Downside Variance Risk Premium

Downside Variance Risk Premium Abstract We propose a new decomposition of the variance risk premium (VRP) in terms of upside and downside VRPs. These components reflect market compensation for changes in good and bad uncertainties. Empirically, we establish that the downside VRP is the main component of the VRP. We find a positive and significant link between the downside VRP and the equity premium, and a negative but statistically insignificant link between the upside VRP and the equity premium. The opposite relationships between these two components and the equity premium explains the stronger link found between the downside VRP and the equity premium compared with the well-established relationship between VRP and the equity premium. A simple equilibrium consumption-based asset pricing model, fitted to the U.S. data, supports our decomposition. A fundamental relationship in asset pricing posits a positive relation between risk and asset returns; see Merton (1973). This relationship holds over long horizons as shown empirically by Bandi and Perron (2008) and Jacquier and Okou (2014), and theoretically by Bandi et al. (2016) through decomposing the predictive relationship at different time scales. In short horizons, however, measures of risk based on historical asset prices lead to inconclusive estimates of this relation.1 Two recent strands of the asset pricing literature have had success in addressing these conflicting short-horizon results caused by using historical measures of risk in empirical exercises. In one line of research, several studies have documented the asymmetry of risk-return trade-off in response to negative and positive realizations in the financial markets. Among them, studies by Bonomo et al. (2011), Feunou, Jahan-Parvar, and Tédongap (2013), and Rossi and Timmermann (2015) are related to our work. In particular, Feunou, Jahan-Parvar, and Tédongap (2013) explicitly model the upside and downside volatilities (the risk borne by market participants, if realized returns exceed or fall below a certain threshold), document the impact of asymmetries on risk-return trade-off, and highlight the role of downside risk. They develop a methodology that delivers an empirically robust, positive relation between risk and returns by allowing a time-varying market price of risk and asymmetric return distributions. In a second line of research, studies such as Bakshi and Kapadia (2003), Carr and Wu (2009), and Bollerslev, Tauchen, and Zhou (2009) rely on the information in option prices to measure time-varying risk compensations in the data. In particular, Bollerslev, Tauchen, and Zhou (2009) (henceforth, BTZ) study the variance risk premium (VRP), defined as the difference between the risk-neutral and physical expectations of return variation. The VRP, as formalized and studied by BTZ, is a robust predictor of asset returns at maturities of three to six months. Because of its significant predictive power for short-term asset returns, the VRP is often viewed as reflecting investors’ appraisals of changes in near-future volatility. In this article, we bridge the gap between these two strands of the literature. We propose a new decomposition of the VRP in terms of upside and downside variance risk premia (VRPU and VRPD, respectively).2 Our proposed decomposition is motivated by a simple assumption: investors like good uncertainty as it increases the potential of substantial gains, but dislike bad uncertainty as it increases the likelihood of severe losses. We define “good uncertainty” and “bad uncertainty” as volatility associated with positive and negative stock market returns, respectively. Given that investors dislike bad uncertainty, they are willing to pay a premium (the VRPD) to hedge against variation in bad uncertainty. Hence, we expect the VRPD to be generally positive valued. Conversely, because investors like good uncertainty, they should be willing to pay a premium (VRPU) to be exposed to variation in good uncertainty. Thus, we expect a mostly negative-valued VRPU. Thus, the (total) VRP that sums these two components lumps together market participants’ (asymmetric) views about good and bad uncertainties. As a result, a positive (total) VRP reflects the investors’ willingness to pay more in order to hedge against changes in bad uncertainty than for exposure to variations in good uncertainty. Hence, focusing on the (total) VRP does not provide a clear view of the trade-off between good and bad uncertainties, as a small positive VRP quantity does not necessarily imply a lower level of risk and/or risk aversion. Rather, it is an indication of a smaller difference between what agents are willing to pay for downside variation hedging versus upside variation exposure. Using a nonparametric framework, we measure VRPD and VRPU and confirm our assertion. We further investigate whether disentangling the VRP improves the equity premium predictability. We find a positive and significant link between the VRPD and the equity premium, and a negative but statistically insignificant link between the VRPU and the equity premium. The stronger empirical link between the VRPD and the equity premium (compared to the well-established relationship between VRP and the equity premium) is accounted by these opposite-signed relationships between VRPD and VRPU and the equity premium. Theoretically, we support our empirical findings with a simple consumption-based equilibrium asset pricing model, where the representative agent is endowed with Epstein and Zin (1989) preferences, and where the consumption growth process is affected by distinct upside and downside shocks. Our model shares some features with Bansal and Yaron (2004), BTZ, Segal, Shaliastovich, and Yaron (2015), and Bekaert and Engstrom (2015), among others. We fit this model to the U.S. data and show that its implications are consistent with the documented salient regularities. Related literature. This article is related to the mounting literature on the properties of the VRP, as discussed in earlier works by Bakshi and Kapadia (2003), Vilkov (2008), and Carr and Wu (2009), among others. Theoretical attempts to rationalize the observed dynamics of the VRP have led to both reduced-form and general equilibrium models in the literature. Within the reduced-form framework, two papers have made significant contributions. An early study by Todorov (2010) focuses on the temporal dependence of continuous versus discontinuous VRP components within a semiparametric stochastic volatility model. He documents that both components exhibit nontrivial dynamics driven by ex ante volatility changes over time, coupled with unanticipated extreme swings in the market. Recently, Bandi and Renò (2016) focused on the role of a particular source of skewness in stock returns: the co-jumps (the dependence between discontinuous changes in asset prices and contemporaneous, discontinuous changes in volatility). The authors find that the co-jumps modify the mean of the return distribution while also inducing a VRP. In a general equilibrium setting, BTZ design a simple model where time-varying volatility-of-volatility of consumption growth is the key determinant of the VRP. Drechsler and Yaron (2011) provide an equilibrium specification that features long-run risks and discontinuities in the stochastic volatility process governing the level of uncertainty about the cash flows. They extend the model of Bansal and Yaron (2004) by introducing a compound Poisson jump process in the state variable specification, thus departing from BTZ’s assumption of Gaussian economic shocks. Our theoretical framework also extends BTZ’s model, as we specify asymmetric predictable consumption growth components and differences of centered Gamma shocks to fundamentals. Another strand of the literature explores the explanatory ability of the VRP. Along the time series dimension, BTZ, Drechsler and Yaron (2011), and Kelly and Jiang (2014), among others, show that the VRP can help forecast the temporal variation in the aggregate stock market returns with high (low) premia predicting high (low) future returns, especially on a within-the-year timescale. Ang et al. (2006) and Cremers, Halling, and Weinbaum (2015), among others, find that the price of variance risk successfully explains a large set of expected stock returns in the cross-section of assets. Drawing on the decomposition of the quadratic variation of stock returns in terms of continuous and discontinuous variation, Bollerslev and Todorov (2011) decompose the VRP in terms of the diffusive and jump-risk compensations. The authors show that the contribution of the jump tail risk premium is sizable. Our work is based on the alternative decomposition of the quadratic variation proposed by Barndorff-Nielsen, Kinnebrock, and Shephard (2010). The authors decompose the realized variance in terms of upside and downside semi-variances obtained by summing high-frequency positive and negative squared returns, respectively. Other authors have used the same decomposition of the realized variance with a focus on either realized variance predictability (Patton and Sheppard, 2015) or on equity risk premium predictability (Guo, Wang, and Zhou, 2015). These studies focus exclusively on realized measures and do not use option prices to infer the risk-neutral counterparts and deduce the corresponding premia. In comparison, our work clearly evaluates the premia associated with upside and downside semi-variances, both realized and risk-neutral. In an independent and concurrent study, Kilic and Shaliastovich (2015) consider an alternative decomposition of the variance premium into the components associated with good and bad events and provide an economic model that explains their empirical findings. Other related studies aim at decomposing the variance of macroeconomic variables. Segal, Shaliastovich, and Yaron (2015) study the impact of changes in good versus bad uncertainty on aggregate consumption growth and asset values. These authors demonstrate that these different types of uncertainties have opposite effects, with good (bad) economic risk implying a rise (decline) in future wealth or consumption growth. They characterize the role of asymmetric uncertainties in the determination of the economic activity level. Similarly, we develop and estimate a consumption-based equilibrium asset pricing model to highlight the roles that upside and downside variances play in pricing a risky asset in an otherwise standard model. The rest of the article proceeds as follows. In Section 1, we present our decomposition of the VRP and the method for construction of risk-neutral and realized semi-variances. Section 2 details the data used in this study and the empirical construction of predictive variables used in our analysis. We present and discuss our main empirical results in Section 3. Specifically, we intuitively describe the components of VRP, discuss their predictive ability, and explore the robustness of our findings. In Section 4, we introduce and estimate a simple equilibrium consumption-based asset pricing model that supports our empirical results. We evaluate the predictive ability of the difference between the VRPU and VRPD (labeled the skewness risk premium) in Section 5. Section 6 concludes. Further results regarding the extraction of risk-neutral quantities from options and robustness analysis are contained in the Online Appendix. 1 Decomposition of the Variance Risk Premium In what follows, we decompose equity price changes into positive and negative returns with respect to a suitably chosen threshold. We set the threshold, κ, to zero in this study. It can assume other values, given the questions to be answered. We sequentially build measures for upside and downside semi-variances and for skewness. When taken to data, these measures are constructed nonparametrically. We posit that equity market indices such as the S&P 500, S, are defined over the physical probability space characterized by (Ω,ℙ,F), where {Ft}t=0∞∈F are progressive filters on F. The risk-neutral probability measure ℚ is related to the physical measure ℙ through Girsanov’s change of measure relation dℚdℙ|FT=ZT,T<∞. At time t, we denote total equity returns as Rte=(St+Dt−St−1)/St−1, where Dt is the dividend paid out in period [t−1,t]. In sufficiently high-sampling frequencies, Dt is equal to zero. Then, we denote the log of prices by st=ln⁡St, log-returns by rt=st−st−1, and excess log-returns by rte=rt−rtf, where rtf is the risk-free rate observed at time t – 1. We obtain cumulative excess returns by summing one-period excess returns, rt→t+ke=∑j=0krt+je, where k is our prediction horizon. We build the VRP components following the steps in BTZ as the difference between option-implied and realized variances. Alternatively, these two components could be viewed as variances under risk-neutral and physical measures, respectively. In our approach, this construction requires three distinct steps: building the upside and downside realized variances, computing their expectations under the physical measure, and then doing the same under the risk-neutral measure. 1.1 Construction of the Realized Variance Components Following Andersen et al. (2003, 2001), we construct the realized variance of returns on any given trading day t as RVt=∑j=1ntrj,t2, where rj,t2 is the jth intraday squared log-return and nt is the number of intraday returns recorded on that day. We add the squared overnight log-return (the difference in log price between when the market opens at t and when it closes at t – 1), and we scale the RVt series to ensure that the sample average realized variance equals the sample variance of daily logarithmic returns. For a given threshold κ, we decompose the realized variance into upside and downside realized variances following Barndorff-Nielsen, Kinnebrock, and Shephard (2010):   RVtU(κ)=∑j=1ntrj,t2I[rj,t>κ], (1)  RVtD(κ)=∑j=1ntrj,t2I[rj,t≤κ]. (2) We add the squared overnight “positive” log-return (exceeding the threshold κ) to the upside realized variance RVtU, and the squared overnight “negative” log-return (falling below the threshold κ) to the downside realized variance RVtD. As the daily realized variance sums the upside and the downside realized variances, we apply the same scale to the two components of the realized variance. Specifically, we multiply both components by the ratio of the sample variance of daily log-returns over the sample average of the (pre-scaled) realized variance. For a given horizon h, we obtain the cumulative realized quantities by summing the one-day realized quantities over h periods:   RVt,hU(κ)=∑j=1hRVt+jU(κ),RVt,hD(κ)=∑j=1hRVt+jD(κ),RVt,h=∑j=1hRVt+j(κ). (3) By construction, the cumulative realized variance adds up the cumulative realized upside and downside variances:   RVt,h≡RVt,hU(κ)+RVt,hD(κ). (4) Barndorff-Nielsen, Kinnebrock, and Shephard (2010) laid down the theoretical underpinning of this decomposition by relying on a jump-diffusion process for the stock price   dst=μtdt+σtdWt+Δst, where dWt is an increment of the standard Brownian motion and Δst≡st−st− refers to the jump component. The instantaneous variance can be defined as σ˜t2=σt2+(Δst)2. Under this general assumption on the instantaneous return process, and for κ = 0, the authors use infill asymptotics to demonstrate that:   RVt,hU(0)→p12∫tt+hσυ2dυ+∑t≤υ≤t+h(Δsυ)2I[Δsυ>0],RVt,hD(0)→p12∫tt+hσυ2dυ+∑t≤υ≤t+h(Δsυ)2I[Δsυ≤0]. 1.2 Construction of the VRP Components Next, we characterize the VRP of BTZ through premia accrued to bearing upside and downside variance risks, following these steps:   VRPt,h=Etℚ[RVt,h]−Etℙ[RVt,h],=(Etℚ[RVt,hU(κ)]−Etℙ[RVt,hU(κ)])+(Etℚ[RVt,hD(κ)]−Etℙ[RVt,hD(κ)]),VRPt,h≡VRPt,hU(κ)+VRPt,hD(κ). (5)Equation (5) represents the decomposition of the VRP of BTZ into upside and downside variance risk premia— VRPt,hU(κ) and VRPt,hD(κ), respectively—that lies at the heart of our analysis. 