Disposed to Be Overconfident

Katrin Gödker, Terrance Odean, and Paul Smeets ∗

September 25, 2024

ABSTRACT

We hypothesize that individuals learn about their investment ability based on realized gains and losses

rather than overall portfolio performance. Thus, how investors sell their stocks, or how they remember

those sales, impacts their confidence. The disposition effect and self-serving memory leads to investor

overconfidence. We provide empirical evidence for this in (i) survey data and transaction records of Dutch

retail investors and (ii) an experiment for causality. In a final step, we outline a model that formalizes the

learning mechanism and how it leads to overconfidence as well as lower trading profits and higher volume.

JEL classification: D01, G4

Keywords: Investor Beliefs, Overconfidence, Disposition Effect, Selective Recall, Experiment

∗ Katrin Gödker: Bocconi University, CESifo, katrin.goedker@unibocconi.it; Terrance Odean: Haas School of Business, the

University of California at Berkeley, odean@berkeley.edu; Paul Smeets: University of Amsterdam, p.m.a.smeets@uva.nl.

We are grateful to Lawrence J. Jin for his valuable input. We thank Brad Barber, Kai Barron, Stefano Cassella, Sebastian
Ebert, Carlo Favero, Cary Frydman, Nicola Gennaioli, Simon Gervais, Andrej Gill, Sam Hartzmark, Paul Heidhues, Zwetelina
Iliewa, Mats Köster, Steffen Meyer, William Mullins, Pietro Ortoleva, Michaela Pagel, Cameron Peng, Julien Sauvagnat,
Philipp Strack, and Roberto Weber for very helpful comments and suggestions and Francesco Bilotta and JD O’Hea for great
research assistance. We thank seminar participants at the Berlin Behavioral Economics Seminar, CEPR Advanced Forum
in Financial Economics (CAFFE) seminar series, CEU Vienna, Cheung Kong Graduate School of Business, ESSEC Business
School, Innsbruck University, LMU Munich, NYU, Radboud University, São Paulo School of Economics, and participants
at the CESifo Area Conference on Behavioral Economics 2023, Citrus Finance Conference at UC Riverside, DFG (Deutsche
Forschungsgemeinschaft) Workshop on Consumer Preferences, Consumer Mistakes, and Firms’ Response at Frankfurt School
2024, Experimental Finance Conference 2022, Helsinki Finance Summit 2022, IZA Beliefs Workshop 2024, Journal of Investment
Management Conference 2023, Research in Behavioral Finance Conference (RBFC) 2022, SAFE Household Finance Workshop
2021, SITE Conference 2022, SFS Cavalcade 2022, WFA 2022, and Zurich Workshop on Economics and Psychology 2023 for
helpful comments.
 I. Introduction

How do investors learn about their ability? Investment skill is difficult to evaluate. Even academics

disagree about which measures of performance investors do, or should, use (Berk and van Binsbergen (2016);

Barber, Huang, and Odean (2016)) when assessing skill. Furthermore, the multi-factor alphas an econome-

trician might favor require data and statistical analysis not available to most retail investors. We believe

that when informally evaluating their investment skill, investors use a metric of past performance that: 1)

is cognitively tractable, 2) uses information that easily comes to mind, and 3) is similar to metrics used in

other domains of life. Specifically, we argue that investors use a metric that is based on realizations.

For example, suppose that you ask a friend how good their favorite sports team performs this season.

One metric they will likely quote you is the team’s win-loss record, e.g., the San Francisco Giants baseball

team currently has a middling record of 53 wins and 55 losses. For a retail stock investor, an analog to

this win-loss record is the number of stocks they sold for a gain versus the number of stocks sold for a

loss. Retrieving these two numbers from memory and comparing them is cognitively easy. Furthermore, for

the investor who trades occasionally, realized gains and losses are likely to be more salient than daily price

movements in their portfolio (Frydman, Barberis, Camerer, Bossaerts, and Rangel, 2014).

This realized gains vs realized losses metric is appealing, but imperfect. One shortcoming is that the

metric discards information from unrealized gains and losses and is thus vulnerable to two pervasive biases:

i) Investors are likely to realize gains more readily than losses relative to opportunities, that is, investors

are likely to have a disposition effect (Shefrin and Statman, 1985; Odean, 1998a; Weber and Camerer, 1998;

Frazzini, 2006) 1 ; ii) Investors are likely to misremember their realized gains and losses in a self-serving

fashion (Gödker, Jiao, and Smeets, forthcoming). This paper documents how such a biased metric can lead

investors to form positively biased beliefs about their abilities.

We first show that net realized gains (i.e., the number of realized gains minus the number of realized

losses) predict the self-assessed relative performance of Dutch retail investors even after controlling for

actual relative performance. Our field evidence is based on portfolio and trading records of retail investors

at a Dutch financial institution as well as a survey conducted among these retail investors. We show that

investors who realize more gains than losses believe they have higher annual portfolio performance relative

to other investors, even after controlling for actual annual portfolio performance. Further, we document that

retail investors selectively recall their realized gains and losses. They recall a higher number of gains than

actually realized, but do not significantly misremember losses. Thus separately or together the tendencies
1 The disposition effect has been found in the United States, Israel, Finland, China, and Sweden by Odean (1998a), Shapira

and Venezia (2001), Grinblatt and Keloharju (2001), Feng and Seasholes (2005), and Calvet, Campbell, and Sodini (2009). It
is also documented for the real estate market and the option market by Genesove and Mayer (2001) and Heath, Huddart, and
Lang (1999).

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to realize gains more readily than losses and to remember gains for readily than losses leads investors who

use net realized gains as a performance metric to overestimate their past performance. Overestimating past

performance can lead to overconfidence about ability and future performance.

While this analysis documents a positive correlation between net realized gains and how investors rank

their own abilities, it does not establish causality. The correlation may be due–as we claim–to investors using

net gains as a measure of their ability. However, it is also possible that an unobserved trait—such as the

desire to maintain a positive self-image—leads some investors to realize more gains (remember more gains)

and rank themselves more highly.

To explore causality, we run an experiment. In our experiment, subjects participate in two trials of an

investment task. The task has five investment periods. At the beginning of the first investment period,

subjects choose an initial portfolio of five risky stocks from a group of twenty. Stocks differ in quality and

stock prices change stochastically. There are two types of stocks, a high type and an ordinary type, which

differ in the probability of experiencing a price increase or decrease each period. Subjects do not directly

observe stock types but can make inferences from previous price changes. After observing three prior price

changes for each of the twenty stocks, subjects choose five stocks. At the beginning of each subsequent

investment period, one of the stocks in each subject’s portfolio is sold automatically by the computer. The

subject then purchases a stock from a newly generated set of four stocks for which the subject also observes

three prior price changes. At the end of each period, stock prices change stochastically. Thus subjects hold

five stocks in their portfolio throughout the task which ends at the end of five investment periods.

We test for the causal effect of gain or loss realizations on subjects’ beliefs about their ability to choose

high-type stocks. To do so, we exogenously manipulate whether subjects sell for a gain or loss. By exoge-

nously imposing sales, we avoid the possibility that an unobserved psychological mechanism is both driving

the decisions to realize gains or losses and subjects’ confidence in their ability. We impose two conditions.

In the Selling Gains condition, a winning stock—a stock whose current price is above its purchase price—is

randomly selected and sold. In the Selling Losses condition, a losing stock is randomly selected and sold. 2

We randomly assign subjects to one of these two conditions. Each subject is assigned to the same condition

for both task trials. For stocks in their portfolio, subjects observe both realized gains and losses as well

as paper gains and losses each period. At the end of the final period, we measure subjects’ self-reported

confidence in their ability to select high-type stocks.

The experiment has four main findings. First, after making investment choices and observing outcomes,

subjects in the Selling Gains condition are more confident than subjects in the Selling Losses condition that
2 If the portfolio has no winning stocks in the Selling Gains condition or has no losing stocks in the Selling Losses condition,

then an arbitrary stock is randomly selected and sold.

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they would outperform most other subjects if the trading experiment were repeated. Second, subjects form

beliefs about their ability to select high-type stocks based on realized gains and losses rather than overall

portfolio performance (including unrealized gains and losses). Third, the treatment effect on confidence

is of similar magnitude to a gender effect on confidence, which has been documented to be a meaningful

predictor of investor overconfidence in many studies. Fourth, after making investment choices and observing

outcomes, subjects in the Selling Gains condition are over confident. They significantly overestimate their

ability relative to others, which is in line with the “better-than-average effect” (Moore and Healy, 2008)

and believe that they have selected significantly more high-type stocks than they actually have, which is in

line with overplacement (Moore and Healy, 2008). In reality, subjects in both treatments select, on average,

a similar number of high-type stocks (5.80 in the Selling Gains and 5.76 in the Selling Losses treatment

(t-test, p = 0.873)). 3

The experiment is designed to identify the causal effect of gain/loss realizations on subjects’ confidence

levels. It is worth noting that it is not designed to test whether investors exhibit the disposition effect,

which is well established (Odean, 1998a), or why. Previous studies have offered several explanations for the

disposition effect. We discuss the most prominent explanations in Appendix C. These explanations rely on

preferences, emotions, beliefs, or institutional factors (e.g., transactions costs). Our documented learning

mechanism is not dependent upon the reason why people exhibit a disposition effect.

To formalize the investors’ biased learning process and its implications, we develop a theoretical model.

In line with our empirical results, the model makes the critical assumption that investors form their beliefs

about their investment ability by counting the number of gains and losses they have realized. The model

considers two types of investors: a high ability investor whose ability to choose stocks is high, and a low

ability investor whose ability to choose stocks is low. Investors vary in how strong their metric is biased.

The model produces three predictions. First, investor overconfidence increases with investors’ biased learning

input: investors who are more likely to realize gains than losses overestimate their abilities to select stocks.

This result is consistent with the findings from our field evidence and experiment. Second, the effect on

overconfidence is greater for low ability investors. Third, investor overconfidence, generated by the biased

learning process, leads to both excessive trading and lower trading profits.

Why is it important to study how investors learn about their ability? A large empirical and theoretical

literature argues that retail investors are overconfident. They systematically believe that their own invest-

ment ability is better than it actually is. In theoretical settings, experimental markets, and actual markets,

overconfident investors trade more than is in their own best interest and contribute to price volatility (Odean,

1998b; Deaves, Lüders, and Luo, 2009; Barber and Odean, 2001; Daniel and Hirshleifer, 2015). Thus under-
3 Subjects’ profit from the investment task was also similar across treatments (t-test, p = 0.957).

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standing the mechanisms that increase investor overconfidence may lead to improvements in investor welfare

and to a better understanding of market dynamics.

Most studies in financial economics treat investor overconfidence as a static personal trait and do not ex-

amine the processes through which investors become more—or less—overconfident. The economics literature

on motivated reasoning investigates overconfidence in a more dynamic way. Several papers argue that people

derive utility from overconfidence and other self-serving beliefs (Bénabou and Tirole, 2002; Köszegi, 2006;

Brunnermeier and Parker, 2005). Consistent with these studies, recent experimental work provides evidence

that people become overconfident by forming and updating optimistic beliefs in ego-relevant domains, such as

intelligence (Zimmermann, 2020), beauty (Eil and Rao, 2011), and generosity (Saucet and Villeval, 2019; Di

Tella, Perez-Truglia, Babino, and Sigman (2015); Carlson, Maréchal, Oud, Fehr, and Crockett, 2020). In

this paper, we document a learning process that increases investor confidence about their ability to invest. If

investors realized gains and losses proportionately to opportunities and remembered realizations accurately,

the net realized gains metric would be unbiased (though not optimal). However, most investors display a

disposition effect and most have self-serving memory bias. Because these biases are widespread, their effect

on investors’ self-assessment is likely to be systematic and to influence mean self-assessments. Thus most

investors will rate themselves as above average and, in aggregate, be overconfident.

