Anticipated Regret and Equity Returns

Y. Eser Arisoy† Turan G. Bali‡ Yi Tang§

Abstract

We investigate the cross-sectional asset pricing implications of anticipated regret for mean-variance

investors. We propose a theoretical framework in which mean-variance investors anticipate regret due

to deviations from the best possible return-volatility synthesis. We show that equity portfolios with

low anticipated regret generate 7.6% more annualized alpha than portfolios with high anticipated

regret. The anticipated regret premium is not explained by costly arbitrage, investor inattention, or

prospect theory. We further decompose anticipated regret into two components, return-regret and

variance-regret, and show that both components play a significant role in mean-variance investors’

financial decision making under anticipated regret.

Key words: Regret theory; mean-variance investors; cross-section of stock returns.

JEL classification: G11, G12, G41.

This Draft: October 2019

† Associate Professor of Finance, NEOMA Business School, Finance Department, 59 rue Pierre Taittinger,

51100 Reims, France. Phone: +33 (0)3 26 35 09 51, Email: eser.arisoy@neoma-bs.fr.‡ Corresponding author. Robert S. Parker Chair Professor of Finance, McDonough School of Business, George-

town University, Washington, D.C. 20057. Phone: +1 (202) 687-5388, Email: Turan.Bali@georgetown.edu.§ Associate Professor of Finance, Gabelli School of Business, Fordham University, New York, N.Y. 10023.

Phone: +1 (646) 312-8292, Email: ytang@fordham.edu.

*We thank Kevin Aretz, Yigit Atilgan, David E. Bell, Michael J. Brennan, Enrico Diecidue, Richard

Engelbrecht-Wiggans, Michael Halling, Fabian Hollstein, Graham Loomes, John Quiggin, Bruno Solnik,

Raman Uppal, and Quan Wen for their insightful comments and constructive suggestions. We also benefited

from discussions with seminar participants at ESCP Europe, EDHEC Business School, Manchester Business

School, NEOMA Business School, TED University, 35th Annual Conference of the French Finance Associ-

ation, 8th Financial Engineering and Banking Society International Conference, 3rd Research in Behavioral

Finance Conference, 2018 FMA Annual Meeting, 11th Annual Meeting of Academy of Behavioral Finance

and Economics, and 5th Inter-Business School Finance Seminar.

1 Introduction

“I visualized my grief if the stock market went way up and I wasn’t in it – or it went way

down and I was completely in it. My intention was to minimize my future regret.”

Harry M. Markowitz (1998)

Regret has a long-standing root in cognitive psychology and is defined as the negative emotion

(a mixture of pain and anger) that people experience when realizing that their present

situation would have been better had they decided or acted differently. Regret originates

from a comparison between outcomes of a chosen action and the foregone alternatives in

which the latter outperforms the former. Although regret is a concept that affects utility ex

post, it does influence decisions ex ante.1 A typical example is that agents who exhibit regret

aversion avoid taking decisive actions because they fear that whatever action they take, it will

be suboptimal.2 Hence, from a financial perspective, regret aversion might affect investors’

decision making process. As expressed in our opening quote, regret aversion can lead even

an investor as sophisticated as Harry Markowitz, founder of the modern portfolio theory, to

divert away from his own theory when deciding on his pension fund investment plan.3

Being such a powerful concept in cognitive psychology and having strong axiomatic

foundations in addressing several violations of expected utility theory, regret theory and

its implications for decision making have been studied in various settings.4 Regret is also

1See, e.g., Gilovich and Medvec (1995) and Zeelenberg and Pieters (2007) for studies in psychologyliterature documenting that people alter their choices in response to the anticipation of regret.

2Imagine you decided to sell a share and after selling the share you observed that it rose in price. The nexttime you consider selling a share, it might be more appealing not to sell despite negative signals about theshare, because you typically feel more regret about your previous action as opposed to actions you did nottake. Regret aversion can also lead to excessive risk taking. Suppose you did not buy an asset whose marketvalue has recently surged and you regret this decision. Next time you shape your investment decision in sucha way that your regret aversion may lead you to invest in extremely risky assets, which could potentiallyresult in significant losses.

3According to a recent survey conducted by Harris Poll in September 2017, 71% of Americans regret theway they handled their financial decisions in managing their own money.

4See, e.g., Bell (1983), Engelbrecht-Wiggans (1989), Orphanides and Zervos (1995), Leland (1998), Camilleet al. (2004), Coricelli et al. (2005), Hayashi (2008), Sarver (2008), Bleichrodt et al. (2010), Nasiry andPopescu (2012), Bikhchandani and Segal (2014), Buturak and Evren (2017), Jiang et al. (2017), and Diecidueand Somasundaram (2017) for different application areas of regret in decision making.

1

found to be an important factor in shaping investor behavior and in financial decision making.

Fogel and Berry (2006) examine the relation between regret and disposition effect, and find

that investors experience more regret about holding on to a losing stock too long than about

selling a winning stock too soon. Muermann et al. (2006) investigate how asset allocation

decisions in a defined contribution pension plan vary with participants’ attitudes toward risk

and regret, and show that investors who take regret into account hold more (less) equity in

their optimal portfolios when the equity premium is low (high). Filiz-Ozbay and Ozbay (2007)

examine auctions with bidders who incorporate anticipated regret in their decision making

and find that overbidding in first price auctions is derived from the anticipation of loser regret.

Their experimental results suggest that bidders can indeed anticipate loser regret. Michenaud

and Solnik (2008) study optimal currency hedging choices for regret-averse investors and

develop a two-factor model that consists of variance risk and regret risk. Solnik and Zuo

(2012) develop a global equilibrium asset pricing model using a utility formulation inspired by

regret theory and provide a regret-aversion based explanation to home bias. Hazan and Kale

(2015) introduce an optimal algorithm for investors’ portfolio selection problem under regret.

Qin (2015) examines the effects of regret on investor behavior and market turbulence by

using a model in which investors not only regret wrong actions, but also regret inaction. He

shows that regret aversion can lead investors to ride a bubble, exit and re-enter the market,

or choose not to trade. Frydman and Camerer (2016) use neural data collected from an

experimental asset market setting to measure regret preferences, while subjects trade stocks.

They observe that subjects are unwilling to repurchase stocks that have recently increased

in price due to their previous experience of regret. Fioretti et al. (2018) examine the role

of anticipated regret in an experimental dynamic stock market setting and find that when

agents are informed that they will observe the future price of an asset after they sell the asset,

they keep the asset longer when they expect a high future maximum price, confirming the

role of anticipated regret in the selling decision. Gollier (2019) shows that when presented

with a one-risky-one-safe-lottery menu, regret-risk-averse agents are more willing to choose

the risky act, implying a preference for positively skewed lotteries.

2

Our paper contributes to the literature by examining the implications of regret theory for

mean-variance investors and for cross-sectional pricing of equities. In particular, we focus

on the modified utility function of regret-averse investors and investigate the implications

of regret that investors anticipate from not achieving their expected return and variance

objectives. We show that the deviation of investors’ expected utility from the best possible

foregone utility is an important factor in investors’ decision making process, and it has

significant implications for the cross-sectional pricing of individual equities. Due to the role of

maximum return and minimum variance in agents’ modified utility functions, our regret-based

framework implies that investors anticipate more regret, and hence expect a decrease in their

modified utility functions, for not holding stocks with large deviations between their expected

return and highest return that could have been obtained within a month and with large

deviations between their expected variance and lowest variance that could have been obtained

within the same month. Stocks with such large deviations between their expected return

(variance) and best possible foregone return (variance) would be viewed as stocks with high

anticipated regret. In order to avoid this regret, mean-variance, regret-averse investors would

increase their demand for such stocks, pushing the prices up and generating lower future

returns in equilibrium.

The contribution of our paper is fourfold. First, we provide a theoretical framework in

which anticipated regret impacts mean-variance investors’ expected utility functions and

their decision-making process. Second, we operationalize our regret-based framework by

introducing a novel measure of anticipated regret, denoted by REG, based on the conditional

mean and conditional variance of daily returns in a month. Third, we decompose anticipated

regret into two components, return-regret (RREG) and variance-regret (VREG), and show

that both components play a significant role in investors’ decision-making under anticipated

regret. Finally, we examine whether and how anticipated regret is priced in the cross-section

of equity returns over the 1963-2018 period.

We start our analyses by investigating the cross-sectional pricing implications of anticipated

regret (REG), which measures mean-variance investors’ anticipated regret resulting from

3

deviations between their monthly expected return and variance objectives and the best foregone

return and variance that could have been attained in the same month. By construction,

stocks with high-REG represent large deviations from mean-variance investors’ expected

return and variance objectives, thus resulting in a higher expected decrease in investors’

modified utility functions under regret. Using REG as a measure of mean-variance investors’

anticipated regret, we first form decile portfolios each month by sorting stocks according to

their REG. Value-weighted univariate portfolio sorts indicate that stocks in the lowest REG

decile outperform stocks in the highest REG decile by 0.76% per month (or 9.12% per annum).

This result is robust to controlling for factors that are documented to be strong predictors of

future returns, using equal-weighted portfolios, removing small and illiquid stocks, using only

NYSE stocks, imposing price, size, and illiquidity screens, across different industries, across

different business cycles and market conditions, and using risk-adjusted returns (alphas) with

respect to the CAPM, the 3-factor model of Fama and French (1993) (FF3), the 4-factor

model of Fama and French (1993) and Carhart (1997) (FFC), the 5-factor model of Fama

and French (1993), Carhart (1997), and Pastor and Stambaugh (2003) (FFCPS), the 5-factor

model of Fama and French (2015) (FF5), and the Q-factor model of Hou et al. (2015). The

difference in risk-adjusted returns (alphas) of portfolios with the highest and lowest REG

remains negative and highly significant in all specifications and stock samples.

Our regret-based framework suggests that mean-variance, regret-averse investors view

high-REG stocks as those with high regret prospects because they represent large deviations

from investors’ expected return and variance objectives as a result of not achieving the highest

possible daily return and the lowest possible daily variance in a month. Thus, investors

anticipate more regret for not buying or missing the opportunity to invest in high-REG

stocks. Because of their high regret prospects, investors increase their demand and pay high

prices for such stocks in order to minimize their future regret, which leads to overpricing of

high-REG stocks and low future returns in equilibrium.

To ensure that the significant premium of REG does not simply capture the risk premia

that have been identified by earlier studies, we conduct bivariate portfolio sorts based on a

4

battery of stock characteristics that have been documented to be strong predictors of future

returns. Bivariate portfolio sorts do not change the significantly negative relation between

REG and future stock returns. Consistent with the univariate and bivariate portfolio-level

analyses, Fama and MacBeth (1973) cross-sectional regressions provide corroborating evidence

that REG commands a significantly negative premium after controlling for a large set of firm

characteristics and risk factors. The results indicate that anticipated regret is a systematically

and distinctly priced factor in the cross-section of stock returns.

We next investigate whether costly arbitrage, investor inattention, and/or prospect theory

provide a complementary explanation for the anticipated regret premium. To do so, we

conduct dependent bivariate portfolio sorts, in which stocks are first sorted into terciles based

on their arbitrage cost index (COST ), investor attention index (ATT ), and lottery demand

index (LTRY ), respectively. Next, within each tercile, we form ten decile portfolios sorted

by REG. Then, we average each of the REG-sorted decile portfolios across the three COST,

ATT, and LTRY terciles, and report the FF5 alpha spread on the value-weighted zero-cost

portfolio that is long in the resulting high-REG portfolio and short in the resulting low-REG

portfolio. The results indicate that the negative relation between anticipated regret and

future equity returns remains significant after controlling for the three potential explanations.

Hence, neither costly arbitrage, nor investor inattention and prospect theory can fully explain

the anticipated regret premium.

Having established the role of anticipated regret in the cross-section of stock returns,

we turn our attention to the two components of anticipated regret, i.e., return-regret and

variance-regret. Univariate portfolio sorts suggest that investors anticipate regret for not

holding stocks with high return-regret and high variance-regret. Hence, the two components

play an important role in investors’ decision-making under anticipated regret. Finally, we

propose an alternative measure of anticipated regret based on a cross-sectional return-variance

comparison. Using this alternative measure of anticipated regret based on the cross-sectional

averages of daily returns and daily variances of all stocks as the return and variance benchmarks

in investors’ anticipated regret function, we find results similar to those obtained from our

5

main proxy for anticipated regret.

The remainder of the paper is organized as follows. Section 2 presents the theoretical

framework that establishes the link between regret theory and expected return-variance

tradeoff for mean-variance, regret-averse investors. Section 3 describes the data and variables

used in our analyses. Section 4 conducts empirical tests to examine the cross-sectional

relation between anticipated regret and future equity returns. Section 5 investigates whether

alternative explanations such as costly arbitrage, investor inattention, and prospect theory

can provide a complementary explanation for the anticipated regret premium. Section 6

offers additional analyses and a battery of robustness checks. Section 7 concludes the paper.

2 Theoretical Framework

Under subjective expected utility framework, if preferences satisfy certain axioms, there are

numerical probabilities and utilities that represent agents’ decisions under uncertainty. The

standard expected utility framework asserts that only the realized outcome matters and agents

choose the alternative with the highest utility in expectation. However, it is well-known that

while making decisions, agents do not only care about the actual outcome of their decision,

but they also account for the foregone alternatives by comparing what they could have gotten,

had they chosen differently, to what they actually get.5 Expected utility theory has been

challenged both theoretically and experimentally by studies showing significant violations

of its axioms such as, loss aversion, ambiguity aversion, and Allais and Ellsberg paradoxes.

These challenges led to new alternative theories such as the cumulative prospect theory of

Kahneman and Tversky (1979), the regret theory of Bell (1982) and Loomes and Sugden

(1982), and the disappointment aversion model of Gul (1991).

5Both prospect and regret theories assume that agents compare their well-being (consumption, wealth,portfolio return, etc.) with some benchmark. Prospect theory assumes that this benchmark is defined by thepast, i.e., an investor’s utility depends on her gains/losses in comparison to her wealth in the previous period,while in regret theory the benchmark is not fixed ex ante but rather depends on the future states of theworld. Hence, the main assumption of regret theory is that people, after making decisions under uncertainty,may have regrets if their decisions turn out to be wrong even if they appeared correct with the informationavailable ex ante.

6

Although the notion of regret in decision making goes back to Savage (1951) and to his

minimax regret algorithm, it is Bell (1982) and Loomes and Sugden (1982) who formally derive

a normative theory of choices under uncertainty that addresses many empirical violations of

the expected utility framework. In their model, they introduce a modified utility function

with a final payoff x, where regret is measured by the distance between the payoff x of the

chosen act X and the payoff y that could have been obtained had another action Y been

selected:

u(x, y) = v(x) + f(v(x)− v(y)), (1)

where u(x, y) is the modified utility of having a final payoff of x knowing that y could have been

achieved, v(x) is the standard von Neumann-Morgenstern utility function which determines

risk attitudes toward known outcomes (also known as choiceless utility), v(x)− v(y) is utility

loss/gain of having chosen act X that delivers a payoff of x rather than another choice Y

that could have paid off y, and f(v(x)− v(y)) is the regret of having chosen X when Y could

have been chosen. The regret function f(·) is monotonically increasing and concave with

f(0) = 0. The regret function implies that regret-averse agents differentiate between small

and large regret, and large intensities of regret are weighted disproportionally heavier than

small ones (Zeelenberg and Pieters, 2007). Another important characteristic of regret theory

is that although the modified utility function is defined over ex post outcomes of investment

choices, investors make choices ex ante by maximizing the expected value of the modified

utility function.

Quiggin (1994) extends the above objective function that is defined for pairwise choices

to general choice sets where investors could select among various investments, with outcomes

xi. For general choice problems, the modified utility function is defined as:

u(xi) = v(xi) + f(v(xi)− v(max[xi])), (2)

7

where max[xi] is the best ex post outcome that could have been obtained among all possible

investments, i.e., the best foregone alternative. The distance between the two value functions,

v(xi)− v(max[xi]), is the non-positive regret term. Regret-averse investors anticipate this

regret and take it into account when making investment decisions and choose the optimal

portfolio by maximizing their modified expected utility across all possible investment choices:

E[u(xi)] = E[v(xi)] + E[f(v(xi)− v(max[xi]))]. (3)

We extend Quiggin’s (1994) framework to mean-variance investors who take anticipated

regret into account when making portfolio selection decisions. Assuming that investors care

about the first two moments of the return distribution with an expected utility function of

the form, E[v(µi, σi)] = µi− b2σ2i , regret-averse investors’ modified utility function is given as:

E[u(µi, σi)] = E[v(µi, σi)] + f(E[v(µi, σi)− v(max[µi, σi])]

)], (4)

where µi is the expected return on the risky asset i, σi is the variance of its returns, b is

the investors’ risk-aversion parameter, max[µi, σi] represents the best expected return and

variance outcome for a given risky asset i, and v(µi, σi)− v(max[µi, σi]) is the non-positive

regret term, which captures the degree of anticipated regret (or decrease in the value of

the choiceless utility) stemming from not attaining the best expected return and variance.

Plugging mean-variance investors’ expected utility function, E[v(µi, σi)] = µi − b2σ2i , into Eq.

(4), we obtain:

E[u(µi, σi)] = µi −b

2σ2i + f

((µi −

b

2σ2i )−max[µi −

b

2σ2i ]),

= µi −b

2σ2i + f

((µi −max[µi])− (

b

2σ2i −max[− b

2σ2i ])),

= µi −b

2σ2i + f

((µi −max[µi]

)− b

2(σ2

i −min[σ2i ])), (5)

The modified expected utility function in Eq. (5) implies that, as opposed to the standard

8

von Neumann-Morgenstern utility function, the optimal investment choice is not only defined

over the expected return and variance of the asset, but it further incorporates a comparison

with the maximum return and minimum variance that could have been attained by the

investor, which forms investors’ anticipated regret. Hence, mean-variance and regret-averse

investors do still take into account the first two moments of the return distribution in their

decision making, but they also care about deviations from the best foregone investment

alternative (i.e., how the expected return on the stock deviates from the maximum return that

could have been attained and how the variance of the stock deviates from the minimum level

of variance that could have been attained), and would like to minimize this anticipated regret

in their financial decision-making. The combined expression in investors’ regret function f(·),

(µi−max[µi])− b2(σ2

i −min[σ2i ]), in Eq. (5) represents the deviation from investors’ expected

return and variance objectives under anticipated regret, which we term as REG. Our first

result leads to the following hypothesis:

H1 : High-REG stocks are associated with more anticipated regret because they are

expected to decrease investors’ modified utility function more than other stocks. Investors

anticipate more regret for not holding stocks with large deviations between their expected

return and the maximum daily return in a month and with large deviations between their

expected variance and the minimum daily variance in the same month. Because of the large

differences between their foregone and future return and variance expectations, investors will

increase their demand and be willing to pay high prices for high-REG stocks in order to

minimize their future regret. As a consequence, overpricing leads to lower future returns for

high-REG stocks.

Under fairly mild conditions (i.e., additive, continuous and monotonically increasing) for

the utility and regret functions, the modified expected utility function in Eq. (5) can be

9

rewritten as:

E[u(µi, σi)] = µi −b

2σ2i + f

((µi −max[µi]

)+ f(− b

2(σ2

i −min[σ2i ]),

= µi + f(µi −max[µi]

)− b

2

(σ2i + f(σ2

i −min[σ2i ])). (6)

The modified expected utility function in Eq. (6) reveals two new attributes for mean-

variance, regret-averse investors: return-regret, denoted by µi −max[µi], and variance-regret,

denoted by σ2i − min[σ2

i ]. Return-regret (RREG) and variance-regret (VREG) can be

viewed as two distinct components of anticipated regret (REG). The implications of the two

components of anticipated regret are as follows. Stocks with high anticipated return-regret,

i.e., stocks with a large spread of µi −max[µi], would be considered as lottery stocks that

investors regret not buying. Because of their regret aversion, investors will increase their

demand and be willing to pay high prices for stocks with high anticipated return-regret, and

as a consequence, overpricing will lead to lower future returns for such stocks. Similarly,

mean-variance, regret-averse investors would view stocks with high variance-regret, i.e., stocks

with a large spread of σ2i −min[σ2

i ], as more attractive and regret not buying them. Because of

this anticipated regret, regret-averse investors will increase their demand and be willing to pay

high prices for high variance-regret stocks over the next period, which in turn will lead to lower

future returns for such stocks. The separate effects of return-regret and variance-regret in

investors’ modified utility functions, as captured essentially by Eq. (6), lead to two additional

hypotheses:6

H2 : Stocks with high-RREG are expected to decrease investors’ modified utility function

more than other stocks. Investors anticipate more return-regret for not holding stocks with

large deviations between their expected return and the highest daily return that could have

6Expected returns and variances jointly determine mean-variance investors’ portfolio decision, hence froma theoretical viewpoint it makes more sense to look into the joint effect of the two components via REG.Furthermore, REG as defined in Eq. (5) does not rely on the additivity assumption for the regret and utilityfunctions, which is required to decompose REG into its return-regret and variance-regret components as in Eq.(6). We nevertheless investigate the impact of return-regret and variance-regret in the cross-sectional pricingof equities separately, as it gives us further insight about the role that each component plays in investors’decision making under anticipated regret. The results are presented in Section 6.4.

10

been attained in a month. Because of the large differences between their foregone and future

return expectations, investors will increase their demand and be willing to pay high prices for

high-RREG stocks in order to minimize their future regret. As a result, overpricing leads to

lower future returns for high-RREG stocks.

H3 : Stocks with high-VREG are expected to decrease investors’ modified utility function

more than other stocks. Investors anticipate more variance-regret for not holding stocks with

large deviations between their expected variance and the lowest daily variance that could

have been attained in a month. Because of the large differences between their foregone and

future variance expectations, investors will increase their demand and be willing to pay high

prices for high-VREG stocks in order to minimize their future regret. As a consequence,

overpricing leads to lower future returns for high-VREG stocks.

3 Data and Variables

In this section, we first describe the data used in our empirical analyses. Next, we explain

the construction of our main proxy for anticipated regret (REG) and its two components,

return-regret (RREG) and variance-regret (VREG). Finally, we provide a description of the

control variables.

3.1 Data sources

Market variables are obtained from the Center for Research in Security Prices (CRSP) and

accounting variables from COMPUSTAT. To ensure that the accounting data are available

to investors in real time, we use accounting values from the fiscal year ending in calendar year

t− 1 to run out-of-sample asset pricing tests for the period from July of calendar year t to

June of calendar year t+1. The monthly returns on the one-month Treasury bill (the risk-free

rate), the equity market (MKT ), size (SMB), book-to-market (HML), momentum (MOM),

profitability (RMW ), and investment (CMA) factors are obtained from Kenneth French’s

11

online data library.7 The Pastor and Stambaugh (2003) liquidity factor (LIQ) is obtained

from Lubos Pastor’s website.8 The monthly data on Q factors, size (RME), profitability

(RROE), and investment (RI/A) are obtained from the authors of Hou, Xue, and Zhang (2015).

The sample period is from July 1963 to December 2018. All tests are based on individual

stocks trading on the New York Stock Exchange (NYSE), American Stock Exchange (AMEX),

and NASDAQ with share codes 10 and 11. Furthermore, we exclude stocks with share prices

less than $5 and more than $1,000 from our analysis to ensure that the results are not driven

by small and illiquid stocks. The final sample contains an average of 3,225 equity observations

per month. Our univariate tests investigating the relation between anticipated regret and

future equity returns use a total of about 2.15 million firm-month observations.9

3.2 Construction of anticipated regret variables

Our main proxy to measure anticipated regret is REG, which is based on the modified

expected utility function given in Eq. (5). An increasing and concave value function f(·)

implies that mean-variance investors’ anticipated regret in a given month t is captured by:

REGi,t = (µi,t −max[ri,d])−b

2(σ2

i,t −min[σ2i,d]), d = 1, 2, . . . , T (7)

where d denotes trading days in month t, b is investors’ risk aversion parameter, µi,t is the

average daily return of stock i in month t, which measures the expected return that an

investor would earn in month t, σ2i,t is the average daily conditional variance of stock i in

month t, proxying for the expected variance, and max[ri,d] and min[σ2i,d] are the maximum

daily return and the minimum conditional daily variance attained in month t, which capture

the best return and variance alternatives that could have been attained during the same

7http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data library.html8https://faculty.chicagobooth.edu/lubos.pastor/research/liq data 1962 2017.txt9Table A.1 in the online appendix provides a detailed description of the control variables used in our tests.

We require that at least 15 (24) non-missing daily (monthly) return observations exist in a month (over thepast five years) when we calculate variables with daily (monthly) data.

