Give me a break: What does the equity premium compensate for?

Patrizia Perras

Niklas Wagner

University of Passau, School of Business, Economics and Information Systems

Tokyo Metropolitan University
Research Center for Quantitative Finance
August 13, 2024

Agenda• Motivation• Contribution & Key Findings• Model Setting• Empirical Analysis• Conclusion

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Motivation

Motivation I

Unresolved puzzle on the overall intertemporal risk-return relation• Significant positive relation(e.g. Harvey 1989, Bali/Peng 2006, Hedegaard/Hodrick 2016 among many others).

• Weak or negative relation(e.g. Campbell 1987, Glosten et al. 1993, Hedegaard/Hodrick 2014 among many others).

What might be reasons for the controversial results?
• Specification of volatility (e.g. Guo/Neely 2008, Leon et al. 2007).
• Omission of Merton’s (1973) hedging component (e.g. Sruggs 1998).
• Others, such as irrational investor behavior (Yu/Yuan 2011, Nagel/Xu 2022,
Lochstoer/Muir 2022).

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Motivation II

Why do trading breaks matter?
• Empirics:• Market volatility is lower during periodic trading breaks(e.g. French/Roll 1980, Stoll/Whaley 1990, Kelly/Clark 2011).

• Significant contribution to the overall equity premium(e.g. Cliff et al. 2008, Perras/Wagner 2020, Hendershott et al. 2020, Boyarchenko et
al. 2023).

• Controversial theories:• Investors are averse of holding stocks over non-trading periods due tothe lack of marketability (Longstaff 1995) and ‘clientele effects’ (Lou et al.
2019, Bogusslavsky 2021, Akbas et al. 2022).

• Decrease in information asymmetry over the trading day increases thestock price resulting in a lower expected return over trading breaks(Hong/Wang 2000).

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Contribution & Key Findings

Contribution & Key Findings

How does this study contribute to literature?
⇒ Evidence on the risk-return relation considering trading breaks.
⇒ The impact of trading breaks on the market risk premium (MRP).
⇒ Distinct pricing of trading and non-trading conditional volatility/illiquidity.
Main empirical findings:
1. Positive relation between the overall MRP and conditional risk andilliquidity during market closures.
2. Conditional risk and illiquidity during trading hours play a minor role inexplaining the MRP.
3. Lack of market functionality and liquidity during closures results in a
premium demanded by investors.
4. Analysis of intraday quotes relates the trading break premium to a decline
before the official market close.
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Model SettingModel Setting
Assume a price process {Pt }t≥0 with jumps at deterministic points in time:dPt= µdt + σdBt + dXt
Pt−
where Xt =

(1)

Nt
X

(e rco,j − 1),

j=1{Bt }t≥0 is a standard Brownian motion, dXt is the discontinuous part of theprocess, Nt is a deterministic process counting the number of scheduled market
closures at {τj }j≥1 , up to time t.
Asset prices hence are:





X
1
rco,j  .
Pt = P0 exp (µ − σ 2 )t + σBt +
2

(2)

τj ≤t

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Intertemporal Pricing Equation“In equilibrium, investors are compensated in terms of expected return, for bearing(systematic) market risk, and for bearing the risk of unfavorable shifts in theinvestment opportunity set.”— Merton (1973, p. 882)

Our basic model at the aggregate market level is given by2
Et [rM,t+1 ] − rf ,t+1 = λM Et [σM,t+1] + λL Et [LM,t+1 ] + λH,x Et [σM∆x ,t+1 ]. (3)• λM implies that investors require compensation for bearing systematic
market risk.
• λL represents the compensation for bearing aggregate market illiquidity.
• λH,x accounts for the pricing of unfavorable shifts in the investment
opportunity set associated with state variables x .

