated with VRP. The insignificant alpha estimates indicate that the conditional asset pricing model proposed in the paper also explains the industry, size, and value premiums.

JEL Classification:G10, G11, C13.

1 Introduction

This paper investigates whether the market price of risk and the market price of uncertainty are significantly positive and whether they predict the time-series and cross-sectional variation in stock returns. Although the literature has so far shown how uncertainty impacts optimal allocation decisions and asset prices, the results have been provided based on a theoretical model.⁴ Earlier studies do not pay much attention to empirical testing of whether the exposures of equity portfolios and individual stocks to market and uncertainty factors predict their future returns. Extending on the original consumption-based CAPM, we show that in the presence of volatility uncertainty, the traditional risk-return regression needs to be augmented because both market risk and volatility uncertainty carry a positive premium. We introduce a conditional asset pricing model in which the consumption growth and its volatility follow the joint dynamics. According to our model with time-varying volatility of the consumption growth and the volatility uncertainty in the consumption growth process, the premium on equity is composed of two separate terms; the first term compensates for the classic consumption risk in a standard consumption-based CAPM and the second term represents a true premium for variance risk. The model's parameter restrictions imply that the variance risk premium embedded in the equity risk premium is always positive.

Following Britten-Jones and Neuberger (2000), Jiang and Tian (2005), and Carr and Wu (2009), we define the variance risk premium (VRP) as the difference between expected variance under the risk-neutral measure and expected variance under the objective measure.⁵ We generate several proxies for financial and economic uncertainty and then compute the correlations between uncertainty variables and VRP. The first set of measures can be viewed as macroeconomic uncertainty proxied by the conditional variance of the US output growth and the conditional variance of the Chicago Fed National Activity Index (CFNAI). The second set of uncertainty measures is based on the extreme downside risk of financial institutions obtained from the left tail of the time-series and cross-sectional distribution of financial firms' returns. The third uncertainty variable is related to the health of the financial sector proxied by the credit default swap (CDS) index. The last uncertainty variable is based on the aggregate measure of investors' disagreement about individual stocks trading at NYSE, AMEX, and NASDAQ. We find that the variance risk premium is strongly and positively correlated with all measures of uncertainty considered in the paper. Our results indicate that VRP can be viewed as a sound proxy for financial and economic uncertainty.⁶

Anderson, Ghysels, and Juergens (2009) introduce a model in which the volatility, skewness and higher order moments of all returns are known exactly, whereas there is uncertainty about mean returns. In other words, asset returns are uncertain only because mean returns are not known. In their model, investors' uncertainty in mean returns is defined as the dispersion of predictions of mean market returns obtained from the forecasts of aggregate corporate profits. They find that the price of uncertainty is significantly positive and explains the cross-sectional variation in stock returns. Bekaert, Engstrom, and Xing (2009) investigate the relative importance of economic uncertainty and changes in risk aversion in the determination of equity prices. Different from Knightian uncertainty or uncertainty originated from disagreement of professional forecasters, Bekaert, Engstrom, and Xing (2009) focus on economic uncertainty proxied by the conditional volatility of dividend growth, and find that both the conditional volatility of cash flow growth and time-varying risk aversion are important determinants of equity returns.

Different from the aforementioned studies, we propose a model in which volatility uncertainty (proxied by VRP) plays a significant role along with the standard consumption risk. After introducing a two-factor model with risk and uncertainty, we investigate the significance of risk-return and uncertainty-return coefficients using the time-series and cross-sectional data. Our empirical analyses are based on the size, book-to-market, and industry portfolios as well as individual stocks. We first use the dynamic conditional correlation (DCC) model of Engle (2002) to estimate equity portfolios' (individual stocks') conditional covariances with the market portfolio and then test whether the conditional covariances predict future returns on equity portfolios (individual stocks). We find the risk-return coefficients to be positive and highly significant, implying a strongly positive link between expected return and market risk. Similarly, we use the DCC model to estimate equity portfolios' (individual stocks') conditional covariances with the variance risk premia and then test whether the conditional covariances with VRP predict future returns on equity portfolios (individual stocks). The results indicate a significantly positive market price of uncertainty. Equity portfolios (individual stocks) that are highly correlated with uncertainty (proxied by VRP) carry a significant premium relative to portfolios (stocks) that are uncorrelated or lowly correlated with VRP. Such a positive relationship between return and uncertainty is also consistent with our model's implication that the intertemporal elasticity of substitution or IES is larger than one--i.e., agents prefer an earlier resolution of uncertainty, hence uncertainty (proxied by VRP) carries a positive premium.

We also examine the empirical validity of the conditional asset pricing model by testing the hypothesis that the conditional alphas on the size, book-to-market, and industry portfolios are jointly zero. The test statistics fail to reject the null hypothesis, indicating that the two-factor model explains the time-series and cross-sectional variation in equity portfolios. Finally, we investigate whether the model explains the return spreads between the high-return (long) and low-return (short) equity portfolios (Small-Big for the size portfolios; Value-Growth for the book-to-market portfolios; and HiTec-Telcm for the industry portfolios). The results from testing the equality of conditional alphas for high-return and low-return portfolios provide no evidence for a significant alpha for "Small-Big", "Value-Growth", and "HiTec-Telcm" arbitrage portfolios, indicating that the two-factor model proposed in the paper provides both statistical and economic success in explaining stock market anomalies. Overall, the DCC-based conditional covariances capture the time-series and cross-sectional variation in returns on size, book-to-market, and industry portfolios because the essential tests of the model are passed: (i) significantly positive risk-return and uncertainty-return tradeoffs; (ii) the conditional alphas are jointly zero; and (iii) the conditional alphas for high-return and low-return portfolios are not statistically different from each other.⁷ These results are robust to using an alternative specification of the time-varying conditional covariances with an asymmetric GARCH model, using a larger cross-section of equity portfolios in asset pricing tests, and after controlling for a wide variety of macroeconomic variables.

The rest of the paper is organized as follows. Section 2 defines the variance risk premium and provides its empirical measurement. Section 3 presents the consumption-based asset pricing model with risk and uncertainty. Section 4 describes the data. Section 5 outlines the estimation methodology. Section 6 presents the empirical results. Section 7 provides a battery of robustness checks. Section 8 concludes the paper.

2 Variance Risk Premium and Empirical Measurement

The central empirical variable of this paper, as a proxy for economic uncertainty, is the market variance risk premium (VRP)--which is not directly observable but can be estimated from the difference between model-free option-implied variance and the conditional expectation of realized variance (Zhou, 2010). There are emerging evidences that VRP can forecast stock market returns (Bollerslev, Tauchen, and Zhou, 2009; Drechsler and Yaron, 2011), Treasury returns (Zhou, 2010; Mueller, Vedolin, and Zhou, 2011), credit spreads (Wang, Zhou, and Zhou, 2011; Buraschi, Trojani, and Vedolin, 2009), and international stock market returns (Londono, 2010; Bollerslev, Marrone, Xu, and Zhou, 2011). Here we are going to use the covariance of asset returns with the variance risk premium to help explain the time-series and cross-section of stock portfolio and individual stock returns.

2.1 Variance Risk Premium: Definition and Measurement

In order to define the model-free implied variance, let denote the price of a European call option maturing at time with strike price , and denote the price of a time zero-coupon bond maturing at time . As shown by Carr and Madan (1998) and Britten-Jones and Neuberger (2000), among others, the market's risk-neutral expectation of the return variance $\sigma^2_{t+1}$ conditional on the information set $\Omega_t$ , or the implied variance $IV_{t,t+1}$ , can be expressed in a "model-free" fashion as a portfolio of European calls,

$\displaystyle IV_{t,t+1} \ \equiv \ \operatorname{E}^Q \left[\sigma^2_{t+1}\vert\Omega_t\right] \ = \ 2 \int_{0}^{\infty}{C_t\left(t+1,\frac{K}{B(t,t+1)}\right) - C_t\left(t,{K}\right) \over K^2}\,dK,$

(1)

which relies on an ever increasing number of calls with strikes spanning from zero to infinity.⁸ This equation follows directly from the classical result in Breeden and Litzenberger (1978), that the second derivative of the option call price with respect to strike equals the risk-neutral density, such that all risk neutral moments payoff can be replicated by the basic option prices (Bakshi and Madan, 2000).

In order to define the actual return variance, let denote the logarithmic price of the asset. The realized variance over the discrete to time interval can be measured in a "model-free" fashion by

$\displaystyle RV_{t,t+1} \equiv \sum_{j=1}^{n}\left[p_{t+\frac{j}{n}} - p_{t+\frac{j-1}{n}}\right]^2 \longrightarrow\sigma^2_{t+1},$

(2)

where the convergence relies on $n\rightarrow\infty$ ; i.e., an increasing number of within period price observations. As demonstrated in the literature (see, e.g., Barndorff-Nielsen and Shephard, 2002; Andersen, Bollerslev, Diebold, and Ebens, 2001), this "model-free" realized variance measure based on high-frequency intraday data offers a much more accurate ex-post observation of the true (unobserved) return variance than the traditional ones based on daily or coarser frequency returns.

Variance risk premium (VRP) at time is defined as the difference between the ex-ante risk-neutral expectation and the objective or statistical expectation of the return variance over the time interval,

$\displaystyle VRP_{t} ~ \equiv ~ \operatorname{E}^Q \left[\sigma^2_{t+1}\vert\Omega_t \right] ~ - ~ \operatorname{E}^P\left[\sigma^2_{t+1}\vert\Omega_t\right],$

(3)

which is not directly observable in practice.⁹ To construct an empirical proxy for such a VRP concept, one needs to estimate various reduced-form counterparts of the risk neutral and physical expectations. In practice, the risk-neutral expectation ${\operatorname{E}}^Q \left[\sigma^2_{t+1}\vert\Omega_t \right]$ is typically replaced by the CBOE implied variance (VIX

) and the true variance $\sigma^2_{t+1}$ is replaced by realized variance $RV_{t,t+1}$ .

To estimate the objective expectation, ${\operatorname{E}}^P\left[\sigma^2_{t+1}\vert\Omega_t \right]$ , we use a linear forecast of future realized variance as $RV_{t,t+1} =\alpha +\beta IV_{t,t+1}+\gamma RV_{t-1,t}+\epsilon_{t,t+1}$ , with current implied and realized variances. The model-free implied variance from options market is an informationally more efficient forecast for future realized variance than the past realized variance (see, e.g., Jiang and Tian, 2005, among others), while realized variance based on high-frequency data also provides additional power in forecasting future realized variance (Andersen, Bollerslev, Diebold, and Labys, 2003). Therefore, a joint forecast model with one lag of implied variance and one lag of realized variance seems to capture the most forecasting power based on time- available information (Drechsler and Yaron, 2011).

3 Conditional ICAPM and Economic Uncertainty

The time-varying conditional version of the Sharpe (1964) and Lintner (1965) capital asset pricing model (CAPM) relates the conditionally expected excess returns on risky assets to the conditionally expected excess return on the market portfolio:

$\displaystyle E\left[R_{i,t+1}\vert\Omega_t\right]=\frac{E\left[R_{m,t+1}\vert\Omega_t\right]}{\mbox{var}\left[R_{m,t+1}\vert\Omega_t\right]}\cdot \mbox{cov}\left[R_{i,t+1},R_{m,t+1}\vert\Omega_t\right],$

(4)

where $R_{i,t+1}$ and $R_{m,t+1}$ are, respectively, the return on risky asset

and the market portfolio

in excess of the risk-free interest rate, $\Omega_t$ denotes the information set at time

that investors use to form expectations about future returns, $E\left[R_{i,t+1}\vert\Omega_t\right]$ and $E\left[R_{m,t+1}\vert\Omega_t\right]$ are the expected excess return on the risky asset and the market portfolio at time

conditional on the information set at time

, var $\left[R_{m,t+1}\vert\Omega_t\right]$ is the time-

expected conditional variance of excess returns on the market at time

, and cov $\left[R_{i,t+1},R_{m,t+1}\vert\Omega_t\right]$ is the time-

expected conditional covariance between excess returns on the risky asset and the market portfolio at time

In equation (4), the ratio of cov $\left[R_{i,t+1},R_{m,t+1}\vert\Omega_t\right]$ to var $\left[R_{m,t+1}\vert\Omega_t\right]$ is the asset's time- expected conditional beta $E\left[\beta_{i,t+1}\vert\Omega_t\right]=\frac{\mbox{cov}\left[R_{i,t+1},R_{m,t+1}\vert\Omega_t\right]}{\mbox{var}\left[R_{m,t+1}\vert\Omega_t\right]}$ , and the ratio $\frac{E\left[R_{m,t+1}\vert\Omega_t\right]}{\mbox{var}\left[R_{m,t+1}\vert\Omega_t\right]}$ is known as the reward-to-risk ratio that represents the compensation the investor must receive for a unit increase in the conditional variance of the market. As pointed out by Merton (1980), the reward-to-risk ratio can also be interpreted as the relative risk aversion coefficient.

Merton (1973) intertemporal capital asset pricing model (ICAPM) implies the following equilibrium relation between expected return and risk for any risky asset :

$\displaystyle \mu_i = A \cdot \sigma_{im} + B \cdot \sigma_{ix},$

(5)

where $\mu_i$ denotes the unconditional expected excess return on risky asset

, $\sigma_{im}$ denotes the unconditional covariance between the excess returns on the risky asset

and the market portfolio

, and $\sigma_{ix}$ denotes a ( $1 \times k$ ) row of unconditional covariances between the excess returns on the risky asset

and the

-dimensional state variables

is the relative risk aversion of market investors and

measures the market's aggregate reaction to shifts in a

-dimensional state vector that governs the stochastic investment opportunity set. Equation (5) states that in equilibrium, investors are compensated in terms of expected return for bearing market risk and for bearing the risk of unfavorable shifts in the investment opportunity set.

In this paper, we provide a time-series and cross-sectional investigation of the conditional ICAPM:

$\displaystyle E\left[R_{i,t+1}\vert\Omega_t\right] = A \cdot$ cov $\displaystyle \left[R_{i,t+1},R_{m,t+1}\vert\Omega_t\right] + B \cdot$ cov $\displaystyle \left[R_{i,t+1},X_{t+1}\vert\Omega_t\right],$

(6)

where

is the reward-to-risk ratio and interpreted as the Arrow-Pratt relative risk-aversion coefficient in Merton (1973) ICAPM. The difference between the conditional CAPM and the conditional ICAPM is the intertemporal hedging demand component, $B\cdot$ cov $\left[R_{i,t+1},X_{t+1}\vert\Omega_t\right],$ in equation (6). Note that cov $\left[R_{i,t+1},X_{t+1}\vert\Omega_t\right]$ measures the time-

expected conditional covariance between the excess returns on risky asset

and a state variable

. The parameter

represents the price of risk for the state variable

In this section, we first rely on a consumption-based asset pricing model to derive the equivalence between the investment opportunity set $X_{t+1}$ and variance risk premium $VRP_{t+1}$ ; then we use the nonlinear relationship between the slope coefficients and and underlying structural parameters to impose the sign restrictions. After deriving the intertemporal relation between expected return and risk and uncertainty based on the consumption-based asset pricing model, we test whether the market price of risk and the market price of uncertainty are significantly positive and whether they help explain returns in a panel data setting:

(7)

where the time-varying exposure of asset

to changes in the market portfolio is measured by the conditional covariance between the excess return on asset

and the excess return on the aggregate stock market, denoted by cov $\left[R_{i,t+1},R_{m,t+1}\vert\Omega_t\right]$ , and the time-varying exposure of asset

to uncertainty in the stock market is proxied by the conditional covariance between the excess returns on asset

and the variance risk premia, denoted by cov $\left[R_{i,t+1},VRP_{t+1}\vert\Omega_t\right]$ .

