# High-Frequency Tail Risk Premium and Stock Return Predictability

Caio Almeida, Kim Ardison, Gustavo Freire, René Garcia, Piotr Orlowski

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Caio Almeida, Kim Ardison, Gustavo Freire, René Garcia, Piotr Orlowski. High-Frequency Tail Risk Premium and Stock Return Predictability. Journal of Financial and Quantitative Analysis, 2024, 59 (8), pp.3633-3670. 10.1017/S0022109023001199. hal-04927211

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JOURNAL OF FINANCIAL AND QUANTITATIVE ANALYSIS Vol. 59, No. 8, Dec. 2024, pp. 3633–3670
© The Author(s), 2023. Published by Cambridge University Press on behalf of the Michael G. Foster
School of Business, University of Washington. This is an Open Access article, distributed under the terms
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doi:10.1017/S0022109023001199

# High-Frequency Tail Risk Premium and Stock Return Predictability

Caio Almeida ID

Princeton University Department of Economics and Bendheim Center for Finance
calmeida@princeton.edu (corresponding author)

Kym Ardison

SPX Capital
kymmarcel@gmail.com

Gustavo Freire

Erasmus University Rotterdam, Tinbergen Institute and ERIM
freire@ese.eur.nl

René Garcia

Université de Montréal and Toulouse School of Economics
rene.garcia@umontreal.ca

Piotr Orlowski

Department of Finance, HEC Montréal
piotr.orlowski@hec.ca

## Abstract

We propose a novel measure of the market return tail risk premium based on minimum-distance state price densities recovered from high-frequency data. The tail risk premium extracted from intra-day S&P 500 returns predicts the market equity and variance risk premiums and expected excess returns on a cross section of characteristics-sorted portfolios. Additionally, we describe the differential role of the quantity of tail risk, and of the tail premium, in shaping the future distribution of index returns. Our results are robust to controlling for established measures of variance and tail risk, and of risk premiums, in the predictive models.

## I. Introduction

Starting with Bollerslev, Tauchen, and Zhou (2009), extant empirical evidence supports that the variance risk premium (VRP) helps predict future aggregate market returns.<fnref n="1" /> This predictability has been rationalized under different theoretical

We thank an anonymous referee, Thierry Foucault (the editor), Rodrigo Hizmeri, seminar participants at the Kellogg School of Management, and conference participants at the 2017 SoFiE Conference, the 2017 Vienna-Copenhagen Conference on Financial Econometrics, and the 2018 IAAE Meeting for useful comments and suggestions. Ardison acknowledges financial support from ANBIMA and FAPERJ. Garcia thanks the NSERC, the SSHRC, and the FQRSC research grant agencies for their financial support. He is a TSE associate faculty and a research Fellow of CIRANO and CIREQ.

<footnote n="1">¹The VRP is defined as the difference between the conditional expected variance of market returns over a given horizon under the physical and risk-neutral measures. It captures the compensation demanded by investors for bearing variance risk.</footnote>
frameworks.<fnref n="2" /> Focusing on the properties of the VRP in a model-free manner, Bollerslev and Todorov (2011) and Bollerslev, Todorov, and Xu (2015) demonstrate how this premium reflects compensation for two different types of risk: diffusive and jump risk. They show that a large part of the VRP and its predictive power for the equity premium comes from the compensation demanded by investors for extreme negative events. Such asymmetric importance of losses relative to gains has strong theoretical foundations, as in the case of loss aversion (Kahneman and Tversky (1979)) and disappointment aversion (Gul (1991), Routledge and Zin (2010)) preferences. These theories posit that agents are specially averse to downside losses, such that greater compensation (in the form of higher expected returns) is demanded for assets with high downside risk.<fnref n="3" />

In this article, we introduce a new measure of the compensation that investors demand for bearing systematic downside risk. We use it to shed light on the predictability of aggregate and cross-sectional risk premiums that is due to aversion to downside risk in a high-frequency environment. The motivation of our analysis is threefold. First, theory indicates that the VRP should predict market returns, where most of the VRP arises directly from compensation for downside risk. Second, theory also suggests that investors will demand higher returns when their perception of downside risk is high. Third, while the predictability literature focuses on monthly to longer horizons, high-frequency evidence can provide new stylized facts for economic theories to account for, and to be judged against.<fnref n="4" />

We compute the daily tail risk premium, denoted as $\Delta_Q^PES$, as the difference between the expected shortfall calculated under the risk-neutral (ES$^Q$) and physical (ES$^P$) probability measures estimated from intra-day S&P 500 return data. The risk-neutralization is based on a nonparametric adjustment of the raw market returns. Motivated by Ait-Sahalia and Lo (1998), (2000), the risk adjustment puts a higher probability weight on extreme negative returns, reflecting investors' compensation for “bad” states of the world. From an economic viewpoint, ES$^Q$ (ES$^P$) is the expectation under the risk-neutral (physical) measure of the payoff of a hypothetical out-of-the-money (OTM) put option on the market, which is naturally sensitive to negative jumps in returns. In particular, ES$^Q$ – ES$^P$ can be expressed as the expected value of the difference between the put price and its payoff, which allows us to interpret $\Delta_Q^PES$ as the expected gain of selling the put. Therefore, $\Delta_Q^PES$ will be

<footnote n="2">²Bollerslev et al. (2009) and Drechsler and Yaron (2011) extend the long-run risk model of Bansal and Yaron (2004) to make the equity premium a function of time-varying volatility-of-volatility and jump-intensity, respectively. In both cases, the VRP effectively isolates the relevant latent factor (vol-of-vol or jump-intensity), justifying its predictive power for market returns. This is because the representative agent's aversion to a shock in the latent factor makes the risk-neutral variance higher than the physical one, i.e., the VRP increases (in absolute value) with the factor. Alternatively, Bonomo, Garcia, Meddahi, and Tédongap (2015) show that the predictability afforded by the VRP can be generated by incorporating generalized disappointment aversion preferences (Routledge and Zin (2010)) in the long-run risk model. See also Bekaert and Engstrom (2017), who extend the habit formation preferences of Campbell and Cochrane (1999) to show that the VRP can be interpreted as a proxy for aggregate risk aversion.</footnote>

<footnote n="3">³This prediction has been empirically confirmed for the cross section of stock returns by Ang et al. (2006) and Farago and Tédongap (2018).</footnote>

<footnote n="4">⁴For instance, Bonomo et al. (2015) develop a model to reproduce moments and predictability patterns of risk and return across different frequencies.</footnote>
high when investors highly value protection against market downside risk (i.e., when they are willing to pay a high premium relative to the expected put payoff).

We investigate the predictive power of $\Delta_Q^{\mathrm{P}}$ES for risk premiums with a comprehensive set of empirical tests. We focus our predictive exercises on very short horizons: daily, weekly, and monthly. This directly complements the longer-horizon predictability results documented in the literature. In the predictive regressions, we control for a number of established variance and tail risk measures. Among these, three are particularly important: the physical expected shortfall ($ES^P$), the left tail variance (LTV) of Bollerslev, Todorov, and Xu (2015) and the VRP of Bollerslev, Tauchen, and Zhou (2009). Including $ES^P$, which is a measure of realized downside risk, allows us to assess what better predicts risk premiums: the tail risk premium or the quantity of tail risk itself. LTV, which estimates jump risk from OTM options, is a natural benchmark to assess if options contain information beyond that provided by our measure. The VRP is included given its important role as a predictor of the equity premium (Bollerslev et al. (2009)).<fnref n="5" /> We complement these measures with the realized variance (RV) of market returns computed from intra-day data and other measures of realized return variation.

We start with a predictive analysis of excess market returns. We find that our measure $\Delta_Q^{\mathrm{P}}$ES has strong predictive power for 1-day ahead market returns, with a positive coefficient that is highly statistically significant and a non-negligible $R^2$.<fnref n="6" /> This provides new high-frequency evidence that investors require a higher compensation to hold the market when aversion to downside risk increases. This predictive power is robust to controlling for all the measures we consider, across different regression specifications. In particular, the only other predictor that appears as significant is the VRP. We show that the predictability afforded by $\Delta_Q^{\mathrm{P}}$ES also holds out-of-sample across different starting dates and re-estimation frequencies. This indicates that the predictive relation between the tail risk premium and the equity premium is stable and persists under different economic conditions. Such predictive power is both statistically and economically significant.

We further investigate whether $\Delta_Q^{\mathrm{P}}$ES is able to predict the variance risk premium. Since the VRP reflects in large part compensation for extreme negative events, we should expect that the tail risk premium is informative about the future VRP. This is indeed what we find in the predictive regressions, where $\Delta_Q^{\mathrm{P}}$ES is a highly significant predictor across different regression specifications and all horizons. In particular, this predictability remains after including the lagged VRP, which is also highly significant. It is worth noting that the LTV has no predictive power for the equity and variance risk premiums. This can be rationalized by the fact that it is computed from options with maturity between 6 and 31 trading days, which reflect market expectations over relatively long horizons. In contrast, our tail risk premium measure contains information completely conditional on day t.<fnref n="7" />

<footnote n="5">⁵We compute a daily version of the VRP that is analogous to the monthly VRP constructed by Bollerslev et al. (2009).</footnote>

<footnote n="6">⁶For the 1-week and 1-month ahead horizons, no predictor is significant.</footnote>

<footnote n="7">⁷In Appendix A, we show that, similarly to LTV, our tail risk premium measure captures time-variation in the risk-neutral and physical tail shape parameters of the market return distribution. However, the two measures naturally differ in how they are estimated and what data are used.</footnote>
We also analyze the predictive relation between the tail risk premium and 1-day ahead excess returns on portfolios comprised of stocks sorted on various characteristics: size, book-to-market (BM), profitability, investment, momentum, reversal, and industry. These portfolios reflect compensation for risks beyond those related to the market portfolio. Given the important role of downside risk factors in explaining the cross section of returns (Ang, Chen, and Xing (2006), Farago and Tédongap (2018)), $\Delta_Q^{\mathrm{PES}$}$ can potentially be useful for predicting cross-sectional risk premiums at high frequency. Our results show that the tail premium measure is able to predict the returns on most of the portfolios with high t-statistics and economically significant $R^2$-s. The same is not true for the other variables we consider. In fact, adding each of them at a time to the predictive regressions does not affect the predictive power of $\Delta_Q^{\mathrm{PES}}$. This reinforces the role of aversion to downside risk as captured by our measure as a fundamental determinant of risk premiums at short horizons.

Our analysis indicates that it is the tail risk premium ($\Delta_Q^{\mathrm{PES}}$) rather than the level of tail risk ($ES^P$) that contains relevant predictive information about future market returns.<fnref n="8" /> In order to shed light on the differential role of these variables for explaining the equity premium, we investigate their predictive power for the whole distribution of next-day market returns by estimating the quantile regression model of Koenker and Gilbert (1978). We find that an increase in risk ($ES^P$) leads to a larger probability of observing both extreme negative and positive market returns, whereas an increase in the aversion to downside risk ($\Delta_Q^{\mathrm{PES}}$) shifts the quantiles around the median and the whole right tail toward more positive values. That is, a positive shock in expected shortfall means a more volatile market, such that it is usually followed by either a large decrease or increase of the S&P 500 index. These extreme effects cancel out when predicting directly the market returns. In contrast, a positive shock in the tail risk premium signals that investors are more averse to extreme negative outcomes, requiring a higher compensation to hold the market. This is reflected in the positive effect of $\Delta_Q^{\mathrm{PES}}$ on essentially all quantiles of the market return distribution. Such unambiguous effect translates to the significant positive relation between $\Delta_Q^{\mathrm{PES}}$ and future market returns.<fnref n="9" />

The rest of the article is organized as follows: We close the introduction with a brief review of the related literature. Section II describes how we define and estimate our tail risk premium measure based on high-frequency market returns. Section III presents our empirical predictability results for risk premiums and quantiles of the S&P 500 excess return distribution. Section IV contains several robustness tests considering different specifications of our tail premium measure. Section V concludes the article.

<footnote n="8">⁸This result is largely in line with Bollerslev et al. (2009), Bollerslev and Todorov (2011), Bollerslev et al. (2015), and Andersen et al. (2017), among others, who study the risk-return trade-off concluding that it is the variance risk premium rather than the quantity of risk that is informative of future risk premiums.</footnote>

<footnote n="9">⁹We then conduct a thorough out-of-sample evaluation of interval forecasts using all of the tests prescribed by Christoffersen (1998), which confirms that the in-sample predictive power of the estimated quantile model translates to out-of-sample performance.</footnote>
## Related Literature

Our article is related to the rich and growing literature on the estimation of tail risk and systematic risk measures and their use to predict the equity and variance risk premiums. This includes, among others, Bali, Demirtas, and Levy (2009), Allen, Bali, and Tang (2012), Siriwardane (2013), Kelly and Jiang (2014), Adrian and Brunnermeier (2016), and Brownlees and Engle (2017). In particular, Bollerslev et al. (2015) decompose the variance risk premium and examine the importance of the diffusive and jump components for return predictability. Andersen, Fusari, and Todorov (2017) estimate the variation in the left tail of the return distribution from short-maturity options with significant predictive power for future short-term returns. Andersen, Fusari, and Todorov (2020) extend this evidence to international equity markets. Vilkov and Xiao (2013), Ghysels and Wang (2014), and Huggenberger, Zhang, and Zhou (2018) use daily index options to model forward looking tail risk based on value-at-risk (VaR) and expected shortfall measures. In contrast to these articles, we are the first to provide a methodology to compute the tail risk premium in high-frequency environments that is applicable to virtually any set of returns.

To estimate the risk-neutral leg of our tail risk premium measure, we build on Almeida, Ardison, and Garcia (2017), who compute a nonparametric risk-neutral expected shortfall based on a cross section of daily portfolio or security returns.<fnref n="10" /> In contrast to that article, we rely solely on returns on a broad market index and we use high-frequency intra-day data to obtain more information about the tail. Moreover, while in the aforementioned article, the authors propose the risk-neutral expected shortfall as a new tail risk measure, we concentrate on measuring the tail risk premium as the difference between expected shortfalls under the risk-neutral and physical measures.<fnref n="11" />

Our article is close in spirit to Weller (2019), who develops a real-time tail risk measure based on intra-day bid and ask quotes. While Weller (2019) focuses on the natural relation between tail risk and jumps, providing substantial evidence of jump realization predictability using intra-day data, we focus on the broader relation between aversion to tail risk and aggregate and cross-sectional risk premiums. Additionally, Weller (2019)'s measure considers a panel of 2,800 firms for its baseline 1-factor model, while our tail measure necessitates only 1 day of intra-daily observations on one stock index. In fact, the measure we propose can be easily applied to different markets and assets for which intra-day returns are available, avoiding the need to rely on option prices or bid-ask spreads.

<footnote n="10">¹⁰Kelly and Jiang (2014) also use a large cross section of observed returns to compute a tail risk measure by assuming that asset return tails follow a power law. With particular emphasis on the financial sector, Allen et al. (2012) and Brownlees and Engle (2017) adopted VaR and expected shortfall measures to estimate systemic risks. In this literature, the estimated tail risk measures are calculated on a monthly or weekly basis rather than daily.</footnote>

<footnote n="11">¹¹In addition, by focusing on high-frequency individual returns instead of cross-sectional returns, supplementary economic restrictions like the non-negativity of the equity premium can be naturally imposed on the Euler equations as advocated by Campbell and Thompson (2008) and Pettenuzzo, Timmermann, and Valkanov (2014).</footnote>
Our article also complements the findings obtained with the use of high-frequency factor models in Bollerslev, Todorov, and Li (2013), Bollerslev, Li, and Todorov (2016), and Aït-Sahalia, Jacod, and Xiu (2020). Bollerslev et al. (2013) use a large high-frequency data set on the cross section of stock returns to measure the quantity of tail risk under the physical probability measure. In contrast, we identify a measure of the high-frequency tail risk premium that captures the wedge between the risk-neutral and physical worlds. Bollerslev et al. (2016) consider an extension of the CAPM with separate betas for the jump component and the continuous component of the market return. They find that only the jump component beta entails significant premiums. We offer a new tail risk premium measure that is not dependent on option data and does not rely on any parametric dynamics for the market return. Aït-Sahalia et al. (2020) estimate a multi-factor model at high frequency. They show that a large part of the market equity premium is due to exposures to the market's jump risk component and that jump risks in Fama-French factors supersede their continuous counterparts. The predictive power of our tail risk premium for the expected returns of the characteristic-based portfolios is consistent with their findings.

## II. The Nonparametric Tail Risk Premium

### A. Background

Let $(\Omega, \mathcal{F}, \mathcal{P})$ be a probability space (with $\mathcal{P}$ the physical probability measure), where $R$ and $R_F$ are random variables denoting, respectively, the return of a primitive basis asset (the stock index), and a risk-free rate. An admissible risk-neutral distribution (RND) $\mathcal{Q}$ is represented by a density $q$, a nonnegative random variable with unitary mean satisfying the Euler pricing equation for the index returns:<fnref n="12" />

$$ (1) \qquad \begin{gathered} \mathbb{E}^{\mathcal{Q}}[R - R_F] = \mathbb{E}^{\mathcal{P}}[q(R - R_F)] = 0, \\ \text{with } \mathbb{E}^{\mathcal{P}}[q] = 1, q \ge 0. \end{gathered} $$

At each day $t$, we observe a sample $\{R_i^t\}_{i=1,...,T}$ of high-frequency stock index returns ($T>1$). We use this high-frequency time-series to identify a conditional RND $\mathcal{Q}_t$ via its density $q^t$ and the empirical conditional physical distribution $\mathbb{P}_t$, with density $p_i^t = 1/T$, $i=1,...,T$, for all $t$.<fnref n="13" />

### B. Definition

Our objective is to build a simple tail risk premium measure that depends solely on the returns of the single stock index observed at a high frequency. To that end, for each date $t$, we estimate daily expected shortfalls under the conditional physical and risk-neutral measures:

<footnote n="12">¹²A RND is a probability distribution $\mathcal{Q}$ equivalent to the physical distribution $\mathbb{P}$, under which the basis assets are correctly priced, i.e., it satisfies the Euler equation. It can be represented one-to-one with its corresponding density $q$. In this article, we use these two definitions interchangeably.</footnote>

<footnote n="13">¹³We assume stationarity and ergodicity of the composite process $(q_i^t, R_i^t)_{(i=1,...,T)}$, such that it satisfies a time-series version of the law of large numbers (Hansen and Richard (1987)).</footnote>
$$ (2) \qquad \mathrm{ES}_{t}^{\mathbb{P}_{t}} := \mathbb{E}^{\mathbb{P}_{t}}[(s_{\alpha} - R)^{+}], $$

$$ (3) \qquad \mathrm{ES}_{t}^{\mathbb{Q}_{t}} := \mathbb{E}^{\mathbb{Q}_{t}}[(s_{\alpha} - R)^{+}]. $$

In equations (2) and (3), $t$ is the estimation date, $\alpha$ is a confidence level, $s_\alpha$ is the $\alpha$-quantile of $R$ under the physical probability, and $\mathbb{P}_t$ and $\mathbb{Q}_t$ indicate the physical and risk-neutral conditional probability densities at time $t$, respectively. We adopt ES to measure risk since it is a coherent measure of risk (Artzner, Delbaen, Eber, and Heath (1999)), which overcomes the main deficiencies of the VaR measure. In particular, while VaR completely ignores the behavior of returns in the tail beyond its confidence level, ES takes an average of these tail returns, being thus highly sensitive to what happens in the tail.<fnref n="14" />

Note that we use a version of ES that can also be interpreted as the expectation of the payoff $(s_\alpha - R)^+$ of a put option with strike $s_\alpha$. $\mathrm{ES}_t^{\mathbb{Q}_t}$ is then the option price computed under the RND incorporating investors' preferences. The put payoff is positive in the states of nature where $R < s_\alpha$, which for $\alpha < 0.5$, at least for symmetric distributions, occurs for returns $R$ smaller than the risk-free rate ($R_F$, i.e., negative excess returns). The smaller the $\alpha$, the more negative the $s_\alpha$ and the farther we are in the left tail of $R$. To strike a good balance between capturing behavior in the left tail and guaranteeing enough return observations to extract information from, we set $\alpha = 0.2$.<fnref n="15" /> This essentially means that $(s_\alpha - R)^+$ is the payoff of an OTM put option, whose price $\mathrm{ES}_t^{\mathbb{Q}_t}$ reflects how protection against downside risk (the risk of large negative returns below $s_\alpha$) is valued according to $\mathbb{Q}_t$.