Our study is closely related to the theory of Gervais and Odean (2001), who argue that investors attribute

positive portfolio performance to their own ability rather than luck and become overconfident. 4 The inno-

vation of our study is that we show that investor overconfidence is not only influenced by past performance

but by how investors evaluate and perceive their past performance. We propose a distinct learning process

in which investors assess their ability by counting their number of realized gains and losses. Thus perceived

performance can differ substantially from actual performance and separately influence the beliefs investors

form about their ability.

Many behavioral finance papers focus on a single bias or behavior, yet biases and behaviors can interact

in ways that magnify or offset their effects (Barberis and Thaler, 2003; Benjamin, 2019). 5 We document a

biased learning mechanism through which the disposition effect as well as selective recall can lead to investor

overconfidence. Thus, how investors sell their stocks, or how they remember those sales, matters for their

confidence levels.

This link between the disposition effect and investor overconfidence could help explain why retail investors

tend to re-invest more after realized gains and tend to reduce their risk and stock market participation after
4 Barberis and Thaler (2003) conjecture that overconfidence may also arise from hindsight bias, but they do not provide tests

of this conjecture.
5 There are few studies in economics examining how biases can affect overconfidence. For instance, Bénabou and Tirole

(2002) investigate the interaction between individuals’ present bias and overconfidence and Jehiel (2018); Barron, Huck, and
Jehiel (2024) study the interaction between individuals’ selection neglect and overconfidence.

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realized losses (Meyer and Pagel, 2022). In addition, the connection between the disposition effect and

investor overconfidence suggests that reducing the disposition effect might also reduce overconfidence. Thus

interventions that have been shown to decrease the disposition effect through using limit orders (Fischbacher,

Hoffmann, and Schudy, 2017), decreasing the salience of purchase price (Frydman and Rangel, 2014), and

transferring or “rolling” assets (Shefrin and Statman, 1985; Frydman, Hartzmark, and Solomon, 2018), could

mitigate investor overconfidence.

Our results contribute to the large literature on the influence of attention on belief formation. It is often

argued that more attention improves belief accuracy and subsequent decision quality (see Gabaix (2019)

for review). In contrast, others argue that in some situations – if greater attention is combined with an

incorrect mental model of the problem – attention can lead agents to overweight specific features of the

decision problem relative to the normative benchmark when updating beliefs (Dawes, 1979; Dawes, Faust,

and Meehl, 1989). For example, Hartzmark, Hirshman, and Imas (2021) have recently shown that people

focus and react more strongly to signals about the quality of a good when they own it. We document that

when people own a good, in our case an investment, they focus on a subset of signals, namely the realized

gains and losses, when updating about the quality of the investment and when assessing their ability to

select high-quality investments.

Our paper adds to a literature that examines how past experienced outcomes affect subsequent invest-

ment decisions and risk taking behavior (Thaler and Johnson, 1990; Weber and Camerer, 1998; Kaustia

and Knüpfer, 2008; Choi, Laibson, Madrian, and Metrick, 2009; Strahilevitz, Odean, and Barber, 2011; Mal-

mendier and Nagel, 2011; Campbell, Ramadorai, and Ranish, 2014; Imas, 2016; Kuhnen, Rudorf, and Weber,

2017; Du, Niessen-Ruenzi, and Odean, 2024). In particular, Imas (2016) documents that a ‘realization ef-

fect’ in prior losses and gains can affect future risk taking: Following a realized loss, individuals avoid risk;

following the same loss that has not been realized individuals take on greater risk. One limitation of this

literature is that it often does not clearly identify whether past experiences affect future behavior through

a beliefs, emotions, or preferences channel. We show that investors’ prior gain and loss realizations directly

causally affect their beliefs about their investment ability.

How investors measure their ability is likely to depend on how actively they trade. In our Dutch retail

data and our experiment, the number of realized gains versus losses matters. Investors who trade much more

actively might aggregate trading outcomes. For example, Brazilian day traders respond to the number of

days on which they were or were not profitable (Chague, Giovannetti, Guimaraes, and Maciel (2023)). Given

the large number of trades made by many day traders, counting days is likely more cognitively tractable

than counting the number of profitable or unprofitable trades.

The paper proceeds as follows. Section II outlines our field evidence for Dutch retail investors based on

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survey results and individual-level transaction analyses. Section III describes the experimental design and

discusses our main findings. Section IV presents a theory and analyzes its implications. Section V concludes.

Additional details of the experimental instructions are in the Appendix.

II. Evidence from the Field

We analyze a unique data set that links confidence measures from an online survey among Dutch retail

investors to their actual realizations and performance from individual portfolio and trades data.

A. Investor Portfolio and Trades Data

We obtain individual portfolio and transaction-level data of the financial institution’s retail clients. The

portfolio data include quarterly holdings during the period of January 01, 2013 to May 31, 2020 and monthly

portfolio returns during the period of January 01, 2013 to May 31, 2020. The portfolio level return includes

the return of all investments in the respective account. Our trades data include information on each trans-

action executed in the clients’ investment accounts during that period, including date and time, asset ID

(ISIN), price in Euros, number of shares, and type of transaction. The study is limited to common stocks

for which this information is available. Multiple buys or sells of the same stock, in the same account, on the

same day, are aggregated.

Investors’ performance. Based on each retail client’s monthly portfolio returns in 2019, we calculate

the geometric average annual return in 2019. We then calculate for each client a performance-based percentile

rank, which we calculate based on their annual portfolio return in 2019 compared to all other retail investment

clients’ annual portfolio return in 2019 at that financial institution (Actual Percentile Rank ).

Investors’ realizations. We calculate client’s realized gains and losses based on their trading records

and count them. Each day that a sale takes place in an investor’s portfolio of at least two stocks, we compare

the selling price of each stock sold to its weighted average purchase price to determine whether that stock

is sold for a gain or a loss. The weighted average purchase price is calculated based on previous purchases

of that specific stock made by the investor during the time period of our data set (January 01, 2013 to May

31, 2020). If there is no corresponding purchase in our data, no realized gain or loss is counted. We adjust

for stock splits if stock splits occurred. We obtain information on stock splits from FactSet. We calculate

the difference between the number of realized gains and losses (Net Gains):

Net Gains = Realized Number of Gains - Realized Number of Losses. (1)

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 A positive difference indicates that the investor realized more gains than losses; a negative difference

means that the investor realized fewer gains than losses; a zero difference indicated that the investor realized

the same number of gains and losses.

For clients who participated in the survey, we can link the portfolio and transactions data to the client’s

survey responses at the individual level using a pseudoanonymized identification number. We limit our

analysis to investment accounts that are i) in the name of the respective survey participant and ii) allow for

own execution of transactions (rather than investment accounts managed by the institution). If a survey

participant holds more than one investment account that satisfy these criteria, we merge the accounts.

B. Survey Data

We conducted an online survey among clients of a Dutch financial institution. 6 The survey was sent

out to the financial institution’s retail clients aged 18 years or older holding an investment account at

the financial institution, excluding very high net worth individuals and those who did not want to be

contacted by email. In total, we sent the survey to 120,865 clients. The survey took about 15 minutes to

complete and by participating respondents had a chance of winning 300 EUR. In addition, the questionnaire

contained incentivized tasks and questions that gave participants the chance of winning extra money, up

to an amount of 120 EUR extra. 5,282 clients completed the survey (response rate of 4.4%). The survey

was conducted between July 1, 2020 and July 16, 2020. The survey consisted of the following parts: 1)

introductory questions for screening, 2) confidence elicitation and elicitation of recalled realizations (in

randomized order), 3) questions about decision-making style and financial knowledge, 4) elicitation of risk

perception, 5) questions about demographics. Our analyses in this paper use clients’ responses to two survey

questions as key variables.

Investors’ beliefs about performance (confidence level). We measure clients’ beliefs about their

relative investment performance by asking for self-reported portfolio performance relative to other retail

investors in the year before the survey, 2019. In particular, we ask survey participants to indicate an

estimate of the percentage of retail investors at the financial institution who achieved a higher annual

portfolio return in 2019 than themselves (between 0% and 100%). We use this response to calculate the

respondents’ performance-based percentile rank among the financial institution’s retail clients by subtracting

the reported percentage from 100 (Elicited Percentile Rank ).

Investors’ recalled realizations. We elicit clients’ recollection of their realizations in 2019 in an

incentivized way. Survey participants could earn 10 EUR for each recall question they answered correctly.
6 The survey and its procedure were approved under ethical approval code ERCIC 187 06 05 2020 by the Ethical Review

Committee Inner City Faculties (ERCIC) of Maastricht University. We obtained subjects’ informed consent before they par-
ticipated in the survey.

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We ask survey participants how many stocks they sold for a gain in the year 2019. We explain to participants

that ”selling for a gain” means selling for a price higher than the average purchase price of that stock.

Participants are asked to indicate the number of stocks sold for a gain. Similarly, we ask survey participants

how many stocks they sold for a loss in the year 2019. We explain to participants that ”selling for a loss”

means selling for a price lower than the average purchase price of that stock. Participants are asked to

indicate the number of stocks sold for a loss. We ask them to indicate their answers without looking up

their actual sales online. We calculate the difference between the recalled number of realized gains and losses

(Recalled Net Gains):

Recalled Net Gains = Recalled Number of Gains - Recalled Number of Losses. (2)

A positive difference indicates that the investor recalls more gains than losses; a negative difference means

that the investor recalls less gains than losses; a zero difference reflects that the investor recalls exactly the

same number of gains and losses. If an investor did not report any sales in 2019, we drop this observation

from the analysis.

C. Sample Demographics and Summary Statistics

Table I reports descriptive statistics for our survey participants. We limit our sample to investors who i)

responded to the confidence question in the survey and ii) can be linked to portfolio and trade data (1,540

investors). We further exclude observations if the participant’s recalled number of realizations deviates by

more than +/- 20 from participant’s number of total sales in 2019 according to their trading records, which

results in a sample of 1,479 retail investors. This investor sample made 8,314 transactions in 2019, i.e., on

average each investor made 5.62 trades in 2019. 11% of the sample is female and 89% of the sample is male.

On average, survey participants were 55.10 years old (min. 18 years). Our investor sample holds average

portfolios of the size of 35,588.52 Euro. The investors earned an average annual portfolio return of 34.70%

in 2019. The average elicited percentile rank of investors’ portfolio performance in 2019 is 55.70 and the

average actual percentile rank is 54.70 (relative to all other retail clients, not only other survey participants).

Investors realized on average 1.41 gains and 0.53 losses in 2019. The Net Gains are on average 0.88. Investors

recall, on average, that they realized 4.22 gains and 0.82 losses, with recalled Net Gains of 3.40 on average.

D. Results

The results based on our field data provide supportive evidence for the experimentally documented

realization effect on investors’ level of confidence.

