Give me a break:
What does the equity premium compensate for?∗Patrizia Perras†

Niklas Wagner‡

Abstract
We provide evidence that the equity premium does not simply compensate investors forbearing market risk per se and contribute to an adequate modeling of the intertemporal risk-return relationship. Our model captures the relationship between conditional
expected excess stock market returns, conditional market volatility, and conditional
market illiquidity, while taking scheduled trading breaks into account. We distinguishbetween two distinct sources of market risk, namely continuous diffusive risk during
trading hours and a discontinuous component representing random overnight pricejumps. Utilizing high-frequency data, we estimate specific premia for trading and nontrading components in terms of conditional volatility as well as conditional illiquidity.
Our findings reveal that the equity premium primarily compensates for bearing risk
and illiquidity during market closures. Conditional volatility and illiquidity during
trading hours play only a minor role in explaining the equity premium and in shaping
the intertemporal risk-return relationship.
Keywords: asset pricing, equity premium, illiquidity premium, jump diffusion, overnight
premium, trading breaks;
JEL Classification: G11, G12, E30;∗

The authors would like to thank Agostino Capponi, Hui Chen, Barbara Dömötör, Darrell Duffie, Garrison Song, James Steeley, Alexander Szimayer as well as participants at the 13th Annual Financial Market
Liquidity Conference and at research seminars at Farmingdale and Wollongong for helpful comments andsuggestions on earlier drafts of the paper. All remaining omissions and errors are the authors’ sole responsibility.
†
University of Passau, School of Business, Economics and Information Systems, E-mail:patrizia.perras@uni-passau.de.
‡Corresponding author. University of Passau, School of Business, Economics and Information Systems,
94030 Passau, Germany. Phone: +49 851 509 3241, Fax: +49 851 509 3242, E-mail: niklas.wagner@unipassau.de, niklas.wagner@berkeley.edu.

1Introduction

The question of our study may seem trivial. The prevalent answer is given by the SharpeLintner-Mossin Capital Asset Pricing Model (CAPM) and known to any Finance 101 student:
A positive equity premium compensates investors for bearing market risk. We argue thatthe underlying dynamics make the issue more complex. While the CAPM is static, its
intertemporal extension (ICAPM) accounts for dynamics and the possibility of evolving
investment conditions.1 However, even the ICAPM does not account for scheduled trading
breaks in equity markets. These breaks affect how risk and illiquidity is priced in the market.
They should therefore have a distinct impact on how the equity premium compensates forrisk and illiquidity.
Considering scheduled trading breaks in financial markets, we provide new evidence on
the risk-return relation by addressing two major questions: (i) given that the equity premiumis a compensation for bearing market risk, what type of market risk does the equity premiumcompensate for?, and (ii) can an identification of the type of market risk help to explain theintertemporal risk-return relation? We thereby add to the existing literature by examining
risk and illiquidity that stand behind the equity premium.
Our findings reveal that the market risk premium is positively related to both conditional trading and non-trading volatility. However, non-trading risk plays a more significant
role in explaining variations in expected market excess returns. The relationship between
conditional non-trading risk and the expected market risk premium is robust and remains
significant even after including state variables that predict the investment opportunity set.
1
Merton’s (1973) intertemporal extension of the CAPM, known as the ICAPM, suggests that the conditional expectation of the market risk premium is positively related not only to conditional market risk but
also to the conditional covariance with relevant state variables that affect the investment opportunity set.
The empirical validation of the predicted positive intertemporal risk-return relation has been a challenging
task in the literature. Numerous papers examine the sign and significance of the intertemporal risk-return
relation. Harvey (1989), Scruggs (1998), Ghysels et al. (2005), Bali and Peng (2006), Guo and Whitelaw(2006), and Hedegaard and Hodrick (2016), for example, support the theoretical presumption of a significant
positive relation. A weak, or even negative, relation is supported by Campbell (1987), Glosten et al. (1993),
Brandt and Kang (2004) and Hedegaard and Hodrick (2014), among others. The mixed evidence may be due
to technical differences regarding the specification of conditional risk, differences in data set specification or
the simplifying assumption of a constant investment opportunity set. Some studies have argued that both,
positive and negative relationships, are consistent with general equilibrium asset pricing models (see e.g.
Gennotte and Marsh (1993)). Other studies consider for example aspects regarding investor sentiment (seee.g. Yu and Yuan (2011)) or dynamics in subjective beliefs (see e.g. Lochstoer and Muir (2022)).

2

Furthermore, our results indicate that the lack of market functionality and liquidity during
closures results in a premium demanded by investors. We conclude that investors exhibit ahigh aversion to illiquidity when the market is closed, due to increased costs of reacting tonews and the potential for a complete loss of trading ability. This underscores the importanceof marketability and immediate reaction capabilities.
These primary conclusions are supported by observed intraday patterns of market quotes.
The required non-trading premium is associated with a decrease in market prices prior tomarket closures, resulting in positive expected market returns over the trading break. Thisis consistent with the clientele effect proposed by Bogousslavsky (2021) and Lou et al. (2019,2023). The authors suggest that institutional investors are reluctant to hold large positions
over trading breaks due to overnight risk and associated capital constraints. As a result,arbitrageurs tend to reduce or close their positions towards the end of the trading day,leading to the observed price pressure effect before the market closes.
We are the first to propose and test a model that explicitly considers conditional risk andconditional illiquidity during periodic market closures and their implications for aggregate
asset prices.2 Our model incorporates two different processes that drive market returns.
For the trading period, we use a pure diffusion process, while a separate process considers
price jumps that occur at fixed points in time due to scheduled market closures.3 Within
this framework, we estimate an intertemporal model where conditional trading day volatility,
conditional volatility due to overnight jumps, and conditional illiquidity are priced separately.
2
While asset pricing remained an open question, there have been two theoretical attempts to model andpredict the implications of periodic market closures for equilibrium asset prices. The model of Hong andWang (2000) predicts a lower return during non-trading periods relative to the trading day. The authors
argue that during the trading day, information asymmetry decreases, which causes the stock price to increase
resulting in a higher return over the trading period than over the non-trading period. The model of Longstaff(1995) suggests a discount for a lack of marketability during periods of reduced liquidity. Hence, the closing
price should impound a discount for the prospective non-marketability over the closure and decline justbefore the market closure.
3
The return process during trading hours differs from the return process during non-trading periods.
During the trading day, the stock price fluctuates as transactions are executed. Once the market officiallycloses, marketability is reduced and costly. However, overnight information matters and is embedded in the
opening price that reflects potential revised investor expectations due to overnight news (French and Roll
(1986)). The resulting close-to-open price change can be viewed as a deterministic jump in the asset pricethat is of random magnitude. As a matter of fact, the notion of an “overnight jump” corresponds to Oldfieldand Rogalski (1980) and Boes et al. (2007) and refers to returns from market close to open. In a similar
fashion, we use the terms “overnight”, “trading break”, “non-trading period” to denote the period fromofficial close at 4:00 PM to the official market opening at 9:30 AM.