We define our tail risk premium measure as the difference between the expected shortfalls under the risk-neutral and physical distributions:

$$ (4) \qquad \Delta_{\mathbb{Q}}^{\mathbb{P}} \mathrm{ES}_t = \mathrm{ES}_t^{\mathbb{Q}_t} - \mathrm{ES}_t^{\mathbb{P}_t}. $$

Given a positive equity premium, negative market return states in which the OTM put pays off are deemed more likely to happen under $\mathbb{Q}_t$ than under $\mathbb{P}_t$ due to risk aversion. By keeping the same threshold $s_\alpha$ for the two ES's, the tail risk premium depends only on how investors' preferences encoded in risk-neutral probabilities make $\mathrm{ES}_t^{\mathbb{Q}_t}$ exceed its physical counterpart $\mathrm{ES}_t^{\mathbb{P}_t}$. In particular, $\Delta_{\mathbb{Q}}^{\mathbb{P}} \mathrm{ES}_t$ can be interpreted as the expected gain of selling the put option by noting that $\mathrm{ES}_t^{\mathbb{Q}_t} - \mathrm{ES}_t^{\mathbb{P}_t} = \mathbb{E}^{\mathbb{P}_t} [\mathbb{E}^{\mathbb{Q}_t}[(s_\alpha - R)^+] - (s_\alpha - R)^+]$ is the expected value of the difference between the put price and its payoff. Therefore, $\Delta_{\mathbb{Q}}^{\mathbb{P}} \mathrm{ES}_t$ is expected to be high when investors highly value protection against market downside risk (i.e., when they are willing to pay a high premium relative to the expected put payoff).<fnref n="16" />

In our empirical application, we compute $\Delta_{\mathbb{Q}}^{\mathbb{P}} \mathrm{ES}_t$ at a daily frequency using the intra-day stock index returns. Since there is no overlapping of data when calculating our measure, it avoids spurious persistence and is able to quickly react to the arrival

<footnote n="14">¹⁴See Berkowitz and O’Brien (2002) and Jorion (2019) for examples in which VaR leads to over-estimation of risks in calm periods, and underestimation during crises.</footnote>

<footnote n="15">¹⁵In Section IV.C, we show that our results are very similar if we define $\alpha$ to be 0.1 instead.</footnote>

<footnote n="16">¹⁶In Appendix A, we also show that our tail premium measure captures time-variation in the risk-neutral and physical tail shape parameters of the market return distribution.</footnote>
of new information embedded in market returns. At this point, our main challenge lies in finding a way of identifying the conditional risk-neutral probability distribution $Q_t$ without using options in the estimation process. Following Almeida et al. (2017), who compute risk-neutral probability distributions from a cross section of portfolio returns, we obtain $Q_t$ at each date $t$ by solving a specific minimum distance problem between the conditional physical probability distribution $P_t$ and the family of risk neutral distributions that correctly price the S&P 500 returns $\{R_i^t\}_{i=1,...,T}$ within day $t$. The details of this procedure are developed in Section II.C.

### C. Identifying Conditional Risk-Neutral Densities from S&P 500 Returns

We work with a sequence of repeated 1-period models indexed by time $t$, characterized by the high-frequency returns $\{R_i^t\}_{i=1,...,T}$ sampled at $t$.<fnref n="17" /> Since $T > 1$, the market within each of these models is inherently incomplete and, under the assumption of no-arbitrage, there exists an infinity of RNDs pricing the index returns at each date $t$. Almeida and Garcia (2017) suggest identifying a subset of RNDs by minimizing functions in the Cressie-Read family of discrepancies $\phi^\gamma(\pi) = \frac{\pi^{\gamma+1}-1}{\gamma(\gamma+1)}$, $\gamma \in \mathbb{R}$, that measure the distance between admissible RNDs $Q$ with density $q$ and the physical probability distribution $P$ with density $p$. Each discrepancy $\phi^\gamma(\pi)$ allows for the identification of a specific RND $\hat{q}^\gamma$ with unique sensitivity to higher-order moments of the stock index returns.<fnref n="18" />

The minimum-discrepancy (MD) problem, which was originally proposed by Almeida and Garcia (2017) for SDFs and later adapted by Almeida and Freire (2022) for RNDs, can be stated in its sample version for a single basis asset as follows:

$$ (5) \qquad \begin{aligned} \hat{q}^\gamma &= \underset{\{q_1, \dots, q_T\}}{\operatorname{arg min}} \sum_{i=1}^{T} p_i \phi^\gamma \left( \frac{q_i}{p_i} \right) \\ &\text{subject to} \quad \sum_{i=1}^{T} q_i (R_i^T - R_F) = 0 \\ & \qquad \sum_{i=1}^{T} q_i = 1 \\ & \qquad q_i \ge 0 (\text{or } q_i > 0) \quad \forall i, \end{aligned} $$

where the last inequality depends on the discrepancy $\phi^\gamma(\cdot)$ chosen to measure the distance between the RND $q$ and the physical density $p$: if $\gamma > 0$, then $q \ge 0$, otherwise $q > 0$. We consider homogeneous empirical probabilities $p_i = \frac{1}{T}$, $i = 1, 2, \dots, T$, to represent the physical distribution, for all dates $t$.<fnref n="19" /> This allows us to exchange the expectation $\mathbb{E}P$ with its sample counterpart $\frac{1}{T} \sum_{i=1}^{T} p_i = \sum_{i=1}^{T} p_i$.

<footnote n="17">¹⁷We often omit the time $t$ dependence for ease of notation.</footnote>

<footnote n="18">¹⁸The original analysis in Almeida and Garcia (2017) is performed in terms of Stochastic Discount Factors (SDFs). Almeida and Freire (2022) adapt it in details to consider RNDs instead. Most of the technical details following below were originally derived in these two articles.</footnote>

<footnote n="19">¹⁹For a set of empirically observed returns $\{R_i^t\}_{i=1,...,T}$, where each $R_i^t$ is independent and identically distributed according to $P$, $p_i = \frac{1}{T}$ is an optimal nonparametric estimator for the physical density $p$ (or, equivalently, the empirical measure $P_T = \frac{1}{T} \sum_{i=1}^{T} \delta_{R_i^t}$ is an optimal nonparametric estimator for $P$, where</footnote>
The MD problem (5) is computationally simpler and faster to solve in its dual formulation:

$$ (6) \qquad \hat{\lambda}^{\gamma} = \underset{\theta \in \mathbb{R}, \lambda \in \Lambda}{\operatorname{argsup}} \theta + \sum_{i=1}^{T} p_i \phi^{*,\gamma}(\theta + \lambda(R_i^t - R_F)) $$

where $\Lambda \subseteq \mathbb{R}$ and $\phi^{*,\gamma}$ denotes the convex conjugate of $\phi^{\gamma}$, restricted to a subset of the nonnegative real line:

$$ (7) \qquad \phi^{*,\gamma}(z) = \sup_{w \in [0, \infty) \cap \text{domain}\phi^{\gamma}} zw - \phi(w). $$

In this dual problem, $\theta$ and $\lambda$ are Lagrange multipliers arising from the restrictions defining an admissible RND in (5). The multiplier $\theta$ determining that $q$ sums to 1 (i.e., that $q$ is a discrete probability distribution) can be concentrated out of the problem. On the other hand, the multiplier $\lambda$ enforcing the Euler equation for the returns is the main one completely characterizing the RND. More specifically, for $\gamma < 0$, we can solve the following dual optimization problem to obtain $\lambda$:

$$ (8) \qquad \hat{\lambda}^{\gamma} = \underset{\lambda \in \Lambda}{\operatorname{argsup}} - \sum_{i=1}^{T} p_i \frac{1}{(\gamma + 1)} (1 + \gamma \lambda (R_i^t - R_F))^{\frac{\gamma+1}{\gamma}}, $$

where $q$ can be recovered from the first-order condition of (8) with respect to $\lambda$:

$$ (9) \qquad \hat{q}_i^{\gamma} = \frac{\left(1 + \gamma \hat{\lambda}^{\gamma} (R_i^t - R_F)\right)^{\frac{1}{\gamma}}}{\sum_{i=1}^{T} p_i \left(1 + \gamma \hat{\lambda}^{\gamma} (R_i^t - R_F)\right)^{\frac{1}{\gamma}}}. $$

For each $\gamma$, the resulting MD RND is different. To help illustrate how $\hat{q}^{\gamma}$ depends on $\gamma$, we replicate the Taylor expansion of the expected value of the discrepancy $\phi(\pi) = \frac{\pi^{\gamma+1}-1}{\gamma(\gamma+1)}$ around 1 executed in Almeida and Freire (2022):

$$ (10) \qquad \mathbb{E}(\phi^{\gamma}(\pi)) = \frac{1}{2} \mathbb{E}(\pi - 1)^2 + \frac{\gamma - 1}{3!} \mathbb{E}(\pi - 1)^3 + \frac{(\gamma - 1)(\gamma - 2)}{4!} \mathbb{E}(\pi - 1)^4 + \dots $$

As can be seen, minimizing the expected value of a given discrepancy amounts to minimizing a particular combination of higher-order moments of the RND determined by $\gamma$. In particular, for $\gamma < 1$, the weight given by the discrepancy to the skewness of the RND is negative, such that skewness is maximized, while the weight assigned to kurtosis is positive, such that kurtosis is minimized. This is consistent with a “preference” for positive skewness and “aversion” to kurtosis of stock index returns, characteristics that are in line with the findings of Kraus and Litzenberger (1976) and

<footnote>$\delta_{R_i^t}$ denotes a unit mass at $R_i^t$). See Kitamura (2007) for more details. This essentially amounts to using the histogram of returns as the empirical physical measure. Alternatively, one could use a kernel density estimator to smooth the histogram and obtain the physical probabilities. This would again amount to setting $p_i = \frac{1}{T}$ but for $T'$ returns drawn from the estimated kernel density, which does not bring any further insights.</footnote>
Backus, Chernov, and Martin (2011). The lower the $\gamma$, the higher is the relative importance of skewness and kurtosis in the estimation of the RND.

Since Cressie-Read discrepancies indexed by negative $\gamma$s have higher sensitivity to higher moments of basis assets returns, they constitute an adequate choice for our goal of identifying an RND to calculate the tail risk premium. Almeida et al. (2017) work with the Hellinger discrepancy ($\gamma = -\frac{1}{2}$) to identify the RND from a cross section of stock returns, due to the robustness of this estimator reported in former econometric studies. We choose instead to conduct our baseline analysis using the $\gamma = -3$ estimator since it is more sensitive to the skewness and the kurtosis of returns.<fnref n="20" /> In Section IV.A, we show that our results are robust to using the $\gamma = -\frac{1}{2}$ estimator.

In Section II.D, we associate the dual problem (8) to an optimal portfolio problem where a marginal investor in the S&P 500 market selects portfolio weights on the risk-free rate and the risky asset. This provides further interpretation to our approach.

### D. Portfolio Interpretation

Consider a standard optimal portfolio problem for an investor with utility function within the hyperbolic absolute risk aversion (HARA) class:

$$ (11) \qquad u'(W) = -\frac{1}{\gamma+1} (b - \alpha\gamma W)^{\frac{\gamma+1}{\gamma}}, $$

where $a > 0$ and $b - \alpha\gamma W > 0$, which guarantees that the function $u'$ is well-defined, concave and strictly increasing. The investor chooses how to allocate initial wealth $W_0$ by investing $\tilde{\lambda}$ units of wealth on the risky asset $R$ and the remaining $W_0 - \tilde{\lambda}$ in a risk-free asset paying $R_F$. The optimal allocation is such that the expected utility of end-of-period wealth $W(\tilde{\lambda}) = W_0 R_F + \tilde{\lambda}(R - R_F)$ is maximized:

$$ (12) \qquad \tilde{\lambda}^{\gamma} = \max_{\tilde{\lambda} \in \mathbb{R}} \mathbb{E}[u'(W(\tilde{\lambda}))]. $$

Almeida and Freire (2022) show that there is a one-to-one mapping between problem (12) and the population version of (8) for a given $\gamma$. This can be easily seen via the first-order condition of (12):

$$ (13) \qquad \begin{aligned} & \mathbb{E}\left\{ (R - R_F) \left[b - \alpha\gamma \left(W_0 R_F + \tilde{\lambda}^{\gamma} (R - R_F)\right)\right]^{\frac{1}{\gamma}} \right\} = 0 \\ & \Leftarrow \mathbb{E}\left\{ (R - R_F) \left[(b - \alpha\gamma W_0 R_F) (1 + \gamma \tilde{\lambda}^{\gamma} (R - R_F))\right]^{\frac{1}{\gamma}} \right\} = 0 \\ & \Leftarrow (b - \alpha\gamma W_0 R_F)^{\frac{1}{\gamma}} \mathbb{E}\left\{ (R - R_F) \left(1 + \gamma \tilde{\lambda}^{\gamma} (R - R_F)\right)^{\frac{1}{\gamma}} \right\} = 0 \\ & \Leftarrow \mathbb{E}\left\{ (R - R_F) \left(1 + \gamma \tilde{\lambda}^{\gamma} (R - R_F)\right)^{\frac{1}{\gamma}} \right\} = 0, \end{aligned} $$

<footnote n="20">²⁰Even though $\gamma$ can in principle attain values in the whole real line, Almeida and Freire (2022) show that the constrained optimization in the dual problem may not have a solution for extreme negative $\gamma$s (usually below -5). To guarantee an admissible RND with high sensitivity to downside risk for which a solution exists, we choose to work with $\gamma = -3$.</footnote>
where $\tilde{\lambda}' = -\tilde{\lambda}'a/(b - a\gamma W_0 R_f)$. The above shows that the RND $\tilde{q}''$ in (9) is proportional to the marginal utility of the HARA investor with concavity parameter $\gamma$, and that the optimal Lagrange multiplier $\tilde{\lambda}'$ is proportional (with opposite sign) to the optimal portfolio weight $\tilde{\lambda}'$ in the risky asset.

Provided that the equity premium is positive (i.e., $E(R - R_f) > 0$), the optimal portfolio solution will contemplate buying a certain amount of the stock index (i.e., $\tilde{\lambda}' > 0$). In such case, the Lagrange multiplier $\tilde{\lambda}'$ is negative and the RND $\tilde{q}''$ will distort the original physical distribution $p$ by putting higher (lower) probability mass to any state of negative (positive) index excess return. Intuitively, the marginal utility of the HARA investor is high (low) for negative (positive) realizations of the optimal portfolio due to risk aversion. Since negative return states are the ones that matter for the expected shortfall, this means that the risk-neutral leg of our tail risk premium measure, $ES^Q$, is greater than the corresponding physical leg, $ES^P$, implying that $\Delta_Q^P ES_I \ge 0$.

In practice, on a given day $t$, it may happen that the sample average of the high-frequency excess market returns is negative. This would imply that the HARA investor short-sells the stock index and has low marginal utility for negative market return states, such that $ES^Q$ would be actually below $ES^P$. Arguably, this is not economically sound, as the marginal investor selling the market would be inconsistent with a representative agent equilibrium model. In fact, the average intra-day return on day $t$ is a noisy estimate of the conditional equity premium, which is often considered to be nonnegative (Campbell and Thompson (2008), Martin (2017)). Therefore, in our baseline analysis, we restrict the equity premium to be nonnegative, which guarantees that our tail risk premium will be a nonnegative measure.<fnref n="21" /> In Section IV.B, we consider the unrestricted case for robustness and show that this restriction has no material impact on our empirical results.

### III. Empirical Analysis

Our empirical analysis of the predictability of risk premiums associated with the U.S. aggregate market, identified by the S&P 500 index, is composed of three parts. The first part focuses on the predictive power of our tail risk premium ($\Delta_Q^P ES$) and tail risk realization ($ES^P$) measures for the equity premium and the variance risk premium. In part two, we extend the study to the cross-sectional predictive ability of our measures. The third part examines the broader implications for the predictability of different quantiles of the distribution of excess market returns.

#### A. Data Description

Our data set is compiled from a number of data sources and covers the period from Jan. 2004 to Dec. 2018. First, we obtain high-frequency data on the S&P 500 index from www tickdata com, and downsample it to the 5-minute frequency by registering the last observation in each 5-minute window. This data are used not only to estimate $\Delta_Q^P ES$ and $ES^P$, but also a number of high-frequency return variation

<footnote n="21">²¹That is, for each day $t$, if the average of the intra-day excess market returns $\{R_t^f\}_{t=1:T}$ is negative, we shift the mean of the return distribution to zero by transforming returns to $\tilde{R}_t = R_t^f - \frac{T}{T}\sum_{i=1}^T R_t^f$. Note that this transformation does not affect the higher (centered) moments of the return distribution.</footnote>
measures, such as RV, integrated quadratic variation (IV) and jump quadratic variation (JV).<sup>22</sup> Data on the VIX index are obtained from the CBOE via WRDS, whereas the data on daily close-to-close S&P 500 returns, inclusive of dividends, are obtained from CRSP. Similarly to Bollerslev et al. (2009), we define the variance risk premium as the RV minus the VIX squared. The risk-free rate and data on the cross-sectional stock portfolios are obtained from the Kenneth French data library (https://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html). High-frequency data on options on the S&P 500 index are sourced from the CBOE in the form of best available bid and ask price quotes at the end of every 1-minute period, and then downsampled to the 5-minute frequency (the procedure to calculate option returns is described in Section IV.D).<sup>23</sup> Finally, the daily time-series of the LTV is obtained from www.tailindex.com. Appendix B collects all the variables' definitions.