12

month.10

Eq. (7) indicates that the higher the difference between the average return (average

variance) and the maximum return (minimum variance) during the month, the more negative

REG is, and hence the lower the investors’ regret-based modified utility is, as outlined in

Eq. (5). Hence, REG captures the joint impact of two distinct regret-based risk factors in

mean-variance investors’ decision making, i.e., the regret that investors anticipate from not

achieving the best possible (maximum) return and the regret that investors anticipate from

not achieving the best possible (minimum) level of variance.

We estimate a stock’s daily conditional variance based on the GJR-GARCH (1,1) model

proposed by Glosten, Jagannathan, and Runkle (1993).11 Specifically, we estimate the following

model using daily excess returns for each stock i over the past five years with a minimum of

200 daily observations:12

ri,d+1 = αi + εi,d+1, (8)

σ2i,d+1 = β0 + (β1 + γIi,d)ε

2i,d + β2σ

2i,d, (9)

where ri,d+1 is the excess return of stock i on day d+ 1, εi,d+1 can be viewed as unexpected

news or information shocks to stock i, Ii,d is an indicator equal to one if εi,d is negative and

zero otherwise, and σ2i,d+1 is the one-day-ahead conditional variance of stock i defined as an

asymmetric function of the lagged squared unexpected news (or information shocks) and the

lagged conditional variance. According to the GJR-GARCH model in Eq. (9), negative shocks

10Our main measure of anticipated regret uses a risk aversion parameter, b = 5, which is in line with studiesdocumenting that investors’ risk aversion parameters change between two and ten (Ghysels, Santa-Clara,and Valkanov, 2005; Guo and Whitelaw, 2006; Bali, 2008; Bali and Engle, 2010). We also try alternativemeasures of anticipated regret based on the risk aversion parameters, b = 2 and b = 10, and verify that theresults are not sensitive to the different values of risk aversion parameters. The results based on alternativemeasures of anticipated regret using the risk aversion parameters b = 2 and b = 10 are presented in SectionA.1.1 and Table A.2 in the online appendix.

11Engle and Ng (1993), Bekaert and Harvey (1997), and Hansen and Lunde (2005) show that the GJR-GARCH is superior to the GARCH model of Bollerslev (1986) in predicting future realized variance.

12We further test the robustness of our results using daily conditional variance estimates based on expandingdaily windows. Table A.3 in the online appendix presents the results of REG-sorted portfolios based on dailyconditional variance estimates using expanding windows.

13

(εi,d < 0) are allowed to increase conditional variance more than positive shocks (εi,d > 0)

of the same magnitude. The parameters are estimated simultaneously by maximizing the

following conditional log-likelihood function:

L(β0, β1, γ, β2) =∞∑n=1

1

2

[− log(2π)− log(σ2

d)− ε2dσ2d

]. (10)

Note that the one-day-ahead variance estimates are based on investors’ true information

set available to them as of the portfolio formation month. Hence, there is no look-ahead bias

in our empirical analysis.

In Section 2, we show that under mild conditions, i.e., additive, continuous, and monoton-

ically increasing utility and regret functions, it is possible to further decompose anticipated

regret into two distinct components: i) anticipated regret that arises from missing the highest

possible average return, i.e., return-regret, and ii) anticipated regret due to missing the

lowest possible variance, i.e., variance-regret. Hence, we further investigate the impact of

return-regret and variance-regret in the cross-sectional pricing of equities separately, as it

provides further insight about the role that each component plays in investors’ decision

making under anticipated regret.

Following Eq. (6), we define return-regret (RREG) and variance-regret (VREG) as the

terms inside the regret function f(·):

RREGi,t = µi,t −max[ri,d], d = 1, 2, . . . , T (11)

V REGi,t = σ2i,t −min[σ2

i,d], d = 1, 2, . . . , T (12)

where µi,t, σ2i,t , max[ri,d], and min[σ2

i,d] are defined as previously.

The anticipated regret measure in Eq. (7) and its two components in Eqs. (11) and (12),

respectively, assume that investors use average daily return in a month as a benchmark for

return-regret and average daily conditional variance in a month as a benchmark for variance-

regret, and then compare these benchmarks with the same stock’s best possible return and

14

variance estimates, respectively. In other words, when making decisions across stocks, the

benchmark is the time-series averages of the same stock (i.e., stock’s own mean-variance

performance). We further propose an alternative measure of anticipated regret, where the

benchmark for investors is the cross-sectional averages of daily returns and daily variance of

all stocks rather than the time-series averages of daily return and daily variance of the same

stock. Using this alternative benchmark, we have the following:

CRREGi,t = µt −max[ri,d], d = 1, 2, . . . , T (13)

CV REGi,t = σ2t −min[σ2

i,d], d = 1, 2, . . . , T (14)

CREGi,t = (µt −max[ri,d])−b

2(σ2

t −min[σ2i,d]), d = 1, 2, . . . , T (15)

where d denotes trading days in month t, µt is the cross-sectional equal-weighted average of

daily returns of all stocks in month t, σ2t is the cross-sectional equal-weighted average of daily

conditional variances of all stocks in month t, and max[ri,d] and min[σ2i,d] are the maximum

daily return and minimum daily variance of stock i attained in month t.

In this framework, mean-variance, regret-averse investors will view stock i as more

attractive than stock j if stock i’s MAX-return performance with respect to all other stocks’

average return is higher than stock j’s MAX-return performance with respect to all other

stocks’ average return. Similarly, stock i will be viewed as more attractive than stock j if

stock i’s MIN-variance performance with respect to all other stocks’ cross-sectional average

variance is better (lower) than stock j’s MIN-variance performance with respect to all other

stocks’ cross-sectional average variance.13

13Our main measure of anticipated regret, REG, is defined in Eq. (7). We further test the pricingimplications of the two components of REG, RREG and VREG, given in Eq. (11) and Eq. (12), and thealternative proxy for anticipated regret based on the cross-sectional average returns and variances, denotedby CREG given in Eq. (15). The results are presented in Sections 6.4 and 6.5, respectively.

15

3.3 Control variables

A significant relation between anticipated regret and future equity returns can potentially

be explained by the correlation between anticipated regret and another firm-specific charac-

teristic, which is known to predict the cross-section of equity returns. Alternatively, the lack

of a relation between anticipated regret and future equity returns can be attributed to the

possibility that anticipated regret and another firm-specific attribute which is correlated with

anticipated regret both impact expected returns, but the effects are in the opposite direction

and subsume each other. Thus, we use as controls several firm-specific characteristics that

have been shown to affect equity returns by earlier studies.

Following Fama and French (1992), we control for the market beta, size, and book-to-

market equity ratio of a firm. We estimate the market beta (BETA) of individual stocks

using monthly returns over the past five years with a minimum of 24 monthly observations.

We calculate the natural logarithm of each stock’s market capitalization (SIZE ) and its

book-to-market equity (BM ) ratio at the end of each month. To control for the medium-term

momentum effect of Jegadeesh and Titman (1993), we measure the momentum return (MOM )

of each stock as its cumulative return during the past 12 months after skipping the most

recent month. We also control for the short-term reversal (STR) effect of Jegadeesh (1990)

by controlling for the one-month lagged stock return. Amihud (2002) shows that there

exists a positive premium to more illiquid stocks, thus, we calculate the Amihud illiquidity

measure (ILLIQ), defined as the absolute daily return divided by the daily dollar trading

volume averaged over all trading days in each month for each stock. We calculate each

stock’s co-skewness (COSKEW ) following Harvey and Siddique (2000) using monthly return

observations over the past five years. Following Ang, Hodrick, Xing and Zhang (2006) who

uncover a negative relation between idiosyncratic volatility (IVOL) and future equity returns,

we calculate idiosyncratic volatility as the standard deviation of the residuals from a regression

of excess stock returns on the excess market return (MKT ), size (SMB), and value (HML)

factors in each month. Following Kumar (2009), we use an index of lottery-likeliness that

16

defines lottery-like stocks as low-priced stocks with high idiosyncratic volatility and high

idiosyncratic skewness. Specifically, we construct a lottery index (LTRY ) by sorting stocks

for each month into 50 bins by price per share (PRC ) in descending order such that stocks in

the lowest bin (i.e., a PRC portfolio rank of 1) have the highest price per share and those in

the highest bin (i.e., a PRC portfolio rank of 50) have the lowest price per share. We also

independently sort stocks into 50 bins by idiosyncratic volatility (IVOL) and idiosyncratic

skewness (ISKEW ) in ascending order. IVOL and ISKEW are, respectively, the standard

deviation and the skewness of residuals from the time series regression of daily stock returns

against the MKT, SMB, and HML factors in a month. We then construct LTRY by summing

up the ranks of the PRC, IVOL, and ISKEW portfolios. The lottery index thus has an integer

value in the range of 3 to 150 that increases with a stock’s lottery feature. Following Fama

and French (2015) and Hou et al. (2015), we control for the cross-sectional pricing effects

of investment (IA) and profitability (OP). We further control for the number of analysts

covering the stock (CVRG) and the quarterly fractional institutional ownership (INST ).

4 Anticipated Regret and the Cross-Section of Equity Returns

This paper is the first to investigate the cross-sectional relation between anticipated regret

and future equity returns. In this section, we conduct parametric and nonparametric tests to

assess the predictive power of anticipated regret (REG) over future stock returns. First, we

present results from univariate portfolio sorts. Second, we report average stock characteristics

to have a clear picture of the composition of anticipated regret portfolios. Third, we conduct

bivariate portfolio-level analyses to examine the predictive power of anticipated regret after

controlling for well-known stock characteristics and risk factors. Fourth, we present stock-

level cross-sectional regression results. Finally, we examine the cross-sectional persistence of

anticipated regret.

17

4.1 Univariate portfolios of stocks sorted by REG

We start our analysis by examining whether anticipated regret (REG) predicts the cross-

sectional differences in future stock returns. For each month from July 1963 to December

2018, stocks are sorted into decile portfolios based on their anticipated regret, REG, where

decile 1 (decile 10) contains stocks with the lowest (highest) REG.14 Next, we calculate the

one-month-ahead value-weighted average portfolio returns, and this procedure is repeated

each month until the sample is exhausted.

Table 1 reports for each decile the average REG, the next-month average excess return,

and the risk-adjusted returns (alphas) based on the CAPM, the 3-factor model of Fama and

French (1993) (FF3), the 4-factor model of Fama and French (1993) and Carhart (1997)

(FFC), the 5-factor model of Fama and French (1993), Carhart (1997), and Pastor and

Stambaugh (2003) (FFCPS), the 5-factor model of Fama and French (2015) (FF5), and the

Q-factor model of Hou et al. (2015). The last row in Table 1 presents the 10−1 average

return and alpha spreads for the hedge portfolio that is long in the decile of stocks with the

highest REG and short in the decile of stocks with the lowest REG.

Univariate portfolio sorts indicate a significantly negative relation between anticipated

regret and next-month average returns. The value-weighted portfolio of stocks with the lowest

REG (decile 1) earns an average excess return of 0.46% per month, whereas the average

excess return on the value-weighted portfolio of stocks with the highest REG (decile 10) is

–0.30% per month. The arbitrage portfolio with a long position in the highest REG stocks

and a short position in the lowest REG stocks (High−Low REG) loses on average 0.76% per

month (or 9.12% per annum) with a Newey-West (1987) t-statistic of –2.69. The last row in

Table 1 presents the next month’s risk-adjusted returns for the High−Low REG portfolio.

The CAPM, FF3, FFC, FFCPS, FF5, and Q-factor alpha spreads for the long-short portfolio

14Note that anticipated regret defined in Eq. (7) takes on non-positive values by construction. Without lossof generality, we use the absolute value of anticipated regret in our tests, so that stocks with high anticipatedregret have a larger (positive) distance between their foregone (i.e., the maximum daily return and theminimum daily variance in a month) and the expected return-variance combination (i.e., average daily returnand average daily conditional variance in a month). Using the absolute value of anticipated regret does notaffect the implications and interpretation of our theoretical framework presented in Section 2.

18

are all negative, economically large, ranging from –0.57% to –1.23% per month, and highly

significant with t-statistics in the range of –2.61 and –5.55.15

Next, we investigate the source of the economically large risk-adjusted return spreads

between the high-REG and low-REG portfolios: Is it due to outperformance by low-REG

stocks, underperformance by high-REG stocks, or both? For this, we focus on the economic

and statistical significance of the risk-adjusted returns of decile 1 versus decile 10 of the

value-weighted portfolios. As reported in Table 1, the CAPM, FF3, FFC, FFCPS, FF5,

and Q alphas of decile 10 (high-REG stocks) are all significantly negative, whereas the

corresponding alphas of decile 1 (low-REG stocks) are all insignificant. Thus, we conclude

that the significantly negative alpha spread between the high-REG and low-REG stocks

is due to the underperformance by stocks in the highest REG decile, but not due to the

outperformance by stocks in the lowest REG decile.

The results indicate that regret-averse investors view stocks with high-REG as those

with high regret prospects because of the large differences between their foregone and future

return and variance expectations. Hence, investors anticipate more regret for not buying or

missing the opportunity to increase their holdings in such stocks. Accordingly, holding high-

REG stocks minimizes investors’ future regret and could even lead to rejoicing. Therefore,

regret-averse investors overvalue high-REG equities whose future return distributions are

more appealing under regret theory and accept lower future returns, which is confirmed by

the significantly negative alphas on high-REG stocks in decile 10.

Table 1 shows that the return predictability is driven by the short leg of the arbitrage

portfolio. Thus, one may think that the anticipated regret premium is explained by short-sale

constraints. To test this conjecture, we focus on alternative subsamples of stocks that would

be less prone to such limits to arbitrage. Following D’Avolio (2002), we perform univariate

portfolio analysis using samples that exclude small and illiquid stocks. First, we replicate

15The significantly negative relation between anticipated regret and the next-month risk-adjusted returnsis robust across different factor models in the value-weighted portfolios of REG. We find slightly strongerresults for the equal-weighted portfolios of stocks sorted by REG. The corresponding results are presented inTable A.4 of the Online Appendix.

19

Table 1 using the NYSE stocks only, screening relatively small and illiquid stocks trading

at AMEX and NASDAQ. Second, we apply an additional price screen by removing the

low-priced NYSE stocks trading below $5 per share. Third, we implement a size screen by

removing the smaller NYSE stocks with market capitalizations that place them in the smallest

NYSE size decile. Fourth, we perform a liquidity screen based on the illiquidity measure of

Amihud (2002) by excluding the NYSE stocks that belong to the lowest NYSE liquidity decile.

Finally, we impose the three screens simultaneously by removing the relatively low-priced,

small, and illiquid NYSE stocks.

The first column in Table 2 presents the FF5 alphas on the value-weighted decile portfolios

of stocks trading at the NYSE. Similar to our earlier findings, the 10−1 alpha spread for

the arbitrage portfolio that is long in the decile of stocks with the highest REG and short in

the decile of stocks with the lowest REG is negative and statistically significant; –0.56% per

month with a t-statistic of –2.59. For the individual price, size, and liquidity screened samples

of NYSE stocks, the FF5 alpha spreads between the high-REG and low-REG deciles remain

economically large, in the range of –0.54% and –0.55% per month, and highly significant with

t-statistics ranging from –2.82 to –2.87. The last column in Table 2 presents a somewhat

lower but still statistically significant alpha spread of –0.48% per month (t-stat. = –2.57) for

the subsample of NYSE stocks that excludes low-priced, small, and illiquid stocks altogether.

As presented in Table 1, the FF5 alpha spread between the high-REG and low-REG

portfolios is –0.63% per month (t-stat. = –3.87) for the NYSE/AMEX/NASDAQ sample

that excludes low-priced stocks, whereas in Table 2 the corresponding alpha spread reduces

to –0.48% per month (t-stat. = –2.57) for the NYSE sample that excludes low-priced, small,

and illiquid NYSE stocks simultaneously (i.e., part of the NYSE sample that is much less

prone to short-sale constraints). Overall, these results suggest that the anticipated regret

premium is not fully explained by short-sale constraints.

20

4.2 Average stock characteristics of REG-sorted portfolios

Having documented significant return differences between the high-REG and low-REG

portfolios, it is important to understand what kind of stocks have high vs. low regret prospects.

We are especially interested in the composition of the high-REG portfolio since the anticipated

regret premium is driven by the underperformance of high-REG stocks. Table 3 presents the

average characteristics of the stocks in the deciles. Specifically, the table reports the market

share of each portfolio (SHR), the number of analysts covering the stock (CVRG), the share

price (PRC ), the fractional institutional ownership (INST ), the market beta (BETA), the

market value of equity measured in billions of dollars (SIZE ), the book-to-market equity ratio

(BM ), the return momentum (MOM ), the past one-month return (STR), the co-skewness

(COSKEW ), the illiquidity (ILLIQ), the idiosyncratic volatility (IVOL), the asset growth

(IA), the operating profitability (OP), and the lottery index (LTRY ).

Table 3 demonstrates that there are significant differences between all firm characteristics

of high-REG vs. low-REG portfolios, except for the book-to-market ratio. In particular,

compared to low-REG stocks, high-REG stocks are lower-priced, smaller, less liquid, more

volatile, and they have higher market beta and stronger lottery-like features. Table 3 also

shows that high-REG stocks are largely held by retail investors and have lower analyst

coverage.16 Moreover, high-REG stocks are momentum and short-term winners and they

have larger negative co-skewness, higher asset growth, and lower profitability.

Considering the prior findings in the literature and the patterns of the firm-specific

attributes in REG-sorted deciles, one may think that some of these characteristics drive

the significantly negative relation between anticipated regret and future equity returns. For

example, the short-term reversal, market beta, idiosyncratic volatility, and lottery demand

are positively related to anticipated regret and negatively related to future returns, which may

be the cause of the anticipated regret premium (see, e.g., Jegadeesh (1990), Ang, Hodrick,

Xing, and Zhang (2006), Kumar (2009), Bali, Cakici, and Whitelaw (2011), and Frazzini and

16These results are consistent with earlier studies (see, e.g., Kumar, 2009; Bali, Cakici, and Whitelaw, 2011;Han and Kumar, 2013; Kumar, Page, and Spalt, 2016; Bali et al., 2017).

21

Pedersen (2014)). Furthermore, equities with higher anticipated regret have higher (lower)

investment (profitability), and thus the negative (positive) relation between investment

(profitability) and expected returns may drive the negative relation between anticipated regret

and expected returns (see, e.g., Fama and French (2015) and Hou, Xue, and Zhang (2015)).

In the following two subsections, we test whether the predictive power of anticipated regret

remains significant after controlling for these well-known, robust return predictors in the

bivariate portfolios and multivariate Fama-MacBeth regressions.

4.3 Bivariate portfolios of REG and the control variables

To verify that the negative relation between anticipated regret and future returns is not

explained by previously documented anomalies, we now conduct bivariate portfolio sorts

using the 13 control variables. Specifically, we form value-weighted decile portfolios at the

end of each month by independently sorting all stocks based on an ascending sort of REG

and on each of the control variables. The intersections of the ten REG and ten portfolios

sorted by a control variable generate a total of 100 portfolios. Subsequently, we average each

of the REG-sorted portfolios across the ten deciles, producing portfolios with dispersion in

anticipated regret that are similar in terms of the control variable. In addition, we form

an arbitrage portfolio (High−Low REG portfolio) that is long in the resulting high-REG

portfolio and short in the resulting low-REG portfolio.

Table 4 reports the one-month-ahead FF 5-factor risk-adjusted returns for each of these

ten value-weighted portfolios averaged across the control deciles. After controlling for 13 stock

characteristics that have been documented to be significant predictors in the literature, the last

column in Table 4 shows that all of the spread portfolios which are long in the highest-REG

stocks and short in the lowest-REG stocks (High−Low REG portfolio) command significantly

negative next-month returns, with the FF5 alpha spreads ranging from –0.50% to –0.83% per

month with t-statistics in the range of –2.88 to –6.80. A notable point in Table 4 is that even

when controlling for short-term reversal, market beta, idiosyncratic volatility, lottery demand,

22

investment, and profitability (potential drivers of anticipated regret premium according to

Table 3), the negative relation between anticipated regret and future returns remains highly

significant.

Harvey, Liu, and Zhu (2016) investigate 316 documented factors related to cross-sectional

pricing effects and find that many of the documented predictors of stock returns capture the

same underlying economic phenomena. Thus, the number of orthogonal drivers of expected

stock returns is likely to be substantially lower. Harvey et al. (2016) indicate that due to data

mining and the large amount of research examining the cross-section of expected returns, a

five percent level of significance is too low a threshold, and argue in favor of using much more

stringent requirements for accepting empirical results as evident of true economic phenomena.

Specifically, Harvey et al. (2016) emphasize that a new return predictor needs to clear a

much higher hurdle, with a t-statistic greater than 3.0. Besides the theoretical foundation

of anticipated regret as described in Section 2, the t-statistics of the alpha spreads on the

value-weighted univariate and bivariate portfolios presented in Tables 1 and 4 are generally

above 3 so that the newly proposed anticipated regret passes the more demanding significance

thresholds arising from correlated multiple testing, data mining and publication bias concerns

highlighted by Harvey et al. (2016).

4.4 Firm-level cross-sectional regressions

We should note that the univariate and bivariate sort analyses are performed at the

portfolio level and could suffer from the aggregation effect due to restraining individual

stock-level information in the cross-section. To mitigate the aggregation effects and to control

for the potential impact of other stock characteristics simultaneously, we run Fama and

MacBeth (1973) regressions at the individual stock level while controlling for a large set of

firm characteristics. In particular, we estimate the following regression (and subsets of it):

ri,t+1 = λ0,t + λREG,tREGi,t + λX,tXi,t + εi,t+1, (16)

23

where ri,t+1 is the excess return of stock i in month t+1, REG is anticipated regret of stock i in

month t defined in Eq. (7), and Xi denotes the set of controls representing the characteristics

of stock i in month t, which are market beta (βMKT ), log of market capitalization measured in

millions of dollars (SIZE ), log of book-to-market ratio (BM ), momentum (MOM ), short-term

return reversal (STR), illiquidity (ILLIQ) calculated following Amihud (2002), co-skewness

(COSKEW ) calculated following Harvey and Siddique (2000), idiosyncratic volatility (IVOL)

calculated following Ang et al. (2006), lottery demand index (LTRY ) calculated following

Kumar (2009), operating profitability (OP), annual growth rate of total assets (IA), analyst

coverage (CVRG), and institutional holdings (INST ).

Panel A of Table 5 presents the time-series average of the cross-sectional intercepts and

slope coefficients from the monthly cross-sectional regressions of one-month-ahead excess

returns on REG and different sets of stock characteristics for the period from July 1963 to

December 2018. The t-statistics reported in parentheses are adjusted for autocorrelation and

heteroskedasticity following Newey and West (1987) with six lags. The first specification

examines the cross-sectional relation between anticipated regret and one-month-ahead stock

returns without any controls. Consistent with our findings from the univariate portfolio

sorts, Column 1 in Panel A of Table 5 provides evidence of a negative and significant relation

between REG and one-month-ahead returns, with an average slope of –0.025 and a t-statistic

of –5.89. The economic magnitude of the associated effect is greater than that documented

in Table 1 for the value-weighted univariate portfolios of REG. The spread in average REG

between decile 1 and 10 is 47.02 (= 49.19 – 2.17), and multiplying this spread by the average

slope of –0.025 yields an estimated monthly return spread of –1.18%.17

Having confirmed the significantly negative relation at the individual stock level via

univariate Fama and MacBeth (1973) regressions, we next control for a battery of stock

characteristics. Following Fama and French (1992, 1993), Jegadeesh and Titman (1993), and

17Note that the ordinary least squares (OLS) methodology used in the Fama-MacBeth regressions givesan equal weight to each cross-sectional observation so that the regression results are more aligned with theequal-weighted portfolios. That is why the economic significance of REG obtained from Fama-MacBethregressions, –1.18% per month, is higher than the –0.76% per month obtained from the value-weightedportfolios (see Table 1).

24

Carhart (1997), we first control for the market beta, size, and book-to-market, and then

add momentum, which becomes our baseline multivariate regression specification. Then, we

sequentially add the remaining nine controls one by one. Several observations are worth

mentioning regarding the control variables. Consistent with earlier studies, the size effect

is negative and significant, the value effect is positive and significant, and stocks exhibit

intermediate-term momentum and short-term reversals. The negative idiosyncratic volatility

(IVOL) effect remains a puzzle, whereas lottery demand (LTRY ) becomes statistically

insignificant due to a possible multicollinearity problem between IVOL and LTRY.18 In

line with Fama and French (2015) and Hou, Xue, and Zhang (2015), stocks with high

(low) profitability (asset growth) generate high (low) future returns. Finally, the other

firm-specific characteristics, namely the market beta, illiquidity, and co-skewness, do not

display a significant relation with future equity returns.