3

The present study contributes to the existing literature in several key ways.4 We decompose market returns into their overnight and intraday components and relate the equity
market premium to differences in conditional volatility dynamics. To achieve this, we develop
a customized GARCH model that captures asymmetric effects arising from past intraday andovernight shocks, which differentially affect conditional trading and non-trading volatility.
Our data sample is based on trade-and-quote data from the aggregate U.S. stock market,
covering high-frequency data not only during official trading hours from 9:30 AM to 4:00
PM but also during pre-trading hours from 4:00 AM to 9:30 AM and post-trading hours
from 4:00 PM to 8:00 PM. This comprehensive dataset allows us to draw conclusions about
illiquidity and intraday price patterns during both official trading hours and non-trading
hours when the market is officially closed. Specifically, our intraday analysis indicates thata decrease in the market price prior to market closure yields the observed non-trading premium. Consequently, the compensation for bearing non-trading risk over the trading breakis generally generated by a discount in the closing price, which occurs due to the expected
costs of non-marketability. Overall, our paper provides significant insights into market dynamics and offers a clearer understanding of how trading conditions and market illiquidity
influence expected market returns and the risk-return relationship at higher frequency.
The remainder of the paper is structured as follows. Section 2 provides the general model
setting and presents the discrete-time intertemporal model with distinct premia for trading,non-trading (overnight) risk, and illiquidity. Section 3 presents the empirical methodology
and results. This section includes descriptive statistics of returns and intraday price andilliquidity patterns, the derivation of models for conditional volatility and illiquidity, and themain results on the model estimation. A series of robustness tests is then applied to control
4
Growing empirical evidence motivates the contribution of non-trading returns to the equity premium.
Studies by Cliff et al. (2008), Kelly and Clark (2011), Branch and Ma (2012), Berkman et al. (2012),
Perras and Wagner (2020) and Boyarchenko et al. (2023) indicate that the overall U.S. equity premium is
primarily attributed to returns that are generated over the trading break. Closest to us is Perras and Wagner(2020), who also decompose market returns into their overnight and intraday components and relate themto expected market returns. Hendershott et al. (2020) and Bogousslavsky (2021) find a positive (negative)
cross-sectional relation between systematic risk and overnight (intraday) returns. In a different approach,Lou et al. (2023) estimate a conditional beta that varies with past overnight returns. Their results showthat the return decomposition may resurrect the conditional CAPM. In contrast to those studies, we directlymodel and test the implications of the trading and non-trading components of conditional market risk andilliquidity on the expected market risk premium.

4

for additional state variables that might affect the observed risk-return relationship. The
paper concludes with Section 4.

2

Model

2.1

Model Setting

We study a Merton economy where a representative investor with information set F demands
risk premia for two different sources of risk, namely, diffusive risk that occurs during the
trading day as well as the risk of stochastic price changes from market closing to market
opening. These price changes are induced by market closures that are scheduled and knowna priori. However, the size of the price changes is of random magnitude and sign. They do
not necessarily have to be very large but lead to observable changes in the asset’s price at
specific points in time and are therefore characterized as jumps.
We assume the following price process {Pt }t≥0 of a risky asset with jumps at deterministic
points in time:dPt= µdt + σdBt + dXt ,Pt−
NtX(erco,τj − 1).
where Xt =

(2.1)

j=1

The first part, µdt + σdBt , is the continuous part of the price process, where the increment of
the standard Brownian motion, dBt , models continuous changes generated by diffusive risk.
The second part, dXt , is the discontinuous part of the process with deterministic process,Nt , counting the number of scheduled market closures at times {τj }j≥1 up to time t. The
j-th close-to-open log return that is generated over the market closure is represented byrco,j . In more general words, we could say that the market closure schedules a deterministic“stopping time” of the process. The asset price follows the familiar geometric Brownian
motion, while at the market closure the price process is not longer observable and a change

5

in the asset price from close to open induces a jump.5
To derive a solution to the Stochastic Differential Equation (2.1), we first apply Itô’s
formula to the function f (t, Pt ) = ln(Pt ) that yields
X
1(ln(Pτj ) − ln(Pτj − )).
ln(Pt ) = ln(P0 ) + (µ − σ 2 )t + σBt +
2
τ ≤t

(2.2)

j

From Equation (2.2), we then have a stock price jump from market closure to market openingby random size∆Pτj = Pτj − Pτj− = Pτj− (erco − 1),(2.3)

ln(Pτj ) − ln(Pτj− ) = ln(Pτj /Pτj− ) = rco .

(2.4)

such that

Solving the SDE in Equation (2.2) yields the dynamics of the asset price that is1
rco,j  .
Pt = P0 exp (µ − σ 2 )t + σBt +
2
τ ≤t
X

(2.5)

j

We can represent the total return of the asset as1ln(Pτj /Pτj−1 ) = (µ − σ 2 )(τj− − τj−1 ) + σψj + rco,j .
2

(2.6)

Equation (2.6) shows that the total close-to-close asset return can be decomposed into its
trading part, (µ − 21 σ 2 )(τj− ) + σψj , where the Brownian motion ψj operates from market
opening to market closure, and the non-trading part, rco,j .6
5

An alternative way is to model a common diffusion component that operates during trading and nontrading periods and separate jump processes that represent the random percent change in the share priceresulting from market closures similarly to Oldfield and Rogalski (1980). We could extend the model torandomly timed jumps in prices, i.e. sudden and unpredictable market moves, that can be modeled by a
Poisson random variable such as in the seminal work of Merton (1976). However, the purpose of this paper
is to exclusively focus on risk related to trading breaks induced by stock market closures.
6
We assume that trajectories of the geometric Brownian motion are distributed log-normal and that the
distribution function F of the jump sizes rco,j has unbounded real support.

6

2.2

Discrete-Time Representation

The discrete-time trading and non-trading return can be calculated straightforward fromdaily opening and closing quotes, Po,t and Pc,t , respectively, that are usually available. The
open-to-close trading return is then defined as roc,t = ln(Pc,t /Po,t ), while the close-to-open
non-trading return can be represented as rco,t = ln(Po,t /Pc,t−1 ). We follow common convention and calculate the total return from close-to-close such that the unconditional mean andvolatility of the total asset return are then given by

E [ln(Pc,t /Pc,t−1 )] = E [ln(Pc,t /Po,t )] + E [ln(Po,t /Pc,t−1 ] ;(2.7)

V ar [ln(Pc,t /Pc,t−1 )] = V ar [ln(Pc,t /Po,t )] + V ar [ln(Po,t /Pc,t−1 )](2.8)+ 2Cov [ln(Pc,t /Po,t ), ln(Po,t /Pc,t−1 )] .
Here, Pc,t and Po,t represent the closing and opening prices at time t, respectively. The ratio(Po,t /Pc,t−1 ) indicates the random change in share price from the market closure at timet − 1 to the market open at time t. If the returns from open to close and the non-trading
returns from close to open are positively (negatively) correlated, the covariance term willconsequently increase (decrease) the overall variance of the total return.7

2.3

Pricing Equation

Motivated by Merton (1973), we propose a framework in which the conditional expectedexcess market return is modeled as a linear function of the conditional expected market
variance and a hedging component. Our framework additionally incorporates the impact
of conditional expected market illiquidity. This approach yields the following theoreticalrepresentation for the pricing relation at the aggregate market level:
7

Implicitly, this analysis assumes that the return over a period τ , denoted as rt = ln[Pt+τ /Pt ], incorporates N jumps due to overnight price changes within the interval [t, t+τ ]. For instance, the return-generatinginterval from the market close on Friday this week, Pt , to the market close on Friday the following week,
Pt+τ , will experience N = τ − h jumps due to trading breaks, where h represents the number of holidays
within the interval. Thus, with h = 0 during a typical week, there are N = 5 jumps.