## B. Properties of the Tail Risk Premium and Predictors

Our tail risk premium measure $\Delta_Q^P ES$ is based on the risk adjustment provided by the RND methodology described in Section II.C. Figure 1 presents the time-series of the physical expected shortfall ES$^P$, the tail premium $\Delta_Q^P ES$, and the Lagrange multiplier (LM) for the S&P 500 returns coming from the dual problem solved to obtain the RND on each day. A negative LM means that the marginal investor is long in the index. Graph C shows that the investor is always long in the index, which is a direct consequence of the economic restriction of a nonnegative equity premium that we impose. This restriction also guarantees that our tail risk premium is always above or equal to 0.<sup>24</sup> Graphs A and B further reveal that both

<caption>FIGURE 1
Time Series of the Tail Measures Implied by the S&P 500 Intraday Data</caption>

Figure 1 shows the daily time series of Tail Risk, Tail Risk Premium, and Lagrange Multiplier for the baseline case with quantile $\alpha=0.2$ and Cressie Read $\gamma=-3$.

<figure><img src="image_3.png" /></figure>

<footnote n="22"><sup>22</sup>RV is estimated as in Andersen, Bollerslev, Diebold, and Labys (2003). IV is estimated as in Mancini (2009). JV is estimated as $max\{RV - IV, 0\}$.</footnote>

<footnote n="23"><sup>23</sup>For each day, we use the options with maturity closest to 1 month, taking the shorter maturity as a tie breaker. We remove all quotes with 0 bid prices and those where the ask price is more than five times the bid price.</footnote>

<footnote n="24"><sup>24</sup>We relax this restriction in Section IV.B and compare predictability results. In the unrestricted case, for a significant number of days the average of intra-day S&P500 excess returns is negative. Nonetheless, this does not affect the predictive ability of our tail premium measure.</footnote>
ES$^P$ and $\Delta_Q^P$ ES achieve their peaks during the 2008–2009 subprime crisis. On the other hand, the marginal investor’s position in the S&P 500 index, which is proportional to minus the LM, is smaller during times of crisis as compared with non-crisis periods. This is also intuitive and consistent with flight-to-safety: The agent invests less in the risky asset during crises.

Table 1 reports the persistence of the tail risk premium and the predictor variables and their correlations. Naturally, measures of risk such as ES$^P$, LTV, RV, IV, and JV are strongly persistent and can be predicted by many of their lags. This is in contrast to risk premium measures. The VRP is only significantly predicted by its first lag, whereas the tail risk premium $\Delta_Q^P$ ES can only be predicted by the sum of lags 11 through 22. As for the correlation matrix in Panel B, it can be seen that our tail risk premium measure is not strongly correlated with any other variable. In particular, it has a 0.26 and 0.04 correlation with ES$^P$ and VRP, respectively. This suggests that our measure captures distinct information relative to the physical expected shortfall and the variance risk premium. On the other hand, ES$^P$ has large correlations with the volatility and jump risk variables. In what follows, we test the ability of these measures to forecast risk premiums and the future distribution of S&P 500 returns.

## C. Predicting Risk Premiums

### 1. Equity Premium

Table 2 reports the forecasting results for 1-day ahead excess returns on the S&P 500 index based on daily predictive regressions. Excess returns are obtained over the 3-month risk-free rate reported on Kenneth French’s data website, appropriately pro-rated. We investigate the market return predictability afforded by our tail risk premium while controlling for several sets of predictors. More specifically, we include as controls the physical expected shortfall ES$^P$, the variance risk premium VRP, the LTV of Bollerslev et al. (2015) and the RV (as is or decomposed into JV and IV).

The first column of Table 2 shows that the physical expected shortfall does not significantly predict 1 day-ahead excess market returns. When we include the tail risk premium in the regression in column 2, ES$^P$ remains insignificant. In contrast, $\Delta_Q^P$ ES has strong predictive power for the equity premium. This can be seen from a regression coefficient that is statistically significant at the 1% level and an adjusted $R^2$ that is relatively high for the 1-day horizon. The positive coefficient indicates that investors require a higher compensation to hold the market (i.e., a higher excess return) when the tail risk premium increases. These findings provide new high-frequency evidence that aversion to downside risk (but not downside risk itself, as captured by ES$^P$) is an important determinant of the equity premium.

Column 3 of Table 2 further includes the VRP as control. The VRP is statistically insignificant and only marginally increases the adjusted $R^2$. Importantly, such inclusion does not affect the predictive power of $\Delta_Q^P$ ES for excess market returns. Results are very similar in column 4 when the expected shortfall is removed from the regression. This suggests that our tail risk premium reflects distinct information from that contained in the VRP. In fact, the fifth column shows that if we orthogonalize $\Delta_Q^P$ ES with respect to the VRP, it still retains most of its
<caption>TABLE 1
Persistence and Co-dependence of the Predictors of Risk Premia</caption>

Panel A of Table 1 presents the estimates of HAR-type (see Corsi (2009)) predictive models for each risk premium predictor. The labels “lag 1” to “lag 5” denote the appropriately lagged regressand. The label “lag 1W” denotes the sum of lags 6 to 10 of the regressand, and the label “lag 1M” denotes the sum of lags 11 to 22. All reported coefficients and standard errors are rounded to 2 decimal places. Those significant at the 0.05 confidence level are printed in bold and those significant at the 0.01 level are additionally marked with a *. We report Andrews standard errors calculated with the use of the sandwich 3.0.0 package for R 4.0.3. Panel B presents the correlation matrix of the predictors. ES<sup>B</sup> is the nonparametric estimate of realized (physical) expected shortfall of intra-day S&P 500 returns before the close, on the day when each return calculation commences, ES<sup>P</sup> = -E<sub>t</sub>|R<sub>t</sub>|R<sub>t-1</sub> ≤ F<sub>t</sub><sup>-1</sup>(0.2). The tail risk premium Δ<sup>P</sup><sub>0</sub>ES, is the difference between the risk-neutral and the physical ES. VRP is the variance risk premium calculated as the difference between the day's realized variance and the (appropriately scaled) VIX<sup>2</sup> index. LTV is the left tail variance of Bollerslev et al. (2015) obtained from www.talindex.com. RV is the realized variance of intra-day returns on the S&P 500 index. IV is an estimate of integrated quadratic variation, the continuous component of realized variance, estimated as in Mancini and Gobbi (2012). JV is the jump component of realized variance, calculated as max{RV – IV, 0}. RV, IV, and JV are calculated from return data sampled at the 5-minute frequency.

<table><caption>Panel A. Persistence of the Predictors of Risk Premia</caption><thead><tr><th></th><th>Δ<sup>P</sup><sub>0</sub>ES</th><th>ES<sup>P</sup></th><th>VRP</th><th>LTV</th><th>RV</th><th>IV</th><th>JV</th></tr><tr><th></th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th></tr></thead><tbody><tr><th>Lag 1</th><td>0.00<br>(0.03)</td><td><strong>0.41*</strong><br>(0.04)</td><td><strong>0.15*</strong><br>(0.06)</td><td><strong>0.86*</strong><br>(0.05)</td><td><strong>0.37*</strong><br>(0.09)</td><td><strong>0.38*</strong><br>(0.10)</td><td><strong>0.16*</strong><br>(0.06)</td></tr><tr><th>Lag 2</th><td>0.01<br>(0.11)</td><td><strong>0.29*</strong><br>(0.09)</td><td>0.36<br>(0.20)</td><td>0.05<br>(0.06)</td><td><strong>0.42</strong><br>(0.18)</td><td><strong>0.45*</strong><br>(0.17)</td><td><strong>0.22*</strong><br>(0.08)</td></tr><tr><th>Lag 3</th><td>-0.01<br>(0.06)</td><td>0.05<br>(0.07)</td><td>0.05<br>(0.09)</td><td><strong>0.28*</strong><br>(0.10)</td><td>0.03<br>(0.06)</td><td>-0.03<br>(0.07)</td><td><strong>0.20*</strong><br>(0.08)</td></tr><tr><th>Lag 4</th><td>0.05<br>(0.07)</td><td>0.10<br>(0.10)</td><td>0.27<br>(0.14)</td><td>0.01<br>(0.09)</td><td><strong>0.36*</strong><br>(0.14)</td><td>0.35<br>(0.19)</td><td>0.03<br>(0.04)</td></tr><tr><th>Lag 5</th><td>-0.03<br>(0.08)</td><td>0.10<br>(0.08)</td><td>0.19<br>(0.14)</td><td><strong>0.21*</strong><br>(0.08)</td><td>0.24<br>(0.14)</td><td>0.33<br>(0.18)</td><td>0.14<br>(0.05)</td></tr><tr><th>Lag 1W</th><td>0.08<br>(0.07)</td><td>-0.03<br>(0.05)</td><td>-0.10<br>(0.09)</td><td><strong>-0.12*</strong><br>(0.04)</td><td>-0.14<br>(0.07)</td><td>-0.14<br>(0.10)</td><td><strong>-0.12*</strong><br>(0.04)</td></tr><tr><th>Lag 1M</th><td><strong>0.02*</strong><br>(0.01)</td><td><strong>0.01*</strong><br>(0.00)</td><td>0.02<br>(0.01)</td><td><strong>0.01*</strong><br>(0.00)</td><td><strong>0.02</strong><br>(0.01)</td><td>0.01<br>(0.01)</td><td><strong>0.06*</strong><br>(0.02)</td></tr><tr><th>Constant</th><td><strong>0.00*</strong><br>(0.00)</td><td><strong>0.00</strong><br>(0.00)</td><td>0.00<br>(0.00)</td><td><strong>0.00*</strong><br>(0.00)</td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td></tr><tr><th>R<sup>2</sup> (adj)</th><td>0.09</td><td>0.72</td><td>0.22</td><td>0.85</td><td>0.61</td><td>0.63</td><td>0.19</td></tr><tr><th>N</th><td>3748</td><td>3748</td><td>3748</td><td>3691</td><td>3748</td><td>3748</td><td>3748</td></tr></tbody></table>

<table><caption>Panel B. Correlation Matrix of the Predictors of Risk Premia</caption><thead><tr><th></th><th>ES<sup>P</sup></th><th>VRP</th><th>LTV</th><th>RV</th><th>IV</th><th>JV</th></tr></thead><tbody><tr><th>Δ<sup>P</sup><sub>0</sub>ES</th><td>0.26</td><td>0.04</td><td>0.31</td><td>0.30</td><td>0.28</td><td>0.26</td></tr><tr><th>ES<sup>P</sup></th><td></td><td>0.41</td><td>0.70</td><td>0.88</td><td>0.88</td><td>0.51</td></tr><tr><th>VRP</th><td></td><td></td><td>0.05</td><td>0.67</td><td>0.64</td><td>0.56</td></tr><tr><th>LTV</th><td></td><td></td><td></td><td>0.71</td><td>0.71</td><td>0.35</td></tr><tr><th>RV</th><td></td><td></td><td></td><td></td><td>0.98</td><td>0.66</td></tr><tr><th>IV</th><td></td><td></td><td></td><td></td><td></td><td>0.51</td></tr></tbody></table>

predictive power for the equity premium. While the variance risk premium is an important predictor of future market returns at relatively low-frequency horizons (Bollerslev et al. (2009)), we find that the same is not true in a high-frequency environment.

Finally, in the last 2 columns of Table 2, we include realized and option-implied measures of volatility and jump risk. The risk measures are all insignificant, but increase substantially the R<sup>2</sup>. In particular, the LTV has no predictive power. This can be rationalized by the fact that it is computed from options with maturity between 6 and 31 trading days, which reflect market expectations over relatively long horizons. When these additional variables are included, the VRP becomes significant at the 5% level, with the expected sign: investors demand higher market returns when they are more averse to volatility risk (i.e., when
<caption>TABLE 2
Predictive Regressions: S&P 500 Excess Returns</caption>

Table 2 reports regression coefficients and their standard deviations (in parentheses) of predictive regressions of the close-to-close S&P 500 excess returns at the 1-day horizon. All reported coefficients and standard errors are rounded to 2 decimal places. Those significant at the 0.05 confidence level are printed in bold and those significant at the 0.01 level are additionally highlighted with a *. We report Andrews (1991) standard errors calculated with the use of the sandwich 3.0.0 package for R 4.0.3 (Zeileis, Koll, and Graham (2020)). ES<sup>P</sup> is the nonparametric estimate of realized (physical) expected shortfall of intra-day S&P 500 returns before the close, on the day when each return calculation commences, ES<sup>P</sup> = -E<sub>t</sub> [R<sub>t</sub>IR<sub>t</sub> ≤ F<sub>P</sub><sup>t</sup> (0.2)]. Our tail risk premium measure, Δ<sub>Q</sub><sup>P</sup>ES, is the difference between the risk-neutral and the physical ES. VRP is the variance risk premium calculated as the difference between the day's RV and the (appropriately scaled) VIX<sup>2</sup> index. Δ<sub>Q</sub><sup>P</sup>ES<sup>⊥</sup> is the component of the tail risk measure that is orthogonal to the variance risk premium. LTV is the left tail variance of Bollerslev et al. (2015) obtained from www.tailindex.com. RV is the realized variance of intra-day returns on the S&P 500 index. IV is an estimate of integrated quadratic variation, the continuous component of realized variance, estimated as in Mancini and Gobbi (2012). JV is the jump component of realized variance, calculated as max{RV - IV, 0}. RV, IV, and JV are calculated from return data sampled at the 5-minute frequency.

<table><thead><tr><th></th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th></tr></thead><tbody><tr><td>ES<sup>P</sup></td><td>0.17<br>(0.77)</td><td>-0.16<br>(0.87)</td><td>-0.32<br>(0.58)</td><td></td><td></td><td>-2.04<br>(1.15)</td><td>-2.01<br>(1.13)</td></tr><tr><td>Δ<sub>Q</sub><sup>P</sup>ES</td><td></td><td><strong>7.50*</strong><br>(2.90)</td><td><strong>7.65*</strong><br>(2.63)</td><td><strong>7.19*</strong><br>(2.68)</td><td></td><td><strong>8.83*</strong><br>(2.83)</td><td><strong>8.75*</strong><br>(2.66)</td></tr><tr><td>VRP</td><td></td><td></td><td>2.35<br>(8.79)</td><td>1.53<br>(9.07)</td><td>0.94<br>(9.25)</td><td>-15.90<br>(7.74)</td><td>-16.05<br>(7.79)</td></tr><tr><td>Δ<sub>Q</sub><sup>P</sup>ES<sup>⊥</sup></td><td></td><td></td><td></td><td></td><td><strong>6.63</strong><br>(2.76)</td><td></td><td></td></tr><tr><td>RV</td><td></td><td></td><td></td><td></td><td></td><td>15.25<br>(10.36)</td><td></td></tr><tr><td>LTV</td><td></td><td></td><td></td><td></td><td></td><td>-0.10<br>(0.07)</td><td>-0.10<br>(0.07)</td></tr><tr><td>JV</td><td></td><td></td><td></td><td></td><td></td><td></td><td>17.73<br>(14.14)</td></tr><tr><td>IV</td><td></td><td></td><td></td><td></td><td></td><td></td><td>14.79<br>(10.81)</td></tr><tr><td>Constant</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td><td>0.00</td></tr><tr><td>(0.00)</td><td></td><td>(0.00)</td><td>(0.00)</td><td>(0.00)</td><td>(0.00)</td><td>(0.00)</td><td>(0.00)</td></tr><tr><td>R<sup>2</sup> (adj %)</td><td>0.00</td><td>0.73</td><td>0.76</td><td>0.75</td><td>0.62</td><td>2.03</td><td>2.01</td></tr></tbody></table>

VRP = RV - VIX<sup>2</sup> is more negative). As for our tail premium measure, it remains significant at the 1% level. In Table C.1, we report the market return predictability results for 5- and 21-day horizons. For these horizons, no variable is able to significantly predict market returns. This suggests that the primary effect of Δ<sub>Q</sub><sup>P</sup>ES on market returns is concentrated on the day following a shock in tail risk premium.

We also investigate whether the market return predictability afforded by Δ<sub>Q</sub><sup>P</sup>ES is robust out-of-sample. According to a model update schedule, we estimate 1-day-ahead predictive regressions using only data available up to a specific date. Then, keeping parameters unchanged until the next model update, we predict the excess market return for day t + 1, based on predictor values for day t. We consider different start dates for the out-of-sample period: Jan. 01, 2008, Jan. 01, 2012, and Apr. 08, 2015 (i.e., we vary the initial sample length). We also vary the model re-estimation frequency, ranging from every month (adding the most recent data) to a single estimation that only uses the initial sample and keeps the parameters unchanged until the end of our sample on Dec. 31, 2018. We provide context for assessing the importance of our results using two additional predictive models: i) a univariate model with VRP, the only relevant control in Table 2, and ii) a bivariate model with both Δ<sub>Q</sub><sup>P</sup>ES and VRP. Finally, we benchmark the predictive power of the
three models against the historical mean of the market excess return, estimated following the same update schedule, as commonly done in the literature (see, e.g., Campbell and Thompson (2008)).

Table 3 contains the results of the out-of-sample exercise. Panel A reports the out-of-sample predictive $R^2$ relative to the historical mean for each model, starting date and update frequency.<fnref n="25" /> A positive value means that a predictor generates better forecasts, in terms of lower mean-squared error, than the historical average return. The univariate model based on our tail risk premium measure almost always generates a positive $R^2$, regardless of when the out-of-sample period starts and how often we re-estimate the model. This indicates that the predictive relation between $\Delta_Q^{PES}$ and future excess market returns is stable and robust to different economic conditions. Remarkably, these $R^2$s for daily market returns are of similar magnitude to (and often larger than) those documented by Campbell and Thompson (2008) for monthly returns using well-known predictors of the equity premium. In contrast, the VRP is unable to outperform the historical mean, leading to $R^2$s that are generally negative. As a result, the performance of the univariate tail premium model is nearly always better than that of the bivariate model including both $\Delta_Q^{PES}$ and VRP, reinforcing the superiority of our measure for predicting the equity premium.

To assess the statistical significance of the superior out-of-sample predictive ability of $\Delta_Q^{PES}$, we implement the commonly used Clark and West (2007) statistic. In Panel B of Table 3, we test whether the forecasts based on the tail risk premium decrease the mean-squared error relative to the historical mean benchmark. For all starting dates and update frequencies, the null hypothesis of equal predictive accuracy is rejected at the 1% significance level. The same is not true when we consider the VRP instead of $\Delta_Q^{PES}$. In this case, improvements are essentially never statistically significant. We further test whether the tail premium improves the predictive accuracy relative to the VRP. Again, the null hypothesis is strongly rejected across all specifications. These results show that the superior predictive ability of our tail premium measure is robust and statistically significant.