To guide our economic interpretation of these empirical exercises, we rely on a structural model with recursive preference and consumption uncertainty (Bollerslev, Tauchen, and Zhou, 2009). One can show that the model-implied market compensations for risk and uncertainty are both positive, under reasonable parameter settings that agents are more risk averse than the log utility and that agents prefer an early resolution of economic uncertainty. In essence, the two risk factors--market return and variance risk premium--span all systematic variations in any risky assets.

3.1 An Economic Model of Return-Uncertainty Tradeoff

The representative agent in the economy is endowed with Epstein-Zin-Weil recursive preferences, and has the value function of her life-time utility as

$\displaystyle V_t ~ = ~ \left[\left(1-\delta\right)C_t^{\frac{1-\gamma}{\theta}} +\delta\left(E_t\left[V_{t+1}^{1-\gamma}\right]\right)^{\frac{1}{\theta}}\right]^{\frac{\theta}{1-\gamma}},$

(8)

where

is consumption at time

, $\delta$ denotes the subjective discount factor, $\gamma$ refers to the coefficient of risk aversion, $\theta = \frac{1-\gamma}{1 - \frac{1}{\psi}}$ , and $\psi$ equals the intertemporal elasticity of substitution (IES). The key assumptions are that $\gamma>1$ , implying that the agents are more risk averse than the log utility investors; and $\psi>1$ hence $\theta <0$ , implying that agents prefer an earlier resolution of economic uncertainty.

Suppose that log consumption growth and its volatility follow the joint dynamics

$\displaystyle g_{t+1}$	$\displaystyle =$	$\displaystyle \mu_{g} + \sigma_{g,t} z_{g,t+1} ,$	(9)
$\displaystyle \sigma^2_{g,t+1}$	$\displaystyle =$	$\displaystyle a_{\sigma} + \rho_{\sigma} \sigma^2_{g,t} + \sqrt{q_t} z_{\sigma, t+1} ,$	(10)
$\displaystyle q_{t+1}$	$\displaystyle =$	$\displaystyle a_q + \rho_q q_t + \varphi_q \sqrt{q_t}z_{q,t+1} ,$	(11)

where $\mu_{g}>0$ denotes the constant mean growth rate, $\sigma^2_{g,t+1}$ represents time-varying volatility in consumption growth, and

introduces the volatility uncertainty process in the consumption growth process.¹⁰

Let denote the logarithm of the price-dividend or wealth-consumption ratio, of the asset that pays the consumption endowment, $\{C_{t+i}\}^\infty_{i=1}$ ; and conjecture a solution for as an affine function of the state variables, $\sigma^2_{g,t}$ and ,

$\displaystyle w_t = A_0 + A_\sigma \sigma^2_{g,t} + A_q q_t .$

(12)

One can solve for the coefficients $A_0,\, A_\sigma$ and

using the standard Campbell and Shiller (1988) approximation $r_{t+1} = \kappa_0 + \kappa_1 w_{t+1} - w_{t} + g_{t+1}$ , where $r_{t+1}$ is the return on the asset that pays the consumption endowment flow. The restrictions that $\gamma>1$ and $\psi>1$ , hence $\theta <0$ , imply that the impact coefficients associated with both volatility and uncertainty state variables are negative; i.e., $A_{\sigma}<0$ and

. So if consumption risk and economic uncertainty are high, the price-dividend ratio is low, hence risk premia are high.

Given the solution of price-dividend ratio, and assume that dividend equals consumption, the model-implied premium of the market portfolio can be shown as

$\displaystyle E\left[R_{m,t+1}\vert\Omega_t\right] ~ =~ \gamma \sigma_{g,t}^2 \, + \, (1-\theta)\kappa_1^2(A_q^2 \varphi_q^2 + A_{\sigma}^2)q_t .$

(13)

The premium is composed of two separate terms. The first term, $\gamma \sigma_{g,t}^2$ , is compensating for the classic consumption risk as in a standard consumption-based CAPM model. The second term, $(1-\theta)\kappa_1^2(A_q^2 \varphi_q^2 + A_{\sigma}^2)q_t$ , represents a true premium for variance risk or economic uncertainty. The restrictions that $\gamma>1$ and $\psi>1$ implies that the uncertainty or variance risk premium is always positive by construction.

The conditional variance of the time to market return, $\sigma^2_{m,t} \equiv$ Var $_t( r_{t+1} )$ , can be shown as $\sigma^2_{m,t} = \sigma_{g,t}^2 + \kappa_1^2 \left( A_{\sigma}^2 + A_q^2 \varphi_q^2 \right) q_t.$ The variance risk premium can be defined as the difference between risk-neutral and objective expectations of the return variance,¹¹

$\displaystyle VRP_{t} \equiv \operatorname{E}^Q\left[\sigma_{m,t+1}^2\vert\Omega_t\right]- \operatorname{E}^P\left[\sigma_{m,t+1}^2\vert\Omega_t\right] \approx (\theta-1)\kappa_1\left[ A_{\sigma} + A_q \kappa_1^2 \left( A_{\sigma}^2 + A_q^2 \varphi_q^2 \right) \varphi_q^2\right]q_t .$

Moreover, provided that $\theta <0$ , $A_{\sigma}<0$ , and

, as would be implied by the agents' preference of an earlier resolution of economic uncertainty, this difference between the risk-neutral and objective expectations of return variances is guaranteed to be positive.

However, due to the measurement difficulty in consumption data and its volatility, we will use market return volatility and variance risk premium to substitute fundamental risk and uncertainty that are harder to pin down accurately,

$\displaystyle E\left[R_{m,t+1}\vert\Omega_t\right] ~ =~ \gamma \sigma^2_{m,t} \, + \, \frac{(1-\theta-\gamma)\kappa_1^2( A_{\sigma}^2 + A_q^2 \varphi_q^2)} {(\theta-1)\kappa_1\left[ A_{\sigma} + A_q \kappa_1^2 \left( A_{\sigma}^2 + A_q^2 \varphi_q^2 \right) \varphi_q^2\right]}VRP_{t} \, .$

(14)

Therefore the risk-return trade-off identified by $\gamma$ is always positive. However, the uncertainty-return trade-off depends on the sign of $(1-\theta-\gamma)$ . Under typical preference parameter setting, as in Bansal and Yaron (2004) and Bollerslev, Tauchen, and Zhou (2009), $\theta$ tends to be a large negative number, and one always has $(1-\theta-\gamma)>0$ . In other words, the model implied uncertainty-return tradeoffs should always be positive.

3.2 Calibrating Uncertainty-Return Tradeoff

To give some empirical guidance on how such a modeling framework with two risk drivers--consumption risk and volatility uncertainty--can play out in empirically testing the time-series and cross-sectional stock returns, we provide some calibration evidence based on the model parameter settings used by Bollerslev, Tauchen, and Zhou (2009, or BTZ2009 for short) focusing on equity return predictability and Zhou (2010) also considering bond return and credit spread predictability. As shown in Table 1, consistent with the analytical characterization above, the risk-return trade-off coefficient or should be equal to the risk-aversion coefficient, which is 10 or 2 under the two model parameter choices. On the other hand, the uncertainty-return coefficient or should be equal to 10.24 or 0.08, which is a highly non-linear function of both the underlying preference and structural parameters. The model implied uncertainty-return trade-off is positive.

More importantly, the positive relationship between variance risk premium and excess market return is fairly robust. There are two key preference parameters--intertemporal elasticity of substitution (IES) and risk aversion coefficient that may materially affect the sign and magnitude of the return-uncertainty trade-off. However, as shown in the top two panels of Figure 1, as long as IES-- $\psi$ is larger than one and risk aversion-- $\gamma$ is larger than one, the model-implied linkage between return and uncertainty should remain positive. In contrast, when agents prefer a late resolution of uncertainty or $0<\psi<1$ (bottom left panel), the model implied return-uncertainty trade-off swings between positive and negative values with a bifurcation towards infinities near $\psi=0.5$ . Similarly, if agents are less risk averse than log investor or $0<\gamma<1$ (bottom right panel), the uncertainty-return trade-off also swings between large positive and negative values near $\gamma=0.67$ . The empirical implication is rather sharp--if we find that the exposures to variance risk premium are positively priced in stock returns, it would be consistent with our modeling framework that both IES and risk aversion are larger than one.

There is a long debate in literature whether the intertemporal elasticity of substitution or IES is larger than one. As emphasized by Beeler and Campbell (2009), IES bigger than one--around 1.5--is key to the success of long-run risks model (Bansal and Yaron, 2004). Although earlier time series evidence by Hall (1988) and Campbell (1999) tend to suggest that IES is small and close to zero, their argument is typically based on measuring the slope coefficient from regressing consumption growth on the real rate, which can be downward biased if consumption volatility is time-varying (Bansal, Kiku, and Yaron, 2007). On the other hand, financial market evidence on IES being less than one are also found by examining market return predictability in both long term and short term (Kandel and Stambaugh, 1991) and by relating the momentum and contrarian profits of individual stocks or portfolios to macroeconomic fundamentals (Liu, Zhang, and Fan, 2011).

Our empirical approach on estimating the risk-return and uncertainty-return trade-off from time-series and cross-section stock returns provides an alternative reduced-form angle to judge whether IES is bigger than one. Such an angle needs not to specify the underlying preference structure or fundamental endowment dynamics. Our sharp finding of a positive uncertainty-return trade-off is consistent with an IES larger than one but without any parametric restriction, not even relying on the Euler equations or GMM estimation as in Bansal, Kiku, and Yaron (2009) and Chen, Favilukis, and Ludvigson (2011).¹²

4 Data on Uncertainty Measures and Equity Portfolios

4.1 Variance Risk Premia and Economic Uncertainty Measures

For the option-implied variance of the S&P500 market return, we use the end-of-month Chicago Board of Options Exchange (CBOE) volatility index on a monthly basis (VIX/12). Following earlier studies, the monthly realized variance for the S&P500 index is calculated as the summation of the 78 intra-day five-minute squared log returns from 9:30am to 4:00pm including the close-to-open interval. As shown in equation (3), variance risk premium (VRP) at time is defined as the difference between the ex-ante risk-neutral expectation and the objective or statistical expectation of the return variance over the time interval. The monthly VRP data are available from January 1990 to December 2010.

To give a visual illustration, Figure 2 plots the monthly time series of variance risk premium (VRP), implied variance, and expected variance. The VRP proxy is moderately high around the 1990 and 2001 economic recessions but much higher during the 2008 financial crisis and to a lessor degree around 1997-1998 Asia-Russia-LTCM crisis. The variance spike during October 2008 already surpasses the initial shock of the Great Depression in October 1929. The huge run-up of VRP in the fourth quarter of 2008 leads the equity market bottom reached in March 2009. The sample mean of VRP is 18.75 (in percentages squared, monthly basis), with a standard deviation of 22.15. Notice that the extraordinary skewness (3.81) and kurtosis (27.46) signal a highly non-Gaussian process for VRP.

According to our model in Section 3, VRP can be viewed as a proxy for uncertainty. To test whether VRP is in fact associated with alternative measures of uncertainty, we generate some proxies for financial and economic uncertainty. We obtain monthly values of the US industrial production index from G.17 database of the Federal Reserve Board and monthly values of the Chicago Fed National Activity Index (CFNAI) from the Federal Reserve Bank of Chicago for the period January 1990 - December 2010.¹³ We use a GARCH(1,1) model of Bollerslev (1986) to estimate the conditional variance of the growth rate of industrial production and the conditional variance of the CFNAI index. These two measures can be viewed as macroeconomic uncertainty. The sample correlation between VRP and economic uncertainty variables is positive and significant; sample correlation is 33.20% with the variance of output growth and 31.82% with the variance of CFNAI index.

Our second set of uncertainty measures is based on the downside risk of financial institutions obtained from the left tail of the time-series and cross-sectional distribution of financial firms' returns. Specifically, we obtain monthly returns for financial firms (6000 $\le$ SIC code $\le$ 6999) for the sample period January 1990 to December 2010. Then, the 1% nonparametric Value-at-Risk (VaR) measure in a given month is measured as the cut-off point for the lower one percentile of the monthly returns on financial firms.¹⁴ For each month, we determine the one percentile of the cross-section of returns on financial firms, and obtain an aggregate 1% VaR measure of the financial system for the period 1990-2010. In addition to the cross-sectional distribution, we use the time-series daily return distribution to estimate 1% VaR of the financial system. For each month from January 1990 to December 2010, we first determine the lowest daily returns on financial institutions over the past 1 to 12 months. The catastrophic risk of financial institutions is then computed by taking the average of these lowest daily returns obtained from alternative measurement windows. The estimation windows are fixed at 1 to 12 months, and each fixed estimation window is updated on a monthly basis. These two downside risk measures can be viewed as a proxy for uncertainty in the financial sector. The sample correlations between VRP and financial uncertainty variables are positive and significant: 47.37% with the cross-sectional VaR measure and 37.01% with the time-series VaR measure.

The third uncertainty variable is related to the health of the financial sector proxied by the credit default swap (CDS) index. We download the monthly CDS data from Bloomberg. For the sample period January 2004 - December 2010, we obtain monthly CDS data for Bank of America (BOA), Citigroup (CICN), Goldman Sachs (GS), JP Morgan (JPM), Morgan Stanley (MS), Wells Fargo (WFC), and American Express (AXP). Then, we standardized all CDS data to have zero mean and unit standard deviation. Finally, we formed the standardized CDS index (EWCDS) based on the equal-weighted average of standardized CDS values for the 7 major financial firms. For the common sample period 2004-2010, the correlation between VRP and EWCDS is positive, 42.99%, and highly significant.

The last uncertainty variable is based on the aggregate measure of investors' disagreement about individual stocks trading at NYSE, AMEX, and NASDAQ. Following Diether, Malloy, and Scherbina (2002), we use dispersion in analysts' earnings forecasts as a proxy for divergence of opinion. It is likely that investors partly form their expectations about a particular stock based on the analysts' earnings forecasts. If all analysts are in agreement about expected returns, uncertainty is likely to be low. However, if analysts provide very different estimates, investors are likely to be unclear about future returns, and uncertainty is high. The sample correlation between VRP and the aggregate measure of dispersion is about 14.92%. Overall, these results indicate that the variance risk premia is strongly and positively correlated with all measures of uncertainty considered here. Hence, VRP can be viewed as a sound proxy for financial and economic uncertainty.

4.2 Equity Portfolios

We use the monthly excess returns on the value-weighted aggregate market portfolio and the monthly excess returns on the 10 value-weighted size, book-to-market, and industry portfolios. The aggregate market portfolio is represented by the value-weighted NYSE-AMEX-NASDAQ index. Excess returns on portfolios are obtained by subtracting the returns on the one-month Treasury bill from the raw returns on equity portfolios. The data are obtained from Kenneth French's online data library. We use the longest common sample period available, from January 1990 to December 2010, yielding a total of 252 monthly observations.