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Intertemporal Pricing Equation
A modified version of the model captures the dynamics of trading andnon-trading periods:

Et [rM,cc,t+1 ] − rf ,t+1 = λM,oc Vart [rM,oc,t+1 ] + λM,co Vart [rM,co,t+1 ]+ λM,occo Covt [rM,oc,t+1 , rM,co,t+1 ](4)+ λL,oc Et [LM,oc,t+1 ] + λL,co Et [LM,co,t+1 ]+ λH,x Covt [rM,cc,t+1 , ∆xt+1 ] .
The expected equity market premium is then a linear function of• Conditional trading volatility and conditional volatility of jumps induced bytrading breaks,• Conditional covariance between trading and non-trading returns,• Conditional expectations associated with illiquidity in trading andnon-trading periods,• Conditional covariance associated with innovations in state variables xthat predict changes in the investment opportunity set.
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Empirical Analysis

Empirical Analysis: Returns• TAQ data from January 3, 2007 to December 29, 2023
• Major U.S. ETF: SPDR S&P 500 ETF• Daily return series:• Total return (rM,cc ): Log-return from official market closure at 4:00

PM to official market closure at 4:00 PM the next day.
• Trading return (rM,oc ): Log-return from official market opening at
9:30 AM to official market closure at 4:00 PM.
• Non-trading return (rM,co ): Log-return from official market closure at
4:00 PM to the official market opening at 9:30 AM the next day.

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Empirical Analysis: Illiquidity• Per-minute bid-ask spreads during three distinct periods each day:• Pre-trading hours (pre): This period covers the time before theofficial market opening from 4:00 AM to 9:30 AM.
• Market-trading hours (market): This period represents the official
trading hours when the market is open from 9:30 AM to 4.00 PM.
• Post-trading hours (post): This period includes the time after themarket officially closes from 4:00 PM to 8:00 PM.

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Empirical Analysis: Control Variables• Macroeconomic variables• Default spread (DEF): Difference between the yields on U.S.

high-yield corporate bonds (Moody’s Baa) and low-yield (Moody’sAaa) corporate bonds.
• Term spread (TERM): Difference between the yields on the 10-year
Treasury bond and 3-month Treasury bill.
• Option-implied volatility: VIX index Chicago Board Options Exchange• Equity factors• Size (SMB) and value (HML) factor (Fama/French 1993)• Profitability (RMW) and investment (CMA) factor (Fama/French
2015)

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Empirical Results: Return Statistics
Table 1: Descriptive Statistics
Panel A: Descriptive Statistics
RM,cc

RM,oc

RM,co

-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)

RM,oc

RM,co

Min (%)AB Statistic8752895(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)

1

Descriptive statistics for daily total market returns (RM,cc ), trading returns (RM,oc ) and non-trading returns (RM,co ) from January 3,
2007 to December 29, 2023 (4278 daily observations). Return calculation is based on the midquote between bid and ask prices. The
Ansari-Bradley (AB) rank sum test of dispersion tests for differences in trading and non-trading return variance. P-values in parentheses.

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Empirical Results: Intraday Price PatternFigure 1: Intraday Price Pattern

Average Midquote

234.60234.58

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

234.56

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

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Empirical Results: Intraday Illiquidity PatternFigure 2: Intraday Illiquidity Pattern (Pre-trading)Pre-trading Hours (4.00 AM - 9.30 AM)

0.050

09:30

09:00

08:30

08:00

07:30

07:00

06:30

06:00

05:30

05:00

04:30

0.025

04:00

Average BAS (%)

0.075

Intraday bid-ask spreads during pre-trading hours. Spreads are sampled in 1-minute intervals.

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Empirical Results: Intraday Illiquidity PatternFigure 3: Intraday Illiquidity Pattern (Market-trading)Market-trading Hours (9.30 AM - 4.00 PM)

0.0059

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

0.005709:30

Average BAS (%)

0.0061

Intraday bid-ask spreads during market-trading hours. Spreads are sampled in 1-minute intervals.

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Empirical Results: Intraday Illiquidity PatternFigure 4: Intraday Illiquidity Pattern (Post-trading)

Post-trading Hours (4.00 PM - 8.00 PM)

0.02

20:00

19:30

19:00

18:30

18:00

17:30

17:00

16:30

0.01

16:00

Average BAS (%)

0.03

Intraday bid-ask spreads during post-trading hours. Spreads are sampled in 1-minute intervals.

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Empirical Results: Model Estimation
Table 2: Estimates of Risk Premia in Equation (4)Model I

Model II

Model III

λ0

λM,cc

Adj.R 20.00030.8353

10.53%(0.0000)(0.0001)

λ0

λM,oc

0.0003

1.2791λM,coAdj.R 2λM,occo-9.61%(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)

9.91%

10.66%

λ0

λM,oc

λM,co

λM,occo

λL,oc

λL,co

Adj.R 20.00090.67801.8844-0.0366

0.00780.0468

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

Modified ICAPM estimation results with λH,x = 0. Estimation is done by Maximum-likelihood, p-values based onheteroskedasticity and autocorrelation robust standard errors are in parentheses.