The last column in Table 5, Panel A, presents results from the most comprehensive

regression specification combining all of the 13 return predictors examined. After controlling

for a wide range of stock characteristics, we still document a negative and robust relation

between stocks’ regret prospects and their future returns. The negative average slope of

–0.009 (t-stat. = –3.01) on REG in the last column represents a sizable economic effect of

0.42% per month, controlling for everything else. Overall, these results show that anticipated

regret has distinct, significant information beyond long-established firm characteristics, and

it is a strong and robust predictor of future equity returns, confirming our main finding that

anticipated regret is priced in the cross-section of individual stocks.

Panel B of Table 5 presents the time-series averages of the cross-sectional intercepts

and slope coefficients of Fama-MacBeth regressions while controlling for the industry effect

via industry dummies. Using 4-digit Standard Industrial Classification codes, we identify

ten industries as outlined in Fama and French (1997), which are Consumer Non-Durables,

Consumer Durables, Manufacturing, Energy, High Tech, Telecom, Shops, Health, Utilities,

18Note that following Kumar (2009), the lottery index (LTRY ) of an individual stock is defined as acombination of low price, high idiosyncratic volatility, and high idiosyncratic skewness characteristics. Thus,LTRY and IVOL are highly correlated.

25

and Others. For each stock i, we define a dummy variable for each industry classification,

taking on the value of one if the stock operates in the industry, and zero otherwise. The

economically large and statistically significant average slope coefficients on REG presented

in Panel B of Table 5 indicate that the anticipated regret premium remains robust after

controlling for the industry effect as well.

4.5 Transition matrix

In this section, we investigate the cross-sectional persistence of anticipated regret. Table

6 presents the average one-month-ahead portfolio transition matrix for our sample firms.

Specifically, we report the average probability that a stock in decile i (defined by the rows)

in one month will be in decile j (defined by the columns) in the subsequent month. All of

the probabilities in the transition matrix should be approximately 10% if the evolution for

anticipated regret for each stock is random and the relative magnitude of anticipated regret

in one period has no implication for the relative anticipated regret values in the subsequent

period. However, Table 6 shows that 47% of the stocks in the lowest REG decile in a certain

month continue to be in the same decile one month later. Similarly, 41% of the stocks in

the highest REG decile in a certain month continue to be in the same decile one month

later. Moreover, the stocks have a 65% probability of being in deciles 9 and 10, which exhibit

higher anticipated regret in the portfolio formation month and lower returns in the subsequent

month. These results collectively suggest that anticipated regret is a highly persistent equity

characteristic.19

Regret theory suggests that investors would pay higher prices for stocks that have exhibited

more attractive return and variance (i.e., higher return and lower variance) in the past with

the expectation that this appealing return distribution will persist in the future. The results

19Table A.5 of the online appendix presents three-month-, six-month- and 12-month-ahead transitionmatrices for anticipated regret. Similar to our findings in Table 6, 37% (27%) of stocks in the lowest (highest)REG decile in a certain month continue to be in the same decile 12 months later. Moreover, the stocks have a48% probability of being in deciles 9 and 10, which exhibit higher anticipated regret in the portfolio formationmonth and lower returns in the future. The results provide evidence for the strong cross-sectional persistenceof anticipated regret even after a 12-month gap is established between the lagged and lead REG variables.

26

indicate that the estimated historical REG values successfully predict the future REG values

and, thus, our proxy for anticipated regret does say something about the future upside

potential (in terms of higher expected return and lower expected variance) of individual

stocks.

5 Alternative Explanations of the Anticipated Regret Premium

In this section, we investigate whether costly arbitrage, investor inattention, and/or

prospect theory provide a complementary explanation for the anticipated regret premium.

5.1 Costly arbitrage

Our main result is that investors overprice equities with higher regret prospects, and

therefore, securities with high anticipated regret experience abnormally low future returns

until the mispricing vanishes. One would expect competitive arbitrageurs to identify this

opportunity and drive prices to their fundamental values. However, Shleifer and Vishny

(1997) and Pontiff (2006) indicate that real-life arbitrage is typically risky and requires capital.

Thus, limits to arbitrage, rather than a broad measure of systematic risk, may be behind

the pricing anomalies observed in the market. The prior literature generally relies on firm

size, idiosyncratic risk, and illiquidity to capture arbitrage costs (see, e.g., Amihud (2002),

Pontiff (2006), Stambaugh, Yu, and Yuan (2015)). Thus, we test if costly arbitrage (or

arbitrage risk) provides an explanation for the anticipated regret premium by investigating

the interactions between REG and firm size, idiosyncratic risk, and illiquidity. As discussed

earlier, we form value-weighted bivariate portfolios of REG and size, idiosyncratic volatility,

and illiquidity.20 Table 4 shows that after controlling for size, idiosyncratic volatility, and

illiquidity in independent sorts, the FF5 alpha spreads between the high- and low-REG

20We further investigate the impact of other variables that are associated with limits to arbitrage in theonline appendix. In particular, we form 3×10 portfolios sorted by age, institutional holdings, analyst coverage,and dispersion in analyst forecasts and anticipated regret and examine if the anticipated regret effect continuesafter controlling for these variables associated with limits to arbitrage. The results are presented in Panel Bof Table A.6.

27

portfolios remain economically large, in the range of –0.55% to –0.75% per month, and highly

significant with t-statistics ranging from –2.88 to –6.80.

In addition to the bivariate portfolio analysis, we conduct multiple size and liquidity

screens and replicate the univariate portfolio results based on the following five subsamples of

big and liquid stocks: i) the 1,000 most liquid stocks with the lowest Amihud (2002) illiquidity

measure, ii) more liquid stocks with lower than the median NYSE Amihud (2002) illiquidity

measure, iii) the 1,000 largest stocks according to their market capitalization, iv) larger stocks

with greater than the median NYSE market capitalization, and v) the S&P 500 stocks.

For each subsample of big and liquid stocks, Table 7 presents the next-month FF5 alphas

of the decile portfolios sorted by REG as well as the alpha spreads for the hedge portfolio

that is long in the decile of stocks with high-REG and short in the decile of stocks with

low-REG. The results indicate that for all five subsamples of big and liquid stocks, the

negative relation between anticipated regret and next-month returns remains robust and

significant. Specifically, the FF5 alpha spreads on the value-weighted portfolios are somewhat

lower, compared to the full CRSP sample, but they are still economically large, in the range

of –0.59% to –0.79% per month, and highly significant with t-statistics ranging from –2.54 to

–3.34. These results collectively suggest that the anticipated regret premium is not driven by

small or illiquid stocks that are costlier to arbitrage.21

The prior literature singles out idiosyncratic risk as the primary arbitrage cost. In our

analysis, rather than relying on idiosyncratic risk as the sole cost of arbitrage, we create an

arbitrage index that utilizes six additional variables that capture other dimensions of limits to

arbitrage. First, we use firm size and Amihud (2002) illiquidity measure to capture transaction

costs. Second, following Nagel (2005), we use level of institutional ownership to capture short-

sale constraints since it is easier to borrow stocks with higher institutional ownership. Third,

21Hou, Xue, and Zhang (2019) investigate 452 equity market anomalies and find that most of these anomaliesare significant due to microcap stocks with market capitalizations smaller than the 20th NYSE size percentile.When they remove microcap stocks and use value-weighted portfolios, Hou et al. (2019) find that 65% of the452 anomalies cannot pass the standard hurdle of the absolute t-value of 1.96. In response to the findings ofHou et al. (2019), our results in Table 7 show that anticipated regret premium is strong in the sample of verylarge and liquid stocks. Moreover, Table 2 provides confirming evidence by reporting a significant relationbetween anticipated regret and future returns on relatively big and liquid stocks trading at the NYSE.

28

to capture information uncertainty that may deter arbitrage, we follow Zhang (2006) and use

analyst coverage, firm age, and dispersion of analyst earnings forecasts.22 To construct the

arbitrage cost index, we sort stocks in increasing order based on their idiosyncratic volatility,

illiquidity, and dispersion of analyst forecasts into deciles since higher values of these variables

indicate higher arbitrage costs. Similarly, we sort stocks in decreasing order based on their

level of institutional ownership, analyst coverage, size, and age since lower values of these

variables indicate higher arbitrage costs. Each stock is given the corresponding score of its

decile rank for all seven variables. The arbitrage cost index (COST ) is the sum of the seven

scores so that it ranges from 7 to 70, with higher values indicating stricter limits to arbitrage.

We first sort stocks into terciles every month based on the arbitrage cost index (COST ).

Next, we divide each COST tercile into deciles based on REG to generate 3× 10 portfolios

of COST and REG. We test whether the abnormal return to the value-weighted zero-cost

portfolio that buys (sells) stocks with higher (lower) REG has a more negative one-month-

ahead alpha among the stocks that are costlier to arbitrage. Panel A of Table 8 shows that

for the low-COST tercile, the FF5 alpha is –19 basis points and insignificant with a t-statistic

of –1.02. As the value of the arbitrage cost index increases or arbitrage constraints are more

binding, the magnitude of the return to the zero-cost portfolio becomes more negative. For

the highest arbitrage cost portfolio, the FF5 alpha spread between the high- and low-REG

stocks is –0.56% with a t-statistic of –2.12. Thus, our hypothesis that the anticipated regret

premium is more pronounced among stocks with higher arbitrage costs is confirmed in the

data.

Finally, we average each of the REG-sorted decile portfolios across the three COST

terciles, producing portfolios with dispersion in REG that are similar in terms of COST. As

shown in the last column of Table 8, Panel A, controlling for the broad index of arbitrage

costs, the FF5 alpha spread on the value-weighted zero-cost portfolio that is long in the

22Firm age is defined as the number of months that the stock has been listed on the CRSP database.Following Diether, Malloy, and Scherbina (2002), analyst earnings forecast dispersion (DISP) is the standarddeviation of annual earnings-per-share forecasts scaled by the absolute value of the average outstandingforecast.

29

resulting high-REG portfolio and short in the resulting low-REG portfolio is economically

smaller but still remains statistically significant at –0.34% (t-stat. = –2.15). Hence, we

conclude that costly arbitrage provides a partial explanation to our main finding.

5.2 Investor inattention

Two market frictions that prevent public information from being incorporated into security

prices are limited investor attention and illiquidity. There is substantial empirical evidence

that investor inattention can lead to underreaction to information. These studies show that,

due to limited investor attention, stock prices underreact to public information about stock

fundamentals and characteristics.23 In this section, we investigate whether investor inattention

can provide a complementary explanation to our main findings.

Market reactions to large movements in individual stock returns and variance can generate

important insights on how the market processes information about these positive and negative

price shocks on the information efficiency of the equity market. We conjecture that large

positive and negative price shocks that generate high anticipated regret as defined in Eq.

(7) are harder to interpret by average investors compared to the direct and well-defined

information events studied in the previous literature. Thus, consistent with Hirshleifer,

Hsu, and Li (2013), who emphasize that investors would have more difficulty in processing

information that is less tangible, we conjecture that the elusive nature of anticipated regret

thus makes the investor attention constraints more likely to be binding. These constraints

would be even more binding for retail investors who are more active in equities with high

regret prospects, compared to institutional investors. As a result, the stock market can

underreact to persistence in our regret proxy, REG. Moreover, as indicated in the model of

Peng and Xiong (2006), an investor who optimizes the amount of attention would allocate

more attention to systematic shocks and less to stock-specific shocks (in some cases even

23See, e.g., Huberman and Regev (2001), Hirshleifer and Teoh (2003), Hirshleifer, Hou, Teoh, and Zhang(2004), Hou and Moskowitz (2005), Peng (2005), Barber and Odean (2008), Cohen and Frazzini (2008),Hirshleifer, Lim, and Teoh (2009), Da, Engelberg, and Gao (2011), Hirshleifer, Hsu, and Li (2013), Bali,Peng, Shen, and Tang (2014), Da, Gurun and Warachka (2014), and Han, Hirshleifer, and Walden (2017).

30

completely ignoring them). Thus, based on theories of investor attention, a case can be made

for underreaction to anticipated regret driven by stock-level price and variance shocks.

Table 3 shows that high-REG stocks have statistically lower institutional ownership than

low-REG stocks, implying that stocks with high regret prospects are largely held by retail

investors and are more likely to earn negative returns in the next month. We investigate this

hypothesis by testing whether the magnitude of the negative relation between anticipated

regret and expected returns is larger for those stocks in which retail investors are more active

compared to those stocks in which institutional investors are more active. Specifically, we sort

stocks into terciles every month based on an ascending order of the percentage of institutional

ownership (INST ). Then, we divide each INST tercile into deciles based on REG to generate

3× 10 portfolios of INST and REG. Panel A of Table A.6 in the online appendix reports the

FF5 alpha for each of the 3× 10 portfolios of INST and REG and the alpha spread between

the extreme REG deciles in each INST group. The last row in Panel A of Table A.6 shows

that the FF5 alpha of the zero-cost REG portfolio is –0.91% per month (t-stat. = –2.62) for

the low-INST portfolio, –0.50% per month (t-stat. = –1.98) for the medium-INST portfolio,

and –0.44% per month (t-stat. = –1.96) for the high-INST portfolio. The FF5 alpha for the

low-INST portfolio is more than twice as high as the FF5 alpha for the high-INST portfolio.

Thus, we conclude that the anticipated regret premium is much stronger for equities that are

more likely to be held by retail investors.

Following the attention literature, we use institutional holdings (INST ), analyst coverage

(CVRG), abnormal trading volume (ABTURN ), and firm size (SIZE ) as proxies for investor

attention. Abnormal trading volume is defined as each stock’s share turnover in month t

minus the average monthly share turnover over the past 12 months from month t − 12 to

t− 1. Stocks with low (high) institutional ownership, low (high) analyst coverage, low (high)

abnormal trading volume, and small (big) market cap are shown by earlier studies to receive

less (more) investor attention. The investor inattention hypothesis predicts that the degree of

underreaction to anticipated regret, as measured by its return predictability, should be more

pronounced for stocks that receive less investor attention. To investigate this conjecture, we

31

first sort stocks into terciles every month based on an investor attention proxy (INST, CVRG,

ABTURN, or SIZE ). Next, we divide each attention tercile into deciles based on REG to

generate 3× 10 portfolios of attention and REG. Panel A of Table A.6 in the online appendix

shows that the FF5 alpha spreads between the high- and low-REG deciles are negative, highly

significant and economically larger for stocks with weak attention-grabbing characteristics

(i.e., low institutional ownership, low analyst coverage, low abnormal trading volume, and

small market cap), compared to stocks with strong attention-grabbing characteristics. Overall,

these results indicate that the anticipated regret premium is stronger for stocks that are more

likely to be exposed to limited investor attention and held by retail investors.

We also build an index of investor attention by sorting stocks in increasing order based

on their institutional ownership, analyst coverage, abnormal trading volume, and market cap

into deciles since higher values of these variables indicate higher investor attention. Each

stock is given the corresponding score of its decile rank for all four variables. The attention

index (ATT ) is the sum of the four scores so that it ranges from 4 to 40 and its higher values

indicate stronger attention-grabbing characteristics. We first sort stocks into terciles every

month based on the attention index (ATT ). Next, we divide each ATT tercile into deciles

based on REG to generate 3× 10 portfolios of ATT and REG. We test whether the abnormal

return to the zero-cost portfolio that buys (sells) stocks with higher (lower) REG has a more

negative one-month-ahead alpha among the stocks that receive less investor attention.

Panel B of Table 8 shows that the FF5 alpha to the zero-cost REG portfolio is –1.33%

per month (t-stat. = –6.31) for the low-ATT portfolio, –0.38% per month (t-stat. = –1.98)

for the medium-ATT portfolio, and –0.24% per month (t-stat. = –0.97) for the high-ATT

portfolio. The FF5 alpha for the low-ATT portfolio is economically and statistically higher

than the FF5 alpha for the high-ATT portfolio. Hence, the anticipated regret premium is

much stronger for equities that receive less investor attention. Finally, we average each of

the REG-sorted decile portfolios across the three ATT terciles and show in the last column

of Table 8, Panel B, that controlling for the broad index of investor attention, the FF5

alpha spread on the value-weighted zero-cost portfolio that is long in the resulting high-REG

32

portfolio and short in the resulting low-REG portfolio remains economically large and highly

significant at –0.65% (t-stat. = –4.03). Thus, we conclude that investor inattention provides

a limited explanation to the underperformance of high-REG stocks.

5.3 Prospect theory

There is strong evidence that investors have a preference for lottery-like assets with a

small probability of a large payoff, which results in overpricing of such assets and low future

returns (e.g., Kumar 2009; Bali, Cakici, and Whitelaw 2011). Preference for lottery-like

payoffs is consistent with cumulative prospect theory (Tversky and Kahneman (1992)) as

modeled by Barberis and Huang (2008), which predicts that errors in investors’ probability

weighting cause them to overvalue stocks that have a small probability of a large positive

return.24 Brunnermeier and Parker (2005), Brunnermeier, Gollier, and Parker (2007), and

Barberis and Huang (2008) show that investors’ preference for positive skewness can cause

high valuations and low future returns for lottery-like securities. These theories are also

supported by empirical evidence. Bali, Cakici, and Whitelaw (2011) demonstrate that the

maximum daily return in a month (a proxy for lottery-like payoffs) is negatively related to

future raw and risk-adjusted stock returns. Green and Hwang (2012) find that IPOs with high

expected skewness experience significantly greater first-day returns, but earn more negative

abnormal returns in the following one to five years. Barberis, Mukherjee, and Wang (2016)

present empirical evidence supporting the predictions of prospect theory. Wang, Yan, and

Yu (2017) and An, Wang, Wang, and Yu (2018) investigate the relation between lottery

characteristics and the future returns of individual stocks based on reference-dependent

preferences. These studies provide evidence that investors’ preference can induce them to pay

high prices for securities with strong lottery-like features, leading to negative future abnormal

returns for such securities.

In this section, we test whether the negative relation between anticipated regret and future

24A notion of lottery demand can be found in the seminal work of Kahneman and Tversky (1979) and isalso consistent with realization theory proposed by Barberis and Xiong (2012).

33

equity returns is consistent with prospect theory and if it remains significant after controlling

for alternative measures of lottery demand. Our first proxy for lottery-like payoffs, adapted

from Kumar (2009), is the lottery index (LTRY ) that defines lottery stocks as low-priced

stocks with high idiosyncratic volatility and high idiosyncratic skewness, as described in

Section 3.3. Our second measure of lottery demand, borrowed from Bali, Cakici, and Whitelaw

(2011), is the maximum daily return of a stock in the previous month (MAX ). Our third

proxy is acquired from Barberis, Mukherjee, and Wang (2016), who compute the prospect

theory value of the stock’s historical return distribution, denoted TK, using the past five years

of monthly data.25 Finally, we follow Bali, Brown, Murray, and Tang (2017) and measure the

stock’s ex-ante lottery-like feature with the average of the five highest daily returns in each

month (MAX5 ).

The prospect theory hypothesis predicts that the anticipated regret premium should be

more pronounced for stocks with high lottery-like payoffs. To investigate this conjecture,

we build an index of lottery demand by sorting stocks in increasing order based on their

LTRY, MAX, TK, and MAX5 values into deciles since higher values of these variables

indicate stronger lottery payoffs (or higher prospect theory values). Each stock is given the

corresponding score of its decile rank for all four variables. The lottery demand index (LDI )

is the sum of the four scores so that it ranges from 4 to 40 and its higher values indicate

stronger lottery features. We first sort stocks into terciles every month based on the lottery

demand index (LDI ). Next, we divide each LDI tercile into deciles based on REG to generate

3 × 10 portfolios of LDI and REG. Panel C of Table 8 shows that the zero-cost portfolio

that buys (sells) stocks with higher (lower) REG has a more negative FF5 alpha among the

stocks with higher lottery payoffs, indicating that the anticipated regret premium is much

stronger for equities with stronger lottery-like features.

Finally, we average each of the REG-sorted decile portfolios across the three LDI terciles

and report in the last column of Table 8, Panel C, controlling for the extended index of

25See Barberis, Mukherjee, and Wang (2016) for the computation of the prospect theory value of anindividual stock. Barberis et al. (2016) show that a stock whose past return distribution has a high (low)prospect theory value earns a low (high) future average return.

34

lottery demand. The FF5 alpha spread on the value-weighted zero-cost portfolio that is long

in the resulting high-REG portfolio and short in the resulting low-REG portfolio remains

significant at –0.36% (t-stat. = –2.41). Overall, these results indicate that the negative

relation between anticipated regret and future equity returns is consistent with the prospect

theory. However, prospect theory does not fully explain the anticipated regret premium,

which remains significant after controlling for prospect theory values and lottery-like payoffs.

Prospect theory assumes that the benchmark that investors use to compare their wealth

is defined by the past, i.e., an investor’s utility depends on her gains/losses in comparison to

her wealth in the previous period, while in regret theory the benchmark is not fixed ex-ante

but rather depends on the future states of the world. Hence, the main assumption of regret

theory is that people, after making decisions under uncertainty, may have regrets if their

decisions turn out to be wrong even if they appeared correct with the information available

ex-ante.26 Our findings collectively suggest that if a stock’s past return distribution (relevant

for prospect theory) is different from its expected future return distribution (relevant for

regret theory), regret and prospect theories can make different empirical predictions.

6 Robustness Checks and Additional Analyses

In this section, we perform a battery of robustness checks and additional analyses. First,

we conduct subperiod analyses to examine whether the predictive power of anticipated regret

is driven by good vs. bad states of the economy. Second, we test the significance of the

anticipated regret premium across different industries. Third, we examine the long-term

predictive power of anticipated regret. Finally, we investigate whether alternative measures

of anticipated regret yield comparable results.

26In our framework, when making investment decisions, regret-averse investors think about future gainsand losses, whereas in Barberis et al. (2016), investors apply prospect theory to stock-level gains and losses(narrow framing), and react to past gains and losses. In other words, in our framework, regret-averse investorsoverprice equities whose future return distributions are appealing under regret theory, whereas in Barberis etal. (2016), investors overprice equities whose past return distributions are appealing under prospect theory.

35

6.1 Subperiod analysis

This section explores whether the negative relation between anticipated regret and next-

month stock returns is driven by different characteristics of the overall market. In particular,

we divide our full sample period from July 1963 to December 2018 into two subperiods defined

by economic activity (high vs. low economic activity), stock market moves (up vs. down

markets), and economic uncertainty (high vs. low economic uncertainty).

We classify different cycles of the economy using the three-month moving average of the

Chicago Fed National Activity Index (CFNAIMA3).27 We identify months with CFNAIMA3

index less than or equal to (greater than) –0.7 as months with low (high) economic activity

corresponding to recessions (expansions). We further define up (down) markets with months

when the excess market return is above (below) zero. Finally, our definition of economic

uncertainty follows Jurado, Ludvigson, and Ng (2015) (hereafter JLN). We define months

with high (low) economic uncertainty when the JLN economic uncertainty index is greater

(lower) than its median over the full sample period.

For each subperiod (high vs. low economic activity, up vs. down market, high vs. low

economic uncertainty), Table 9 presents the next-month FF5 alphas of the decile portfolios

sorted by REG as well as the alpha spreads for the hedge portfolio that is long in the decile

of stocks with the highest REG and short in the decile of stocks with the lowest REG. We

find that the negative relation between anticipated regret and next-month returns is most

significant in periods of low economic activity (recessions) and in periods with high economic

uncertainty. The alpha spread of High−Low REG portfolios is much larger in magnitude

during bad states of the economy: –1.46% vs. –0.50% during recessions vs. expansions, and

–0.68% vs. –0.45% during high vs. low economic uncertainty periods. Our results suggest that

the effect of anticipated regret on the cross-sectional pricing of equities is amplified during

bad states of the economy.

27https://www.chicagofed.org/research/data/cfnai/historical-data

36

6.2 Industry analysis

Investors might view stocks in different industries with different regret prospects. Stocks

in certain industries might exhibit higher (lower) REG, and thus investors might regret more

(less) holding stocks in those industries. This section explores whether the negative relation

between anticipated regret and next-month stock returns vary across different industries.

Using the industry breakpoints of Fama and French (1997), we identify ten industries as

Consumer Non-Durables, Consumer Durables, Manufacturing, Energy, High Tech, Telecom,

Shops, Health, Utilities, and Others. Table 10 presents the next-month FF5 alphas of the

decile portfolios sorted by REG as well as the alpha spreads for the hedge portfolio that is

long in the decile of stocks with the highest REG and short in the decile of stocks with the

lowest REG across the ten industries. Consistent with our main results, we find that the alpha

spreads of High−Low REG portfolios are negative and significant across most industries.