As described in Fama and French (1993), the size portfolios are constructed at the end of each June using the June market equity (ME) and NYSE breakpoints. The portfolios for July of year to June of include all NYSE, AMEX, and NASDAQ stocks for which we have market equity data for June of . The book-to-market portfolios are formed based on the ratio of book value equity (BE) to market value of equity (ME), BE/ME, at the end of each June using NYSE breakpoints. The BE used in June of year is the book equity for the last fiscal year end in . ME is price times shares outstanding at the end of December of . The book-to-market portfolios include all NYSE, AMEX, and NASDAQ stocks for which we have ME for December of and June of , and BE for . The breakpoints and portfolios include Compustat firms plus the firms hand-collected from the Moody's Industrial, Utilities, Transportation, and Financial Manuals. In addition to the size and book-to-market portfolios, we use the 10 value-weighted industry portfolios of Kenneth French. Fama and French (1997) assign each NYSE, AMEX, and NASDAQ stock to an industry portfolio at the end of June of year based on its four-digit SIC code at that time. They use Compustat SIC codes for the fiscal year ending in calendar year . Whenever Compustat SIC codes are not available, they use CRSP SIC codes for June of year . They then compute returns from July of to June of .

Table 4 presents the monthly raw return and CAPM Alpha differences between high-return (long) and low-return (short) equity portfolios. The results are reported for the size, book-to-market (BM), and industry portfolios for the period January 1990 - December 2010.¹⁵ The OLS -statistics are reported in parentheses. The Newey-West -statistics are given in square brackets.

For the ten size portfolios, "Small" (Decile 1) is the portfolio of stocks with the smallest market capitalization and "Big" (Decile 10) is the portfolio of stocks with the biggest market capitalization. For the 1990-2010 period, the average return difference between the Small and Big portfolios is 0.40% per month with the OLS -statistic of 1.22 and the Newey-West (1987) -statistic of 1.13, implying that small stocks on average do not generate higher returns than big stocks. In addition to the average raw returns, Table 4 presents the intercept (CAPM alpha) from the regression of Small-Big portfolio return difference on a constant and the excess market return. The CAPM Alpha (or abnormal return) for the long-short size portfolio is 0.35% per month with the OLS -statistic of 1.06 and the Newey-West -statistic of 0.98. This economically and statistically insignificant Alpha indicates that the static CAPM does explain the size effect for the 1990-2010 period.

For the ten book-to-market portfolios, "Growth" is the portfolio of stocks with the lowest book-to-market ratios and "Value" is the portfolio of stocks with the highest book-to-market ratios. For the sample period January 1990 - December 2010, the average return difference between the Value and Growth portfolios is economically and statistically insignificant; 0.29% per month with the OLS -statistic of 0.92 and the Newey-West -statistic of 0.79, implying that value stocks on average do not generate higher returns than growth stocks. Similar to our findings for the size portfolios, the unconditional CAPM can explain the value premium for the 1990-2010 period; the CAPM Alpha (or abnormal return) for the long-short book-to-market portfolio is only 0.28% per month with the OLS -statistic of 0.86 and the Newey-West -statistic of 0.71.¹⁶

Interestingly, industry effects in the US equity market are economically and statistically strong over the past two decades although size and value premiums are not. The average raw and risk-adjusted return differences between the high-return and low-return industry portfolios are significant for the sample period 1990-2010. The high-return and low-return portfolios of 48 and 49 industries generate highly significant return differences, 30 and 38 industry portfolios generate marginally significant return differences, whereas the average return differences and Alphas for the high-return and low-return portfolios of 10 and 17 industries are insignificant. Specifically, for 30-, 48- and 49-industry portfolios of Kenneth French, "Coal" industry has the highest average monthly return, whereas "Other" industry has the lowest return, yielding an average raw and risk-adjusted return differences of 1.54% to 1.79% per month and statistically significant. The static CAPM cannot explain these economically and statistically strong industry effects either.

Earlier studies starting with Fama and French (1992,1993) provide evidence for the significant size and value premiums for the post-1963 period. Some readers may find the insignificant size and value premiums for the 1990-2010 period controversial. Hence, in Appendix A, we examine the significance of size and book-to-market effects for the longest sample period July 1926 - December 2010 and the subsample period July 1963 - December 2010. The results indicate significant raw return difference between the Value and Growth portfolios for both sample periods and significant risk-adjusted return difference (Alpha) only for the post-1963 period. Consistent with the findings of earlier studies, we find significant raw return difference between the Small and Big stock portfolios for the 1926-2010 period, which becomes very weak for the post-1963 period. The CAPM Alpha (or abnormal return) for the long-short size portfolio is economically and statistically insignificant for both sample periods.

5 Estimation Methodology

Following Bali (2008) and Bali and Engle (2010), our estimation approach proceeds in steps.

1): We take out any autoregressive elements in returns and VRP and estimate univariate GARCH models for all returns and VRP.
2): We construct standardized returns and compute bivariate DCC estimates of the correlations between each portfolio and the market and between each portfolio and VRP using the bivariate likelihood function.
3): We estimate the expected return equation as a panel with the conditional covariances as regressors. The error covariance matrix specified as seemingly unrelated regression (SUR). The panel estimation methodology with SUR takes into account heteroskedasticity and autocorrelation as well as contemporaneous cross-correlations in the error terms.

The following subsections provide details about the estimation of time-varying covariances and the estimation of time-series and cross-sectional relation between expected returns and risk and uncertainty.

5.1 Estimating Time-Varying Conditional Covariances

We estimate the conditional covariance between excess returns on equity portfolio and the market portfolio based on the mean-reverting dynamic conditional correlation (DCC) model:

$\displaystyle R_{i,t+1}=\alpha^i_0+\alpha^i_1 R_{i,t} + \varepsilon_{i,t+1}$

(15)

$\displaystyle R_{m,t+1}=\alpha^m_0+\alpha^m_1 R_{m,t} + \varepsilon_{m,t+1}$

(16)

$\displaystyle E_t\left[ \varepsilon_{i,t+1}^2 \right] \equiv \sigma_{i,t+1}^2 =\beta^i_0+\beta^i_1 \varepsilon_{i,t}^2 + \beta^i_2 \sigma_{i,t}^2$

(17)

$\displaystyle E_t\left[ \varepsilon_{m,t+1}^2 \right] \equiv \sigma_{m,t+1}^2 =\beta^m_0+\beta^m_1 \varepsilon_{m,t}^2 + \beta^m_2 \sigma_{m,t}^2$

(18)

$\displaystyle E_t\left[ \varepsilon_{i,t+1} \varepsilon_{m,t+1}\right] \equiv \sigma_{im,t+1} = \rho_{im,t+1} \cdot \sigma_{i,t+1} \cdot \sigma_{m,t+1}$

(19)

$\displaystyle \rho_{im,t+1} = \frac {q_{im,t+1}}{\sqrt{q_{ii,t+1} \cdot q_{mm,t+1}}}, \ q_{im,t+1} =\bar \rho_{im} + a_1 \cdot \left( u_{i,t} \cdot u_{m,t} - \bar \rho_{im} \right) + a_2 \cdot \left( q_{im,t}-\bar \rho_{im}\right)$

(20)

where $R_{i,t+1}$ and $R_{m,t+1}$ denote the time (

) excess return on equity portfolio

and the market portfolio

over a risk-free rate, respectively, and $E_t\left[ \cdot \right]$ denotes the expectation operator conditional on time

information. $\sigma_{i,t+1}^2$ is the time-

expected conditional variance of $R_{i,t+1}$ , $\sigma_{m,t+1}^2$ is the time-

expected conditional variance of $R_{m,t+1}$ , and $\sigma_{im,t+1}$ is the time-

expected conditional covariance between $R_{i,t+1}$ and $R_{m,t+1}$ . $\rho_{im,t+1} = {q_{im,t+1}}/{\sqrt{q_{ii,t+1} \cdot q_{mm,t+1}}}$ is the time-

expected conditional correlation between $R_{i,t+1}$ and $R_{m,t+1}$ , and $\bar \rho_{im}$ is the unconditional correlation. To ease the parameter convergence, we use correlation targeting assuming that the time-varying correlations mean reverts to the sample correlations $\bar \rho_{im}$ .

We estimate the conditional covariance between each equity portfolio and the variance risk premia , $\sigma_{i,VRP}$ , using an analogous DCC model:

$\displaystyle R_{i,t+1}=\alpha^i_0+\alpha^i_1 R_{i,t} + \varepsilon_{i,t+1}$

(21)

$\displaystyle VRP_{t+1}=\alpha^{VRP}_0+\alpha^{VRP}_1 VRP_{t} + \varepsilon_{VRP,t+1}$

(22)

$\displaystyle E_t\left[ \varepsilon_{i,t+1}^2 \right] \equiv \sigma_{i,t+1}^2 =\beta^i_0+\beta^i_1 \varepsilon_{i,t}^2 + \beta^i_2 \sigma_{i,t}^2$

(23)

$\displaystyle E_t\left[ \varepsilon_{VRP,t+1}^2 \right] \equiv \sigma_{VRP,t+1}^2 =\beta^{VRP}_0+\beta^{VRP}_1 \varepsilon_{VRP,t}^2 + \beta^{VRP}_2 \sigma_{VRP,t}^2$

(24)

$\displaystyle E_t\left[ \varepsilon_{i,t+1} \varepsilon_{VRP,t+1}\right] \equiv \sigma_{i,VRP,t+1} = \rho_{i,VRP,t+1} \cdot \sigma_{i,t+1} \cdot \sigma_{VRP,t+1}$

(25)

where $\sigma_{i,VRP,t+1}$ is the time-

expected conditional covariance between $R_{i,t+1}$ and $VRP_{t+1}$ . $\rho_{i,VRP,t+1}$ is the time-

expected conditional correlation between $R_{i,t+1}$ and $VRP_{t+1}$ . We use the same DCC model to estimate the conditional covariance between the market portfolio

and the variance risk premia

, $\sigma_{m,VRP}$ .¹⁷

We estimate the conditional covariances of each equity portfolio with the market portfolio and with using the maximum likelihood method described in Appendix B. Then, as discussed in the following section, we estimate the time-series and cross-sectional relation between expected return and risk and uncertainty as a panel with the conditional covariances as regressors.

5.2 Estimating Risk-Uncertainty-Return Tradeoff

Given the conditional covariances, we estimate the portfolio-specific intercepts and the common slope estimates from the following panel regression:

$\displaystyle R_{i,t+1}$	$\displaystyle =$	$\displaystyle \alpha_i+A\cdot Cov_t\left(R_{i,t+1},R_{m,t+1}\right)+B\cdot Cov_t\left(R_{i,t+1},VRP_{t+1}\right)+\varepsilon_{i,t+1}$	(26)
$\displaystyle R_{m,t+1}$	$\displaystyle =$	$\displaystyle \alpha_m+A\cdot Var_t\left(R_{m,t+1}\right)+B\cdot Cov_t\left(R_{m,t+1},VRP_{t+1}\right)+\varepsilon_{m,t+1}$	(27)

where $Cov_t\left(R_{i,t+1},R_{m,t+1}\right)$ is the time-

expected conditional covariance between the excess return on portfolio

( $R_{i,t+1}$ ) and the excess return on the market portfolio ( $R_{m,t+1}$ ), $Cov_t\left(R_{i,t+1},VRP_{t+1}\right)$ is the time-

expected conditional covariance between the excess return on portfolio

and the variance risk premia ( $VRP_{t+1}$ ), $Cov_t\left(R_{m,t+1},VRP_{t+1}\right)$ is the time-

expected conditional covariance between the excess return on the market portfolio

and the variance risk premia ( $VRP_{t+1}$ ), and $Var_t\left(R_{m,t+1}\right)$ is the time-

expected conditional variance of excess returns on the market portfolio.

We estimate the system of equations in (26)-(27) using a weighted least square method that allows us to place constraints on coefficients across equations. We compute the -statistics of the parameter estimates accounting for heteroskedasticity and autocorrelation as well as contemporaneous cross-correlations in the errors from different equations. The estimation methodology can be regarded as an extension of the seemingly unrelated regression (SUR) method, the details of which are in Appendix C.

6 Empirical Results

In this section we first present results from the 10 decile portfolios of size, book-to-market, and industry. Second, we discuss the economic significance of risk and uncertainty compensations and the underlying economic intuition. Third, we compare the relative performances of conditional CAPM and ICAPM with both risk and uncertainty.

6.1 Ten Decile Portfolios of Size, Book-to-Market, and Industry

The common slopes and the intercepts are estimated using the monthly excess returns on the 10 value-weighted size, book-to-market, and industry portfolios for the sample period January 1990 to December 2010. The aggregate stock market portfolio is measured by the value-weighted CRSP index. Table 5 reports the common slope estimates (), the abnormal returns or conditional alphas for each equity portfolio ( $\alpha_i$ ) and the market portfolio ( $\alpha_m$ ), and the -statistics of the parameter estimates. The last two rows, respectively, show the Wald statistics; Wald from testing the joint hypothesis $H_0: \alpha_1=...=\alpha_{10}=\alpha_m=0$ , and Wald from testing the equality of conditional alphas for high-return and low-return portfolios (Small vs. Big; Value vs. Growth; and HiTec vs. Telcm). The -values of Wald and Wald statistics are given in square brackets.

The risk aversion coefficient is estimated to be positive and highly significant for all equity portfolios: with the -statistic of 3.12 for the size portfolios, with the -statistic of 2.53 for the book-to-market portfolios, and with the -statistic of 2.35 for the industry portfolios.¹⁸ These results imply a positive and significant relation between expected return and market risk.¹⁹ Consistent with our theoretical model in equation (14), the uncertainty aversion coefficient is also estimated to be positive and highly significant for all equity portfolios: with the -statistic of 2.97 for the size portfolios, with the -statistic of 2.27 for the book-to-market portfolios, and with the -statistic of 2.78 for the industry portfolios. These results indicate a significantly positive market price of uncertainty in the aggregate stock market. Equity portfolios with higher sensitivity to increases in the variance risk premia are expected to generate higher returns next period.

One implication of the conditional asset pricing model in equation (14) is that the intercepts ( $\alpha_i$ , $\alpha_m$ ) are not jointly different from zero assuming that the conditional covariances of equity portfolios with the market portfolio and the variance risk premia have enough predictive power for expected future returns. To examine the empirical validity of the conditional asset pricing model, we test the joint hypothesis $H_0: \alpha_1=...=\alpha_{10}=\alpha_m=0$ . As presented in Table 5, the Wald statistics for the size, book-to-market, and industry portfolios are, respectively, 16.74, 8.88, and 14.35 with the corresponding -values of 0.12, 0.63, and 0.21. The significantly positive risk and uncertainty aversion coefficients and the insignificant Wald statistics indicate that the two-factor model proposed in the paper is empirically sound.

We also investigate whether the model explains the return spreads between Small and Big; Value and Growth; and HiTec and Telcm portfolios. The last row in Table 5 reports Wald statistics from testing the equality of conditional alphas for high-return and low-return portfolios ( $H_0: \alpha_1=\alpha_{10}$ ). These intercepts capture the monthly abnormal returns on each portfolio that cannot be explained by the conditional covariances with the market portfolio and the variance risk premia.