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Empirical Results: Stochastic Investment Opportunities

Table 3: Risk-Return Trade-off under Stochastic Investment Opportunities
Panel A: Macro-economic Variables
λ0

λM,oc

λM,co

λM,occo

λL,oc

λL,co

λH,DefλH,Term

Adj.R 2 (%)

0.0001

0.5244

1.7279-0.03620.00740.0473-0.6889-

11.42(0.0000)(0.4170)(0.0006)(0.0000)(0.2549)(0.0025)(0.7534)-0.00010.6686

1.8984-0.0367

0.00710.0478--0.0186(0.0000)(0.2309)(0.0000)(0.0000)(0.2621)(0.0053)-(0.9216)0.0001

0.5222

1.7145-0.36200.00750.0469-0.71180.0088(0.0000)(0.4219)(0.0106)(0.0000)(0.2626)(0.0086)(0.7730)(0.9675)

11.37

11.40

Panel B: Option-implied Volatility
λ0

λM,oc

λM,co

λM,occo

λL,oc

λL,co

λH,VIX

Adj.R 2 (%)

0.0001

1.195232.8231-0.0468

0.00890.0356

0.055813.73(0.0000)(0.1572)(0.0000)(0.0000)(0.1749)(0.0228)(0.0154)

Modified ICAPM estimation results with λH,x 6= 0. Conditional covariances between market returns andinnovations in state variables are estimated by the DCC model of Engle (2002). Robust p-values in parentheses.

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Empirical Results: Stochastic Investment Opportunities

Table 3 Cont.: Risk-Return Trade-off under Stochastic Investment Opportunities
Panel C: Equity Factors
λ0

λM,oc

λM,co

λM,occo

λL,oc

λL,co

λH,SMB

λH,HML

λH,RMW

λH,CMA

Adj.R 2 (%)

0.0001

1.18232.4231-0.0468

0.00900.03630.0580

-

-

-

12.73

(0.0000)(0.0572)(0.0000)(0.0000)(0.1749)(0.0228)(0.0154)---0.00011.38542.7709-0.0379

0.00740.0680---(0.0000)(0.0095)(0.0000)(0.0000)(0.3254)(0.0002)-(0.0520)--0.0001-0.0509

1.4750-0.0386

0.00890.0368---0.4498-(0.0000)(0.9315)(0.0003)(0.0000)(0.1744)(0.0165)--(0.1176)-0.00010.30601.7333-0.0406

0.00700.0587----0.0367(0.0000)(0.5942)(0.0015)(0.0000)(0.2725)(0.0020)---(0.2621)0.00011.97393.4078-0.04620.0086

0.05790.0626-0.0246--(0.0000)(0.0001)(0.0000)(0.0000)(0.2418)(0.0030)(0.0093)(0.0447)--0.00011.31043.1497-0.0456

0.00871

0.05920.0523-0.0246-0.0236-0.0165(0.0000)(0.1727)(0.0000)(0.0000)(0.2128)(0.0012)(0.1147)(0.0479)(0.5701)(0.5701)-0.0234

13.11

12.16

11.68

14.67

14.90

Estimation results with λH,x 6= 0. Conditional covariances between market returns and innovations in statevariables are estimated by the DCC model of Engle (2002). Fama-French (1993, 2015) factors : Small-minus-big(SMB), high-minus-low (HML), robust-minus-weak (RMW), conservative-minus-aggressive (CMA). Robust
p-values in parentheses.

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Summary & Conclusions

Summary and Conclusions

The equity premium compensates primarily for bearing risk
and illiquidity during market closures.
⇒ Conditional volatility and illiquidity during trading hours play only a minor
role in shaping the intertemporal risk-return relationship.

Why?
• Expected costs of non-marketability induce a discount prior to the official
closure in line with Longstaff’s (1995) theory.
• Higher risk aversion of investors during market closures with reduced
marketability and increased information asymmetry.
• Intraday patterns suggest a potential ‘clientele effect’: Assets change
owners.

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Thank you for your attention!

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