1.2.1 Construction of ℙ-expectation The goal here is to evaluate Etℙ[RVt,hU(κ)] and Etℙ[RVt,hD(κ)]. To this end, we consider three specifications: random walk (RW), upside/downside heteroskedastic autoregressive realized variance (U/D-HAR), and multivariate heteroskedastic autoregressive realized variance (M-HAR). Both U/D-HAR and M-HAR specifications mimic Corsi (2009)’s HAR-RV model. To get genuine expected values for realized measures that are not contaminated by forward bias or the use of contemporaneous data, we perform an out-of-sample forecasting exercise to predict the three realized variances, at different horizons, corresponding to 1, 2, 3, 6, 9, 12, 18, and 24 months ahead. We find that these alternative specifications provide qualitatively similar results, likely due to persistence in volatility. Hence, for simplicity and to save space, we only report the results based on the RW model. The RW model is specified as:   Etℙ[RVt,hU/D(κ)]=RVt−h,hU/D(κ), where U/D stands for “U or D”. This is the model used in BTZ.3 1.2.2 Construction of ℚ-expectation Let Ft=Stexp⁡(rtfh) denote the price of a future contract at time t, with maturity h. To build the risk-neutral expectation of RVt,h, we follow the methodology of Andersen and Bondarenko (2007):   Etℚ[RVt,hU(κ)]≈Etℚ[∫tt+hσ˜υ2I[ln⁡(Fυ/Ft)>κF]dυ],=Etℚ[∫tt+hσ˜υ2I[Fυ>Ftexp⁡(κF)]dυ], where κF is a threshold used to compute risk-neutral expectations of semi-variances.4 Thus,   Etℚ[RVt,hU(κ)]≈2∫Ftexp⁡(κF)∞M0(S̲)S̲2dS̲,M0(S̲)=min⁡(Pt(S̲),Ct(S̲)), (6) where, Pt(S̲),Ct(S̲), and S̲ are prices of European put and call options (with maturity h), and the strike price of the underlying asset, respectively. Similarly for Etℚ[RVt,hD(κ)], we get:   Etℚ[RVt,hD(κ)]≈2∫0Ftexp⁡(κF)M0(S̲)S̲2dS̲. (7) Intuitively, one can view the above equalities as replicating option portfolios weighted across moneyness above (upside) and below (downside) a given threshold. We simplify our notation by renaming Etℚ[RVt,hU(κ)] and Etℚ[RVt,hD(κ)] as:   IVt,hU=Etℚ[RVt,hU(κ)], (8)  IVt,hD=Etℚ[RVt,hD(κ)]. (9) We refer to IVt,hU/D as the “risk-neutral semi-variance” or “implied semi-variance” of returns. These quantities are conditioned on the threshold value κ, which we suppress to simplify notation. As evident in this section, our measures of realized and implied volatility are model free. A number of recent studies have raised concerns about the accuracy of traditional methods for evaluating risk-neutral expectations of realized higher moments. For instance, papers by Orlowski (2017) and Schneider and Trojani (2015) document that a nontrivial fraction of information appears to be lost at short maturities (typically one week), as the traditional approach uses a feasible incomplete-market replicating option portfolio with a discrete set of options to approximate a complete market form, which requires a compact set of options.5 In our study, we focus on maturities of one month or longer. In the Online Appendix, we document that both approaches yield similar (highly correlated, with correlation above 0.95) risk-neutral times series dynamics at a monthly horizon and beyond. Thus, we expect our construction of upside and downside VRP components (at a one-month horizon and beyond) as well as their links to the equity premium to be robust to the incomplete-market discretization error. 2 Data In this study, we adapt BTZ’s methodology and use modified measures introduced in Section 1.2. We thus need suitable data to construct excess returns, realized semi-variances ( RVU/D), and risk-neutral semi-variances ( IVU/D). Throughout the study, we set κ = 0. 2.1 Excess Returns We first compute the excess returns by subtracting three-month U.S. Treasury bill rates from log-differences in the S&P 500 composite index, sampled at the end of each month. We downloaded monthly S&P 500 index returns and three-month T-Bill rates from the Center for Research in Security Prices (CRSP) database. As our study requires reliable high-frequency data and option-implied volatilities, our sample runs from September 1996 to March 2015. Panel A of Table 1 reports the descriptive statistics of monthly S&P 500 excess return series. Table 1. Summary statistics   Mean (%)  Median (%)  Std. Dev. (%)  Skewness  Kurtosis  AR(1)  Panel A: Excess returns  Equity  1.9771  14.5157  20.9463  −0.1531  10.5559  −0.0819  Equity (1996–2007)  3.0724  12.5824  17.6474  −0.1379  5.9656  −0.0165    Panel B: Risk-neutral  Variance  19.3544  18.7174  6.6110  1.5650  7.6100  0.9466  Downside variance  16.9766  16.2104  5.8727  1.6746  8.0637  0.9548  Upside variance  9.2570  9.1825  3.1295  1.1479  6.0030  0.8991  Skewness  −7.7196  −7.0090  3.0039  −2.0380  9.6242  0.7323    Panel C: Realized  Variance  16.7137  15.3429  5.5216  3.6748  25.6985  0.9667  Downside variance  11.7677  10.8670  3.9857  3.9042  29.4323  0.9603  Upside variance  11.8550  10.8683  3.8639  3.6288  25.3706  0.9609  Skewness  0.0872  0.1315  1.0911  −6.3619  170.4998  0.6319    Panel D: Risk premium  Variance  2.6407  2.3932  4.2538  −0.3083  6.8325  0.9265  Downside variance  5.2089  4.8693  3.8159  0.2019  4.6310  0.9444  Upside variance  −2.5979  −2.5730  2.5876  −2.2178  22.7198  0.8877  Skewness  −7.8068  −6.9942  3.0606  −2.0696  10.6270  0.9345    Panel E: Macroeconomic  Consumption growth  1.4733  1.4443  1.2808  0.6384  5.1933  −0.2915  Real interest rate  1.0707  0.9607  1.9070  0.7341  5.0354  0.9878    Mean (%)  Median (%)  Std. Dev. (%)  Skewness  Kurtosis  AR(1)  Panel A: Excess returns  Equity  1.9771  14.5157  20.9463  −0.1531  10.5559  −0.0819  Equity (1996–2007)  3.0724  12.5824  17.6474  −0.1379  5.9656  −0.0165    Panel B: Risk-neutral  Variance  19.3544  18.7174  6.6110  1.5650  7.6100  0.9466  Downside variance  16.9766  16.2104  5.8727  1.6746  8.0637  0.9548  Upside variance  9.2570  9.1825  3.1295  1.1479  6.0030  0.8991  Skewness  −7.7196  −7.0090  3.0039  −2.0380  9.6242  0.7323    Panel C: Realized  Variance  16.7137  15.3429  5.5216  3.6748  25.6985  0.9667  Downside variance  11.7677  10.8670  3.9857  3.9042  29.4323  0.9603  Upside variance  11.8550  10.8683  3.8639  3.6288  25.3706  0.9609  Skewness  0.0872  0.1315  1.0911  −6.3619  170.4998  0.6319    Panel D: Risk premium  Variance  2.6407  2.3932  4.2538  −0.3083  6.8325  0.9265  Downside variance  5.2089  4.8693  3.8159  0.2019  4.6310  0.9444  Upside variance  −2.5979  −2.5730  2.5876  −2.2178  22.7198  0.8877  Skewness  −7.8068  −6.9942  3.0606  −2.0696  10.6270  0.9345    Panel E: Macroeconomic  Consumption growth  1.4733  1.4443  1.2808  0.6384  5.1933  −0.2915  Real interest rate  1.0707  0.9607  1.9070  0.7341  5.0354  0.9878  Notes: This table reports the summary statistics for the quantities investigated in this study. Mean, median, and standard deviation values are annualized and in percentages. We report excess kurtosis values. AR(1) represents the values for the first autocorrelation coefficient. The full sample is from September 1996 to March 2015. We also consider a sub-sample ending in December 2007. The data for seasonally adjusted, real per capita consumption growth (based on 2009 chained prices) starts in February 1999 and ends in March 2015. We report annualized mean, median, and standard deviation values for consumption growth and real interest rates. Table 1. Summary statistics   Mean (%)  Median (%)  Std. Dev. (%)  Skewness  Kurtosis  AR(1)  Panel A: Excess returns  Equity  1.9771  14.5157  20.9463  −0.1531  10.5559  −0.0819  Equity (1996–2007)  3.0724  12.5824  17.6474  −0.1379  5.9656  −0.0165    Panel B: Risk-neutral  Variance  19.3544  18.7174  6.6110  1.5650  7.6100  0.9466  Downside variance  16.9766  16.2104  5.8727  1.6746  8.0637  0.9548  Upside variance  9.2570  9.1825  3.1295  1.1479  6.0030  0.8991  Skewness  −7.7196  −7.0090  3.0039  −2.0380  9.6242  0.7323    Panel C: Realized  Variance  16.7137  15.3429  5.5216  3.6748  25.6985  0.9667  Downside variance  11.7677  10.8670  3.9857  3.9042  29.4323  0.9603  Upside variance  11.8550  10.8683  3.8639  3.6288  25.3706  0.9609  Skewness  0.0872  0.1315  1.0911  −6.3619  170.4998  0.6319    Panel D: Risk premium  Variance  2.6407  2.3932  4.2538  −0.3083  6.8325  0.9265  Downside variance  5.2089  4.8693  3.8159  0.2019  4.6310  0.9444  Upside variance  −2.5979  −2.5730  2.5876  −2.2178  22.7198  0.8877  Skewness  −7.8068  −6.9942  3.0606  −2.0696  10.6270  0.9345    Panel E: Macroeconomic  Consumption growth  1.4733  1.4443  1.2808  0.6384  5.1933  −0.2915  Real interest rate  1.0707  0.9607  1.9070  0.7341  5.0354  0.9878    Mean (%)  Median (%)  Std. Dev. (%)  Skewness  Kurtosis  AR(1)  Panel A: Excess returns  Equity  1.9771  14.5157  20.9463  −0.1531  10.5559  −0.0819  Equity (1996–2007)  3.0724  12.5824  17.6474  −0.1379  5.9656  −0.0165    Panel B: Risk-neutral  Variance  19.3544  18.7174  6.6110  1.5650  7.6100  0.9466  Downside variance  16.9766  16.2104  5.8727  1.6746  8.0637  0.9548  Upside variance  9.2570  9.1825  3.1295  1.1479  6.0030  0.8991  Skewness  −7.7196  −7.0090  3.0039  −2.0380  9.6242  0.7323    Panel C: Realized  Variance  16.7137  15.3429  5.5216  3.6748  25.6985  0.9667  Downside variance  11.7677  10.8670  3.9857  3.9042  29.4323  0.9603  Upside variance  11.8550  10.8683  3.8639  3.6288  25.3706  0.9609  Skewness  0.0872  0.1315  1.0911  −6.3619  170.4998  0.6319    Panel D: Risk premium  Variance  2.6407  2.3932  4.2538  −0.3083  6.8325  0.9265  Downside variance  5.2089  4.8693  3.8159  0.2019  4.6310  0.9444  Upside variance  −2.5979  −2.5730  2.5876  −2.2178  22.7198  0.8877  Skewness  −7.8068  −6.9942  3.0606  −2.0696  10.6270  0.9345    Panel E: Macroeconomic  Consumption growth  1.4733  1.4443  1.2808  0.6384  5.1933  −0.2915  Real interest rate  1.0707  0.9607  1.9070  0.7341  5.0354  0.9878  Notes: This table reports the summary statistics for the quantities investigated in this study. Mean, median, and standard deviation values are annualized and in percentages. We report excess kurtosis values. AR(1) represents the values for the first autocorrelation coefficient. The full sample is from September 1996 to March 2015. We also consider a sub-sample ending in December 2007. The data for seasonally adjusted, real per capita consumption growth (based on 2009 chained prices) starts in February 1999 and ends in March 2015. We report annualized mean, median, and standard deviation values for consumption growth and real interest rates. 2.2 High-Frequency Data and Realized Variance Components We then use intraday S&P 500 data downloaded from the Institute of Financial Markets, to construct the daily RVU/D series. We sum the five-minute squared negative returns for the downside realized variance (RVD) and the five-minute squared positive returns for the upside realized variance (RVU). We next add the daily squared overnight negative returns to the downside semi-variance, and the daily squared overnight positive returns to the upside realized variance. The overnight returns are computed for 4:00 p.m. to 9:30 a.m. The total realized variance is obtained by adding the downside and the upside realized variances.6 To construct physical expectations of volatility measures, we use a RW model to forecast the three realized variances at horizons ranging between 1 and 24 months ahead. 2.3 Options Data and Risk-Neutral Variances We extract risk-neutral quantities from daily data of European-style put and call options on the S&P 500 index, available through the OptionMetrics Ivy database. To obtain consistent risk-neutral moments, we preprocess the data by applying the same filters as in Chang, Christoffersen, and Jacobs (2013). Risk-neutral upside and downside variances ( IVU/D) are constructed using the model-free “corridor” implied volatility methodology discussed in Andersen, Bondarenko, and Gonzalez-Perez (2015), Andersen and Bondarenko (2007), and Carr and Madan (1999), among others.7 Panel B of Table 1 shows that risk-neutral volatility measures are persistent—AR(1) parameters are all above 0.60—and demonstrate significant skewness and excess kurtosis. It is also clear that the main factor behind volatility behavior is the downside variance. The contribution of upside volatility to risk-neutral volatility is considerably less than that of downside volatility. 2.4 Data for Structural Estimation In Section 4.5, we fit the theoretical model developed throughout Section 4 to the U.S. macroeconomic and financial data. We use data sampled at a monthly frequency. Excess returns, realized and risk-neutral volatility components, and (upside and downside) VRP are constructed as described earlier in this section. We use seasonally adjusted, monthly, per capita consumption growth, deflated based on chained 2009 prices and downloaded from the Federal Reserve Bank of St. Louis FRED II data bank, to compute log-growth rates of consumption. This series spans January 1999 to March 2015. Our measure for the risk-free rate is the three-month U.S. Treasury Bill rate. To construct the real risk-free rate, we regress the expost real three-month Treasury Bill yield on the nominal rate and past annualized inflation, available from the Wharton Research Data Services (WRDS) Treasury and Inflation database. The fitted values from this regression are the proxy for the exante real interest rates. Panel E of Table 1 reports the descriptive statistics of consumption growth data. 3 Empirical Results In this section, we provide economic intuition and empirical support for our proposed decomposition of the VRP. First, we describe the intuitively expected behavior of the components of the VRP, as well as some salient features of the size and variability of these components. As our approach is nonparametric, these facts stand as guidelines for realistic models (reduced-form and general equilibrium). Second, we provide an extensive investigation of the predictability of the equity premium based on the VRP and its components. We empirically demonstrate the contribution of downside risk premium and characterize the sources of VRP predictability documented by BTZ. Third, we provide a comprehensive robustness study. 3.1 Description of the VRP components We view the VRP as the premium that a market participant is willing to pay to hedge against variations in future realized volatilities. It is expected to be positive, as rationalized within equilibrium frameworks in section ‘Related literature’. We confirm these findings by reporting the summary statistics for the VRP in Table 1. We also plot the time series of the VRP and its components in Figure 1. We present measures based on RW and univariate U/D-HAR forecasts of realized volatility and its components under the physical measure. Constructions of quantities based on multivariate HAR (M-HAR) are virtually identical to those obtained from univariate HAR.8 To save space, and because the results obtained from either the RW or HAR methods are qualitatively similar, we only report findings based on RW forecasts of realized volatility. The series plotted in Figure 1 demonstrate that while HAR-based quantities are more volatile than time series based on the RW, the difference is mainly due to the magnitude of fluctuations and not the fluctuations themselves. This observation may explain the similarities in empirical findings. As expected, from 1996 to 2015, we can see that the VRP is positive most of the time and remains high in uncertain episodes. It is important to emphasize that the economics of the VRP may be richer than that induced by the volatility-of-volatility. Alternative approaches reveal additional VRP drivers such as co-jumps, as formalized in Bandi and Renò (2016). Using high-frequency data and infinitesimal cross-moments, the authors elicit both theoretically and empirically, how contemporaneous (anti-correlated) large discontinuities in the joint dynamics of return and volatility contribute to the time variation of the VRP. Figure 1. View largeDownload slide Time series for variance and skewness risk premia. These figures plot the paths of annualized monthly values ( ×103) for the VRP, upside VRP, downside VRP, and skewness risk premium, extracted from U.S. financial markets data for September 1996 to March 2015. Solid lines represent premia constructed from RW forecasts of the realized volatility and components. The dotted lines represent the same for M-HAR forecasts of the realized volatility and components. The shaded areas represent NBER recessions. Figure 1. View largeDownload slide Time series for variance and skewness risk premia. These figures plot the paths of annualized monthly values ( ×103) for the VRP, upside VRP, downside VRP, and skewness risk premium, extracted from U.S. financial markets data for September 1996 to March 2015. Solid lines represent premia constructed from RW forecasts of the realized volatility and components. The dotted lines represent the same for M-HAR forecasts of the realized volatility and components. The shaded areas represent NBER recessions. We argued at the beginning of the article that we intuitively expect negative-valued VRPU and positive-valued VRPD. Table 1 clearly illustrates these intuitions, as the average VRPU is about –2.60%. Moreover, Figure 1 confirms that VRPU is usually negative through our sample period. The same table reports an average VRPD of roughly 5.21%, and in Figure 1 we observe that VRPD is usually positive. In Section 4, we show that under mild assumptions these expectations about VRP components are supported by our theoretical model. Table 1 also reveals highly persistent, negatively skewed, and fat-tailed empirical distributions for (downside/upside) variance premia. Nonetheless, upside variance appears more left-skewed and leptokurtic compared with (total) variance and downside VRP. 3.2 Predictability of the Equity Premium BTZ derive a theoretical model where the VRP emerges as the main driver of time variation in the equity premium. They show both theoretically and empirically that a higher VRP predicts higher future excess returns. Because the VRP sums downside and upside VRP, BTZ’s framework entails imposing the same coefficient on both (upside and downside) components of the VRP when they are jointly included in a predictive regression of excess returns. However, such a constraint seems very restrictive given the asymmetric views of investors on good uncertainty (preference for upward variability) versus bad uncertainty (aversion to downward variability). Sections 3.1 and 4 document that both VRPD and VRPU have intrinsically different features. Our results are based on a simple linear regression of k-steps-ahead cumulative S&P 500 excess returns on values of a set of predictors that include the VRP, VRPU, and VRPD. Following the results of Ang and Bekaert (2007), reported Student’s t-statistics