However, the effect of anticipated regret is weak (insignificant) for energy (utility) industries.

Given our previous finding that the anticipated regret premium is driven by stocks in the

highest REG decile and that stocks in the highest REG decile of energy (utility) industries

exhibit weak (insignificant) returns, the results suggest that investors do not anticipate as

much regret as in other industries while investing in stocks in energy and utility sectors.

6.3 Long-term predictability

Fama-MacBeth regressions in Section 4.4 produce a negative and strong relation between

anticipated regret and one-month-ahead stock returns. We now investigate whether this

negative relation holds for longer-term returns. To that end, we run monthly cross-sectional

regressions of Eq. (16) using the full set of control variables, this time using two- to 12-month

lagged anticipated regret as the main explanatory variable. Other controls are lagged by one

month.

Table 11 presents the time-series average of the cross-sectional intercepts and slope

coefficients of these longer-term regressions together with the one-month lagged anticipated

37

regret results documented in column 13 of Table 5. The results show that the negative

and significant relation between REG and future returns is not limited to one-month-ahead

returns, but also extends to longer-term returns. Anticipated regret can predict excess returns

up to eight months into the future. These results together with the high persistence of

the anticipated regret effect documented in Section 4.5 suggest that the anticipated regret

premium is not a one-month affair, but lasts all the way up to eight months.

6.4 Components of anticipated regret

In Section 2, under the assumption of additivity for utility and regret functions, we identify

two components of anticipated regret; return-regret (RREG) and variance-regret (VREG).

In this section, using the RREG and VREG variables defined in Eq. (11) and Eq. (12)

respectively, we test hypotheses H2 and H3 as outlined in Section 2.

For each month from July 1963 to December 2018, we sort stocks into decile portfolios

based on their RREG (VREG), where decile 1 (decile 10) contains stocks with the lowest

(highest) RREG (VREG). Next, we calculate the one-month-ahead value-weighted average

portfolio returns.28

Columns 2–4 (5–7) in Panel A of Table 12 report for each decile the average RREG

(VREG), the next-month average excess return, and the FF5 alpha. The last row in columns

3–4 (6–7) of Panel A presents the average return and alpha spreads for the hedge portfolio

that is long in the decile of stocks with the highest RREG (VREG) and short in the decile of

stocks with the lowest RREG (VREG).

In line with our main results documented in Section 4, Table 12 shows that investors

anticipate more regret for not holding stocks with high return-regret in their portfolios.

Columns 3 and 4 in Panel A of Table 12 document that the raw return (FF5 alpha) spread

28Note that RREG defined in Eq. (11) takes on non-positive values by construction. Without loss ofgenerality, in line with our methodology for anticipated regret (REG), we use the absolute value of return-regret(RREG) in our tests so that stocks with high return-regret have a larger (positive) distance between theirforegone (i.e., the maximum daily return in a month) and expected (i.e., the average daily return in a month)return. Using the absolute value of return-regret (RREG) does not affect the implications and interpretationof our theoretical framework presented in Section 2.

38

on the High−Low RREG portfolio is negative and significant, and the significance of the

alpha spread of the arbitrage portfolio is driven by stocks in the highest RREG portfolio.

Furthermore, we document that mean-variance, regret-averse investors anticipate more regret

for not holding stocks with low variance-regret in their portfolios. Columns 6 and 7 in Panel A

of Table 12 show that the raw return (FF5 alpha) spread on the High−Low VREG portfolio

is negative and significant, and the significance of the alpha spread of the arbitrage portfolio

is driven by stocks in the highest VREG portfolio.

The results suggest that investors view stocks with high return-regret, i.e. stocks with a

large spread of µi −max[µi], as lottery stocks that investors regret not buying because of

their persistently high positive returns. Due to their regret aversion, investors increase their

demand and pay high prices for stocks with high anticipated return-regret. As a consequence,

overpricing leads to negative future returns for such stocks. Similarly, mean-variance, regret-

averse investors view stocks with high variance-regret, i.e. stocks with a large spread of

σ2i −min[σ2

i ], as more attractive and regret not buying them. Because of this anticipated regret,

regret-averse investors increase their demand and pay high prices for high variance-regret

stocks, which in turn leads to low subsequent returns for such stocks.

We next investigate whether return-regret and variance-regret capture similar or different

information regarding investors’ anticipated regret. To do that, we independently sort stocks

into 100 (10×10) portfolios based on their RREG and VREG. Subsequently, we average each

of the RREG-sorted portfolios across the ten deciles sorted by VREG producing portfolios

with dispersion in return-regret that are similar in terms of variance-regret. We repeat the

corresponding procedure and average each of the VREG-sorted portfolios across the ten deciles

sorted by RREG producing portfolios with dispersion in variance-regret that are similar in

terms of return-regret. In addition, we form two arbitrage portfolios, i.e., the High−Low

RREG (VREG) portfolios, that are long in the resulting high-RREG (high-VREG) portfolio

and short in the resulting low-RREG (low-VREG) portfolio.

Panel B of Table 12 shows that the FF5 alpha spread on the High−Low RREG portfolio

is negative and significant after controlling for variance-regret. Furthermore, the FF5 alpha

39

spread on the High−Low VREG portfolio remains negative and significant after controlling for

return-regret. The results suggest that both the return-regret and variance-regret components

are important drivers of anticipated regret premium for mean-variance, regret-averse investors.

6.5 An alternative measure of anticipated regret

The anticipated regret measure in Eq. (7) and its two components in Eq. (11) and (12),

respectively, assume that investors use average daily return in a month as a benchmark for

return-regret and they use average daily conditional variance in a month as a benchmark

for variance-regret, and then compare these benchmarks with the same stock’s best possible

return and variance, respectively. In other words, when making decisions across stocks, the

benchmark is the time-series average of the same stock’s own mean-variance performance. In

Section 3.2, we propose an alternative measure of anticipated regret, where the benchmark for

investors is the cross-sectional average of daily return and daily variance of all stocks. Using

this alternative benchmark and a risk-aversion coefficient of b = 5, we have the following

measure of anticipated regret:

CREGi,t = (µt −max[ri,d])−5

2(σ2

t −min[σ2i,d]), d = 1, 2, . . . , T (17)

where d denotes trading days in month t, µt is the cross-sectional equal-weighted average of

the daily returns of all stocks in month t, σ2t is the cross-sectional equal-weighted average of

the average daily conditional variances of all stocks in month t, and max[ri,d] and min[σ2i,d]

are the maximum daily return and the minimum daily variance of stock i attained in month

t.

Similar to our analysis in Section 4.1, we form univariate portfolios of stocks sorted by

CREG over the period from July 1963 to December 2018, where decile 1 (decile 10) contains

stocks with the lowest (highest) CREG. Columns 1 and 2 of Table 13 report for each decile

the average CREG and the FF5 alpha. The last row presents the alpha spread for the hedge

portfolio that is long in the decile of stocks with the highest CREG and short in the decile of

40

stocks with the lowest CREG.

Using an alternative measure of anticipated regret based on the cross-sectional averages

of the daily return and the daily conditional variance of all stocks as the return and variance

benchmarks in investors’ anticipated regret function, we find similar results. Univariate

sorts based on all stocks in our sample indicate a significantly negative relation between

anticipated regret and next-month average returns. The value-weighted arbitrage portfolio

with a long position in the highest CREG stocks and a short position in the lowest CREG

stocks (High−Low CREG) loses on average 0.41% risk-adjusted return per month with a

t-statistic of –3.09.

In order to make sure that the results are not driven by stocks with short-sale constraints

or with limits to arbitrage, we perform univariate portfolio analysis using samples that exclude

small and illiquid stocks. Columns 3–7 of Table 13 report the results of univariate sorts using

the NYSE stocks only (column 3), removing the low-priced NYSE stocks trading below $5 per

share (column 4), removing the smaller NYSE stocks with market capitalizations that place

them in the smallest NYSE size decile (column 5), excluding the NYSE stocks that belong to

the lowest NYSE liquidity decile (column 6), and imposing the three screens simultaneously

by removing the low-priced, small, and illiquid NYSE stocks from our sample (column 7).

Screening for low-priced, small, and illiquid stocks does not change the negative and

significant relation between CREG and next-month stock returns. The FF5 alpha spread

between the high-CREG and low-CREG portfolios ranges from –0.44% to –0.52% per month

with t-statistics ranging from –2.52 to –3.19, respectively. Even when we exclude low-priced,

small, and illiquid stocks altogether from the NYSE sample (i.e., the part of the NYSE sample

that is much less prone to short-sale constraints), the corresponding alpha spread is still

significantly negative at –0.35% per month (t-stat. = –2.52). Overall, these results suggest

that the anticipated regret premium is not fully explained by short-sale constraints or limits

to arbitrage.

The cross-sectional return-variance benchmark in investors’ anticipated regret function

implies that mean-variance, regret-averse investors view stock i as more attractive than stock

41

j if stock i’s MAX-return performance with respect to all other stocks’ average daily return

is higher than stock j’s MAX-return performance with respect to all other stocks’ average

daily return. Similarly, stock i will be viewed as more attractive than stock j if stock i’s

MIN-variance performance with respect to all other stocks’ cross-sectional average variance

is better (lower) than stock j’s MIN-variance performance with respect to all other stocks’

cross-sectional average variance. Hence, using an alternative cross-sectional return-variance

benchmark, the results suggest not only that anticipated regret arising from the comparison

of the stock’s return and variance with the same stock’s own maximum return and minimum

variance is important, but also a cross-sectional comparison of the maximum return and

minimum variance of the stock is an important determinant of anticipated regret, and hence

investors’ financial decision making.

7 Conclusion

This paper examines the cross-sectional asset pricing implications of anticipated regret for

mean-variance investors. We propose a theoretical framework based on an extension of the

modified expected utility function a la Bell (1982) and Loomes and Sugden (1982) and show

that not only the expected return and variance of returns matters for regret-averse investors,

but also the best foregone return and variance become an important factor in determining

investors’ financial decision making. Using a novel measure of anticipated regret (REG), we

investigate whether anticipated regret predicts the cross-sectional variation in future stock

returns.

First, we document that anticipated regret is negatively related to the cross-section of

future equity returns. Sorting stocks into value-weighted portfolios based on their REG,

we show that stocks with high anticipated regret underperform stocks with low anticipated

regret. The results indicate that mean-variance, regret-averse investors anticipate more regret

from not investing or missing the opportunity to invest in stocks with high regret prospects.

Thus, they demand stocks with high regret prospects because not having such stocks in their

42

portfolio is expected to decrease investors’ modified utility function more than other stocks.

As a result, stocks with high regret prospects earn lower returns in equilibrium.

Second, the negative relation between anticipated regret and future returns is robust

to using alternative factor models in the calculation of risk-adjusted returns (alphas), an

alternative portfolio weighting scheme, controlling for different stock characteristics, screening

out small, illiquid, and low-priced stocks, and over different periods and stock samples.

Multivariate Fama and MacBeth (1973) regressions that simultaneously control for individual

stock characteristics further corroborate our main finding that anticipated regret is an

important determinant of the cross-sectional dispersion in equity returns.

Finally, we show that the two components of anticipated regret, return-regret and variance-

regret, are also important determinants of investors’ financial decision-making. The results

suggest that mean-variance, regret-averse investors anticipate regret for not holding stocks

with high return-regret and high variance-regret, confirming the hypothesis that comparing

individual stocks’ average return (average variance) with their maximum return (minimum

variance) is an important element of the decision making process of regret-averse investors.

43

References

Amihud, Y., 2002. “Illiquidity and stock returns: Cross-section and time-series effects.”Journal of Financial Markets 5, 31–56.

An, L., Wang, H., Wang, J., Yu, J., 2018. “Lottery-related anomalies: The role of referencedependent preferences.” Management Science forthcoming.

Ang, A., Hodrick, R. J., Xing, Y., Zhang, X., 2006. “The cross-section of volatility andexpected returns.” Journal of Finance 61, 259–299.

Bali, T. G., 2008. “The intertemporal relation between expected returns and risk.” Journalof Financial Economics 87, 101–131.

Bali, T. G., Engle, R. F., 2010. “The intertemporal capital asset pricing model withdynamic conditional correlations.” Journal of Monetary Economics 57, 377–390.

Bali, T. G., Cakici, N., Whitelaw, R. F., 2011. “Maxing out: Stocks as lotteries and thecross-section of stock returns.” Journal of Financial Economics 99, 427–446.

Bali, T. G., Peng, L., Shen, Y., Tang, Y., 2014. “Liquidity shocks and stock marketreactions.” Review of Financial Studies 27, 1434-1485.

Bali, T. G., Brown, S., Murray, S., Tang, Y., 2017. ““A lottery demand-based explanationof the beta anomaly.” Journal of Financial and Quantitative Analysis 52, 2369–2397.

Barber, B., Odean, T., 2008. “All that glitters: The effect of attention on the buyingbehavior of individual and institutional investors.” Review of Financial Studies 21,785–818.

Barberis, N., Huang, M., 2008. “Stocks as lotteries: The implications of probability weightingfor security prices.” American Economic Review 98, 2066-2100.

Barberis, N., Xiong, W., 2012. “Realization Utility.” Journal of Financial Economics 104,251-271.

Barberis, N., Mukherjee, A., Wang, B., 2016. “Prospect theory and stock returns: Anempirical test.” Review of Financial Studies 29, 3068-3107.

Bekaert, G., Harvey, C. R., 1997. “Emerging equity market volatility.” Journal of FinancialEconomics 43, 29–77.

Bell, D. E., 1982. “Regret in decision making under uncertainty.” Operations Research 29,1156–1166.

Bell, D. E., 1983. “Risk premiums for decision regret.” Management Science 56, 961–981.

Bikhchandani, S., Segal, U., 2014. “Transitive regret over statistically independent lotteries.”Journal of Economic Theory 152, 237–248.

Bleichrodt, H., Cillo, A., Diecidue, E., 2010. “A quantitative measurement of regret theory.”Management Science 56, 161–175.

Bollerslev, T., 1986. “Generalized autoregressive conditional heteroskedasticity.” Journal ofEconometrics 31, 307–327.

Brunnermeier, M. K., Parker, J. A., 2005. “Optimal expectations.” American EconomicReview 95, 1092-1118.

44

Brunnermeier, M. K., Gollier, C., Parker, J. A., 2007. “Optimal beliefs, asset prices, andthe preference for skewed returns.” American Economic Review 97, 159-165.

Buturak, G., Evren, O., 2017. “Choice overload and asymmetric regret.” TheoreticalEconomics 12, 1029–1056.

Camille, N., Coricelli, G., Sallet, J., Pradat-Diehl, P., Duhamel, J. R., Sirigu, A., 2004. “Theinvolvement of orbitofrontal cortex in the experience of regret.” Science 304, 1167–1170.

Carhart, M., 1997. “On persistence in mutual fund performance.” Journal of Finance 52,57–82.

Cohen, L., Frazzini, A., 2008. “Economic links and predictable returns.” Journal of Finance63, 1977–2011.

Coricelli, G., Critchley, H. D., Joffily, M., Sirigu, A., Dolan, R. J., 2005. “Regret andits avoidance: A neuroimaging study of choice behavior.” Nature Neuroscience 8,1255–1262.

Da, Z., Engelberg, J., Gao, P., 2011. “In search for attention.” Journal of Finance 66,1461–1499.

Da, Z., Gurun, U. G., Warachka, M., 2011. “Frog in the pan: Continuous information andmomentum.” Review of Financial Studies 27, 2171–2218.

D’Avolio, G., 2002. “The market for borrowing stock.” Journal of Financial Economics 66,271–306.

Diecidue, E., Somasundaram, J., 2017. “Regret theory: A new foundation.” Journal ofEconomic Theory 172, 88–119.

Diether, K. B., Malloy, C. J., Scherbina, A., 2002. “Differences of opinion and the crosssection of stock returns.” Journal of Finance 57, 2113-2141.

Engelbrecht-Wiggans, R., 1989. “The effects of regret on optimal bidding in auctions.”Management Science 35, 685–692.

Engelbrecht-Wiggans, R., Katok, E., 2008. “Regret and feedback information in first-pricesealed-bid auctions.” Management Science 54, 808–819.

Engle, R. F., Ng, V. K., 1993. “Measuring and testing the impact of news on volatility.”Journal of Finance 48, 1749-1778.

Fama, E. F., MacBeth, J., 1973. “Risk, return and equilibrium: Empirical tests.” Journalof Political Economy 81, 607–636.

Fama, E. F., French, K. R., 1992. “The cross-section of expected stock returns.” Journalof Finance 46, 427–466.

Fama, E. F., French, K. R., 1993. “Common risk factors in the returns on stocks andbonds.” Journal of Financial Economics 33, 3–56.

Fama, E. F., French, K. R., 1997. “Industry costs of equity.” Journal of Financial Economics43, 153–193.

Fama, E. F., French, K. R., 2015. “A five-factor asset pricing model.” Journal of FinancialEconomics 116, 1–22.

Fama, E. F., French, K. R., 2016. “Dissecting anomalies with a five-factor model.” Reviewof Financial Studies 29, 69–103.

45

Filiz-Ozbay, E., Ozbay, E. Y., 2007. “Auctions with anticipated regret: Theory andexperiment.” American Economic Review 97, 1407–1418.

Fioretti, M. Vostroknutov, A., Coricelli, G., 2018. “Dynamic regret avoidance.” WorkingPaper, University of Southern California.

Fogel, S. O., Berry, T., 2006. “The disposition effect and individual investor decisions:The roles of regret and counterfactual alternatives.” Journal of Behavioral Finance 7,107–116.

Frazzini, A., Pedersen, L. H., 2014. “Betting against beta.” Journal of Financial Economics111, 1–25.

Frydman, C., Camerer, C., 2016. “Neural evidence of regret and its implications for investorbehavior.” Review of Financial Studies 29, 3108–3139.

Ghysels, E., Santa-Clara, P., Valkanov, R., 2005. “There is a risk-return trade-off after all.”Journal of Financial Economics 76, 509–548.

Gilovich, T., Medvec, V. H., 1995. “The experience of regret: What, when, and why.”Psychological Review 102, 379–395

Glosten, L. R., Jagannathan, R., Runkle, D.E., 1993. “On the relation between the expectedvalue and the volatility of the nominal excess return on stocks.” Journal of Finance 48,1779–1801.

Gollier, C., 2019. “Aversion to risk of regret and preference for positively skewed risks.”Economic Theory, forthcoming.

Green, T. C., Hwang, B.-H., 2012. “IPOs as lotteries: Skewness preference and first-dayreturns.” Management Science 58, 432–444.

Gul, F., 1991. “A theory of disappointment aversion.” Econometrica 59, 667–686.

Guo, H., Whitelaw, R. F., 2006. “Uncovering the risk-return relation in the stock market.”Journal of Finance 61, 1433–1463.

Han, B., Hirshleifer, D. A., Walden, J., 2019. “Social transmission bias and investorbehavior.” Working paper, Rotman School of Management.

Han, B., Kumar, A., 2008. “Speculative retail trading and asset prices.” Journal ofFinancial and Quantitative Analysis 48, 377–404.

Hansen, P. R. Lunde, A. 2005. “A forecast comparison of volatility models: Does anythingbeat a GARCH (1,1)?” Journal of Applied Econometrics 20, 873-899.

Harvey, C. R., Siddique, A., 2000. “Conditional skewness in asset pricing tests.” Journal ofFinance 55, 1263–1295.

Harvey, C. R., Liu, Y., Zhu, H., 2016. “... and the cross-section of expected returns.”Review of Financial Studies 29, 5–68.

Hayashi, T., 2008. “Regret aversion and opportunity dependence.” Journal of EconomicTheory 139, 242–268.

Hazan, E., Kale, S., 2015. “An online portfolio selection algorithm with regret logarithmicin price variation.” Mathematical Finance 25, 288–310.

Hirshleifer, D., Teoh, S. H., 2003. “Limited attention, information disclosure, and financialreporting” Journal of Accounting and Economics 36, 337–386.

46

Hirshleifer, D., Hou, K., Teoh, S. H., Zhang, Y., 2004. “Do investors overvalue firms withbloated balance sheets?” Journal of Accounting and Economics 38, 297–331.

Hirshleifer, D., Lim, S., Teoh, S. H., 2009. “Driven to distraction: Extraneous events andunderreaction to earnings news.” Journal of Finance 64, 2289–2325.

Hirshleifer, D., Hsu, P.-H., Li, D., 2013. “Innovative efficiency and stock returns.” Journalof Financial Economics 107, 632–654.

Hou, K., Moskowitz, T., 2005. ““Market frictions, price delay, and the cross-section ofexpected returns” Review of Financial Studies 18, 981–1020.

Hou, K., Xue, C., Zhang, L., 2015. “Digesting anomalies: An investment approach.” Reviewof Financial Studies 28, 650–705.

Hou, K., Xue, C., Zhang, L., 2019. “Replicating anomalies.” Review of Financial Studiesforthcoming.

Jegadeesh, N., 1990. “Evidence of predictable behavior of security returns.” Journal ofFinance 45, 881–898.

Huberman, G., Regev, T., 2001. “Contagious speculation and a cure for cancer: A non-eventthat made stock prices soar.” Journal of Finance 56, 387–396.

Jegadeesh, N., Titman, S., 1993. “Returns to buying winners and selling losers: Implicationsfor stock market efficiency.” Journal of Finance 48, 65–91.

Jiang, B., Narasimhan, C., Turut, O., 2017 “Anticipated regret and product innovation.”Management Science 63, 4308–4323.

Jurado, K., Ludvigson, S., Ng, S., 2015. “Measuring uncertainty.” American EconomicReview 105, 1177–1216.

Kahneman, D., Tversky, A., 1979. “Prospect theory: An analysis of decision under risk.”Econometrica 47, 263–291.

Kumar, A., 2009. “Who gambles in the stock market?” Journal of Finance 64, 1889–1933.

Kumar, A., Page, J. K., Spalt, O. G., 2016. “Gambling and comovement.” Journal ofFinancial and Quantitative Analysis 51, 85–111.

Leland, J. W., 1998. “Similarity judgments in choice under uncertainty: A reinterpretationof the predictions of regret theory.” Management Science 44, 659–672.

Loomes, G., Sugden, R., 1982. “Regret theory: An alternative theory of rational choiceunder uncertainty.” Economic Journal 92, 805–824.

Michenaud, S., Solnik, B., 2008. “Applying regret theory to investment choices: Currencyhedging decisions.” Journal of International Money and Finance 27, 677–694.

Muermann, A., Mitchell, O., Volkman, J., 2006. Regret, portfolio choice, and guarantees indefined contribution schemes.” Insurance: Mathematics and Economics 39, 219–229.

Nagel, S., 2005. “Short sales, institutional investors and the cross-section of stock returns.”Journal of Financial Economics 78, 277–309.

Nasiry, J., Popescu, I., 2012. “Advance selling when consumers regret.” Management Science58, 1160–1177.

47

Newey, W. K., West, K. D., 1987. “A simple, positive semi-definite, heteroscedasticity andautocorrelation consistent covariance matrix.” Econometrica 55, 703–708.

Orphanides, A., Zervos, D., 1995. “Rational addiction with learning and regret.” Journalof Political Economy 103, 739–758.

Pastor, L., Stambaugh, R. F., 2003. “Liquidity risk and expected stock returns.” Journalof Political Economy 111, 642–685.

Peng, L., 2005. “Learning with information capacity constraints.” Journal of Financial andQuantitative Analysis 40, 307–330.

Pontiff, J., 2006. “Costly arbitrage and the myth of idiosyncratic risk.” Journal of Accountingand Economics 42, 35–52.

Qin, J., 2015. “A model of regret, investor behavior, and market turbulence.” Journal ofEconomic Theory 160, 150–174.

Quiggin, J., 1994. “Regret theory with general choices.” Journal of Risk and Uncertainty 8,153–165.

Sarver, T., 2008. “Anticipating regret: Why fewer options may be better.” Econometrica76, 263–305.

Savage, L. J., 1951. “The theory of statistical decision.” Journal of the American StatisticalAssociation 46, 55–67.

Shleifer, A., Vishny, R. W., 1997. “The limits of arbitrage” Journal of Finance 52, 35–55.

Solnik, B., Zuo, L., 2012. “A global equilibrium asset pricing model with home preference.”Management Science 58, 273–292.

Somasundaram, J., Diecidue, E., 2018. “Regret theory and risk attitudes.” Journal of Riskand Uncertainty, forthcoming.