7

2] + λL Et [LM,t+1 ] + λH,x Et [σM ∆x,t+1 ].
Et [rM,t+1 ] − rf,t+1 = λM Et [σM,t+1(2.9)

In the model, E[·] denotes the expectation operator, rM represents the aggregate market
return, and rf,t+1 is the risk-free rate of return, which is known at time t. The first termon the right-hand side accounts for the linear relationship between the conditional expectedexcess market return and conditional market variance. A positive λM implies that investors
require compensation for bearing systematic market risk reflected in conditional market
volatility. The second term addresses the relationship between conditional expected market
returns and the conditional expectations of market illiquidity, denoted by Et [LM,t+1 ]. Given
that investors generally demand a premium for holding illiquid assets, the model recognizes
that investors are compensated for bearing aggregate market illiquidity. The third term in themodel reflects the hedging component of Merton’s (1973) ICAPM. In this regard, σM ∆x,t+1
represents the conditional covariance of market returns with innovations in state variable
x and λH,x accounts for the pricing of unfavorable shifts in the investment opportunity set
associated with state variable x.
To capture the dynamics of trading and non-trading periods, as discussed in Subsection
2.2, we decompose market volatility into its distinct components: Conditional market volatility during the trading day, conditional market volatility induced by overnight price changes,
and conditional covariance of return components. Similarly, we differentiate between conditional expected illiquidity during trading and non-trading periods and present the following
relation:Et [rM,cc,t+1 ] − rf,t+1 = λM,oc V art [rM,oc,t+1 ] + λM,co V art [rM,co,t+1 ]+ λM,occo Covt [rM,oc,t+1 , rM,co,t+1 ](2.10)+ λL,oc Et [LM,oc,t+1 ] + λL,co Et [LM,co,t+1 ]+ λH,x Covt [rM,cc,t+1 , ∆xt+1 ] .
This model distinguishes between the pricing of trading and non-trading components plus the8hedging component. The coefficients λM,oc and λM,co represent the compensation an investor
receives for a unit increase in conditional trading volatility and conditional variability inovernight price changes while λM,occo captures effects induced by conditional correlations
between trading and non-trading returns. Further, λL,oc and λL,co , respectively, denote thecompensation for conditional trading illiquidity and conditional expectations associated with
illiquidity in non-trading periods.

3

Empirical Analysis

3.1

Preliminary Results

3.1.1

Data and Descriptive Statistics

The empirical analysis of this study utilizes trade and quote data spanning from January 3,
2007 to December 29, 2023, focusing on the SPDR S&P 500 ETF. As a traded security, the
SPDR S&P 500 ETF not only closely mirrors the performance of the S&P 500 index but alsosignificantly reduces the likelihood of encountering stale price biases commonly associatedwith index quotes. Data is sampled at 1-minute intervals and the full data set contains a
total of 4278 days. The primary focus of our analysis is on three daily return series:• The close-to-close return (total return rM,cc ): The return from official market closure
at 4:00 PM to official market closure at 4:00 PM the next day.
• The open-to-close return (trading return rM,oc ): The return from official market opening at 9:30 AM to official market closure at 4:00 PM.
• The close-to-open return (non-trading return rM,co ): Return from official market closure at 4:00 PM to the official market opening at 9:30 AM the next day.
In constructing these return series, we follow the suggestions of Lease et al. (1991), Branchand Ma (2012) and Berkman et al. (2012) to utilize bid and ask quotes in order to mitigate

9bid-ask bounce effects and associated artificial overnight price changes in the quote data.8
Specifically, we rely on the midpoint of the first valid bid price and ask price observed at
the official market opening at 9.30 AM and the midpoint of the last valid bid price and ask
price observed prior to the official market closing at 4.00 PM.9
Panel A of Table 1 provides descriptive statistics for the constructed series of the total,
trading and non-trading returns. The unconditional mean of the close-to-open non-trading
return is positive and slightly higher than the unconditional mean of the open-to-close trading return. In line with early literature on return patterns such as French and Roll (1986)
and Stoll and Whaley (1990), our results show that non-trading volatility (i.e. unconditional
standard deviation) is lower than trading volatility. Further, the distribution of the trading
returns appears to be closer to the normal distribution than that of non-trading returns.
In particular, the higher value of kurtosis of the non-trading returns can be explained bysignificant news that arrive during market closures. Any news that arrives overnight accumulates and is directly incorporated in the opening price of the next day. This results in aneither positive or negative overnight price change. To determine whether there is a significant difference in the variance of trading and non-trading returns, we report the results on
a one-sided version of the non-parametric Ansari-Bradley (AB) test. The strong rejectionof the null suggests that trading and non-trading returns significantly differ regarding theirvariance. Overall, these preliminary results indicate that the extra risk during the trading
day measured in terms of higher volatility may not be fully rewarded on the stock market.
Pearson correlation coefficients of the return series are presented in Panel B of Table 1. The
correlation between close-to-close, trading, and non-trading returns is positive and high, asboth trading and non-trading returns contribute to the overall market return. Considering
the correlation between non-trading and trading returns, the coefficient is positive, albeit
modest in magnitude, suggesting a limited degree of predictability from non-trading returns
to subsequent trading returns.
8

Bid-ask bounce effects might induce spurious negative auto-correlation and a positive overnight return
if the stock quote bounces from the bid price at closing to the ask price at market opening, even if theunderlying quote has not changed.
9
Following Berkman et al. (2012), a ‘valid’ quote is defined as a quote with a non-zero trade size on
both, the bid and the ask side avoiding a potential bias in quotes associated with zero-size trades.

10[Table 1]
Our trade-based data sample further allows to investigate the source of positive expected
overnight returns. In particular, we examine the evolution of intraday midquotes over timeto answer whether the expected overnight return can be attributed to a high average priceat the market opening or to a low average price at the market closing. For this, we gather
midquotes on a per-minute basis and average the quotes over the full sample of trading days.
Figure 1 displays the average intraday midquotes. After the market opening at 9:30 AM, theaverage price initially rises during the very first minutes of trading, followed by a subsequentdecline and large price moves in the first hour of trading when institutional investors usually
adjust their initial positions. The positive throughout the trading day. This aligns with thereturn statistics reported in Table 1 and the positive average returns observed from market
opening to market closure.
[Figure 1]
Most notably, a sharp price decrease occurs towards the end of the trading day. Hence, thepositive average overnight return is not driven by high average opening quotes, as proposedby Berkman et al. (2012), but rather can be attributed to a price decline just before thetrading break. This observation aligns with the theory of Longstaff (1995), who posits
that restricted marketability impacts security values. Drawing on option-pricing theory,
Longstaff demonstrates that discounts for lack of marketability can be significant even overshort periods, and that security prices are substantially reduced when trading is restricted
and liquidity is compromised. In this context, stock market closures can be viewed as trading
halts, during which the marketability of securities is temporarily restricted. A price decline
just before the market closure then reflects the discount for the reduced marketability andliquidity.
3.1.2