are based on heteroskedasticity and serial correlation consistent standard errors that explicitly take into account the overlap in the regressions, as advocated by Hodrick (1992). The model used for our analysis is simply:   rt→t+ke=β0+β1xt(h)+εt→t+k, (10) where rt→t+ke is the cumulative excess returns between time t and t + k, xt(h) is one of the predictors discussed in Section 1.2 at time t, h is the construction horizon of xt(h), and εt→t+k is a zero-mean error term. We focus our discussion on the significance of the estimated slope coefficients (β1s), determined by the robust Student’s t-statistics. We report the predictive ability of regressions, measured by the corresponding adjusted R2s. For highly persistent predictor variables, the R2s for the overlapping multi-period return regressions must be interpreted with caution, as noted by BTZ and Jacquier and Okou (2014), among others.9 We decompose the contribution of our predictors to show that (i) predictability results documented by BTZ are driven by the VRPD and (ii) predictability results are mainly driven by risk-neutral expectations—thus, risk-neutral measures contribute more than realized measures. Our empirical findings, presented in Tables 2–5, support our claims. In Panel A of Table 2, we show that the two main regularities uncovered by BTZ, the hump-shaped increase in robust Student’s t-statistics and adjusted R2s reaching their maximum at k = 3 (one-quarter ahead), are present in the data. These effects, however, weaken as the aggregation horizon (h) increases from one month to three months or more; the predictability pattern weakens and then largely disappears for h > 6. Table 2. Predictive content of premium measure h  1   3   6   12     t-Stat  R¯2  t-Stat  R¯2  t-Stat  R¯2  t-Stat  R¯2  k  Panel A: Variance risk premium    1  2.43  2.61  2.51  2.83  1.02  0.02  0.68  −0.30  2  2.84  3.76  3.42  5.58  1.50  0.68  1.04  0.05  3  4.11  8.13  3.58  6.18  1.78  1.19  1.56  0.78  6  2.78  3.65  2.24  2.22  1.57  0.82  2.09  1.87  9  1.98  1.65  1.94  1.57  1.47  0.66  1.98  1.64  12  1.96  1.64  1.43  0.61  1.53  0.77  1.73  1.14    k  Panel B: Downside variance risk premium    1  2.57  2.99  2.68  3.30  1.27  0.34  0.95  −0.06  2  3.22  4.92  4.08  7.95  2.07  1.78  1.54  0.74  3  4.76  10.72  4.46  9.50  2.61  3.12  2.32  2.37  6  3.72  6.75  3.42  5.70  2.84  3.85  3.21  4.98  9  2.96  4.27  3.14  4.86  2.82  3.86  2.99  4.35  12  3.04  4.60  2.65  3.39  2.81  3.86  2.80  3.84    k  Panel C: Upside variance risk premium    1  2.08  1.79  1.91  1.44  0.44  −0.44  −0.04  −0.55  2  2.15  1.96  2.15  1.96  0.39  −0.47  −0.18  −0.54  3  3.05  4.41  2.07  1.79  0.26  −0.52  −0.27  −0.52  6  1.57  0.82  0.61  −0.36  −0.40  −0.48  −0.27  −0.53  9  0.83  −0.18  0.36  −0.50  −0.52  −0.42  −0.14  −0.57  12  0.74  −0.27  −0.05  −0.59  −0.33  −0.52  −0.41  −0.49    k  Panel D: Skewness risk premium    1  −0.10  −0.55  0.41  −0.46  0.96  −0.04  1.25  0.30  2  0.61  −0.35  1.67  0.98  1.98  1.59  2.16  1.98  3  1.03  0.04  2.24  2.18  2.81  3.70  3.29  5.17  6  2.27  2.30  3.33  5.38  4.05  8.00  4.45  9.59  9  2.57  3.13  3.39  5.70  4.20  8.73  3.98  10.59  12  2.83  3.95  3.43  5.93  3.88  7.60  4.07  8.34  h  1   3   6   12     t-Stat  R¯2  t-Stat  R¯2  t-Stat  R¯2  t-Stat  R¯2  k  Panel A: Variance risk premium    1  2.43  2.61  2.51  2.83  1.02  0.02  0.68  −0.30  2  2.84  3.76  3.42  5.58  1.50  0.68  1.04  0.05  3  4.11  8.13  3.58  6.18  1.78  1.19  1.56  0.78  6  2.78  3.65  2.24  2.22  1.57  0.82  2.09  1.87  9  1.98  1.65  1.94  1.57  1.47  0.66  1.98  1.64  12  1.96  1.64  1.43  0.61  1.53  0.77  1.73  1.14    k  Panel B: Downside variance risk premium    1  2.57  2.99  2.68  3.30  1.27  0.34  0.95  −0.06  2  3.22  4.92  4.08  7.95  2.07  1.78  1.54  0.74  3  4.76  10.72  4.46  9.50  2.61  3.12  2.32  2.37  6  3.72  6.75  3.42  5.70  2.84  3.85  3.21  4.98  9  2.96  4.27  3.14  4.86  2.82  3.86  2.99  4.35  12  3.04  4.60  2.65  3.39  2.81  3.86  2.80  3.84    k  Panel C: Upside variance risk premium    1  2.08  1.79  1.91  1.44  0.44  −0.44  −0.04  −0.55  2  2.15  1.96  2.15  1.96  0.39  −0.47  −0.18  −0.54  3  3.05  4.41  2.07  1.79  0.26  −0.52  −0.27  −0.52  6  1.57  0.82  0.61  −0.36  −0.40  −0.48  −0.27  −0.53  9  0.83  −0.18  0.36  −0.50  −0.52  −0.42  −0.14  −0.57  12  0.74  −0.27  −0.05  −0.59  −0.33  −0.52  −0.41  −0.49    k  Panel D: Skewness risk premium    1  −0.10  −0.55  0.41  −0.46  0.96  −0.04  1.25  0.30  2  0.61  −0.35  1.67  0.98  1.98  1.59  2.16  1.98  3  1.03  0.04  2.24  2.18  2.81  3.70  3.29  5.17  6  2.27  2.30  3.33  5.38  4.05  8.00  4.45  9.59  9  2.57  3.13  3.39  5.70  4.20  8.73  3.98  10.59  12  2.83  3.95  3.43  5.93  3.88  7.60  4.07  8.34  Notes: This table reports predictive regression results for prediction horizons (k) between 1 and 12 months ahead, and aggregation levels (h) between 1 and 12 months, based on a predictive regression model of the form rt→t+k=β0+β1xt(h)+ɛt→t+k. In this regression model, rt→t+k is the cumulative excess returns between t and t + k; xt(h) is the proposed variance or skewness risk premia component that takes the values from variance risk, upside variance risk, downside variance risk, or skewness risk premia measures; and ɛt→t+k is a zero-mean error term. The reported Student’s t-statistics for slope parameters are constructed from heteroskedasticity and serial correlation consistent standard errors that explicitly take account of the overlap in the regressions, following Hodrick (1992). R¯2 represents adjusted R2s. Table 2. Predictive content of premium measure h  1   3   6   12     t-Stat  R¯2  t-Stat  R¯2  t-Stat  R¯2  t-Stat  R¯2  k  Panel A: Variance risk premium    1  2.43  2.61  2.51  2.83  1.02  0.02  0.68  −0.30  2  2.84  3.76  3.42  5.58  1.50  0.68  1.04  0.05  3  4.11  8.13  3.58  6.18  1.78  1.19  1.56  0.78  6  2.78  3.65  2.24  2.22  1.57  0.82  2.09  1.87  9  1.98  1.65  1.94  1.57  1.47  0.66  1.98  1.64  12  1.96  1.64  1.43  0.61  1.53  0.77  1.73  1.14    k  Panel B: Downside variance risk premium    1  2.57  2.99  2.68  3.30  1.27  0.34  0.95  −0.06  2  3.22  4.92  4.08  7.95  2.07  1.78  1.54  0.74  3  4.76  10.72  4.46  9.50  2.61  3.12  2.32  2.37  6  3.72  6.75  3.42  5.70  2.84  3.85  3.21  4.98  9  2.96  4.27  3.14  4.86  2.82  3.86  2.99  4.35  12  3.04  4.60  2.65  3.39  2.81  3.86  2.80  3.84    k  Panel C: Upside variance risk premium    1  2.08  1.79  1.91  1.44  0.44  −0.44  −0.04  −0.55  2  2.15  1.96  2.15  1.96  0.39  −0.47  −0.18  −0.54  3  3.05  4.41  2.07  1.79  0.26  −0.52  −0.27  −0.52  6  1.57  0.82  0.61  −0.36  −0.40  −0.48  −0.27  −0.53  9  0.83  −0.18  0.36  −0.50  −0.52  −0.42  −0.14  −0.57  12  0.74  −0.27  −0.05  −0.59  −0.33  −0.52  −0.41  −0.49    k  Panel D: Skewness risk premium    1  −0.10  −0.55  0.41  −0.46  0.96  −0.04  1.25  0.30  2  0.61  −0.35  1.67  0.98  1.98  1.59  2.16  1.98  3  1.03  0.04  2.24  2.18  2.81  3.70  3.29  5.17  6  2.27  2.30  3.33  5.38  4.05  8.00  4.45  9.59  9  2.57  3.13  3.39  5.70  4.20  8.73  3.98  10.59  12  2.83  3.95  3.43  5.93  3.88  7.60  4.07  8.34  h  1   3   6   12     t-Stat  R¯2  t-Stat  R¯2  t-Stat  R¯2  t-Stat  R¯2  k  Panel A: Variance risk premium    1  2.43  2.61  2.51  2.83  1.02  0.02  0.68  −0.30  2  2.84  3.76  3.42  5.58  1.50  0.68  1.04  0.05  3  4.11  8.13  3.58  6.18  1.78  1.19  1.56  0.78  6  2.78  3.65  2.24  2.22  1.57  0.82  2.09  1.87  9  1.98  1.65  1.94  1.57  1.47  0.66  1.98  1.64  12  1.96  1.64  1.43  0.61  1.53  0.77  1.73  1.14    k  Panel B: Downside variance risk premium    1  2.57  2.99  2.68  3.30  1.27  0.34  0.95  −0.06  2  3.22  4.92  4.08  7.95  2.07  1.78  1.54  0.74  3  4.76  10.72  4.46  9.50  2.61  3.12  2.32  2.37  6  3.72  6.75  3.42  5.70  2.84  3.85  3.21  4.98  9  2.96  4.27  3.14  4.86  2.82  3.86  2.99  4.35  12  3.04  4.60  2.65  3.39  2.81  3.86  2.80  3.84    k  Panel C: Upside variance risk premium    1  2.08  1.79  1.91  1.44  0.44  −0.44  −0.04  −0.55  2  2.15  1.96  2.15  1.96  0.39  −0.47  −0.18  −0.54  3  3.05  4.41  2.07  1.79  0.26  −0.52  −0.27  −0.52  6  1.57  0.82  0.61  −0.36  −0.40  −0.48  −0.27  −0.53  9  0.83  −0.18  0.36  −0.50  −0.52  −0.42  −0.14  −0.57  12  0.74  −0.27  −0.05  −0.59  −0.33  −0.52  −0.41  −0.49    k  Panel D: Skewness risk premium    1  −0.10  −0.55  0.41  −0.46  0.96  −0.04  1.25  0.30  2  0.61  −0.35  1.67  0.98  1.98  1.59  2.16  1.98  3  1.03  0.04  2.24  2.18  2.81  3.70  3.29  5.17  6  2.27  2.30  3.33  5.38  4.05  8.00  4.45  9.59  9  2.57  3.13  3.39  5.70  4.20  8.73  3.98  10.59  12  2.83  3.95  3.43  5.93  3.88  7.60  4.07  8.34  Notes: This table reports predictive regression results for prediction horizons (k) between 1 and 12 months ahead, and aggregation levels (h) between 1 and 12 months, based on a predictive regression model of the form rt→t+k=β0+β1xt(h)+ɛt→t+k. In this regression model, rt→t+k is the cumulative excess returns between t and t + k; xt(h) is the proposed variance or skewness risk premia component that takes the values from variance risk, upside variance risk, downside variance risk, or skewness risk premia measures; and ɛt→t+k is a zero-mean error term. The reported Student’s t-statistics for slope parameters are constructed from heteroskedasticity and serial correlation consistent standard errors that explicitly take account of the overlap in the regressions, following Hodrick (1992). R¯2 represents adjusted R2s. Panel B of Table 2 reports the predictability results based on using VRPD as the predictor. We observe the hump-shaped pattern for Student’s t-statistics and the adjusted R2s reaching their maximum between k = 3 or k = 6 months. Moreover, these results are more robust to the aggregation horizon of the predictor. We notice that, in contrast to the VRP results—where predictability is only present for monthly or quarterly constructed risk premia—the VRPD results are largely robust to aggregation horizons; the slope parameters are statistically different from zero even for annually constructed downside VRP (h = 12). Moreover, the VRPD results yield higher adjusted R2s compared with the VRP regressions at similar prediction horizons, an observation that we interpret as the superior ability of the VRPD to explain the variation in aggregate excess returns. Finally, we notice a gradual shift in prediction results from the familiar one-quarter-ahead peak of predictability documented by BTZ to 9-to-12-month-ahead peaks once we increase the aggregation horizon h. Based on these results, we infer that the VRPD is the likely candidate to explain the predictive power of VRP, documented by BTZ. Our results for predictability based on the VRPU, reported in Panel C of Table 2, are weak. The hump-shaped pattern in both robust Student’s t-statistics and in adjusted R2s, while present, is significantly weaker than the results reported by BTZ. Once we increase the aggregation horizon, h, these results are lost. We conclude that bearing upside variance risk does not appear to be an important contributor to the equity premium and, hence, is not a good predictor of this quantity. In addition, we interpret these findings as the VRPU having a low contribution to overall VRP. At this point, it is natural to inquire about including both VRP components in a predictive regression. We present the empirical evidence from this estimation in Panel A of Table 5. After inclusion of the VRPU and VRPD in the same regression, the statistical significance of the VRPU’s slope parameters are broadly lost. We also notice a sign change in Student’s t-statistics associated with the estimated slope parameters of the VRPU and VRPD. This observation is not surprising. As we show in our equilibrium model, and also intuitively, risk-averse investors like variability in positive outcomes of returns but dislike it in negative outcomes. Hence, in a joint regression, we expect the coefficient of VRPD to be positive and that of VRPU to be negative. We claim that the patterns discussed earlier—and, hence, the predictive power of the VRP, and VRPD—are rooted in expectations. That is, the driving force behind our results, as well as those of BTZ, is expected risk-neutral measures of the volatility components. To show the empirical findings supporting our claim, we run predictive regressions using Equation (10). Instead of using the premia employed so far, we use realized and risk-neutral measures of variances, up- and downside variances, and skewness for xt based on our discussions in Section 1. Our empirical findings using risk-neutral volatility measures are available in Table 3. In Panel A, we report the results of running a predictive regression when the predictor is the risk-neutral variance obtained from direct application of the Andersen and Bondarenko (2007) method. It is clear that the estimated slope parameters are statistically different from zero for k≥3 at most construction horizons h. The reported adjusted R2s also imply that the predictive regressions have explanatory power for aggregate excess return variations at k≥3. The same patterns are discernible for risk-neutral downside and upside variances (Panels B and C). Reported adjusted R2s are lower than those reported in Table 2, and these measures of variation yield statistically significant slope parameters at longer prediction horizons than what we observe for the VRP and its components. Taken together, these observations imply that using the premium (rather than the risk-neutral variation) yields better predictions. Table 3. Predictive content of risk-neutral measure h  1   3   6   12     t-Stat  R¯2  t-Stat  R¯2  t-Stat  R¯2  t-Stat  R¯2  k  Panel A: Risk-neutral variance    1  0.28  −0.51  0.50  −0.41  0.69  −0.29  0.75  −0.24  2  1.14  0.17  1.24  0.30  1.35  0.44  1.39  0.51  3  1.30  0.38  1.52  0.72  1.83  1.28  2.13  1.92  6  2.10  1.88  2.33  2.43  2.76  3.57  3.21  4.95  9  2.32  2.44  2.55  3.05  2.95  4.22  3.15  4.85  12  2.21  2.20  2.45  2.82  2.89  4.11  3.30  5.45    k  Panel B: Risk-neutral downside variance    1  0.27  −0.51  0.57  −0.37  0.77  −0.23  0.87  −0.14  2  1.22  0.27  1.39  0.51  1.49  0.66  1.54  0.74  3  1.42  0.56  1.70  1.04  2.03  1.70  2.35  2.44  6  2.23  2.17  2.52  2.91  2.97  4.21  3.42  5.67  9  2.43  2.71  2.68  3.42  3.10  4.70  3.26  5.22  12  2.32  2.48  2.55  3.09  2.99  4.42  3.42  5.84    k  Panel C: Risk-neutral upside variance    1  0.29  −0.50  0.27  −0.51  0.36  −0.48  0.20  −0.53  2  0.93  −0.07  0.76  −0.23  0.78  −0.21  0.60  −0.35  3  0.99  −0.02  0.94  −0.06  1.04  0.05  1.00  0.00  6  1.74  1.12  1.72  1.09  1.92  1.48  2.05  1.77  9  2.01  1.71  2.09  1.89  2.31  2.42  2.38  2.59  12  1.90  1.49  2.09  1.92  2.45  2.83  2.57  3.17    k  Panel D: Risk-neutral skewness    1  0.22  −0.52  0.87  −0.14  1.10  0.11  1.27  0.34  2  1.51  0.70  2.02  1.67  2.06  1.76  2.08  1.79  3  1.93  1.48  2.47  2.74  2.85  3.80  3.13  4.65  6  2.70  3.42  3.28  5.21  3.80  7.02  4.11  8.18  9  2.76  3.64  3.17  4.92  3.64  6.54  3.56  6.26  12  2.67  3.44  2.88  4.07  3.27  5.35  3.66  6.73  h  1   3   6   12     t-Stat  R¯2  t-Stat  R¯2  t-Stat  R¯2  t-Stat  R¯2  k  Panel A: Risk-neutral variance    1  0.28  −0.51  0.50  −0.41  0.69  −0.29  0.75  −0.24  2  1.14  0.17  1.24  0.30  1.35  0.44  1.39  0.51  3  1.30  0.38  1.52  0.72  1.83  1.28  2.13  1.92  6  2.10  1.88  2.33  2.43  2.76  3.57  3.21  4.95  9  2.32  2.44  2.55  3.05  2.95  4.22  3.15  4.85  12  2.21  2.20  2.45  2.82  2.89  4.11  3.30  5.45    k  Panel B: Risk-neutral downside variance    1  0.27  −0.51  0.57  −0.37  0.77  −0.23  0.87  −0.14  2  1.22  0.27  1.39  0.51  1.49  0.66  1.54  0.74  3  1.42  0.56  1.70  1.04  2.03  1.70  2.35  2.44  6  2.23  2.17  2.52  2.91  2.97  4.21  3.42  5.67  9  2.43  2.71  2.68  3.42  3.10  4.70  3.26  5.22  12  2.32  2.48  2.55  3.09  2.99  4.42  3.42  5.84    k  Panel C: Risk-neutral upside variance    1  0.29  −0.50  0.27  −0.51  0.36  −0.48  0.20  −0.53  2  0.93  −0.07  0.76  −0.23  0.78  −0.21  0.60  −0.35  3  0.99  −0.02  0.94  −0.06  1.04  0.05  1.00  0.00  6  1.74  1.12  1.72  1.09  1.92  1.48  2.05  1.77  9  2.01  1.71  2.09  1.89  2.31  2.42  2.38  2.59  12  1.90  1.49  2.09  1.92  2.45  2.83  2.57  3.17    k  Panel D: Risk-neutral skewness    1  0.22  −0.52  0.87  −0.14  1.10  0.11  1.27  0.34  2  1.51  0.70  2.02  1.67  2.06  1.76  2.08  1.79  3  1.93  1.48  2.47  2.74  2.85  3.80  3.13  4.65  6  2.70  3.42  3.28  5.21  3.80  7.02  4.11  8.18  9  2.76  3.64  3.17  4.92  3.64  6.54  3.56  6.26  12  2.67  3.44  2.88  4.07  3.27  5.35  3.66  6.73  Notes: This table reports predictive regression results for risk-neutral variance and skewness measures. The predictive regression model, prediction horizons, aggregation levels, and notation are the same as in the results reported in Table 2. The difference is in the definition of xt(h): instead of risk premia, we use risk-neutral measures for variance, upside variance, downside variance, and skewness. The reported Student’s t-statistics for slope parameters are constructed from heteroskedasticity and serial correlation consistent standard errors that explicitly take account of the overlap in the regressions, following Hodrick (1992). R¯2 represents adjusted R2s. Table 3. Predictive content of risk-neutral measure h  1   3   6   12     t-Stat  R¯2  t-Stat  R¯2  t-Stat  R¯2  t-Stat  R¯2  k  Panel A: Risk-neutral variance    1  0.28  −0.51  