Finally, we investigate the economic significance of the predictive power for excess market returns afforded by $\Delta_Q^{PES}$. We follow Campbell and Thompson (2008) in considering an investor with mean-variance preferences who optimally chooses the portfolio weight on the market based on the return forecast. We impose realistic portfolio constraints that prevent the investor from shorting the market or taking more than 100% leverage. Panel C of Table 3 reports the annualized certainty equivalent return associated with each predictive model. As can be seen, the values corresponding to $\Delta_Q^{PES}$ are generally large and always positive. For instance, starting in 2008, a mean-variance investor would require a certain return of 9.60% each year (in excess of the risk-free rate) to be indifferent with respect to the risky strategy exploiting forecasts from the tail premium model updated every

<footnote n="25">²⁵If the historical mean of market returns is a poor predictor of future returns, this could inflate this measure of $R^2$. In Table C.2, we report an alternative measure of predictive $R^2$ without subtracting the historical mean in the denominator, i.e., reflecting just the amount of variation explained by the tail risk premium. While the predictive $R^2$s calculated in this manner are actually slightly higher than these reported in Table 3, the results are overall very similar.</footnote>
<caption>TABLE 3
Out-of-Sample Predictive Power for S&P 500 Excess Returns</caption>

Table 3 contains results on the out-of-sample predictive performance of linear predictive models for S&P 500 excess returns. The following model specifications are considered: i) $\Delta_Q^P$ES, our measure of the tail risk premium; ii) $\hat{\mu}$ (constant) (i.e., the historical average excess return); iii) VRP, the variance risk premium; and iv) “both,” containing $\Delta_Q^P$ES and VRP. In Panel A, we report the predictive $R^2$, calculated against the benchmark of $\hat{\mu}$ as in Campbell and Thompson (2008), $R^2 = 1 - \frac{\sum_t (R_t - \hat{R}_t)}{\sum_t (R_t - \hat{\mu})^2}$, where $\hat{R}_t$ is the model's predicted return in period t. In Panel B, we report the Clark and West (2007) statistic for the test of forecast accuracy. Univariate models in columns 1, 2, 4, 5, 7, and 8 are tested against the benchmark of $\hat{\mu}$ while the bivariate model in columns 3, 6, and 9 is tested against the benchmark of VRP. The CW test statistic is approximately normally distributed under the null that the additional regressors do not have predictive power. The test is 1-sided (i.e., the alternative hypothesis is that the more complex model has better predictive power). The critical values are 1.28, 1.64, and 2.33 at the 10%, 5%, and 1% confidence levels. In Panel C, we report the certainty equivalent (CE) of a quadratic utility investor with relative risk aversion of 3. The CE is reported in excess of the risk-free rate, in percent, and annualized. Following Campbell and Thompson (2008), the investor allocates the fraction $\alpha_t = \hat{R}_t/(\sigma\alpha_n)$ to the S&P 500, where $\sigma\alpha_n$ is the historical volatility of the returns at last model update; the remaining wealth is allocated to the risk-free asset. The risky asset weight is constrained to lie between 0 and 2 (i.e., we restrict short sales and excessive leverage). Predictive regressions are estimated on expanding samples starting in Jan. 2008 (columns 1–3), Jan. 2012 (columns 4–6), and Apr. 2015 (columns 7–9). In Panels A and B, in the first 2 columns of each sub-panel, we consider univariate predictive models, and in each respective third column, a model containing both predictors. In Panel C, we focus on univariate models. Across the rows of each panel, we consider different parameter update frequencies, from every month in the top row to no updates (i.e., the parameters are estimated on data prior to the starting date and never updated) in the final row.

<table><caption>Panel A: Out-of-Sample Predictive $R^2$ (%)</caption><thead><tr><th rowspan="2">Update Freq.</th><th colspan="3">OOS from 2008-01-01</th><th colspan="3">OOS from 2012-01-01</th><th colspan="3">OOS from 2015-04-08</th></tr><tr><th>VRP</th><th>Δ<sup>P</sup><sub>Q</sub>ES</th><th>Both</th><th>VRP</th><th>Δ<sup>P</sup><sub>Q</sub>ES</th><th>Both</th><th>VRP</th><th>Δ<sup>P</sup><sub>Q</sub>ES</th><th>Both</th></tr></thead><tbody><tr><td>1 months</td><td>-1.14</td><td>0.10</td><td>-0.35</td><td>-0.10</td><td>0.70</td><td>0.61</td><td>-0.05</td><td>1.40</td><td>1.34</td></tr><tr><td>3 months</td><td>-1.40</td><td>-0.01</td><td>-0.34</td><td>0.00</td><td>0.52</td><td>0.52</td><td>-0.02</td><td>1.47</td><td>1.45</td></tr><tr><td>6 months</td><td>0.00</td><td>0.64</td><td>0.60</td><td>0.00</td><td>0.52</td><td>0.53</td><td>-0.02</td><td>1.46</td><td>1.44</td></tr><tr><td>12 months</td><td>-0.14</td><td>0.60</td><td>0.32</td><td>-0.05</td><td>0.28</td><td>0.26</td><td>-0.03</td><td>1.51</td><td>1.48</td></tr><tr><td>24 months</td><td>-0.04</td><td>0.64</td><td>0.48</td><td>-0.05</td><td>0.26</td><td>0.24</td><td>-0.15</td><td>1.53</td><td>1.40</td></tr><tr><td>Never</td><td>0.04</td><td>0.76</td><td>0.79</td><td>-0.08</td><td>0.59</td><td>0.52</td><td>-0.17</td><td>1.56</td><td>1.39</td></tr></tbody></table>

<table><caption>Panel B. Clark and West (2007) Test</caption><thead><tr><th rowspan="3">Update Freq.</th><th colspan="3">OOS from 2008-01-01</th><th colspan="3">OOS from 2012-01-01</th><th colspan="3">OOS from 2015-04-08</th></tr><tr><th colspan="2">vs. μ̂</th><th rowspan="2">VRP</th><th colspan="2">vs. μ̂</th><th rowspan="2">VRP</th><th colspan="2">vs. μ̂</th><th rowspan="2">VRP</th></tr><tr><th>VRP</th><th>Δ<sup>P</sup><sub>Q</sub>ES</th><th>VRP</th><th>Δ<sup>P</sup><sub>Q</sub>ES</th><th>VRP</th><th>Δ<sup>P</sup><sub>Q</sub>ES</th></tr><tr><td></td><td>1</td><td>2</td><td>Both</td><td>4</td><td>5</td><td>Both</td><td>7</td><td>8</td><td>Both</td></tr></thead><tbody><tr><td>1 months</td><td>1.42</td><td>4.43</td><td>5.52</td><td>-1.56</td><td>5.07</td><td>5.09</td><td>-0.66</td><td>4.95</td><td>4.97</td></tr><tr><td>3 months</td><td>1.30</td><td>3.89</td><td>4.68</td><td>0.41</td><td>4.20</td><td>4.23</td><td>0.05</td><td>4.80</td><td>4.83</td></tr><tr><td>6 months</td><td>0.94</td><td>6.01</td><td>5.61</td><td>0.40</td><td>4.13</td><td>4.17</td><td>0.04</td><td>4.83</td><td>4.86</td></tr><tr><td>12 months</td><td>0.01</td><td>5.71</td><td>4.72</td><td>-1.23</td><td>3.06</td><td>3.13</td><td>-0.22</td><td>4.59</td><td>4.62</td></tr><tr><td>24 months</td><td>0.34</td><td>5.81</td><td>4.92</td><td>-1.16</td><td>2.98</td><td>3.05</td><td>-2.09</td><td>3.85</td><td>3.90</td></tr><tr><td>Never</td><td>0.66</td><td>6.75</td><td>5.95</td><td>-1.29</td><td>4.21</td><td>4.25</td><td>-1.87</td><td>5.04</td><td>5.06</td></tr></tbody></table>

<table><caption>Panel C. Certainty Equivalent (annualized, %)</caption><thead><tr><th rowspan="2">Update Freq.</th><th colspan="3">OOS from 2008-01-01</th><th colspan="3">OOS from 2012-01-01</th><th colspan="3">OOS from 2015-04-08</th></tr><tr><th>μ̂</th><th>VRP</th><th>Δ<sup>P</sup><sub>Q</sub>ES</th><th>μ̂</th><th>VRP</th><th>Δ<sup>P</sup><sub>Q</sub>ES</th><th>μ̂</th><th>VRP</th><th>Δ<sup>P</sup><sub>Q</sub>ES</th></tr></thead><tbody><tr><td>1 months</td><td>0.56</td><td>-5.78</td><td>9.60</td><td>2.53</td><td>1.81</td><td>4.43</td><td>1.95</td><td>2.62</td><td>7.17</td></tr><tr><td>3 months</td><td>1.47</td><td>-4.49</td><td>12.08</td><td>4.39</td><td>5.58</td><td>7.31</td><td>4.10</td><td>5.74</td><td>8.19</td></tr><tr><td>6 months</td><td>0.91</td><td>-4.91</td><td>11.93</td><td>3.99</td><td>5.29</td><td>6.94</td><td>4.17</td><td>5.81</td><td>8.22</td></tr><tr><td>12 months</td><td>-2.77</td><td>-6.79</td><td>12.61</td><td>4.08</td><td>4.71</td><td>7.10</td><td>3.15</td><td>4.72</td><td>6.73</td></tr><tr><td>24 months</td><td>-1.96</td><td>-7.48</td><td>10.14</td><td>3.54</td><td>4.10</td><td>6.82</td><td>2.74</td><td>2.89</td><td>8.28</td></tr><tr><td>Never</td><td>0.75</td><td>-7.50</td><td>5.35</td><td>1.23</td><td>1.22</td><td>3.05</td><td>1.53</td><td>-0.13</td><td>3.56</td></tr></tbody></table>

month. At the same time, this investor would require only 0.56% to not exploit information from the historical mean, and would be willing to actually pay a certain return of 5.78% to avoid adopting the VRP-based forecasts. Importantly, $\Delta_Q^P$ES outperforms the historical mean and the VRP in all cases, often by a large extent. This confirms that the predictive power of the tail risk premium is not only statistically, but also economically significant.
In sum, we document that our tail risk premium measure is a strong predictor of future excess market returns at the 1-day horizon. This predictability is robust to controlling for the expected shortfall, variance risk premium and several volatility and jump risk variables. Our findings are consistent with the idea that investors require a higher compensation to hold the market following an increase in aversion to downside risk as captured by $\Delta_Q^PES$. The predictive power of the tail premium also holds out-of-sample, with strong statistical and economic significance, above and beyond different benchmarks.

## 2. Variance Risk Premium

Table 4 reports the forecasting results for the 1-day ahead market variance risk premium based on daily predictive regressions. We consider the same controls as in Table 2. The first 2 columns show that, while downside risk as captured by $ES^P$ has no predictive power for the variance risk premium, aversion to downside risk reflected in $\Delta_Q^PES$ appears as a statistically significant predictor associated with a high $R^2$. This provides new evidence in a high-frequency environment that part of the variance risk premium can be explained by aversion to left tail risk. As can be seen from columns 3 and 4, the lagged VRP also affords strong predictive power.

<caption>TABLE 4
Predictive Regressions: Short-Term Variance Risk Premium</caption>

Table 4 reports regression coefficients and their standard deviations (in parentheses) for predictive regressions of the variance risk premium at the 1-day horizon. All reported coefficients and standard errors are rounded to 2 decimal places. Those significant at the 0.05 confidence level are printed in bold and those significant at the 0.01 level are additionally highlighted with a *. We report Andrews (1991) standard errors calculated with the use of the sandwich 3.0.0 package for R 4.0.3 (Zeileis et al. (2020)). $ES^P$ is the nonparametric estimate of realized (physical) expected shortfall of intra-day S&P 500 returns before the close, on the day when each return calculation commences, $ES^P = -E_i|R_{it}|R_{it} \le F_{R_i}^{1/2}$. Our tail risk premium measure, $\Delta_Q^PES$, is the difference between the risk-neutral and the physical ES. VRP is the variance risk premium calculated as the difference between the day's realized variance and the (appropriately scaled) VIX² index. $\Delta_Q^PES^+$ is the component of the tail risk measure that is orthogonal to the variance risk premium. LTV is the left tail variance of Bollerslev et al. (2015) obtained from www.tailindex.com. RV is the realized variance of intra-day returns on the S&P 500 index. IV is an estimate of integrated quadratic variation, the continuous component of realized variance, estimated as in Mancini and Gobbi (2012). JV is the jump component of realized variance, calculated as $max(RV - IV, 0)$. RV, IV, and JV are calculated from return data sampled at the 5-minute frequency.

<table><thead><tr><th></th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th></tr></thead><tbody><tr><td>ES<sup>P</sup></td><td>0.02<br>(0.01)</td><td>0.00<br>(0.01)</td><td><strong>-0.02*</strong><br>(0.01)</td><td></td><td></td><td>-0.03<br>(0.02)</td><td>-0.03<br>(0.01)</td></tr><tr><td>Δ<sub>Q</sub><sup>P</sup>ES</td><td></td><td><strong>0.38*</strong><br>(0.14)</td><td><strong>0.40*</strong><br>(0.14)</td><td><strong>0.36*</strong><br>(0.13)</td><td></td><td><strong>0.22*</strong><br>(0.06)</td><td><strong>0.22*</strong><br>(0.06)</td></tr><tr><td>VRP</td><td></td><td></td><td><strong>0.32*</strong><br>(0.07)</td><td><strong>0.26*</strong><br>(0.06)</td><td><strong>0.23*</strong><br>(0.07)</td><td><strong>0.38*</strong><br>(0.10)</td><td><strong>0.39*</strong><br>(0.10)</td></tr><tr><td>Δ<sub>Q</sub><sup>P</sup>ES<sup>⊥</sup></td><td></td><td></td><td></td><td></td><td><strong>0.38*</strong><br>(0.14)</td><td></td><td></td></tr><tr><td>RV</td><td></td><td></td><td></td><td></td><td></td><td>0.01<br>(0.11)</td><td></td></tr><tr><td>LTV</td><td></td><td></td><td></td><td></td><td></td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td></tr><tr><td>JV</td><td></td><td></td><td></td><td></td><td></td><td></td><td>-0.09<br>(0.26)</td></tr><tr><td>IV</td><td></td><td></td><td></td><td></td><td></td><td></td><td>0.01<br>(0.10)</td></tr><tr><td>Constant</td><td><strong>0.00*</strong><br>(0.00)</td><td><strong>0.00*</strong><br>(0.00)</td><td><strong>0.00<br>(0.00)</strong></td><td><strong>0.00*</strong><br>(0.00)</td><td><strong>0.00*</strong><br>(0.00)</td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td></tr><tr><td>R<sup>2</sup> (adj %)</td><td>9.24</td><td>23.31</td><td>30.93</td><td>29.56</td><td>30.34</td><td>29.91</td><td>29.95</td></tr></tbody></table>
Importantly, the significance at the 1% level of our tail risk premium measure is robust to the inclusion of the lagged VRP, even when we consider its orthogonalized version in the fifth column. In fact, both the $\Delta_Q^PES$ and lagged VRP are relevant to forecast the variance risk premium, having a similar contribution to the $R^2$s. The last 2 columns reveal that the volatility and jump risk measures are insignificant and have no impact in the previous results. In Table C.3, we further report the VRP predictability for 5- and 21-day horizons. Results are very similar to those of Table 4, where $\Delta_Q^PES$ robustly predicts the variance risk premium across virtually all horizons and regression specifications.

### 3. The Cross Section of Characteristic-Sorted Portfolio Returns

Subsections IIIC.1 and IIIC.2 show that our tail risk premium is an important determinant of aggregate market risk premiums at high frequency. In this subsection, we investigate whether aversion to downside risk, as captured by $\Delta_Q^PES$, also commands risk premium in a cross section of characteristic-sorted portfolios. Portfolios sorted by specific characteristics reflect compensation for different types of risk beyond those related to the market portfolio. More specifically, we consider several sets of portfolios sorted according to the often-used characteristics of size, BM, profitability, investment, momentum, reversal, and industry. Most of these characteristics form the basis of factor pricing models such as those of Fama and French (1993), (2015) and Hou, Xue, and Zhang (2014).

In Figure 2, we report the results of 1-day ahead predictive regressions of characteristic-sorted portfolios. We forecast the daily excess returns for each decile portfolio obtained from sorting stocks on a given characteristic. We consider five sets of daily predictors: i) $\Delta_Q^PES$, ii) $\Delta_Q^PES$ and $ES^P$, iii) $\Delta_Q^PES$ and RV, iv) $\Delta_Q^PES$ and LTV, and v) $\Delta_Q^PES$ and VRP. For each decile portfolio, we plot a corresponding group of five t-statistics on $\Delta_Q^PES$ calculated with Andrews (1991) standard errors and the adjusted $R^2$ of the regression. For comparison, Figure 3 shows the same results for univariate predictive regressions based on each alternative predictor ($ES^P$, RV, LTV, and VRP).

Across all sets of predictors and characteristics, the (absolute value of the) t-statistics on $\Delta_Q^PES$ reported in Graph A of Figure 2 are mostly above 2 and often above 3. This indicates that our tail risk premium measure has strong predictive power for cross-sectional risk premiums at high frequency. Such predictive power cannot be explained by the different controls we consider. In fact, the reported t-statistics are very similar across the different sets of predictors. Graph B further shows that nearly all of the $R^2$s are between 0.5% and 1.3%, which is of the same order of magnitude of those obtained for market returns in Table 2. The alternative predictors also do not affect the $R^2$, with the exception of VRP which usually leads to a slight improvement. In Figure 3, we can see that the alternative predictors are individually statistically insignificant for predicting characteristic-sorted portfolios, yielding mostly negligible $R^2$s. This reinforces that our tail risk premium measure is the only predictor of characteristic-portfolios at high frequency among the variables we consider.
<caption>FIGURE 2
Predictive Regressions: Fama-French Portfolios</caption>

Figure 2 presents a summary of predictive regressions of the 1-day ahead excess return on Fama-French portfolios of stocks sorted on size, book/market, profitability, investment, momentum, reversal, and the 10 industry portfolios. Five regression specifications are considered, using the following regressors: i) $\Delta_Q^P ES$, ii) $\Delta_Q^P ES$ and $ES^P$, iii) $\Delta_Q^P ES$ and RV, iv) $\Delta_Q^P ES$ and LTV, v) $\Delta_Q^P ES$ and VRP. We report Andrews (1991) standard errors calculated with the use of the sandwich 3.0.0 package for R 4.0.3 (Zeileis et al. (2020)).