The first column of Table 5 shows that the abnormal return on the small-stock portfolio is $\alpha_1=0.41\%$ per month with a -statistic of 0.94, whereas the abnormal return on the big-stock portfolio is $\alpha_{10}=0.01\%$ per month with a -statistic of 0.01. The Wald statistic from testing the equality of alphas on the Small and Big portfolios is 1.56 with a -value of 0.21, indicating that there is no significant risk-adjusted return difference between the small-stock and big-stock portfolios. The second column provides the conditional alphas on the Value and Growth portfolios: $\alpha_1=0.36\%$ per month with a -statistic of 0.90, and $\alpha_{10}=0.82\%$ per month with a -statistic of 1.90. The Wald statistic from testing $H_0: \alpha_1=\alpha_{10}$ is 1.79 with a -value of 0.18, implying that the conditional asset pricing model explains the value premium, i.e., the risk-adjusted return difference between value and growth stocks is statistically insignificant. The last column shows that the conditional alphas on HiTec and Telcm portfolios are, respectively, 0.26% and -0.05% per month, generating a risk-adjusted return spread of 31 basis points per month. As reported in the last row, the Wald statistic from testing the significance of this return spread is 0.40 with a -value of 0.53, yielding insignificant industry effect over the sample period 1990-2010.

The differences in conditional alphas are both economically and statistically insignificant, indicating that the two-factor model proposed in the paper provides both statistical and economic success in explaining stock market anomalies. Overall, the DCC-based conditional covariances capture the time-series and cross-sectional variation in returns on size, book-to-market, and industry portfolios because the essential tests of the conditional CAPM are passed: (i) significantly positive risk-return and uncertainty-return tradeoffs; (ii) the conditional alphas are jointly zero; and (iii) the conditional alphas for high-return and low-return portfolios are not statistically different from each other.

6.2 Economic Significance of Uncertainty-Return Tradeoff

In this section, we test whether the risk-return () and uncertainty-return () coefficients are sensible and whether the uncertainty measure is associated with macroeconomic state variables.

Specifically, we rely on equation (27) and compute the expected excess return on the market portfolio based on the estimated prices of risk and uncertainty as well as the sample averages of the conditional covariance measures:

$\displaystyle E_t\left[R_{m,t+1}\right]=\alpha_m+A\cdot Var_t\left(R_{m,t+1}\right)+B\cdot Cov_t\left(R_{m,t+1},VRP_{t+1}\right)$

(28)

where $\alpha_m=0.0008$ ,

, and

for the 10 size portfolios; $\alpha_m=0.0032$ ,

, and

for the 10 book-to-market portfolios; and $\alpha_m=0.0019$ ,

, and

for the 10 industry portfolios (see Table 5). The sample averages of $Var_t\left(R_{m,t+1}\right)$ and $Cov_t\left(R_{m,t+1},VRP_{t+1}\right)$ are 0.002187 and -0.7026, respectively. These values produce $E_t\left[R_{m,t+1}\right]=0.54\%$ per month when the parameters are estimated using the 10 size portfolios, $E_t\left[R_{m,t+1}\right]=0.52\%$ per month when the parameters are estimated using the 10 book-to-market portfolios, and $E_t\left[R_{m,t+1}\right]=0.51\%$ when the parameters are estimated using the 10 industry portfolios.

To evaluate the performance of our model with risk and uncertainty, we calculate the sample average of excess returns on the market portfolio, which is a standard benchmark for the market risk premium. The sample average of $R_{m,t+1}$ is found to be 0.52% per month for the period January 1990 - December 2010, indicating that the estimated market risk premiums of 0.51% - 0.54% are very close to the benchmark. This again shows outstanding performance of the two-factor model introduced in the paper.

To further appreciate the economics behind the apparent connection between the variance risk premium (VRP) and the time-series and cross-sectional variations in expected stock returns, Figure 3 plots the VRP together with the quarterly growth rate in GDP. As seen from the figure, there is a tendency for VRP to rise in the quarter before a decline in GDP, while it typically narrows ahead of an increase in GDP. Indeed, the sample correlation equals -0.17 between lag VRP and current GDP (as first reported in Bollerslev and Zhou, 2007). In other words, VRP as a proxy for economic uncertainty does seem to negatively relate to future macroeconomic performance.

Thus, not only the difference between the implied and realized variances positively covaries with stock returns, it also covaries negatively with future growth rates in GDP. Intuitively, when VRP is high (low), it generally signals a high (low) degree of aggregate economic uncertainty. Consequently agents tend to simultaneously cut (increase) their consumption and investment expenditures and shift their portfolios from more (less) to less (more) risky assets. This in turn results in a rise (decrease) in expected excess returns for stock portfolios that covaries more (less) with the macroeconomic uncertainty, as proxied by VRP. In essence, our finding of a positive significant relation between economic uncertainty measure and stock expected returns, is consistent with our model's setting when agents prefer an earlier resolution of uncertainty. Therefore economic uncertainty carries a positive premium, and heightened VRP does signal a worsening of macroeconomic fundamentals.

6.3 Relative Performance of Conditional ICAPM with Uncertainty

We now assess the relative performance of the newly proposed model in predicting the cross-section of expected returns on equity portfolios. Specifically, we test whether the conditional ICAPM with the market and uncertainty factors outperforms the conditional CAPM with the market factor in terms of statistical fit. The goodness of fit of an asset pricing model describes how well it fits a set of realized return observations. Measures of goodness of fit typically summarize the discrepancy between observed values and the values expected under the model in question. Hence, we focus on the cross-section of realized average returns on equity portfolios (as a benchmark) and the portfolios' expected returns implied by the two competing models.

Using equation (26), we compute the expected excess return on equity portfolios based on the estimated prices of risk and uncertainty () and the sample averages of the conditional covariance measures, $Cov_t\left(R_{i,t+1},R_{m,t+1}\right)$ and $Cov_t\left(R_{i,t+1},VRP_{t+1}\right)$ :

$\displaystyle E_t\left[R_{i,t+1}\right]=\alpha_i+A\cdot Cov_t\left(R_{i,t+1},R_{m,t+1}\right)+B\cdot Cov_t\left(R_{i,t+1},VRP_{t+1}\right).$

(29)

Table 8 presents the realized monthly average excess returns on the size, book-to-market, and industry portfolios and the cross-section of expected excess returns generated by the Conditional CAPM and the Conditional ICAPM models. Clearly the newly proposed model with risk and uncertainty provides much more accurate estimates of expected returns on the size, book-to-market, and industry portfolios. Especially for the size and industry portfolios, expected returns implied by the Conditional ICAPM with the market and VRP factors are almost identical to the realized average returns. The last row in Table 8 reports the Mean Absolute Percentage Errors (MAPE) for the two competing models:

$\displaystyle MAPE = \frac{\left\vert\mbox{Realized}-\mbox{Expected}\right\vert}{\mbox{Expected}},$

(30)

where "Realized" is the realized monthly average excess return on each equity portfolio and "Expected" is the expected excess return implied by equation (29). For the conditional CAPM with the market factor, MAPE equals 5.20% for the size portfolios, 5.37% for the book-to-market portfolios, and 6.32% for the industry portfolios. Accounting for the variance risk premium improves the cross-sectional fitting significantly: MAPE reduces to 0.61% for the size portfolios, 1.66% for the book-to-market portfolios, and 0.55% for the industry portfolios.

Figure 4 provides a visual depiction of the realized and expected returns for the size, book-to-market, and industry portfolios. It is clear that our conditional ICAPM with uncertainty nails down the realized returns of the size, book-to-market, and industrial portfolios, while the conditional CAPM systematically over-predicts these portfolio returns. In other words, if there were size premium and value premium puzzles, our ICAPM model with market and VRP factors would have resolved these puzzles, which is indeed the case for the industry portfolios. Overall, the results indicate superior performance of the conditional asset pricing model introduced in the paper.

7 Robustness Check

In this section we first test whether using an alternative specification of the DCC-based conditional covariances with an asymmetric GARCH model influences our main findings. Second, we examine whether the model's performance changes when we use a larger cross-section of equity portfolios. Third, we provide robustness analysis when controlling for popular macroeconomic and financial variables. Finally, we provide results from individual stocks.

7.1 DCC with Asymmetric GARCH

Because the conditional variance and covariance of stock market returns are not observable, different approaches and specifications used in estimating the conditional variance and covariance could lead to different conclusions. We have so far used the bivariate GARCH(1,1) model of Bollerslev (1986) in equations (17)-(18) and (23)-(24) to obtain conditional variance and covariance estimates. In this section, we investigate whether changing these specifications influences our main findings.

The current volatility in the GARCH(1,1) model is defined as a symmetric, linear function of the last period's unexpected news and the last period's volatility. Since, in a symmetric GARCH process, positive and negative information shocks of the same magnitude produce the same amount of volatility, the symmetric GARCH model cannot cope with the skewness of stock return distribution. If a negative return shock causes more volatility than a positive return shock of the same size, the symmetric GARCH model underpredicts the amount of volatility following negative shocks and overpredicts the amount of volatility following positive shocks. Furthermore, if large return shocks cause more volatility than a quadratic function allows, then the symmetric GARCH model underpredicts volatility after a large return shock and overpredicts volatility after a small return shock.

In this section we use an asymmetric GARCH model of Glosten, Jagannathan, and Runkle (1993) that explicitly takes account of skewed distributions and allows good news and bad news to have different impacts on the conditional volatility forecasts. To test whether such variations in the variance forecasting specification alter our conclusion, we re-estimate the DCC-based conditional covariances using the following alternative specification:

$\begin{displaymath}\begin{array}{l} R_{i,t+1}=\alpha^i_0+\alpha^i_1 R_{i,t} + \v... ...RP,t+1} \cdot \sigma_{m,t+1} \cdot \sigma_{VRP,t+1} \end{array}\end{displaymath}$

(31)

where $D_{i,t}^-$ , $D_{m,t}^-$ , and $D_{VRP,t}^-$ are indicator functions that equals one when $\varepsilon_{i,t+1}$ , $\varepsilon_{m,t+1}$ , and $\varepsilon_{VRP,t+1}$ are negative and zero otherwise. The indicator function generates an asymmetric GARCH effect between positive and negative shocks. $\rho_{im,t+1}$ , $\rho_{i,VRP,t+1}$ , and $\rho_{m,VRP,t+1}$ are the time-

expected conditional correlations estimated using the mean-reverting DCC model of Engle (2002).

A notable point in Table 9 is that the main findings from an asymmetric GARCH specification of the conditional covariances are very similar to those reported in Table 5. Specifically, the risk aversion coefficients are estimated to be positive and highly significant for all equity portfolios; is in the range of 2.53 to 3.54 with the -statistics ranging from 2.58 to 3.11, implying a significantly positive link between expected return and risk. Similar to our results from GARCH(1,1) specification, asymmetric GARCH model of Glosten et al. (1993) yields positive and significant coefficient estimates on the covariance between equity portfolios and the variance risk premia. Specifically, the uncertainty aversion coefficients (B) are in the range of 0.0054 to 0.0075 with the -statistics between 2.68 and 3.30. These results show that equity portfolios that are highly correlated with uncertainty (proxied by VRP) carry a significant premium relative to portfolios that are uncorrelated or lowly correlated with VRP.

With this alternative covariance specification, we also examine the empirical validity of the conditional asset pricing model by testing the joint hypothesis. As shown in Table 9, the Wald statistics for the size, book-to-market, and industry portfolios are, respectively, 16.91, 7.89, and 14.41 with the corresponding -values of 0.11, 0.72, and 0.21. The significantly positive risk and uncertainty aversion coefficients and the insignificant Wald statistics indicate that the two-factor model explains the time-series and cross-sectional variation in equity portfolios. Finally, we investigate whether the model with asymmetric GARCH specification explains the return spreads between Small and Big; Value and Growth; and HiTec and Telcm portfolios. The last row in Table 9 reports Wald statistics from testing the equality of conditional alphas for high-return and low-return portfolios ( $H_0: \alpha_1=\alpha_{10}$ ). For the size, book-to-market, and industry portfolios, the Wald statistics provide no evidence for a significant conditional alpha for "Small-Big", "Value-Growth", and "HiTec-Telcm" arbitrage portfolios. Overall, the DCC-based conditional covariances from the asymmetric GARCH model captures the time-series and cross-sectional variation in returns on size, book-to-market, and industry portfolios and generates significantly positive risk-return and uncertainty-return tradeoffs.²⁰

7.2 Results from Larger Cross-Section of Industry Portfolios

Given the positive risk-return and positive uncertainty-return coefficient estimates from the three data sets and the success of the conditional asset pricing model in explaining the industry, size, and value premia, we now examine how the model performs when we use a larger cross-section of equity portfolios.

The robustness of our findings is investigated using the monthly excess returns on the value-weighted 17-, 30-, 38-, 48-, and 49-industry portfolios for the sample period January 1990 - December 2010. Table 12 reports the common slope estimates (, ), their -statistics in parentheses, and the Wald and Wald statistics along with their -values in square brackets. For the industry portfolios, the risk aversion coefficients () are estimated to be positive, in the range of 2.20 to 2.78, and highly significant with the -statistics ranging from 2.31 to 3.34. Consistent with our earlier findings from the 10 size, 10 book-to-market, and 10 industry portfolios, the results from the larger cross-section of industry portfolios (17 to 49) imply a positive and significant relation between expected return and market risk. Again similar to our findings from 10 decile portfolios, the uncertainty aversion coefficients are estimated to be positive, in the range of 0.0036 to 0.0041, and highly significant with the -statistics ranging from 2.44 to 4.21. These results provide evidence for a significantly positive market price of uncertainty and show that assets with higher correlation with the variance risk premia generate higher returns next month.

Not surprisingly, the Wald statistics for all industry portfolios have -values in the range of 0.20 to 0.75, indicating that the two-factor asset pricing model can explain the time-series and cross-sectional variation in larger number of equity portfolios. The last row shows that the Wald statistics from testing the equality of conditional alphas on the high-return and low-return industry portfolios have -values ranging from 0.44 to 0.80, implying that there is no significant risk-adjusted return difference between the extreme portfolios of 17, 30, 38, 48, and 49 industries. The differences in conditional alphas are both economically and statistically insignificant, showing that the two-factor model introduced in the paper provides success in explaining industry effects.

7.3 Controlling for Macroeconomic Variables

A series of papers argue that the stock market can be predicted by financial and/or macroeconomic variables associated with business cycle fluctuations. The commonly chosen variables include default spread (DEF), term spread (TERM), dividend price ratio (DIV), and the de-trended riskless rate or the relative T-bill rate (RREL).²¹ We define DEF as the difference between the yields on BAA- and AAA-rated corporate bonds, and TERM as the difference between the yields on the 10-year Treasury bond and the 3-month Treasury bill. RREL is defined as the difference between 3-month T-bill rate and its 12-month backward moving average.²² We obtain the aggregate dividend yield using the CRSP value-weighted index return with and without dividends based on the formula given in Fama and French (1988). In addition to these financial variables, we use some fundamental variables affecting the state of the US economy: Monthly inflation rate based on the US Consumer Price Index (INF); Monthly growth rate of the US industrial production (IP) obtained from the G.17 database of the Federal Reserve Board; and Monthly US unemployment rate (UNEMP) obtained from the Bureau of Labor Statistics.