0.50  −0.41  0.69  −0.29  0.75  −0.24  2  1.14  0.17  1.24  0.30  1.35  0.44  1.39  0.51  3  1.30  0.38  1.52  0.72  1.83  1.28  2.13  1.92  6  2.10  1.88  2.33  2.43  2.76  3.57  3.21  4.95  9  2.32  2.44  2.55  3.05  2.95  4.22  3.15  4.85  12  2.21  2.20  2.45  2.82  2.89  4.11  3.30  5.45    k  Panel B: Risk-neutral downside variance    1  0.27  −0.51  0.57  −0.37  0.77  −0.23  0.87  −0.14  2  1.22  0.27  1.39  0.51  1.49  0.66  1.54  0.74  3  1.42  0.56  1.70  1.04  2.03  1.70  2.35  2.44  6  2.23  2.17  2.52  2.91  2.97  4.21  3.42  5.67  9  2.43  2.71  2.68  3.42  3.10  4.70  3.26  5.22  12  2.32  2.48  2.55  3.09  2.99  4.42  3.42  5.84    k  Panel C: Risk-neutral upside variance    1  0.29  −0.50  0.27  −0.51  0.36  −0.48  0.20  −0.53  2  0.93  −0.07  0.76  −0.23  0.78  −0.21  0.60  −0.35  3  0.99  −0.02  0.94  −0.06  1.04  0.05  1.00  0.00  6  1.74  1.12  1.72  1.09  1.92  1.48  2.05  1.77  9  2.01  1.71  2.09  1.89  2.31  2.42  2.38  2.59  12  1.90  1.49  2.09  1.92  2.45  2.83  2.57  3.17    k  Panel D: Risk-neutral skewness    1  0.22  −0.52  0.87  −0.14  1.10  0.11  1.27  0.34  2  1.51  0.70  2.02  1.67  2.06  1.76  2.08  1.79  3  1.93  1.48  2.47  2.74  2.85  3.80  3.13  4.65  6  2.70  3.42  3.28  5.21  3.80  7.02  4.11  8.18  9  2.76  3.64  3.17  4.92  3.64  6.54  3.56  6.26  12  2.67  3.44  2.88  4.07  3.27  5.35  3.66  6.73  h  1   3   6   12     t-Stat  R¯2  t-Stat  R¯2  t-Stat  R¯2  t-Stat  R¯2  k  Panel A: Risk-neutral variance    1  0.28  −0.51  0.50  −0.41  0.69  −0.29  0.75  −0.24  2  1.14  0.17  1.24  0.30  1.35  0.44  1.39  0.51  3  1.30  0.38  1.52  0.72  1.83  1.28  2.13  1.92  6  2.10  1.88  2.33  2.43  2.76  3.57  3.21  4.95  9  2.32  2.44  2.55  3.05  2.95  4.22  3.15  4.85  12  2.21  2.20  2.45  2.82  2.89  4.11  3.30  5.45    k  Panel B: Risk-neutral downside variance    1  0.27  −0.51  0.57  −0.37  0.77  −0.23  0.87  −0.14  2  1.22  0.27  1.39  0.51  1.49  0.66  1.54  0.74  3  1.42  0.56  1.70  1.04  2.03  1.70  2.35  2.44  6  2.23  2.17  2.52  2.91  2.97  4.21  3.42  5.67  9  2.43  2.71  2.68  3.42  3.10  4.70  3.26  5.22  12  2.32  2.48  2.55  3.09  2.99  4.42  3.42  5.84    k  Panel C: Risk-neutral upside variance    1  0.29  −0.50  0.27  −0.51  0.36  −0.48  0.20  −0.53  2  0.93  −0.07  0.76  −0.23  0.78  −0.21  0.60  −0.35  3  0.99  −0.02  0.94  −0.06  1.04  0.05  1.00  0.00  6  1.74  1.12  1.72  1.09  1.92  1.48  2.05  1.77  9  2.01  1.71  2.09  1.89  2.31  2.42  2.38  2.59  12  1.90  1.49  2.09  1.92  2.45  2.83  2.57  3.17    k  Panel D: Risk-neutral skewness    1  0.22  −0.52  0.87  −0.14  1.10  0.11  1.27  0.34  2  1.51  0.70  2.02  1.67  2.06  1.76  2.08  1.79  3  1.93  1.48  2.47  2.74  2.85  3.80  3.13  4.65  6  2.70  3.42  3.28  5.21  3.80  7.02  4.11  8.18  9  2.76  3.64  3.17  4.92  3.64  6.54  3.56  6.26  12  2.67  3.44  2.88  4.07  3.27  5.35  3.66  6.73  Notes: This table reports predictive regression results for risk-neutral variance and skewness measures. The predictive regression model, prediction horizons, aggregation levels, and notation are the same as in the results reported in Table 2. The difference is in the definition of xt(h): instead of risk premia, we use risk-neutral measures for variance, upside variance, downside variance, and skewness. The reported Student’s t-statistics for slope parameters are constructed from heteroskedasticity and serial correlation consistent standard errors that explicitly take account of the overlap in the regressions, following Hodrick (1992). R¯2 represents adjusted R2s. However, in comparison with realized (physical) variation measures, risk-neutral measures yield better results. The analysis using realized variation measures is available in Table 4. It is obvious that, by themselves, the realized measures do not yield reasonable predictability, an observation corroborated by the empirical findings of Bekaert, Engstrom, and Ermolov (2015). The majority of the estimated slope parameters are statistically indistinguishable from zero, and the adjusted R2s are low, especially at short horizons. Inclusion of both risk-neutral and realized variance components does not change our findings dramatically, as demonstrated in Panels B and C of Table 5. Table 4. Predictive content of realized (physical) measure h  1   3   6   12     t-Stat  R¯2  t-Stat  R¯2  t-Stat  R¯2  t-Stat  R¯2  k  Panel A: Realized variance    1  −1.10  0.12  −0.99  −0.01  −0.10  −0.55  0.09  −0.55  2  −0.67  −0.30  −0.86  −0.15  0.18  −0.54  0.40  −0.47  3  −1.18  0.22  −0.75  −0.25  0.36  −0.48  0.62  −0.34  6  0.01  −0.57  0.48  −0.44  1.13  0.15  1.07  0.09  9  0.55  −0.40  0.78  −0.22  1.33  0.44  1.12  0.15  12  0.49  −0.45  0.98  −0.02  1.27  0.35  1.40  0.56    k  Panel B: Realized downside variance    1  −1.05  0.06  −0.90  −0.10  −0.08  −0.55  0.09  −0.55  2  −0.53  −0.40  −0.76  −0.23  0.21  −0.53  0.39  −0.47  3  −1.04  0.05  −0.68  −0.30  0.39  −0.48  0.59  −0.36  6  0.05  −0.57  0.48  −0.44  1.08  0.10  0.99  −0.01  9  0.54  −0.41  0.75  −0.25  1.21  0.27  1.01  0.01  12  0.44  −0.48  0.89  −0.12  1.14  0.18  1.30  0.40    k  Panel C: Realized upside variance    1  −1.15  0.18  −1.09  0.10  −0.13  −0.54  0.10  −0.55  2  −0.82  −0.18  −0.95  −0.05  0.14  −0.54  0.41  −0.46  3  −1.33  0.43  −0.82  −0.18  0.34  −0.49  0.66  −0.32  6  −0.05  −0.57  0.48  −0.44  1.17  0.21  1.16  0.19  9  0.39  −0.41  0.81  −0.19  1.44  0.61  1.23  0.29  12  0.54  −0.42  1.08  0.09  1.39  0.54  1.51  0.74    k  Panel D: Realized skewness    1  0.44  −0.45  1.58  0.81  0.63  −0.33  −0.06  −0.55  2  1.51  0.70  1.67  0.99  0.96  −0.05  −0.26  −0.52  3  1.45  0.61  1.19  0.23  0.71  −0.28  −0.89  −0.12  6  0.54  −0.40  0.11  −0.56  −1.01  0.01  −2.64  3.25  9  0.07  −0.58  −0.54  −0.41  −3.01  4.41  −3.67  6.67  12  −0.55  −0.41  −1.82  1.33  −3.37  5.70  −3.36  5.67  h  1   3   6   12     t-Stat  R¯2  t-Stat  R¯2  t-Stat  R¯2  t-Stat  R¯2  k  Panel A: Realized variance    1  −1.10  0.12  −0.99  −0.01  −0.10  −0.55  0.09  −0.55  2  −0.67  −0.30  −0.86  −0.15  0.18  −0.54  0.40  −0.47  3  −1.18  0.22  −0.75  −0.25  0.36  −0.48  0.62  −0.34  6  0.01  −0.57  0.48  −0.44  1.13  0.15  1.07  0.09  9  0.55  −0.40  0.78  −0.22  1.33  0.44  1.12  0.15  12  0.49  −0.45  0.98  −0.02  1.27  0.35  1.40  0.56    k  Panel B: Realized downside variance    1  −1.05  0.06  −0.90  −0.10  −0.08  −0.55  0.09  −0.55  2  −0.53  −0.40  −0.76  −0.23  0.21  −0.53  0.39  −0.47  3  −1.04  0.05  −0.68  −0.30  0.39  −0.48  0.59  −0.36  6  0.05  −0.57  0.48  −0.44  1.08  0.10  0.99  −0.01  9  0.54  −0.41  0.75  −0.25  1.21  0.27  1.01  0.01  12  0.44  −0.48  0.89  −0.12  1.14  0.18  1.30  0.40    k  Panel C: Realized upside variance    1  −1.15  0.18  −1.09  0.10  −0.13  −0.54  0.10  −0.55  2  −0.82  −0.18  −0.95  −0.05  0.14  −0.54  0.41  −0.46  3  −1.33  0.43  −0.82  −0.18  0.34  −0.49  0.66  −0.32  6  −0.05  −0.57  0.48  −0.44  1.17  0.21  1.16  0.19  9  0.39  −0.41  0.81  −0.19  1.44  0.61  1.23  0.29  12  0.54  −0.42  1.08  0.09  1.39  0.54  1.51  0.74    k  Panel D: Realized skewness    1  0.44  −0.45  1.58  0.81  0.63  −0.33  −0.06  −0.55  2  1.51  0.70  1.67  0.99  0.96  −0.05  −0.26  −0.52  3  1.45  0.61  1.19  0.23  0.71  −0.28  −0.89  −0.12  6  0.54  −0.40  0.11  −0.56  −1.01  0.01  −2.64  3.25  9  0.07  −0.58  −0.54  −0.41  −3.01  4.41  −3.67  6.67  12  −0.55  −0.41  −1.82  1.33  −3.37  5.70  −3.36  5.67  Notes: This table reports predictive regression results for realized variance and skewness measures. The predictive regression model, prediction horizons, aggregation levels, and notation are the same as in the results reported in Table 2. The difference is in the definition of xt(h): instead of risk premia, we use realized (historical) measures for variance, upside variance, downside variance, and skewness. The reported Student’s t-statistics for slope parameters are constructed from heteroskedasticity and serial correlation consistent standard errors that explicitly take account of the overlap in the regressions, following Hodrick (1992). R¯2 represents adjusted R2s. Table 4. Predictive content of realized (physical) measure h  1   3   6   12     t-Stat  R¯2  t-Stat  R¯2  t-Stat  R¯2  t-Stat  R¯2  k  Panel A: Realized variance    1  −1.10  0.12  −0.99  −0.01  −0.10  −0.55  0.09  −0.55  2  −0.67  −0.30  −0.86  −0.15  0.18  −0.54  0.40  −0.47  3  −1.18  0.22  −0.75  −0.25  0.36  −0.48  0.62  −0.34  6  0.01  −0.57  0.48  −0.44  1.13  0.15  1.07  0.09  9  0.55  −0.40  0.78  −0.22  1.33  0.44  1.12  0.15  12  0.49  −0.45  0.98  −0.02  1.27  0.35  1.40  0.56    k  Panel B: Realized downside variance    1  −1.05  0.06  −0.90  −0.10  −0.08  −0.55  0.09  −0.55  2  −0.53  −0.40  −0.76  −0.23  0.21  −0.53  0.39  −0.47  3  −1.04  0.05  −0.68  −0.30  0.39  −0.48  0.59  −0.36  6  0.05  −0.57  0.48  −0.44  1.08  0.10  0.99  −0.01  9  0.54  −0.41  0.75  −0.25  1.21  0.27  1.01  0.01  12  0.44  −0.48  0.89  −0.12  1.14  0.18  1.30  0.40    k  Panel C: Realized upside variance    1  −1.15  0.18  −1.09  0.10  −0.13  −0.54  0.10  −0.55  2  −0.82  −0.18  −0.95  −0.05  0.14  −0.54  0.41  −0.46  3  −1.33  0.43  −0.82  −0.18  0.34  −0.49  0.66  −0.32  6  −0.05  −0.57  0.48  −0.44  1.17  0.21  1.16  0.19  9  0.39  −0.41  0.81  −0.19  1.44  0.61  1.23  0.29  12  0.54  −0.42  1.08  0.09  1.39  0.54  1.51  0.74    k  Panel D: Realized skewness    1  0.44  −0.45  1.58  0.81  0.63  −0.33  −0.06  −0.55  2  1.51  0.70  1.67  0.99  0.96  −0.05  −0.26  −0.52  3  1.45  0.61  1.19  0.23  0.71  −0.28  −0.89  −0.12  6  0.54  −0.40  0.11  −0.56  −1.01  0.01  −2.64  3.25  9  0.07  −0.58  −0.54  −0.41  −3.01  4.41  −3.67  6.67  12  −0.55  −0.41  −1.82  1.33  −3.37  5.70  −3.36  5.67  h  1   3   6   12     t-Stat  R¯2  t-Stat  R¯2  t-Stat  R¯2  t-Stat  R¯2  k  Panel A: Realized variance    1  −1.10  0.12  −0.99  −0.01  −0.10  −0.55  0.09  −0.55  2  −0.67  −0.30  −0.86  −0.15  0.18  −0.54  0.40  −0.47  3  −1.18  0.22  −0.75  −0.25  0.36  −0.48  0.62  −0.34  6  0.01  −0.57  0.48  −0.44  1.13  0.15  1.07  0.09  9  0.55  −0.40  0.78  −0.22  1.33  0.44  1.12  0.15  12  0.49  −0.45  0.98  −0.02  1.27  0.35  1.40  0.56    k  Panel B: Realized downside variance    1  −1.05  0.06  −0.90  −0.10  −0.08  −0.55  0.09  −0.55  2  −0.53  −0.40  −0.76  −0.23  0.21  −0.53  0.39  −0.47  3  −1.04  0.05  −0.68  −0.30  0.39  −0.48  0.59  −0.36  6  0.05  −0.57  0.48  −0.44  1.08  0.10  0.99  −0.01  9  0.54  −0.41  0.75  −0.25  1.21  0.27  1.01  0.01  12  0.44  −0.48  0.89  −0.12  1.14  0.18  1.30  0.40    k  Panel C: Realized upside variance    1  −1.15  0.18  −1.09  0.10  −0.13  −0.54  0.10  −0.55  2  −0.82  −0.18  −0.95  −0.05  0.14  −0.54  0.41  −0.46  3  −1.33  0.43  −0.82  −0.18  0.34  −0.49  0.66  −0.32  6  −0.05  −0.57  0.48  −0.44  1.17  0.21  1.16  0.19  9  0.39  −0.41  0.81  −0.19  1.44  0.61  1.23  0.29  12  0.54  −0.42  1.08  0.09  1.39  0.54  1.51  0.74    k  Panel D: Realized skewness    1  0.44  −0.45  1.58  0.81  0.63  −0.33  −0.06  −0.55  2  1.51  0.70  1.67  0.99  0.96  −0.05  −0.26  −0.52  3  1.45  0.61  1.19  0.23  0.71  −0.28  −0.89  −0.12  6  0.54  −0.40  0.11  −0.56  −1.01  0.01  −2.64  3.25  9  0.07  −0.58  −0.54  −0.41  −3.01  4.41  −3.67  6.67  12  −0.55  −0.41  −1.82  1.33  −3.37  5.70  −3.36  5.67  Notes: This table reports predictive regression results for realized variance and skewness measures. The predictive regression model, prediction horizons, aggregation levels, and notation are the same as in the results reported in Table 2. The difference is in the definition of xt(h): instead of risk premia, we use realized (historical) measures for variance, upside variance, downside variance, and skewness. The reported Student’s t-statistics for slope parameters are constructed from heteroskedasticity and serial correlation consistent standard errors that explicitly take account of the overlap in the regressions, following Hodrick (1992). R¯2 represents adjusted R2s. Table 5. Joint regression results h  1   3   6   12     t-Stat   R¯2  t-Stat   R¯2  t-Stat   R¯2  t-Stat   R¯2  k  Up  Down    Up  Down    Up  Down    Up  Down    Panel A: Risk premium  1  −0.01  1.49  2.45  −0.12  1.86  2.77  −0.58  1.32  −0.03  −0.85  1.27  −0.21  2  −0.78  2.49  4.72  −1.28  3.66  8.28  −1.38  2.46  2.26  −1.54  2.17  1.49  3  −1.28  3.79  11.04  −1.81  4.32  10.63  −2.06  3.33  4.84  −2.34  3.31  4.77  6  −2.46  4.19  9.36  −3.00  4.56  9.80  −3.31  4.39  8.98  −3.14  4.54  9.54  9  −2.75  3.99  7.76  −3.10  4.45  9.36  −3.46  4.49  9.50  −2.74  4.09  7.83  12  −3.06  4.29  9.06  −3.18  4.18  8.30  −3.13  4.24  8.61  −2.97  4.10  8.06    Panel B: Risk-neutral measures  1  0.08  −0.01  −1.06  −1.13  1.24  −0.22  −1.30  1.46  0.15  −1.48  1.70  0.51  2  −1.00  1.27  0.27  −2.42  2.69  3.10  −2.27  2.61  2.90  −2.00  2.45  2.35  3  −1.59  1.89  1.39  −2.92  3.26  5.01  −3.22  3.68  6.55  −2.87  3.59  6.21  6  −1.64  2.14  3.09  −2.93  3.47  6.89  −3.25  3.99  9.12  −2.61  3.79  8.66  9  −1.32  1.88  3.12  −2.02  2.62  5.09  −2.25  3.05  6.88  −1.42  2.62  5.78  12  −1.35  1.89  2.94  −1.49  2.07  3.77  −1.39  2.18  4.93  −1.28  2.55  6.20    Panel C: Realized (physical) measures  1  −0.63  0.42  −0.28  −1.82  1.72  1.17  −0.69  0.68  −0.84  0.11  −0.10  −1.10  2  −1.62  1.50  0.51  −1.92  1.83  1.24  −0.93  0.95  −0.60  0.48  −0.46  −0.90  3  −1.68  1.46  1.05  −1.41  1.34  0.25  −0.62  0.64  −0.82  1.27  −1.24  −0.02  6  −0.54  0.54  −0.97  −0.01  0.06  −1.01  1.39  −1.31  0.61  3.54  −3.49  6.14  9  0.01  0.08  −0.99  0.72  −0.64  −0.54  3.61  −3.52  6.77  4.82  −4.76  11.39  12  0.63  −0.55  −0.83  2.07  −1.98  1.77  3.99  −3.91  8.24  4.66  −4.59  11.22  h  1   3   6   12     t-Stat   R¯2  t-Stat   R¯2  t-Stat   R¯2  t-Stat   R¯2  k  Up  Down    Up  Down    Up  Down    Up  Down    Panel A: Risk premium  1  −0.01  1.49  2.45  −0.12  1.86  2.77  −0.58  1.32  −0.03  −0.85  1.27  −0.21  2  −0.78  2.49  4.72  −1.28  3.66  8.28  −1.38  2.46  2.26  −1.54  2.17  1.49  3  −1.28  3.79  11.04  −1.81  4.32  10.63  −2.06  3.33  4.84  −2.34  3.31  4.77  6  −2.46  4.19  9.36  −3.00  4.56  9.80  −3.31  4.39  8.98  −3.14  4.54  9.54  9  −2.75  3.99  7.76  −3.10  4.45  9.36  −3.46  4.49  9.50  −2.74  4.09  7.83  12  −3.06  4.29  9.06  −3.18  4.18  8.30  −3.13  4.24  8.61  −2.97  4.10  8.06    Panel B: Risk-neutral measures  1  0.08  −0.01  −1.06  −1.13  1.24  −0.22  −1.30  1.46  0.15  −1.48  1.70  0.51  2  −1.00  1.27  0.27  −2.42  2.69  3.10  −2.27  2.61  2.90  −2.00  2.45  2.35  3  −1.59  1.89  1.39  −2.92  3.26  5.01  −3.22  3.68  6.55  −2.87  3.59  6.21  6  −1.64  2.14  3.09  −2.93  3.47  6.89  −3.25  3.99  9.12  −2.61  3.79  8.66  9  −1.32  1.88  3.12  −2.02  2.62  5.09  −2.25  3.05  6.88  −1.42  2.62  5.78  12  −1.35  1.89  2.94  −1.49  2.07  3.77  −1.39  2.18  4.93  −1.28  2.55  6.20    Panel C: Realized (physical) measures  1  −0.63  0.42  −0.28  −1.82  1.72  1.17  −0.69  0.68  −0.84  0.11  −0.10  −1.10  2  −1.62  1.50  0.51  −1.92  1.83  1.24  −0.93  0.95  −0.60  0.48  −0.46  −0.90  3  −1.68  1.46  1.05  −1.41  1.34  0.25  −0.62  0.64  −0.82  1.27  −1.24  −0.02  6  −0.54  0.54  −0.97  −0.01  0.06  −1.01  1.39  −1.31  0.61  3.54  −3.49  6.14  9  0.01  0.08  −0.99  0.72  −0.64  −0.54  3.61  −3.52  6.77  4.82  −4.76  11.39  12  0.63  −0.55  −0.83  2.07  −1.98  1.77  3.99  −3.91  8.24  4.66  −4.59  11.22  Notes: This table reports predictive regression results when multiple variance components (risk premia, risk-neutral, and realized measures) are included in the regression model. The prediction horizons, aggregation levels, and notation are the same as in the results reported in Table 2. The difference is in the regression model. Both upside and downside variance components are in the model: rt→t+k=β0+β1x1,t(h)+β2x2,t(h)+ɛt→t+k. x1,t(h) pertains to upside measures and x2,t(h) represents the downside measures used in the analysis. The reported Student’s t-statistics for slope parameters are constructed from heteroskedasticity and serial correlation consistent standard errors that explicitly take account of the overlap in the regressions, following Hodrick (1992). R¯2 represents adjusted R2s. Table 5. Joint regression results h  1   3   6   12     t-Stat   R¯2  t-Stat   R¯2  t-Stat   R¯2  t-Stat   R¯2  k  Up  Down    Up  Down    Up  Down    Up  Down    Panel A: Risk premium  1  −0.01  1.49  2.45  −0.12  1.86  2.77  −0.58  1.32  −0.03  −0.85  1.27  −0.21  2  −0.78  2.49  4.72  −1.28  3.66  8.28  −1.38  2.46  2.26  −1.54  2.17  1.49  3  −1.28  3.79  11.04  −1.81  4.32  10.63  −2.06  3.33  4.84  −2.34  3.31  4.77  6  −2.46  4.19  9.36  −3.00  4.56  9.80  −3.31  4.39  8.98  −3.14  4.54  9.54  9  −2.75  3.99  7.76  −3.10  4.45  9.36  −3.46  4.49  9.50  −2.74  4.09  7.83  12  −3.06  4.29  9.06  −3.18  4.18  8.30  −3.13  4.24  8.61  −2.97  4.10  8.06    Panel B: Risk-neutral measures  1  0.08  −0.01  −1.06  −1.13  1.24  −0.22  −1.30  1.46  0.15  −1.48  1.70  0.51  2  −1.00  1.27  0.27  −2.42  2.69  3.10  −2.27  2.61  2.90  −2.00  2.45  2.35  3  −1.59  1.89  1.39  −2.92  3.26  5.01  −3.22  3.68  6.55  −2.87  3.59  6.21  6  −1.64  2.14  3.09  −2.93  3.47  6.89  −3.25  3.99  9.12  −2.61  3.79  8.66  9  −1.32  1.88  3.12  −2.02  2.62  5.09  −2.25  3.05  6.88  −1.42  2.62  5.78  12  −1.35  1.89  2.94  −1.49  2.07  3.77  −1.39  2.18  4.93  −1.28  2.55  6.20    Panel C: Realized (physical) measures  1  −0.63  0.42  −0.28  −1.82  1.72  1.17  −0.69  0.68  −0.84  0.11  −0.10  −1.10  2  −1.62  1.50  0.51  −1.92  1.83  1.24  −0.93  0.95  −0.60  0.48  −0.46  −0.90  