Stambaugh, R. F., Yu, J., Yuan, Y., 2015. “Arbitrage asymmetry and the idiosyncraticvolatility puzzle.” Journal of Finance 70, 1903–1948.

Tversky, A., Kahneman, D., 1992. “Advance in prospect theory: Cumulative representationof uncertainty.” Journal of Risk and Uncertainty 5, 297–323.

Wang, H., Yan, J., Yu, J., 2017. “Reference-dependent preferences and the risk-returntradeoff.” Journal of Financial Economics 123, 395–414.

Zeelenberg, M., Pieters, R., 2007. “A theory of regret regulation 1.0.” Journal of ConsumerPsychology 17, 3–18.

Zhang, X. F., 2006. “Information uncertainty and stock returns.” Journal of Finance 61,105–137.

48

Table 1Univariate portfolios of stocks sorted by REG

Each month from July 1963 to December 2018, stocks are sorted into decile portfolios basedon their REG. This table reports the average REG values, the next-month value-weightedaverage excess returns (Mean), the CAPM, Fama-French 3-factor (FF3), Fama-French-Carhart4-factor (FFC), Fama-French-Carhart-Pastor-Stambaugh 5-factor (FFCPS), Fama-French5-factor (FF5), and q-factor (Q) risk-adjusted returns (alphas) for the 10 decile portfolios ofstocks sorted by REG. The last row presents the 10–1 average raw and risk-adjusted returndifferences of the arbitrage portfolio with a long position in the decile of stocks with thehighest REG (decile 10) and a short position in the decile of stocks with the lowest REG(decile 1). The Newey-West (1987) adjusted t-statistics (with six lags) are given in squarebrackets.

Decile REG Mean CAPM FF3 FFC FFCPS FF5 Q

1 (Low) 2.17 0.46 0.14 0.07 0.02 0.02 −0.04 −0.03

[3.35] [1.89] [1.00] [0.30] [0.29] [−0.67] [−0.29]

2 3.71 0.56 0.14 0.10 0.06 0.07 −0.01 0.00

[3.50] [2.21] [1.78] [0.96] [0.99] [−0.11] [0.01]

3 4.90 0.52 0.05 0.02 0.00 −0.01 −0.09 −0.09

[2.96] [0.94] [0.30] [0.03] [−0.16] [−1.79] [−1.40]

4 6.12 0.59 0.07 0.03 0.04 0.02 −0.02 −0.06

[3.09] [1.16] [0.59] [0.81] [0.38] [−0.43] [−1.01]

5 7.53 0.71 0.14 0.13 0.14 0.12 0.11 0.08

[3.46] [2.09] [2.06] [2.35] [1.74] [1.50] [0.92]

6 9.27 0.64 0.04 0.01 0.04 0.05 0.03 0.05

[2.89] [0.56] [0.15] [0.61] [0.61] [0.50] [0.63]

7 11.56 0.52 −0.14 −0.13 −0.09 −0.07 −0.01 0.03

[2.15] [−1.44] [−1.62] [−1.12] [−0.80] [−0.08] [0.34]

8 14.94 0.47 −0.25 −0.24 −0.12 −0.13 −0.09 −0.08

[1.70] [−2.09] [−2.76] [−1.41] [−1.39] [−0.93] [−0.63]

9 21.04 0.35 −0.41 −0.40 −0.32 −0.32 −0.10 0.01

[1.15] [−2.85] [−3.44] [−2.94] [−2.72] [−1.03] [0.04]

10 (High) 49.19 −0.30 −1.09 −1.08 −0.91 −0.94 −0.68 −0.60

[−0.86] [−5.83] [−6.28] [−5.51] [−5.14] [−4.74] [−3.46]

High−Low 47.01 −0.76 −1.23 −1.15 −0.93 −0.97 −0.63 −0.57

[−2.69] [−5.31] [−5.55] [−4.67] [−4.42] [−3.87] [−2.61]

49

Table 2Univariate portfolios of NYSE stocks with price, size, and liquidity screens

This table reports the results of univariate portfolio of stocks by REG based on differentsubsamples of NYSE stocks. The subsamples include i) all NYSE stocks, ii) NYSE stockswith prices greater than $5, iii) NYSE stocks with market capitalizations greater than 10th

NYSE market capitalization breakpoint, iv) NYSE stocks with Amihud (2002) illiquiditymeasure less than 90th NYSE Amihud (2002) illiquidity breakpoint, and v) NYSE stocks withprices greater than $5, market capitalizations greater than 10th NYSE market capitalizationbreakpoint, and Amihud (2002) illiquidity measure less than 90th NYSE Amihud (2002)illiquidity breakpoint. For each subsample and each month from July 1963 to December2018, stocks are sorted into decile portfolios based on their REG. The table presents thenext-month value-weighted FF5 alphas of REG-sorted decile portfolios for each subsample,and the alpha spreads for the hedge portfolio that is long in the decile of stocks with thehighest REG and short in the decile of stocks with the lowest REG. The Newey-West (1987)adjusted t-statistics (with six lags) are given in square brackets.

NYSE, > 10th NYSE, < 90th

NYSE SIZE NYSE ILLIQ NYSE, AllDecile NYSE NYSE, > $5 breakpoint breakpoint restrictions

1(Low) −0.09 −0.08 −0.08 −0.07 −0.08[−1.23] [−1.16] [−1.09] [−1.02] [−1.12]

2 −0.12 −0.12 −0.12 −0.11 −0.11[−2.05] [−2.00] [−1.89] [−1.82] [−1.69]

3 0.01 −0.01 −0.04 −0.04 −0.03[0.21] [−0.19] [−0.74] [−0.72] [−0.50]

4 −0.06 −0.07 −0.03 −0.05 −0.04[−1.15] [−1.22] [−0.45] [−0.94] [−0.69]

5 −0.10 −0.07 −0.12 −0.08 −0.09[−1.50] [−0.97] [−1.64] [−1.25] [−1.33]

6 −0.05 −0.08 −0.05 −0.04 −0.04[−0.59] [−0.97] [−0.61] [−0.47] [−0.48]

7 −0.12 −0.16 −0.07 −0.07 −0.15[−1.23] [−1.69] [−0.81] [−0.70] [−1.52]

8 −0.24 −0.19 −0.19 −0.22 −0.17[−2.70] [−2.04] [−2.03] [−2.45] [−1.80]

9 −0.34 −0.30 −0.29 −0.31 −0.30[−2.87] [−3.19] [−2.83] [−2.91] [−3.15]

10 (High) −0.65 −0.63 −0.62 −0.62 −0.56[−3.43] [−3.91] [−3.82] [−3.73] [−3.47]

High−Low −0.56 −0.55 −0.54 −0.55 −0.48[−2.59] [−2.87] [−2.87] [−2.82] [−2.57]

50

Table 3Average stock characteristics of REG-sorted portfolios

This table reports the average stock characteristics of REG-sorted portfolios for the sample period from July 1963 to December2018. The stock characteristics are the market share (SHR), the analyst coverage (CVRG), the share price (PRC), the institutionalownership (INST), the market beta (BETA), the market value of equity measured in millions of dollars (SIZE), the book-to-marketratio (BM), the return momentum (MOM), short-term return reversal (STR), the co-skewness (COSKEW), the illiquidity(ILLIQ), the idiosyncratic volatility (IVOL), the asset growth (IA), the operating profitability (OP), and the lottery index (LTRY),respectively.

Decile REG SHR CVRG PRC INST BETA SIZE BM MOM STR COSKEW ILLIQ IVOL IA OP LTRY

1 2.17 21% 9 35.83 0.40 0.78 5.22 0.89 16.33 0.10 -0.039 0.64 0.75 0.11 0.04 382 3.71 21% 10 34.99 0.47 0.95 4.24 0.83 15.43 0.41 -0.023 0.62 1.07 0.12 0.00 443 4.90 16% 9 32.18 0.47 1.07 3.22 0.82 16.00 0.60 -0.024 0.72 1.29 0.14 0.03 504 6.12 12% 8 29.33 0.46 1.17 2.45 0.81 17.16 0.80 -0.028 0.86 1.50 0.15 0.02 565 7.53 9% 8 26.75 0.45 1.26 1.95 0.82 18.32 0.95 -0.032 1.05 1.71 0.17 0.02 626 9.27 7% 7 24.09 0.44 1.34 1.54 0.82 19.62 1.18 -0.034 1.29 1.95 0.19 -0.01 687 11.56 5% 6 21.84 0.42 1.43 1.21 0.83 21.33 1.57 -0.035 1.55 2.22 0.20 0.02 748 14.94 4% 6 19.59 0.39 1.51 0.91 0.84 23.75 2.09 -0.038 1.94 2.56 0.22 0.03 819 21.04 3% 5 17.41 0.36 1.58 0.66 0.86 26.64 3.03 -0.043 2.80 3.07 0.24 0.00 8910 49.19 2% 4 14.84 0.30 1.64 0.41 0.92 34.17 7.45 -0.050 5.90 4.53 0.25 -0.07 99

51

Table 4Independent bivariate sorts based on REG and the control variables

Each month from July 1963 to December 2018, stocks are sorted into 10×10 (100) portfolios basedon their REG and a control variable. Control variables are the stock’s market beta (BETA), marketvalue of equity (SIZE), book-to-market ratio (BM), return momentum (MOM), short-term returnreversal (STR), co-skewness (COSKEW), illiquidity (ILLIQ), idiosyncratic volatility (IVOL), lotterydemand (LTRY), operating profitability (OP), asset growth (IA), analyst coverage (CVRG), andinstitutional holdings (INST), respectively. Subsequently, we average each of the REG-sortedportfolios across the ten deciles producing portfolios with dispersion in anticipated regret that aresimilar in terms of the control variables. In addition, we form an arbitrage portfolio (High−LowREG portfolio) that is long in the resulting high-REG portfolio and short in the resulting low-REGportfolio. The table reports the one-month-ahead FF 5-factor risk-adjusted returns for each of theseten portfolios and the last column presents the 10–1 (High–Low) FF5 alpha spreads of REG-sortedportfolios averaged across the control deciles. The Newey-West (1987) adjusted t-statistics (with sixlags) are given in square brackets.

Decile

1 2 3 4 5 6 7 8 9 10 High-(Low) (High) Low

BETA −0.10 −0.03 −0.04 −0.02 0.01 −0.04 −0.05 −0.18 −0.18 −0.71 −0.60[−1.49] [−0.47] [−0.62] [−0.36] [0.20] [−0.73] [−0.76] [−2.26] [−2.18] [−6.37] [−4.52]

SIZE 0.10 0.17 0.21 0.19 0.14 0.13 −0.01 −0.11 −0.23 −0.65 −0.75[1.46] [2.68] [3.33] [3.11] [2.53] 2.51] [−0.20] [−2.12] [−3.96] [−7.98] [−6.80]

BM −0.05 −0.01 −0.09 0.02 0.01 0.00 −0.07 −0.16 −0.16 −0.55 −0.51[−0.84] [−0.24] [−1.70] [0.32] [0.17] [−0.07] [−1.03] [−2.34] [−1.95] [−5.80] [−4.39]

MOM −0.03 −0.02 −0.06 −0.04 −0.05 −0.12 −0.16 −0.26 −0.33 −0.61 −0.58[−0.42] [−0.40] [−1.16] [−0.70] [−0.77] [−1.83] [−2.04] [−2.97] [−3.29] [−5.92] [−4.36]

STR 0.04 −0.03 0.01 0.00 0.06 0.08 −0.04 −0.07 −0.12 −0.79 −0.83[0.62] [−0.54] [0.23] [−0.08] [0.95] [1.37] [−0.50] [−0.84] [−1.39] [−6.16] [−5.85]

COSKEW −0.02 0.00 −0.03 0.03 0.07 −0.03 −0.03 −0.12 −0.12 −0.67 −0.65[−0.36] [0.02] [−0.53] [0.53] [1.03] [−0.46] [−0.36] [−1.57] [−1.43] [−6.74] [−5.31]

ILLIQ 0.01 0.09 0.01 0.03 0.01 −0.05 −0.13 −0.21 −0.33 −0.73 −0.74[0.16] [1.51] [0.23] [0.56] [0.11] [−1.03] [−2.38] [−3.69] [−5.49] [−8.73] [−6.68]

IVOL 0.03 −0.14 −0.11 −0.07 0.03 −0.11 −0.08 −0.10 −0.14 −0.52 −0.55[0.27] [−1.76] [−1.29] [−0.87] [0.40] [−1.21] [−0.92] [−1.05] [−1.45] [−3.04] [−2.88]

LTRY 0.16 0.03 0.11 0.15 0.14 0.06 −0.09 −0.12 −0.07 −0.61 −0.78[1.73] [0.42] [1.40] [2.16] [1.95] [0.93] [−1.09] [−1.46] [−0.79] [−4.33] [−4.56]

OP −0.06 −0.12 −0.13 −0.08 −0.01 −0.07 −0.07 −0.12 −0.12 −0.56 −0.50[−0.97] [−1.96] [−2.41] [−1.36] [−0.13] [−0.99] [−0.81] [−1.43] [−1.29] [−5.03] [−3.75]

IA −0.07 −0.05 −0.06 0.01 0.01 0.03 −0.03 −0.14 −0.02 −0.59 −0.52[−1.02] [−0.79] [−0.93] [0.14] [0.13] [0.39] [−0.37] [−1.56] [−0.21] [−5.18] [−3.62]

CVRG 0.01 −0.02 −0.03 −0.04 −0.08 −0.08 −0.04 −0.11 −0.28 −0.53 −0.53[0.12] [−0.27] [−0.49] [−0.69] [−1.27] [−1.41] [−0.73] [−1.65] [−2.98] [−4.12] [−3.44]

INST 0.11 0.20 0.11 0.10 0.15 0.16 0.03 −0.07 −0.02 −0.64 −0.75[1.64] [3.06] [1.95] [1.47] [1.75] [1.91] [0.29] [−0.79] [−0.16] [−4.87] [−5.16]

52

Table 5Stock-level Fama and MacBeth regressions

This table reports time-series averages of the intercepts and slope coefficients from the monthly cross-sectional regressions of one-month-aheadexcess stock returns on REG and a large set of stock characteristics for the period of July 1963 – December 2018. The regressions are runwithout (with) industry dummies in Panel A (Panel B). Industry dummy is is an indicator variable which takes on value 1 if the stock is inone of the ten industries as determined by Fama and French (1997), and 0 otherwise. The Newey-West (1987) adjusted t-statistics (with sixlags) are given in square brackets.

Panel A: Fama and MacBeth regressions without industry dummies(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13)

REG −0.025 −0.026 −0.028 −0.022 −0.030 −0.028 −0.021 −0.030 −0.025 −0.024 −0.025 −0.021 −0.009[−5.89] [−9.74] [−11.03] [−8.36] [−11.15] [−11.02] [−5.36] [−11.35] [−10.13] [−10.86] [−9.18] [−9.83] [−3.01]

BETA 0.127 0.078 0.072 0.082 0.087 0.098 0.065 0.050 0.064 0.037 0.079 0.091[1.46] [0.96] [0.80] [0.99] [1.06] [1.25] [0.84] [0.55] [0.73] [0.41] [0.87] [0.93]

SIZE −0.095 −0.105 −0.086 −0.124 −0.103 −0.119 −0.102 −0.080 −0.078 −0.154 −0.041 −0.100[−3.34] [−3.84] [−3.13] [−4.46] [−3.76] [−4.47] [−4.05] [−3.18] [−3.14] [−4.20] [−1.56] [−2.84]

BM 0.164 0.158 0.187 0.157 0.158 0.144 0.153 0.218 0.153 0.104 0.158 0.114[2.87] [2.90] [3.31] [2.80] [2.95] [2.70] [2.85] [3.37] [2.60] [1.61] [2.49] [1.74]

MOM 0.008 0.007 0.008 0.008 0.008 0.008 0.006 0.007 0.008 0.006 0.007[6.37] [6.03] [6.21] [6.40] [6.40] [6.35] [4.79] [5.58] [5.48] [4.61] [4.53]

STR −0.038 −0.027[−9.71] [−6.56]

ILLIQ −0.012 −0.051[−0.94] [−3.00]

COSKEW −0.118 −0.122[−1.32] [−1.10]

IVOL −0.134 −0.173[−3.61] [−4.93]

LTRY 0.001 0.004[0.46] [2.89]

OP 3.084 1.836[5.08] [4.22]

IA −0.269 −0.284[−4.69] [−5.25]

CVRG 0.016 0.015[2.81] [2.89]

INST −0.173 −0.412[−1.05] [−2.35]

Intercept 0.973 1.380 1.358 1.257 1.487 1.348 1.579 1.336 1.218 1.292 1.643 1.243 1.553[4.94] [5.75] [5.84] [5.43] [6.24] [5.79] [7.03] [6.24] [5.16] [5.52] [5.56] [4.85] [5.80]

Adj. R2 0.01 0.04 0.05 0.06 0.06 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.07

(continued on next page)

53

Table 5Stock-level Fama and MacBeth regressions (cont.)

Panel B: Fama and MacBeth regressions with industry dummies

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) (12) (13)

REG −0.026 −0.029 −0.031 −0.023 −0.033 −0.031 −0.022 −0.032 −0.026 −0.026 −0.027 −0.022 −0.010

[−6.93] [−10.74] [−11.37] [−9.16] [−11.58] [−11.41] [−4.95] [−10.71] [−11.02] [−11.65] [−10.07] [−11.01] [−3.60]

BETA 0.106 0.061 0.056 0.062 0.070 0.085 0.054 0.054 0.071 0.045 0.079 0.090

[1.41] [0.87] [0.71] [0.87] [0.98] [1.24] [0.81] [0.68] [0.92] [0.59] [0.98] [1.04]

SIZE −0.096 −0.105 −0.083 −0.125 −0.103 −0.122 −0.108 −0.080 −0.075 −0.140 −0.023 −0.065

[−3.39] [−3.82] [−3.01] [−4.41] [−3.77] [−4.53] [−4.23] [−3.19] [−3.05] [−4.22] [−0.97] [−2.20]

BM 0.220 0.213 0.249 0.214 0.216 0.202 0.209 0.286 0.220 0.172 0.225 0.202

[4.48] [4.53] [5.01] [4.33] [4.64] [4.31] [4.45] [4.99] [4.39] [3.14] [4.28] [3.53]

MOM 0.007 0.007 0.007 0.007 0.007 0.007 0.005 0.006 0.007 0.005 0.006

[6.45] [5.95] [6.29] [6.45] [6.45] [6.43] [4.62] [5.39] [5.37] [4.32] [4.19]

STR −0.044 −0.033

[−11.08] [−7.99]

ILLIQ −0.011 −0.053

[−0.84] [−3.06]

COSKEW −0.069 −0.032

[−0.91] [−0.35]

IVOL −0.159 −0.188

[−4.53] [−5.93]

LTRY 0.000 0.004

[−0.19] [2.93]

OP 3.340 2.001

[5.20] [4.49]

IA −0.289 −0.273

[−5.05] [−5.53]

CVRG 0.012 0.011

[2.38] [2.42]

INST −0.294 −0.590

[−1.99] [−3.75]

Intercept 0.979 1.477 1.447 1.333 1.610 1.442 1.703 1.496 1.231 1.300 1.635 1.196 1.496

[4.62] [5.62] [5.66] [5.24] [6.00] [5.60] [6.91] [6.57] [4.76] [5.11] [5.21] [4.22] [5.05]

Adj. R2 0.04 0.07 0.08 0.08 0.08 0.08 0.08 0.08 0.07 0.07 0.08 0.07 0.09

54

Table 6One-month-ahead transition matrix for anticipated regret

This table reports the percentage of stocks moving from one REG-sorted decile portfolio in month t to the same or anotherREG-sorted decile portfolio in month t+ 1. Portfolios 1 and 10 indicate portfolio of stocks with the lowest REG and the highestREG, respectively.

Portfolio rank in month t

Portfolio rank 1 2 3 4 5 6 7 8 9 10in month t+ 1 (LOW) (HIGH)

1 (LOW) 46.7% 22.2% 12.0% 6.9% 4.2% 2.8% 2.0% 1.5% 1.4% 1.7%2 20.4% 24.6% 19.3% 13.6% 9.0% 5.7% 3.6% 2.1% 1.5% 1.2%3 11.1% 17.5% 18.9% 16.7% 13.3% 9.4% 6.2% 3.9% 2.4% 1.5%4 6.8% 12.0% 15.4% 16.7% 15.3% 12.8% 9.6% 6.4% 3.7% 2.1%5 4.7% 8.1% 11.4% 14.2% 15.5% 15.2% 12.9% 9.7% 6.1% 3.0%6 3.1% 5.6% 8.4% 11.0% 13.8% 15.3% 15.5% 13.5% 9.7% 4.7%7 2.3% 3.7% 5.9% 8.2% 11.2% 14.0% 16.4% 16.9% 14.1% 7.8%8 1.8% 2.6% 4.2% 5.9% 8.4% 11.3% 15.0% 18.2% 19.5% 13.2%9 1.5% 1.9% 2.8% 4.0% 5.8% 8.4% 11.6% 16.8% 23.0% 23.5%10 (HIGH) 1.6% 1.5% 2.0% 2.6% 3.6% 5.0% 7.3% 10.8% 18.6% 41.3%

Sum 100.0% 100.0% 100.0% 100.0% 100.0% 100.0% 100.0% 100.0% 100.0% 100.0%

55

Table 7Subsample analysis

This table reports the results of univariate portfolio of stocks sorted by REG based on differentsubsamples of stocks. The subsamples include i) 1,000 most liquid stocks (1,000 stocks withthe lowest Amihud (2002) illiquidity measure), ii) more liquid stocks (stocks with Amihud(2002) illiquidity measure lower than the median NYSE Amihud (2002) illiquidity measure),iii) 1,000 largest stocks (1,000 stocks with the largest market capitalizations), iv) larger stocks(stocks with market capitalizations greater than the median NYSE market capitalization),and v) S&P 500 stocks. For each subsample and each month from July 1963 to December2018, stocks are sorted into decile portfolios based on their REG. The table presents thenext-month value-weighted FF5 alphas of REG-sorted decile portfolios for each subsample,and the alpha spreads for the hedge portfolio that is long in the decile of stocks with thehighest REG and short in the decile of stocks with the lowest REG. The Newey-West (1987)adjusted t-statistics (with six lags) are given in square brackets.

1,000 most More 1,000Decile liquid liquid largest Larger S&P 500

1(Low) −0.04 −0.03 −0.05 −0.04 −0.08[−0.66] [−0.55] [−0.71] [−0.59] [−1.07]

2 −0.01 −0.01 −0.01 −0.01 −0.02[−0.10] [−0.16] [−0.12] [−0.20] [−0.29]

3 −0.09 −0.09 −0.10 −0.10 −0.11[−1.71] [−1.85] [−1.89] [−1.99] [−1.91]

4 −0.03 −0.03 −0.03 −0.05 −0.03[−0.52] [−0.49] [−0.56] [−0.77] [−0.52]

5 0.13 0.11 0.12 0.12 0.09[1.64] [1.50] [1.49] [1.51] [0.91]

6 0.06 0.04 0.05 0.04 0.02[0.79] [0.54] [0.65] [0.54] [0.24]

7 0.04 −0.01 0.03 0.01 0.07[0.39] [−0.11] [0.27] [0.07] [0.55]

8 −0.02 −0.07 −0.05 −0.03 −0.14[−0.18] [−0.59] [−0.46] [−0.23] [−0.83]

9 −0.05 −0.12 −0.01 0.01 0.00[−0.35] [−1.05] [−0.10] [0.06] [0.00]

10 (High) −0.69 −0.69 −0.63 −0.71 −0.87[−3.19] [−3.91] [−2.95] [−3.04] [−2.93]

High−Low −0.65 −0.65 −0.59 −0.68 −0.79[−2.80] [−3.34] [−2.56] [−2.71] [−2.54]

56

Table 83x10 sorts by arbitrage cost, investor attention, lottery demand and REG

Each month from July 1963 to December 2018, stocks are first sorted into three portfolios (Low, Medium, High) by one of thefollowing characteristics: arbitrage cost index (ATT), investor attention index (ATT), and lottery demand index (LDI), and thensorted by their anticipated regret, measured by REG. The table presents the next-month FF5 alphas for each of the COST-, ATT-,and LDI-sorted tercile portfolio and REG-sorted decile portfolio, and the alpha spreads for the hedge portfolio that is long in thedecile of stocks with the highest REG and short in the decile of stocks with the lowest REG for each of the COST-, ATT-, andLDI-sorted tercile portfolio. The last column of each panel reports the average FF5 alpha of the REG-sorted decile portfolios andthe zero-cost arbitrage portfolio across the three terciles. The Newey-West (1987) adjusted t-statistics (with six lags) are given insquare brackets.