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 Table I. Descriptive statistics for survey participants.

Investor sample (N = 1,479)

Mean Median St. Dev.

Female 0.11 0.00 0.31

Age (in years) 55.10 57.00 14.48
Stock portfolio size (in Euro) 35,588.52 16,860.87 51,321.14
Annual portfolio return (in 2019, in %) 34.69 20.37 447.57
Actual percentile rank (in 2019) 54.70 53.40 32.91
Elicited percentile rank 55.70 50.00 23.21
Number of transactions (in 2019) 5.62 1.00 19.21
Net Gains 0.88 0.00 3.58
Recalled Net Gains 3.40 2.00 5.23

Result 1. Retail investors who realized more gains than losses during a year, self-report higher performance

relative to other investors during that year after controlling for actual performance.

Table II provides coefficients from linear regression estimates of investors’ beliefs about their performance-

based rank among other retail investors (Elicited Percentile Rank ). As explanatory variables, the models

include investors’ difference in the number of gains and losses realized (Net Gains) as well as their recollection

of it (Recalled Net Gains). We restrict our analysis to investors who at least had one realization in 2019.

The results are robust to restricting the sample to investors who at least had two realizations in 2019 (see

Appendix D). We control for investors’ actual performance-based percentile rank among all retail clients at

the financial institution (Actual Percentile Rank ). In addition we control for the order of elicitation of the

two survey measures, i.e., whether participants were first asked to recollect their realized gains and losses

and then about their annual portfolio performance relative to other retail investors or in opposite order. In

total, 415 of the 1,479 survey participants responded to the recall questions (28.1%).

The results in column 1 show that participants form their beliefs about own performance relative to others

based on the number of realized gains versus losses. The coefficient is significantly positive (p < 0.001). The

more gains rather than losses an investor realized, the higher the investor’s confidence measured by the self-

reported performance-based percentile rank. This finding holds when controlling for the investor’s actual

performance-based percentile rank (column 2). Each additional realized gain over a realized loss increases

an investor’s percentile rank belief by 0.36 (p < 0.05).

Result 2. Selective recall of realized gains is associated with even higher self-reports of performance relative

to other investors.

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Table II. Beliefs about own performance of Dutch retail investors. This table contains the co-
efficients and robust standard errors (in parentheses) of OLS regressions. The dependent variable is the
investors’ beliefs about their performance-based rank among other retail investors, Elicited Percentile Rank,
(between 0 and 100). Net Gains is the difference in the number of investors’ realized gains and losses in
2019. Recalled Net Gains is the difference in the recalled number of investors’ realized gains and losses in
2019. Actual Percentile Rank is investors’ actual percentile rank among all retail clients at the financial
institution based on annual portfolio performance in 2019 (from 0 to 100). Order of Elicitation is a dummy
variable indicating the order in which our two survey items were elicited (1 = recall of realizations first and
0 = otherwise). *, **, and *** denote significance at the 10%, the 5%, and the 1% level, respectively.

(1) (2) (3) (4)

Elicited Perc. Rank Elicited Perc. Rank Elicited Perc. Rank Elicited Perc. Rank

Net Gains 0.487∗∗∗ 0.362∗∗

(0.15) (0.15)
Recalled Net Gains 0.726∗∗∗ 0.625∗∗∗
(0.17) (0.17)
Actual Perc. Rank 0.159∗∗∗ 0.133∗∗∗
(0.02) (0.03)
Order of Elicitation -1.335
(2.04)
Constant 55.275∗∗∗ 46.685∗∗∗ 55.883∗∗∗ 50.494∗∗∗
(0.62) (1.21) (1.22) (3.87)

N 1,479 1,479 415 415

R2 0.01 0.06 0.03 0.07

Investors in our sample selectively recall their realized gains and losses, which is in line with previous

findings from the lab (Gödker et al., forthcoming). As depicted in Table III, investors recall a significantly

higher number of gains than actually realized (p < 0.001), but do not significantly misremember losses. This

suggests that selective recall can reinforce our documented learning mechanism. Selective recall can bias

the metric investors use to assess their ability further towards including more gains than losses, leading to a

more optimistic belief about own ability.

Table III. Recall of gains and losses. This table reports t-test statistics for participants’ recalled number
of realized gains or losses and their actual number of realized gains or losses based on their transaction data.

N = 415 Mean St. Dev. t-statistic

Recalled # of Realized Gains 4.22 5.34
Actual # of Realized Gains 2.17 4.39
Difference (Memory Bias) 2.05 3.78 11.02
Recalled # of Realized Losses 0.82 1.77
Actual # of Realized Losses 0.73 2.12
Difference (Memory Bias) 0.09 2.04 0.94

Indeed the effect on confidence is stronger if we test for the effect of investors’ recalled difference in

realized gains and losses – those gains and losses that stick to investors’ minds (Table II, column 3 and 4).

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Controlling for the actual performance-based percentile rank and order of elicitation, each additional recalled

gain over a loss increases investors rank belief by 0.63 (p < 0.001).

III. The Experiment

A. Experimental Design

To investigate how gain and loss realizations causally affect an investor’s belief about her ability to

select stocks, we adopt an experimental set-up with (i) a decision that generates investment outcomes, (ii)

exogenous variation in realized investment outcomes (gains and losses), (iii) an environment that facilitates

learning about own ability, and (iv) a direct elicitation of beliefs about own ability. In this section, we outline

these features in more detail. The experiment instructions are provided in Appendix A.

In our experiment, subjects make investment decisions. They participate in two investment task trials

of five periods. Before each period t, subjects select the stock(s) to invest in from a set of risky stocks. The

purchase price of each stock is the same ($30). Each period, the stock price increases or decreases.

There are two types of stocks, an ordinary type and a high type. These two stock types differ in the

probability that their price increases or decreases. An ordinary-type stock has a 40% probability of a price

increase and a 60% probability of a price decrease. A high-type stock has a 60% probability of a price increase

and a 40% probability of a price decrease. The price change is drawn from {–3, –1, 2, 6}. In particular,

if the price increases, the price change is $2 or $6, with equal probability. If the price decreases, the price

change is –$1 or –$3, with equal probability. Subjects are not told which stock is of which type. However,

they know the return generating process for the two types of stocks. And, before each choice, subjects view

three recent outcomes of each of the stocks from which they select.

Subjects begin with an endowment of $180 and must buy a portfolio of five stocks from a list of 20

available stocks. The list contains exactly 15 ordinary-type stocks and five high-type stocks, which is known

to subjects. This gives us enough room to measure overestimation of the number of high-type stocks selected.

After the initial portfolio selection, the investment periods (1-5) begin. In periods 1 through 4, subjects (i)

observe the period price change for each of the stocks in their portfolio. After the new prices are displayed,

(ii) one of the stocks is automatically sold by the computer program at the stock’s current price. Subjects

accumulate earnings from sales from period to period. After the sale of one stock, (iii) subjects must choose

an additional stock to buy from a new list of four stocks. Each new list contains exactly three ordinary-type

stocks and one high-type stock, which is known to subjects. Subjects do not know the stocks’ types, but

observe three previous price changes. In period 5, observe the period price change for each of the stocks

11
in their portfolio but no stocks are sold or purchased. Thus each subject chooses nine stocks to purchase

and, for each subject, four stocks are automatically sold in each of the two task trials for a total of eighteen

purchases and eight sales.

Importantly, in each period, subjects are provided with information on both realized gains and losses as

well as paper gains and losses. Subjects view the information on two separate screens: One screen shows the

realized gain or loss from the sale and one screen shows the paper gains and losses of the holding positions

(i.e., stocks) in subject’s portfolio (see Appendix B).

We follow convention in randomly generating the price paths at the beginning of the experiment for

both treatments (Fischbacher et al., 2017). This facilitates between-subject analyses since it reduces noise in

response data that stems from different price paths across treatments. We draw seven sets of price paths. 7

We exogenously manipulate whether the computer sells winning stocks or losing stocks from subjects’

portfolios. We have two between-subjects conditions. Subjects are randomly assigned to one of the two

conditions. In Selling Gains, in each period, a winning stock is liquidated if the portfolio contains at least

one winning stock and otherwise a random stock is liquidated. In Selling Losses, in each period, a losing stock

is liquidated if the portfolio contains at least one losing stock and otherwise a random stock is liquidated.

In this experiment, a winning stock is a stock with its current price higher than the initial purchase price of

$30, and a losing stock is a stock with its current price lower than the initial purchase price of $30.

Note that we are not inducing a disposition effect–i.e., a preference for selling winners. Rather, by forcing

half of the subjects to sell winners, we create patterns of realized winners and losers similar to those generated

by the disposition effect.

Further note, because of the momentum in each stock, selling winner stocks can hurt performance in

the long run. In our experiment, we all but eliminate this effect by setting a short time period. In actual

markets however, the effect can be substantial. Odean (1998a) estimates that stocks sold by retail investors

for a gain outperform stocks they continued to hold for a loss, by 3.4% over the next year. This difference

in the returns to realized winners and paper losses is likely driven, at least partially, by momentum.

Our main outcome variable is subjects’ confidence. We tell each subject to imagine that they are going to

participate in another trial of the investment task and we will compare their performance to the performance

of 9 other randomly selected people who were invited to participate in this study. We asked subjects to

indicate the likelihood that they would be ranked in the upper half of that group (Zimmermann, 2020).

With this measure, we elicit subjects’ perception of own ability independent of whether they are more or

less willing to participate in future investment rounds. This helps to isolate the treatment effect on subjects’

confidence levels apart from any treatment effects–such as the house money effect or the realization effect in
7 That is, we draw seven price sets of 72 stocks each (each of the two task trials includes 36 available stocks).

12
risk taking (Thaler and Johnson, 1990; Imas, 2016)–on subsequent risk aversion and risk taking.

In addition, we elicit subjects’ beliefs about how many high-type stocks they selected during the invest-

ment task. This measure helps us determine whether subjects’ beliefs that they selected more high-type

stocks than they actually did is a possible mechanism leading to overconfidence about their investment

ability.

In this setting, a price decrease is a negative signal about the selected stock’s quality while a price

increase is a positive signal. Across both treatments a Bayesian agent would learn from all price increases

and decreases, realized or unrealized.

B. Procedure, Incentives, and Sample

The experiment was conducted online with U.S. residents of the Prolific subject pool in May 2021. 8 It

was organized into two parts. The first part consisted of the investment task. In both treatments, subjects

had to participate in two trials of the investment task. Their payoff depended on their choices and on

the randomly generated price changes for stocks in their portfolio in one of the two task trials. In each

period, subjects accumulated the proceeds from stock sales in their cash holdings and paid the cost of stocks

purchased. Their potential payoff for each trial was 1/100 of their final holdings at the end of the task

trial; that is, the sum of final cash holdings and the value (i.e., the current prices) of the stocks in the final

portfolio after period 5. Thus payment to subjects was based on both realized and unrealized gains and

losses. Part 2 of the experiment consisted of the belief elicitation, which was not incentivized. At the end of

the experiment, one of the two task trials was randomly selected for payment. In addition, subjects received

a fixed participation fee of $1.

We designed comprehension questions to test subjects’ understanding of the experimental instructions.

Subjects had to answer five comprehension questions after reading the instructions and before participating

in the first part of the experiment. We excluded subjects from the experiment who gave an incorrect answer

to more than one comprehension question.