Intraday Liquidity Pattern

We next investigate the dynamics of illiquidity over the trading day. Following Amihud and
Mendelson (1986), we measure the degree of illiquidity by the cost of immediate execution.
11

Therefore, we rely on intradaily quoted ask prices that include a premium for immediate buys
and the corresponding quoted bid prices that include a concession for immediate sales. The
estimate of illiquidity is then the sum of the buying premium and the selling concession, thatis the spread between bid and ask prices. In our investigations, we then rely on the relative
bid-ask spread (BAS), measured as the bid-ask spread divided by the midpoint between
the bid and ask quotes. Per-minute bid-ask spreads are collected during the three distinct
periods each day:• Pre-trading hours (pre): This period covers the time before the official market opening
from 4:00 AM to 9:30 AM.
• Market-trading hours (market): This period represents the official trading hours whenthe market is open from 9:30 AM to 4.00 PM.
• Post-trading hours (post): This period includes the time after the market officiallycloses from 4:00 PM to 8:00 PM.
Figure 2 illustrates the time-series of average spreads for each period. Four noteworthy features are observed: (i) Pre-trading and post-trading periods exhibit intervals with significant
increases in illiquidity, potentially due to market dysfunctions at the beginning of the sample period; (ii) relative spreads have generally narrowed over time due to advancements inmarket structure and trading technology, leading to enhanced market transparency and efficiency; (iii) spreads increase sharply during periods of market turmoil; and (iv) pre-trading
relative spreads (mean = 0.0240%) are, on average, higher than post-trading spreads (mean= 0.0150%) and market-trading spreads (mean = 0.0056%).
[Figure 2 and 3]
Figure 3 plots the evolution of the averaged intraday spreads over the trading day. In the
beginning of the trading day, the spreads sharply decline, followed by dramatic rises around10.00 AM and noon. Despite those peaks, the bid-ask spreads tend to be constant and low
until around 1.30 PM and then show a modest increase, accompanied by a decrease just prior12

to the closure. Referring to Figure 1, the decrease in the spread at the end of the trading
day tends to be accompanied by a fall in midquotes. As shown by Amihud and Mendelson(1986), returns are positively related to the bid-ask spread and hence, a decline in the spreaddecreases prices and returns. On an intradaily basis, the positive relation between prices andspreads then provides an additional explanation for the decrease in the average asset pricetowards the market closure and the resulting positive average overnight return.

3.2

Trading and Non-Trading Risk and Illiquidity Premia

3.2.1

Modeling Conditional Trading and Non-Trading Volatility

A major part of our analysis is focused on conditional market return volatility which is2= var(rM,t |Ft−1 ), where Ft−1 is the information set. Since conditional
defined by σM,tvolatility of market returns is not directly observable, we consider GARCH class models
that are commonly used in empirical studies to characterize the process of volatility. To
accommodate leptokurtosis and tail thickness in the distribution of errors, the process isdefined such that the innovations follow a skewed student-t distribution. First, we focus onconditional volatility that is described by:
2
2
2
2
σM,
••,t = var(rM,••,t |F••,t−1 ) = ω•• + α•• ••,t−1 + β•• σ••,t−1 + γ•• St−1 ••,t−1
1 if ••,t−1 ≤ 0,with S =0 otherwise,

(3.1)

2
where •• ∈ {cc, oc, co} and σM,••,t denotes the conditional variance of rM,••,t based on infor-

mation up to time t − 1, i.e. F••,t−1 is the sigma field generated by r••,t−j with j ≥ 1. The
parameter γ allows for a different response of the conditional volatility to good and bad news
that arrive the stock market and thus captures asymmetric effects of news on conditional
volatility as in the Glosten et al. (1993) GJR-GARCH(1,1) model. If an asymmetric effectis not present in the return series, γ takes on the value zero and the model reduces to the13standard GARCH specification of Bollerslev (1986).
For conditional trading variance, the information set is based on the past information
set Foc,t−1 = σ(rM,oc,t−j , j ≥ 1). Accordingly, conditional non-trading variance is based on
the information set Fco,t−1 = σ(rM,co,t−j , j ≥ 1). Given that investors make their investment
decision based on past realized values of all relevant variables, we extend the underlying
information set for modeling conditional trading and non-trading volatility. In particular,
we include information on past non-trading returns to predict conditional trading volatility.
2:This extension then yields the following model for conditional trading volatility σoc,t2
σoc,t= var(rM,oc,t |Foc,t−1 ∪ Fco,t−1 )

(3.2)

2
= ωoc + αoc 2oc,t−1 + αco 2co,t−1 + βoc σoc,t−1+ γoc Soc,t−1 2oc,t−1 + γco Sco,t−1 2co,t−1 .

The model allows us to obtain the effects of past daytime shocks (αoc ) as well as the effects ofpast overnight shocks (αco ) on current trading volatility. The parameters γoc and γco capture
asymmetric effects of past daytime and overnight news on adjacent trading and non-trading
volatility. With αco = 0 and γco = 0, the model reduces to the classical GJR-GARCH(1,1)
model for modeling conditional trading volatility.
Correspondingly, the modified non-trading GJR-GARCH(1,1) model is defined as:
2
σco,t= var(rM,co,t |Fco,t−1 ∪ Foc,t )

(3.3)

2
= ωco + αco 2co,t−1 + αoc 2oc,t + βco σco,t−1+ γco Sco,t−1 2co,t−1 + γoc Soc,t 2oc,t .

Next, we apply the GARCH models and nested versions of it to the daily return series
in order to infer conditional return volatility. Table 2 reports quasi-maximum likelihood
estimates of the parameters with robust standard errors based on the method of White(1982) and several diagnostic tests for a set of nested models. To account for potential
serial correlation in the daily return series, autoregressive and moving average terms arechosen according to the BIC information criteria. For the total (close-to-close) return series,
estimation of the standard GARCH (s-GARCH) model reveals that there is a significant
impact of past squared return shocks on conditional volatility. The GJR-GARCH estimation
14

further shows a positive and significant parameter γcc , indicating that there is an asymmetricresponse of conditional return volatility to positive and negative return shocks. Notably, once
the asymmetric term γ is included, the estimation yields negative (and even significant)values for αcc . In case of the trading and non-trading return series, the result regarding
sign and significance on the coefficients αoc and αco is mixed while the leverage coefficient γ
appears to be consistently positive and highly significant.
[Table 2]
In general, a negative value of α in conjunction with the corresponding positive valueof γ suggests a pure form of asymmetric volatility where past negative return shocks cause
higher conditional volatility and past positive return shocks even decrease conditional return
volatility. This phenomenon can be explained by a volatility feedback and leverage mechanism generating asymmetry on the market level (Bekaert and Wu (2000)). The volatility
feedback effect on the market is directly implied by the intertemporal CAPM stating thathigher conditional market volatility should be rewarded with higher expected returns. Asa negative return shock leads to a rise of conditional volatility, this increases the requiredrisk premium which causes an immediate and steady fall of market prices until the expected
return is sufficiently high. The price decline further leads to an increased level of leverageon the market and thus, higher conditional volatility. In case of a positive shock, leverage isreduced on the market level which, vice versa, may decrease conditional volatility. However,as news in general increase current market volatility, the positive price move in responseto good news is likely to be absorbed by the price decline due to the higher risk premium
required to compensate for increasing conditional volatility. Hence, there is an offsetting
effect that makes the impact of positive news on conditional volatility unclear and may explain the observed mixed results on the coefficient α. It has to be noted that the instancesof α < 0, in principle, may create problems with the non-negativity requirement for theestimated conditional variance.Our estimated models are valid as the series at hand all meetthe condition, (α + γ) > 0, which ensures positive estimates of conditional volatility.10
10Imposing a non-negativity constraint on α yields slightly positive and not statistically significant values