According to Merton's (1973) ICAPM, state variables that are correlated with changes in consumption and investment opportunities are priced in capital markets in the sense that an asset's covariance with those state variables affects its expected returns. Merton (1973) also indicates that securities affected by such state variables (or systematic risk factors) should earn risk premia in a risk-averse economy. Macroeconomic variables used in the literature are excellent candidates for these systematic risk factors because innovations in macroeconomic variables can generate global impact on firm's fundamentals, such as their cash flows, risk-adjusted discount factors and/or investment opportunities. Following the existing literature, we use the aforementioned financial and macroeconomic variables as a proxy for state variables capturing shifts in the investment opportunity set.

We now investigate whether incorporating these variables into the predictive regressions affects the significance of the market prices of risk and uncertainty. Specifically, we estimate the portfolio-specific intercepts and the common slope coefficients from the following panel regression:

$\displaystyle R_{i,t+1}$	$\displaystyle =$	$\displaystyle \alpha_i+A\cdot Cov_t\left(R_{i,t+1},R_{m,t+1}\right)+B\cdot Cov_t\left(R_{i,t+1},VRP_{t+1}\right)+\lambda\cdot X_t+\varepsilon_{i,t+1}$
$\displaystyle R_{m,t+1}$	$\displaystyle =$	$\displaystyle \alpha_m+A\cdot Var_t\left(R_{m,t+1}\right)+B\cdot Cov_t\left(R_{m,t+1},VRP_{t+1}\right)+\lambda\cdot X_t+\varepsilon_{m,t+1}$

where

denotes a vector of lagged control variables; default spread (DEF), term spread (TERM), relative T-bill rate (RREL), aggregate dividend yield (DIV), inflation rate (INF), growth rate of industrial production (IP), and unemployment rate (UNEMP). The common slope coefficients (

, and $\lambda$ ) and their

-statistics are estimated using the monthly excess returns on the market portfolio and the ten size, book-to-market, and industry portfolios.

As presented in Table 17, after controlling for a wide variety of financial and macroeconomic variables, our main findings remain intact for all equity portfolios. The common slope estimates on the conditional covariances of equity portfolios with the market factor () remain positive and highly significant, indicating a positive and significant relation between expected return and market risk. Similar to our earlier findings, the common slopes on the conditional covariances of equity portfolios with the uncertainty factor () remain significantly positive as well, showing that assets with higher correlation with the variance risk premium generate higher returns next month. Among the control variables, the growth rate of industrial production is the only variable predicting future returns on equity portfolios; $\lambda_{IP}$ turns out to be positive and significant--especially for the industry portfolios. The positive relation between expected stock returns and innovations in output makes economic sense. Increases in real economic activity (proxied by the growth rate of industrial production) increase investors' expectations of future growth. Overall, the results in Table 17 indicate that after controlling for variables associated with business conditions, the time-varying exposures of equity portfolios to the market and uncertainty factors carry positive risk premiums.²³

7.4 Results from Individual Stocks

We have so far investigated the significance of risk, uncertainty, and return tradeoffs using equity portfolios. In this section, we replicate our analyses using individual stocks trading at NYSE, AMEX, and NASDAQ. First, we generate a dataset for the largest 500 common stocks (share code = 10 or 11) traded at NYSE/AMEX/NASDAQ. Following Shumway (1997), we adjust for stock de-listing to avoid survivorship bias.²⁴ Firms with missing observations on beginning-of-month market cap or monthly returns over the period January 1990 - December 2010 are eliminated. Due to the fact that the list of 500 firms changes over time as a result of changes in firms' market capitalizations, we obtain more than 500 firms over the period 1990-2010. Specifically, the largest 500 firms are determined based on their end-of-month market cap as of the end of each month from January 1990 to December 2010. There are 738 unique firms in our first dataset. In our second dataset, the largest 500 firms are determined based on their market cap at the end of December 2010. Our last dataset contains stocks in the S&P 500 index. Since the stock composition of the S&P 500 index changes through time, we rely on the most recent sample (as of December 2010). We also restrict our S&P 500 sample to 318 stocks with non-missing monthly return observations for the period January 1990 - December 2010.

Table 18 presents the common slope estimates (, ) and their -statistics for the individual stocks in the aforementioned data sets. The risk aversion coefficient is estimated to be positive and highly significant for all stock samples considered in the paper: with the -statistic of 8.04 for the first dataset containing 738 stocks (largest 500 stocks as of the end of each month from January 1990 to December 2010); with the -statistic of 8.70 for the second dataset containing largest 500 stocks as of the end of December 2010; and with the -statistic of 6.79 for the last dataset containing 318 stocks with non-missing monthly return observations for the period 1990-2010. Confirming our findings from equity portfolios, the results from individual stocks imply a positive and significant relation between expected return and market risk. Similarly, consistent with our earlier findings from equity portfolios, the uncertainty aversion coefficient is also estimated to be positive and highly significant for all data sets: with the -statistic of 3.61 for the first dataset, with the -statistic of 3.67 for the second dataset, and with the -statistic of 3.52 for the last dataset. These results indicate a significantly positive market price of uncertainty for large stocks trading in the U.S. stock market.

8 Conclusion

Although uncertainty is more common in decision-making process than risk, relatively little attention is paid to the phenomenon of uncertainty in asset pricing literature. This paper focuses on economic uncertainty and augments the original consumption-based CAPM to incorporate the time-varying volatility of the consumption growth and the volatility uncertainty in the consumption growth process. According to the augmented asset pricing model, the premium on equity is composed of two separate terms; the first term compensating for the standard consumption risk and the second term representing a true premium for variance risk. We find that in the presence of volatility uncertainty, both market risk and volatility uncertainty carry a positive premium, which is consistent with an economy where the intertemporal elasticity of substitution (IES) is larger than one.

Since information about consumption volatility uncertainty is too imprecise to measure with available data, we have to come up with a proxy for volatility uncertainty that should be consistent with our underlying economic model. Following Zhou (2010), we measure volatility uncertainty with the variance risk premium (VRP) of the aggregate stock market portfolio. Different from earlier studies, we provide empirical evidence that VRP is indeed closely related to economic and financial market uncertainty. Specifically, we generate several proxies for uncertainty based on the macroeconomic variables, return distributions of financial firms, credit default swap market, and investors' disagreement about individual stocks. We show that VRP is highly correlated with all measures of uncertainty.

Based on the two-factor asset pricing model, we investigate whether the market prices of risk and uncertainty are economically and statistically significant in the US equity market. Using the dynamic conditional correlation (DCC) model of Engle (2002), we estimate equity portfolios' conditional covariances with the market portfolio and VRP factors and then test whether these dynamic conditional covariances predict future returns on equity portfolios. The empirical results from the size, book-to-market, and industry portfolios indicate that the DCC-based conditional covariances of equity portfolios with the market and VRP factors predict the time-series and cross-sectional variation in stock returns. We find the risk-return coefficients to be positive and highly significant, implying a strongly positive link between expected return and market risk. Similarly, the results indicate a significantly positive market price of uncertainty. That is, equity portfolios that are highly correlated with uncertainty (proxied by VRP) carry a significant premium relative to portfolios that are uncorrelated or lowly correlated with VRP. In addition to the size, book-to-market, and industry portfolios, we investigate the significance of risk, uncertainty, and return tradeoffs using the largest 500 stocks trading at NYSE, AMEX, and NASDAQ as well as stocks in the S&P 500 index. Consistent with our findings from equity portfolios, we find significantly positive market prices of risk and uncertainty for large stocks trading in the US equity market.

We also examine whether the conditional asset pricing model proposed in the paper can provide risk-uncertainty based explanation of stock market anomalies. We test whether the conditional alphas on the size, book-to-market, and industry portfolios are zero. The test statistics indicate that the conditional alphas are jointly zero, implying economically and statistically insignificant abnormal returns on equity portfolios. Finally, we investigate whether the model explains the return spreads between the high-return (long) and low-return (short) equity portfolios (Small-Big for the size portfolios; Value-Growth for the book-to-market portfolios; and HiTec-Telcm for the industry portfolios). The results from testing the equality of conditional alphas for high-return and low-return portfolios provide no evidence for a significant alpha for "Small-Big", "Value-Growth", and "HiTec-Telcm" arbitrage portfolios, indicating that the newly proposed model with risk and uncertainty provides both statistical and economic success in explaining well-known stock market anomalies.

This table reports the calibration parameter values for the stochastic volatility-of-volatility model used in this paper. BTZ2009 refers to the calibration setting of Bollerslev, Tauchen, and Zhou (2009), with an emphasis on equity risk premium and its short-run predictability, while the setting of Zhou (2010) also considers bond risk premium and credit spread and their forecastability from variance risk premium. The Campbell-Shiller linearization constants are $\kappa_1=0.9$ and $\kappa_0=0.3251$ .

Table 1: Model Calibration Parameter Setting
Calibration Parameters	BTZ2009	Zhou (2010)
Preference Parameters:	$\delta=0.997$	$\delta=0.997$
Preference Parameters:	$\gamma=10$	$\gamma=2$
Preference Parameters:	$\psi=1.5$	$\psi=1.5$
Endowment Parameters:	$\mu_g=0.0015$	$\mu_g=0.0015$
Endowment Parameters:	$a_{\sigma}=1.34\times10^{-6}$	$a_{\sigma}=0.002$
Endowment Parameters:	$\rho_{\sigma}=0.978$	$\rho_{\sigma}=0.1$
Endowment Parameters:	$a_q=2\times10^{-7}$	$a_q=1.4\times10^{-5}$
Endowment Parameters:	$\rho_q=0.8$	$\rho_q=0.98$
Endowment Parameters:	$\varphi_q=0.001$	$\varphi_q=0.008$
Risk-Return Trade-off ()	10	2
Uncertainty-Return Trade-off ()	10.24	0.08

This table presents the monthly raw return and CAPM Alpha differences between high-return (long) and low-return (short) equity portfolios. The results are reported for the size, book-to-market (BM), and industry portfolios for the sample periods July 1926 - December 2010. The OLS -statistics are reported in parentheses. The Newey-West -statistics are given in square brackets.

Table 2a: Monthly Raw Returns and CAPM Alphas for the Long-Short Equity Portfolios: July 1926 - December 2010
Portfolio	Long-Short	Return Diff.	Alpha
10 Size	Small-Big	0.60%	0.27%
10 Size OLS t-statistics	Small-Big OLS t-statistics	(2.49)	(1.22)
10 Size Newey-West t-statistics	Small-Big Newey-West t-statistics	[2.36]	[1.38]
10 BM	Value-Growth	0.53%	0.24%
10 BM OLS t-statistics	Value-Growth OLS t-statistics	(2.52)	(1.25)
10 BM Newey-West t-statistics	Value-Growth Newey-West t-statistics	[2.46]	[1.26]
10 Industry	Durbl-Telcm	0.27%	-0.09%
10 Industry OLS t-statistics	Durbl-Telcm OLS t-statistics	(1.44)	(-0.56)
10 Industry Newey-West t-statistics	Durbl-Telcm Newey-West t-statistics	[1.31]	[-0.54]
17 Industry	Cars-Other	0.28%	0.07%
17 Industry OLS t-statistics	Cars-Other OLS t-statistics	(1.65)	(0.44)
17 Industry Newey-West t-statistics	Cars-Other Newey-West t-statistics	[1.55]	[0.44]
30 Industry	Coal-Other	0.51%	0.65%
30 Industry OLS t-statistics	Coal-Other OLS t-statistics	(1.79)	(2.26)
30 Industry Newey-West t-statistics	Coal-Other Newey-West t-statistic	[1.71]	[2.28]
38 Industry	Oil-Whlsl	0.42%	0.49%
38 Industry OLS t-statistics	Oil-Whlsl OLS t-statistics	(1.90)	(2.18)
38 Industry Newey-West t-statistics	Oil-Whlsl Newey-West t-statistics	[1.85]	[2.06]
48 Industry	Aero-Other	0.66%	0.50%
48 Industry OLS t-statistics	Aero-Other OLS t-statistics	(2.66)	(2.02)
48 Industry Newey-West t-statistics	Aero-Other Newey-West t-statistics	[2.55]	[2.04]
49 Industry	Aero-Other	0.66%	0.50%
49 Industry OLS t-statistics	Aero-Other OLS t-statistics	(2.66)	(2.02)
49 Industry Newey-West t-statistics	Aero-Other Newey-West t-statistics	[2.55]	[2.04]

This table presents the monthly raw return and CAPM Alpha differences between high-return (long) and low-return (short) equity portfolios. The results are reported for the size, book-to-market (BM), and industry portfolios for the sample period July 1963 - December 2010. The OLS -statistics are reported in parentheses. The Newey-West -statistics are given in square brackets.

Table 2b: Monthly Raw Returns and CAPM Alphas for the Long-Short Equity Portfolios: July 1963 - December 2010
Portfolio	Long-Short	Return Diff.	Alpha
10 Size	Small-Big	0.38%	0.29%
10 Size OLS t-statistics	Small-Big OLS t-statistics	(1.58)	(1.41)
10 Size Newey-West t-statistics	Small-Big Newey-West t-statistics	[1.81]	[1.26]
10 BM	Value-Growth	0.55%	0.55%
10 BM OLS t-statistics	Value-Growth OLS t-statistics	(2.77)	(2.76)
10 BM Newey-West t-statistics	Value-Growth Newey-West t-statistics	[2.48]	[2.40]
10 Industry	Utils-Enrgy	0.28%	0.17%
10 Industry OLS t-statistics	Utils-Enrgy OLS t-statistics	(1.50)	(0.93)
10 Industry Newey-West t-statistics	Utils-Enrgy Newey-West t-statistics	[1.50]	[0.91]
17 Industry	Mines-Durbl	0.43%	0.45%
17 Industry OLS t-statistics	Mines-Durbl OLS t-statistics	(1.54)	(1.58)
17 Industry Newey-West t-statistics	Mines-Durbl Newey-West t-statistics	[1.48]	[1.51]
30 Industry	Coal-Other	0.82%	0.79%
30 Industry OLS t-statistics	Coal-Other OLS t-statistics	(2.14)	(2.05)
30 Industry Newey-West t-statistics	Coal-Other Newey-West t-statistics	[1.95]	[1.84]
38 Industry	Smoke-Govt	1.06%	1.07%
38 Industry OLS t-statistics	Smoke-Govt OLS t-statistics	(2.38)	(2.40)
38 Industry Newey-West t-statistics	Smoke-Govt Newey-West t-statistics	[2.61]	[2.69]
48 Industry	Coal-Other	0.92%	0.92%
48 Industry OLS t-statistics	Coal-Other OLS t-statistics	(2.24)	(2.25)
48 Industry Newey-West t-statistics	Coal-Other Newey-West t-statistics	[2.06]	[2.04]
49 Industry	Coal-Other	0.92%	0.92%
49 Industry OLS t-statistics	Coal-Other OLS t-statistics	(2.24)	(2.25)
49 Industry Newey-West t-statistics	Coal-Other Newey-West t-statistics	[2.06]	[2.04]

This table presents the monthly raw return and CAPM Alpha differences between high-return (long) and low-return (short) equity portfolios. The results are reported for the size, book-to-market (BM), and industry portfolios for the sample period January 1990 - December 2010. The OLS -statistics are reported in parentheses. The Newey-West -statistics are given in square brackets.