A total of 301 subjects participated in the experiment, 139 subjects in treatment Selling Gains and 162 in

treatment Selling Losses. Participating in the experiment took on average 12 minutes and 42 seconds. The

experiment was programmed and conducted with oTree (Chen, Schonger, and Wickens, 2016). This study

was pre-registered at AsPredicted under ID 66925 (https://aspredicted.org/ia5zy.pdf). Table IV reports

descriptive statistics. Our sample consists of 167 female (55% of the sample) and 134 male (45% of the

sample) subjects. On average, subjects were 33 years old (min. 18 years and max. 70 years). As intended,
8 The experiment and its procedure were approved under ethical approval code ERCIC 212 28 09 2020 by the Ethical Re-

view Committee Inner City Faculties (ERCIC) of Maastricht University. We obtained subjects’ informed consent before they
participated in the experiment.

13
subjects’ number of realized gains differed significantly between our two conditions (t-test, p = 0.000). In the

Selling Gains treatment, subjects realized on average 7.74 gains (out of a possible maximum of 8), whereas

subjects in the Selling Losses treatment realized on average 0.98 gains (out of a possible maximum of 8).

Subjects’ profit from the investment task was similar across treatments (t-test, p = 0.957) and subjects

selected on average a similar number of high-type stocks across treatments, namely 5.80 in the Selling Gains

and 5.76 in the Selling Losses treatment (t-test, p = 0.873) out of 18 stocks they select in total. The average

payment was $2.98, which translates to $14 per hour.

Table IV. Descriptive statistics for the experiment’s subjects.

Full sample Selling Gains Selling Losses

(N = 301) (N = 139) (N = 162)

Mean Median St. Dev. Mean Median St. Dev. Mean Median St. Dev.

Female 0.55 1.00 0.50 0.53 1.00 0.50 0.58 1.00 0.50

Age (in years) 33.10 31.00 11.55 32.83 30.00 11.56 33.34 31.00 11.57

Total number of
4.10 3.00 3.48 7.74 8.00 0.50 0.98 1.00 1.03
realized gains

Average portfolio
17.97 18.00 11.34 17.92 17.00 10.82 18.01 18.50 11.81
performance (profit in $)

Total number of high-type

5.78 6.00 2.12 5.80 6.00 2.11 5.76 6.00 2.13
stocks selected

Subject payment (in $) 2.98 2.97 0.18 2.98 2.97 0.19 2.98 2.98 0.17

C. Results

The results from our experiment provide evidence for a learning process which biases subjects’ level of

confidence about their ability to invest.

Result 3. Subjects report significantly higher confidence in their own ability to invest if more gains were

realized than if more losses were realized.

The key outcome measure for our subsequent analysis is subjects’ forward-looking belief about their group

rank based on investment performance. Subjects reported the likelihood that they would be ranked in the

upper half of a group of 10 subjects if they were to participate in another investment trial. Subjects had to

provide their answer as a percentage, and every integer between 0 and 100 was admissible.

14
Figure 1. Average beliefs about own ability. This figure displays mean values of subjects’ belief about
own ability measured by subjects’ elicited likelihood in percent of being ranked in the upper half of a group
of 10 participants based on performance in the task (from 0% to 100%). The bars represent the mean values
by treatment. Error bars indicate 95% confidence intervals. The red reference line represents the average
belief if all subjects report an accurate belief about their own ability relative to others in the group (50%).

Figure 1 shows subjects’ average beliefs for the two treatments. The figure confirms the basic pattern we

hypothesized. On average, subjects in the Selling Gains treatment report significantly higher confidence in

their investment ability than subjects in the Selling Losses treatment (t-test, p = 0.000). Subjects for whom

mainly gains were realized, indicate a mean belief of 59.27%, whereas subjects for whom mainly losses were

realized, indicate a mean belief of 47.40%.

This finding is supported in a regression analysis. Table V provides coefficients from linear estimates

of subjects’ beliefs. Column 1 documents the treatment effect. The coefficient of the treatment dummy is

significantly positive. Subjects’ average beliefs are 11.87% higher in the Selling Gains treatment compared

to the Selling Losses treatment. 9

Result 4. Subjects form beliefs about their own ability to invest based on realized gains and losses rather

than overall portfolio performance.

We further analyze whether subjects’ actual past performance in the two investment trials is related to

subsequent belief reports and how it compares to our treatment effect. Subjects’ past portfolio performance

includes paper gains and losses and the cash position at the end of the investment trials from realized gains
9 Note that we analyze the difference across the two exogenously varied experimental conditions, not the effect of the number

of realized gains. There is only very little variation in number of realized gains within the conditions, see Table IV.

15
and losses. We take the average portfolio performance across both investment trials.

Column 2 of Table V provides coefficients from linear estimates of subjects’ beliefs with the treatment

dummy as well as their past average portfolio performance as explanatory variables. The results show that

subjects in our experiment form their beliefs about own investment ability based on realized gains and losses

and not based on overall portfolio performance, including paper gains and losses. The coefficient of the

treatment dummy is significantly positive, however, subjects’ portfolio performance with the added position

of paper gains and losses has no significant association with subjects’ beliefs about own ability.

Table V. Beliefs about own ability. This table contains the coefficients and robust standard errors
(in parentheses) of OLS regression. The dependent variable is the subjective likelihood in percent of being
ranked in the upper half of a group of 10 participants based on performance in the task (from 0% to
100%). Treatment is a dummy variable representing our treatment with 1 = Selling Gains and 0 = Selling
Losses. Portfolio Performance is subjects’ average portfolio performance of both investment trials, including
paper gains and losses and the cash position and excluding initial endowment. Female is a dummy variable
indicating subjects’ gender with 1 = subject is female and 0 otherwise. Individual Skill is subjects’ skill
to select potential high type stocks. It indicates the number of price increases in pre-periods of selected
stocks of subjects’ initial portfolio in both investment trials. Belief High Types Selected is subjects’ reported
number of high-type stocks selected (from 0 to 18). *, **, and *** denote significance at the 10%, the 5%,
and the 1% level, respectively.

(1) (2) (3) (4) (5)

Belief Ability Belief Ability Belief Ability Belief Ability Belief Ability

Treatment 11.872∗∗∗ 11.889∗∗∗ 11.455∗∗∗ 11.384∗∗∗

(2.79) (2.79) (2.78) (2.79)
Portfolio Performance 0.194 0.165
(0.13) (0.13)
Female -7.836∗∗∗ -7.868∗∗∗ -8.463∗∗∗
(2.86) (2.91) (2.61)
Individual Skill 0.159
(0.26)
Belief High Types Selected 2.694∗∗∗
(0.32)
Constant 47.401∗∗∗ 43.904∗∗∗ 48.981∗∗∗ 49.004∗∗∗ 37.901∗∗∗
(2.02) (3.12) (3.83) (5.90) (3.33)

N 301 301 301 301 301

R2 0.06 0.06 0.09 0.08 0.21

Result 5. In our sample the treatment effect on confidence is of similar magnitude to the gender effect.

Many studies find that, while both men and women tend to exhibit overconfidence in many domains, men

are generally more overconfident than women (Taylor and Brown, 1988; Lundeberg, Fox, and Punćcohaŕ,

1994), especially so in areas, such as finance and investing, that are perceived as masculine, such as finance

(Deaux and Farris, 1977; Beyer and Bowden, 1997; Prince, 1993; Barber and Odean, 2001). Our experimental

data is in line with this established pattern. In our sample, females’ reported confidence in their ability to

invest is on average 8.79 percentage points lower than males’ confidence (t-test, p = 0.002). Yet, our

16
treatment effect is robust to differences in gender. Figure 2 illustrates subjects’ average beliefs for the two

treatments by gender. For both genders, subjects’ average belief is significantly higher in the Selling Gains

treatment than in the Selling Losses treatment.

Figure 2. Average beliefs about own ability by gender. This figure displays mean values of subjects’
belief about own ability measured by subjects’ elicited likelihood in percent of being ranked in the upper
half of a group of 10 participants based on performance in the task (from 0% to 100%). The bars represent
the mean values by treatment and gender. Error bars indicate 95% confidence intervals.

In Table V, Column 3, we regress belief in ability against the treatment dummy, the portfolio performance,

and a female dummy variable. The coefficient on the treatment dummy, 11.46, is hardly affected. Portfolio

performance is insignificant and the female dummy variable is significant with a coefficient of -7.84. Note

that the gender effect on confidence is slightly smaller, though of similar magnitude, than our treatment

effect.

Subjects’ portfolio performance depends on their ability to select high-type stocks as well as luck. In

Column 4 of Table V, we control for subjects’ individual skill to select potential high-type stocks. It reflects

the normative Bayesian rule subjects should follow when selecting stocks in the experiments. It indicates the

number of price increases in pre-periods of selected stocks of subjects’ initial portfolio in both investment

trials. This variable captures subjects’ skill to select high-type stocks while ruling out any luck component.

The coefficient of the treatment dummy remains significantly positive.

Result 6. Realizing more gains leads to overconfidence.

17
Besides providing evidence for a treatment effect on subjects’ confidence, the experimental results document

significant overconfidence in the Selling Gains treatment. We test for overconfidence in two ways to capture

both the ”better-than-average effect” as well as overplacement (Moore and Healy, 2008). In Figure 1, the red

reference line represents the average belief if all subjects reported an accurate belief about their own ability

relative to others in the group (50%). Yet, subjects’ average belief is larger than 50% in the Selling Gains

treatment. The figure illustrates that subjects for whom mainly gains were realized significantly overestimate

their ability relative to the others, which is in line with the ”better-than-average effect” (Moore and Healy,

2008). Subjects’ mean belief is significantly different from and larger than 50% (t-test, p = 0.000).

In addition, the observed confidence patterns are related to subjects’ beliefs about how many high-type

stocks they have selected. In total, subjects selected 18 stocks during the two trials of the investment task.

We let them report their belief about how many high-type stocks they selected during the investment task.

The measure ranges from 0 to 18.

We find that subjects’ confidence in own ability is positively correlated with their reports of how many

high-type stocks they believe they have selected. Column 5 of Table V shows a significantly positive associ-

ation between the two variables. Subjects’ confidence increases by 2.7 percentage points for each additional

high-type stock that they believe they have selected.

Table VI. Beliefs about number of selected high-type stocks. This table contains the coefficients
and robust standard errors (in parentheses) of OLS regression. The dependent variable is subjects’ reported
number of high-type stocks selected (from 0 to 18). Treatment is a dummy variable representing our treat-
ment with 1 = Selling Gains and 0 = Selling Losses. Portfolio Performance is subjects’ average portfolio
performance of both investment trials, including paper gains and losses and the cash position and excluding
initial endowment. Actual High Types Selected is subjects’ actual number of high-type stocks selected (from
0 to 18). Female is a dummy variable indicating subjects’ gender with 1 = subject is female and 0 otherwise.
*, **, and *** denote significance at the 10%, the 5%, and the 1% level, respectively.