15Regarding the leverage term and its effect on trading and non-trading volatility, our results show that overnight return shocks captured by γco have, on average, a stronger impact
on volatility than daytime return shocks captured by γoc . This observation indicates thatvolatility responses stronger to negative overnight news than to negative news announced
during the day. Economically, this finding might be explained by investors’ reaction to different sources of information as suggested by Masulis and Ng (1995). In general, investors
are faced with two different sources of information, public information and the release of private information through the trading of informed investors. During the non-trading period,
investors are faced with public announcements but have practically no access to trading
information which results in a stronger immediate impact of negative overnight news onvolatility. During the trading day, the information flow is obscured by the activities ofliquidity (or “noise”) traders and hence, uninformed investors tend to process private information provided by the trading activities of informed investors much slower than public
information.
In order to conduct a specification check of the fitted models, we conduct several diagnostic checks. In particular, we use weighted versions of the Ljung-Box and ARCH LM
test, derived by Fisher and Gallagher (2012), that are shown to have better power in detecting a potential underfitting. As reported in Table 2, the weighted Ljung-Box statistics
does not detect any significant serial correlation in the standardized residual at first, fifthor ninth-order. According to the weighted ARCH LM test, the residuals do not show any
remaining arch-effects at lag two, four and six which reveals an adequate fit of the ARCH(1)process. To compare the nested model, we rely on values of log-likelihood (L) as well ason the Akaike (AIC) and Bayesian (BIC) information criteria. The results show that theasymmetric GJR-GARCH model outperforms the s-GARCH model in modeling conditional
volatility of total market returns. Likewise, the modified GJR-GARCH model outperforms
the classical GJR-GARCH model in capturing the dynamic behavior of conditional trading
and non-trading volatility.
on the parameter. Estimating conditional volatility under a non-negativity restriction of parameters does
not change the main results of the subsequent analysis.

16

3.2.2Modeling Conditional Trading and Non-Trading Illiquidity

Similar to the definition of conditional volatility, we define conditional illiquidity based on
past information set Ft−1 . We then have the following models for conditional expectationsof trading and non-trading illiquidity:
E[Loc,t |Foc,t−1 ∪ Fco,t−1 ] = µ + φLoc,t−1 + πLco,t−1(3.4)= µ + πpre cpre,t + φmarket cmarket,t−1 + πpost cpost,t−1 ;E[Lco,t |Fco,t−1 ∪ Foc,t−1 ] = µ + πLco,t−1 + φLoc,t−1(3.5)= µ + πpost cpost,t−1 + φmarket cmarket,t−1 + πpre cpre,t−1 .
The three variables that determine conditional trading illiquidity (Loc,t ) and non-trading
illiquidity (Loc,t ) in the model are: (i) cpre , the average bid-ask spread during the preceding
pre-trading hours, (ii) cmarket , the average bid-ask spread during preceding official markettrading hours and (iii), cpost , the average bid-ask spread during the preceding after-hourtrading period.

3.2.3

Model Estimates of Risk Premia

In this section, we examine the risk-return relation on the aggregate stock market. For this,
we initially set λHx = 0 and λL = 0 in Equation 2.9. Under these two basic assumptions,the time-variation in expected market excess returns is linearly related to conditional market
variance. To get a sense of the relation between daily conditional volatility and excess market
returns, we then have:

Model I: Et [rM,cc,t+1 ] − rf,t+1 = λ0 + λM,cc V art [rM,cc,t+1 ].

(3.6)

The daily market return at time t + 1 is denoted by rM,cc,t+1 , rf,t+1 is the risk-free rate andV art {rM,cc,t+1 } denotes the time-t expected conditional total market variance. Following
Scruggs (1998), the intercept term λ0 is included in the model estimations in order to ac-

17count for market imperfections such as taxes or transaction costs. To assess the relative
contribution of trading and non-trading conditional volatility in explaining the market risk
premium, we estimate a nested version of Equation 2.10:
fModel II: Et [rM,cc,t+1 ] − rt+1
= λ0 + λM,oc V art [rM,oc,t+1 ] + λM,co V art [rM,co,t+1 ](3.7)+ λM,occo Covt [rM,oc,t+1 , rM,co,t+1 ],where V art {rM,oc,t+1 } denotes the time-t expected conditional trading variance and V art {rco,t+1 }denotes the time-t conditional non-trading variance. Correspondingly, the model that additionally accounts for risk premia of trading and non-trading illiquidity is then given by:fModel III: Et [rM,cc,t+1 ] − rt+1
= λ0 + λM,oc V art [rM,oc,t+1 ] + λM,co V art [rM,co,t+1 ]+ λM,occo Covt [rM,oc,t+1 , rM,co,t+1 ](3.8)+ λL,oc Et [LM,oc,t+1 ] + λL,co Et [LM,co,t+1 ].
Table 3 presents estimation results for the three models. In order to assess the goodnessof-fit of the estimated models, we observe the adjusted ordinary least squares coefficient ofdetermination (Adj. R2 ). Estimation results of Model I in Table 3 reveal that the market
risk premium is positively related to conditional market variance which is in line with thetheoretical hypothesis of a positive risk-return relation according to Merton (1973). We further report estimation results of Model II with conditional trading and non-trading volatility
estimated by the volatility models as described in Section 3.2.1.11[Table 3]
Single estimation results on the coefficients λM,oc and λM,co show that the market risk
premium is positively related to conditional trading volatility and conditional non-trading
volatility. The positive estimated signs are both economically meaningful and consistent
with theory as investors are risk-averse in general and require a higher premium when the11

Trading and non-trading volatility and illiquidity are conditioned on an overlapping information set
and exhibit, by construction, some degree of correlation. To avoid potential variance inflation due to multicollinearity, the Gram-Schmidt algorithm is applied to orthogonalize the covariates in the multivariate
regressions.