Table 2c: Monthly Raw Returns and CAPM Alphas for the Long-Short Equity Portfolios: January 1990 - December 2010
Portfolio	Long-Short	Return Diff.	Alpha
10 Size	Small-Big	0.40%	0.35%
10 Size OLS t-statistics	Small-Big OLS t-statistics	(1.22)	(1.06)
10 Size Newey-West t-statistics	Small-Big Newey-West t-statistics	[1.13]	[0.98]
10 BM	Value-Growth	0.29%	0.28%
10 BM OLS t-statistics	Value-Growth OLS t-statistics	(0.92)	(0.86)
10 BM Newey-West t-statistics	Value-Growth Newey-West t-statistics	[0.79]	[0.71]
10 Industry	HiTech-Telcm	0.56%	0.31%
10 Industry OLS t-statistics	HiTech-Telcm OLS t-statistics	(1.55)	(0.92)
10 Industry Newey-West t-statistics	HiTech-Telcm Newey-West t-statistics	[1.64]	[0.95]
17 Industry	Mines-Durbl	0.56%	0.61%
17 Industry OLS t-statistics	Mines-Durbl OLS t-statistics	(1.15)	(1.23)
17 Industry Newey-West t-statistics	Mines-Durbl Newey-West t-statistics	[1.16]	[1.22]
30 Industry	Coal-Other	1.66%	1.54%
30 Industry OLS t-statistics	Coal-Other OLS t-statistics	(2.25)	(2.08)
30 Industry Newey-West t-statistics	Coal-Other Newey-West t-statistics	[2.12]	[1.90]
38 Industry	Mines-Govt	1.34%	1.45%
38 Industry OLS t-statistics	Mines-Govt OLS t-statistics	(1.48)	(1.59)
38 Industry Newey-West t-statistics	Mines-Govt Newey-West t-statistics	[1.75]	[1.90]
48 Industry	Coal-Other	1.79%	1.71%
48 Industry OLS t-statistics	Coal-Other OLS t-statistics	(2.38)	(2.26)
48 Industry Newey-West t-statistics	Coal-Other Newey-West t-statistics	[2.33]	[2.13]
49 Industry	Coal-Other	1.79%	1.71%
49 Industry OLS t-statistics	Coal-Other OLS t-statistics	(2.38)	(2.26)
49 Industry Newey-West t-statistics	Coal-Other Newey-West t-statistics	[2.33]	[2.13]

This table reports the portfolio-specific intercepts and the common slope estimates from the following panel regression:

$\displaystyle R_{i,t+1}$	$\displaystyle =$	$\displaystyle \alpha_i+A\cdot Cov_t\left(R_{i,t+1},R_{m,t+1}\right)+B\cdot Cov_t\left(R_{i,t+1},VRP_{t+1}\right)+\varepsilon_{i,t+1}$
$\displaystyle R_{m,t+1}$	$\displaystyle =$	$\displaystyle \alpha_m+A\cdot Var_t\left(R_{m,t+1}\right)+B\cdot Cov_t\left(R_{m,t+1},VRP_{t+1}\right)+\varepsilon_{m,t+1}$

where $Cov_t\left(R_{i,t+1},R_{m,t+1}\right)$ is the time-

expected conditional covariance between the excess return on portfolio

( $R_{i,t+1}$ ) and the excess return on the market portfolio ( $R_{m,t+1}$ ), $Cov_t\left(R_{i,t+1},VRP_{t+1}\right)$ is the time-

expected conditional covariance between the excess return on portfolio

and the variance risk premia ( $VRP_{t+1}$ ), $Cov_t\left(R_{m,t+1},VRP_{t+1}\right)$ is the time-

expected conditional covariance between the excess return on the market portfolio

and the variance risk premia ( $VRP_{t+1}$ ), and $Var_t\left(R_{m,t+1}\right)$ is the time-

expected conditional variance of excess returns on the market portfolio. The parameters and their

-statistics are estimated using the monthly excess returns on the market portfolio and the ten decile size, book-to-market, and industry portfolios for the sample period from January 1990 to December 2010. The alphas ( $\alpha_i$ ) are reported for each equity portfolio and the

-statistics are presented in parentheses. The

-statistics are adjusted for heteroskedasticity and autocorrelation for each series and cross-correlations among the portfolios. The last four rows, respectively, show the common slope coefficients (

and

), the Wald

statistics from testing the joint hypothesis $H_0: \alpha_1=\alpha_2=...\alpha_m=0$ , and the Wald

statistics from testing the equality of Alphas for high-return and low-return portfolios (Small vs. Big; Value vs. Growth; and HiTec vs. Telcm). The

-values of Wald

and Wald

statistics are given in square brackets.

Table 3a: Results from Ten Decile Size
	$\alpha_i, \alpha_m$
Small	0.0041
Small t-statistic	(0.94)
2	0.0022
2 t-statistic	(0.48)
3	0.0025
3 t-statistic	(0.58)
4	0.0015
4 t-statistic	(0.35)
5	0.0023
5 t-statistic	(0.54)
6	0.0023
6 t-statistic	(0.61)
7	0.0028
7 t-statistic	(0.76)
8	0.0020
8 t-statistic	(0.53)
9	0.0023
9 t-statistic	(0.67)
Big	0.0001
Big t-statistic	(0.01)
Market	0.0008
Market t-statistic	(0.17)
	3.9562
t-statistic	(3.12)
	0.0058
t-statistic	(2.97)
Wald	16.74
Wald $\rho$ -statistic	[0.12]
Wald	1.56
Wald $\rho$ -statistic	[0.21]

Table 3b: Results from Book-to-Market
	$\alpha_m, \alpha_m$
Growth	0.0036
Growth t-statistic	(0.90)
2	0.0048
2 t-statistic	(1.38)
3	0.0053
3 t-statistic	(1.58)
4	0.0065
4 t-statistic	(1.88)
5	0.0057
5 t-statistic	(1.74)
6	0.0051
6 t-statistic	(1.51)
7	0.0058
7 t-statistic	(1.78)
8	0.0059
8 t-statistic	(1.76)
9	0.0067
9 t-statistic	(1.94)
Value	0.0082
Value t-statistic	(1.90)
Market	0.0032
Market t-statistic	(1.23)
	2.5101
t-statistic	(2.53)
	0.0050
t-statistic	(2.27)
Wald	8.88
Wald $\rho$ -statistic	[0.63]
Wald	1.79
Wald $\rho$ -statistic	[0.18]

Table 3c: Results from Industry Portfolios
	$\alpha_i, \alpha_m$
NoDur	0.0043
NoDur t-statistic	(1.53)
Durbl	0.0019
Durbl t-statistic	(0.37)
Manuf	0.0046
Manuf t-statistic	(1.26)
Enrgy	0.0059
Enrgy t-statistic	(1.88)
HiTec	0.0026
HiTec t-statistic	(0.45)
Telcm	-0.0005
Telcm t-statistic	(-0.13)
Shops	0.0028
Shops t-statistic	(0.80)
Hlth	0.0036
Hlth t-statistic	(1.13)
Utils	0.0038
Utils t-statistic	(1.39)
Other	0.0018
Other t-statistic	(0.47)
Market	0.0019
Market t-statistic	(0.55)
	3.4055
t-statistic	(2.35)
	0.0060
t-statistic	(2.78)
Wald	14.35
Wald $\rho$ -statistic	[0.21]
Wald	0.40
Wald $\rho$ -statistic	[0.53]

This table presents the realized monthly average excess returns on the size, book-to-market, and industry portfolios and the cross-section of expected excess returns generated by the Conditional CAPM with the market factor and the Conditional ICAPM with the market and VRP factors. The last row reports the Mean Absolute Percentage Errors (MAPE) for the two competing models.

Table 4: Relative Performance of Conditional ICAPM with Risk and Uncertainty
	Realized Return Benchmark: Average Excess Returns	Conditional ICAPM with VRP: Expected Excess Returns	Conditional CAPM: Expected Excess Returns
Size: Small	0.8464%	0.8461%	0.8742%
Size: 2	0.7737%	0.7677%	0.8110%
Size: 3	0.7690%	0.7647%	0.8093%
Size: 4	0.6632%	0.6637%	0.7032%
Size: 5	0.7525%	0.7550%	0.7943%
Size: 6	0.7055%	0.7025%	0.7406%
Size: 7	0.7409%	0.7379%	0.7749%
Size: 8	0.6837%	0.6810%	0.7221%
Size: 9	0.6670%	0.6643%	0.7000%
Size: Big	0.4479%	0.4598%	0.4789%
Size: MAPE		0.61%	5.20%
Book-to-Market: Growth	0.5286%	0.5327%	0.5645%
Book-to-Market: 2	0.5614%	0.5658%	0.5961%
Book-to-Market: 3	0.6140%	0.6039%	0.6488%
Book-to-Market: 4	0.6752%	0.6559%	0.6960%
Book-to-Market: 5	0.6119%	0.6017%	0.6423%
Book-to-Market: 6	0.5439%	0.5547%	0.5803%
Book-to-Market: 7	0.6014%	0.5979%	0.6360%
Book-to-Market: 8	0.5885%	0.5956%	0.6233%
Book-to-Market: 9	0.6827%	0.6666%	0.7133%
Book-to-Market: Value	0.8221%	0.7994%	0.8564%
Book-to-Market: MAPE		1.66%	5.37%
Industry: Telcm	0.2727%	0.2747%	0.3280%
Industry: Utils	0.4712%	0.4727%	0.4965%
Industry: Other	0.4965%	0.4910%	0.5366%
Industry: Durbl	0.5313%	0.5315%	0.5513%
Industry: Shops	0.5954%	0.5912%	0.6247%
Industry: Hlth	0.6138%	0.6088%	0.6478%
Industry: NoDur	0.6110%	0.6152%	0.6534%
Industry: Manuf	0.7172%	0.7206%	0.7474%
Industry: Enrgy	0.7606%	0.7643%	0.7824%
Industry: HiTec	0.8358%	0.8350%	0.8466%
Industry: MAPE		0.55%	6.32%

This table reports the portfolio-specific intercepts and the common slope estimates from the following panel regression:

$\displaystyle R_{i,t+1}$	$\displaystyle =$	$\displaystyle \alpha_i+A\cdot Cov_t\left(R_{i,t+1},R_{m,t+1}\right)+B\cdot Cov_t\left(R_{i,t+1},VRP_{t+1}\right)+\varepsilon_{i,t+1}$
$\displaystyle R_{m,t+1}$	$\displaystyle =$	$\displaystyle \alpha_m+A\cdot Var_t\left(R_{m,t+1}\right)+B\cdot Cov_t\left(R_{m,t+1},VRP_{t+1}\right)+\varepsilon_{m,t+1}$

where the conditional variance and covariances are estimated using the asymmetric GARCH model of Glosten, Jagannathan, and Runkle (1993). The parameters and their

-statistics are presented in parentheses. The

-statistics are adjusted for heteroskedasticity and autocorrelation for each series and cross-correlations among the portfolios. The last four rows, respectively, show the common slope coefficients (

and

), the Wald

statistics from testing the joint hypothesis $H_0: \alpha_1=\alpha_2=...\alpha_m=0$ , and the Wald

statistics from testing the equality of Alphas for high-return and low-return portfolios (Small vs. Big; Value vs. Growth; and HiTec vs. Telcm). The

-values of Wald

and Wald

statistics are given in square brackets.

Table 5a: Results from Asymmetric GARCH Model with the Size Portfolio
Size	$\alpha_i, \alpha_m$
Small	0.0052
Small t-statistics	(1.23)
2	0.0037
2 t-statistics	(0.85)
3	0.0040
3 t-statistics	(0.99)
4	0.0030
4 t-statistics	(0.75)
5	0.0038
5 t-statistics	(0.97)
6	0.0037
6 t-statistics	(1.05)
7	0.0041
7 t-statistics	(1.19)
8	0.0034
8 t-statistics	(0.97)
9	0.0036
9 t-statistics	(1.11)
Big	0.0012
Big t-statistics	(0.38)
Market	0.0018
Market t-statistics	(0.57)
	3.2927
t-statistics	(3.11)
	0.0054
t-statistics	(3.12)
Wald	16.91
Wald $\rho$ -values	[0.11]
Wald	1.48
Wald $\rho$ -values	[0.22]

Table 5b: Results from Asymmetric GARCH Model with the Book-to-Market Portfolio
	$\alpha_m, \alpha_m$
Growth	0.0035
Growth t-statistics	(0.87)
2	0.0047
2 t-statistics	(1.35)
3	0.0052
3 t-statistics	(1.55)
4	0.0064
4 t-statistics	(1.85)
5	0.0056
5 t-statistics	(1.71)
6	0.0050
6 t-statistics	(1.48)
7	0.0057
7 t-statistics	(1.76)
8	0.0058
8 t-statistics	(1.74)
9	0.0066
9 t-statistics	(1.92)
Value	0.0081
Value t-statistics	(1.88)
Market	0.0033
Market t-statistics	(1.20)
	2.5303
t-statistics	(2.62)
	0.0060
t-statistics	(2.68)
Wald	7.89
Wald $\rho$ -values	[0.72]
Wald	1.99
Wald $\rho$ -values	[0.16]

Table 5c: Results from Asymmetric GARCH Model with the Industry Portfolio
	$\alpha_i, \alpha_m$
NoDur	0.0051
NoDur t-statistics	(1.94)
Durbl	0.0028
Durbl t-statistics	(0.57)
Manuf	0.0055
Manuf t-statistics	(1.61)
Enrgy	0.0064
Enrgy t-statistics	(1.85)
HiTec	0.0029
HiTec t-statistics	(0.52)
Telcm	0.0004
Telcm t-statistics	(0.11)
Shops	0.0036
Shops t-statistics	(1.04)
Hlth	0.0043
Hlth t-statistics	(1.37)
Utils	0.0042
Utils t-statistics	(1.58)
Other	0.0030
Other t-statistics	(0.81)
Market	0.0028
Market t-statistics	(0.82)
	3.5369
t-statistics	(2.58)
	0.0075
t-statistics	(3.30)
Wald	14.41
Wald $\rho$ -values	[0.21]
Wald	0.46
Wald $\rho$ -values	[0.50]

This table presents the common slope estimates (, ) from the following panel regression:

$\displaystyle R_{i,t+1}$	$\displaystyle =$	$\displaystyle \alpha_i+A\cdot Cov_t\left(R_{i,t+1},R_{m,t+1}\right)+B\cdot Cov_t\left(R_{i,t+1},VRP_{t+1}\right)+\varepsilon_{i,t+1}$
$\displaystyle R_{m,t+1}$	$\displaystyle =$	$\displaystyle \alpha_m+A\cdot Var_t\left(R_{m,t+1}\right)+B\cdot Cov_t\left(R_{m,t+1},VRP_{t+1}\right)+\varepsilon_{m,t+1}$

where $Cov_t\left(R_{i,t+1},R_{m,t+1}\right)$ is the time-

expected conditional covariance between the excess return on portfolio

( $R_{i,t+1}$ ) and the excess return on the market portfolio ( $R_{m,t+1}$ ), $Cov_t\left(R_{i,t+1},VRP_{t+1}\right)$ is the time-

expected conditional covariance between the excess return on portfolio

and the variance risk premia ( $VRP_{t+1}$ ), $Cov_t\left(R_{m,t+1},VRP_{t+1}\right)$ is the time-

expected conditional covariance between the excess return on the market portfolio

and the variance risk premia ( $VRP_{t+1}$ ), and $Var_t\left(R_{m,t+1}\right)$ is the time-

expected conditional variance of excess returns on the market portfolio. The parameters and their

-statistics are estimated using the monthly excess returns on the market portfolio and the 17, 30, 38, 48, and 49 industry portfolios for the sample period from January 1990 to December 2010. The alphas ( $\alpha_i$ ) are reported for each equity portfolio and the

-statistics are presented in parentheses. The

-statistics are adjusted for heteroskedasticity and autocorrelation for each series and cross-correlations among the portfolios. The last four rows, respectively, show the common slope coefficients (

and

), the Wald

statistics from testing the joint hypothesis $H_0: \alpha_1=\alpha_2=...\alpha_m=0$ , and the Wald

statistics from testing the equality of Alphas for high-return and low-return portfolios (Small vs. Big; Value vs. Growth; and HiTec vs. Telcm). The

-values of Wald

and Wald

statistics are given in square brackets.