3  −1.68  1.46  1.05  −1.41  1.34  0.25  −0.62  0.64  −0.82  1.27  −1.24  −0.02  6  −0.54  0.54  −0.97  −0.01  0.06  −1.01  1.39  −1.31  0.61  3.54  −3.49  6.14  9  0.01  0.08  −0.99  0.72  −0.64  −0.54  3.61  −3.52  6.77  4.82  −4.76  11.39  12  0.63  −0.55  −0.83  2.07  −1.98  1.77  3.99  −3.91  8.24  4.66  −4.59  11.22  h  1   3   6   12     t-Stat   R¯2  t-Stat   R¯2  t-Stat   R¯2  t-Stat   R¯2  k  Up  Down    Up  Down    Up  Down    Up  Down    Panel A: Risk premium  1  −0.01  1.49  2.45  −0.12  1.86  2.77  −0.58  1.32  −0.03  −0.85  1.27  −0.21  2  −0.78  2.49  4.72  −1.28  3.66  8.28  −1.38  2.46  2.26  −1.54  2.17  1.49  3  −1.28  3.79  11.04  −1.81  4.32  10.63  −2.06  3.33  4.84  −2.34  3.31  4.77  6  −2.46  4.19  9.36  −3.00  4.56  9.80  −3.31  4.39  8.98  −3.14  4.54  9.54  9  −2.75  3.99  7.76  −3.10  4.45  9.36  −3.46  4.49  9.50  −2.74  4.09  7.83  12  −3.06  4.29  9.06  −3.18  4.18  8.30  −3.13  4.24  8.61  −2.97  4.10  8.06    Panel B: Risk-neutral measures  1  0.08  −0.01  −1.06  −1.13  1.24  −0.22  −1.30  1.46  0.15  −1.48  1.70  0.51  2  −1.00  1.27  0.27  −2.42  2.69  3.10  −2.27  2.61  2.90  −2.00  2.45  2.35  3  −1.59  1.89  1.39  −2.92  3.26  5.01  −3.22  3.68  6.55  −2.87  3.59  6.21  6  −1.64  2.14  3.09  −2.93  3.47  6.89  −3.25  3.99  9.12  −2.61  3.79  8.66  9  −1.32  1.88  3.12  −2.02  2.62  5.09  −2.25  3.05  6.88  −1.42  2.62  5.78  12  −1.35  1.89  2.94  −1.49  2.07  3.77  −1.39  2.18  4.93  −1.28  2.55  6.20    Panel C: Realized (physical) measures  1  −0.63  0.42  −0.28  −1.82  1.72  1.17  −0.69  0.68  −0.84  0.11  −0.10  −1.10  2  −1.62  1.50  0.51  −1.92  1.83  1.24  −0.93  0.95  −0.60  0.48  −0.46  −0.90  3  −1.68  1.46  1.05  −1.41  1.34  0.25  −0.62  0.64  −0.82  1.27  −1.24  −0.02  6  −0.54  0.54  −0.97  −0.01  0.06  −1.01  1.39  −1.31  0.61  3.54  −3.49  6.14  9  0.01  0.08  −0.99  0.72  −0.64  −0.54  3.61  −3.52  6.77  4.82  −4.76  11.39  12  0.63  −0.55  −0.83  2.07  −1.98  1.77  3.99  −3.91  8.24  4.66  −4.59  11.22  Notes: This table reports predictive regression results when multiple variance components (risk premia, risk-neutral, and realized measures) are included in the regression model. The prediction horizons, aggregation levels, and notation are the same as in the results reported in Table 2. The difference is in the regression model. Both upside and downside variance components are in the model: rt→t+k=β0+β1x1,t(h)+β2x2,t(h)+ɛt→t+k. x1,t(h) pertains to upside measures and x2,t(h) represents the downside measures used in the analysis. The reported Student’s t-statistics for slope parameters are constructed from heteroskedasticity and serial correlation consistent standard errors that explicitly take account of the overlap in the regressions, following Hodrick (1992). R¯2 represents adjusted R2s. Given the weak performance of realized measures, it is easy to conclude that realized variation plays a secondary role to risk-neutral variation measures in driving the predictability results documented by BTZ or in this study. However, we need both elements in the construction of the variance or skewness risk premia, as realized or risk-neutral measures individually possess inferior prediction power. 3.3 Robustness We perform extensive robustness exercises to document the prediction power of the VRPD for aggregate excess returns in the presence of traditional predictor variables. The goal is to highlight the contribution of our proposed variables in a wider empirical context. Simply put, we observe that the predictive power does not disappear when we include other pricing variables, implying that the VRPD is not simply a proxy for other well-known pricing ratios. Following BTZ and Feunou et al. (2014), among many others, we include equity pricing measures such as the log price–dividend ratio ( log⁡(pt/dt)), lagged log price–dividend ratio ( log⁡(pt−1/dt)), and log price–earnings ratio ( log⁡(pt/et)); yield and spread measures such as term spread (tmst), the difference between 10-year U.S. Treasury Bond yields and three-month U.S. Treasury Bill yields; default spread (dfst), defined as the difference between BBB and AAA corporate bond yields; CPI inflation (inflt); the skewness risk premium ( srpt, introduced in Section 5); and, finally, Kelly and Pruitt (2013) partial least-squares-based, cross-sectional in-sample and out-of-sample predictors (kpist and kpost, respectively). We consider two periods for our analysis: our full sample—September 1996 to December 2010—and a pre-Great Recession sample, September 1996 to December 2007. The latter ends at the time as the BTZ sample. We report our empirical findings in Tables 6–9. These results are based on semi-annually aggregated excess returns and estimated for the one-month-ahead prediction horizon.10 In this robustness study, we scale the cumulative excess returns; we use rt→t+6e/6 as the predicted value and regress it on a one-month lagged predictive variable. Full-sample simple predictive regression results are available in Table 6. Among VRP components, only the downside VRP (dvrpt) has slope parameters that are statistically different from zero and adjusted R2s comparable in magnitude with other pricing variables. Once we use dvrpt along with other pricing variables, we observe the following regularities in Table 7, which reports the joint multivariate regression results. First, the estimated slope parameter for dvrpt is statistically different from zero in all cases. Second, these regressions yield adjusted R2s that range between 3.10% (for dvrpt and tmst, in line with findings of BTZ that report weak predictability for tmst) and 25.71% (for dvrpt and inflt).11 The downside VRP in conjunction with the VRP or upside VRP remains statistically significant and yields adjusted R2s that are around 7%. Table 6. Semi-annual simple predictive regressions, September 1996 to March 2015 Intercept  0.0005  −0.0026  −0.0132  0.0012  0.0756  0.0861  0.0514  0.0004  −0.0000  0.0210  −0.0105  −0.0093    (0.1995)  (−1.1632)  (−3.0304)  (0.7230)  (2.9076)  (3.1978)  (2.0974)  (0.1293)  (−0.0067)  (6.3627)  (−3.1169)  (−2.0903)  uvrpt  −0.0179                          (−0.3917)                        dvrpt    0.1135                          (2.5900)                      srpt      −0.1838                          (−3.6007)                    vrpt        0.0516                          (1.4937)                  log(pt/dt)          −0.0419                          (−2.8630)                log(pt−1/dt)            −0.0478                          (−3.1550)              log(pt/et)              −0.0384                          (−2.0487)            tmst                0.7181                          (0.4233)          dfst                  1.6755                          (0.3885)        inflt                    −0.8052                          (−6.7137)      kpist                      0.1472                          (4.0170)    kpost                        0.1203                          (2.5798)  Adj. R2(%)  −0.5157  3.3439  6.7611  0.7406  4.1793  5.1472  1.9009  −0.5000  −0.5172  21.0807  8.4028  3.3140  Intercept  0.0005  −0.0026  −0.0132  0.0012  0.0756  0.0861  0.0514  0.0004  −0.0000  0.0210  −0.0105  −0.0093    (0.1995)  (−1.1632)  (−3.0304)  (0.7230)  (2.9076)  (3.1978)  (2.0974)  (0.1293)  (−0.0067)  (6.3627)  (−3.1169)  (−2.0903)  uvrpt  −0.0179                          (−0.3917)                        dvrpt    0.1135                          (2.5900)                      srpt      −0.1838                          (−3.6007)                    vrpt        0.0516                          (1.4937)                  log(pt/dt)          −0.0419                          (−2.8630)                log(pt−1/dt)            −0.0478                          (−3.1550)              log(pt/et)              −0.0384                          (−2.0487)            tmst                0.7181                          (0.4233)          dfst                  1.6755                          (0.3885)        inflt                    −0.8052                          (−6.7137)      kpist                      0.1472                          (4.0170)    kpost                        0.1203                          (2.5798)  Adj. R2(%)  −0.5157  3.3439  6.7611  0.7406  4.1793  5.1472  1.9009  −0.5000  −0.5172  21.0807  8.4028  3.3140  Notes: This table presents predictive regressions of the semi-annually (scaled) cumulative excess return rt→t+6e/6=∑j=16rt+je/6 on each one-period (one-month) lagged predictor from September 1996 to March 2015. The Student’s t-statistics presented in parentheses below the estimated coefficients are constructed from heteroskedasticity and serial correlation consistent standard errors that explicitly take account of the overlap in the regressions, following Hodrick (1992). Table 6. Semi-annual simple predictive regressions, September 1996 to March 2015 Intercept  0.0005  −0.0026  −0.0132  0.0012  0.0756  0.0861  0.0514  0.0004  −0.0000  0.0210  −0.0105  −0.0093    (0.1995)  (−1.1632)  (−3.0304)  (0.7230)  (2.9076)  (3.1978)  (2.0974)  (0.1293)  (−0.0067)  (6.3627)  (−3.1169)  (−2.0903)  uvrpt  −0.0179                          (−0.3917)                        dvrpt    0.1135                          (2.5900)                      srpt      −0.1838                          (−3.6007)                    vrpt        0.0516                          (1.4937)                  log(pt/dt)          −0.0419                          (−2.8630)                log(pt−1/dt)            −0.0478                          (−3.1550)              log(pt/et)              −0.0384                          (−2.0487)            tmst                0.7181                          (0.4233)          dfst                  1.6755                          (0.3885)        inflt                    −0.8052                          (−6.7137)      kpist                      0.1472                          (4.0170)    kpost                        0.1203                          (2.5798)  Adj. R2(%)  −0.5157  3.3439  6.7611  0.7406  4.1793  5.1472  1.9009  −0.5000  −0.5172  21.0807  8.4028  3.3140  Intercept  0.0005  −0.0026  −0.0132  0.0012  0.0756  0.0861  0.0514  0.0004  −0.0000  0.0210  −0.0105  −0.0093    (0.1995)  (−1.1632)  (−3.0304)  (0.7230)  (2.9076)  (3.1978)  (2.0974)  (0.1293)  (−0.0067)  (6.3627)  (−3.1169)  (−2.0903)  uvrpt  −0.0179                          (−0.3917)                        dvrpt    0.1135                          (2.5900)                      srpt      −0.1838                          (−3.6007)                    vrpt        0.0516                          (1.4937)                  log(pt/dt)          −0.0419                          (−2.8630)                log(pt−1/dt)            −0.0478                          (−3.1550)              log(pt/et)              −0.0384                          (−2.0487)            tmst                0.7181                          (0.4233)          dfst                  1.6755                          (0.3885)        inflt                    −0.8052                          (−6.7137)      kpist                      0.1472                          (4.0170)    kpost                        0.1203                          (2.5798)  Adj. R2(%)  −0.5157  3.3439  6.7611  0.7406  4.1793  5.1472  1.9009  −0.5000  −0.5172  21.0807  8.4028  3.3140  Notes: This table presents predictive regressions of the semi-annually (scaled) cumulative excess return rt→t+6e/6=∑j=16rt+je/6 on each one-period (one-month) lagged predictor from September 1996 to March 2015. The Student’s t-statistics presented in parentheses below the estimated coefficients are constructed from heteroskedasticity and serial correlation consistent standard errors that explicitly take account of the overlap in the regressions, following Hodrick (1992). Table 7. Semi-annual multiple predictive regressions, September 1996 to March 2015 Intercept  −0.0128  −0.0128  −0.0133  0.0787  0.0951  0.0582  −0.0045  −0.0084  0.0171  −0.0137  −0.0131    (−2.9375)  (−2.9375)  (−2.9489)  (3.0953)  (3.6144)  (2.4206)  (−1.3553)  (−1.7752)  (4.9989)  (−3.8625)  (-2.8590)  uvrpt  −0.1545                        (−2.7123)                      dvrpt  0.2100  0.0555  0.4321  0.1272  0.1392  0.1309  0.1178  0.1359  0.1289  0.1048  0.1129    (3.7629)  (1.1549)  (3.4605)  (2.9688)  (3.2550)  (2.9980)  (2.6638)  (2.9172)  (3.3499)  (2.4918)  (2.6228)  srpt    −0.1545                        (−2.7123)                    vrpt      −0.2640                        (−2.7177)                  log(pt/dt)        −0.0462                        (−3.2112)                log(pt−1/dt)          −0.0557                        (−3.7277)              log(pt/et)            −0.0471                        (−2.5413)            tmst              1.2900                        (0.7681)          dfst                6.2346                        (1.3864)        inflt                  −0.8272                        (−7.0977)      kpist                    0.1425                        (3.9439)    kpost                      0.1197                        (2.6128)  Adj. R2(%)  6.9505  6.9505  6.9664  8.5372  10.3900  6.4572  3.1016  3.8843  25.7111  11.2225  6.6602  Intercept  −0.0128  −0.0128  −0.0133  0.0787  0.0951  0.0582  −0.0045  −0.0084  0.0171  −0.0137  −0.0131    (−2.9375)  (−2.9375)  (−2.9489)  (3.0953)  (3.6144)  (2.4206)  (−1.3553)  (−1.7752)  (4.9989)  (−3.8625)  (-2.8590)  uvrpt  −0.1545                        (−2.7123)                      dvrpt  0.2100  0.0555  0.4321  0.1272  0.1392  0.1309  0.1178  0.1359  0.1289  0.1048  0.1129    (3.7629)  (1.1549)  (3.4605)  (2.9688)  (3.2550)  (2.9980)  (2.6638)  (2.9172)  (3.3499)  (2.4918)  (2.6228)  srpt    −0.1545                        (−2.7123)                    vrpt      −0.2640                        (−2.7177)                  log(pt/dt)        −0.0462                        (−3.2112)                log(pt−1/dt)          −0.0557                        (−3.7277)              log(pt/et)            −0.0471                        (−2.5413)            tmst              1.2900                        (0.7681)          dfst                6.2346                        (1.3864)        inflt                  −0.8272                        (−7.0977)      kpist                    0.1425                        (3.9439)    kpost                      0.1197                        (2.6128)  Adj. R2(%)  6.9505  6.9505  6.9664  8.5372  10.3900  6.4572  3.1016  3.8843  25.7111  11.2225  6.6602  Notes: This table presents predictive regressions of the semi-annually (scaled) cumulative excess return rt→t+6e/6=∑j=16rt+je/6 on one-period (one-month) lagged downside VRP dvrp and one alternative predictor in turn from September 1996 to March 2015. The Student’s t-statistics presented in parentheses below the estimated coefficients are constructed from heteroskedasticity and serial correlation consistent standard errors that explicitly take account of the overlap in the regressions, following Hodrick (1992). Table 7. Semi-annual multiple predictive regressions, September 1996 to March 2015 Intercept  −0.0128  −0.0128  −0.0133  0.0787  0.0951  0.0582  −0.0045  −0.0084  0.0171  −0.0137  −0.0131    (−2.9375)  (−2.9375)  (−2.9489)  (3.0953)  (3.6144)  (2.4206)  (−1.3553)  (−1.7752)  (4.9989)  (−3.8625)  (-2.8590)  uvrpt  −0.1545                        (−2.7123)                      dvrpt  0.2100  0.0555  0.4321  0.1272  0.1392  0.1309  0.1178  0.1359  0.1289  0.1048  0.1129    (3.7629)  (1.1549)  (3.4605)  (2.9688)  (3.2550)  (2.9980)  (2.6638)  (2.9172)  (3.3499)  (2.4918)  (2.6228)  srpt    −0.1545                        (−2.7123)                    vrpt      −0.2640                        (−2.7177)                  log(pt/dt)        −0.0462                        (−3.2112)                log(pt−1/dt)          −0.0557                        (−3.7277)              log(pt/et)            −0.0471                        (−2.5413)            tmst              1.2900                        (0.7681)          dfst                6.2346                        (1.3864)        inflt                  −0.8272                        (−7.0977)      kpist                    0.1425                        (3.9439)    kpost                      0.1197                        (2.6128)  Adj. R2(%)  6.9505  6.9505  6.9664  8.5372  10.3900  6.4572  3.1016  3.8843  25.7111  11.2225  6.6602  Intercept  −0.0128  −0.0128  −0.0133  0.0787  0.0951  0.0582  −0.0045  −0.0084  0.0171  −0.0137  −0.0131    (−2.9375)  (−2.9375)  (−2.9489)  (3.0953)  (3.6144)  (2.4206)  (−1.3553)  (−1.7752)  (4.9989)  (−3.8625)  (-2.8590)  uvrpt  −0.1545                        (−2.7123)                      dvrpt  0.2100  0.0555  0.4321  0.1272  0.1392  0.1309  0.1178  0.1359  0.1289  0.1048  0.1129    (3.7629)  (1.1549)  (3.4605)  (2.9688)  (3.2550)  (2.9980)  (2.6638)  (2.9172)  (3.3499)  (2.4918)  (2.6228)  srpt    −0.1545                        (−2.7123)                    vrpt      −0.2640                        (−2.7177)                  log(pt/dt)        −0.0462                        (−3.2112)                log(pt−1/dt)          −0.0557                        (−3.7277)              log(pt/et)            −0.0471                        (−2.5413)            tmst              1.2900                        (0.7681)          dfst                6.2346                        (1.3864)        inflt                  −0.8272                        (−7.0977)      kpist                    0.1425                        (3.9439)    kpost                      0.1197                        (2.6128)  Adj. R2(%)  6.9505  6.9505  6.9664  8.5372  10.3900  6.4572  3.1016  3.8843  25.7111  11.2225  6.6602  Notes: This table presents predictive regressions of the semi-annually (scaled) cumulative excess return rt→t+6e/6=∑j=16rt+je/6 on one-period (one-month) lagged downside VRP dvrp and one alternative predictor in turn from September 1996 to March 2015. The Student’s t-statistics presented in parentheses below the estimated coefficients are constructed from heteroskedasticity and serial correlation consistent standard errors that explicitly take account of the overlap in the regressions, following Hodrick (1992). We obtain adjusted R2s that are lower than those reported by BTZ for quarterly and annually aggregated multivariate regressions. These differences are driven by the inclusion of the Great Recession period data in our full sample. To illustrate this point, we repeat our estimation with the data ending in December 2007. Simple predictive regression results based on these data are available in Table 8. Excluding the Great Recession period data improves even the univariate predictive regression adjusted R2s across the board. The estimated slope parameters are also closer to BTZ estimates and are generally statistically significant. Table 8. Semi-annual simple predictive regressions, September 1996 to December 2007 Intercept  0.0129  −0.0070  −0.0124  0.0014  0.2016  0.2164  0.0913  0.0035  0.0206  0.0052  −0.0059  −0.0047    (5.1388)  (−3.5015)  (−2.6470)  (1.0794)  (6.6115)  (7.0123)  (3.9851)  (1.5568)  (3.4844)  (1.0608)  (−2.2330)  (−1.3202)  uvrpt  0.2679                          (4.5558)                        dvrpt    0.2784                          (6.6863)                      srpt      −0.2156                          (−3.5176)                    vrpt        0.2300                          (6.5106)                  log(pt/dt)          −0.1092                          (−6.5085)                log(pt−1/dt)            −0.1176                          (−6.9106)              log(pt/et)              −0.0662                          (−3.8474)            tmst                −0.1990                          (−0.1295)          dfst                  −25.6836                          (−3.0118)        inflt                    −0.0735                          (−0.4029)      kpist                      0.1217                          (4.0782)    kpost                        0.0940                          (2.4679)  Adj. R2(%)  13.2802  25.3069  8.1025  24.2902  24.2781  26.6030  9.6656  −0.7680  5.8882  −0.6536  10.8080  3.7964  Intercept  0.0129  −0.0070  −0.0124  0.0014  0.2016  0.2164  0.0913  0.0035  0.0206  0.0052  −0.0059  −0.0047    (5.1388)  (−3.5015)  (−2.6470)  (1.0794)  (6.6115)  (7.0123)  (3.9851)  (1.5568)  (3.4844)  (1.0608)  (−2.2330)  (−1.3202)  uvrpt  0.2679                          (4.5558)                        dvrpt    0.2784                          (6.6863)                      srpt      −0.2156                          (−3.5176)                    vrpt        0.2300                          (6.5106)                  log(pt/dt)          −0.1092                          (−6.5085)                log(pt−1/dt)            −0.1176                          (−6.9106)              log(pt/et)              −0.0662                          (−3.8474)            tmst                −0.1990                          (−0.1295)          dfst                  −25.6836                          (−3.0118)        inflt                    −0.0735                          (−0.4029)      kpist                      0.1217                          (4.0782)    kpost                        0.0940                          (2.4679)  Adj. R2(%)  13.2802  25.3069  8.1025  24.2902  24.2781  26.6030  9.6656  −0.7680  5.8882  −0.6536  10.8080  3.7964  Notes: This table presents predictive regressions of the semi-annually (scaled) cumulative excess return rt→t+6e/6=∑j=16rt+je/6 on each one-period (one-month) lagged predictor from September 1996 to December 2007. The Student’s t-statistics presented in parentheses below the estimated coefficients are constructed from heteroskedasticity and serial correlation consistent standard errors that explicitly take account of the overlap in the regressions, following Hodrick (1992). Table 8. Semi-annual simple predictive regressions, September 1996 to December 2007 Intercept  0.0129  −0.0070  −0.0124  0.0014  0.2016  0.2164  0.0913  0.0035  0.0206  0.0052  −0.0059  −0.0047    (5.1388)  (−3.5015)  (−2.6470)  (1.0794)  (6.6115)  (7.0123)  (3.9851)  (1.5568)  (3.4844)  (1.0608)  (−2.2330)  (−1.3202)  uvrpt  0.2679                          (4.5558)                        dvrpt    0.2784                          (6.6863)                      srpt      −0.2156                          (−3.5176)                    vrpt        0.2300                          (6.5106)                  log(pt/dt)          −0.1092                          (−6.5085)                log(pt−1/dt)            −0.1176                          (−6.9106)              log(pt/et)              −0.0662                          (−3.8474)            tmst                −0.1990                          (−0.1295)          dfst                  −25.6836                          (−3.0118)        inflt                    −0.0735                          (−0.4029)      kpist                      0.1217                          (4.0782)    kpost                        0.0940                          (2.4679)  Adj. R2(%)  13.2802  25.3069  8.1025  24.2902  24.2781  26.6030  9.6656  −0.7680  5.8882  −0.6536  10.8080  3.7964  Intercept  0.0129  −0.0070  −0.0124  0.0014  0.2016  0.2164  0.0913  0.0035  0.0206  0.0052  −0.0059  −0.0047    (5.1388)  (−3.5015)  (−2.6470)  (1.0794)  (6.6115)  (7.0123)  (3.9851)  (1.5568)  (3.4844)  (1.0608)  (−2.2330)  (−1.3202)  uvrpt  0.2679                          (4.5558)                        dvrpt    0.2784                          (6.6863)                      srpt      −0.2156                          (−3.5176)                    vrpt        0.2300                          (6.5106)                  log(pt/dt)          −0.1092                          (−6.5085)                log(pt−1/dt)            −0.1176                          (−6.9106)              log(pt/et)              −0.0662                          (−3.8474)            tmst                −0.1990                          (−0.1295)          dfst                  −25.6836                          (−3.0118)        inflt                    −0.0735                          (−0.4029)      kpist                      0.1217                          (4.0782)    kpost                        0.0940                          (2.4679)  Adj. R2(%)  13.2802  25.3069  8.1025  24.2902  24.2781  26.6030  9.6656  −0.7680  5.8882  −0.6536  10.8080  3.7964  Notes: This table presents predictive regressions of the semi-annually (scaled) cumulative excess return rt→t+6e/6=∑j=16rt+je/6 on each one-period (one-month) lagged predictor from September 1996 to December 2007. The Student’s t-statistics presented in parentheses below the estimated coefficients are constructed from heteroskedasticity and serial correlation consistent standard errors that explicitly take account of the overlap in the regressions, following Hodrick (1992). In Table 9, we report multivariate regression results, based on data from 1996 to 2007. We notice that once dvrpt is included in the regression model, the VRP and the upside VRP are no longer statistically significant. Other pricing variables—except for term spread, default spread, and inflation—yield slope parameters that are statistically significant. Thus, inflation seems to lack prediction power in this sub-sample. We do not observe statistically insignificant slope parameters for the downside variance risk premium except when we include vrpt. Across the board, adjusted R2s are high in this sub-sample. Table 9. Semi-annual multiple predictive regressions, September 1996 to December 2007 Intercept  −0.0045  −0.0045  −0.0049  0.1736  0.1954  0.1053  −0.0087  −0.0033  −0.0142  −0.0155  −0.0162    (−1.0099)  (−1.0099)  (−1.0492)  (6.6158)  (7.5813)  (5.7055)  (−3.2688)  (−0.4896)  (−2.8139)  (−5.8453)  (−4.7195)  uvrpt  0.0454                        (0.6200)                      dvrpt  0.2554  0.3008  0.2065  0.2534  0.2629  0.3156  0.2853  0.2669  0.2980  0.2716  0.2847    (4.5711)  (5.4521)  (1.4127)  (7.0696)  (7.6580)  (8.4728)  (6.7545)  (5.7833)  (6.8848)  (6.9931)  (7.0767)  srpt    0.0454                        (0.6200)                    vrpt      0.0632                        (0.5133)                  log(pt/dt)        −0.0990                        (−6.8974)                log(pt−1/dt)          −0.1113                        (−7.8692)              log(pt/et)            −0.0855                        (−6.1130)            tmst              1.3106                        (0.9772)          dfst                −4.9164                        (−0.5838)        inflt                  0.2541                        (1.5553)      kpist                    0.1148                        (4.5066)    kpost                      0.1050                        (3.2382)  Adj. R2(%)  24.9460  24.9460  24.8746  45.2343  49.3937  41.8336  25.2805  24.9203  26.1258  35.0975  30.4605  Intercept  −0.0045  −0.0045  −0.0049  0.1736  0.1954  0.1053  −0.0087  −0.0033  −0.0142  −0.0155  −0.0162    (−1.0099)  (−1.0099)  (−1.0492)  (6.6158)  (7.5813)  (5.7055)  (−3.2688)  (−0.4896)  (−2.8139)  (−5.8453)  (−4.7195)  uvrpt  0.0454                        (0.6200)                      dvrpt  0.2554  0.3008  0.2065  0.2534  0.2629  0.3156  0.2853  0.2669  0.2980  0.2716  0.2847    (4.5711)  (5.4521)  (1.4127)  (7.0696)  (7.6580)  (8.4728)  (6.7545)  (5.7833)  (6.8848)  (6.9931)  (7.0767)  srpt    0.0454                        (0.6200)                    vrpt      0.0632                        (0.5133)                  log(pt/dt)        −0.0990                        (−6.8974)                log(pt−1/dt)          −0.1113                        (−7.8692)              log(pt/et)            −0.0855                        (−6.1130)            tmst              1.3106                        (0.9772)          dfst                −4.9164                        (−0.5838)        inflt                  0.2541                        (1.5553)      kpist                    0.1148                        (4.5066)    kpost                      0.1050                        (3.2382)  Adj. R2(%)  24.9460  24.9460  24.8746  45.2343  49.3937  41.8336  25.2805  24.9203  26.1258  35.0975  30.4605  Notes: This table presents predictive regressions of the semi-annually (scaled) cumulative excess return rt→t+6e/6=∑j=16rt+je/6 on one-period (one-month) lagged downside VRP dvrp and one alternative predictor in turn from September 1996 to December 2007. The Student’s t-statistics presented in parentheses below the estimated coefficients are constructed from heteroskedasticity and serial correlation consistent standard errors that explicitly take account of the overlap in the regressions, following Hodrick (1992). Table 9. Semi-annual multiple predictive regressions, September 1996 to December 2007 Intercept  −0.0045  −0.0045  −0.0049  0.1736  0.1954  0.1053  −0.0087  −0.0033  −0.0142  −0.0155  −0.0162    (−1.0099)  (−1.0099)  (−1.0492)  (6.6158)  (7.5813)  (5.7055)  (−3.2688)  (−0.4896)  (−2.8139)  (−5.8453)  (−4.7195)  uvrpt  0.0454                        (0.6200)                      dvrpt  0.2554  0.3008  0.2065  0.2534  0.2629  0.3156  0.2853  0.2669  0.2980  0.2716  0.2847    (4.5711)  (5.4521)  (1.4127)  (7.0696)  (7.6580)  (8.4728)  (6.7545)  (5.7833)  (6.8848)  (6.9931)  (7.0767)  srpt    0.0454                        (0.6200)                    vrpt      0.0632                        (0.5133)                  log(pt/dt)        −0.0990                        (−6.8974)                log(pt−1/dt)          −0.1113                        (−7.8692)              log(pt/et)            −0.0855                        (−6.1130)            tmst              1.3106                        (0.9772)          dfst                −4.9164                        (−0.5838)        inflt                  0.2541                        (1.5553)      kpist                    0.1148                        (4.5066)    kpost                      0.1050                        (3.2382)  Adj. R2(%)  24.9460  24.9460  24.8746  45.2343  49.3937  41.8336  25.2805  24.9203  26.1258  35.0975  30.4605  Intercept  −0.0045  −0.0045  −0.0049  0.1736  0.1954  0.1053  −0.0087  −0.0033  −0.0142  −0.0155  −0.0162    (−1.0099)  (−1.0099)  (−1.0492)  (6.6158)  (7.5813)  (5.7055)  (−3.2688)  (−0.4896)  (−2.8139)  (−5.8453)  (−4.7195)  uvrpt  0.0454                        (0.6200)                      dvrpt  0.2554  0.3008  0.2065  0.2534  0.2629  0.3156  0.2853  0.2669  0.2980  0.2716  0.2847    (4.5711)  (5.4521)  (1.4127)  (7.0696)  (7.6580)  (8.4728)  (6.7545)  (5.7833)  (6.8848)  (6.9931)  (7.0767)  srpt    0.0454                        (0.6200)                    vrpt      0.0632                        (0.5133)                  log(pt/dt)        −0.0990                        (−6.8974)                log(pt−1/dt)          −0.1113                        (−7.8692)              log(pt/et)            −0.0855                        (−6.1130)            tmst              1.3106                        (0.9772)          dfst                −4.9164                        (−0.5838)        inflt                  0.2541                        (1.5553)      kpist                    0.1148                        (4.5066)    kpost                      0.1050                        (3.2382)  Adj. R2(%)  24.9460  24.9460  24.8746  45.2343  49.3937  41.8336  25.2805  24.9203  26.1258  35.0975  30.4605  Notes: This table presents predictive regressions of the semi-annually (scaled) cumulative excess return rt→t+6e/6=∑j=16rt+je/6 on one-period (one-month) lagged downside VRP dvrp and one alternative predictor in turn from September 1996 to December 2007. The Student’s t-statistics presented in parentheses below the estimated coefficients are constructed from heteroskedasticity and serial correlation consistent standard errors that explicitly take account of the overlap in the regressions, following Hodrick (1992). 4. A Simple Equilibrium Model In this section, we present an equilibrium consumption-based asset pricing model that supports the proposed decomposition of the VRP in terms of upside and downside components. We estimate this model, using a maximum likelihood procedure. We view this exercise as one possible theoretical motivation for our empirical findings, and it is important to stress that this is not the only way. We rely on BTZ, but could have built on Bandi and Renò (2016) or Drechsler and Yaron (2011) as well. Our main objective is to highlight the roles that upside and downside variances play in pricing a risky asset in an otherwise standard asset pricing model. In particular, we show that the model, under standard and mild assumptions yields closed-form solutions for VRP components that align well with empirical regularities. In addition, fitting the model to the data yields empirical results that align well with model predictions, empirical regularities discussed in Section 3, and with BTZ findings. To save space, we only report the main results. Appendix A reports step-by-step derivations of the theoretical findings and the estimation procedure. 