18

payoff is more risky. However, the reported adjusted R2 values indicate that it is conditional
non-trading risk that achieves superior performance in explaining the time-variation in conditionally expected market excess returns. The significance of λM,oc is unstable and vanishes
in the estimation of Model II with additional variables while the coefficient λM,co is positiveand highly significant across all model regressions. This finding is in line with Bollerslev
et al. (2016) who consider the cross-section of expected returns to examine the pricing ofseparate betas for trading day and overnight returns and conclude that betas associated withovernight returns entail a significant risk premium, while the continuous intraday betas donot.
The relationship between the conditional trading and non-trading return covariance andthe expected market risk premium, measured by coefficient λM,occo , is negative and significant. The sign and significance of λM,occo aligns with valid economic explanations. As
suggested by the clientele effect (see Bogousslavski (2021) and Lou et al. (2019, 2023)), institutional investors are reluctant to trade when the market is closed, leading to the overnightmarket being dominated primarily by noise traders. When positive news reaches the market
overnight, noise traders tend to overreact. This overreaction by noise traders results in a
price correction by arbitrageurs after the market opens, leading to a negative return correlation. The higher expected return resulting from a decrease in conditional covariance may
represent a “noise trader premium’, driven by irrational trading by retail investors that affects prices and expected returns. This hypothesis aligns with a related proposition by Abkas
et al. (2021), who argue that a series of positive overnight price increases do not signal good
fundamental news for arbitrageurs. Instead, arbitrageurs believe positive overnight returns
are driven by uninformed trading and overreact in correction of the perceived mispricing
after market opening.
Overall, we conjecture that the risk premium associated with the market closure stems
from investors’ preference for immediacy and their aversion to restricted marketability. During trading hours, investors can respond immediately to market changes and adverse news.
After the market closure, immediate reactions to news become costly due to the reduced19liquidity. This conclusion is reinforced by the results on the illiquidity premia, λL,oc and
λL,co . While both coefficients are positive, only conditional illiquidity during non-trading
periods is significantly priced. When the market is officially closed and liquidity is alreadylow, investors have a high aversion to experience even higher illiquidity that may result in a
potential liquidity meltdown and a complete loss of trading ability.

3.2.4

Controlling for Stochastic Investment Opportunities

Previous research on the intertemporal risk-return relation such as Scruggs (1998) and Guo
and Whitelaw (2006), for example, argue that estimates on risk premia may be biased due tothe omission of state variables that describe changes in the future investment set. Hence, we
control for a set of state variables that are assumed to have predictive power for aggregate
market returns and may act as state variables in the spirit of Merton (1973). Setting λx 6= 0,
we then estimate:fModel IV: Et [rM,cc,t+1 ] − rt+1
= λ0 + λM,oc V art [rM,oc,t+1 ] + λM,co V art [rM,co,t+1 ]+ λM,occo Covt [rM,oc,t+1 , rM,co,t+1 ](3.9)+ λL,oc Et [LM,oc,t+1 ] + λL,co Et [LM,co,t+1 ]+ λH,x Covt [rM,cc,t+1 , ∆xt+1 ] .
In particular, we test whether conditional covariances of market returns with innovations
in certain state variables x earn positive or negative risk premia and whether the inclusionof these additional state variables affects the previous results on risk premia estimates. For
the estimation of conditional covariances of market returns with innovations in certain statevariables, we use the dynamic conditional correlation (DCC) model of Engle (2002). For this,we specify a univariate GARCH process that satisfies appropriate stationarity conditions andnon-negativity constraints for each variable and then use the resulting standardized residuals
to estimate the time-varying correlation matrix.12
Our first set of state variables includes macro-economic variables that are related to risks
12

For details on the DCC estimation procedure, we directly refer to Engle (2002).

20

associated with business cycle fluctuations and known to forecast aggregate stock returns.
Changes in the default spread (Def ) and the term spread (T erm) capture revisions in themarket expectation about future credit market conditions and interest rates. Among others,
Hahn and Lee (2006) argue that these variables represent state variables that determine
the investment opportunity set of investors. The default spread is defined as the spread
between the yield on Moody’s seasoned BAA- and AAA-rated corporate bonds. The term
spread is calculated as the difference between the yields on the 10-year Treasury bond andthe three-month Treasury bill. The macro-economic data are available at daily frequency
and obtained from Fred St. Louis Fed.
[Table 4]
Panel A of Table 4 presents the estimates of the trading and non-trading risk coefficients after incorporating the macro-economic variables. Clearly, the inclusion of the macroeconomic variables does not affect our results on the risk-return relation. Still, the coefficientestimate on conditional trading volatility is positive but not statistically significant while thecoefficient estimate non-trading volatility term is positive and significant at least at the 5%
level. Further, the partial relation between conditional expectations of market excess returns
and conditional covariance of market returns with innovations in Def and T erm is not significant. The estimated signs of λH,Def and λH,T erm are consistent to the results reported byBali and Peng (2006) and Sevi and Baena (2012). Overall, we can conclude that the default
and term spread does not play a role in shaping the intertemporal risk-return relation at
daily frequency.
Next, we include equity factors that are commonly used in empirical asset pricing. Fama
and French (1993) argue that the equity factors SMB and HML act as proxies for statevariables that describe investment opportunities. If so, these variables should have predictive
power for aggregate market returns and foremost, the covariance of the innovation in thesefactors with the market return should be priced on the market level. While there is extensive
empirical research on the cross-sectional relation between equity factors and expected stock(or portfolio) excess returns, there is only scarce evidence on the predictive power and the21pricing of equity factors on the market level. We include four equity factors as control
variables in our model, the size (SM B) and book-to-market (HM L) factor of Fama andFrench (1993) and the profitability (RM W ) and investment (CM A) of Fama and French(2015). SMB is the return on portfolios of small stocks minus the return on portfolios of big
stocks while HML is the difference between the returns on portfolios of high and low bookto-market stocks. RMW is constructed as the difference between the returns on portfolios ofstocks with robust and weak profitability and CMA is constructed as the difference between
the returns on portfolios of the stocks of low (conservative) and high (aggressive) investment
firms. Daily data on the factors can be obtained from Kenneth R. French’s website.
Panel B of Table 4 reports results on additional risk premia induced by the conditional
covariance between the obtained equity factors. The estimation results indicate that theequity factors induce some noise in the risk-return relation but even strengthen the statisticalsignificance of the slope coefficient λM,CO . Still, the slope coefficient for trading risk λM,COalso turns (weakly) statistically significant after including the Fama and French factors. The
estimated coefficients λH,SM B and λH,HM L significantly affect conditionally expected market
excess returns. Hence, the SMB and the HML factor capture information about fundamental
economic risk that is affecting the investment opportunity set.
There is also convincing evidence that option-implied volatility provided by the Chicago
Board of Exchange (CBOE) volatility index (VIX) has forecasting power for stock market
returns and represents a systematic risk factor that is priced in the expected returns ofassets (Bakshi and Kapadia (2003), Ang et al. (2006) and Banerjee et al. (2007), among
others). Further, option-implied volatility is known to capture additional information about
conditional market volatility.This motivates the inclusion of the VIX as an additional predictive state variable that may be priced in the time series of expected market returns. The
estimation results of Panel C in Table 4 show that the results on trading and non-tradingrisk premium remain stable after controlling for option-implied volatility. We obtain a positive sign and a high statistical significance regarding the premium associated with implied
volatility. In line with theory, a higher value of the VIX index induces a higher expected22market excess return. Still, even incorporating implied-volatility does not affect our main
results on the pricing of conditional trading and non-trading risk and illiquidity.