Table 6a: Results from Larger Cross-Section of Industry Portfolios: 17-industry portfolios
	2.6399
t-statistics	(2.31)
	0.0041
t-statistics	(2.44)
Wald	16.41
Wald $\rho$ -values	[0.56]
Wald	0.58
Wald $\rho$ -values	[0.44]

Table 6b: Results from Larger Cross-Section of Industry Portfolios: 30-industry portfolios
	2.1975
t-statistics	(2.52)
	0.0036
t-statistics	(2.98)
Wald	35.11
Wald $\rho$ -values	[0.28]
Wald	0.06
Wald $\rho$ -values	[0.80]

Table 6c: Results from Larger Cross-Section of Industry Portfolios: 38-industry portfolios
	2.2988
t-statistics	(2.47)
	0.0035
t-statistics	(2.45)
Wald	30.89
Wald $\rho$ -values	[0.75]
Wald	0.32
Wald $\rho$ -values	[0.57]

Table 6d: Results from Larger Cross-Section of Industry Portfolios: 48-industry portfolios
	2.3271
t-statistics	(2.97)
	0.0041
t-statistics	(3.47)
Wald	57.20
Wald $\rho$ -values	[0.20]
Wald	0.53
Wald $\rho$ -values	[0.47]

Table 6e: Results from Larger Cross-Section of Industry Portfolios: 49-industry portfolios
	2.7840
t-statistics	(3.34)
	0.0041
t-statistics	(4.21)
Wald	52.04
Wald $\rho$ -values	[0.39]
Wald	0.13
Wald $\rho$ -values	[0.72]

This table presents the common slope estimates from the following panel regression:

$\displaystyle R_{i,t+1}$	$\displaystyle =$	$\displaystyle \alpha_i+A\cdot Cov_t\left(R_{i,t+1},R_{m,t+1}\right)+B\cdot Cov_t\left(R_{i,t+1},VRP_{t+1}\right)+\lambda\cdot X_t+\varepsilon_{i,t+1}$
$\displaystyle R_{m,t+1}$	$\displaystyle =$	$\displaystyle \alpha_m+A\cdot Var_t\left(R_{m,t+1}\right)+B\cdot Cov_t\left(R_{m,t+1},VRP_{t+1}\right)+\lambda\cdot X_t+\varepsilon_{m,t+1}$

where

, and $\lambda$ ) and their

-statistics are estimated using the monthly excess returns on the market portfolio and the ten size, book-to-market, and industry portfolios for the sample period January 1990 to December 2010. The

-statistics are adjusted for heteroskedasticity and autocorrelation for each series and cross-correlations among the portfolios. The last two rows the Wald

statistics from testing the joint hypothesis $H_0: \alpha_1=\alpha_2=...\alpha_m=0$ , and the Wald

statistics from testing the equality of Alphas for high-return and low-return portfolios (Small vs. Big; Value vs. Growth; and HiTec vs. Telcm). The

-values of Wald

and Wald

statistics are given in square brackets.

Table 7: Controlling for Macroeconomic Variables
	Size	Book-to-Market	Industry
	4.2630	2.5763	4.0421
t-statstics	(3.32)	(2.40)	(2.74)
	0.0057	0.0051	0.0066
t-statstics	(2.85)	(2.25)	(2.96)
$\lambda_{DEF}$	-0.3804	-0.0739	0.6243
$\lambda_{DEF}$ t-statstics	(-0.50)	(-0.09)	(1.02)
$\lambda_{TERM}$	-0.1964	-0.5366	-0.5405
$\lambda_{TERM}$ t-statstics	(-0.64)	(-1.69)	(-2.17)
$\lambda_{RREL}$	0.2330	0.1834	0.0104
$\lambda_{RREL}$ t-statstics	(0.68)	(0.52)	(0.04)
$\lambda_{DIV}$	0.0489	0.0228	0.0314
$\lambda_{DIV}$ t-statstics	(1.33)	(0.60)	(1.05)
$\lambda_{INF}$	0.0270	0.7158	-0.1862
$\lambda_{INF}$ t-statstics	(0.04)	(0.93)	(-0.31)
$\lambda_{IP}$	0.7433	0.8689	1.1941
$\lambda_{IP}$ t-statstics	(1.77)	(2.01)	(3.51)
$\lambda_{UNEMP}$	0.0031	0.0047	0.0026
$\lambda_{UNEMP}$ t-statstics	(1.13)	(1.61)	(1.15)
Wald	16.96	7.97	14.78
Wald $\rho$ -values	[0.11]	[0.72]	[0.19]
Wald	1.46	1.63	0.67
Wald $\rho$ -values	[0.23]	[0.20]	[0.41]

This table presents the common slope estimates (, ) from the following panel regression:

$\displaystyle R_{i,t+1}$	$\displaystyle =$	$\displaystyle \alpha_i+A\cdot Cov_t\left(R_{i,t+1},R_{m,t+1}\right)+B\cdot Cov_t\left(R_{i,t+1},VRP_{t+1}\right)+\varepsilon_{i,t+1}$
$\displaystyle R_{m,t+1}$	$\displaystyle =$	$\displaystyle \alpha_m+A\cdot Var_t\left(R_{m,t+1}\right)+B\cdot Cov_t\left(R_{m,t+1},VRP_{t+1}\right)+\varepsilon_{m,t+1}$

where $Cov_t\left(R_{i,t+1},R_{m,t+1}\right)$ is the time-

expected conditional covariance between the excess return on portfolio

( $R_{i,t+1}$ ) and the excess return on the market portfolio ( $R_{m,t+1}$ ), $Cov_t\left(R_{i,t+1},VRP_{t+1}\right)$ is the time-

expected conditional covariance between the excess return on portfolio

and the variance risk premia ( $VRP_{t+1}$ ), $Cov_t\left(R_{m,t+1},VRP_{t+1}\right)$ is the time-

expected conditional covariance between the excess return on the market portfolio

and the variance risk premia ( $VRP_{t+1}$ ), and $Var_t\left(R_{m,t+1}\right)$ is the time-

expected conditional variance of excess returns on the market portfolio. The parameters and their

-statistics are estimated using the monthly excess returns on the market portfolio and the largest 500 stocks trading at NYSE, AMEX, and NASDAQ, and 318 stocks in the S&P 500 index for the sample period from January 1990 to December 2010. First, the largest 500 firms is determined based on their end-of-month market cap as of the end of each month from January 1990 to December 2010. Due to the fact that the list of 500 firms changes over time as a result of changes in firms' market capitalizations, there are 738 unique firms in our first dataset. In our second dataset, the largest 500 firms is determined based on their market cap at the end of December 2010. Our last dataset contains stocks in the S&P 500 index. Since the stock composition of the S&P 500 index changes through time, we rely on the most recent sample. We also restrict our S&P 500 sample to 318 stocks with non-missing monthly return observations for the period January 1990 - December 2010. The

-statistics are adjusted for heteroskedasticity and autocorrelation for each series and cross-correlations among the portfolios.

Table 8a: Results from Individual Stocks: Largest 500 Stocks end-of-month
	6.4237
t-statistics	(8.04)
	0.0043
t-statistics	(3.61)

Table 8b: Results from Individual Stocks: Largest 500 Stocks as of December 2010
	6.8014
t-statistics	(8.70)
	0.0044
t-statistics	(3.67)

Table 8c: Results from Individual Stocks: Largest 500 Stocks S&P 500 Index
	6.0243
t-statistics	(6.79)
	0.0046
t-statistics	(3.52)

Figure 1: Return-Uncertainty Trade-off Coefficient

Figure 2: Variance Risk Premium, Implied and Realized Variances

Figure 3: Variance Risk Premium and GDP Growth

Figure 4: Relative Performance of the Conditional ICAPM with Uncertainty

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Appendix A Equity Portfolios

In addition to the 1990-2010 period, Table 4 presents the monthly raw return and CAPM Alpha differences between high-return (long) and low-return (short) equity portfolios (size, book-to-market, and industry) for the sample periods 1926-2010 and 1963-2010. For the sample period July 1926 - December 2010, the average return difference between the Small and Big portfolios is 0.60% per month with the OLS -statistic of 2.49 and the Newey-West (1987) -statistic of 2.36, implying that small stocks on average generate higher returns than big stocks. The CAPM Alpha (or abnormal return) for the long-short size portfolio is 0.27% per month with the OLS -statistic of 1.22 and the Newey-West -statistic of 1.38. This economically and statistically insignificant Alpha indicates that the static CAPM does explain the size effect for the 1926-2010 period.

For the ten book-to-market portfolios, the average return difference between the Value and Growth portfolios is 0.53% per month with the OLS -statistic of 2.52 and the Newey-West -statistic of 2.46, implying that value stocks on average generate higher returns than growth stocks (the so-called value premium). Similar to our findings for the size portfolios, the unconditional CAPM can explain the value premium for the 1926-2010 period; the CAPM Alpha (or abnormal return) for the long-short book-to-market portfolio is only 0.24% per month with the OLS -statistic of 1.25 and the Newey-West -statistic of 1.26.

The last six rows in Table 4 report average return differences and CAPM Alphas for the industry portfolios (10-, 17-, 30-, 38-, 48-, and 49-industry portfolios). For the long sample period of 1926-2010, only the extreme portfolios of 48 and 49 industries generate significant return differences, whereas the average return differences for the high-return and low-return portfolios of 10, 17, 30, and 38 industries are either statistically insignificant or marginally significant. For 48- and 49-industry portfolios of Kenneth French, "Aero" industry has the highest average monthly return, whereas "Other" industry has the lowest return, yielding an average monthly return difference of 66 basis points with the Newey-West -statistic of 2.55.²⁵ More importantly, the static CAPM cannot explain the industry effect; the CAPM alpha (or abnormal return) for the "Aero-Other" arbitrage portfolio is 0.50% per month and statistically significant with the -statistic of 2.04. Although the average return differences between high-return and low-return portfolios of 30 and 38 industries are marginally significant, the CAPM Alphas are found to be significant. For 30-industry portfolios, the average return difference between "Coal" and "Other" industries is 0.51% per month and marginally significant with the -statistic of 1.71.²⁶ However, the CAPM Alpha for the "Coal-Other" arbitrage portfolio is 0.65% per month with the -statistic of 2.28. For 38-industry portfolios, the average return difference between "Oil" and "Whlsl" industries is 0.42% per month and marginally significant with the -statistic of 1.85.²⁷ However, the CAPM Alpha for the "Oil-Whlsl" arbitrage portfolio is 0.49% per month with the -statistic of 2.06.

Fama and French (1992) identify economically and statistically significant value premium for the post-1963 period. Moreover, Fama and French (1992) find that the post-1963 value premium is not explained by the CAPM. However, Ang and Chen (2007) provide evidence that the value premium is captured by the CAPM for the sample period of 1926-1963. They also show that the conditional CAPM with stochastic betas can explain the return differences between value and growth portfolios even for the post-1963 period. Fama and French (2006) indicate that the performance of the CAPM with regard to the book-to-market effect varies across subperiods. We investigate the significance of size, book-to-market, and industry effects for the sample that generated heated debate on value premium. We compute the average return differences and Alphas for the subsample period of July 1963 - December 2010.

As presented in Table 4, the average return difference between the Small and Big portfolios as well as the CAPM Alpha for "Small-Big" arbitrage portfolio are positive, but they are economically and statistically insignificant, indicating that the size effect disappears for the post-1963 period. Similar to the findings of Ang and Chen (2007) and Fama and French (2006), value premium remains economically and statistically significant for the sample period July 1963 - December 2010; the average raw and risk-adjusted return differences between the Value and Growth portfolios is 0.55% per month and statistically significant, implying that value stocks on average generate higher returns than growth stocks and this value premium cannot be explained by the static CAPM.

The results for the industry portfolios are similar for the post-1963 period. The high-return and low-return portfolios of 30 and 38 industries generate marginally significant, 48 and 49 industries generate significant return differences, whereas the average return differences for the high-return and low-return portfolios of 10 and 17 industries are insignificant. Specifically, for 30-, 48- and 49-industry portfolios of Kenneth French, "Coal" industry has the highest average monthly return, whereas "Other" industry has the lowest return, yielding an average raw and risk-adjusted return differences of 79 to 92 basis points per month and statistically significant. The unconditional CAPM cannot explain these industry effects either. For 38-industry portfolios, the average return and Alpha differences between "Smoke" and "Govt" industries are about 1.06% and 1.07% per month and significant with the Newey-West -statistics of 2.61 and 2.69, respectively.²⁸

Appendix B DCC Model of Engle (2002)

We estimate the conditional covariances of each equity portfolio with the market portfolio and ( $\sigma_{im,t+1}$ , $\sigma_{i,VRP,t+1}$ ) based on the mean-reverting DCC model of Engle (2002). Engle defines the conditional correlation between two random variables and that each has zero mean as

$\displaystyle \rho_{12,t}=\frac{E_{t-1}\left(r_{1,t} \cdot r_{2,t}\right)}{\sqrt{E_{t-1}\left(r^2_{1,t}\right) \cdot E_{t-1}\left(r^2_{1,t}\right)}},$

(A1)

where the returns are defined as the conditional standard deviation times the standardized disturbance:

$\displaystyle \sigma^2_{i,t}=E_{t-1}\left(r^2_{i,t}\right), \ \ \ r_{i,t}=\sigma_{i,t} \cdot u_{i,t}, \ \ \ i=1, 2$

(A2)

where $u_{i,t}$ is a standardized disturbance that has zero mean and variance one for each series. Equations (A1) and (A2) indicate that the conditional correlation is also the conditional covariance between the standardized disturbances:

$\displaystyle \rho_{12,t}=\frac{E_{t-1}\left(u_{1,t} \cdot u_{2,t}\right)}{\sqrt{E_{t-1}\left(u^2_{1,t}\right) \cdot E_{t-1}\left(u^2_{1,t}\right)}} =E_{t-1}\left(u_{1,t} \cdot u_{2,t}\right).$

(A3)