Panel A: COST Panel B: ATT Panel C: LDI

REG Low 2 High Avg. Low 2 High Avg. Low 2 High Avg.

1(Low) 0.01 0.11 −0.02 0.03 0.24 0.15 −0.07 0.11 0.05 −0.23 0.12 −0.02[0.10] [1.08] [−0.17] [0.38] [2.49] [1.68] [−0.70] [1.51] [0.55] [−2.93] [0.87] [−0.29]

2 −0.07 0.15 −0.14 −0.02 0.12 0.26 −0.01 0.12 −0.08 −0.14 0.14 −0.03[−0.82] [1.77] [−1.15] [−0.33] [1.10] [2.23] [−0.17] [1.92] [−1.04] [−1.70] [0.97] [−0.46]

3 0.09 0.07 −0.06 0.04 0.02 0.03 0.11 0.05 −0.03 −0.20 0.21 −0.01[1.17] [0.84] [−0.42] [0.58] [0.20] [0.32] [1.40] [0.99] [−0.37] [−2.05] [1.27] [−0.10]

4 0.16 0.18 −0.02 0.10 −0.13 0.32 −0.16 0.01 0.00 0.10 −0.27 −0.06[1.65] [1.78] [−0.15] [1.90] [−1.12] [2.16] [−1.75] [0.19] [0.05] [1.19] [−1.68] [−0.82]

5 −0.11 0.13 −0.24 −0.07 −0.15 0.29 −0.14 0.00 −0.01 −0.03 −0.31 −0.11[−1.13] [1.20] [−2.10] [−1.05] [−1.35] [1.72] [−1.57] [−0.02] [−0.08] [−0.21] [−2.23] [−1.93]

6 −0.10 0.22 −0.39 −0.09 −0.38 0.20 −0.16 −0.11 0.09 −0.07 0.15 0.06[−1.42] [150] [−2.81] [−1.22] [−2.97] [1.73] [−1.79] [−1.90] [1.09] [−0.69] [0.93] [0.84]

7 −0.22 0.10 −0.27 −0.13 −0.32 0.17 −0.06 −0.07 −0.05 −0.10 −0.13 −0.09[−2.43] [0.80] [−1.92] [−1.93] [−2.25] [1.08] [−0.49] [−0.77] [−0.56] [−0.91] [−0.77] [−1.25]

8 −0.03 −0.05 −0.23 −0.10 −0.43 0.15 0.09 −0.06 −0.07 −0.02 −0.51 −0.20[−0.27] [−0.40] [−1.49] [−1.16] [−3.21] [1.02] [0.67] [−0.71] [−0.88] [−0.19] [−3.27] [−2.51]

9 0.03 0.21 −0.37 −0.04 −0.69 −0.05 −0.02 −0.25 −0.12 0.06 −0.58 −0.22[0.27] [1.14] [−1.96] [−0.36] [−4.82] [−0.33] [−0.17] [−3.10] [−1.12] [0.42] [−3.37] [−2.49]

10 (High) −0.18 −0.16 −0.59 −0.31 −1.09 −0.23 −0.31 −0.54 0.00 −0.36 −0.77 −0.37[−1.40] [−0.86] [−2.60] [−2.58] [−5.78] [−1.41] [−1.49] [−4.05] [0.03] [−2.04] [−3.27] [−3.17]

High−Low −0.19 −0.27 −0.56 −0.34 −1.33 −0.38 −0.24 −0.65 −0.05 −0.14 −0.89 −0.36[−1.02] [−1.35] [−2.12] [−2.15] [−6.31] [−1.98] [−0.97] [−4.03] [−0.26] [−0.63] [−3.38] [−2.41]

57

Table 9Subperiod analysis

This table reports the results of univariate portfolio of stocks sorted by REG over differentsubperiods defined by different states of the economy/market. The first two subperiodsare high and low economic activity periods, where high (low) economic activity periods aredefined by months in which CFNAIMA3 index is greater than or equal to (less than) –0.7.The second two subperiods are up and down markets, where up (down) markets are definedby months in which the excess market returns are greater than or equal to (less than) zero.The final two subperiods are high and low macroeconomic uncertainy periods, where high(low) economic uncertainty is defined by months in which the economic uncertainty indexof Jurado, Ludvigson, and Ng (2015) (hereafter JLN) is greater (lower) than its medianvalue over the full sample period. For each month in the corresponding subperiod, the tablepresents the next-month value-weighted FF5 alphas of REG-sorted decile portfolios, and thealpha spreads for the hedge portfolio that is long in the decile of stocks with the highest REGand short in the decile of stocks with the lowest REG. The Newey-West (1987) adjustedt-statistics (with six lags) are given in square brackets.

ECONOMIC ACTIVITY MARKET UNCERTAINTY

Decile High Low Up Down Low High

1(Low) −0.03 −0.18 0.01 −0.06 −0.05 −0.10[−0.47] [−0.77] [0.11] [−0.75] [−0.57] [−0.90]

2 0.01 −0.05 0.01 0.02 0.02 −0.06[0.18] [−0.22] [0.10] [0.19] [0.26] [−0.63]

3 −0.10 −0.03 −0.16 −0.05 −0.06 −0.10[−2.09] [−0.15] [−2.86] [−0.75] [−1.07] [−1.22]

4 −0.02 −0.37 −0.08 −0.02 −0.01 −0.04[−0.30] [−1.77] [−1.44] [−0.17] [−0.17] [−0.46]

5 0.06 0.39 0.06 0.09 0.13 0.12[0.68] [1.66] [0.60] [0.96] [1.44] [1.20]

6 0.03 0.15 0.07 −0.03 −0.05 0.17[0.37] [0.56] [0.61] [−0.30] [−0.40] [1.66]

7 −0.04 0.58 −0.09 0.02 −0.08 0.10[−0.42] [1.79] [−0.83] [0.14] [−0.64] [0.85]

8 −0.14 0.15 −0.22 −0.07 −0.17 0.08[−1.34] [0.59] [−1.93] [−0.50] [−1.46] [0.51]

9 −0.06 −0.19 −0.14 −0.13 −0.03 −0.12[−0.58] [−0.72] [−1.03] [−0.93] [−0.21] [−0.86]

10 (High) −0.54 −1.63 −0.68 −0.78 −0.50 −0.78[−4.11] [−3.87] [−3.11] [−4.31] [−3.21] [−3.62]

High−Low −0.50 −1.46 −0.69 −0.72 −0.45 −0.68[−3.28] [−2.98] [−2.70] [−3.21] [−2.35] [−2.66]

58

Table 10Industry analysis

This table reports the results of univariate portfolio of stocks sorted by REG based on tenindustries as determined by Fama and French (1997), i.e., Consumer Non-Durables (NDURB),Consumer Durables (DURB), Manufacturing (MANU), Energy (ENRG), High Tech (TECH),Telecom (TLCM), Shops (SHOP), Health (HLTH), Utilities (UTIL), and Others (OTHR).The table presents the next-month value-weighted FF5 alphas of REG-sorted decile portfoliosfor each industry, and the alpha spreads for the hedge portfolio that is long in the decile ofstocks with the highest-REG and short in the decile of stocks with the lowest-REG. TheNewey-West (1987) adjusted t-statistics (with six lags) are given in square brackets.

Decile NDURB DURB MANU ENRG TECH TLCM SHOP HLTH UTIL OTHR

1 (Low) −0.02 −0.52 −0.06 −0.03 0.04 0.17 −0.16 0.55 0.01 −0.06[−0.23] [−2.44] [−0.71] [−0.17] [0.27] [1.11] [−1.14] [3.20] [0.09] [−0.74]

2 0.01 −0.14 −0.31 −0.06 0.29 0.14 0.04 0.09 0.13 −0.10[0.11] [−0.75] [−3.10] [−0.32] [1.68] [0.98] [0.28] [0.62] [0.99] [−1.05]

3 0.10 −0.18 −0.27 −0.12 0.27 −0.15 −0.18 0.26 −0.05 −0.25[0.66] [−1.14] [−2.70] [−0.60] [1.72] [−0.86] [−1.26] [1.74] [−0.36] [−2.63]

4 −0.03 −0.48 −0.33 0.18 0.37 0.37 −0.07 0.22 0.21 −0.09[−0.19] [−2.57] [−2.77] [0.81] [2.00] [1.70] [−0.38] [1.45] [1.26] [−0.88]

5 −0.04 −0.70 −0.16 0.03 0.42 0.14 −0.24 0.58 −0.13 −0.16[−0.26] [−3.08] [−1.26] [0.12] [2.00] [0.59] [−1.68] [2.91] [−0.69] [−1.44]

6 −0.23 −0.36 −0.45 0.32 0.60 0.17 −0.19 0.44 0.04 −0.20[−1.36] [−1.59] [−3.20] [1.01] [3.28] [0.50] [−1.26] [2.17] [0.18] [−1.92]

7 −0.30 −0.56 −0.23 −0.16 0.21 0.72 −0.20 0.31 −0.09 −0.25[−1.53] [−2.74] [−1.47] [−0.54] [1.21] [2.31] [−0.97] [1.30] [−0.33] [−2.02]

8 −0.49 −0.56 −0.28 −0.44 0.38 −0.16 −0.40 −0.06 0.02 −0.29[−2.38] [−2.44] [−1.77] [−1.29] [1.83] [−0.48] [−2.03] [−0.25] [0.06] [−2.16]

9 −0.55 −0.48 −0.16 −0.59 0.14 −0.14 −0.57 0.05 −0.84 −0.21[−2.72] [−1.91] [−0.90] [−1.63] [0.69] [−0.47] [−2.45] [0.18] [−1.96] [−1.47]

10 (High) −1.23 −1.23 −0.84 −0.86 −0.51 −0.69 −0.77 −0.64 −0.23 −0.82[−4.59] [−3.82] [−4.30] [−1.66] [−2.49] [−1.82] [−2.67] [−2.16] [−0.42] [−3.68]

High−Low −1.21 −0.84 −0.78 −0.79 −0.60 −0.81 −0.61 −1.13 −0.26 −0.76[−4.34] [−2.56] [−3.70] [−1.60] [−2.05] [−2.10] [−1.96] [−3.67] [−0.45] [−3.29]

59

Table 11Long-term predictability

This table reports time-series averages of the intercepts and slope coefficients from the monthly cross-sectional regressions ofexcess stock returns on one-month up to 12-month lagged values of REG and one-month lagged values of a large set of stockcharacteristics for the period of July 1963 – December 2018. The Newey-West (1987) adjusted t-statistics (with six lags) are givenin square brackets.

t+ 1 t+ 2 t+ 3 t+ 4 t+ 5 t+ 6 t+ 7 t+ 8 t+ 9 t+ 10 t+ 11 t+ 12

REG −0.009 −0.015 −0.009 −0.006 −0.009 −0.006 −0.005 −0.006 −0.001 −0.005 −0.004 0.000[−3.01] [−6.46] [−3.83] [−2.32] [−3.93] [−2.55] [−2.17] [−2.41] [−0.45] [−2.44] [−2.02] [−0.05]

BETA 0.091 0.105 0.096 0.094 0.087 0.088 0.085 0.093 0.089 0.082 0.080 0.087[0.93] [1.09] [1.01] [0.99] [0.93] [0.93] [0.89] [0.96] [0.92] [0.86] [0.84] [0.91]

SIZE −0.100 −0.120 −0.113 −0.110 −0.113 −0.108 −0.106 −0.110 −0.100 −0.107 −0.106 −0.100[−2.84] [−3.41] [−3.19] [−3.09] [−3.19] [−3.07] [−2.93] [−3.04] [−2.78] [−2.98] [−2.97] [−2.76]

BM 0.114 0.091 0.096 0.102 0.087 0.096 0.098 0.096 0.104 0.099 0.112 0.114[1.74] [1.40] [1.48] [1.57] [1.35] [1.47] [1.50] [1.47] [1.58] [1.50] [1.69] [1.74]

MOM 0.007 0.006 0.007 0.006 0.006 0.006 0.006 0.007 0.006 0.007 0.006 0.007[4.53] [4.24] [4.24] [4.10] [4.01] [4.06] [3.93] [4.08] [4.03] [3.99] [3.85] [3.98]

STR −0.027 −0.027 −0.027 −0.027 −0.028 −0.029 −0.029 −0.029 −0.028 −0.028 −0.029 −0.029[−6.56] [−5.95] [−5.96] [−6.00] [−6.21] [−6.34] [−6.38] [−6.48] [−6.32] [−6.44] [−6.55] [−6.63]

ILLIQ −0.051 −0.051 −0.050 −0.051 −0.046 −0.049 −0.047 −0.047 −0.049 −0.051 −0.045 −0.047[−3.00] [−2.89] [−2.84] [−2.94] [−2.76] [−2.85] [−2.74] [−2.67] [−2.86] [−2.90] [−2.55] [−2.76]

COSKEW −0.122 −0.114 −0.110 −0.114 −0.101 −0.106 −0.120 −0.113 −0.139 −0.125 −0.118 −0.106[−1.10] [−1.04] [−1.00] [−1.04] [−0.94] [−0.99] [−1.11] [−1.04] [−1.25] [−1.16] [−1.11] [−0.99]

IVOL −0.173 −0.226 −0.241 −0.246 −0.244 −0.251 −0.258 −0.259 −0.258 −0.252 −0.258 −0.265[−4.93] [−6.37] [−6.71] [−6.76] [−6.84] [−7.07] [−7.29] [−7.38] [−7.42] [−7.03] [−7.28] [−7.34]

LTRY −0.004 0.005 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004 0.004[2.89] [3.46] [3.21] [3.12] [3.21] [3.16] [3.10] [3.03] [2.98] [2.99] [3.01] [2.91]

OP 1.836 1.856 1.894 2.018 2.051 2.077 1.969 2.037 2.150 2.160 2.370 2.218[4.22] [4.30] [4.50] [4.46] [447] [4.52] [4.34] [4.55] [4.63] [4.57] [4.64] [4.50]

IA −0.284 −0.263 −0.267 −0.268 −0.257 −0.264 −0.265 −0.271 −0.271 −0.268 −0.267 −0.282[−5.25] [−4.85] [−4.70] [−4.81] [−4.63] [−4.79] [−4.84] [−5.04] [−5.05] [−4.94] [−4.95] [−5.26]

CVRG 0.015 0.016 0.016 0.016 0.016 0.016 0.016 0.016 0.015 0.015 0.015 0.015[2.89] [3.09] [2.96] [3.02] [3.02] [2.96] [2.90] [2.94] [2.84] [2.85] [2.91] [2.87]

INST −0.412 −0.438 −0.407 −0.411 −0.403 −0.396 −0.381 −0.383 −0.370 −0.373 −0.376 −0.380[−2.35] [−2.52] [−2.33] [−2.41] [−2.34] [−2.29] [−2.18] [−2.20] [−2.14] [−2.15] [−2.17] [−2.22]

Intercept 1.553 1.761 1.706 1.686 1.707 1.674 1.667 1.691 1.606 1.677 1.660 1.606[5.80] [6.49] [6.33] [6.20] [6.26] [6.20] [6.08] [6.16] [5.78] [6.10] [6.00] [5.71]

Adj. R2 0.067 0.068 0.068 0.069 0.068 0.069 0.069 0.069 0.069 0.069 0.069 0.069

60

Table 12Portfolios of stocks sorted by RREG and VREG

Each month from July 1963 to December 2018, stocks are sorted into decile portfolios basedon the two components of anticipated regret, i.e., return-regret (RREG) and variance-regret(VREG), as defined in Eq. (11) and Eq. (12), respectively. Panel A reports the next-monthvalue-weighted average excess returns, FF5 alphas, and the average RREG (VREG) values forthe 10 univariate decile portfolios of stocks sorted by RREG (VREG). The last row presentsthe 10–1 average raw and risk-adjusted return differences of the arbitrage portfolio with along position in the decile of stocks with the highest RREG (RREG) (decile 10) and a shortposition in the decile of stocks with the lowest RREG (VREG) (decile 1). The Newey-West(1987) adjusted t-statistics (with six lags) are given in square brackets. Panel B reports theFF5 alphas from independent bivariate sorts of stocks by RREG and VREG. The last twocolumns in Panel B report the FF5 alpha of each RREG (VREG) portfolio averaged acrossthe ten VREG (RREG) portfolios.

Panel A: Univariate sorts Panel B: Bivariate sorts

Decile RREG Mean FF5 VREG Mean FF5 Decile RREG VREG

1 (Low) 1.41 0.49 −0.02 0.18 0.50 −0.02 1 (Low) −0.01 0.35

[3.54] [−0.28] [3.50] [−0.30] [−0.09] [2.06]

2 2.34 0.56 −0.02 0.41 0.56 −0.02 2 −0.08 0.09

[3.72] [−0.33] [3.52] [−0.32] [−1.02] [0.93]

3 2.96 0.47 −0.12 0.64 0.57 −0.02 3 0.00 −0.11

[2.75] [−2.17] [3.26] [−0.35] [−0.01] [−1.38]

4 3.55 0.54 −0.05 0.89 0.62 0.03 4 −0.05 −0.14

[2.94] [−1.09] [3.32] [0.47] [−0.80] [−2.22]

5 4.18 0.67 0.08 1.22 0.66 0.03 5 0.06 −0.03

[3.37] [1.10] [3.24] [0.47] [0.77] [−0.43]

6 4.90 0.67 0.13 1.64 0.55 −0.02 6 −0.03 −0.06

[3.17] [1.59] [2.52] [−0.29] [−0.39] [−0.86]

7 5.78 0.56 −0.02 2.25 0.47 −0.08 7 −0.02 0.00

[2.29] [−0.30] [2.02] [−0.96] [−0.26] [−0.01]

8 6.95 0.48 −0.05 3.20 0.47 −0.03 8 −0.06 0.01

[1.72] [−0.47] [1.86] [−0.36] [−0.68] [0.12]

9 8.80 0.42 −0.06 5.05 0.33 −0.16 9 −0.12 −0.12

[1.39] [−0.61] [1.11] [−1.84] [−1.33] [−1.34]

10 (High) 14.65 −0.08 −0.43 14.78 −0.19 −0.58 10 (High) −0.37 −0.26

[−0.50] [−5.54] [−0.54] [−4.01] [−3.02] [−1.72]

High−Low 13.24 −0.57 −0.41 14.60 −0.69 −0.56 High−Low −0.36 −0.35

[−2.05] [−3.01] [−2.50] [−3.33] [−2.58] [−2.06]

61

Table 13Univariate portfolio of stocks sorted by CREG

This table reports the results of univariate portfolio of stocks sorted by an alternative measureof regret, i.e., cross-sectional regret (CREG), as defined in Eq. (17). CREG is based on thecross-sectional averages of daily return and daily conditional variance of all stocks. For eachmonth from July 1963 to December 2018, stocks are sorted into decile portfolios based ontheir CREG. We further use five subsamples; i) all NYSE stocks, ii) NYSE stocks with pricesgreater than $5, iii) NYSE stocks with market capitalizations greater than 10th NYSE marketcapitalization breakpoint, iv) NYSE stocks with Amihud (2002) illiquidity measure less than90th NYSE Amihud (2002) illiquidity breakpoint, and v) NYSE stocks with prices greaterthan $5, market capitalizations greater than 10th NYSE market capitalization breakpoint, andAmihud (2002) illiquidity measure less than 90th NYSE Amihud (2002) illiquidity breakpoint.The table presents the next-month value-weighted FF5 alphas of CREG-sorted decile portfoliosfor each subsample, and the alpha spreads for the hedge portfolio that is long in the decile ofstocks with the highest CREG and short in the decile of stocks with the lowest CREG. TheNewey-West (1987) adjusted t-statistics (with six lags) are given in square brackets.

NYSE, > 10th NYSE, < 90th

NYSE SIZE NYSE ILLIQ NYSE, AllDecile All NYSE NYSE, > $5 breakpoint breakpoint restrictions

1(Low) 0.23 0.18 0.08 0.11 0.12 0.07[2.61] [1.50] [0.81] [1.07] [1.08] [0.68]

2 0.22 0.00 0.00 −0.01 −0.02 −0.01[2.34] [0.06] [0.03] [−0.09] [−0.26] [−0.08]

3 0.05 −0.07 −0.08 −0.06 −0.05 −0.05[0.80] [−1.01] [−1.26] [−0.85] [−0.81] [−0.82]

4 0.07 −0.08 −0.12 −0.10 −0.09 −0.10[1.18] [−1.58] [−2.31] [−2.01] [−1.82] [−2.03]

5 −0.05 −0.07 −0.01 −0.03 −0.03 −0.01[−1.13] [−1.18] [−0.11] [−0.56] [−0.46] [−0.20]

6 −0.02 −0.07 −0.09 −0.10 −0.10 −0.09[−0.29] [−1.04] [−1.53] [−1.61] [−1.58] [−1.45]

7 −0.03 −0.12 −0.11 −0.13 −0.14 −0.12[−0.49] [−1.75] [−1.45] [−1.72] [−1.88] [−1.61]

8 −0.07 −0.17 −0.13 −0.12 −0.10 −0.10[−1.06] [−2.02] [−1.64] [−1.42] [−1.21] [−1.28]

9 −0.07 −0.17 −0.13 −0.12 −0.10 −0.10[−1.06] [−2.02] [−1.64] [−1.42] [−1.21] [−1.28]

10 (High) −0.18 −0.31 −0.30 −0.28 −0.26 −0.28[−1.81] [−3.00] [−3.10] [−2.88] [−2.71] [−2.88]

High−Low −0.41 −0.49 −0.38 −0.39 −0.38 −0.35[−3.09] [−3.19] [−2.73] [−2.67] [−2.55] [−2.52]

62

Internet Appendix for

“Anticipated Regret and Equity Returns”

Y. Eser ARISOY Turan G. BALI Yi TANG

This Internet Appendix presents the definition and construction of variables used in our

analyses as well as the results from supplementary tests not reported in the paper. We start

with a detailed description of the variables used in our analyses (Table A.1). In Section A.1,

we present the value-weighted raw and risk-adjusted returns of portfolios sorted by REG

based on risk aversion parameters b = 2 and b = 10 (Table A.2), the value-weighted raw

and risk-adjusted returns of portfolios sorted by REG based on GJR-GARCH(1,1) using

expanding window with the initial window being past five years of daily data (Table A.3),

and the equal-weighted raw and risk-adjusted returns of univariate portfolios sorted by REG

(Table A.4). In Section A.2, we present the transition matrix of portfolio of stocks sorted

by REG in month t and their corresponding portfolios in months t + 3, t + 6, and t + 12,

respectively (Table A.5), and the results of bivariate portfolio sorts by REG and different

controls for limits to arbitrage and investor inattention (Table A.6).

TABLE A.1 ABOUT HERE

A.1 Further robustness checks using REG

In this section, we present the value-weighted raw and risk-adjusted returns of portfolios

sorted by REG based on risk aversion parameters b = 2 and b = 10 (Section A.1.1), the

value-weighted raw and risk-adjusted returns of portfolios sorted by REG based on GJR-

GARCH(1,1) using expanding window with the initial window being past five years of daily

data (Section A.1.2), and the equal-weighted raw and risk-adjusted returns of univariate

portfolios sorted by REG (Section A.1.3).

1

A.1.1 Univariate portfolios of stocks sorted by REG based on different risk

aversion parameters

The definition of our alternative anticipated regret measure, REG depends on the choice

of investors’ risk-aversion parameter. Eq. (10) estimates REG based on a risk aversion

parameter b equal to five. However, studies document that investors’ risk aversion parameters

change between two and ten. (See Ghysels, Santa-Clara, and Valkanov (2005), Guo and

Whitelaw (2006), Bali (2008), and Bali and Engle (2010) who determine market risk aversion

parameters between two and ten.) In this section, we use two further values of risk aversion

paramater b = 2, b = 10, and estimate anticipated regret as:

REG(b = 2)i,t = (µi,t −max[ri,d])− (σ2i,t −min[σ2

i,d]), d = 1, 2, . . . , T (18)

REG(b = 10)i,t = (µi,t −max[ri,d])− 5(σ2i,t −min[σ2

i,d]), d = 1, 2, . . . , T (19)

where d denotes trading days in month t, b is investors’ risk aversion parameter, µi,t is the

average daily return of stock i in month t, which measures the expected return that an

investor would earn during month t, σ2i,t is the average daily conditional variance of stock

i in month t, which measures the expected variance that an investor would be exposed to

during month t, and max[ri,d] and min[σ2i,d] are the maximum daily return and minimum

daily variance attained during the same month t, which capture the best return and variance

alternatives that could have been attained during the month. The stocks’ daily conditional

variance is calculated using the GJR-GARCH (1,1) model proposed by Glosten, Jagannathan,

and Runkle (1993) as outlined in Section 3.2.