4

Conclusion

In this study, we provide evidence that the equity premium does not merely compensate investors for bearing market risk in general but also specifically addresses risks associated withmarket closures. By incorporating scheduled trading breaks, we enhance the understandingof the intertemporal risk-return relationship. Our model distinguishes between continuous
diffusive risk during trading hours and discontinuous overnight price jumps, capturing therelationship between conditionally expected excess market returns, market volatility, andmarket illiquidity.
Our key findings indicate that the expected market risk premium is primarily driven
by conditional risks and illiquidity during non-trading periods, rather than during trading
hours. The positive relationship between the conditional expected market risk premium andconditional non-trading volatility suggests that investors demand a higher premium for bearing overnight risks, which aligns with theoretical expectations and empirical observations.
The robustness of this relationship, even after accounting for state variables that predictinvestment opportunities, underscores the importance of non-trading risk in shaping therisk-return relation. Additionally, our results reveal that market closures impose significant
costs on due to reduced liquidity and the inability to react immediately to news. This leads
to an illiquidity premium during non-trading periods. The findings are consistent with theeffects of marketability restrictions and a clientele effect in which some market participants
are reluctant to hold assets overnight.
Overall, this study highlights the significant role of non-trading risks and illiquidity andhas important implications for market participants. The risk premium associated with market closures stems from investors’ preference for immediacy and aversion to restricted marketability. The higher costs and reduced liquidity after market closures increases investors’
illiquidity aversion, reinforcing the importance of marketability and immediate reaction ca23

pabilities. This is supported by the significant pricing of conditional illiquidity during nontrading periods, which underscores the impact of market functionality on expected market
returns. Enhancing market functionality during non-trading periods, such as improved extended trading mechanism and platforms, could mitigate some of the adverse effects associated with market closures.

24

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29

Figures
Figure 1: Intraday Price Pattern
Average intraday midquotes from market opening at 9:30 AM to market closure at 4:00 PM. Midquotes are sampled in 1-minuteintervals over the full sample period from January 3, 2007 to December 29, 2023.

234.58

30

16:00

15:30

15:00

14:30

14:00

13:30

13:00

12:30

12:00

11:30

11:00

10:30

10:00

234.56

09:30

Average Midquote

234.60Figure 2: Illiquidity over TimeTime-series of daily average bid-ask spreads during pre-trading hours, market-trading hours and post-trading hours from January
3, 2007 to December 29, 2023.
Pre-trading Hours (4.00 AM - 9.30 AM)

1.00

Daily Average BAS (%)

0.75

0.50

0.25

0.00
2007

2008

2009

2010

2011

2012

2013

2014

2015

2016

2017

2018

2019

2020

2021

2022

2023

2024

2018

2019

2020

2021

2022

2023

2024

2018

2019

2020

2021

2022

2023

2024

Market-trading Hours (9.30 AM - 4.00 PM)

Daily Average BAS (%)

0.04

0.03

0.02

0.01

0.00
2007

2008

2009

2010

2011

2012

2013

2014

2015

2016

2017

Post-trading Hours (4.00 PM - 8.00 PM)

0.4

Daily Average BAS (%)

0.3

0.2

0.1

0.0
2007

2008

2009

2010

2011

2012

2013

2014

2015

2016

31

2017

Figure 3: Intraday Illiquidity Pattern
Intraday bid-ask spreads during pre-trading hours, market-trading hours and post-trading hours. Spreads are sampled in
1-minute intervals. The sample ranges from January 3, 2007 to December 29, 2023.
Pre-trading Hours (4.00 AM - 9.30 AM)

Average BAS (%)

0.075

0.050

09:30

09:00

08:30

08:00

07:30

07:00

06:30

06:00

05:30

05:00

04:30

04:00

0.025

Market-trading Hours (9.30 AM - 4.00 PM)

Average BAS (%)

0.00610.0059

16:00

15:30

15:00

14:30

14:00

13:30

13:00

12:30

12:00

11:30

11:00

10:30

10:00

09:30

0.0057

Post-trading Hours (4.00 PM - 8.00 PM)

0.02

32

20:00

19:30

19:00

18:30

18:00

17:30

17:00

16:30

0.01

16:00

Average BAS (%)

0.03

Tables
Table 1: Descriptive Statistics and Correlations of Returns
Descriptive statistics for daily total market returns (RM,cc ), trading returns (RM,oc ) and non-trading returns (RM,co ) fromJanuary 3, 2007 to December 29, 2023 (4278 daily observations). Return calculation is based on the midquote between bid andask prices. The Ansari-Bradley (AB) rank sum test of dispersion is applied to test for differences in trading and non-trading
return variance (p-values in parentheses).

Panel A: Descriptive Statistics
RM,cc

RM,oc

RM,co

Min (%)-12.3638-8.2173-11.7082

Mean (%)

0.02900.01400.0150Max (%)

10.8385

7.34205.8557

Std. Dev. (%)

1.25730.96980.7591

Skewness-0.4725-0.3057-1.7955

Kurtosis14.5905

11.3805

30.4936

JB Statistics

24517

12801

139376(0.0000)(0.0000)(0.0000)

AB Statistics

8752895

RM,oc

RM,co

(0.0000)
Panel B: Correlations
RM,cc
RM,cc

1

RM,oc

0.79761(0.0000)
RM,co

0.63730.0434(0.0000)(0.0040)

33

1

Table 2: Estimates for Specifications of Conditional Volatility
Parameters of fitted GARCH models described in Section 3.2.1 with skewed student-t innovations to daily total returns (RM,cc ,trading returns (RMo c ) and non-trading returns (RM,co ). Weighted Ljung-Box and ARCH-LM statistics are applied for testing
the null of adequacy of the ARMA fit and the null of an adequately fitted ARCH process. P-values based on robust standard
errors in parentheses.

Mean Equation

RM,cc

µ
φ1
φ2
ϕ1
ϕ2

ω

Variance Equation

αcc
αoc
αco

RM,oc
GJR

Mod. GJR

RM,costandard

GJR

GJR

Mod. GJR

0.00090.0008-(0.0000)(0.0092)

-

-

-

-

-

-

-

-

-

-

-

0.54310.4946

-

-

-

-

-

-

(0.0000)(0.6882)---0.5446--

0.2781--(0.0000)(0.0777)-0.0551-0.0569-0.0437(0.0000)(0.0018)(0.0051)-0.0337

1.49750.4398(0.0359)(0.0000)---0.0161-0.01180.5008(0.7217)-0.3190--(0.2985)(0.4545)[0.0000](0.2578)0.00000.00000.00000.00000.00000.0000(0.7387)(0.0324)(0.4930)(0.0000)(0.0000)(0.3793)0.1511-0.0502

-

-

-

-

(0.0475)(0.0113)

-

-

-

-

-

-

0.07500.0003-0.0001--(0.0000)(0.0000)-(0.0000)----0.0253-0.0692

0-.0048---(0.0391)(0.0000)(0.4192)

0.84780.86590.90000.83580.81830.9057(0.0000)(0.0000)(0.0000)(0.0000)(0.0000)(0.0000)-

0.2396

-

-

-

-

-

(0.0000)

-

-

-

-

-

-

0.05000.1839-0.0771--(0.0002)(0.0000)-(0.0000)