The conditional covariance matrix of returns is defined as

$\displaystyle H_t = D_t \cdot \rho_t \cdot D_t, \$ where $\displaystyle \ D_t =$ diag $\displaystyle \left\{\sqrt{\sigma^2_{i,t}}\right\},$

(A4)

where $\rho_t$ is the time-varying conditional correlation matrix

$\displaystyle E_{t-1}\left(u_{t} \cdot u'_{t}\right)=D_t^{-1} \cdot H_t \cdot D_t^{-1}=\rho_t, \$ where $\displaystyle \ u_t = D_t^{-1} \cdot r_t.$

(A5)

Engle (2002) introduces a mean-reverting DCC model:

$\displaystyle \rho_{ij,t}=\frac{q_{ij,t}}{\sqrt{q_{ii,t} \cdot q_{jj,t}}},$

(A6)

$\displaystyle q_{ij,t} =\bar \rho_{ij} + a_1 \cdot \left( u_{i,t-1} \cdot u_{j,t-1} - \bar \rho_{ij} \right) + a_2 \cdot \left( q_{ij,t-1}-\bar \rho_{ij}\right)$

(A7)

where $\bar \rho_{ij}$ is the unconditional correlation between $u_{i,t}$ and $u_{j,t}$ . Equation (A7) indicates that the conditional correlation is mean reverting towards $\bar \rho_{ij}$ as long as

Engle (2002) assumes that each asset follows a univariate GARCH process and writes the log likelihood function as:

$\displaystyle \notag L$	$\displaystyle =$	$\displaystyle - \frac{1}{2}\sum_{t=1}^T\left(n\log (2\pi)+\log \left\vert H_t \right\vert + r'_t H^{-1}_t r_t \right)$
	$\displaystyle =$	$\displaystyle - \frac{1}{2}\sum_{t=1}^T\left(n\log (2\pi)+2\log \left\vert D_t \right\vert + r'_t D^{-1}_t D^{-1}_t r_t -u'_t u_t +\log \left\vert\rho_t\right\vert + u'_t \rho_t^{-1} u_t \right).$	(A8)

As shown in Engle (2002), letting the parameters in

be denoted by $\theta$ and the additional parameters in $\rho_t$ be denoted by $\varphi$ , equation (A8) can be written as the sum of a volatility part and a correlation part:

$\displaystyle L(\theta, \varphi)=L_V(\theta) + L_C(\theta, \varphi).$

(A9)

The volatility term is

$\displaystyle L_V(\theta) = - \frac{1}{2}\sum_{t=1}^T\left(n\log (2\pi)+\log \left\vert D_t \right\vert^2 + r'_t D^{-2}_t r_t \right),$

(A10)

and the correlation component is

$\displaystyle L_C(\theta, \varphi)=- \frac{1}{2}\sum_{t=1}^T\left(\log \left\vert\rho_t\right\vert + u'_t \rho_t^{-1} u_t-u'_t u_t \right).$

(A11)

The volatility part of the likelihood is the sum of individual GARCH likelihoods:

$\displaystyle L_V(\theta) = - \frac{1}{2}\sum_{t=1}^T\sum_{i=1}^n\left(\log (2\pi)+\log \left( \sigma_{i,t}^2 \right) + \frac{r_{i,t}^2}{\sigma_{i,t}^2} \right),$

(A12)

which is jointly maximized by separately maximizing each term. The second part of the likelihood is used to estimate the correlation parameters. The two-step approach to maximizing the likelihood is to find

$\displaystyle \hat \theta=\arg\max\{ L_V(\theta)\},$

(A13)

and then take this value as given in the second stage:

$\displaystyle \hat \varphi=\arg\max \{ L_C(\hat\theta, \varphi)\}.$

(A14)

Appendix K System of Regression Equations

Consider a system of equations, of which the typical th equation is

$\displaystyle y_i=X_i \beta_i +u_i ,$

(A15)

where

is a $N\times 1$ vector of time-series observations on the

th dependent variable,

is a $N\times k_i$ matrix of observations of

independent variables, $\beta_i$ is a $k_i\times 1$ vector of unknown coefficients to be estimated, and

is a $N\times 1$ vector of random disturbance terms with mean zero. Parks (1967) proposes an estimation procedure that allows the error term to be both serially and cross-sectionally correlated. In particular, he assumes that the elements of the disturbance vector

follow an AR(1) process:

$\displaystyle u_{it}=\rho u_{it-1}+\varepsilon_{it} ; \ \rho_i<1 ,$

(A16)

where $\varepsilon_{it}$ is serially independently but contemporaneously correlated:

$\displaystyle Cov\left(\varepsilon_{it}\varepsilon_{jt}\right)=\sigma_{ij}, \$ for any $\displaystyle \ i, j, \$ and $\displaystyle \ Cov\left(\varepsilon_{it}\varepsilon_{js}\right)=0, \$ for $\displaystyle \ s \ne t$

(A17)

Equation (A15) can then be written as

$\displaystyle y_i=X_i \beta_i + P_i u_i ,$

(A18)

with

$\displaystyle P_i = \left[ \begin{array}{lllll} \left(1-\rho_i^2\right)^{-1/2} & 0 & 0 & ... & 0\\ \rho_i\left(1-\rho_i^2\right)^{-1/2} & 1 & 0 & ... & 0\\ \rho_i^2\left(1-\rho_i^2\right)^{-1/2} & \rho & 1 & ... & 0\\ .\\ .\\ .\\ \rho_i^{N-1}\left(1-\rho_i^2\right)^{-1/2} & \rho^{N-2} & \rho^{N-3} & ... & 1 \end{array} \right] \ .$

(A19)

Under this setup, Parks (1967) presents a consistent and asymptotically efficient three-step estimation technique for the regression coefficients. The first step uses single equation regressions to estimate the parameters of autoregressive model. The second step uses single equation regressions on transformed equations to estimate the contemporaneous covariances. Finally, the Aitken estimator is formed using the estimated covariance,

$\displaystyle \hat\beta=\left(X^T \Omega^{-1} X\right)^{-1} X^T \Omega^{-1}y ,$

(A20)

where $\Omega \equiv E[uu^T]$ denotes the general covariance matrix of the innovation. In our application, we use the aforementioned methodology with the slope coefficients restricted to be the same for all equity portfolios and individual stocks. In particular, we use the same three-step procedure and the same covariance assumptions as in equations (A16) to (A19) to estimate the covariances and to generate the

-statistics for the parameter estimates.

Footnotes

1. We would like to thank Nick Bloom, Tim Bollerslev, John Campbell, John Cochrane, Frank Diebold, Rob Engle, Xavier Gabaix, Hui Guo, Matt Pritsker, Mark Seasholes, George Tauchen, Andrew Vedolin, Robert Whitelaw, Hong Yan, and Harold Zhang for helpful comments. The views presented here are solely those of the authors and do not necessarily represent those of the Federal Reserve Board or its staff. Return to Text

2. Turan G. Bali is the Dean's Research Professor of Finance, Department of Finance, McDonough School of Business, Georgetown University, Washington, D.C. 20057. Phone: (202) 687-5388, Fax: (202) 687-4031, E-mail: [email protected]. Return to Text

3. Hao Zhou is a senior economist with the Risk Analysis Section, Division of Research and Statistics, Federal Reserve Board, Mail Stop 91, Washington, D.C. 20551. Phone: (202) 452-3360, Fax: (202) 728-5887, E-mail: [email protected]. Return to Text

4. The concepts of risk and risk aversion are the basis of a wide variety of models in economics and finance. In contrast, relatively little attention is paid in formal models to the phenomenon of uncertainty that is arguably more prevalent than risk. Although formal understanding of uncertainty and uncertainty aversion is poor, there exists a definition of uncertainty aversion originally introduced by Schmeidler (1989) and Epstein (1999). In recent studies, uncertainty aversion is defined for a large class of preferences and in different economic settings by Epstein and Wang (1994), Epstein and Zhang (2001), Chen and Epstein (2002), Klibanoff, Marinacci, and Mukerji (2005), and Maccheroni, Marinacci, and Rustichini (2006). In addition to these theoretical papers, Ellsberg's (1961) experimental evidence demonstrates that the distinction between risk and uncertainty is meaningful empirically because people prefer to act on known rather than unknown or ambiguous probabilities. Return to Text

5. Earlier studies (e.g., Rosenberg and Engle (2002), Bakshi and Madan (2006), and Bollerslev, Gibson, and Zhou (2010)) interpret the difference between the implied and expected volatilities as an indicator of the representative agent's risk aversion. Bollerslev, Tauchen, and Zhou (2009) and Drechsler and Yaron (2011) relate the variance risk premia to economic uncertainty risk. Return to Text

6. Knight (1921) draws a distinction between risk and true uncertainty and argues that uncertainty is more common in decision-making process. Knight (1921) points out that risk occurs where the future is unknown, but the probability of all possible outcomes is known. Uncertainty occurs where the probability distribution is itself unknown. We use the variance risk premium as a proxy for economic uncertainty, which is different from Knightian uncertainty. Return to Text

7. Alternatively, our empirical result on VRP may be interpreted as compensating for the rare disaster risk (Gabaix, 2010) or tail risk (Kelly, 2011). The finding may also be related to the expected business condition (Campbell and Diebold, 2009) and its cross-sectional implications for stock returns (Goetzmann, Watanabe, and Watanabe, 2009). Return to Text

8. Such a characterization is accurate up to the second order when there are jumps in the underlying asset (Carr and Wu, 2009; Jiang and Tian, 2005), though Martin (2011) has refined the above formulation to make it robust to jumps. Return to Text

9. The difference between option implied and GARCH type filtered volatilities has been associated in existing literature with notions of aggregate market risk aversion (Rosenberg and Engle, 2002; Bollerslev, Gibson, and Zhou, 2010; Bakshi and Madan, 2006). Return to Text

10. The parameters satisfy $a_{\sigma} > 0, a_q > 0$ , $\vert\rho_{\sigma}\vert < 1$ , $\vert\rho_{q}\vert < 1$ , $\varphi_q>0$ ; and $\{z_{g,t}\}$ , $\{z_{\sigma,t}\}$ and $\{z_{q,t}\}$ are iid Normal

processes jointly independent with each other. Return to Text

11. The approximation comes from the fact that the model-implied risk-neutral conditional expectation cannot be computed in closed form, and a log-linear approximation is applied. Return to Text

12. The idea of relying on reduced-form pattern between market return and variance risk premium is conceptually related to the approach of using the observed market leverage effect to infer the underlying IES coefficient (Tauchen, 2005). Return to Text

13. The CFNAI is a monthly index that determines increases and decreases in economic activity and is designed to assess overall economic activity and related inflationary pressure. It is a weighted average of 85 existing monthly indicators of national economic activity, and is constructed to have an average value of zero and a standard deviation of one. Since economic activity tends toward a trend growth rate over time, a positive index reading corresponds to growth above trend and a negative index reading corresponds to growth below trend. Return to Text

14. Assuming that we have 900 financial firms in month t, the nonparametric measure of 1% VaR is the 9th lowest observation in the cross-section of monthly returns. Return to Text

15. Since the monthly data on variance risk premia (VRP) start in January 1990, our empirical analyses with equity portfolios and VRP are based on the sample period January 1990 Ã¢Â€Â“ December 2010. Return to Text

16. Although we report both the OLS and Newey-West

-statistics in Table 4, hereafter we only discuss the Newey-West

-statistics in the text to preserve space. Return to Text

17. We assume that the excess returns on equity portfolios and the market portfolio as well as the variance risk premia follow an autoregressive of order one, AR(1) process, given in equations (15), (16), and (22). At an earlier stage of the study, we consider alternative specifications of the conditional mean. More specifically, the excess returns are assumed to follow a moving average of order one, MA(1) process, ARMA(1,1) process, and a constant. Our main findings are not sensitive to the choice of the conditional mean specification. Return to Text

18. Our risk aversion estimates ranging from 2.51 to 3.41 are very similar to the median level of risk aversion, 2.52, identified by Bekaert, Engstrom, and Xing (2009) in a different model. Return to Text

19. Although the literature is inconclusive on the direction and significance of a risk-return tradeoff, some studies do provide evidence supporting a positive and significant relation between expected return and risk (e.g., Bollerslev, Engle, and Wooldridge (1988), Ghysels, Santa-Clara, and Valkanov (2005), Guo and Whitelaw (2006), Lundblad (2007), Bali (2008), and Bali and Engle (2010)). Return to Text

20. To save space, in our follow-up tables, the results are reported from the more parsimonious GARCH(1,1) specification of the DCC-based conditional covariances. Return to Text

21. See, e.g., Campbell (1987), Fama and French (1989), and Ferson and Harvey (1991) who test the predictive power of these variables for expected stock returns. Return to Text

22. The monthly data on 10-year T-bond yields, 3-month T-bill rates, BAA- and AAA-rated corporate bond yields are available from the Federal Reserve statistics release website. Return to Text

23. At an earlier stage of the study, we replaced term spread (TERM) with TED spread defined as the difference between the interest rates for 3-month US Treasuries contracts and the 3-month Eurodollars contract as represented by the London Interbank Offered Rate (LIBOR). Our results from TED spread, indicator of perceived credit risk in the general economy, turned out to be very similar to those reported in Table 17. We also used "expected business conditions" variable of Campbell and Diebold (2009) and our main findings remain intact for all equity portfolios. To save space, we do not report these results in the paper. They are available upon request. Return to Text

24. Specifically, the last return on stock

used is either the last return available on CRSP, or the de-listing return, if available. Otherwise, a de-listing return of -100% is included in the study, except that the deletion reason is coded as 500 (reason unavailable), 520 (went to OTC), 551-573, 580 (various reason), 574 (bankruptcy), and 584 (does not meet exchange financial guidelines). For these observations, a return of -30% is assigned. Return to Text

25. According to the 48- and 49-industry definitions and four-digit SIC codes reported at Kenneth French's online data library, "Aero" industry includes Aircraft & parts (3720-3720), Aircraft (3721-3721), Aircraft engines, engine parts (3723-3724), Aircraft parts (3725-3725), and Aircraft parts (3728-3729). "Other" industry includes Sanitary services (4950-4959), Steam, air conditioning supplies (4960-4961), Irrigation systems (4970-4971), and Cogeneration - SM power producer (4990-4991). Return to Text

26. According to the 30-industry definitions and four-digit SIC codes reported at Kenneth French's online data library, "Coal" industry includes Bituminous coal (1200-1299). "Other" industry includes Sanitary services (4950-4959), Steam, air conditioning supplies (4960-4961), Irrigation systems (4970-4971), and Cogeneration - SM power producer (4990-4991). Return to Text

27. According to the 38-industry definitions and four-digit SIC codes reported at Kenneth French's online data library, "Oil" industry includes Oil and Gas Extraction (1300-1399) and "Whlsl" industry includes Wholesale (5000-5199). Return to Text

28. According to the 38-industry definitions and four-digit SIC codes reported at Kenneth French's online data library, "Smoke" industry includes Tobacco Products (2100-2199) and "Govt" industry includes Public Administration (9000-9999). Return to Text

^♣ This version is optimized for use by screen readers. Descriptions for all mathematical expressions are provided in LaTex format. A printable pdf version is available. Return to Text

Risk, Uncertainty, and Expected Returns1

Bibliography

Risk, Uncertainty, and Expected Returns¹