4.1 Preferences We consider an endowment economy in discrete time. The representative agent’s preferences over the future consumption stream are characterized by Kreps and Porteus (1978) intertemporal preferences, as formulated by Epstein and Zin (1989) and Weil (1989),   Ut=[(1−δ)Ct1−γθ+δ(EtUt+11−γ)1θ]θ1−γ, (11) where Ct is the consumption bundle at time t, δ is the subjective discount factor, γ is the coefficient of risk aversion, and ψ is the elasticity of intertemporal substitution (IES).12 Parameter θ is defined as θ≡1−γ1−1ψ. If θ = 1, then γ=1/ψ and Kreps and Porteus preferences collapse to the expected power utility, which implies an agent who is indifferent to the timing of the resolution of the consumption path uncertainty. With γ>1/ψ, the agent prefers an early resolution of uncertainty. For γ<1/ψ, the agent prefers a late resolution of uncertainty. Epstein and Zin (1989) show that the logarithm of the stochastic discount factor (SDF) implied by these preferences is given by:   ln Mt+1=mt+1=θ ln⁡ δ−θψΔct+1+(θ−1)rc,t+1, (12) where Δct+1=ln⁡(Ct+1Ct) is the log growth rate of aggregate consumption, and rc,t is the log return of the asset that delivers aggregate consumption as dividends. This asset represents the returns on a wealth portfolio. The Euler equation states that:   Et[exp⁡(mt+1+ri,t+1)]=1, (13) where rc,t represents the log returns for the consumption-generating asset. The risk-free rate, which represents the returns on an asset that delivers a unit of consumption in the next period with certainty, is defined as:   rtf=ln⁡[1Et(Mt+1)]. (14) 4.2 Consumption Dynamics under the Physical Measure Our specification of consumption dynamics incorporates elements from Bansal and Yaron (2004), BTZ, Feunou, Jahan-Parvar, and Tédongap (2013), Segal, Shaliastovich, and Yaron (2015), and Bekaert and Engstrom (2015). Fundamentally, we follow Bansal and Yaron (2004) in assuming that consumption growth has a predictable component. We differ from Bansal and Yaron in assuming that the predictable component is proportional to consumption growth’s upside and downside volatility components. Thus, we are closer to Segal, Shaliastovich, and Yaron (2015).13 As a result, we have   Δct+1=μ0+μ1Vu,t+μ2Vd,t+σc(ɛu,t+1−ɛd,t+1), (15) where μ1,μ2∈ℝ, ɛu,t+1 and ɛd,t+1 are two mean-zero shocks that affect both the realized and expected consumption growth.14 ɛu,t+1 represents upside shocks to consumption growth, and ɛd,t+1 denotes downside shocks. Following Bekaert and Engstrom (2015) and Segal, Shaliastovich, and Yaron (2015), we assume that these shocks have a demeaned Gamma distribution and model them as:   ɛi,t+1=ɛ˜i,t+1−Vi,t, i={u,d}, (16) where ɛ˜i,t+1∼Γ(Vi,t,1).15 These distributional assumptions imply that volatilities of upside and downside shocks are time varying and driven by shape parameters Vu,t and Vd,t. In particular, we have:   Vart[ɛi,t+1]=Vi,t, i={u,d}. (17) Naturally, the total conditional variance of consumption growth when ɛu,t+1 and ɛd,t+1 are conditionally independent is simply σc2(Vu,t+Vd,t). The distribution of the shock to return shares similar properties with the difference of the demeaned Gamma in Equation (15): (i) the variance is the sum of the variances of upside and downside shocks, and (ii) the skewness is (up to a scaling factor) the difference between the variances of upside and downside shocks. The sign and size of μ1 and μ2 matter in this context. With μ1=μ2, we have a stochastic volatility component in the conditional mean of the consumption growth process, similar to the classic GARCH-in-Mean structure for modeling risk-return trade-off in equity returns. With both slope parameters equal to zero, the model yields the BTZ unpredictable consumption growth.16 If |μ1|=|μ2|, with μ1>0 and μ2<0, we have Skewness-in-Mean, similar in spirit to the Feunou, Jahan-Parvar, and Tédongap (2013) formulation for equity returns. With μ1≠μ2, we have free parameters that have an impact on loadings of risk factors on risky asset returns and the stochastic discount factor. Intuitively, we expect μ1>0: a rise in upside volatility at time t implies higher consumption growth at time t + 1, all else being equal. By the same logic, we intuitively expect a negative-valued μ2, implying an expected fall in consumption growth following an uptick in downside volatility—following bad economic outcomes, households curb their consumption. We observe that:   ln⁡Et exp⁡(νɛi,t+1)=f(ν)Vi,t, (18) where f(ν)=−(ln⁡(1−ν)+ν). Both Bekaert, Engstrom, and Ermolov (2015) and Segal, Shaliastovich, and Yaron (2015) use this compact functional form for the Gamma distribution cumulant. It simply follows that f(ν)>0,f′′(ν)>0, and f(ν)>f(−ν) for all ν>0. We assume that Vi,t follows a time-varying, square root process with time-varying volatility-of-volatility, similar to the specification of the volatility process in BTZ:   Vu,t+1=αu+βuVu,t+qu,tzt+1u, (19)  qu,t+1=γu,0+γu,1qu,t+ϕuqu,tzt+11, (20)  Vd,t+1=αd+βdVd,t+qd,tzt+1d, (21)  qd,t+1=γd,0+γd,1qd,t+ϕdqd,tzt+12, (22) where zti are standard normal innovations, and i={u,d,1,2}. The parameters must satisfy the following restrictions: αu>0,αd>0,γu,0>0,γd,0>0,|βu|<1,|βd|<1,|γu,1|<1,|γd,1|<1,ϕu>0, and ϕd>0. In addition, we assume that {ztu},{ztd},{zt1}, and {zt2} are i.i.d.∼N(0,1) and jointly independent from {ɛu,t} and {ɛd,t}. The assumptions above yield time-varying uncertainty and asymmetry in consumption growth. Through volatility-of-volatility processes qu,t and qd,t, the setup induces additional temporal variation in consumption growth. Temporal variation in the volatility-of-volatility process is necessary for generating a sizable VRP. Asymmetry is needed to generate upside and downside VRP. We solve the model following the methodology proposed by Bansal and Yaron (2004), BTZ, and many others. We consider that the logarithm of the wealth–consumption ratio wt or the price–consumption ratio ( pct=ln⁡(PtCt)) for the asset that pays the consumption endowment {Ct+i}i=1∞ is affine with respect to state variables Vi,t and qi,t. We then posit that the consumption-generating returns are approximately linear with respect to the log price-consumption ratio, as popularized by Campbell and Shiller (1988):   rc,t+1=κ0+κ1wt+1−wt+Δct+1,wt=A0+A1Vu,t+A2Vd,t+A3qu,t+A4qd,t, where κ0 and κ1 are log-linearization coefficients, and A0,A1,A2,A3, and A4 are factor-loading coefficients to be determined. We solve for the consumption–generating asset returns, rc,t, using the Euler Equation (13). Following standard arguments (see Appendix A.1), we find the equilibrium values of coefficients A0 to A4:   A1=−f[σc(1−γ)]+(1−γ)μ1θ(κ1βu−1), (23)  A2=−f[−σc(1−γ)]+(1−γ)μ2θ(κ1βd−1), (24)  A3=(1−κ1γu,1)−(1−κ1γu,1)2−θ2ϕu2κ14A12θκ12ϕu2, (25)  A4=(1−κ1γd,1)−(1−κ1γd,1)2−θ2ϕd2κ14A22θκ12ϕd2, (26)  A0=ln⁡δ+(1−1ψ)μ0+κ0+κ1(αuA1+αdA2+γu,0A3+γd,0A4)1−κ1. (27) It is easy to see that while A3 and A4 are negative-valued, the signs of A1 and A2 depend on the signs and sizes of μ1 and μ2. We report the conditions that ensure A1>0 and A2<0 after deriving the dynamic of the model under the risk-neutral measure. 4.3 Risk-Neutral Dynamics and the Premia Combining the historical dynamic and the stochastic discount factor imply the risk-neutral dynamic and closed-form expression for different risk premia (see Appendix A.2 for details on the mathematical derivations). Starting with the equity risk premium, we have   ERPt≡Et[rc,t+1]−Etℚ[rc,t+1]=γσc21+γσcVu,t+γσc21−γσcVd,t+(1−θ)κ12(A12+A32ϕu2)qu,t+(1−θ)κ12(A22+A42ϕd2)qd,t. (28) This expression for the equity risk premium shows that our model implies unequal loadings for upside and downside volatility factors. The slope coefficients for volatility-of-volatility factors are also, in general, unequal. To derive the upside and downside VRP, we need to decompose the total variance of rc,t+1. The total conditional variance ( σr,t2≡Vart[rc,t+1]) is given by:   σr,t2=σc2Vu,t+σc2Vd,t+κ12(A12+A32ϕu2)qu,t+κ12(A22+A42ϕd2)qd,t. The upside and downside variances are:   (σr,tu)2=σc2Vu,t+κ12(A12+A32ϕu2)qu,t, (29)  (σr,td)2=σc2Vd,t+κ12(A22+A42ϕd2)qd,t. (30) Hence, the upside and downside VRP are given by:   VRPtU≡Etℚ[(σr,t+1u)2]−Et[(σr,t+1u)2]=(θ−1)(σc2κ1A1+κ13(A12+A32ϕu2)A3ϕu2)qu,t,VRPtD≡Etℚ[(σr,t+1d)2]−Et[(σr,t+1d)2]=(θ−1)(σc2κ1A2+κ13(A22+A42ϕd2)A4ϕd2)qd,t. As discussed before, we expect VRPtU<0 and VRPtD>0. It follows that:   σc2κ1A1+κ13(A12+A32ϕu2)A3ϕu2>0, (31)  σc2κ1A2+κ13(A22+A42ϕd2)A4ϕd2<0. (32) In Appendix A.3, we discuss the necessary and sufficient conditions ensuring that both inequalities (31) and (32) hold. We can express these conditions in a very simple and intuitive way: A sufficient condition for VRPtd>0 is μ2≤0, A necessary condition for VRPtu<0 is μ1≥f(σc(1−γ))γ−1≥0.Our estimation delivers structural parameter estimates that satisfy all the theoretically expected restrictions—inducing VRPtU<0 and VRPtD>0—and are consistent with the facts discussed in depth in the first part of the paper. Since Equation (43) implies that the equity risk premium loads positively on both qu,t and qd,t, and because VRPtU<0 is negatively proportional to qu,t while VRPtD>0 is positively proportional to qd,t, the equity risk premium loads positively on the downside VRP but negatively on the upside VRP. This feature of the general equilibrium model is also consistent with the empirical regularities documented in the predictability analysis. 4.4 Estimation To appraise the empirical performance of our general equilibrium, we implement a maximum likelihood estimation procedure. Namely, we maximize the joint likelihood of consumption growth, stock market return, and risk-free rate series. The following steps provide a brief description of the estimation. The shocks to consumption growth and stock return are:   Δct+1−Et[Δct+1]=ε1,t+1,rc,t+1−Et(rc,t+1)=ε1,t+1+ε2,t+1, where   ε1,t+1≡σc(ɛu,t+1−ɛd,t+1),ε2,t+1≡κ1[(A1zt+1u+ϕuA3zt+11)qu,t+(A2zt+1d+ϕdA4zt+12)qd,t], and ε1,t+1, ε2,t+1 are conditionally independent random variables. Note that   ε2,t+1∼N(0,κ12(A12+ϕu2A32)qu,t+κ12(A22+ϕd2A42)qd,t). Hence, the joint density function of consumption and return writes   f(c,rc)(Δct+1,rc,t+1)=fε1(Δct+1−Et[Δct+1])fε2(rc,t+1−Δct+1−(Et(rc,t+1)−Et(rc,t+1))), where fε1 and fε2 are the marginal densities of ε1,t+1 and ε2,t+1, respectively. These marginal densities can be computed according to   fε1(ε1,t+1)=1π∫0∞Re[exp⁡(−iνε1,t+1+f(iσcν)Vu,t+f(−iσcν)Vd,t)]dν, and   ln⁡[fε2(ε2,t+1)]=−12ln⁡(2π)−12ln⁡(κ12(A12+ϕu2A32)qu,t+κ12(A22+ϕd2A42)qd,t)−12ε2,t+12κ12(A12+ϕu2A32)qu,t+κ12(A22+ϕd2A42)qd,t. It follows that the joint log-likelihood of consumption and return is calculated as   ln⁡L(C,R)=∑t=0T−1ln⁡(f(c,rc)(Δct+1,rc,t+1)), (33) where T denotes the sample size. The risk-free rate distribution is based on a Gaussian error likelihood   ln⁡LRF∝−12∑t=1T{ln⁡(RFRMSE2)+et2/RFRMSE2}, where the error term is computed as et=rftobserved−rftModel, and the corresponding root-mean-square error is given by RFRMSE≡1T∑t=1Tet2. Finally, all the parameters are estimated by maximizing the joint likelihood of consumption growth, stock return, and risk-free rate   ln⁡L(C,R)+ln⁡LRF. (34) To effectively implement our estimation strategy, we need Vu,t, Vd,t, qu,t and qd,t that are latent factors in the model. To circumvent this challenge, we assume that observed one-month ahead upside VRP ( VRPtU), downside VRP ( VRPtD), conditional upside stock return variance ( Etℙ[RVt+1U]), and conditional downside stock return variance ( Etℙ[RVt+1D]) are measured without error. This assumption entails that the observed quantities exactly match their theoretical counterparts, thus allowing to infer the latent factors as   qu,t=−1(1−θ)κ1(σc2A1+κ12(A12+A32ϕu2)A3ϕu2)VRPtU≡ςuVRPtU,qd,t=−1(1−θ)κ1(σc2A2+κ12(A22+A42ϕd2)A4ϕd2)VRPtD≡ςdVRPtD,Vu,t=−κ12σc2(A12+A32ϕu2)γu,0βu−αuβu+1βuσc2Etℙ[RVt+1U]−κ12σc2(A12+A32ϕu2)γu,1βuςuVRPtU,≡ϖu+ϑuEtℙ[RVt+1U]+ϱuVRPtU,Vd,t=−κ12σc2(A22+A42ϕd2)γd,0βd−αdβd+1βdσc2Etℙ[RVt+1D]−κ12σc2(A22+A42ϕd2)γd,1βdςdVRPtD,≡ϖd+ϑdEtℙ[RVt+1D]+ϱdVRPtD.Table 10 reports the structural parameter estimates of the general equilibrium model and their corresponding standard errors. These results clearly show that our general equilibrium model yields accurate parameter estimates that are consistent with the empirical evidence. Specifically, the estimated values confirm that μ1>0, μ2<0, and both parameters with different magnitudes ( |μ1|<|μ2|) are economically and statistically significant. Table 10. Structural estimation of the theoretical model Parameters  Estimates  Std. Err.  γ  1.01  0.00  θ  −0.04  0.02  δ  1.00  0.00  μ0  0.32  0.10  μ1  0.05  0.01  μ2  −0.10  0.01  σc  0.87  0.04  αu  0.30  0.03  βu  0.99  0.00  αd  0.16  0.01  βd  0.99  0.00  γu,0  8.22E+03  3.07E+03  γu,1  0.53  0.00  ϕu  1.14E+05  4.72E+04  γd,0  1.27E+03  2.01E+02  γd,1  0.55  0.00  ϕd  4.76E+04  1.75E+04  κ0  0.00  0.00  κ1  0.13  0.04  Parameters  Estimates  Std. Err.  γ  1.01  0.00  θ  −0.04  0.02  δ  1.00  0.00  μ0  0.32  0.10  μ1  0.05  0.01  μ2  −0.10  0.01  σc  0.87  0.04  αu  0.30  0.03  βu  0.99  0.00  αd  0.16  0.01  βd  0.99  0.00  γu,0  8.22E+03  3.07E+03  γu,1  0.53  0.00  ϕu  1.14E+05  4.72E+04  γd,0  1.27E+03  2.01E+02  γd,1  0.55  0.00  ϕd  4.76E+04  1.75E+04  κ0  0.00  0.00  κ1  0.13  0.04  Notes: This table reports the structural parameter estimates of the general equilibrium model along with their standard errors (Std. Err.). These estimates are obtained by maximizing the joint likelihood of consumption growth, stock return, and risk-free rate. Table 10. Structural estimation of the theoretical model Parameters  Estimates  Std. Err.  γ  1.01  0.00  θ  −0.04  0.02  δ  1.00  0.00  μ0  0.32  0.10  μ1  0.05  0.01  μ2  −0.10  0.01  σc  0.87  0.04  αu  0.30  0.03  βu  0.99  0.00  αd  0.16  0.01  βd  0.99  0.00  γu,0  8.22E+03  3.07E+03  γu,1  0.53  0.00  ϕu  1.14E+05  4.72E+04  γd,0  1.27E+03  2.01E+02  γd,1  0.55  0.00  ϕd  4.76E+04  1.75E+04  κ0  0.00  0.00  κ1  0.13  0.04  Parameters  Estimates  Std. Err.  γ  1.01  0.00  θ  −0.04  0.02  δ  1.00  0.00  μ0  0.32  0.10  μ1  0.05  0.01  μ2  −0.10  0.01  σc  0.87  0.04  αu  0.30  0.03  βu  0.99  0.00  αd  0.16  0.01  βd  0.99  0.00  γu,0  8.22E+03  3.07E+03  γu,1  0.53  0.00  ϕu  1.14E+05  4.72E+04  γd,0  1.27E+03  2.01E+02  γd,1  0.55  0.00  ϕd  4.76E+04  1.75E+04  κ0  0.00  0.00  κ1  0.13  0.04  Notes: This table reports the structural parameter estimates of the general equilibrium model along with their standard errors (Std. Err.). These estimates are obtained by maximizing the joint likelihood of consumption growth, stock return, and risk-free rate. 4.5 Confronting the Model with Data In implementing our structural estimation procedure, we show that model-implied upside equity returns variance, downside equity returns variance, and their corresponding premia perfectly match the observed series. Thus, there are at least three remaining empirical challenges for the equilibrium model: comparing structural model-implied to observed (i) predictive regression slopes of excess return on variance risk-premium components, (ii) equity returns and consumption growth expectations, and (iii) consumption growth variance dynamics. Figure 2 graphically summarizes additional performance results of the equilibrium model. It shows that the predictive regression slopes as per Equation (10) lie within the observed 95% confidence bounds. We also see that the equilibrium model-implied conditional variance of consumption growth tracks the EGARCH forecasts remarkably well. Overall, the proposed general equilibrium model yields significant empirical support for its theoretical implications. Figure 2. View largeDownload slide Confronting the general equilibrium model with data. The top plots show the slopes of the regressions (as per Equation (10)) of excess return on upside (resp. downside) VRP implied by the general equilibrium model at different maturities (expressed in months), along with the observed 95% confidence intervals. The middle (resp. bottom) plots present monthly model-implied equity return and consumption growth (resp. volatility) paths against the observed corresponding series. Figure 2. View largeDownload slide Confronting the general equilibrium model with data. The top plots show the slopes of the regressions (as per Equation (10)) of excess return on upside (resp. downside) VRP implied by the general equilibrium model at different maturities (expressed in months), along with the observed 95% confidence intervals. The middle (resp. bottom) plots present monthly model-implied equity return and consumption growth (resp. volatility) paths against the observed corresponding series. 5 Skewness or Signed Jump Risk Premium The difference between realized upside and downside variance is also known as the signed jump variation; see Patton and Sheppard (2015). The signed jump variation can be interpreted as a measure of (realized) skewness, see Feunou, Jahan-Parvar, and Tédongap (2013, 2016), since its expectation (under mild conditions) equals the conditional skewness.17 As a result, we use the terms “signed jump” and “realized skewness” interchangeably. The realized skewness between t and t + h is computed as:   RSVt,h(κ)=RVt,hU(κ)−RVt,hD(κ). (35) While the predictive ability of RSVt,h(0) has been studied in the literature (e.g., by Guo, Wang, and Zhou (2015)), to the best of our knowledge, this is the first study to investigate its premium. We define the skewness (or signed jump) risk premium (denoted by SRP) as the difference between the risk-neutral and physical expectations of the realized skewness (or signed jumps). It can be shown that this measure of the skewness risk premium is the spread between the upside and downside components of the VRP:   SRPt,h=Etℚ[RSVt,h]−Etℙ[RSVt,h],SRPt,h=VRPt,hU(κ)−VRPt,hD(κ). (36) If RSVt,h<0, we view SRPt,h as a skewness premium—the compensation for an agent who bears downside risk. Alternatively, if RSVt,h>0, we view SRPt,h as a skewness discount—the amount that the agent is willing to pay to secure a positive return on an investment. This measure of the skewness risk premium is nonparametric and model free. Table 1 reports that the average skewness risk premium is –7.8%. In Figure 1, we observe that SRP is generally negative valued. We study the predictive power of signed jump or skewness risk premium for aggregate returns by estimating Equation (10). The equity premium prediction results using the SRP are reported in Panel D of Table 2. It is clear that the SRP displays a stronger predictive power at longer horizons than the VRP. For monthly excess returns, the SRP slope coefficient is statistically different from zero at prediction horizons of 6 months ahead or longer. At k = 6, the adjusted R2 of the SRP is comparable in size with that of the VRP (2.30% against 3.65%, respectively) and is strictly greater thereafter. At k = 6, the adjusted R2 for the monthly excess return regression based on the SRP is smaller than that of the VRPD. However, their sizes are comparable at k = 9 and k = 12 months ahead. Both trends strengthen as we consider higher aggregation levels for excess returns. At the semi-annual construction level (h = 6), the SRP already has more predictive power than both the VRP and VRPD at a quarter-ahead prediction horizon. The increase in adjusted R2s of the SRP is not monotonic in the construction horizon level. We can detect a maximum at a roughly three-quarters-ahead prediction window for semi-annual and annually constructed SRP. This observation implies that the SRP is the intermediate link between one-quarter-ahead predictability using the VRP uncovered by BTZ and the long-term predictors such as the price–dividend ratio, dividend yield, or consumption–wealth ratio of Lettau and Ludvigson (2001). Given the generally unfavorable findings of Goyal and Welch (2008) regarding long-term predictors of equity premium, our findings regarding the predictive power of the SRP are particularly encouraging. Panel D of Table 4 and Panel C of Table 5 report statistically significant slope parameters and notable adjusted R2s for realized skewness in long prediction horizons ( k≥6) and for construction horizons ( h≥6). Compared with the SRP, the realized skewness lacks predictive power in low construction or prediction horizons. Based on the results presented in Table 2, we argue that the SRP (and not the realized skewness) is a superior predictor, as it overcomes these shortcomings. 