(1) (2) (3) (4)

Belief High Types Selected Belief High Types Selected Belief High Types Selected Belief High Types Selected

Treatment 0.956∗∗ 0.958∗∗ 0.957∗∗ 0.956∗∗

(0.45) (0.45) (0.45) (0.45)
Portfolio
Performance 0.029 0.027 0.027
(0.02) (0.02) (0.02)
Actual High
Types Selected 0.037 0.036
(0.11) (0.11)
Female -0.008
(0.46)
Constant 6.864∗∗∗ 6.348∗∗∗ 6.167∗∗∗ 6.173∗∗∗
(0.31) (0.48) (0.74) (0.81)

N 301 301 301 301

R2 0.01 0.02 0.02 0.02

18
 We investigate whether our treatment leads people to believe that they have selected more or less high-

type stocks. Table VI provides coefficients from linear estimates of subjects’ report of how many high-type

stocks they believe they have selected during the investment task. We find that subjects believe that they

have selected significantly more high-type stocks if more gains were realized than if more losses were realized.

The coefficient of the treatment dummy in Column 1 is significantly positive. Subjects for whom mainly

gains were realized believe they have selected on average 7.82 high-type stocks, whereas subjects for whom

mainly losses were realized believe they have selected on average 6.86 high-type stocks. Note that, as reported

in Table IV, subjects in both treatment groups selected an average of 5.8 (median of 6) high type stocks.

Thus subjects believe they have selected more high-type stocks than they actually did which is in line with

overplacement (Moore and Healy, 2008).

Column 2 of Table VI provides coefficients from linear estimates of subjects’ beliefs with the treatment

dummy as well as their past average portfolio performance as explanatory variables. Similar to subjects’

confidence, we find that subjects form beliefs about how many high-type stocks they have selected based on

realized gains and losses rather than overall portfolio performance. The coefficient of the treatment dummy

is significantly positive, however, subjects’ portfolio performance has no significant association with subjects’

beliefs about selected high-type stocks selected. This result holds when we control for the actual number of

high type stocks the subjects selected (column 3) and gender (column 4).

IV. The Model

In this section, we present a simple model demonstrating how the biased learning process leads to over-

confidence. The model has two assumptions. First, investors assess their trading ability (which will be

defined precisely later) by counting past realized gains and losses. Second, investors are subject to biased

realizations in their selling decisions. We study the implications of such a model. We organize the model’s

implications by three parts: the implications for investor confidence; the implications for trading behavior;

and the implications for the investor’s expected profit.

A. Model Setup

Asset space—We consider a finite-horizon economy with N risky assets which we also refer to as stocks.

Each asset has a liquidating dividend paid at the end of period T ; denote the liquidating dividend for asset

i as Di,T . News about Di,T is sequentially released over time. The incremental news released at the end of

period t about Di,T is denoted as vi,t and the cumulative news released by the end of period t about Di,T

19
is denoted as Di,t . We have

Di,t = Di,0 + vi,1 + vi,2 + · · · + vi,t , 1 ≤ i ≤ N and 1 ≤ t ≤ T. (3)

We further assume

vi,t ∼ N (0, σi2 ), t ≥ 1, ∀i,

(4)
i.i.d. over time and independent across stocks.

Signal structure—At the beginning of period t, a risky asset—asset i, say—is randomly selected from N

risky assets. One type of market participant, a risk-neutral investor, is endowed with the following signal

about asset i

θi,t = δi,t vi,t + (1 − δi,t )εi,t , (5)

where εi,t has an identical distribution as vi,t but is independent from it. The variable δi,t takes the values

of one or zero: when δi,t = 1 the signal is θi,t = vi,t , hence it is fully informative about the quality of the

asset; when δi,t = 0, the signal is θi,t = εi,t and the signal is pure noise. Therefore, it makes sense to posit

that the investor’s ability for correctly anticipating the payoff of asset i is measured by the probability that

δi,t takes the value of one. Denote this probability as ai and refer to it as the investor’s ability. We assume

that there are two possible ability levels: ai = H and ai = L, where 0 < L < H < 1.

We assume that no market participant, including the investor, knows the investor’s ability. Instead, all

market participants are endowed with the correct prior belief that ai = H with probability ϕ0 and ai = L

with probability 1 − ϕ0 , where 0 < ϕ0 < 1. Moreover, we assume that the investor’s ability for correctly

anticipating the payoff of asset i also represents his ability for correctly anticipating the payoff of another

asset; that is, ability is at the investor level, not at the asset level (formally, ai = aj for all i, j ≤ N ). As

such, we abbreviate ai as a. Finally, we assume N ≫ T , so the probability that the investor obtains a signal

for the same asset over two different time periods is negligible (recall that the signal is about a risky asset

chosen at random).

Market participants—There are three market participants: the investor mentioned above, a liquidity

trader, and a market maker. We discuss them in order.

The investor starts with rational prior beliefs about his ability, but he develops biased beliefs over time,

due to a misspecified updating rule. We first describe the rational beliefs about the investor’s ability. We

then discuss how the investor’s beliefs differ from the rational beliefs. As in Gervais and Odean (2001), let

st denote the number of times that the investor’s information about risky assets was true by the end of the

20
 Pt
first t periods: we write st = u=1 δi(u),u , where i(u) denotes the asset about which the investor has a signal

in period u; δi(u),u equals one if θi(u),u = vi(u),u , and δi(u),u equals zero if θi(u),u ̸= vi(u),u . Under rational

beliefs, the investor would correctly understand that the number of times his information was true in the

past is diagnostic of his ability. Hence, at the beginning of period t, after his information was true st−1 = s

times, a rational investor would update beliefs about his ability according to Bayes’ rule,

Pr(a = H) Pr(st−1 = s|a = H)

ϕt−1 (s) ≡ Pr(a = H|st−1 = s) = =
Pr(a = H) Pr(st−1 = s|a = H) + Pr(a = L) Pr(st−1 = s|a = L)
t−1−s
H s (1 − H) ϕ0
= s t−1−s s t−1−s . (6)
H (1 − H) ϕ0 + L (1 − L) (1 − ϕ0 )

To understand the formula, recall that Pr(st−1 = s|a = A) = As (1 − A)t−1−s for A ∈ {H, L}, since (δi,u )tu=1

is a vector of i.i.d Bernoulli random variables for which Pr(δi,u = 1|a = A) = A.

Therefore, the rational expectation about the investor’s ability, computed at the beginning of period t, is

ξt−1 (s) ≡ E(a|st−1 = s) = ϕt−1 (s) · H + (1 − ϕt−1 (s)) · L. (7)

And, since a is the probability that δ = 1, ξt−1 (s) is the rational investor’s expectation that his period t

information is true, after it has been true s times over the last t − 1 observations.

In our model, the investor deviates from rational beliefs as follows. To assess his ability, rather than

using st , the investor uses the number of times that he has sold stocks for a gain, denoted by kt . Gains and

losses for an asset i at time t are defined as the difference between the asset’s price at time t − 1 and its

purchase price in the case of positive share demand; vice versa for negative share demand. Later we will

provide a precise definition of stock-level gains and losses.

Under the investor’s biased beliefs, the probability that a = H at the beginning of period t, is

t−1−k
H k (1 − H) ϕ0
ψt−1 (k) ≡ Prb (a = H|kt−1 = k) := k t−1−k k t−1−k
, (8)
H (1 − H) ϕ0 + L (1 − L) (1 − ϕ0 )

where the subscript “b” denotes biased beliefs.

Accordingly, the investor’s biased expectation of his ability, computed at the beginning of period t, is

Ξt−1 (k) ≡ Eb (a|kt−1 = k) = ψt−1 (k) · H + (1 − ψt−1 (k)) · L. (9)

And, since a is the probability that δ = 1, Ξt−1 (s) is the biased investor’s expectation that his information

is true, after he has sold for a gain k times over the last t − 1 periods.

21
 We now turn to the investor’s buying and selling decisions. Selling decisions follow an exogenous rule. At

any time t < T , the investor holds M ≪ N risky assets in his portfolio. Suppose that M1 out of M stocks

are held at a gain, and that the remaining M − M1 stocks are held at a loss, where gains and losses are

defined as below. With probability χ, the investor will randomly sell one of the M1 stocks, resulting in a gain

realization; with the remaining probability 1 − χ, the investor will randomly sell one of the M − M1 stocks

resulting in a loss realization. 10 At time T , the investor sells all his stocks. The probability χ measures the

intensity of the bias to sell rather for a gain than a loss (i.e., a disposition effect), to which the investor is

subject. For example, for values of χ close to 1, the investor will almost surely sell an asset held at a gain

and keep assets held at a loss.

At the beginning of period t < T , once the investor sells a stock, he receives a signal θi,t about a new

asset i according to the signal structure described above. Given this signal and the investor’s biased belief

about his ability (which is based on kt−1 ), the investor maximizes the expected profit of holding asset i

by choosing xi,t , his share demand for asset i. We further describe this maximization problem in the next

section. Note that, though assets in the investor’s portfolio may be held for multiple periods, the investor

only receives a signal about an asset when he makes the initial purchase decision. At the end of each period,

the price of the asset is set to its fair value, Di,t .

The liquidity trader has a random demand for the asset that the investor buys; denote this demand
2
at the beginning of period t as zi,t . We suppose that zi,t is a Normal random variable: zi,t ∼ N (0, σi,z ),

independent from all other variables.

As in Kyle (1985), we assume that the market maker is risk-neutral and competitive and will therefore

set prices so as to make zero expected profits. Furthermore, the market maker holds rational beliefs. At

the beginning of period t, he observes st−1 , kt−1 , and ωi,t = xi,t + zi,t , which is the total demand from the

investor and the liquidity trader for asset i. The market maker then sets a competitive price pi,t for asset i.

In other words, he sets a price equal to his expectation E[Di,T ] at the beginning of period t. Note that, at

the beginning of period t, the market maker does not observe θi,t or vi,t .

B. Model Implications

We formally solve the model in Appendix E. With the model’s solution in hand, we now examine the

model’s implications through numerical simulations. We examine the implications for investor confidence,

trading behavior, and expected profits.

Specifically, we set M = 10, T = 10, L = 0.4, H = 0.6, σi = 1, σi,z = 1, and ϕ0 = 0.5. With T = 0, the

economy has eleven dates (date t goes from zero to ten) and ten periods. At date 0, the investor’s ability
10 If M1 equals zero or M , then the investor randomly sells one out of all M stocks.

22
level is drawn: with probability ϕ0 , a = H; and with probability 1 − ϕ0 , a = L. The investor is then endowed

with ten stocks. At each date 1 ≤ t ≤ 9, the investor sells one of the ten stocks according to the probability

χ described in the previous section; he is endowed with a signal about a new stock; and he decides his share

demand for the stock. At date t = 10, the investor sells all his stocks.

Investor overconfidence.—We record the investor’s overconfidence level at date t = 9 (the beginning of

the final period, period 10), measured by Ξ(k) − ξ(s). Thus, overconfidence on our model captures how

much an investor overplaces his perceived ability to his true ability (Moore and Healy, 2008). We simulate

the above economy for 10,000 times. We compute the level of overconfidence averaged across the 10,000

investors and then plot it against probability χ. Figure 7 below presents this result in Panel a).

The graph in Panel a) shows that investor overconfidence increases with χ. Moreover, when χ is low,

ξ(s) tends to be greater than Ξ(k) and hence investors exhibit underconfidence. When χ is high, ξ(s) tends

to be lower than Ξ(k) and hence investors exhibit overconfidence.