γco

-

-

-

0.18320.05000.1484---(0.0000)(0.0000)(0.0000)shape

5.5141

6.4772

6.40486.37115.55225.9917(0.0000)(0.0000)(0.0000)(0.0000)(0.0000)(0.0000)-7.6487

β
γcc
γoc

AIC

-6.5143-6.5605-6.9787-7.0077-7.5848

BIC-6.5053-6.5486-6.9683-6.9943-7.5714-7.6323

L

13933.50

14034.4414927.45

14991.39

16225.24

16363.82

Weighted LB Test
Lag[1](0.4655)(0.3449)(0.3929)(0.5532)(0.4358)(0.5344)

Lag[2](0.9552)(0.7864)(0.5526)(0.6509)(0.8523)(0.7256)

Lag[5](0.7510)(0.6775)(0.4137)(0.8936)(0.9704)(0.8876)

Weighted Arch LM Test
Lag[2](0.7382)(0.5519)(0.3988)(0.4862)(0.8033)(0.7575)

Lag[4](0.7500)(0.5795)(0.8576)(0.8665)(0.8188)(0.8378)

Lag[6](0.7603)(0.6825)(0.8109)(0.8451)(0.9440)(0.9314)34

Table 3: Estimates of Risk Premia
Results on the risk premia estimated byModel I: Et [rM,cc,t+1 ] − rf,t+1 = λ0 + λM,cc V art [rM,cc,t+1 ],f= λ0 + λM,oc V art [rM,oc,t+1 ] + λM,co V art [rM,co,t+1 ] + λM,occo Covt [rM,oc,t+1 , rM,co,t+1 ],Model II: Et [rM,cc,t+1 ] − rt+1f
= λ0 + λM,oc V art [rM,oc,t+1 ] + λM,co V art [rM,co,t+1 ] + λM,occo Covt [rM,oc,t+1 , rM,co,t+1 ]Model III: Et [rM,cc,t+1 ] − rt+1

+ λL,oc Et [LM,oc,t+1 ] + λL,co Et [LM,co,t+1 ].
Coefficient λM,cc measures the relation between the expected market risk premium and conditional total market variance.
Conditional total market variance is estimated using the asymmetric GARCH model of Glosten et al. (1993). Coefficients λM,ocand λM,co , respectively, measure the partial relation between the expected market risk premium and conditional trading and nontrading variance. Conditional trading and non-trading variance are estimated using the volatility models described in Section
3.2.1. The coefficient λM,occo captures effects of conditional covariance between trading and non-trading returns. Coefficients
λL,oc and λL,co , respectively, measure the partial relation between the expected market risk premium and conditional trading
and non-trading illiquidity. The estimation of conditional trading and non-trading illiquidity is described in Section 3.2.2.
P-values based on heteroskedasticity and autocorrelation robust standard errors are in parentheses.

Model I

Model II

Model III

λ0

λM,cc

Adj.R20.00030.8353

10.53%(0.0000)(0.0001)

λ0

λM,oc

λM,co

0.0003

1.2791-(0.0003)(0.0000)-

0.0003-1.8758(0.0003)-(0.0000)

0.00030.5294

1.9085-0.0186(0.0003)(0.4021)(0.0000)(0.0173)

λ0

λM,oc

λM,co

λM,occo

λL,oc

λL,co

Adj.R20.00090.67801.8844-0.0366

0.00780.0468

11.39%(0.0000)(0.2178)(0.0000)(0.0000)(0.2462)(0.0030)

Adj.R2λM,occo

9.61%
9.91%

35

10.66%

36

0.5244(0.4170)0.6686(0.2309)

0.5222(0.4219)0.0001(0.0000)

0.0001(0.0000)

0.0001(0.0000)(0.0095)-0.0509(0.9315)

0.3060(0.5942)1.9739(0.0001)

1.3104(0.1727)(0.0000)

0.0001(0.0000)

0.0001(0.0000)

0.0001(0.0000)

0.0001(0.0000)(0.0000)

3.1497(0.0000)

3.4078(0.0015)

1.7333(0.0003)

1.4750(0.0000)

2.7709(0.0000)

2.4231λM,co(0.0106)

1.7145(0.0000)

1.8984(0.0006)

1.7279

λM,co

λM,oc

1.19523(0.1572)λ00.0001(0.0000)(0.0000)

2.8231λM,coPanel C: Option-implied Volatility(0.0572)

1.38540.0001

1.18230.0001(0.0000)

λM,oc

λ0

Panel B: Equity Factors

λM,oc

λ0

Panel A: Macro-economic Variables(0.0000)-0.0468

λM,occo(0.0000)-0.0456(0.0000)-0.0462(0.0000)-0.0406(0.0000)-0.0386(0.0000)-0.0379(0.0000)-0.0468

λM,occo(0.0000)-0.3620(0.0000)-0.0367(0.0000)-0.0362

λM,occo(0.1749)

0.0089λL,oc(0.2128)0.00871(0.2418)0.0086(0.2725)

0.0070(0.1744)0.0089(0.3254)

0.0074(0.1749)

0.0090λL,oc(0.2626)

0.0075(0.2621)0.0071(0.2549)

0.0074λL,oc(0.0228)0.0356

λL,co(0.0012)0.0592(0.0030)

0.0579(0.0020)0.0587(0.0165)

0.0368(0.0002)

0.0680(0.0228)

0.0363λL,co(0.0086)

0.0469(0.0053)0.0478(0.0025)

0.0473λL,co(0.0154)0.0558

λH,V IX(0.1147)

0.0523(0.0093)0.0626

-

-

-

-

-

-

(0.0154)0.0580λH,SM B

(0.7730)-0.7118--(0.7534)-0.6889

λH,Def

13.73

Adj.R2 (%)(0.0479)-0.0246(0.0447)-0.0246

-

-

-

-

(0.0520)-0.0234--

λH,HM L(0.9675)

0.0088(0.9216)-0.0186--

λH,T erm(0.5701)-0.0236

-

-

-

-

(0.1176)-0.4498

-

-

-

-

λH,RM W

11.40

11.37

11.42

Adj.R2 (%)(0.5701)-0.0165--(0.2621)-0.0367

-

-

-

-

-

-

λH,CM A

14.90

14.67

11.68

12.16

13.11

12.73

Adj.R2 (%)

with λH,x 6= 0 and x = {Def, T erm, SM B, HM L, RM W, CM A}. Coefficients λH,x represent the risk premium associated with innovations in the default spread (Def ), term spread(T erm), Fama and French (1993) equity factors (SM B, HM L) and the Fama and French (2015) equity factors (RM W , CM A). P-values based on heteroskedasticity and autocorrelation
robust standard errors are in parentheses.

+ λL,oc Et [LM,oc,t+1 ] + λL,co Et [LM,co,t+1 ]+ λH,x Covt rM,cc,t+1 , ∆xt+1 ,







Model IV: Et [rM,cc,t+1 ] − rf,t+1 = λ0 + λM,oc V art rM,oc,t+1 + λM,co V art rM,co,t+1 + λM,occo Covt rM,oc,t+1 , rM,co,t+1

Slope estimates controlling for additional state variables for:

Table 4: Estimates of Risk and Illiquidity Premia with Stochastic Investment Opportunities