TABLE A.2 ABOUT HERE

Panels A and B of Table A.2 report the value-weighted returns of univariate portfolios

sorted by REG(b = 2) and REG(b = 10), respectively. Similar to the results reported in

Table 1, univariate portfolio sorts based on the two alternative measures of anticipated regret

confirm the previously documented negative relation between anticipated regret and next-

month average returns. The value-weighted portfolio of stocks with the lowest REG(b=2)

(REG(b=10)) earns an average excess return of 0.46% (0.49%) per month, whereas the

average excess return on the value-weighted portfolio of stocks with the highest REG(b=2)

(REG(b=10)) is –0.21% (–0.26%) per month. The arbitrage portfolio with a long position in

the highest REG(b=2) (REG(b=10)) stocks and a short position in the lowest REG(b=2)

2

(REG(b=10)) loses on average 0.67% (0.75%) per month with a t-statistic of –2.33 (–2.61).

The negative and significant next month’s risk-adjusted returns for the High−Low REG(b=2)

and High−Low REG(b=10) portfolios presented in the last row of Panels A and B of Table

A.2, respectively, further corroborate our findings and suggest that negative and significant

relation between anticipated regret and the next-month risk-adjusted returns is robust to

using different values of investors’ risk aversion parameter.

A.1.2 REG based on daily conditional variance estimates using expanding win-

dows

Daily conditional variances that help constitute the variance-regret component of anticipated

regret in Eq. (7) are based on GJR-GARCH(1,1) model estimated using daily excess stock

returns over five-year rolling windows (See Section 3.2 for further details). In this section, we

test the robustness of our results to using a different estimation window, i.e., using expanding

daily windows where the initial window is five years. Specifically, we estimate the following

model using daily excess returns for each stock i using expanding windows with the initial

window being the past five years:

ri,d+1 = αi + εi,d+1, (20)

σ2i,d+1 = β0 + (β1 + γIi,d)ε

2i,d + β2σ

2i,d, (21)

where ri,d+1 is the excess return of stock i on day d+ 1, εi,d+1 can be viewed as unexpected

news or information shocks to stock i, Ii,d is an indicator equal to 1 if εi,d is negative and

zero otherwise, and σ2i,d+1 is the one-day-ahead conditional variance of stock i defined as an

asymmetric function of the lagged squared unexpected news (or information shocks) and the

lagged conditional variance. We require a minimum of 200 daily observations to estimate

the conditional variances. The parameters are estimated simultaneously by maximizing the

following conditional log-likelihood function:

L(β0, β1, γ, β2) =∞∑n=1

1

2

[− log(2π)− log(σ2

d)− ε2dσ2d

]. (22)

TABLE A.3 ABOUT HERE

The results of univariate portfolio sorts based on REG using expanding windows to

3

estimate conditional variances are presented in Table A.3. The results confirm our previous

finding that there is significantly negative relation between anticipated regret and one-month-

ahead retuens. Using a different estimation window to estimate daily conditional variances

and anticipated regret, we find that stocks in the highest anticipated regret decile earn

significantly negative raw and risk-adjusted returns and the zero-cost arbitrage portfolio that

is long in stocks with high anticipated regret and short in stocks with low anticipated regret

loses on average –0.73% per month (t-stat. = –2.52) with alphas ranging between –0.57%

(t-stat. = –2.78)and –1.17% (t-stat. = –4.92).

A.1.3 Univariate equal-weighted portfolios of stocks sorted by REG

Section 4.1 documents a significant negative relation between anticipated regret and one-

month ahead returns of value-weighted portfolio of stocks sorted by REG. In this section, we

test whether the negative relation between anticipated regret and one-month ahead stock

returns is robust to using a different portfolio weighting scheme, i.e., equal-weighting. For

each month from July 1963 to December 2018, stocks are sorted into decile portfolios based

on their anticipated regret, i.e., REG. Decile 1 (10) contains stocks with the lowest (highest)

REG. Next, we calculate the one-month-ahead equal-weighted average portfolio returns, and

this procedure is repeated each month until the sample is exhausted. Table A.4 reports for

each decile the average REG, the next-month average excess returns, and the risk-adjusted

returns (alphas) based on the CAPM, the 3-factor model of Fama and French (1993) (FF3),

the 4-factor model of Fama and French (1993) and Carhart (1997) (FFC), the 5-factor model

of Fama and French (1993), Carhart (1997), and Pastor and Stambaugh (2003) (FFCPS),

the 5-factor model of Fama and French (2015) (FF5), and the q-factor model of Hou et al.

(2015). The last row of Table A.4 presents the average return and alpha spreads for the hedge

portfolio that is long in the decile of stocks with the highest REG and short in the decile of

stocks with the lowest REG.

TABLE A.4 ABOUT HERE

Equal-weighted univariate portfolio sorts confirm the previously documented negative

relation between anticipated regret and next-month average returns. The equal-weighted

portfolio of stocks with the lowest REG (decile 1) earns an average excess return of 0.65% per

month, whereas the average excess return on the value-weighted portfolio of stocks with the

highest REG (decile 10) is –0.11% per month. The arbitrage portfolio with a long position

4

in the highest REG stocks and a short position in the lowest REG stocks (High−Low REG)

loses on average 0.76% per month (9.51% per annum) with a t-statistic of –3.38. The last row

of Table A.4 also presents the next month’s risk-adjusted returns for the High−Low REG

portfolio. Alpha spreads for the long-short portfolio are significantly negative ranging from

–0.68% per month (t-stat. = –4.16) for Q-factor alphas to –1.17% per month (t-stat. = –6.31)

for CAPM alphas, respectively. The negative and significant relation between anticipated

regret and the next-month risk-adjusted returns is robust in all cases.

Similar to the results obtained in Section 4.1, we find that the FF5 and Q alphas of decile

1 (low REG stocks) are not significant, whereas the the CAPM, FF3, FFC, FFCPS, FF5,

and Q alphas of decile 10 (high-REG stocks) are significantly negative. Thus, we confirm our

finding that the significantly negative alpha spread between high REG and low REG stocks

is due to underperformance by stocks in the highest REG decile.

A.2 Further robustness checks

Section 4.5 highlighted the fact that REG is highly persistent over the next-month. Fama

and MacBeth (1973) regressions in Section 6.3 also indicate that the predictive ability of

anticipated regretextends over longer horizons, up to eight months. Furthermore, in Section

5, we investigated whether limits to arbitrage, investor inattention and lottery demand can

explain the negative relation between anticipated regret and future stock returns. In this

section we investigate whether the persistece of anticipated regret extends to longer horizons

(Section A.2.1), and examine whether alternative controls that are associated with limits to

arbitrage and investor inattention can potentially explain the documented anticipated regret

effect in the cross-section (Section A.2.2).

A.2.1 Longer-ahead persistence of anticipated regret

We extend our analysis in Section 4.5 to longer horizons and investigate whether stocks

that are associated with high (low) anticipated regret tend to stay in the same anticipated

regret category three, six, and twelve months ahead. Specifically, Table A.5 reports the

average probability that a stock in decile i (defined by the rows) in one month will be in

decile j (defined by the columns) after three months (Panel A), six months (Panel B), and 12

months (Panel C) following the initial portfolio formation period. All the probabilities in

the transition matrix should be approximately 10% if the evolution for anticipated regret for

each stock is random and the relative magnitude of anticipated regret in one period has no

5

implication about the relative anticipated regret values in the subsequent period.

TABLE A.5 ABOUT HERE

Table A.5 suggets that anticipated regret is highly persistent even after 12 months. Panel

C shows that 36.5% (26.9%) of stocks in the lowest (highest) REG decile in a certain month

continue to be in the same decile 12 months later. Moreover, the stocks have a 47.5%

probability of being in deciles 9 and 10 after 12 months, which exhibit higher anticipated

regret in the portfolio formation month and lower returns in the subsequent month. These

results collectively suggest that anticipated regret is a highly persistent equity characteristic.

The results provide evidence of strong cross-sectional persistence of anticipated regret even

after a 12-month gap is established between the lagged and lead REG variables.

Regret theory suggests that investors would pay higher prices for stocks that have exhibited

more attractive expected return-variance combinations in the past with the expectation that

this appealing future return distribution will persist in the future. The results indicate that

the estimated historical REG values successfully predict the future REG values and thus,

our proxy for anticipated regret does say something about the future upside potential (in

terms of expected return-variance combinations) of individual stocks.

A.2.2 Bivariate portfolios of REG with control variables associated with in-

vestor inattention and limits to arbitrage

To verify that the negative relation between anticipated regret and future returns is not

explained by limits to arbitrage or investor inattention, we further conduct bivariate portfolio

sorts using eight control variables. Following the literature, we use firm size, idiosyncratic

volatility, illiquidity, institutional holdings, analyst coverage, dispersion of analyst earnings

forecasts, and firm age as proxies for limits to arbitrage, and institutional holdings, analyst

coverage, abnormal trading volume, and firm size as proxies for investor attention.29

TABLE A.6 ABOUT HERE

29See, e.g. Huberman and Regev (2001), Amihud (2002), Diether, Malloy, and Scherbina (2002), Hirshleiferand Teoh (2003), Hirshleifer, Hou, Teoh, and Zhang (2004), Nagel (2005), Hou and Moskowitz (2005), Peng(2005), Zhang (2006), Barber and Odean (2008), Cohen and Frazzini (2008), Hirshleifer, Lim, and Teoh(2009), Da, Engelberg, and Gao (2011), Hirshleifer, Hsu, and Li (2013), Bali, Peng, Shen, and Tang (2014),Da, Gurun and Warachka (2014), and Han, Hirshleifer, and Walden (2017).

6

We first sort stocks into terciles every month based on the control variables associated

with investor inattention and limits to arbitrage. Next, we divide each control variable tercile

into deciles based on REG to generate 3 × 10 portfolios of the control variable and REG.

We test whether the abnormal return to the value-weighted zero-cost portfolio that buys

(sells) stocks with higher (lower) REG has a more negative one-month-ahead alpha among

the stocks that are costlier to arbitrage and that has lower investor attention. Panel A (B) of

Table A.6 shows the results of 3× 10 bivariate portfolio sorts using controls associated with

investor inattention (costly arbitrage). Panel A and B of Table A.6 suggest that anticipated

regret is more prominant across stocks which are costlier to arbitrage and with low investor

attention, i.e., small, illiquid, young stocks, with high idiosyncratic volatility, low institutional

holdings, high dispersion in analyst forecasts, low analyst coverage and low abnormal trading

volume.

7

References

Amihud, Y., 2002. “Illiquidity and stock returns: Cross-section and time-series effects.”Journal of Financial Markets 5, 31–56.

Bali, T. G., 2008. “The intertemporal relation between expected returns and risk.” Journalof Financial Economics 87, 101–131.

Bali, T. G., Engle, R. F., 2010. “The intertemporal capital asset pricing model withdynamic conditional correlations.” Journal of Monetary Economics 57, 377–390.

Bali, T. G., Peng, L., Shen, Y., Tang, Y., 2014. “Liquidity shocks and stock marketreactions.” Review of Financial Studies 27, 1434-1485.

Barber, B., Odean, T., 2008. “All that glitters: The effect of attention on the buyingbehavior of individual and institutional investors.” Review of Financial Studies 21,785–818.

Bekaert, G., Harvey, C. R., 1997. “Emerging equity market volatility.” Journal of FinancialEconomics 43, 29–77.

Carhart, M., 1997. “On persistence in mutual fund performance.” Journal of Finance 52,57–82.

Cohen, L., Frazzini, A., 2008. “Economic links and predictable returns.” Journal of Finance63, 1977–2011.

Da, Z., Engelberg, J., Gao, P., 2011. “In search for attention.” Journal of Finance 66,1461–1499.

Da, Z., Gurun, U. G., Warachka, M., 2011. “Frog in the pan: Continuous information andmomentum.” Review of Financial Studies 27, 2171–2218.

Diether, K. B., Malloy, C. J., Scherbina, A., 2002. “Differences of opinion and the crosssection of stock returns.” Journal of Finance 57, 2113-2141.

Engle, R. F., Ng, V. K., 1993. “Measuring and testing the impact of news on volatility.”Journal of Finance 48, 1749-1778.

Fama, E. F., MacBeth, J., 1973. “Risk, return and equilibrium: Empirical tests.” Journalof Political Economy 81, 607–636.

Fama, E. F., French, K. R., 1993. “Common risk factors in the returns on stocks andbonds.” Journal of Financial Economics 33, 3–56. item Fama, E. F., French, K. R.,2015. “A five-factor asset pricing model.” Journal of Financial Economics 116, 1–22.

Ghysels, E., Santa-Clara, P., Valkanov, R., 2005. “There is a risk-return trade-off after all.”Journal of Financial Economics 76, 509–548.

Glosten, L. R., Jagannathan, R., Runkle, D.E., 1993. “On the relation between the expectedvalue and the volatility of the nominal excess return on stocks.” Journal of Finance 48,1779–1801.

Guo, H., Whitelaw, R. F., 2006. “Uncovering the risk-return relation in the stock market.”Journal of Finance 61, 1433–1463.

Han, B., Hirshleifer, D. A., Walden, J., 2019. “Social transmission bias and investorbehavior.” Working paper, Rotman School of Management.

8

Hansen, P. R. Lunde, A. 2005. “A forecast comparison of volatility models: Does anythingbeat a GARCH (1,1)?” Journal of Applied Econometrics 20, 873-899.

Hirshleifer, D., Teoh, S. H., 2003. “Limited attention, information disclosure, and financialreporting” Journal of Accounting and Economics 36, 337–386.

Hirshleifer, D., Hou, K., Teoh, S. H., Zhang, Y., 2004. “Do investors overvalue firms withbloated balance sheets?” Journal of Accounting and Economics 38, 297–331.

Hirshleifer, D., Lim, S., Teoh, S. H., 2009. “Driven to distraction: Extraneous events andunderreaction to earnings news.” Journal of Finance 64, 2289–2325.

Hirshleifer, D., Hsu, P.-H., Li, D., 2013. “Innovative efficiency and stock returns.” Journalof Financial Economics 107, 632–654.

Hou, K., Moskowitz, T., 2005. ““Market frictions, price delay, and the cross-section ofexpected returns” Review of Financial Studies 18, 981–1020.

Hou, K., Xue, C., Zhang, L., 2015. “Digesting anomalies: An investment approach.” Reviewof Financial Studies 28, 650–705.

Huberman, G., Regev, T., 2001. “Contagious speculation and a cure for cancer: A non-eventthat made stock prices soar.” Journal of Finance 56, 387–396.

Nagel, S., 2005. “Short sales, institutional investors and the cross-section of stock returns.”Journal of Financial Economics 78, 277–309.

Newey, W. K., West, K. D., 1987. “A simple, positive semi-definite, heteroscedasticity andautocorrelation consistent covariance matrix.” Econometrica 55, 703–708.

Pastor, L., Stambaugh, R. F., 2003. “Liquidity risk and expected stock returns.” Journalof Political Economy 111, 642–685.

Peng, L., 2005. “Learning with information capacity constraints.” Journal of Financial andQuantitative Analysis 40, 307–330.

Zhang, X. F., 2006. “Information uncertainty and stock returns.” Journal of Finance 61,105–137.

9

Table A.1Variable definitions

This table gives the names and definitions of the variables used in our tests. We calculate variables indexedby an “M” on a monthly basis. We calculate variables indexed by a “Q” on a quarterly basis and convert tomonthly data by using the value at the end of the last fiscal quarter during or before month t. We calculatevariables indexed by an “A” on an annual basis, using the computed values from June of calendar year n toMay of calendar year n+ 1.

Name Description

Panel A: Anticipated regret measures

REG (M) Main anticipated regret measure, calculated using the difference between RREG andb2 × V REG, where b is the investors’ risk aversion parameter.

CREG (M) Alternative anticipated regret measure, calculated using the cross-sectional averages ofmonthly returns and monthly conditional variances of all stocks as return and variancebenchmarks.

Panel B: Components of anticipated regret measure (REG)

RREG (M) Return regret measure, calculated using the difference between the mean return and themaximum return attained in the same month.

VREG (M) Variance regret measure, calculated using the difference between mean variance andminimum variance attained during a given month.

Panel C: Control variables

βMKT (M) Market beta of individual stocks estimated using monthly returns over the past fiveyears with a minimum of 24 monthly observations; (see Fama and French (1992)).

SIZE (M) Log of the market value of equity (abs(prc) × shrout; see Fama and French (1992)).BM (A) Log of the ratio of the book value to the market value of equity (abs(prc) × shrout),

where the book value of equity is total assets (at) less total liabilities (lt) plus deferredtaxes (txditc, zero if missing) less preferred stock (pstkl, pstkrv, prfstck, or zero, in thatorder of availability) and the variables are from the fiscal year end in calendar yearn− 1; (see Fama and French (1992)).

STR (M) Short-term return reversal, measured with past one-month return; (see Jegadeesh(1990)).

MOM (M) The compounded stock return (ret) over the prior twelve months of monthly data, butexcluding the most recent month; (see Jegadeesh and Titman (1993)).

ILLIQ (M) The ratio of the absolute daily stock return (ret) to daily dollar trading volume(abs(prc) × vol), where the average is taken over all trading days over the month; (seeAmihud (2002)).

COSKEW (M) Co-skewness of stock i in month t, i.e.,E[εi,tε

2M,t]√

E[ε2i,t]E[ε2M,t], where εi,t is residual from the

regression of stock’s daily excess return on the contemporaneous market excess return;(see Harvey and Siddique (2000)).

IVOL (M) Monthly standard deviation of the residual term from the time series regression of stocki’ s excess returns on Fama and French (1993) 3-factor model; (see Ang et al. (2006)).

MAX (M) Maximum daily return of the stock in month t; (see Bali et al. (2011)).MAX5 (M) The average of the five highest daily returns of the stock in month t; (see Bali et al.

(2017)).CVRG (M) The number of analysts covering the stock (I/B/E/S).DISP (M) Analyst earnings forecast dispersion, calculated by the standard deviation of annual

earnings-per-share forecasts scaled by the absolute value of the average outstandingforecast.

AGE (M) Number of months that the stock has been listed on the CRSP database.ABTURN (M) Abnormal trading volume, defined as each stock’s share turnover in month t minus the

average monthly share turnover over the past 12 months from month t− 12 to t− 1.

(continued on next page)

10

Table A.1Variable definitions (cont.)

Name Description

COST (M) Arbitrage cost index created by sorting stocks in increasing order into deciles based ontheir idiosyncratic volatility, illiquidity and dispersion of analysts forecasts. Similarly,stocks are also sorted in decreasing order into deciles based on their level of institutionalownership, analyst coverage, size, and age. Each stock is then given the correspondingscore of its decile rank for all seven variables. The arbitrage cost index is the sum of theseven scores so that it ranges from 7 to 70 and its higher values indicate stricter limitsto arbitrage.

ATT (M) Investor attention index created by sorting stocks in increasing order into deciles basedon their institutional ownership, analyst coverage, abnormal trading volume, andmarket capitalization. Each stock is given the corresponding score of its decile rank forall four variables. The attention index is the sum of the four scores so that it rangesfrom 4 to 40 and its higher values indicate stronger attention-grabbing characteristic..

LTRY (M) Lottery index created by sorting stocks for each month into 50 bins by price per share(PRC) in descending order such that stocks in the lowest bin (i.e., a PRC portfolio rankof 1) have the highest price per share and those in the highest bin (i.e., a PRC portfoliorank of 50) have the lowest price per share. We further sort stocks into 50 bins byidiosyncratic volatility (IVOL) and idiosyncratic skewness (ISKEW) in ascending order.IVOL and ISKEW are, respectively, the standard deviation and the skewness ofresiduals from the time series regression of daily stock returns against the MKT, SMB,and HML factors in a month. LTRY is then constructed by summing up the ranks ofthe PRC, IVOL, and ISKEW portfolios. The lottery index thus has an integer value inthe range of 3 to 150 and it increases with a stock’s lottery feature; (see Kumar (2009)).

TK (M) Prospect theory value of the stock’s historical return distribution using past five yearsof monthly data; (see Barberis, Mukherjee, and Wang (2016)).

IA (A) Annual growth rate of total assets (at) for the fiscal year ending in calendar year n-1;(see Fama and French (2015), Hou et al. (2015)).

OP (Q) Income before extraordinary items (ibq) for the most recent quarter prior to theportfolio formation month divided by one-quarter-lagged book equity. Quarterly bookequity is shareholders’ equity, plus balance-sheet deferred taxes, and investment taxcredit (txditcq) if available, minus book value of preferred stock. Depending onavailability, shareholders’ equity is measured by stockholder’s equity (seqq), or commonequity (ceqq) plus the carrying value of preferred stock (pstkq), or total assets (atq)minus total liabilities (ltq). The book value of preferred stock is measured byredemption value (pstkrq) if available or carrying value (pstkq); (see Davis et al.(2008)).

INST (Q) Institutional ownership ratio, calculated by dividing the institutional ownership levelcomputed using equity holdings by institutions that file 13F reports divided by totalshares outstanding at quarter end.

11

Table A.2Univariate portfolio of stocks sorted by REG with b=2 and b=10

Each month from July 1963 to December 2018, stocks are sorted into decile portfolios based ontheir REG. This table reports the average REG values, the next-month value-weighted averageexcess returns, the CAPM, Fama-French 3-factor (FF3), Fama-French-Carhart 4-factor (FFC),Fama-French-Carhart-Pastor-Stambaugh 5-factor (FFCPS), Fama-French 5-factor (FF5), andq-factor (Q) risk-adjusted returns (alphas) for the 10 decile portfolios of stocks sorted byREG. The last row presents the 10–1 average raw and risk-adjusted return differences of thearbitrage portfolio with a long position in the decile of stocks with the highest REG (decile10) and a short position in the decile of stocks with the lowest REG (decile 1). Panel A andB report the results for REG based on the risk aversion parameters b = 2 and b = 10 asdefined in Eq. (18) and Eq. (19), respectively. The Newey-West (1987) adjusted t-statistics(with six lags) are given in square brackets.

Panel A: Portfolio sorts by REG with b=2

Decile REG Mean CAPM FF3 FFC FFCPS FF5 Q

1 (Low) 1.78 0.46 0.14 0.07 0.02 0.02 −0.04 −0.03[3.33] [1.80] [0.95] [0.21] [0.22] [−0.61] [−0.27]

2 2.95 0.59 0.18 0.13 0.09 0.10 0.01 0.00[3.75] [2.76] [2.23] [1.44] [1.42] [0.09] [−0.02]

3 3.80 0.48 0.01 −0.03 −0.06 −0.04 −0.14 −0.15[2.73] [0.11] [−0.61] [−1.13] [−0.80] [−2.86] [−2.25]

4 4.65 0.60 0.09 0.06 0.08 0.04 0.00 −0.02[3.30] [1.75] [1.21] [1.50] [0.75] [0.06] [−0.36]

5 5.59 0.65 0.08 0.05 0.06 0.06 0.04 0.04[3.08] [1.16] [0.79] [1.07] [0.89] [0.51] [0.54]

6 6.72 0.64 0.04 0.02 0.08 0.07 0.05 0.06[3.04] [0.61] [0.33] [1.09] [0.89] [0.74] [0.66]

7 8.15 0.68 0.02 0.02 0.09 0.10 0.15 0.22[2.80] [0.29] [0.23] [1.14] [1.18] [1.73] [2.03]

8 10.17 0.46 −0.24 −0.24 −0.15 −0.13 −0.11 −0.12[1.66] [−1.98] [−2.53] [−1.76] [−1.39] [−1.08] [−1.03]

9 13.62 0.28 −0.47 −0.44 −0.37 −0.38 −0.13 −0.07[0.92] [−3.15] [−3.39] [−3.00] [−2.87] [−1.19] [−0.47]

10 (High) 27.58 −0.21 −1.00 −1.03 −0.86 −0.91 −0.64 −0.52[−0.60] [−5.44] [−6.13] [−5.21] [−5.05] [−4.46] [−3.13]

High−Low 25.80 −0.67 −1.14 −1.09 −0.88 −0.93 −0.60 −0.49[−2.33] [−4.98] [−5.44] [−4.39] [−4.29] [−3.64] [−2.40]

(continued on next page)

12

Table A.2Univariate portfolio of stocks sorted by REG with b=2 and b=10 (cont.)