We further look at investor overconfidence separately for the low-type investors and the high-type in-

vestors. We find that, for all values of χ, the low-type investors tend to be more overconfident than the

high-type investors. In particular, when χ is high, the rational expectation about the low-type investors’

ability, computed at date t = 9, is close to L = 0.4. However, the low-type investors’ subjective expectation

about their ability, after these investors have experienced many gain realizations by date t = 9, is significantly

higher than 0.4.

Next, we examine how investor overconfidence varies over time. For this exercise, we set T = 21 so the

economy has 22 dates (date t goes from zero to 21) and 21 periods. We set χ = 1, so investors always sell

stocks at a gain (so long as there is at least one stock in the portfolio that is held at a gain). Panel b) of

Figure 7 plots, for 1 ≤ t ≤ 20, the level of overconfidence averaged across all investors, across the high-type

investors, and across the low-type investors. Overall, the level of investor overconfidence increases over time:

the increase is particularly significant for the first few periods and then becomes smaller and eventually

negligible.

The dynamics of investor overconfidence depend strongly on the investor’s type. Low-type investors tend

to become more overconfident over time: their subjective expectation about their ability increases as they

experience a higher number of gain realizations, while the rational expectation about their ability decreases

over time towards their true ability a = L. High-type investors, however, tend to become more overconfident

only for the first few periods; subsequently, their level of overconfidence decreases towards zero. For these

investors, their subjective expectation about their trading ability initially increases at a faster pace compared

to the rational expectation, leading to a higher level of overconfidence. As time goes, both the subjective

expectation and the rational expectation converges to the investors’ true ability a = H, and therefore the

23
 0.15 0.15

0.1

0.05 0.1

-0.05 0.05

-0.1

-0.15 0
0 0.5 1 5 10 15 20
a) b)
1 0.251

0.95
0.25
0.9

0.249
0.85

0.8
0.248

0.75
0.247
0.7

0.65 0.246
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
c) d)

Figure 7. Model implications. We simulate the economy 10,000 times for T periods. At date 0, we draw
the investor’s ability level. For all panels, M = 10, L = 0.4, H = 0.6, σi = 1, σi,z = 1, and ϕ0 = 0.5. For
Panels a, c, and d, T = 10. For Panel b, T = 21. For Panel b, χ = 1.
Panel a): The graph plots investor overconfidence as a function of χ. We record the investor’s overconfidence
level at the beginning of the final period, measured by Ξ(k) − ξ(s). We then compute the average level of
investor overconfidence for all 10,000 investors, for low type investors, and for high type investors.
Panel b): The graph plots investor overconfidence as a function of time averaged across all 10,000 investors.
We record the investor’s overconfidence level at the beginning of each period, measured by Ξ(k) − ξ(s).
Panel c): The graph plots the investor’s average trading volume as a function of χ for all 10,000 investors.
We record the investor’s absolute share demand for a risky asset at the beginning of the final period measured
by |x|.
Panel d): The graph plots the investor’s expected profit from the final investment as a function of χ for
all 10,000 investors. We record the objective expectation of the investor’s profit from the final investment
made at the beginning of the final period.

level of overconfidence drops.

Trading behavior —We set T = 10 and simulate the economy for 10,000 times. For each simulation,

we compute the magnitude of the investor’s share demand for the new risky asset at the beginning of the

final period (period 10), measured by |x|, where x is the share demand from equation (E.5). We then

compute the absolute share demand, |x|, averaged across the 10,000 investors and plot the share demand

against probability χ. Panel c) of Figure 7 presents this result. The graph shows that the magnitude of the

investor’s share demand for risky assets increases with χ, the tendency to sell rather for a gain than a loss.

24
For high values of χ, the learning mechanism gives rise to investor overconfidence. This, in turn, generates

excessive trading (|x| > |xR |).

Expected profit.—At the beginning of the final period, the expectation of the investor’s profit from the

final investment is

E[πi,t |θi,t , st−1 , kt−1 , xi,t ] = xi,t · [E[vi,t |θi,t , st−1 ] − λi,t (st−1 , kt−1 ) · xi,t ]
(10)
= xi,t · [ξt−1 (st−1 ) · θi,t − λi,t (st−1 , kt−1 ) · xi,t ] .

Panel d) of Figure 7 plots the expectation of the investor’s profit against probability χ. The graph in Panel

d) shows that the investor’s expected profit is hump shaped in χ. For low values of χ, the investor’s demand

is too low to maximize profits and when χ the investor’s demand is too high to maximize profits. In our

model, the investor is the only market participant with a signal about the future value of a risky asset.

Thus the investor’s expected profits are positive. High values of χ lead to investor overconfidence, which in

turn gives rise to excessive trading. In the model, excessive trading is detrimental to the investor’s gross

profit because such trading reveals too much information to the market maker. In actual markets, excessive

trading further reduces net profit because retail investors incur transaction costs (Barber and Odean, 2000).

V. Conclusion

We document that when learning about their own investment ability investors use a metric based on

realizations. This metric is appealing, since it is cognitively tractable, uses information that easily comes

to mind, and is similar to metrics used in other domains of life. Yet it is biased by investors’ tendency to

sell gains more readily than losses relative to opportunities, i.e., the disposition effect, and by the tendency

to remember realizing more gains than one actually realized. In conjunction with these biases, the learning

mechanism generates investor overconfidence.

In field data we find that Dutch retail investors who realized more gains than losses report higher confi-

dence in ability relative to other retail investors, controlling for actual portfolio performance. Furthermore,

retail investors selectively recall their realized gains and losses. They recall a higher number of gains than

actually realized, but do not significantly misremember losses. Selective recall is associated with an even

more optimistic belief about own ability.

In an experiment, we document the causal effect of gain/loss realizations on individuals’ confidence.

Subjects randomly assigned to a treatment in which gains are sold more readily than losses, become over-

confident in their abilities, while those in a treatment in which losses are sold more readily than gains do

25
not. Controlling for actual performance on the investment task, subjects in our Selling Gains condition,

who realized mainly gains during the task, are more confident in their ability to select high-type stocks than

subjects in the Selling Losses condition. Subjects in the Selling Gains believe that they selected significantly

more high-type stocks than they actually did and overestimate their ability to perform well on a future in-

vestment task relative to others. Many studies find that men are generally more overconfident than women.

We replicate this empirical pattern, but find that both men and women form their beliefs about own ability

based on realized gains and losses rather than overall portfolio performance. In our sample, the treatment

effect on confidence is of similar magnitude to the gender effect.

Building on our empirical results, we develop a theoretical model of how the disposition effect as well

as self-serving memory bias can lead to overconfidence, excessive trading, and lower trading profits. In

our model, investors assess their trading ability by counting past realized gains and losses. Investors who

realize more gains because of the disposition effect or, equivalently, remember realizing more gains because

of memory bias, are disposed to be overconfident.

26
 REFERENCES

Barber, Brad, and Terrance Odean, 2001, Boys will be boys: Gender, overconfidence, and common stock

investment, Quarterly Journal of Economics 116, 261–292.

Barber, Brad M., Xing Huang, and Terrance Odean, 2016, Which factors matter to investors? Evidence

from mutual fund flows, Review of Financial Studies 29, 2600–2642.

Barber, Brad M., and Terrance Odean, 2000, Trading is hazardous to your wealth: The common stock

investment performance of individual investors, Journal of Finance 55, 773–806.

Barber, Brad M, and Terrance Odean, 2013, The behavior of individual investors, in Handbook of the

Economics of Finance, volume 2, 1533–1570 (Elsevier).

Barberis, Nicholas, and Richard Thaler, 2003, A survey of behavioral finance, in George Constantinides,

Milton Harris, and René M. Stulz, eds., Handbook of the Economics of Finance (North Holland, Amster-

dam).

Barberis, Nicholas, and Wei Xiong, 2009, What drives the disposition effect? An analysis of a long-standing

preference-based explanation, Journal of Finance 64, 751–784.

Barberis, Nicholas, and Wei Xiong, 2012, Realization utility, Journal of Financial Economics 104, 251–271.

Barron, Kai, Steffen Huck, and Philippe Jehiel, 2024, Everyday econometricians: Selection neglect and

overoptimism when learning from others, American Economic Journal: Microeconomics 16, 162–198.

Ben-David, Itzhak, and David Hirshleifer, 2012, Are investors really reluctant to realize their losses? Trading

responses to past returns and the disposition effect, Review of Financial Studies 25, 2485–2532.

Bénabou, Roland, and Jean Tirole, 2002, Self-confidence and personal motivation, Quarterly Journal of

Economics 117, 871–915.

Benjamin, Daniel J., 2019, Errors in probabilistic reasoning and judgment biases, in Douglas Bernheim,

Stefano DellaVigna, and David Laibson, eds., Handbook of Behavioral Economics, 69–186 (North Holland,

Amsterdam).

Berk, Jonathan, and Jules van Binsbergen, 2016, Assessing asset pricing models using revealed preference,

Journal of Financial Economics 119, 1– 23.

Beyer, Sylvia, and Edward M. Bowden, 1997, Gender differences in self-perceptions: Convergent evidence

from three measures of accuracy and bias, Personality and Social Psychology Bulletin 23, 157–172.

27
Brunnermeier, Markus K., and Jonathan A. Parker, 2005, Optimal expectations, American Economic Review

95, 1092–1118.

Calvet, Laurent E., John Y. Campbell, and Paolo Sodini, 2009, Fight or flight? Portfolio rebalancing by

individual investors, Quarterly Journal of Economics 124, 301–348.

Campbell, John Y., Tarun Ramadorai, and Benjamin Ranish, 2014, Getting better or feeling better? How

equity investors respond to investment experience, NBER working paper No. 20000.

Carlson, Ryan W., Michel André Maréchal, Bastiaan Oud, Ernst Fehr, and Molly J. Crockett, 2020, Moti-

vated misremembering of selfish decisions, Nature Communications 11, 2100.

Chague, Fernando, Bruno Giovannetti, Bernardo Guimaraes, and Bernardo Maciel, 2023, Why do individuals

keep trading and losing?, Working paper.

Chen, Daniel L., Martin Schonger, and Chris Wickens, 2016, oTree - An open-source platform for laboratory,

online, and field experiments, Journal of Behavioral and Experimental Finance 9, 88–97.

Choi, James J., David Laibson, Brigitte C. Madrian, and Andrew Metrick, 2009, Reinforcement learning

and savings behavior, Journal of Finance 64, 2515–2534.

Daniel, Kent, and David Hirshleifer, 2015, Overconfident investors, predictable returns, and excessive trading,

Journal of Economic Perspectives 29, 61–88.

Dawes, Robyn M, 1979, The robust beauty of improper linear models in decision making, American Psy-

chologist 34, 571–582.

Dawes, Robyn M, David Faust, and Paul E Meehl, 1989, Clinical versus actuarial judgment, Science 243,

1668–1674.

Deaux, Kay, and Elizabeth Farris, 1977, Attributing causes for one’s own performance: The effects of sex,

norms, and outcome, Journal of research in Personality 11, 59–72.

Deaves, Richard, Erik Lüders, and Guo Ying Luo, 2009, An experimental test of the impact of overconfidence

and gender on trading activity, Review of Finance 13, 555–575.

Di Tella, Rafael, Ricardo Perez-Truglia, Andres Babino, and Mariano Sigman, 2015, Conveniently upset:

Avoiding altruism by distorting beliefs about others’ altruism, American Economic Review 105, 3416–

3442.