6 Conclusion In this study, we have decomposed the variance risk premium—arguably one of the most successful short-term predictors of excess equity returns—to show that its prediction power stems from the downside VRP embedded in this measure. Market participants seem more concerned with market downturns and demand a premium for bearing that risk. By contrast, they seem to favor upward uncertainty in the market. We support this intuition through a simple equilibrium consumption-based asset pricing model able to replicate the main stylized facts observed in our empirical investigation. Empirically, the downside VRP demonstrates significant prediction power (that is at least as powerful as the VRP, and often stronger) for excess returns. We also show that the difference between upside and downside VRP—the skewness risk premium—is a powerful predictor of excess returns. The skewness risk premium performs well for intermediate prediction steps beyond the reach of short-run predictors such as downside variance risk or VRP and long-term predictors such as price–dividend or price–earnings ratios alike. Supplementary Data Supplementary data are available at Journal of Financial Econometrics online. Appendix A: DETAILS ON EQUILIBRIUM DERIVATIONS A.1. Deriving the Log Price–Consumption Ratio Coefficients We solve for the consumption-generating asset returns, rc,t, using the Euler Equation (13):   Et[exp⁡[θ ln δ−θψΔct+1+(θ−1)rc,t+1+rc,t+1]]=1 and   ln Et[exp[θ ln δ−θψΔct+1+θrc,t+1]]=0. Substituting for Δct+1,rc,t+1,Vi,t+1 and qi,t+1, we get:   ln⁡Et[exp⁡[θ ln⁡δ+(1−γ)[μ0+μ1Vu,t+μ2Vd,t]+θ[κ0+(κ1−1)A0]+θ(αuA1+αdA2+γu,0A3+γd,0A4)+(1−γ)σc(ɛu,t+1−ɛd,t+1)+θ[A1(κ1βu−1)Vu,t+A2(κ1βd−1)Vd,t+A3(κ1γu,1−1)qu,d+A4(κ1γd,1−1)qd,t]+θκ1(A1qu,tzt+1u+A2qd,tzt+1d+A3ϕuqu,tzt+11+A4ϕdqd,tzt+1d)]]=0, (37) and then proceed to compute the expectations and coefficients, as follows: ln⁡Et[exp⁡[σc(1−γ)ɛu,t+1]]=f[σc(1−γ)]Vu,t, ln⁡Et[exp⁡[−σc(1−γ)ɛd,t+1]]=f[−σc(1−γ)]Vd,t ln⁡Et[exp⁡[θκ1(A1qu,tzt+1u+A2qd,tzt+1d+A3ϕuqu,tzt+11+A4ϕdqd,tzt+1d)]]= 12θ2κ12[(A12+ϕ2A32)qu,t+(A22+ϕ2A42)qd,t], ln⁡Et[exp⁡[(1−γ)[μ1Vu,t+μ2Vd,t]+θ[A1(κ1βu−1)Vu,t+A2(κ1βd−1)Vd,t+A3(κ1γu,1−1)qu,d+A4(κ1γd,1−1)qd,t]]]= (1−γ)[μ1Vu,t+μ2Vd,t]+θ[A1(κ1βu−1)Vu,t+A2(κ1βd−1)Vd,t+A3(κ1γu,1−1)qu,d+A4(κ1γd,1−1)qd,t], ln⁡Et[exp⁡[θln⁡δ+(1−γ)μ0+θ[κ0+(κ1−1)A0]+θ(αuA1+αdA2+γu,0A3+γd,0A4)]]= θ[ln δ+1−γθμ0+κ0+(κ1−1)A0+κ1(αuA1+αdA2+γu,0A3+γd,0A4)]. We gather the terms for Vu,t,Vd,t,qu,t and qd,t, and solve for A0 to A4:   A1=−f[σc(1−γ)]+(1−γ)μ1θ(κ1βu−1), (38)  A2=−f[−σc(1−γ)]+(1−γ)μ2θ(κ1βd−1), (39)  A3=(1−κ1γu,1)−(1−κ1γu,1)2−θ2ϕu2κ14A12θκ12ϕu2, (40)  A4=(1−κ1γd,1)−(1−κ1γd,1)2−θ2ϕd2κ14A22θκ12ϕd2, (41)  A0=ln δ+(1−1ψ)μ0+κ0+κ1(αuA1+αdA2+γu,0A3+γd,0A4)1−κ1. (42) A.2. Analytical Derivations of Equity Premium and VRP Components We begin by deriving the risk-neutral distribution of all the shocks, ɛu,t+1, ɛd,t+1, zt+1u, zt+1d, zt+11, and zt+12. In this computation, we construct the characteristic function for each shock and exploit the properties of characteristic functions to derive the expectations under the risk-neutral measure. Thus, our derivations yield exact equity and risk premia measures, in contrast to approximate values reported, for example, in Equation (15) of BTZ or in Drechsler and Yaron (2011). We start from ɛu,t+1. The SDF is the Radon–Nikodym change of measure and ln⁡Et(Mt+1) is the risk-neutral drift term. We have:   Etℚ(exp⁡(νɛu,t+1))=Et[Mt+1Et(Mt+1)exp⁡(νɛu,t+1)]=Et[exp⁡(νɛu,t+1+mt+1−ln⁡(Et(Mt+1)))]=Et[exp⁡(νɛu,t+1+θln⁡δ−θψΔct+1+(θ−1)rc,t+1−ln⁡(Et(Mt+1)))]=Et[exp⁡(νɛu,t+1+θln⁡δ−θψΔct+1+(θ−1)(κ0+κ1wt+1−wt+Δct+1)−ln⁡(Et(Mt+1)))]≡Et[exp⁡((ν−γσc)ɛu,t+1+Bt∗,1)] where   Bt∗,1=ln⁡[Et[exp⁡(θln⁡δ−γ(μ0+μ1Vu,t+μ2Vd,t−σcɛd,t+1)+(θ−1)(κ0+κ1wt+1−wt)−ln⁡(Et(Mt+1)))]]. Hence, it follows that:   Etℚ(exp⁡(νɛu,t+1))=exp⁡(Bt∗,1+f(ν−γσc)Vu,t). With ν = 0, we have   1=exp⁡(Bt1+f(−γσc)Vu,t), hence, we obtain:   Bt1=−f(−γσc)Vu,t. In conclusion:   Etℚ(exp⁡(νɛu,t+1))=exp⁡((f(ν−γσc)−f(−γσc))Vu,t) Similarly for ɛd,t+1, we deduce that:   Etℚ(exp⁡(νɛd,t+1))=exp⁡((f(ν+γσc)−f(γσc))Vd,t) It follows that:   Etℚ[ɛu,t+1]=f′(−γσc)Vu,t=−γσc1+γσcVu,tEtℚ[ɛd,t+1]=f′(γσc)Vd,t=γσc1−γσcVd,t We now derive the expectation of Gaussian shocks, zt+1u,zt+1d,zt+11 and zt+12, under the risk-neutral measure, ℚ. We start by deriving the characteristic function for zt+1u:   Etℚ(exp⁡(νzt+1u))=Et[Mt+1Et(Mt+1)exp⁡(νzt+1u)]=Et[exp⁡(νzt+1u+mt+1−ln⁡(Et(Mt+1)))]=Et[exp⁡(νzt+1u+θln⁡δ−θψΔct+1+(θ−1)rc,t+1−ln⁡(Et(Mt+1)))]=Et[exp⁡(νzt+1u+θln⁡δ−θψΔct+1+(θ−1)(κ0+κ1wt+1−wt+Δct+1)−ln⁡(Et(Mt+1)))]≡Et[exp⁡((ν+(θ−1)κ1A1qu,t)zt+1u+Bt*,g1)] where   Bt*,g1=ln⁡[Et[exp⁡((θ−1)κ1A1(αu+βuVu,t)+θln⁡δ−γΔct+1+(θ−1)(κ0+κ1(wt+1−A1Vu,t+1)−wt)−ln⁡(Et(Mt+1)))]]. It follows that:   Etℚ(exp⁡(νzt+1u))=exp⁡(Bt*,g1+(ν+(θ−1)κ1A1qu,t)22). Setting ν = 0, we have   1=exp⁡(Btg1+((θ−1)κ1A1qu,t)22), and hence: Btg1=−(θ−1)2κ12A12qu,t2. Thus:   Etℚ(exp⁡(νzt+1u))=exp⁡((ν+(θ−1)κ1A1qu,t)2−((θ−1)κ1A1qu,t)22)=exp⁡(ν22+ν(θ−1)κ1A1qu,t). Based on the last result, we characterize the distribution of zt+1u under the risk-neutral measure as:   zt+1u∼ℚN((θ−1)κ1A1qu,t,1) Similarly, we characterize the distributions for the remaining shocks:   zt+1d∼ℚN((θ−1)κ1A2qd,t,1)zt+11∼ℚN((θ−1)κ1A3ϕuqu,t,1)zt+12∼ℚN((θ−1)κ1A4ϕdqd,t,1) Thus far, we have derived the distribution of shock processes under the risk-neutral measure, ℚ. Since any premium—whether equity, variance risk, or skewness risk premia—can be defined as the difference between the physical and risk-neutral expectations of processes, we are now ready to compute all the premia of interest. We start with the equity risk premium:   ERPt≡Et[rc,t+1]−Etℚ[rc,t+1]=Et[κ0+κ1wt+1−wt+Δct+1]−Etℚ[κ0+κ1wt+1−wt+Δct+1]=κ1(Et[wt+1]−Etℚ[wt+1])+Et[Δct+1]−Etℚ[Δct+1] As it is clear from the expressions above, we need to compute both Et[Δct+1]−Etℚ[Δct+1] and Et[wt+1]−Etℚ[wt+1]. Starting with Et[Δct+1]−Etℚ[Δct+1], we have:   Et[Δct+1]−Etℚ[Δct+1]=−σcEtℚ[ɛu,t+1−ɛd,t+1]=σc(γσc1+γσcVu,t+γσc1−γσcVd,t)=γσc2(11+γσcVu,t+11−γσcVd,t) Similarly, for Et[wt+1]−Etℚ[wt+1] we get:   Et[wt+1]−Etℚ[wt+1]=A1(Et[Vu,t]−Etℚ[Vu,t])+A2(Et[Vd,t]−Etℚ[Vd,t])+A3(Et[qu,t]−Etℚ[qu,t])+A4(Et[qd,t]−Etℚ[qd,t]) At this stage, we need the premia for each risk factor ( Vu,t,Vd,t,qu,t and qd,t) to compute Et[wt+1]−Etℚ[wt+1]. We start with Vu,t:   Et[Vu,t+1]−Etℚ[Vu,t+1]=−qu,tEtℚ[zt+1u]=−qu,t(θ−1)κ1A1qu,t=−(θ−1)κ1A1qu,t. Thus, we derive the premia accrued to each risk factor as:   Et[Vu,t+1]−Etℚ[Vu,t+1]=(1−θ)κ1A1qu,t,Et[Vd,t+1]−Etℚ[Vd,t+1]=(1−θ)κ1A2qd,t,Et[qu,t+1]−Etℚ[qu,t+1]=(1−θ)κ1A3ϕu2qu,t,Et[qd,t+1]−Etℚ[qd,t+1]=(1−θ)κ1A4ϕd2qd,t. Collecting these terms and substituting in the expression for Et[wt+1]−Etℚ[wt+1], we get:   Et[wt+1]−Etℚ[wt+1]=(1−θ)κ1[(A12+A32ϕu2)qu,t+(A22+A42ϕd2)qd,t]. It easily follows that the equity premium in our model is:   ERPt≡γσc21+γσcVu,t+γσc21−γσcVd,t+(1−θ)κ1(A12+A32ϕu2)qu,t+(1−θ)κ1(A22+A42ϕd2)qd,t. (43) We now characterize variance of the consumption-generating asset   σr,t2≡Vart[rc,t+1]=Vart[σc(ɛu,t+1−ɛd,t+1)+κ1[(A1zt+1u+ϕuA3zt+11)qu,t+(A2zt+1d+ϕdA4zt+12)qd,t]]=σc,t2+κ12σw,t2=σc2Vu,t+σc2Vd,t+κ12(A12+A32ϕu2)qu,t+κ12(A22+A42ϕd2)qd,t≡(σr,tu)2+(σr,td)2, where upside and downside variances are defined as:   (σr,tu)2=σc2Vu,t+κ12(A12+A32ϕu2)qu,t, (44)  (σr,td)2=σc2Vd,t+κ12(A22+A42ϕd2)qd,t. (45) Using the definition of VRP, we define upside VRP as:   VRPtu≡Etℚ[(σr,t+1u)2]−Et[(σr,t+1u)2],=σc2(θ−1)κ1A1qu,t+κ12(A12+A32ϕu2)(θ−1)κ1A3ϕu2qu,t,=(θ−1)(σc2κ1A1+κ13(A12+A32ϕu2)A3ϕu2)qu,t. (46) Similarly, we define downside VRP as:   VRPtd≡Etℚ[(σr,t+1d)2]−Et[(σr,t+1d)2]=(θ−1)(σc2κ1A2+κ13(A22+A42ϕd2)A4ϕd2)qd,t. (47) A.3. Restrictions on Parameters We expect VRPtU<0 and VRPtD>0. It follows that   σc2κ1A1+κ13(A12+A32ϕu2)A3ϕu2>0, (48)  σc2κ1A2+κ13(A22+A42ϕd2)A4ϕd2<0. (49) In this Appendix, we discuss the necessary and sufficient conditions ensuring that both inequalities (48) and (49) hold. Since   A3=A3∗θ, A4=A4∗θ, withA3∗=(1−κ1γu,1)−(1−κ1γu,1)2−ϕu2κ14(A1∗)2κ12ϕu2A4∗=(1−κ1γd,1)−(1−κ1γd,1)2−ϕd2κ14(A2∗)2κ12ϕd2, it must be that   0≤ϕu≤1−κ1γu,1−κ12A1∗,0≤ϕd≤1−κ1γd,1κ12A2∗, and   A1∗≡θA1=−f(σc(1−γ))+(1−γ)μ1κ1βu−1, (50)  A2∗≡θA2=−f(−σc(1−γ))+(1−γ)μ2κ1βd−1. (51) Since A4<0, A2<0 is a sufficient condition for σc2κ1A2+κ13(A22+A42ϕd2)A4ϕd2<0. Note that   A2<0⇔μ2<f[−σc(1−γ)]γ−1 In particular, we have   μ2≤0⇒A2<0⇒VRPtd>0. Since A3<0, A1>0 is a necessary condition for σc2κ1A1+κ13(A12+A32ϕu2)A3ϕu2>0. Because we want A1>0, it must be the case that   f(σc(1−γ))γ−1≤μ1. We can show that, the constraint σc2κ1A1+κ13(A12+A32ϕu2)A3ϕu2>0 translates into   θ<−2(κ1γu,1−1)A1∗ϕuA3∗σc. Appendix B: Linking the Asymmetry to the Signed Jump Variation Consider that st=ln⁡St, the stock log-price evolves according to a jump-diffusion process   dst=μtdt+σtdWt+Δst, (52) where dWt is an increment of standard Brownian motion and Δst≡st−st− is the jump component. From Equation (52), the return over a time interval [t,t+h] is   rt,h≡∫tt+hdsυ=∫tt+hμυdυ+∫tt+hσυdWυ+∑t≤υ≤t+hΔsυ (53) The corresponding demeaned return is   ɛt,hr=rt,h−(∫tt+hμυdυ+Et[ɛ˜t,hJ]),=ɛt,hC+ɛt,hJ, (54) where ɛ˜t,hJ=∑t≤υ≤t+hΔsυ denotes the jump component, ɛt,hJ=ɛ˜t,hJ−Et[ɛ˜t,hJ] is the centered conditional jump innovation, and ɛt,hC=∫tt+hσυdWυ stands for the diffusion part. Between time t and t + h, let Nt,h be the total number of jumps drawn from a counting process. Thus, the jump component in Equation (54) can be rewritten as:   ɛ˜t,hJ=∑j=0Nt,hΔsj,=∑j=0Nt,h|Δsj|I[Δsj>0]−∑j=0Nt,h|Δsj|I[Δsj≤0],=∑j=0Nt,hU|Δsj|−∑j=0Nt,hD|Δsj|, (55) where Nt,hU=∑j=0Nt,hI[Δsj>0] and Nt,hD=∑j=0Nt,hI[Δsj≤0] denote the number of upside and downside jumps, respectively. It follows that   ɛt,hJ=ɛt,hU−ɛt,hD, (56) where ɛt,hU=∑j=0Nt,hU|Δsj|−Et[∑j=0Nt,hU|Δsj|], and ɛt,hU=∑j=0Nt,hD|Δsj|−Et[∑j=0Nt,hD|Δsj|]. We further assume that absolute values of jump sizes ( |Δsj|) are independent and identically distributed according to a positive law, and that the number of upside ( Nt,hU∼P(λt,hU)) and downside ( Nt,hD∼P(λt,hD)) jumps are independent Poisson distributed random variables with distinct intensities. In the sequel, we drop time subscripts to ease notation. We now compute the first three cumulants of ɛJ. The cumulant generating function is   ψɛJ(ν)=ψɛU(ν)+ψɛD(−ν). (57) Thus, the kth derivative of ψɛJ(ν) evaluated at 0 yields the kth cumulant μk. We get the following equalities   μ1(ɛJ)=μ1(ɛU)−μ1(ɛD)=0, (58)  μ2(ɛJ)=μ2(ɛU)+μ2(ɛD), (59)  μ3(ɛJ)=μ3(ɛU)−μ3(ɛD). (60) Moreover, the distributional assumptions imply   μ2(ɛU/D)=λU/D(σ¯2+m¯2), (61) and   μ3(ɛU/D)=λU/D(sk¯3+3σ¯2m¯+m¯3), (62) where m¯=μ1(|Δsj|), σ¯2=μ2(|Δsj|), and sk¯=μ3(|Δsj|). Combining Equations (60) and (62), we obtain   μ3(ɛJ)=(λU−λD)(sk¯3+3σ¯2m¯+m¯3). (63) Interestingly, we see from Equation (61) that the difference between variances of ɛU and ɛD is,   μ2(ɛU)−μ2(ɛD)=(λU−λD)(σ¯2+m¯2), (64) which implies that   (λU−λD)=(μ2(ɛU)−μ2(ɛD))(σ¯2+m¯2)−1. (65) Substituting Equation (65) into Equation (60) gives   μ3(ɛJ)=(μ2(ɛU)−μ2(ɛD))(σ¯2+m¯2)−1(sk¯3+3σ¯2m¯+m¯3), (66) which clearly shows that the difference between the variances of ɛU and ɛD drives the asymmetry. Finally, when diffusion and jump components are independent, the skewness of the return is   μ3(r)≡μ3(ɛr)=μ3(ɛC)+μ3(ɛJ), (67) which sums the skewness of diffusive ( μ3(ɛC)) and jump ( μ3(ɛJ)) innovations. In the absence of leverage effect, μ3(ɛC)=0 inducing that the total of returns boils down (up to a constant) to the difference of ɛU and ɛD variances. Obviously, the larger the leverage effect, the wider the wedge between the return skewness and the difference of ɛU and ɛD variances. Footnotes 1 Among many studies, Brandt and Kang (2004) observe a negative relation between realized market risk and returns. Ghysels, Santa-Clara, and Valkanov (2005) and Ludvigson and Ng (2007) find a positive relation. On the other hand, Baillie and DeGennaro (1990) and Bollerslev and Zhou (2006) document mixed results. 2 We define the downside (upside) variance as the realized variance of the stock market returns for negative (positive) returns. The downside (upside) variance risk premium is the difference between option-implied and realized downside (upside) variance. Decomposing variance in this way was pioneered by Barndorff-Nielsen, Kinnebrock, and Shephard (2010). 3 Specifications for U/D-HAR and M-HAR models are available from the authors upon request. 4 Note that κF should be set to (κ−rtf)h to get consistent thresholds when computing realized and option-implied quantities. 5 This is due to the fact that a continuum of traded options in not available in the market. The feasible incomplete market approach in the traditional risk-neutral expectation assessment may imply a sizable discretization error. Instead, the authors argue that ℚ-expectations should be obtained from the prices of dynamically hedged trading strategies that exactly deliver the realized moment (typically, realized variance and its up/down components) as a payoff. This alternative technology dynamically optimizes the option portfolio weights by minimizing the discrepancy between the hedged quantity and the received payoff. 6 For the three series, we use a multiplicative scaling of the average total realized variance series to match the unconditional variance of S&P 500 returns. Hansen and Lunde (2006) discuss various approaches to adjusting open-to-close RVs. 7 Our dataset contains a large set of option contracts. Moreover, the sample features comparable numbers of out-of-the-money (OTM) put and call contracts (especially in longer-horizon maturities from 18 to 24 months) that enable a precise computation of risk-neutral semi-variances. A detailed description of options data is provided in the Online Appendix. 8 U/D-HAR and M-HAR forecasts of realized volatility and its components are based on the methodology of Corsi (2009). 9 We consider alternative measures of model assessment, such as Goyal and Welch (2008) out-of-sample root-mean-squared errors (RMSEOOS) in our robustness study. A complete set of results is available upon request, but is not reported to save space. 10 A complete set of robustness checks, including monthly, quarterly, and annually aggregated excess returns results, are available in the Online Appendix. 11 The dynamics of inflation during the Great Recession period mimic the behavior of our variance risk premia. Gilchrist et al. (2017) meticulously study the behavior of this variable in the period from 2007 to 2009. According to their study, both full and matched producer price index (PPI) inflation in their model display an aggregate drop in 2008 and 2009, while the reactions of financially sound and weak firms are asymmetric, with the former lowering prices and the latter raising prices in this period. Thus, the predictive power of this variable, given the inherent asymmetric responses, is not surprising. 12 Our theoretical findings do not depend on our choice of preferences. We use Epstein and Zin (1989) preferences in order to make our results comparable with BTZ, Segal, Shaliastovich, and Yaron (2015), or other similar papers. We have not attempted this exercise, but one can follow Bekaert and Engstrom (2015) and use habit-based preferences to obtain very similar theoretical results. The only additional ingredient needed in the Bekaert and Engstrom “BE-GE” setup is time-varying dynamics for both volatility-of-volatility components. 13 Feunou, Jahan-Parvar, and Tédongap (2013), and Bekaert and Engstrom (2015) consider variations of this assumption in their studies. 14 This assumption is for the sake of brevity. Violating this assumption adds to algebraic complexity but does not affect our analytical findings. 15 We use demeaned Gamma distributions for the sake of tractability. One can use a number of alternative distributions with positive support and fat tails. For example, one may choose from compound Poisson, χ2, inverse Gaussian, or Lévy distributions. 16 A consumption-based asset pricing model with a representative agent endowed with Epstein and Zin (1989) preferences and an unpredictable consumption growth process does not support the existence of distinct upside and downside variance risk premia with the expected signs. 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