<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /></figure>

D. Predicting the Distribution of Market Returns

Section III.C demonstrates that it is the aversion to downside risk (as captured by $\Delta_Q^P ES$) rather than the level of downside risk itself (as captured by $ES^P$) that contains relevant predictive information for risk premiums at high
<caption>FIGURE 3

Predictive Regressions: Fama-French Portfolios, Alternative Predictors</caption>

Figure 3 shows a summary of predictive regressions of the 1-day ahead excess return on Fama-French portfolios of stocks sorted on size, book/market, profitability, investment, momentum, reversal, and the 10 industry portfolios. Four regression specifications are considered, using the following regressors: i) ES<sup>P</sup>, ii) RV, iii) LTV, and iv) VRP. We report Andrews (1991) standard errors calculated with the use of the sandwich 3.0.0 package for R 4.0.3 (Zeileis et al. (2020)). In order to facilitate the comparison with the main predictability results for Fama-French portfolios, the axis scales are set the same as in Figure 2.

<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /></figure>

frequency. In this section, we turn to an analysis of the predictive power of Δ<sup>P</sup><sub>Q</sub>ES and ES<sup>P</sup> for the whole distribution of future excess market returns in order to shed light on the differential role of these variables for explaining the equity premium.
# 1. Quantiles of the Distribution of Market Returns

To understand how different variables affect the distribution of future S&P 500 returns, we adopt the conditional quantile regression framework introduced by Koenker and Gilbert (1978). The conditional quantile model for S&P 500 daily excess returns {$r_t$}$_{t=1,...,T_q}$, reads as follows:

$$ (14) \qquad Q_{r_{t+h}}(\tau|p_t) = \theta_0(\tau) + \theta_1(\tau) \Delta_Q^{\mathrm{P}} ES_t + \theta_2(\tau) ES_t + \theta_3(\tau) VRP_t + \theta_4(\tau) RV_t, $$

where $p_t = \{1, \Delta_Q^{\mathrm{P}} ES_t, ES_t^{\mathrm{P}}, VRP_t, RV_t\}$ and the $\theta_{js}$ are functions mapping $\tau \in [0, 1]$ into $\mathbb{R}$. This equation states that the conditional $\tau$-quantile of the daily S&P 500 excess return distribution at time $t+h$ is a linear function of the tail risk premium and control variables at time $t$. In the quantile regression of $r_{t+h}$ on the variables in $p_t$, the regression coefficients $\theta_{\tau}$ are estimated by minimizing the quantile-weighted absolute value of errors:

$$ (15) \qquad \hat{\theta}_{\tau} = \arg \min_{\theta_{\tau} \in \mathbb{R}^s} \sum_{t=1}^{T_q-h} (\tau \cdot \mathbf{1}(r_{t+h} \ge p_t \theta_{\tau}) |r_{t+h} - p_t \theta_{\tau}| + (1-\tau) \cdot \mathbf{1}(r_{t+h} < p_t \theta_{\tau}) |r_{t+h} - p_t \theta_{\tau}|), $$

where $\mathbf{1}(\cdot)$ is the indicator function. The predicted quantile conditional on $p_t$ is:

$$ (16) \qquad \hat{Q}_{r_{t+h}}(\tau|p_t) = p_t \hat{\theta}_{\tau}. $$

We estimate predictive quantile regressions for the 1-day ahead ($h=1$) distribution of S&P 500 excess returns over the full sample from Jan. 2, 2004 to Dec. 31, 2018. The regressions are estimated for the 5th through the 95th percentiles in 10-percentage-point increments, and for the median. We report the estimated coefficients and their standard deviations for all quantiles in Table 5.

We first focus on the differential contribution of $\Delta_Q^{\mathrm{P}} ES$ and $ES^{\mathrm{P}}$ for predicting the distribution of excess market returns. The statistical significance of the estimated coefficients indicates that an increase in risk ($ES^{\mathrm{P}}$) leads to a larger probability of observing both extreme negative and positive market returns, whereas an increase in the aversion to downside risk ($\Delta_Q^{\mathrm{P}} ES$) shifts the quantiles around the median and the whole right tail toward more positive values. These findings can be interpreted as follows: A positive shock in expected shortfall means a more volatile market, such that it is usually followed by either a large decrease or increase of the S&P 500 index. These extreme effects cancel out when predicting directly the market returns, such that $ES^{\mathrm{P}}$ is insignificant in Table 2. In contrast, a positive shock in the tail risk premium signals that investors are more averse to extreme negative outcomes, requiring a higher compensation to hold the market. This is reflected in the positive effect of $\Delta_Q^{\mathrm{P}} ES$ on essentially all quantiles of the market return distribution. Such unambiguous effect translates to the significant positive relation between our tail premium and future market returns observed in Table 2.

Table 5 also helps understand why VRP and RV lack predictive power for future excess market returns. A decrease in the VRP (i.e., an increase in the compensation required by investors to bear variance risk) leads to a positive shift
<caption>TABLE 5
Predictive Quantile Regressions: S&P 500 Excess Returns</caption>

Table 5 reports quantile regression coefficients and their standard deviations (in parentheses) of quantile regressions predicting the distribution of the excess returns on the S&P 500 index at the 1-day horizon. The sample is from 2004-01-02 to 2018-12-31. All reported coefficients and standard errors are rounded to 2 decimal places. Those significant at the 0.05 confidence level are printed in bold and those significant at the 0.01 level are additionally highlighted with a *. Those significant at the 0.01 level are additionally marked with a *. The standard errors are computed by pairwise bootstrap with the use of the quantreg 5.83 (Koenker (2013)) package for R 4.0.3. The R¹ goodness-of-fit measure is calculated as in Koenker and Machado (1999). ES<sup>P</sup> is the nonparametric estimate of realized (physical) expected shortfall of intra-day S&P 500 returns before the close, on the day when each return calculation commences, ES<sup>P</sup><sub>t</sub> = -E<sub>t</sub>|R<sub>it</sub>|R<sub>it</sub> ≤ F<sub>R<sub>i</sub></sub><sup>-1</sup>(0.2). Our tail risk premium measure, Δ<sub>Q</sub><sup>P</sup>ES, is the difference between the risk-neutral and the physical ES, RV is the realized variance of intra-day returns on the S&P 500 index. The daily value of the VRP is defined as VRP = RV × 365 - (VIX/100)<sup>2</sup>.

<table><thead><tr><th></th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th>8</th><th>9</th><th>10</th><th>11</th></tr></thead><tbody><tr><td>Quantile:</td><td>0.05</td><td>0.15</td><td>0.25</td><td>0.35</td><td>0.45</td><td>0.5</td><td>0.55</td><td>0.65</td><td>0.75</td><td>0.85</td><td>0.95</td></tr><tr><td>Δ<sup>P</sup><sub>Q</sub>ES</td><td>1.32<br>(4.81)</td><td>4.05<br>(4.81)</td><td>4.05<br>(2.48)</td><td><strong>5.99*</strong><br>(2.24)</td><td><strong>7.39*</strong><br>(2.21)</td><td><strong>7.65*</strong><br>(2.14)</td><td><strong>8.05*</strong><br>(2.18)</td><td><strong>7.81*</strong><br>(2.39)</td><td><strong>7.41</strong><br>(2.89)</td><td><strong>10.83</strong><br>(4.40)</td><td><strong>12.27*</strong><br>(3.03)</td></tr><tr><td>ES</td><td><strong>-9.90*</strong><br>(2.22)</td><td><strong>-6.40*</strong><br>(1.65)</td><td><strong>-4.31*</strong><br>(1.18)</td><td>-1.10<br>(1.17)</td><td>0.00<br>(0.96)</td><td>-0.05<br>(1.02)</td><td>0.66<br>(0.95)</td><td>1.11<br>(1.04)</td><td><strong>3.12*</strong><br>(1.15)</td><td><strong>4.20*</strong><br>(1.18)</td><td><strong>7.39*</strong><br>(1.46)</td></tr><tr><td>VRP</td><td><strong>34.43*</strong><br>(11.32)</td><td><strong>19.63</strong><br>(9.21)</td><td>8.04<br>(7.89)</td><td>1.96<br>(5.81)</td><td>-4.94<br>(5.24)</td><td>-8.65<br>(4.66)</td><td><strong>-11.02</strong><br>(5.46)</td><td><strong>-14.84</strong><br>(6.65)</td><td><strong>-22.85*</strong><br>(8.83)</td><td><strong>-35.73*</strong><br>(8.10)</td><td><strong>-53.82*</strong><br>(7.94)</td></tr><tr><td>RV</td><td>-25.57<br>(15.07)</td><td>-17.39<br>(13.98)</td><td>-5.65<br>(9.94)</td><td>-10.83<br>(8.08)</td><td>-5.26<br>(8.59)</td><td>1.77<br>(8.99)</td><td>2.23<br>(8.56)</td><td>12.79<br>(10.59)</td><td>16.46<br>(9.01)</td><td><strong>24.11*</strong><br>(9.02)</td><td><strong>31.67*</strong><br>(10.02)</td></tr><tr><td>Constant</td><td><strong>0.00</strong><br>(0.00)</td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td><td><strong>0.00</strong><br>(0.00)</td><td><strong>0.00*</strong><br>(0.00)</td></tr><tr><td>R<sup>1</sup>(%)</td><td>21.55</td><td>9.36</td><td>4.04</td><td>1.45</td><td>0.56</td><td>0.63</td><td>1.09</td><td>2.85</td><td>6.16</td><td>13.45</td><td>29.82</td></tr></tbody></table>

in right tail quantiles, but also to a negative shift in the left tail quantiles. Similarly, RV has a positive (negative) effect on the right (left) tail quantiles. This contributes to make the predictive relation with respect to excess market returns weak, as effects cancel out. In contrast, our tail premium measure has a positive effect on all quantiles of the S&P 500 return distribution. This reinforces the predominant role that aversion to downside risk, as measured by Δ<sup>P</sup><sub>Q</sub>ES, plays on explaining future market returns at high frequency. In Section III.D.2, we evaluate the quality of the estimated conditional market return distributions through an out-of-sample study of conditional interval forecasts.

## 2. Out-of-Sample Forecasts of the S&P 500 Return Distribution

To assess the out-of-sample predictive power of the different models for various parts of the return distribution, we rely on the framework developed by Christoffersen (1998) to evaluate conditional interval forecasts. Namely, we consider the likelihood ratio tests of interval forecast conditional coverage (CC), which are comprised of the joint tests of interval forecast error independence (ID) and unconditional coverage (UC). For model evaluation, we split our data into the estimation sample which contains 75% of the data (2,820 observations starting on Jan. 2, 2004 and ending on Apr. 7, 2015), and the evaluation sample (the remaining 940 observations ending on Dec. 31, 2018).

Table 6 reports the results. For all intervals as previously defined, we report the p-values of the CC, ID, and UC tests for several conditioning tests based on different sets of predictors. The predictor sets are: i) ES<sup>P</sup>, ii) Δ<sup>P</sup><sub>Q</sub>ES, iii) ES<sup>P</sup> and Δ<sup>P</sup><sub>Q</sub>ES, iv) VRP, v) VRP and ES<sup>P</sup>, vi) VRP and Δ<sup>P</sup><sub>Q</sub>ES, and vii) VRP, ES<sup>P</sup>, and
<caption>TABLE 6
Out-of-Sample Interval Forecast Tests of S&P 500 Excess Return Distribution Models</caption>

Table 6 reports the results of evaluating out-of-sample interval forecasts for the excess returns on the S&P 500 index obtained with different quantile predictive model specifications. Each row contains the p-values of the Christoffersen (1998) tests of interval forecast conditional coverage (CC), interval forecast error independence (ID), and unconditional coverage (UC) for the quantile interval indicated in the first column. The row denoted "joint" contains the result of a joint evaluation of interval forecasts which follows Section IV.B in Christoffersen (1998) and aggregates the interval forecasts to six intervals spanning the quantiles (0.0 to 0.05), (0.05 to 0.25), (0.25 to 0.5), (0.5 to 0.75), (0.75 to 0.95], and (0.95 to 1.00). For model evaluation purposes, we split our data into the estimation sample which contains 75% of the data (2,820 observations starting on Jan. 2, 2004 and ending on Apr. 7, 2015), and the evaluation sample (the remaining 940 observations ending on Dec. 31, 2018). The p-values which indicate the rejection of the null hypotheses at the 0.05 level are printed in bold face. Those that indicate rejections at the 0.01 level are further marked with a * . All model specifications contain a constant term and the regressor set indicated on the label. ES<sup>D</sup> is the nonparametric estimate of realized (physical) expected shortfall of intra-day S&P 500 returns before the close, on the day when each return calculation commences, ES<sup>D</sup> = -E<sub>t</sub> |R<sub>t</sub>|R<sub>t-1</sub> ≤ F<sub>t-1</sub><sup>-1</sup> (0.2). Our tail risk premium measure, Δ<sub>t</sub><sup>D</sup>ES, is the difference between the risk-neutral and the physical ES. The daily value of the VRP is defined as VRP = RV × 365 - (VIX/100)<sup>2</sup>. "Both" indicates that the predictive model contains both the ES<sup>D</sup> and Δ<sub>t</sub><sup>D</sup>ES terms.

<table><thead><tr><th rowspan="2">Interval</th><th colspan="3">ES</th><th colspan="3">Δ<sup>R<sub>g</sub></sup><sub>t</sub>ES</th><th colspan="3">Both</th><th colspan="3">VRP</th><th colspan="3">VRP+ES</th><th colspan="3">VRP+Δ<sup>R<sub>g</sub></sup><sub>t</sub>ES</th><th colspan="3">VRP+Both</th></tr><tr><th>CC</th><th>ID</th><th>UC</th><th>CC</th><th>ID</th><th>UC</th><th>CC</th><th>ID</th><th>UC</th><th>CC</th><th>ID</th><th>UC</th><th>CC</th><th>ID</th><th>UC</th><th>CC</th><th>ID</th><th>UC</th><th>CC</th><th>ID</th><th>UC</th></tr></thead><tbody><tr><td>Joint</td><td><strong>0.05</strong></td><td>0.25</td><td><strong>0.01</strong></td><td><strong>0.00*</strong></td><td>0.07</td><td><strong>0.00*</strong></td><td>0.19</td><td>0.80</td><td><strong>0.00*</strong></td><td><strong>0.00*</strong></td><td><strong>0.00*</strong></td><td><strong>0.00*</strong></td><td>0.12</td><td>0.65</td><td><strong>0.00*</strong></td><td><strong>0.00*</strong></td><td>0.09</td><td><strong>0.00*</strong></td><td>0.66</td><td>0.80</td><td>0.21</td></tr><tr><td>0.0 – 0.05</td><td>0.10</td><td>0.08</td><td>0.20</td><td><strong>0.00*</strong></td><td><strong>0.02</strong></td><td><strong>0.01*</strong></td><td>0.25</td><td>0.28</td><td>0.20</td><td><strong>0.00*</strong></td><td><strong>0.00*</strong></td><td><strong>0.02</strong></td><td>0.70</td><td>0.49</td><td>0.62</td><td><strong>0.00*</strong></td><td>0.06</td><td><strong>0.00*</strong></td><td>0.41</td><td>0.36</td><td>0.33</td></tr><tr><td>0.05 – 0.15</td><td>0.93</td><td>0.72</td><td>0.87</td><td><strong>0.00*</strong></td><td>0.05</td><td><strong>0.00*</strong></td><td>0.80</td><td>0.51</td><td>0.96</td><td><strong>0.00*</strong></td><td><strong>0.03</strong></td><td><strong>0.00*</strong></td><td>0.75</td><td>0.64</td><td>0.54</td><td><strong>0.00*</strong></td><td><strong>0.02</strong></td><td><strong>0.00*</strong></td><td>0.95</td><td>0.76</td><td>0.96</td></tr><tr><td>0.15 – 0.25</td><td>0.99</td><td>0.99</td><td>0.87</td><td>0.14</td><td>0.20</td><td>0.13</td><td>0.36</td><td>0.33</td><td>0.29</td><td>0.07</td><td><strong>0.04</strong></td><td>0.29</td><td>0.96</td><td>0.93</td><td>0.78</td><td>0.08</td><td>0.10</td><td>0.13</td><td>0.96</td><td>0.93</td><td>0.78</td></tr><tr><td>0.25 – 0.35</td><td><strong>0.01*</strong></td><td>0.24</td><td><strong>0.00*</strong></td><td>0.17</td><td>0.98</td><td>0.06</td><td><strong>0.03</strong></td><td>0.58</td><td><strong>0.01*</strong></td><td>0.23</td><td>0.66</td><td>0.10</td><td>0.32</td><td>0.51</td><td>0.18</td><td>0.71</td><td>0.95</td><td>0.42</td><td>0.93</td><td>0.99</td><td>0.70</td></tr><tr><td>0.35 – 0.45</td><td>0.91</td><td>0.66</td><td>0.96</td><td>0.00*</td><td>0.87</td><td>0.00*</td><td>0.39</td><td>0.57</td><td>0.22</td><td><strong>0.00*</strong></td><td>0.80</td><td><strong>0.00*</strong></td><td>0.56</td><td>0.42</td><td>0.47</td><td><strong>0.00*</strong></td><td>0.57</td><td><strong>0.00*</strong></td><td>0.76</td><td>0.83</td><td>0.48</td></tr><tr><td>0.45 – 0.5</td><td>0.69</td><td>0.94</td><td>0.39</td><td><strong>0.02</strong></td><td>0.17</td><td><strong>0.01</strong></td><td>0.33</td><td>0.71</td><td>0.15</td><td><strong>0.03</strong></td><td>0.21</td><td><strong>0.02</strong></td><td><strong>0.04</strong></td><td>0.85</td><td><strong>0.01</strong></td><td>0.25</td><td>0.39</td><td>0.15</td><td>0.32</td><td>0.42</td><td>0.20</td></tr><tr><td>0.5 – 0.55</td><td>0.39</td><td>0.46</td><td>0.25</td><td>0.06</td><td>0.52</td><td><strong>0.02</strong></td><td>0.10</td><td>0.07</td><td>0.26</td><td>0.08</td><td>0.96</td><td><strong>0.02</strong></td><td>0.61</td><td>0.45</td><td>0.52</td><td><strong>0.00*</strong></td><td>0.75</td><td><strong>0.00*</strong></td><td>0.12</td><td><strong>0.05</strong></td><td>0.62</td></tr><tr><td>0.55 – 0.65</td><td>0.23</td><td>0.42</td><td>0.16</td><td>0.88</td><td>0.75</td><td>0.62</td><td>0.44</td><td>0.94</td><td>0.24</td><td>0.38</td><td>0.73</td><td>0.15</td><td>0.55</td><td>0.33</td><td>0.70</td><td>0.28</td><td>0.14</td><td>0.48</td><td>0.71</td><td>0.94</td><td>0.41</td></tr><tr><td>0.65 – 0.75</td><td><strong>0.02</strong></td><td>0.24</td><td><strong>0.01</strong></td><td>0.68</td><td>0.74</td><td>0.41</td><td><strong>0.03</strong></td><td><strong>0.02</strong></td><td>0.16</td><td>0.71</td><td>0.94</td><td>0.41</td><td><strong>0.01*</strong></td><td>0.37</td><td><strong>0.00*</strong></td><td>0.54</td><td>0.28</td><td>0.78</td><td>0.30</td><td>0.31</td><td>0.29</td></tr><tr><td>0.75 – 0.85</td><td>0.95</td><td>0.76</td><td>0.96</td><td>0.77</td><td>0.59</td><td>0.62</td><td>0.83</td><td>0.94</td><td>0.54</td><td>0.64</td><td>0.65</td><td>0.41</td><td>0.39</td><td>0.17</td><td>0.87</td><td>0.68</td><td>0.74</td><td>0.41</td><td>0.24</td><td>0.44</td><td>0.13</td></tr><tr><td>0.85 – 0.95</td><td>0.56</td><td>0.83</td><td>0.29</td><td>0.29</td><td>0.29</td><td>0.24</td><td>0.87</td><td>0.65</td><td>0.78</td><td>0.33</td><td>0.45</td><td>0.20</td><td>0.59</td><td>0.45</td><td>0.47</td><td>0.42</td><td>0.26</td><td>0.47</td><td>0.25</td><td>0.33</td><td>0.18</td></tr><tr><td>0.95 – 1.0</td><td>0.05</td><td>0.24</td><td><strong>0.03</strong></td><td><strong>0.00*</strong></td><td>0.09</td><td><strong>0.00*</strong></td><td>0.19</td><td>0.35</td><td>0.12</td><td><strong>0.01*</strong></td><td>0.27</td><td><strong>0.00*</strong></td><td><strong>0.01*</strong></td><td>0.62</td><td><strong>0.00*</strong></td><td><strong>0.00*</strong></td><td><strong>0.02</strong></td><td><strong>0.00*</strong></td><td>0.21</td><td>0.71</td><td>0.09</td></tr></tbody></table>
$\Delta_Q^P$ES.<fnref n="26" /> We also conduct a joint evaluation of interval forecasts that aggregates forecasts to six intervals spanning the quantiles (0.0 to 0.05], (0.05 to 0.25], (0.25 to 0.5], (0.5 to 0.75], (0.75 to 0.95], and (0.95 to 1.00].