28
Du, Mengqiao, Alexandra Niessen-Ruenzi, and Terrance Odean, 2024, Stock repurchasing bias of mutual

funds, Review of Finance 28, 699–728.

Eil, David, and Justin M. Rao, 2011, The good news-bad news effect: Asymmetric processing of objective

information about yourself, American Economic Journal: Microecenomics 3, 114–138.

Feng, Lei, and Mark S. Seasholes, 2005, Do investor sophistication and trading experience eliminate behav-

ioral biases in financial markets?, Review of Finance 9, 305–351.

Fischbacher, Urs, Gerson Hoffmann, and Simeon Schudy, 2017, The causal effect of stop-loss and take-gain

orders on the disposition effect, Review of Financial Studies 30, 2110–2129.

Frazzini, Andrea, 2006, The disposition effect and underreaction to news, Journal of Finance 61, 2017–2046.

Frydman, Cary, Nicholas Barberis, Colin Camerer, Peter Bossaerts, and Antonio Rangel, 2014, Using neural

data to test a theory of investor behavior: An application to realization utility, Journal of Finance 69,

907–946.

Frydman, Cary, Samuel M Hartzmark, and David H Solomon, 2018, Rolling mental accounts, Review of

Financial Studies 31, 362–397.

Frydman, Cary, and Antonio Rangel, 2014, Debiasing the disposition effect by reducing the saliency of

information about a stock’s purchase price, Journal of Economic Behavior & Organization 107, 541–552.

Gabaix, Xavier, 2019, Behavioral inattention, in Handbook of Behavioral Economics: Applications and Foun-

dations 1 , volume 2, 261–343 (Elsevier).

Genesove, David, and Christopher Mayer, 2001, Loss aversion and seller behavior: Evidence from the housing

market, Quarterly Journal of Economics 116, 1233–1260.

Gervais, Simon, and Terrance Odean, 2001, Learning to be overconfident, Review of Financial Studies 14,

1–27.

Gödker, Katrin, Peiran Jiao, and Paul Smeets, forthcoming, Investor memory, in Review of Financial Studies.

Grinblatt, Mark, and Matti Keloharju, 2001, What makes investors trade?, Journal of Finance 56, 589–616.

Hartzmark, Samuel M, Samuel D Hirshman, and Alex Imas, 2021, Ownership, learning, and beliefs, Quarterly

Journal of Economics 136, 1665–1717.

Heath, Chip, Steven Huddart, and Mark Lang, 1999, Psychological factors and stock option exercise, Quar-

terly Journal of Economics 114, 601–627.

29
Hens, Thorsten, and Martin Vlcek, 2011, Does prospect theory explain the disposition effect?, Journal of

Behavioral Finance 12, 141–157.

Imas, Alex, 2016, The realization effect: Risk-taking after realized versus paper losses, American Economic

Review 106, 2086–2109.

Jehiel, Philippe, 2018, Investment strategy and selection bias: An equilibrium perspective on overoptimism,

American Economic Review 108, 1582–97.

Kaustia, Markku, 2010a, Disposition effect, Behavioral Finance: Investors, Corporations, and Markets 169–

189.

Kaustia, Markku, 2010b, Prospect theory and the disposition effect, Journal of Financial and Quantitative

Analysis 45, 791–812.

Kaustia, Markku, and Samuli Knüpfer, 2008, Do investors overweight personal experience? Evidence from

IPO subscriptions, Journal of Finance 63, 2679–2702.

Köszegi, Botond, 2006, Ego utility, overconfidence, and task choice, Journal of the European Economic

Association 4, 673–707.

Kuhnen, Camelia M., Sarah Rudorf, and Bernd Weber, 2017, The effect of prior choices on expectations and

subsequent portfolio decisions, NBER working paper No. 23438.

Kyle, Albert S, 1985, Continuous auctions and insider trading, Econometrica 53, 1315–1335.

Lundeberg, Mary A., Paul W. Fox, and Judith Punćcohaŕ, 1994, Highly confident but wrong: Gender

differences and similarities in confidence judgments., Journal of Educational Psychology 86, 114–121.

Malmendier, Ulrike, and Stefan Nagel, 2011, Depression babies: Do macroeconomic experiences affect risk

taking?, Quarterly Journal of Economics 126, 373–416.

Meng, Juanjuan, and Xi Weng, 2018, Can prospect theory explain the disposition effect? A new perspective

on reference points, Management Science 64, 3331–3351.

Meyer, Steffen, and Michaela Pagel, 2022, Fully closed: Individual responses to realized gains and losses,

Journal of Finance 77, 1529–1585.

Moore, Don A, and Paul J Healy, 2008, The trouble with overconfidence., Psychological Review 115, 502.

Muermann, Alexander, and Jacqueline Volkman Wise, 2006, Regret, pride, and the disposition effect, Avail-

able at SSRN 930675 .

30
Odean, Terrance, 1998a, Are investors reluctant to realize their losses?, Journal of Finance 53, 1775–1798.

Odean, Terrance, 1998b, Volume, volatility, price, and profit when all traders are above average, Journal of

Finance 53, 1887–1934.

Prince, Melvin, 1993, Women, men, and money styles., Journal of Economic Psychology 14, 175–182.

Saucet, Charlotte, and Marie Claire Villeval, 2019, Motivated memory in dictator games, Games and Eco-

nomic Behavior 117, 250–275.

Seru, Amit, Tyler Shumway, and Noah Stoffman, 2010, Learning by trading, Review of Financial Studies

23, 705–739.

Shapira, Zur, and Itzhak Venezia, 2001, Patterns of behavior of professionally managed and independent

investors, Journal of Banking and Finance 25, 1573–1587.

Shefrin, Hersh, and Meir Statman, 1985, The disposition to sell winners too early and ride losers too long:

Theory and evidence, Journal of Finance 40, 777–790.

Strahilevitz, Michal Ann, Terrance Odean, and Brad M. Barber, 2011, Once burned, twice shy: How naive

learning, counterfactuals, and regret affect the repurchase of stocks previously sold, Journal of Marketing

Research 48, S102–S120.

Summers, Barbara, and Darren Duxbury, 2012, Decision-dependent emotions and behavioral anomalies,

Organizational Behavior and Human Decision Processes 118, 226–238.

Taylor, Shelley E., and Jonathon D. Brown, 1988, Illusion and well-being: a social psychological perspective

on mental health., Psychological Bulletin 103, 193–210.

Thaler, Richard, and Eric Johnson, 1990, Gambling with the house money and trying to break even: The

effects of prior outcomes on risky choice, Management Science 36, 643–660.

Weber, Martin, and Colin F Camerer, 1998, The disposition effect in securities trading: An experimental

analysis, Journal of Economic Behavior & Organization 33, 167–184.

Weber, Martin, and Frank Welfens, 2011, The follow-on purchase and repurchase behavior of individual

investors: An experimental investigation, Die Betriebswirtschaft 71, 139–154.

Zimmermann, Florian, 2020, The dynamics of motivated beliefs, American Economic Review 110, 337–61.

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 Appendices

A. Experimental Instructions

Instructions of Investment Task

In the investment task, you select stocks. Each period a stock can either increase or decrease in price.

There are two types of stock: an ordinary type and a high type. These two stock types differ in the

probability that their price increases or decreases. An ordinary-type stock has a 40% probability of

a price increase and a 60% probability of a price decrease. A high-type stock has a 60% probability of a

price increase and a 40% probability of a price decrease.

If the price increases, the price change is + $2 or + $6, with equal probability.

If the price decreases, the price change is - $1 or - $3, with equal probability.

The purchase price of a stock is always $30.

NEXT SCREEN

The investment task consists of 5 periods. Each period, a stock’s price either increases or decreases.

First investment choice

You begin the investment task with $180 and must buy a portfolio of 5 stocks from a list of 20 available

stocks. Each stock is priced at $30.

The list contains exactly 15 ordinary-type stocks and exactly 5 high-type stocks. You are not told which

stock is of which type. However, for each of the 20 stocks, you are shown price changes from the three

previous periods.

The picture below shows you how your choice screen will look like. You will be asked to choose 5 stocks of

such a list.

32
 Example Screen.

NEXT SCREEN

First period

After the initial portfolio selection, you observe the first period price changes for each of the stocks in your

portfolio. After the new prices are displayed, one of the stocks is automatically sold by the computer

program at the stock’s current price. After the sale of one stock, you must buy an additional stock from

a new list of four stocks. Once again, you observe the previous three price changes for each of the four

stocks. The purchase price of each stock is always $30.

33
The list contains exactly 3 ordinary-type stocks and exactly 1 high-type stock. You are not told which stock is

of which type. However, for each of the 4 stocks, you are shown price changes from the three previous periods.

Periods 2-4

You observe the next period’s new prices for each of the stocks in your portfolio. After the new prices are

displayed, one of the stocks is automatically sold by the computer program at the stock’s current price.

After the sale of one stock, you must buy an additional stock from a new list of four stocks. Once

again, you observe the previous three price changes for each of the four stocks. The purchase price of each

stock is always $30.

As in period 1, the list contains exactly 3 ordinary-type stocks and exactly 1 high-type stock. You are not

told which stock is of which type. However, for each of the 4 stocks, you are shown price changes from the

three previous periods.

Period 5

You observe the new prices for each of the stocks in your portfolio. The investment task is now over.

34
 B. Example Screen

In each period, subjects observed the purchase price, the price from one period ago, and the current price

of all stock positions in the portfolio.

Example Screen: Overview of portfolio positions.

35
 C. Explanations for the Disposition Effect

Shefrin and Statman (1985) coin the term “Disposition Effect” and attribute the behavior to a combina-

tion of Prospect Theory; mental accounting; regret aversion; and self-control. They discuss the emotions of

regret and pride pointing out that “While closing a stock account at a loss induces regret, closing at a gain

induces pride.” Weber and Camerer (1998) find the disposition effect in an experimental setting and Odean

(1998a) finds it for individual investors.

Odean (1998a) considers and rejects several potential explanations for the disposition effect including

rebalancing, transactions costs, and superior information. The author notes that the disposition effect could

be driven by a belief that stock prices mean revert but that retail investors tend to buy stocks that have

been going up, which is inconsistent with a belief in mean reversion. One exception to this tendency is that

investors are more likely to buy additional shares of a stock that they already own if it has gone down since

they bought it (which is consistent with a belief in mean reversion). Odean (1998a) also points out that

reference points are likely to be path dependent and not simply equal to purchase price.

Several subsequent papers argue that the disposition effect cannot be explained by Prospect Theory

(Barberis and Xiong, 2009; Kaustia, 2010b; Hens and Vlcek, 2011; Meng and Weng, 2018). Barberis and

Xiong (2012) generate a disposition effect in a theoretical setting by assuming that investors derive utility

from the act of selling for a gain (realization utility). Frydman et al. (2014) find neurological activity

consistent with realization utility in an investment like setting.

Seru, Shumway, and Stoffman (2010) document that, controlling for time held, the propensity to sell is

V-shaped in a stock’s return since purchase.

Ben-David and Hirshleifer (2012) point out that realization utility fails to account for this V-shaped

selling pattern or for the propensity to purchase additional shares of a stock already owned at prices below

the purchase price. Ben-David and Hirshleifer (2012) propose that the disposition effect, and related trading

behaviors, can be explained by overconfident beliefs.