Panel A: Portfolio sorts by REG with b=10

Decile REG Mean CAPM FF3 FFC FFCPS FF5 Q

1 (Low) 2.74 0.49 0.16 0.09 0.04 0.04 −0.03 0.00

[3.54] [2.18] [1.31] [0.62] [0.62] [−0.48] [0.01]

2 4.86 0.54 0.11 0.08 0.04 0.05 −0.03 −0.03

[3.30] [1.88] [1.53] [0.72] [0.78] [−0.55] [−0.40]

3 6.59 0.58 0.11 0.08 0.07 0.06 −0.01 −0.03

[3.42] [2.03] [1.51] [1.26] [1.04] [−0.21] [−0.45]

4 8.45 0.61 0.09 0.06 0.07 0.05 0.00 −0.03

[3.17] [1.61] [1.16] [1.40] [0.89] [0.01] [−0.39]

5 10.64 0.69 0.12 0.11 0.13 0.13 0.07 0.08

[3.33] [1.74] [1.71] [1.99] [1.92] [1.05] [1.07]

6 13.41 0.61 0.00 −0.03 −0.02 −0.01 0.00 −0.03

[2.70] [0.01] [−0.45] [−0.34] [−0.15] [0.04] [−0.38]

7 17.17 0.47 −0.18 −0.18 −0.12 −0.13 −0.04 −0.01

[1.94] [−1.75] [−2.06] [−1.51] [−1.41] [−0.48] [−0.12]

8 22.88 0.46 −0.25 −0.25 −0.14 −0.14 −0.10 −0.08

[1.72] [−2.38] [−2.96] [−1.57] [−1.49] [−1.03] [−0.66]

9 33.53 0.39 −0.36 −0.35 −0.27 −0.27 −0.09 0.00

[1.28] [−2.57] [−3.29] [−2.65] [−2.45] [−0.93] [−0.04]

10 (High) 85.90 −0.26 −1.05 −1.05 −0.87 −0.88 −0.64 −0.53

[−0.75] [−5.65] [−6.02] [−5.17] [−4.67] [−4.46] [−3.05]

High−Low 83.16 −0.75 −1.21 −1.13 −0.91 −0.93 −0.61 −0.53

[−2.61] [−5.26] [−5.51] [−4.51] [−4.14] [−3.75] [−2.42]

13

Table A.3Univariate portfolios of stocks sorted by REG based on daily conditional varianceestimates using expanding windows

Each month from July 1963 to December 2018, stocks are sorted into decile portfolios basedon their REG. The variance-regret component of REG in Eq.(10) is estimated from aGJR-GARCH(1,1) model using expanding daily windows, where the initial window is fiveyears. This table reports the average REG values, the next-month equal-weighted averageexcess returns, the CAPM, Fama-French 3-factor (FF3), Fama-French-Carhart 4-factor (FFC),Fama-French-Carhart-Pastor-Stambaugh 5-factor (FFCPS), Fama-French 5-factor (FF5), andq-factor (Q) risk-adjusted returns (alphas) for the 10 decile portfolios of stocks sorted byREG. The last row presents the 10–1 average raw and risk-adjusted return differences ofthe arbitrage portfolio with a long position in the decile of stocks with the highest REG(decile 10) and a short position in the decile of stocks with the lowest REG (decile 1). TheNewey-West (1987) adjusted t-statistics (with six lags) are given in square brackets.

Decile REG Mean CAPM FF3 FFC FFCPS FF5 Q

1 (Low) 2.22 0.49 0.17 0.10 0.05 0.04 0.00 0.03

[3.52] [2.20] [1.39] [0.68] [0.54] [−0.02] [0.29]

2 3.76 0.54 0.13 0.09 0.06 0.07 −0.01 −0.02

[3.30] [2.09] [1.67] [0.94] [0.97] [−0.24] [−0.23]

3 4.94 0.51 0.06 0.02 0.01 0.01 −0.11 −0.10

[2.99] [0.92] [0.35] [0.11] [0.19] [−2.08] [−1.69]

4 6.17 0.52 0.00 −0.04 −0.02 −0.06 −0.06 −0.10

[2.57] [−0.03] [−0.66] [−0.30] [−0.98] [−0.96] [−1.32]

5 7.58 0.76 0.20 0.18 0.20 0.19 0.18 0.21

[3.59] [2.83] [2.55] [2.73] [2.56] [2.46] [2.39]

6 9.30 0.58 −0.03 −0.05 0.03 0.05 −0.02 0.01

[2.50] [−0.36] [−0.60] [0.36] [0.61] [−0.25] [0.10]

7 11.58 0.60 −0.05 −0.07 −0.02 −0.01 0.01 0.05

[2.38] [−0.53] [−0.94] [−0.29] [−0.09] [0.08] [0.50]

8 14.93 0.46 −0.25 −0.25 −0.13 −0.15 −0.07 −0.04

[1.62] [−2.10] [−2.74] [−1.46] [−1.46] [−0.76] [−0.36]

9 20.95 0.26 −0.47 −0.46 −0.39 −0.39 −0.18 −0.12

[0.88] [−3.25] [−4.18] [−3.59] [−3.35] [−1.76] [−0.85]

10 (High) 48.33 −0.24 −1.01 −0.99 −0.83 −0.85 −0.59 −0.54

[−0.68] [−5.36] [−6.38] [−5.59] [−5.35] [−4.61] [−3.50]

High−Low 46.12 −0.73 −1.17 −1.09 −0.88 −0.89 −0.59 −0.57

[−2.52] [−4.92] [−5.61] [−4.76] [−4.50] [−3.71] [−2.78]

14

Table A.4Univariate equal-weighted portfolios of stocks sorted by REG

Each month from July 1963 to December 2018, stocks are sorted into decile portfolios based ontheir REG. This table reports the average REG values, the next-month equal-weighted averageexcess returns, the CAPM, Fama-French 3-factor (FF3), Fama-French-Carhart 4-factor (FFC),Fama-French-Carhart-Pastor-Stambaugh 5-factor (FFCPS), Fama-French 5-factor (FF5), andq-factor (Q) risk-adjusted returns (alphas) for the 10 decile portfolios of stocks sorted byREG. The last row presents the 10–1 average raw and risk-adjusted return differences ofthe arbitrage portfolio with a long position in the decile of stocks with the highest REG(decile 10) and a short position in the decile of stocks with the lowest REG (decile 1). TheNewey-West (1987) adjusted t-statistics (with six lags) are given in square brackets.

Decile REG Mean CAPM FF3 FFC FFCPS FF5 Q

1 (Low) 2.17 0.65 0.35 0.21 0.19 0.20 0.07 0.14

[4.34] [3.63] [2.67] [2.61] [2.54] [1.07] [1.50]

2 3.71 0.82 0.40 0.23 0.25 0.25 0.09 0.15

[4.65] [4.27] [3.63] [4.44] [4.22] [1.61] [1.63]

3 4.90 0.90 0.43 0.24 0.27 0.27 0.11 0.14

[4.62] [4.26] [4.03] [4.74] [4.38] [2.06] [1.67]

4 6.12 0.95 0.45 0.26 0.29 0.29 0.13 0.16

[4.59] [4.24] [4.50] [5.21] [4.83] [2.42] [1.96]

5 7.53 0.93 0.38 0.19 0.23 0.23 0.07 0.10

[4.21] [3.41] [3.47] [4.35] [4.16] [1.30] [1.38]

6 9.27 0.93 0.35 0.18 0.22 0.22 0.11 0.16

[3.92] [2.99] [3.72] [4.96] [4.70] [2.27] [2.55]

7 11.56 0.80 0.18 0.01 0.07 0.09 −0.02 0.04

[3.14] [1.44] [0.24] [1.50] [1.75] [−0.38] [0.57]

8 14.94 0.67 0.00 −0.14 −0.07 −0.07 −0.10 −0.05

[2.44] [0.02] [−3.00] [−1.50] [−1.26] [−1.97] [−0.75]

9 21.04 0.48 −0.21 −0.35 −0.26 −0.24 −0.22 −0.12

[1.64] [−1.43] [−5.89] [−4.48] [−3.92] [−3.63] [−1.54]

10 (High) 49.19 −0.11 −0.81 −0.91 −0.81 −0.82 −0.64 −0.53

[−0.35] [−4.71] [−10.48] [−8.62] [−8.13] [−7.89] [−4.48]

High−Low 47.01 −0.76 −1.17 −1.12 −1.00 −1.02 −0.71 −0.68

[−3.38] [−6.31] [−8.52] [−7.83] [−7.48] [−7.09] [−4.16]

15

Table A.5Longer-term portfolio transition matrices for anticipated regret

This table reports the percentage of stocks moving from one REG-sorted decile portfolio in month t to the same or anotherREG-sorted decile portfolio in month t+ 3 (Panel A), in month t+ 6 (Panel B), and in month t+ 12 (Panel C). Portfolios 1 and10 indicate portfolio of stocks with the lowest REG and the highest REG, respectively.

Panel A: Three-month ahead portfolio transition matrix

Portfolio rank in month t

Portfolio rank 1 2 3 4 5 6 7 8 9 10

in month t+ 3 (Low) (High)

1 (Low) 41.4% 21.1% 12.5% 8.1% 5.5% 3.8% 2.9% 2.3% 1.9% 2.3%

2 20.4% 22.5% 18.0% 13.2% 9.6% 6.7% 4.6% 3.1% 2.1% 1.7%

3 12.1% 17.0% 17.5% 15.5% 12.6% 9.7% 7.1% 4.9% 3.3% 2.1%

4 7.7% 12.6% 14.6% 15.1% 14.1% 12.4% 9.8% 7.4% 4.9% 2.9%

5 5.4% 8.8% 11.6% 13.6% 14.3% 13.7% 12.3% 10.0% 7.5% 4.3%

6 3.8% 6.3% 8.9% 11.3% 13.1% 14.0% 14.2% 12.8% 10.3% 6.3%

7 2.8% 4.5% 6.4% 8.8% 11.2% 13.2% 14.9% 15.2% 13.9% 9.5%

8 2.2% 3.2% 4.7% 6.5% 9.0% 11.5% 14.3% 16.7% 17.4% 14.3%

9 1.9% 2.3% 3.2% 4.7% 6.5% 9.0% 12.0% 16.0% 20.4% 22.0%

10 (High) 2.2% 1.8% 2.4% 3.1% 4.2% 5.9% 8.1% 11.6% 18.4% 34.5%

Sum 100.0% 100.0% 100.0% 100.0% 100.0% 100.0% 100.0% 100.0% 100.0% 100.0%

(continued on next page)

16

Table A.5Longer-term portfolio transition matrices for anticipated regret (cont.)

Panel B: Six-month ahead portfolio transition matrix

Portfolio rank in month t

Portfolio rank 1 2 3 4 5 6 7 8 9 10

in month t+ 6 (Low) (High)

1 (Low) 39.1% 20.7% 12.9% 8.7% 5.9% 4.3% 3.2% 2.6% 2.2% 2.4%

2 20.2% 21.5% 17.6% 13.4% 9.8% 7.3% 5.1% 3.6% 2.7% 2.0%

3 12.1% 16.8% 16.8% 15.2% 12.6% 9.9% 7.5% 5.4% 3.7% 2.5%

4 8.1% 12.4% 14.5% 14.5% 13.8% 12.2% 10.0% 7.9% 5.6% 3.5%

5 5.7% 9.0% 11.6% 13.0% 13.8% 13.4% 12.3% 10.4% 7.8% 4.9%

6 4.2% 6.6% 8.8% 11.1% 12.7% 13.4% 13.8% 12.9% 10.8% 7.2%

7 3.2% 4.8% 6.6% 8.8% 11.1% 12.8% 14.2% 14.8% 13.0% 10.3%

8 2.6% 3.5% 4.9% 6.8% 9.0% 11.4% 13.7% 15.7% 17.0% 14.7%

9 2.3% 2.6% 3.6% 4.9% 6.8% 9.1% 11.8% 15.1% 19.1% 21.3%

10 (High) 2.6% 2.1% 2.7% 3.5% 4.5% 6.1% 8.4% 11.5% 17.3% 31.1%

Sum 100.0% 100.0% 100.0% 100.0% 100.0% 100.0% 100.0% 100.0% 100.0% 100.0%

(continued on next page)

17

Table A.5Longer-term portfolio transition matrices for anticipated regret (cont.)

Panel C: Twelve-month ahead portfolio transition matrix

Portfolio rank in month t

Portfolio rank 1 2 3 4 5 6 7 8 9 10

in month t+ 12 (Low) (High)

1 (Low) 36.5% 20.8% 13.2% 9.4% 6.5% 4.8% 3.7% 2.9% 2.5% 2.5%

2 19.7% 20.9% 17.3% 13.8% 10.2% 7.8% 5.8% 4.1% 3.1% 2.2%

3 12.4% 16.2% 16.4% 14.8% 12.8% 10.4% 8.1% 6.2% 4.5% 3.0%

4 8.5% 12.3% 14.0% 14.2% 13.5% 12.2% 10.4% 8.5% 6.4% 4.1%

5 6.0% 9.1% 11.3% 12.7% 13.3% 13.0% 12.2% 10.9% 8.7% 5.9%

6 4.5% 6.8% 8.8% 10.6% 12.3% 13.1% 13.6% 12.7% 11.4% 8.2%

7 3.6% 4.9% 6.8% 8.8% 10.7% 12.3% 13.5% 14.2% 13.9% 11.4%

8 3.1% 3.8% 5.3% 6.9% 8.9% 10.9% 13.0% 15.1% 16.1% 15.2%

9 2.6% 2.9% 3.9% 5.1% 6.9% 9.1% 11.4% 14.2% 17.6% 20.6%

10 (High) 3.1% 2.4% 2.9% 3.7% 4.9% 6.3% 8.5% 11.2% 15.8% 26.9%

Sum 100.0% 100.0% 100.0% 100.0% 100.0% 100.0% 100.0% 100.0% 100.0% 100.0%

18

Table A.63x10 sorts by investor attention, limits to arbitrage, and REG

Each month from July 1963 to December 2018, stocks are first sorted into three portfolios (Low, Medium, High) by control variablesassociated with investor inattention and limits to arbitrage. The control variables are institutional holdings, analyst coverage, abnormaltrading volume, and firm size for investor attention, and firm size, idiosyncratic volatility, illiquidity, institutional holdings, analyst coverage,dispersion of analyst earnings forecasts, and firm age for limits to arbitrage. In the next step, stocks are sorted by their anticipated regretmeasure REG. Panel A (Panel B) presents the next-month FF5 alphas for each of the stock characteristic-sorted tercile portfolio andREG-sorted decile portfolio, and the alpha spreads for the hedge portfolio that is long in the decile of stocks with the highest REG andshort in the decile of stocks with the lowest REG for each of the tercile portfolio sorted by the stock characteristic associated with investorinattention (limits to arbitrage). The last column of each panel reports the average FF5 alpha of the REG-sorted decile portfolios andthe zero-cost arbitrage portfolio across the three terciles. The Newey-West (1987) adjusted t-statistics (with six lags) are given in squarebrackets.

Panel A: Investor inattentionSIZE INST CVRG ABTURN

REG Low 2 High Low 2 High Low 2 High Low 2 High

1(Low) 0.25 0.07 −0.06 0.21 0.16 −0.15 0.02 0.04 −0.06 −0.26 −0.03 0.14[2.30] [1.04] [−0.72] [1.61] [1.66] [−1.58] [−0.28] [0.51] [−0.69] [−2.34] [−0.43] [1.73]

2 0.42 0.10 −0.05 0.44 0.30 −0.10 0.02 0.00 −0.01 −0.17 0.00 0.18[4.06] [1.61] [−0.66] [2.86] [3.23] [−1.08] [0.18] [−0.06] [−0.08] [−1.48] [−0.05] [2.15]

3 0.37 0.07 0.03 0.34 0.27 −0.33 0.00 0.01 0.12 −0.25 −0.08 −0.06[3.22] [1.06] [0.50] [1.96] [2.81] [−4.01] [−0.04] [0.14] [1.34] [−2.58] [−1.05] [−0.68]

4 0.24 0.08 −0.08 0.41 0.20 −0.19 0.06 −0.13 −0.11 −0.23 −0.03 0.07[2.30] [1.14] [−1.10] [1.89] [2.02] [−2.08] [0.69] [−1.42] [−1.26] [−2.08] [−0.38] [0.75]

5 0.10 0.02 −0.03 0.47 0.37 −0.44 −0.13 −0.12 −0.13 0.06 −0.09 0.13[1.11] [0.27] [−0.63] [1.68] [2.92] [−4.10] [−1.20] [−1.47] [−1.74] [0.50] [−0.95] [1.03]

6 0.00 −0.04 0.05 0.49 0.22 −0.49 −0.03 −0.08 0.01 −0.01 0.08 0.08[0.05] [−0.66] [0.75] [1.49] [1.78] [−3.97] [−0.34] [−0.86] [0.09] [−0.08] [0.94] [0.64]

7 −0.11 −0.14 0.12 0.43 0.10 −0.25 −0.15 −0.13 0.03 −0.12 −0.16 0.15[−1.16] [−2.02] [1.72] [1.23] [0.75] [−2.25] [−1.54] [−1.30] [0.27] [−0.86] [−1.47] [1.02]

8 −0.31 −0.26 0.17 0.21 −0.03 −0.27 −0.17 −0.13 0.21 −0.37 −0.19 0.03[−2.99] [−3.96] [1.69] [0.74] [−0.24] [−1.94] [−1.20] [−1.27] [1.69] [−2.52] [−1.97] [0.21]

9 −0.59 −0.29 0.02 −0.13 −0.19 −0.31 −0.54 −0.27 −0.02 −0.50 −0.28 −0.19[−5.86] [−3.62] [0.22] [−0.38] [−1.18] [−1.94] [−4.35] [−2.66] [−0.19] [−3.13] [−2.16] [−1.42]

10 (High) −1.14 −0.48 −0.15 −0.70 −0.34 −0.58 −0.80 −0.18 −0.16 −1.10 −0.42 −0.62[−10.03] [−4.04] [−1.27] [−2.09] [−1.59] [−2.97] [−4.86] [−1.04] [−1.01] [−5.49] [−3.12] [−3.05]

High−Low −1.40 −0.55 −0.09 −0.91 −0.50 −0.44 −0.83 −0.22 −0.10 −0.84 −0.38 −0.75[−10.36] [−3.67] [−0.58] [−2.62] [−1.98] [−1.96] [−4.13] [−1.07] [−0.49] [−3.93] [−2.41] [−3.37]

(continued on next page)

19

Table A.63x10 sorts by investor attention, limits to arbitrage, and REG (cont.)

Panel B: Limits to arbitrage

SIZE INST CVRG ILLIQ IVOL DISP AGE

REG Low 2 High Low 2 High Low 2 High Low 2 High Low 2 High Low 2 High Low 2 High

1(Low) 0.25 0.07 −0.06 0.21 0.16 −0.15 0.02 0.04 −0.06 −0.05 −0.05 0.23 0.03 −0.05 −0.15 0.07 −0.09 −0.13 0.26 −0.02 −0.08[2.30] [1.04] [−0.72] [1.61] [1.66] [−1.58] [−0.28] [0.51] [−0.69] [−0.63] [−0.70] [2.26] [0.28] [−0.56] [−1.13] [0.69] [−1.06] [−0.93] [2.71] [−0.23] [−1.02]

2 0.42 0.10 −0.05 0.44 0.30 −0.10 0.02 0.00 −0.01 −0.07 −0.01 0.19 −0.03 −0.08 −0.25 0.10 0.04 −0.12 0.05 0.08 −0.10[4.06] [1.61] [−0.66] [2.86] [3.23] [−1.08] [0.18] [−0.06] [−0.08] [−0.95] [−0.18] [1.86] [−0.33] [−0.82] [−1.71] [1.06] [0.40] [−1.14] [0.61] [0.87] [−1.35]

3 0.37 0.07 0.03 0.34 0.27 −0.33 0.00 0.01 0.12 0.05 −0.05 0.11 0.02 0.07 −0.26 −0.03 −0.04 −0.21 0.24 −0.07 −0.02[3.22] [1.06] [0.50] [1.96] [2.81] [−4.01] [−0.04] [0.14] [1.34] [0.81] [−0.79] [1.22] [0.31] [0.82] [−1.87] [−0.32] [−0.39] [−1.48] [2.32] [−0.82] [−0.30]

4 0.24 0.08 −0.08 0.41 0.20 −0.19 0.06 −0.13 −0.11 −0.01 0.03 0.01 0.00 0.04 −0.07 0.00 −0.14 −0.01 0.35 0.03 −0.07[2.30] [1.14] [−1.10] [1.89] [2.02] [−2.08] [0.69] [−1.42] [−1.26] [−0.19] [0.44] [0.14] [−0.02] [0.34] [−0.51] [−0.03] [−1.32] [−0.08] [3.23] [0.30] [−1.15]

5 0.10 0.02 −0.03 0.47 0.37 −0.44 −0.13 −0.12 −0.13 −0.07 −0.01 −0.10 −0.02 0.01 −0.10 −0.03 0.10 −0.14 0.18 0.18 −0.08[1.11] [0.27] [−0.63] [1.68] [2.92] [−4.10] [−1.20] [−1.47] [−1.74] [−1.30] [−0.19] [−1.06] [−0.18] [0.08] [−0.60] [−0.34] [0.92] [−0.95] [1.60] [1.43] [−1.08]

6 0.00 −0.04 0.05 0.49 0.22 −0.49 −0.03 −0.08 0.01 0.13 −0.10 −0.18 −0.04 0.01 −0.15 −0.03 0.08 0.21 0.02 0.09 −0.09[0.05] [−0.66] [0.75] [1.49] [1.78] [−3.97] [−0.34] [−0.86] [0.09] [1.79] [−1.70] [−1.68] [−0.51] [0.12] [−1.20] [−0.29] [0.68] [1.04] [0.15] [0.75] [−1.12]

7 −0.11 −0.14 0.12 0.43 0.10 −0.25 −0.15 −0.13 0.03 0.03 −0.13 −0.32 −0.16 0.11 −0.29 −0.01 0.10 −0.26 0.00 0.05 −0.07[−1.16] [−2.02] [1.72] [1.23] [0.75] [−2.25] [−1.54] [−1.30] [0.27] [0.41] [−2.05] [−2.79] [−2.14] [1.43] [−1.77] [−0.12] [0.73] [−1.38] [−0.03] [0.39] [−0.72]

8 −0.31 −0.26 0.17 0.21 −0.03 −0.27 −0.17 −0.13 0.21 0.23 −0.21 −0.44 −0.08 0.01 −0.62 −0.10 0.02 −0.12 −0.11 −0.13 −0.13[−2.99] [−3.96] [1.69] [0.74] [−0.24] [−1.94] [−1.20] [−1.27] [1.69] [2.22] [−2.60] [−3.67] [−1.09] [0.10] [−4.27] [−0.89] [0.18] [−0.53] [−0.77] [−1.02] [−1.30]

9 −0.59 −0.29 0.02 −0.13 −0.19 −0.31 −0.54 −0.27 −0.02 −0.01 −0.34 −0.80 −0.03 −0.09 −0.51 −0.19 0.13 −0.43 −0.22 −0.07 −0.31[−5.86] [−3.62] [0.22] [−0.38] [−1.18] [−1.94] [−4.35] [−2.66] [−0.19] [−0.11] [−4.27] [−7.38] [−0.38] [−0.78] [−2.93] [−1.20] [0.98] [−2.17] [−1.47] [−0.48] [−2.34]

10 (High) −1.14 −0.48 −0.15 −0.70 −0.34 −0.58 −0.80 −0.18 −0.16 −0.21 −0.41 −1.38 −0.05 0.00 −0.93 −0.18 −0.14 −0.76 −0.70 −0.62 −0.41[−10.03] [−4.04] [−1.27] [−2.09] [−1.59] [−2.97] [−4.86] [−1.04] [−1.01] [−1.65] [−3.51] [−10.98] [−0.43] [0.03] [−4.44] [−1.16] [−0.80] [−2.76] [−4.24] [−3.87] [−2.56]

High−Low −1.40 −0.55 −0.09 −0.91 −0.50 −0.44 −0.83 −0.22 −0.10 −0.16 −0.36 −1.61 −0.08 0.05 −0.77 −0.25 −0.06 −0.63 −0.96 −0.60 −0.33[−10.36] [−3.67] [−0.58] [−2.62] [−1.98] [−1.96] [−4.13] [−1.07] [−0.49] [−0.98] [−2.38] [−11.03] [−0.44] [0.33] [−3.23] [−1.35] [−0.29] [−2.09] [−4.90] [−3.09] [−1.67]

20