For ES$^P$, the null hypothesis of correct conditional coverage is rejected only for two intervals, while the independence of prediction errors is never rejected. For the joint interval test, CC and UC are rejected. Results for $\Delta_Q^P$ES are similar, but there is less efficiency for intervals covering the left tail. When $\Delta_Q^P$ES is added to ES$^P$ in the column titled “both,” efficiency improves as there is no rejection in the CC joint interval test. In fact, across nearly all intervals, results are supportive of interval forecast efficiency. VRP is the predictor that leads to most rejections of the out-of-sample coverage tests. Even so, the best performance is obtained with the model including the three predictors, for which there are essentially no rejections of CC, ID, and UC in the joint test and across intervals. Overall, the results support that the in-sample predictive power of the estimated quantile models containing our tail premium measure translates to out-of-sample performance.

# IV. Robustness Analysis

Our tail risk premium measure depends on a number of choices regarding the estimation of the RND and the expected shortfall. In this section, we perturb the method to learn about its sensitivity to those choices and compare its performance to the baseline case analyzed in Section III.

## A. Risk-Neutralization with the Hellinger Discrepancy

Our baseline tail premium measure relies on the particular value of $\gamma = -3$ for the parameter indexing the discrepancy in the Cressie–Read family used for estimating the RND. As explained in Section II.C, this value is chosen to tilt the distribution of market returns toward large negative returns, as an investor with high aversion to downside risk does. In this subsection, we check whether this choice is essential to our findings by investigating if a less drastic risk-adjustment, implied by the Hellinger discrepancy ($\gamma = -\frac{1}{2}$), produces similar results. This discrepancy is still consistent with aversion to downside risk (i.e., large negative returns are still overweighted), but gives less probability mass to the left tail compared with $\gamma = -3$.

In Appendix D.1, we plot the time-series of the $\Delta_Q^P$ES obtained using the Hellinger discrepancy and the associated Lagrange multiplier in Figure D.1. The dynamics of the new $\Delta_Q^P$ES is very similar to that obtained for $\gamma = -3$ in Figure 1. The main difference between the two is that the new $\Delta_Q^P$ES attains smaller values, which is natural as the expected shortfall under a RND giving less probability mass to the left tail is smaller. The Lagrange multipliers, on the other hand, are larger (in absolute value) for $\gamma = -\frac{1}{2}$. This is because the investor associated with the portfolio optimization problem buys a larger amount of the risky asset (the market index) as her aversion to downside risk is smaller. Table D.1 reproduces the main predictability analysis for the equity premium and variance risk premium using the

<footnote n="26">²⁶Due to the fact that VRP has a better performance than RV in the previous quantile regressions, we decided to drop RV. Results including RV are similar.</footnote>
new $\Delta_Q^P$ES. The results are very similar to the baseline ones for $\gamma = -3$, where the tail risk premium measure has strong predictive power across all regression specifications. This confirms that our findings are robust to a change of discrepancy measure within the Cressie-Read family.

## B. Risk-Neutralization Without Equity Premium Restrictions

Our baseline estimation of the RND imposes a non-negativity constraint for the conditional equity premium (i.e., for the average of the high-frequency excess market returns on day $t$). This is to prevent the marginal investor solving the dual portfolio problem from shorting the market index, which would imply that marginal utility is low for negative market return states, such that $ES^Q$ would be below $ES^P$. In this subsection, we drop this restriction and, after re-calculating $\Delta_Q^P$ES, conclude that this has no material impact on our predictability results.

In Appendix D.2, we plot the time-series of the $\Delta_Q^P$ES obtained without imposing the restriction on the equity premium and the associated Lagrange multiplier in Figure D.2 in the Appendix D. The immediate consequence of allowing for a negative equity premium is that the estimates of the Lagrange multiplier turn positive on a significant part of the sample (meaning that the investor sells the market index). This, in turn, makes the implied RND to put higher probability weights in states of nature where the index has large positive returns (which represent negative returns of the investor's portfolio). As a consequence, ES becomes smaller under $Q$ than under $P$ during those dates, as visible in the central panel of Figure D.2. This difference notwithstanding, a visual comparison with our main Figure 1 uncovers similar dynamics of $\Delta_Q^P$ES with and without restrictions.

Table D.2 reproduces the main predictability analysis for the equity premium and variance risk premium using the $\Delta_Q^P$ES without restrictions. A comparison with the baseline results in Tables 2 and 4 suggests that removing the positive equity premium constraint renders the estimated $\Delta_Q^P$ES noisier, as the associated $t$-statistics and $R^2$'s of the 1-day horizon predictive regressions become lower. Even so, the tail risk premium measure still retains statistical significance and is the strongest predictor of aggregate market risk premiums among the variables we consider. We therefore conclude that our findings are robust to removing the equity premium constraint.

## C. Lower Expected Shortfall Threshold

In our baseline analysis, we set $\alpha = 0.2$ as the confidence level of the $\alpha$-quantile of the return distribution ($s_\alpha$) for calculating the expected shortfalls in equations (2) and (3). In this subsection, we examine the sensitivity of our results to this choice by setting $\alpha = 0.1$, meaning the expected shortfall threshold is farther in the left tail of the returns and there are less return observations to extract information from.

The results are collected in Appendix D.3. First, decreasing $\alpha$ mechanically translates to an increase in the $ES^P$ measure, which is evident when comparing the left graphs of Figure D.3 in the Appendix D and Figure 1. Somewhat less intuitively, we also observe changes to the estimates of the tail risk premium comparing the central panels of the aforementioned figures. With $\alpha = 0.1$, $\Delta_Q^P$ES becomes
significantly higher than with $\alpha = 0.2$ in periods of market distress. This is compatible with heterogeneous investors' attitudes toward downside risk observed under different extreme quantiles of the distribution of returns.<fnref n="27" />

Table D.3 contains the main predictability results for risk premiums using the expected shortfall and tail premium measures estimated with $\alpha = 0.1$. We do not observe any changes relative to our baseline analysis regarding the predictive power of ES$^P$ and $\Delta_Q^P$ES for the equity and variance risk premiums. That is, aversion to downside risk (as captured by our tail premium measure) has strong predictive power, while downside risk itself (as captured by the expected shortfall) remains a statistically insignificant predictor.

## D. Including Option Returns in the Estimation of the Risk-Neutral Distribution

Finally, our baseline specification to compute the risk-neutral expected shortfall relies on a nonparametric RND extracted solely from high-frequency market returns. However, in principle, the risk-neutral measure can be estimated using information from any security for which high-frequency return data are available. In particular, given that options are informative about higher-order risks of the underlying asset, a natural question is whether adding index option returns to the estimation of the RND helps the resulting tail risk premium measure better predict the equity premium. We use high-frequency data on S&P 500 options to test this possibility.

Using our high-frequency option data, we calculate the index option returns as follows: In each 5-minute window, we sort OTM call (put) options into five portfolios based on the absolute value of their Black-Scholes delta, denoted as $\Delta_{01}$ (deep OTM options with absolute deltas ranging from 0.0 to 0.1) through $\Delta_{05}$ (close to at-the-money (ATM) options with absolute deltas ranging from 0.4 to 0.5). Next, we calculate the mid price for each option. In the following step, we match the observations in a given 5-minute window to those in the subsequent window, and we discard options with no match. We further drop from the return calculation the options whose mid prices did not change between two observation windows. Finally, from the remaining data, we calculate the equal-weighted return on each option portfolio.

We estimate the RND by solving the minimum discrepancy problem in (5) including as basis assets the S&P 500 market index and one option portfolio at a time.<fnref n="28" /> The excess market return predictability results for the tail premium are reported in Table 7, where each column corresponds to the estimation including option returns grouped by the indicated $\Delta$. We include ES$^P$ as control and consider the 1-day horizon. The tail risk premium $\Delta_Q^P$ES is statistically significant only for the call groups $\Delta_{01}$ and $\Delta_{03}$. In terms of $R^2$, this model performs worse than the equivalent one without options in Table 2 (0.63 and 0.65 for $\Delta_{01}$ and $\Delta_{03}$,

<footnote n="27">²⁷See Castro and Galvao (2019) for the development of a rational dynamic model based on quantile utility preferences.</footnote>

<footnote n="28">²⁸We refrain from including all option portfolios at once in the estimation as the number of basis assets would be too large relative to the number of high-frequency observations, which can lead to unstable results.</footnote>
<caption>TABLE 7
Predictive Regressions: S&P 500 Excess Returns with Option-Implied Tail Premium</caption>

Table 7 reports regression coefficients and their standard deviations (in parentheses) for predictive regressions of the close-to-close S&P 500 excess returns at the 1-day horizon with Δ<sub>Q</sub> ES calculated with the use of risk-neutral probabilities obtained from the joint risk-neutralization of high-frequency returns on the S&P 500 index and on options thereon, with options grouped into five portfolios based on their type and Δ. Δ01 corresponds to deep out of the money options while Δ05 corresponds to at the money options. All reported coefficients and standard errors are rounded to 2 decimal places. Those significant at the 0.05 confidence level are printed in bold and those significant at the 0.01 level are additionally highlighted with a *. We report Andrews (1991) standard errors calculated with the use of the sandwich 3.0.0 package for R 4.0.3 (Zeileis et al. (2020)). ES<sup>P</sup> is the nonparametric estimate of realized (physical) expected shortfall of intra-day S&P 500 returns before the close, on the day when each return calculation commences. ES<sup>S</sup> = -E<sub>t</sub>|R<sub>t</sub>|R<sub>t-1</sub> ≤ F<sub>R<sub>t</sub></sub> (0.2)]. Our tail risk premium measure, Δ<sub>Q</sub><sup>S</sup>ES, is the difference between the risk-neutral and the physical ES.

<table><thead><tr><th rowspan="3"></th><th colspan="5">Call Options</th><th colspan="5">Put Options</th></tr><tr><th>Δ01</th><th>Δ02</th><th>Δ03</th><th>Δ04</th><th>Δ05</th><th>Δ01</th><th>Δ02</th><th>Δ03</th><th>Δ04</th><th>Δ05</th></tr><tr><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th>8</th><th>9</th><th>10</th></tr></thead><tbody><tr><th>Δ<sub>Q</sub><sup>P</sup>ES</th><td>6.85*<br>(2.46)</td><td>3.87<br>(2.22)</td><td>6.09*<br>(1.90)</td><td>2.49<br>(1.62)</td><td>2.22<br>(1.68)</td><td>4.21<br>(2.93)</td><td>3.37<br>(2.71)</td><td>3.09<br>(2.43)</td><td>1.98<br>(2.78)</td><td>2.67<br>(2.73)</td></tr><tr><th>ES<sup>P</sup></th><td>-0.07<br>(0.78)</td><td>0.02<br>(0.77)</td><td>-0.04<br>(0.79)</td><td>0.10<br>(0.81)</td><td>0.04<br>(0.80)</td><td>0.00<br>(0.82)</td><td>-0.02<br>(0.84)</td><td>0.00<br>(0.81)</td><td>0.02<br>(0.86)</td><td>0.04<br>(0.84)</td></tr><tr><th>Constant</th><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td></tr><tr><th>R<sup>2</sup> (adj %)</th><td>0.63</td><td>0.25</td><td>0.65</td><td>0.14</td><td>0.09</td><td>0.25</td><td>0.17</td><td>0.15</td><td>0.06</td><td>0.16</td></tr></tbody></table>

respectively, against 0.73). For all other option portfolios, there is little predictive power coming from the associated tail risk premium.

The previous results can be rationalized by the following. When we include option and index returns together in the estimation of the risk-neutral measure, we lose the interpretation of how the risk-neutral measure acts on index returns. When the S&P 500 index is the unique asset, the RND overweights market negative returns and underweights positive ones, provided that the equity premium is positive. This is such that the risk-neutral expected shortfall appropriately reflects downside risk of the market index. In contrast, when the index and an option portfolio are basis assets, the RND overweights negative returns coming from the optimal solution of the dual portfolio problem (i.e., a linear combination of index and option returns), which are not necessarily negative realizations of the market. Table 7 reports that this hurts performance, giving support to our baseline specification using only market returns for the estimation.

## V. Conclusion

In this article, we propose a new method to compute tail risk premium at high frequency using solely intra-day market returns and a risk-neutralization algorithm. Empirically, we show that our tail risk premium measure has strong predictive power for aggregate market and cross-sectional risk premiums at short horizons. Such predictability is robust to controlling for established measures of risk and risk premiums and to different specifications of our measure. Our findings provide new high-frequency evidence that aversion to downside risk is fundamental to explain asset pricing behavior. A natural extension of our method would be to use a cross section of high-frequency returns to investigate to what extent this would improve the model forecasting ability.
# Appendix

## Appendix A. Extreme Value Theory, Expected Shortfall, and the Tail Shape Parameter

Extreme value theory (EVT) shows that once we are looking far in the tail of a random variable $X$ representing losses (negative of the returns) with a given distribution $F$, at values larger than an exogenous threshold $u$, the conditional distribution function $F(X \le x | x \ge u)$ can be well approximated by a generalized Pareto distribution (GPD) $G(\xi, \beta)$.<fnref n="29" /> We use this result to identify a direct link between our ES measures of risk defined under $\mathbb{P}_t$ and $\mathbb{Q}_t$ and the corresponding shape parameters $\xi_t^P$ and $\xi_t^Q$ that determine the thickness of the GPD's capturing the behavior of the tails of $\mathbb{P}_t$ and $\mathbb{Q}_t$.

The cumulative GPD distribution function is given by $G_{\xi, \beta}(x) = 1 - (1 + \frac{\xi x}{\beta})^{-\frac{1}{\beta}}$, $x \ge 0$, $\beta > 0$, $0 < \xi < 1$, where $\beta$ is a scaling parameter and $\xi$ the shape parameter. The larger the $\xi$, the thicker the tail is. It is well-known that given a sample of returns $\mathcal{R} = \{R_1, \dots, R_T\}$ whose conditional tail distribution is approximated by a GPD and a fixed confidence level $\alpha$, there is a direct relation between the expected shortfall measure with confidence level $\alpha$, ES$^\alpha(\mathcal{R})$, the value-at-risk based on the same confidence level, VaR$^\alpha(\mathcal{R})$, and the shape parameter $\xi$:

$$ (A-1) \qquad \mathrm{ES}^\alpha(\mathcal{R}) = \frac{\mathrm{VaR}^\alpha(\mathcal{R})}{1 - \xi} + \frac{\beta - \xi u}{1 - \xi}. $$

Our tail risk measures ($\mathrm{ES}_t^P$, $\mathrm{ES}_t^Q$) and tail risk premium ($\Delta_Q^P \mathrm{ES}_t$) are calculated on a daily frequency. Following Bollerslev and Todorov (2014), we allow the tail's shape parameters ($\xi_t^P$, $\xi_t^Q$) to be time-varying. Assuming for simplicity that the threshold $u$ and scale $\beta$ parameters are time-invariant and common to both physical and risk-neutral distributions, we can invert (A-1) to obtain the time-varying shape parameters as hyperbolic functions of our ES measures:

$$ (A-2) \qquad \xi_t^P = 1 + \frac{\mathrm{VaR}_t^{P, \alpha} + \beta - u}{u - \mathrm{ES}_t^{P, \alpha}}, $$

$$ (A-3) \qquad \xi_t^Q = 1 + \frac{\mathrm{VaR}_t^{Q, \alpha} + \beta - u}{u - \mathrm{ES}_t^{Q, \alpha}}. $$

Equation (A-1) also implies that our tail risk premium is a continuous function $H(\cdot, \cdot)$ of the time-varying tail shape parameters of both physical and risk-neutral distributions:

$$ (A-4) \qquad \Delta_Q^P \mathrm{ES}_t = H(\xi_t^P, \xi_t^Q). $$

Bollerslev and Todorov (2014) and Bollerslev et al. (2015) propose the left jump tail variance (LTV), a risk-neutral jump tail risk measure for the S&P 500 returns. They build a new dynamic model for the S&P 500 prices under the risk neutral measure $\mathbb{Q}$, whose dynamics is decomposed into a continuous stochastic volatility component and a

<footnote n="29">²⁹This threshold-exceedance distribution, or distribution of the tail, is usually identified based on a sample of observed random variables with the same distribution $F$. See Chapter 7 in McNeil, Rudiger, and Paul (2005) for more details on EVT.</footnote>
jump component with time-varying stochastic jump-intensity. The key novelty with respect to the previous literature is the use of EVT to model the tails (both left and right) of the jump-intensity measure, suggesting that they follow Frechet distributions with time-varying shape/decay parameters $F(x) = x^{-\alpha_t}, \alpha_t > 0$. They adopt the inverse of this shape parameter, $\xi_t = \frac{1}{\alpha_t}$, as a measure of tail risk. Since the Frechet distribution is directly comparable to the GPD when the shape parameter $\xi$ is positive (i.e., $\xi_t > 0$), equations (A-3) and (A-4) directly connect our risk-neutral expected shortfall (ES$^Q$) and tail risk premium ($\Delta_Q^P$ES) measures to LTV.

Moreover, since ES$_t^Q$ is an invertible function of $\xi_t^Q$ and $\Delta_Q^P$ES$_t$ is linear in ES$_t^Q$, LTV, and our tail measures, at least from a theoretical point of view, induce the same probability filtration (i.e., the same information sets) along the time dimension. However, they naturally differ in practice in terms of their estimation procedure and data used. LTV is estimated with the use of options. It represents the expected (risk-neutral) return volatility that stems from large negative price jumps. To identify and separate the diffusive part and the jump component of the return process, Bollerslev et al. (2015) use index options with maturity between 6 and 31 trading days, reflecting market expectations over these horizons. In contrast, our tail risk premium measure relies on high-frequency market return data, containing information completely conditional on day $t$.

## Appendix B. Definitions of Risk Premiums Predictors

ES$^P$: Estimate of the expected shortfall $\Delta_Q^P$ES$_t = -E_t[R_{it} | R_{it} \le F_{R_t}^{-1}(0.2)]$ obtained from the empirical distribution of 5-minute returns on the S&P 500 index during market opening hours on day $t$. See equation (2).

$\Delta_Q^P$ES: Difference between ES$^P$ and its $Q$-measure counterpart; the change of measure is described in equation (1) and $\Delta_Q^P$ES is defined in equation (4).

$\Delta_Q^P$ES$^\perp$: Residual from the full-sample regression of $\Delta_Q^P$ES on VRP.

VRP: $RV - (VIX)^2/100/365$.

RV: Realized variance of 5-minute intraday returns on the S&P 500 index during market opening hours calculated as $\sum_{j=1}^{N_t} (\log r_{jt})^2$.

LTV: Left tail variance the S&P 500 index returns as defined in Bollerslev et al. (2015) obtained from www.tailindex.com.

IV: Integrated quadratic variation of 5-minute intraday returns on the S&P 500 index during market opening hours, the continuous component of RV, estimated as in Mancini and Gobbi (2012).

IV: The jump component of realized variance, calculated as $\max\{RV - IV, 0\}$
Appendix C. Supplementary Results

<caption>TABLE C.1
Predictive Regressions: Medium- and Long-Term S&P 500 Excess Returns</caption>

Table C.1 reports regression coefficients and their standard deviations (in parentheses) of predictive regressions of the close-to-close S&P 500 excess returns at the 5- and 21-day horizons. All reported coefficients and standard errors are rounded to 2 decimal places. Those significant at the 0.05 confidence level are printed in bold and those significant at the 0.01 level are additionally highlighted with a *. We calculate standard errors for overlapping observations using the procedure of Britten-Jones, Neuberger, and Nolte (2011). ES<sup>†</sup> is the nonparametric estimate of realized (physical) expected shortfall of intra-day S&P 500 returns before the close, on the day when each return calculation commences. ES<sup>†</sup> = -E<sub>t</sub>|R<sub>t</sub>|R<sub>t</sub> ≤ F<sub>t</sub><sup>-1</sup> (0.2). Our tail risk premium measure, Δ<sub>Q</sub><sup>ES</sup>, is the difference between the risk-neutral and the physical ES. VRP is the variance risk premium calculated as the difference between the day's realized variance and the (appropriately scaled) VIX<sup>2</sup> index. Δ<sub>Q</sub><sup>ES</sup><sup>⊥</sup> is the component of the tail risk measure that is orthogonal to the variance risk premium. LTV is the left tail variance of Bollerslev et al. (2015) obtained from www tailindex com. RV is the realized variance of intra-day returns on the S&P 500 index. IV is an estimate of integrated quadratic variation, the continuous component of realized variance, estimated as in Mancini and Gobbi (2012). JV is the jump component of realized variance, calculated as max (RV - IV, 0). RV, IV, and JV are calculated from return data sampled at the 5-minute frequency.

<table><caption>TABLE C.2
Out-of-Sample Predictive R<sup>2</sup> for S&P 500 Excess Returns</caption><thead><tr><th></th><th colspan="2">5 Days</th><th colspan="2">21 Days</th><th colspan="2">5 Days</th><th colspan="2">21 Days</th><th colspan="2">5 Days</th><th colspan="2">21 Days</th><th colspan="2">5 Days</th><th colspan="2">21 Days</th><th colspan="2">5 Days</th><th colspan="2">21 Days</th><th colspan="2">5 Days</th><th colspan="2">21 Days</th><th colspan="2">5 Days</th><th colspan="2">21 Days</th></tr><tr><th></th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th>8</th><th>9</th><th>10</th><th>11</th><th>12</th><th>13</th><th>14</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th>8</th><th>9</th><th>10</th><th>11</th><th>12</th><th>13</th><th>14</th></tr></thead><tbody><tr><td>ES<sup>Q</sup></td><td>0.51<br>(2.88)</td><td>-1.66<br>(10.08)</td><td>0.03<br>(2.77)</td><td>-2.14<br>(9.47)</td><td>0.19<br>(2.64)</td><td>-0.37<br>(9.86)</td><td>-2.61<br>(4.48)</td><td>-5.10<br>(13.96)</td><td>-2.63<br>(4.41)</td><td>-4.98<br>(13.88)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Δ<sub>Q</sub><sup>ES</sup></td><td></td><td></td><td>11.17<br>(8.33)</td><td>11.33<br>(19.69)</td><td>11.02<br>(9.13)</td><td>9.69<br>(21.73)</td><td>11.29<br>(10.60)</td><td>9.16<br>(30.49)</td><td></td><td></td><td>8.82<br>(7.65)</td><td>3.81<br>(18.27)</td><td>8.81<br>(7.25)</td><td>6.73<br>(18.28)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>VRP</td><td></td><td></td><td></td><td></td><td>-2.43<br>(16.95)</td><td>-26.50<br>(42.86)</td><td>-1.95<br>(17.68)</td><td>-27.47<br>(42.73)</td><td>-2.53<br>(17.12)</td><td>-27.73<br>(40.36)</td><td>-26.59<br>(25.60)</td><td>-82.43<br>(82.05)</td><td>-27.06<br>(25.61)</td><td>-80.94<br>(83.73)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Δ<sub>Q</sub><sup>ES</sup><sup>⊥</sup></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>8.69<br>(10.77)</td><td>4.72<br>(31.82)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>RV</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>28.22<br>(39.78)</td><td>50.48<br>(122.31)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>LTV</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>-0.24<br>(0.27)</td><td>-0.48<br>(0.70)</td><td>-0.24<br>(0.27)</td><td>-0.49<br>(0.70)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>JV</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>29.57<br>(45.40)</td><td>1.10<br>(150.93)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>IV</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>28.25<br>(40.45)</td><td>54.15<br>(121.18)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Constant</td><td>0.00<br>(0.00)</td><td>0.01<br>(0.01)</td><td>0.00<br>(0.00)</td><td>0.01<br>(0.01)</td><td>0.00<br>(0.00)</td><td>0.00<br>(0.01)</td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td><td>0.01<br>(0.01)</td><td>0.01<br>(0.00)</td><td>0.00<br>(0.00)</td><td>0.01<br>(0.01)</td><td>0.00<br>(0.00)</td><td>0.01<br>(0.00)</td><td>0.00<br>(0.00)</td><td>0.01<br>(0.00)</td><td>0.00<br>(0.00)</td><td>0.01<br>(0.00)</td><td>0.00<br>(0.00)</td><td>0.01<br>(0.00)</td><td>0.00<br>(0.00)</td><td>0.01<br>(0.00)</td><td>0.00<br>(0.00)</td></tr><tr><td>R<sup>2</sup> (adj<br>%)</td><td>-0.01</td><td>-0.01</td><td>0.32</td><td>0.09</td><td>0.36</td><td>0.47</td><td>0.27</td><td>0.11</td><td>0.15</td><td>0.09</td><td>0.51</td><td>0.60</td><td>0.50</td><td>0.58</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr></tbody></table>

Table C.2 reports the predictive R<sup>2</sup>, calculated as R<sup>2</sup> = 1 - Σ<sub>t</sub>(R<sub>t</sub> - Ŷ<sub>t</sub>)<sup>2</sup>/Σ<sub>t</sub>(R<sub>t</sub>)<sup>2</sup>, where Ŷ<sub>t</sub> is the model's predicted return in period t. (This is in contrast to Table 3, where Ŷ<sub>t</sub> is subtracted from the return in the denominator.) The following model specifications are considered: i) Δ<sub>Q</sub><sup>ES</sup>, our measure of the tail risk premium; ii) Ŷ<sub>t</sub> (constant) (i.e., the historical average excess return); iii) VRP, the variance risk premium; and iv) “both,” containing Δ<sub>Q</sub><sup>ES</sup> and VRP. Predictive regressions are estimated on expanding samples starting in Jan. 2008 (columns 1–3), Jan. 2012 (columns 4–6), and Apr. 2015 (columns 7–9). In the first 2 columns of each sub-panel, we consider univariate predictive models, and in each respective third column, a model containing both predictors. Across the rows of each panel we consider different parameter update frequencies, from every month in the top row to no updates (i.e., the parameters are estimated on data prior to the starting date and never updated) in the final row.

<table><thead><tr><th rowspan="2">Update Freq.</th><th colspan="4">OOS from Jan. 1, 2008</th><th colspan="4">OOS from Jan. 1, 2012</th><th colspan="4">OOS from Apr. 15, 2015</th></tr><tr><th>μ̂</th><th>VRP</th><th>Δ<sub>Q</sub><sup>ES</sup></th><th>Both</th><th>μ̂</th><th>VRP</th><th>Δ<sub>Q</sub><sup>ES</sup></th><th>Both</th><th>μ̂</th><th>VRP</th><th>Δ<sub>Q</sub><sup>ES</sup></th><th>Both</th></tr></thead><tbody><tr><td>1 months</td><td>0.00</td><td>-1.15</td><td>0.10</td><td>-0.35</td><td>0.13</td><td>0.03</td><td>0.83</td><td>0.74</td><td>0.08</td><td>0.02</td><td>1.48</td><td>1.42</td></tr><tr><td>3 months</td><td>0.02</td><td>-1.38</td><td>0.01</td><td>-0.32</td><td>0.24</td><td>0.24</td><td>0.75</td><td>0.76</td><td>0.19</td><td>0.17</td><td>1.65</td><td>1.64</td></tr><tr><td>6 months</td><td>0.01</td><td>0.01</td><td>0.65</td><td>0.61</td><td>0.22</td><td>0.21</td><td>0.73</td><td>0.74</td><td>0.19</td><td>0.17</td><td>1.65</td><td>1.63</td></tr><tr><td>12 months</td><td>0.00</td><td>-0.14</td><td>0.60</td><td>0.32</td><td>0.24</td><td>0.19</td><td>0.52</td><td>0.49</td><td>0.13</td><td>0.10</td><td>1.64</td><td>1.61</td></tr><tr><td>24 months</td><td>0.03</td><td>-0.01</td><td>0.67</td><td>0.51</td><td>0.21</td><td>0.16</td><td>0.47</td><td>0.45</td><td>0.10</td><td>-0.05</td><td>1.63</td><td>1.50</td></tr><tr><td>Never</td><td>0.03</td><td>0.07</td><td>0.80</td><td>0.82</td><td>0.08</td><td>0.00</td><td>0.67</td><td>0.61</td><td>0.06</td><td>-0.12</td><td>1.62</td><td>1.45</td></tr></tbody></table>
<caption>TABLE C.3
Predictive Regressions: Medium- and Long-Term Variance Risk Premium</caption>

Table C.3 reports regression coefficients and their standard deviations (in parentheses) for predictive regressions of the variance risk premium at the 5- and 21-day horizon. All reported coefficients and standard errors are rounded to 2 decimal places. Those significant at the 0.05 confidence level are printed in bold and those significant at the 0.01 level are additionally highlighted with a *. We calculate standard errors for overlapping observations using the procedure of Britten-Jones et al. (2011). We report Andrews (1991) standard errors calculated with the use of the sandwich 3.0.0 package for R 4.0.3 (Zeileis et al. (2020)). ES<sup>o</sup> is the nonparametric estimate of realized (physical) expected shortfall of intra-day S&P 500 returns before the close, on the day when each return calculation commences, ES<sup>o</sup> = -E<sub>t</sub> |R<sub>t</sub>|R<sub>t-1</sub> ≤ F<sub>t-1</sub><sup>o</sup> (0.2). Our tail risk premium measure, Δ<sub>Q</sub><sup>P</sup>ES, is the difference between the risk-neutral and the physical ES. VRP is the variance risk premium calculated as the difference between the day's realized variance and the (appropriately scaled) VIX<sup>2</sup> index. Δ<sub>Q</sub><sup>P</sup>ES<sup>+</sup> is the component of the tail risk measure that is orthogonal to the variance risk premium. LTV is the left tail variance of Bollerslev et al. (2015) obtained from www.tailindex.com. RV is the realized variance of intra-day returns on the S&P 500 index. IV is an estimate of integrated quadratic variation, the continuous component of realized variance, estimated as in Mancini and Gobbi (2012). JV is the jump component of realized variance, calculated as max{RV - IV, 0}. RV, IV, and JV are calculated from return data sampled at the 5-minute frequency.

<table><thead><tr><th></th><th>5 Days</th><th>21 Days</th><th>5 Days</th><th>21 Days</th><th>5 Days</th><th>21 Days</th><th>5 Days</th><th>21 Days</th><th>5 Days</th><th>21 Days</th><th>5 Days</th><th>21 Days</th><th>5 Days</th><th>21 Days</th><th>5 Days</th><th>21 Days</th></tr></thead><tbody><tr><td>Horizon:</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>ES<sup>o</sup></td><td>0.05<br>(0.07)</td><td>-0.16<br>(0.24)</td><td>0.01<br>(0.05)</td><td>-0.22<br>(0.17)</td><td><strong>-0.10</strong><br>(0.04)</td><td><strong>-0.57</strong><sup>*</sup><br>(0.17)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Δ<sub>Q</sub><sup>P</sup>ES</td><td></td><td></td><td><strong>0.84</strong><sup>*</sup><br>(0.27)</td><td><strong>1.43</strong><sup>*</sup><br>(0.51)</td><td><strong>0.93</strong><sup>*</sup><br>(0.23)</td><td><strong>1.75</strong><sup>*</sup><br>(0.53)</td><td><strong>0.80</strong><sup>*</sup><br>(0.26)</td><td>0.95<br>(0.76)</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>VRP</td><td></td><td></td><td></td><td></td><td><strong>1.59</strong><sup>*</sup><br>(0.18)</td><td><strong>5.13</strong><sup>*</sup><br>(0.58)</td><td><strong>1.34</strong><sup>*</sup><br>(0.21)</td><td><strong>3.68</strong><sup>*</sup><br>(0.93)</td><td><strong>1.26</strong><sup>*</sup><br>(0.19)</td><td><strong>3.55</strong><sup>*</sup><br>(0.86)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>Δ<sub>Q</sub><sup>P</sup>ES<sup>⊥</sup></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td><strong>0.85</strong><sup>*</sup><br>(0.27)</td><td>1.20<br>(0.80)</td><td></td><td></td><td></td><td></td><td></td></tr><tr><td>RV</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>-0.22<br>(0.59)</td><td>-1.95<br>(2.06)</td><td></td><td></td><td></td><td></td></tr><tr><td>LTV</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>0.00<br>(0.00)</td><td>-0.01<br>(0.01)</td><td>0.00<br>(0.00)</td><td>-0.01<br>(0.01)</td><td></td></tr><tr><td>JV</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>-1.14<br>(0.59)</td><td>-1.11<br>(2.40)</td><td></td></tr><tr><td>IV</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td>-0.14<br>(0.66)</td><td>-2.03<br>(1.85)</td><td></td></tr><tr><td>Constant</td><td><strong>0.00</strong><sup>*</sup><br>(0.00)</td><td><strong>0.00</strong><sup>*</sup><br>(0.00)</td><td><strong>0.00</strong><sup>*</sup><br>(0.00)</td><td><strong>0.00</strong><sup>*</sup><br>(0.00)</td><td><strong>0.00</strong><sup>*</sup><br>(0.00)</td><td><strong>0.00</strong><sup>*</sup><br>(0.00)</td><td><strong>0.00</strong><sup>*</sup><br>(0.00)</td><td><strong>0.00</strong><sup>*</sup><br>(0.00)</td><td><strong>0.00</strong><sup>*</sup><br>(0.00)</td><td><strong>0.00</strong><sup>*</sup><br>(0.00)</td><td><strong>0.00</strong><sup>*</sup><br>(0.00)</td><td><strong>0.00</strong><sup>*</sup><br>(0.00)</td><td><strong>0.00</strong><sup>*</sup><br>(0.00)</td><td><strong>0.00</strong><sup>*</sup><br>(0.00)</td></tr><tr><td>R<sup>2</sup> (adj %)</td><td>8.68</td><td>8.69</td><td>23.26</td><td>22.53</td><td>32.46</td><td>24.92</td><td>27.55</td><td>19.97</td><td>27.54</td><td>19.95</td><td>32.00</td><td>29.98</td><td>32.21</td><td>31.10</td><td></td></tr></tbody></table>

## Appendix D. Robustness Results

### D.1 Alternative CR Discrepancy

<caption>FIGURE D.1
Time Series of the Tail Measures Implied by the S&P 500 Intraday Data Calculated with the Hellinger Divergence</caption>

Figure D.1 shows the daily time series of Tail Risk, Tail Risk Premium, and Lagrange Multiplier for the Hellinger case with quantile α=0.2 and Cressie Read γ = -½.

<figure><img src="image_8.png" /></figure>
<caption>TABLE D.1
Predictive Regressions: S&P 500 Excess Returns and Variance Risk Premium with Hellinger Discrepancy</caption>

Table D.1 reports regression coefficients and their standard deviations (in parentheses) of predictive regressions of the close-to-close S&P 500 excess returns at the 1-day horizon. All reported coefficients and standard errors are rounded to 2 decimal places. Those significant at the 0.05 confidence level are printed in bold and those significant at the 0.01 level are additionally highlighted with a *. We report Andrews (1991) standard errors calculated with the use of the sandwich 3.0.0 package for R 4.0.3 (Zeileis et al. (2020)). ES<sup>P</sup> is the nonparametric estimate of realized (physical) expected shortfall of intra-day S&P 500 returns before the close, on the day when each return calculation commences, ES<sup>⊥</sup> = -E<sub>t</sub>|R<sub>it</sub>|R<sub>it</sub> ≤ F<sub>Rt</sub><sup>⊥</sup> (0.2). Our tail risk premium measure, Δ<sub>Q</sub><sup>P</sup>ES, is the difference between the risk-neutral and the physical ES. VRP is the variance/risk premium calculated as the difference between the day's realized variance and the (appropriately scaled) VIX<sup>2</sup> index. Δ<sub>Q</sub><sup>P</sup>ES<sup>⊥</sup> is the component of the tail risk measure that is orthogonal to the variance risk premium. LTV is the left tail variance of Bollerslev et al. (2015) obtained from www.tailindex.com. RV is the realized variance of intra-day returns on the S&P 500 index. IV is an estimate of integrated quadratic variation, the continuous component of realized variance, estimated as in Mancini and Gobbi (2012). JV is the jump component of realized variance, calculated as max{RV - IV, 0}. RV, IV, and JV are calculated from return data sampled at the 5-minute frequency.

<table><thead><tr><th></th><th>1 Day</th><th>1 Day</th><th>1 Day</th><th>1 Day</th><th>1 Day</th><th>1 Day</th><th>1 Day</th></tr><tr><th></th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th></tr></thead><tbody><tr><td colspan="8"><strong>Panel A. S&P 500 Excess Returns</strong></td></tr><tr><td>ES<sup>P</sup></td><td>0.17<br>(0.77)</td><td>-0.19<br>(0.87)</td><td>-0.35<br>(0.58)</td><td></td><td></td><td>-2.15<br>(1.14)</td><td>-2.12<br>(1.12)</td></tr><tr><td>Δ<sub>Q</sub><sup>P</sup>ES</td><td></td><td><strong>11.01</strong><br>(4.44)</td><td><strong>11.26*</strong><br>(4.00)</td><td><strong>10.42*</strong><br>(4.04)</td><td></td><td><strong>12.57*</strong><br>(4.18)</td><td><strong>12.42*</strong><br>(4.00)</td></tr><tr><td>VRP</td><td></td><td></td><td>2.38<br>(8.97)</td><td>1.48<br>(9.06)</td><td>0.91<br>(9.17)</td><td>-16.18<br>(7.79)</td><td>-16.35<br>(7.81)</td></tr><tr><td>Δ<sub>Q</sub><sup>P</sup>ES<sup>⊥</sup></td><td></td><td></td><td></td><td></td><td><strong>9.54</strong><br>(4.22)</td><td></td><td></td></tr><tr><td>RV</td><td></td><td></td><td></td><td></td><td></td><td>15.68<br>(10.36)</td><td></td></tr><tr><td>LTV</td><td></td><td></td><td></td><td></td><td></td><td>-0.10<br>(0.07)</td><td>-0.10<br>(0.07)</td></tr><tr><td>JV</td><td></td><td></td><td></td><td></td><td></td><td></td><td>18.76<br>(14.39)</td></tr><tr><td>IV</td><td></td><td></td><td></td><td></td><td></td><td></td><td>15.16<br>(10.75)</td></tr><tr><td>Constant</td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td></tr><tr><td>R<sup>2</sup> (adj %)</td><td>0.00</td><td>0.67</td><td>0.70</td><td>0.68</td><td>0.56</td><td>1.95</td><td>1.93</td></tr><tr><td colspan="8"><strong>Panel B. Variance Risk Premium</strong></td></tr><tr><td>ES<sup>P</sup></td><td>0.02<br>(0.01)</td><td>0.00<br>(0.01)</td><td>-0.02*</td><td></td><td></td><td>-0.03<br>(0.02)</td><td>-0.03<br>(0.02)</td></tr><tr><td>Δ<sub>Q</sub><sup>P</sup>ES</td><td></td><td><strong>0.54*</strong><br>(0.20)</td><td><strong>0.57*</strong><br>(0.20)</td><td><strong>0.52*</strong><br>(0.20)</td><td></td><td><strong>0.31*</strong><br>(0.09)</td><td><strong>0.32*</strong><br>(0.09)</td></tr><tr><td>VRP</td><td></td><td></td><td><strong>0.32*</strong><br>(0.07)</td><td><strong>0.26*</strong><br>(0.06)</td><td><strong>0.22*</strong><br>(0.06)</td><td><strong>0.37*</strong><br>(0.10)</td><td><strong>0.38*</strong><br>(0.10)</td></tr><tr><td>Δ<sub>Q</sub><sup>P</sup>ES<sup>⊥</sup></td><td></td><td></td><td></td><td></td><td><strong>0.54*</strong><br>(0.20)</td><td></td><td></td></tr><tr><td>RV</td><td></td><td></td><td></td><td></td><td></td><td>0.02<br>(0.11)</td><td></td></tr><tr><td>LTV</td><td></td><td></td><td></td><td></td><td></td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td></tr><tr><td>JV</td><td></td><td></td><td></td><td></td><td></td><td></td><td>-0.06<br>(0.27)</td></tr><tr><td>IV</td><td></td><td></td><td></td><td></td><td></td><td></td><td>0.02<br>(0.10)</td></tr><tr><td>Constant</td><td>0.00*</td><td>0.00*</td><td>0.00<br>(0.00)</td><td>0.00*</td><td>0.00*</td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td></tr><tr><td>R<sup>2</sup> (adj %)</td><td>9.24</td><td>21.73</td><td>29.38</td><td>27.88</td><td>28.63</td><td>29.53</td><td>29.55</td></tr></tbody></table>
## D.2 No Equity Risk Premium Constraints

<caption>FIGURE D.2</caption>

Time Series of the Tail Measures Implied by the S&P 500 Intraday Data Calculated Without Imposing the Positive Equity Risk Premium Constraint

Figure D.2 shows the daily time series of Tail Risk, Tail Risk Premium, and Lagrange Multiplier with quantile α=0.2, Cressie Read γ=-3 with unconstrained Equity Risk Premium.

<figure><img src="image_9.png" /><figcaption>TABLE D.2</figcaption></figure>

Predictive Regressions: S&P 500 Excess Returns and Variance Risk Premium Without Imposing Ex Ante Equity Premium Constraints

Table D.2 reports regression coefficients and their standard deviations (in parentheses) of predictive regressions of the close-to-close S&P 500 excess returns at the 1-day horizon. All reported coefficients and standard errors are rounded to 2 decimal places. Those significant at the 0.05 confidence level are printed in bold and those significant at the 0.01 level are additionally highlighted with a *. We report Andrews (1991) standard errors calculated with the use of the sandwich 3.0.0 package for R 4.0.3 (Zeileis et al. (2020)). ES<sup>P</sup> is the nonparametric estimate of realized (physical) expected shortfall of intra-day S&P 500 returns before the close, on the day when each return calculation commences, ES<sup>P</sup> = -E<sub>t</sub> [R<sub>it</sub>/R<sub>it</sub> ≤ F<sub>t</sub><sup>-1</sup> (0.2)]. Our tail risk premium measure, Δ<sub>Q</sub><sup>P</sup>ES, is the difference between the risk-neutral and the physical ES. VRP is the variance risk premium calculated as the difference between the day's realized variance and the (appropriately scaled) VIX<sup>2</sup> index. Δ<sub>Q</sub><sup>P</sup>ES<sup>⊥</sup> is the component of the tail risk measure that is orthogonal to the variance risk premium. LTV is the left tail variance of Bollerslev et al. (2015) obtained from www.tailindex.com. RV is the realized variance of intra-day returns on the S&P 500 index. IV is an estimate of integrated quadratic variation, the continuous component of realized variance, estimated as in Mancini and Gobbi (2012). JV is the jump component of realized variance, calculated as max{RV - IV, 0}. RV, IV, and JV are calculated from return data sampled at the 5-minute frequency.

Panel A. S&P 500 Excess Returns

<table><thead><tr><th></th><th>1 Day</th><th>1 Day</th><th>1 Day</th><th>1 Day</th><th>1 Day</th><th>1 Day</th><th>1 Day</th></tr><tr><th></th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th></tr></thead><tbody><tr><td>ES<sup>P</sup></td><td>0.17<br>(0.77)</td><td>0.13<br>(0.84)</td><td>-0.02<br>(0.57)</td><td></td><td></td><td>-1.68<br>(1.21)</td><td>-1.65<br>(1.19)</td></tr><tr><td>Δ<sub>Q</sub><sup>P</sup>ES</td><td></td><td><strong>5.44</strong><br>(2.39)</td><td><strong>5.55</strong><br>(2.19)</td><td><strong>5.55</strong><br>(2.22)</td><td></td><td><strong>6.28</strong><sup>*</sup><br>(2.22)</td><td><strong>6.18</strong><sup>*</sup><br>(2.08)</td></tr><tr><td>VRP</td><td></td><td></td><td>2.28<br>(8.89)</td><td>2.22<br>(9.01)</td><td>1.38<br>(9.31)</td><td><strong>-16.05</strong><br>(7.72)</td><td><strong>-16.25</strong><br>(7.79)</td></tr><tr><td>Δ<sub>Q</sub><sup>P</sup>ES<sup>⊥</sup></td><td></td><td></td><td></td><td></td><td><strong>4.76</strong><br>(2.32)</td><td></td><td></td></tr><tr><td>RV</td><td></td><td></td><td></td><td></td><td></td><td>15.23<br>(10.61)</td><td></td></tr><tr><td>LTV</td><td></td><td></td><td></td><td></td><td></td><td>-0.11<br>(0.07)</td><td>-0.11<br>(0.08)</td></tr><tr><td>JV</td><td></td><td></td><td></td><td></td><td></td><td></td><td>20.21<br>(13.77)</td></tr><tr><td>IV</td><td></td><td></td><td></td><td></td><td></td><td></td><td>14.47<br>(11.24)</td></tr><tr><td>Constant</td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td></tr><tr><td>R<sup>2</sup> (adj %)</td><td>0.00</td><td>0.58</td><td>0.61</td><td>0.63</td><td>0.45</td><td>1.86</td><td>1.84</td></tr></tbody></table>

(continued on next page)
<table><caption>TABLE D.2 (continued)
Predictive Regressions: S&P 500 Excess Returns and Variance Risk Premium Without Imposing Ex Ante Equity Premium Constraints</caption><caption>Panel B. Variance Risk Premium</caption><thead><tr><th></th><th>1 Day</th><th>1 Day</th><th>1 Day</th><th>1 Day</th><th>1 Day</th><th>1 Day</th><th>1 Day</th></tr><tr><th></th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th></tr></thead><tbody><tr><th>ES<sup>P</sup></th><td>0.02<br>(0.01)</td><td>0.01<br>(0.01)</td><td>-0.01<br>(0.01)</td><td></td><td></td><td>-0.02<br>(0.01)</td><td>-0.02<br>(0.01)</td></tr><tr><th>Δ<sup>P</sup><sub>Q</sub>ES</th><td></td><td><strong>0.27</strong><br>(0.11)</td><td><strong>0.28*</strong><br>(0.10)</td><td><strong>0.28*</strong><br>(0.10)</td><td></td><td><strong>0.14*</strong><br>(0.05)</td><td><strong>0.15*</strong><br>(0.05)</td></tr><tr><th>VRP</th><td></td><td></td><td><strong>0.31*</strong><br>(0.07)</td><td><strong>0.30*</strong><br>(0.05)</td><td><strong>0.25*</strong><br>(0.06)</td><td><strong>0.37*</strong><br>(0.10)</td><td><strong>0.38*</strong><br>(0.10)</td></tr><tr><th>Δ<sup>P</sup><sub>Q</sub>ES<sup>⊥</sup></th><td></td><td></td><td></td><td></td><td><strong>0.30*</strong><br>(0.10)</td><td></td><td></td></tr><tr><th>RV</th><td></td><td></td><td></td><td></td><td></td><td>0.02<br>(0.11)</td><td></td></tr><tr><th>LTV</th><td></td><td></td><td></td><td></td><td></td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td></tr><tr><th>JV</th><td></td><td></td><td></td><td></td><td></td><td></td><td>-0.01<br>(0.28)</td></tr><tr><th>IV</th><td></td><td></td><td></td><td></td><td></td><td></td><td>0.01<br>(0.10)</td></tr><tr><th>Constant</th><td>0.00*<br>(0.00)</td><td>0.00*<br>(0.00)</td><td>0.00<br>(0.00)</td><td>0.00*<br>(0.00)</td><td>0.00*<br>(0.00)</td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td></tr><tr><th>R<sup>2</sup> (adj %)</th><td>9.24</td><td>19.99</td><td>27.40</td><td>27.29</td><td>28.20</td><td>28.22</td><td>28.19</td></tr></tbody></table>

## D.3 Alternative Threshold for ES Calculation

<caption>FIGURE D.3
Time Series of the Tail Measures Implied by the S&P 500 Intraday Data with $\alpha = 0.10$</caption>

Figure D.3 shows the daily time series of Tail Risk, Tail Risk Premium, and Lagrange Multiplier for the case with quantile $\alpha=0.1$ and Cressie Read $\gamma=-3$.

<figure><img src="image_10.png" /></figure>
<caption>TABLE D.3
Predictive Regressions: S&P 500 Excess Returns and Variance Risk Premium with Tail Measures Calculated at the 10th Percentile of Intraday Returns</caption>

Table D.3 reports regression coefficients and their standard deviations (in parentheses) of predictive regressions of the close-to-close S&P 500 excess returns at the 1-day horizon. All reported coefficients and standard errors are rounded to 2 decimal places. Those significant at the 0.05 confidence level are printed in bold and those significant at the 0.01 level are additionally highlighted with a *. We report Andrews (1991) standard errors calculated with the use of the sandwich 3.0.0 package for R 4.0.3 (Zeileis et al. (2020)). ES<sup>B</sup> is the nonparametric estimate of realized (physical) expected shortfall of intra-day S&P 500 returns before the close, on the day when each return calculation commences, ES<sup>⊥</sup> = -E<sub>t</sub> [R<sub>it</sub> | R<sub>it</sub> ≤ F<sub>t</sub>]. Our tail risk premium measure, Δ<sub>Q</sub><sup>B</sup>ES, is the difference between the risk-neutral and the physical ES. VRP is the variance risk premium calculated as the difference between the day's realized variance and the (appropriately scaled) VIX<sup>2</sup> index. Δ<sub>Q</sub><sup>B</sup>ES<sup>⊥</sup> is the component of the tail risk measure that is orthogonal to the variance risk premium. LTV is the left tail variance of Bollerslev et al. (2015) obtained from www.tailindex.com. RV is the realized variance of intra-day returns on the S&P 500 index. IV is an estimate of integrated quadratic variation, the continuous component of realized variance, estimated as in Mancini and Gobbi (2012). JV is the jump component of realized variance, calculated as max{RV - IV, 0}. RV, IV, and JV are calculated from return data sampled at the 5-minute frequency.

<table><thead><tr><th></th><th>1 Day</th><th>1 Day</th><th>1 Day</th><th>1 Day</th><th>1 Day</th><th>1 Day</th><th>1 Day</th></tr><tr><th></th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th></tr></thead><tbody><tr><td colspan="8"><em>Panel A. S&amp;P 500 Excess Returns</em></td></tr><tr><td>ES<sup>B</sup></td><td>0.11<br>(0.61)</td><td>-0.13<br>(0.67)</td><td>-0.26<br>(0.44)</td><td></td><td></td><td><strong>-1.76</strong><br>(0.87)</td><td><strong>-1.75</strong><br>(0.86)</td></tr><tr><td>Δ<sub>Q</sub><sup>B</sup>ES</td><td></td><td><strong>5.80*</strong><br>(2.17)</td><td><strong>5.91*</strong><br>(2.00)</td><td><strong>5.57*</strong><br>(2.04)</td><td></td><td><strong>6.97*</strong><br>(2.25)</td><td><strong>6.85*</strong><br>(2.11)</td></tr><tr><td>VRP</td><td></td><td></td><td>2.43<br>(8.84)</td><td>1.54<br>(9.14)</td><td>1.01<br>(9.27)</td><td><strong>-16.39</strong><br>(7.48)</td><td><strong>-16.63</strong><br>(7.71)</td></tr><tr><td>Δ<sub>Q</sub><sup>P</sup>ES<sup>⊥</sup></td><td></td><td></td><td></td><td></td><td><strong>5.16</strong><br>(2.08)</td><td></td><td></td></tr><tr><td>RV</td><td></td><td></td><td></td><td></td><td></td><td>16.60<br>(9.86)</td><td></td></tr><tr><td>LTV</td><td></td><td></td><td></td><td></td><td></td><td>-0.11<br>(0.07)</td><td>-0.11<br>(0.07)</td></tr><tr><td>JV</td><td></td><td></td><td></td><td></td><td></td><td></td><td>20.12<br>(14.14)</td></tr><tr><td>IV</td><td></td><td></td><td></td><td></td><td></td><td></td><td>16.14<br>(10.25)</td></tr><tr><td>Constant</td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td></tr><tr><td>R<sup>2</sup> (adj %)</td><td>0.00</td><td>0.74</td><td>0.77</td><td>0.75</td><td>0.63</td><td>2.16</td><td>2.14</td></tr><tr><td colspan="8"><em>Panel B. Variance Risk Premium</em></td></tr><tr><td>ES<sup>B</sup></td><td>0.01<br>(0.01)</td><td>0.00<br>(0.01)</td><td><strong>-0.01*</strong><br>(0.01)</td><td></td><td></td><td><strong>-0.02</strong><br>(0.01)</td><td><strong>-0.02</strong><br>(0.01)</td></tr><tr><td>Δ<sub>Q</sub><sup>B</sup>ES</td><td></td><td><strong>0.27*</strong><br>(0.10)</td><td><strong>0.29*</strong><br>(0.10)</td><td><strong>0.27*</strong><br>(0.10)</td><td></td><td><strong>0.16*</strong><br>(0.05)</td><td><strong>0.16*</strong><br>(0.05)</td></tr><tr><td>VRP</td><td></td><td></td><td><strong>0.31*</strong><br>(0.07)</td><td><strong>0.26*</strong><br>(0.06)</td><td><strong>0.23*</strong><br>(0.06)</td><td><strong>0.37*</strong><br>(0.10)</td><td><strong>0.38*</strong><br>(0.10)</td></tr><tr><td>Δ<sub>Q</sub><sup>P</sup>ES<sup>⊥</sup></td><td></td><td></td><td></td><td></td><td><strong>0.28*</strong><br>(0.10)</td><td></td><td></td></tr><tr><td>RV</td><td></td><td></td><td></td><td></td><td></td><td>0.03<br>(0.10)</td><td></td></tr><tr><td>LTV</td><td></td><td></td><td></td><td></td><td></td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td></tr><tr><td>JV</td><td></td><td></td><td></td><td></td><td></td><td></td><td>-0.06<br>(0.25)</td></tr><tr><td>IV</td><td></td><td></td><td></td><td></td><td></td><td></td><td>0.03<br>(0.09)</td></tr><tr><td>Constant</td><td><strong>0.00*</strong><br>(0.00)</td><td><strong>0.00*</strong><br>(0.00)</td><td><strong>0.00</strong><br>(0.00)</td><td><strong>0.00*</strong><br>(0.00)</td><td><strong>0.00*</strong><br>(0.00)</td><td>0.00<br>(0.00)</td><td>0.00<br>(0.00)</td></tr><tr><td>R<sup>2</sup> (adj %)</td><td>9.41</td><td>22.02</td><td>29.34</td><td>28.28</td><td>28.91</td><td>29.47</td><td>29.49</td></tr></tbody></table>
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