Finally, several papers argue that the investors’ selling and repurchase decisions are the result of investors

choosing behaviors that avoid or postpone the negative emotion of regret (Muermann and Volkman Wise,

2006; Summers and Duxbury, 2012; Strahilevitz et al., 2011; Weber and Welfens, 2011).

See Kaustia (2010a) and Barber and Odean (2013) for more detailed discussions of this literature.

36
 D. Robustness

Table AI. Beliefs about own performance of Dutch retail investors. This table shows the results
of Table II, restricting the sample to investors who at least had 2 realizations in 2019. The table contains
the coefficients and robust standard errors (in parentheses) of OLS regressions. The dependent variable is
the investors’ beliefs about their performance-based rank among other retail investors, Elicited Percentile
Rank, (between 0 and 100). Net Gains is the difference in the number of investors’ realized gains and losses
in 2019. Recalled Net Gains is the difference in the recalled number of investors’ realized gains and losses
in 2019. Actual Percentile Rank is investors’ actual percentile rank among all retail clients at the financial
institution based on annual portfolio performance in 2019 (from 0 to 100). Order of Elicitation is a dummy
variable indicating the order in which our two survey items were elicited (1 = recall of realizations first and
0 = otherwise). *, **, and *** denote significance at the 10%, the 5%, and the 1% level, respectively.

(1) (2) (3) (4)

Elicited Perc. Rank Elicited Perc. Rank Elicited Perc. Rank Elicited Perc. Rank

Net Gains 0.476∗∗∗ 0.345∗∗

(0.15) (0.15)
Recalled Net Gains 0.613∗∗∗ 0.537∗∗∗
(0.17) (0.17)
Actual Perc. Rank 0.171∗∗∗ 0.121∗∗∗
(0.02) (0.04)
Order of Elicitation -1.474
(2.24)
Constant 55.420∗∗∗ 46.176∗∗∗ 56.653∗∗∗ 52.033∗∗∗
(0.66) (1.31) (1.33) (4.33)

N 1,320 1,320 348 348

R2 0.01 0.06 0.03 0.06

37
 E. Solution to Model

We now describe the procedure for solving the model. We conjecture that, at the beginning of period t,

the equilibrium for asset i that is being traded has the following linear structure:

pi,t (ωi,t , st−1 , kt−1 ) = Di,t−1 + λi,t (st−1 , kt−1 ) · ωi,t ,

(E.1)
xi,t (θi,t , st−1 , kt−1 ) = βi,t (st−1 , kt−1 ) · θi,t ,

where pi,t is the price for asset i, xi,t is the investor’s demand for asset i, ωi,t = xi,t + zi,t is the total demand

from the investor and the liquidity trader, st−1 is the number of times investor’s information was true in the

past, and kt−1 is the number of times the investor sold an asset for a gain.

We first solve the investor’s problem, taking the conjectured equilibrium price rule as given. The investor

is risk neutral and chooses the demand, xi,t , that maximizes his biased expectation of the profits he will

make on asset i, given his information (θi,t , kt−1 ).11 Under the conjectured equilibrium structure, the investor

anticipates that profits when demanding share xi,t of asset i will be

πi,t ≡ xi,t ·(Di,t −pi,t ) = xi,t ·(Di,t −Di,t−1 −λi,t (st−1 , kt−1 )ωi,t = xi,t ·(vi,t −λi,t (st−1 , kt−1 )(xi,t +zi,t )) (E.2)

where we used the fact that vi,t = Di,t − Di,t−1 and ωi,t = xi,t + zi,t . The investor’s objective is to maximize

Eb [πi,t |θi,t , kt−1 ] = Eb [xi,t · (vi,t − λi,t (st−1 , kt−1 )(xi,t + zi,t ))|θi,t , kt−1 ]
(E.3)
= −x2i,t λ(st−1 , kt−1 ) + xi,t [Eb [vi,t |θi,t , kt−1 ] + λ(st−1 , kt−1 ) Eb [zi,t |θi,t , kt−1 ]].
| {z }
=0

Note that the expectation is linear, that zi,t is independent from a, and that the investor has correct

expectations about zi,t , therefore Eb [zi,t |θi,t , kt−1 ] = E[zi,t ] = 0. To find xi,t that maximizes Eb [πi,t |θi,t , kt−1 ],

we set the first order derivative equal to 0

Eb [vi,t |θi,t , kt−1 ] − 2λi,t (st−1 , kt−1 )xi,t = 0 (E.4)

and solve for xi,t

Eb [vi,t |θi,t , kt−1 ] Ξt−1 (kt−1 )
xi,t (θi,t , st−1 , kt−1 ) = = · θi,t (E.5)
2λi,t (st−1 , kt−1 ) 2λi,t (st−1 , kt−1 )

Note that Prb (δi,t = 1|θi,t , kt−1 ) = Ξ(kt−1 ) and Eb [vi,t |θi,t , kt−1 , δi,t = 0] = E[vi,t ] = 0. Therefore, by the
11 The investor also observes st−1 , but deems this quantity irrelevant in updating his beliefs

38
law of iterated expectations,12

Eb [vi,t |θi,t , kt−1 ] = Eb [vi,t |θi,t , kt−1 , δi,t = 1] Pr(δi,t = 1|θi,t , kt−1 )+Eb [vi,t |θi,t , kt−1 , δi,t = 0] Pr(δi,t = 0|θi,t , kt−1 )
b b
(E.6)

Next we solve the market maker’s problem which is to set the fair price for asset i at the beginning of

period t, i.e., the market maker imposes a price equal to his unbiased expectation of Di,t at the beginning of

period t. We assume that the investor chooses his demand as conjectured in Equations (E.1) and that the

market maker solves for:

pi,t = E[Di,t |ωi,t , st−1 , kt−1 ]

Recalling that Di,t = Di,t−1 + vt we have

pi,t = Di,t−1 + E[vt |ωi,t , st−1 , kt−1 ]

As before, we can use the law of iterated expectations to evaluate the second quantity. Begin with:

E[vt |ωi,t , st−1 , kt−1 ] = E[vt |ωi,t , st−1 , kt−1 , δi,t = 1] Pr(δi,t = 1) + E[vt |ωi,t , st−1 , kt−1 , δi,t = 0] Pr(δi,t = 0)

(E.7)

Now recall that, under our conjecture of the equilibrium structure, xi,t = βi,t (st−1 , kt−1 ) · θi,t . As a

consequence, the market maker anticipates ωi,t = βi,t (st−1 , kt−1 ) · θi,t + zi,t . Then, conditional on δi,t = 1,

we have vi,t = θi,t and, in this case, the vector (vi,t , ωi,t ) is the vector (vi,t , βi,t (st−1 , kt−1 ) · vi,t + zi,t ) which

is jointly normal with mean vector and variance-covariance matrix

 
 σi2 2 2
βi,t σi 
µ = (0, 0) Σ=

.
 (E.8)
2 2 2 2 2
βi,t σi βi,t σi + σi,z

These are easily computed recalling that vi,t ∼ N (0, σi2 ), zi,t ∼ N (0, σi,z
2
) and vi,t ⊥ zi,t . It follows from the

formulas for the conditional expectation of normal random vectors that

βi,t (st−1 , kt−1 )σi2

E[vi,t |ωi,t , st−1 , kt−1 , δi,t = 1] = 2 ωi,t . (E.9)
βi,t (st−1 , kt−1 )σi2 + σi,z
2

12 Here and in the following, when we condition on δ

i,t we are not assuming that the variable is the agent’s information.
Indeed, nobody in the economy knows δi,t . Rather, we are using the law of iterated expectation, which, from the point of view
of the agent, is reasoning by cases. In other words, to asses his expectation conditional on some information set I, the agent
partitions I in I ∩ δi,t = 1 and I ∩ δi,t = 0 and computes the average of his expectations conditional on each information
subset, weighted by their probability conditional on I.

39
Note that conditional on δi,t = 0, θi,t = εi,t . In this case, we have (vi,t , ωi,t ) = (vi,t , βi,t (st−1 , kt−1 ) · εi,t + zi,t )

and ωi,t is a sum of variables that are independent from vi,t . It follows that

E[vi,t |ωi,t , st−1 , kt−1 , δi,t = 0] = E[vi,t ] = 0 (E.10)

Finally, as noted after Equation (7), we have ξ(st−1 ) = Pr(δi,t = 1). Substituting all these pieces into

Equation (E.7) we have:

ξt−1 (st−1 ) · βi,t (st−1 , kt−1 )σi2

pi,t = Di,t−1 + 2 (s 2 2 ωi,t (E.11)
βi,t t−1 , kt−1 )σi + σi,z

Finally, substituting in our solutions for pi,t (ωi,t , st−1 , kt−1 ) and xi,t (θi,t , st−1 , kt−1 ) from Equations (E.5)

and (E.11) into our conjectured linear equilibrium in Equation (E.1), we obtain the following system of two

equations in two unknowns (βi,t (st−1 , kt−1 ) and λi,t (st−1 , kt−1 ))



βi,t (st−1 , kt−1 ) Ξt−1 (kt−1 )
 = 2λi,t (st−1 ,kt−1 )
(E.12)

λi,t (st−1 , kt−1 ) ξt−1 (st−1 )·βi,t (st−1 ,kt−1 )σi2
 = 2 (s
βi,t 2 2
t−1 ,kt−1 )σi +σi,z

This system has a solution if and only if an equilibrium of the conjectured form exists. To find a closed

form solution, obtain λi,t (st−1 , kt−1 ) from the first equation in the system and equate it to the second,

obtaining
Ξt−1 (kt−1 ) ξt−1 (st−1 ) · βi,t (st−1 , kt−1 )σi2
= 2 (s 2 2 (E.13)
2βi,t (st−1 , kt−1 ) βi,t t−1 , kt−1 )σi + σi,z

Rearranging, we obtain

[2ξt−1 (st−1 ) − Ξt−1 (kt−1 )]σi2 βi,t

2 2
(st−1 , kt−1 ) = Ξt−1 (kt−1 )σi,z (E.14)

Solving the equation for βi,t (st−1 , kt−1 ) and substituting in the expression for λi,t (st−1 , kt−1 ) we obtain

s
2
σi,z Ξt−1 (kt−1 )
βi,t (st−1 , kt−1 ) ≡ 2 · ,
σ 2ξt−1 (st−1 ) − Ξt−1 (kt−1 )
si (E.15)
1 σi2
λi,t (st−1 , kt−1 ) ≡ 2 · Ξt−1 (kt−1 ) · [2ξt−1 (st−1 ) − Ξt−1 (kt−1 )] .
2 σi,z

40
 Equation (E.15) makes it clear that the existence of equilibrium requires

2ξt−1 (st−1 ) > Ξt−1 (kt−1 ). (E.16)

And it is easy to show that

2L > H (E.17)

is sufficient to guarantee (E.16) and hence the existence of equilibrium. To see this notice that E.16 can be

re-written as

2(ϕt−1 (st−1 )H + (1 − ϕt−1 (st−1 ))L) > ψt−1 (kt−1 )H + (1 − ψt−1 (kt−1 )
(E.18)
L + (2ϕt−1 (st−1 ) − ψt−1 (kt−1 ))(H − L) > 0

Since H > L > 0, the only case in which this condition may fail is when 2ϕt−1 (st−1 ) − ψt−1 (kt−1 ) < 0. The

smallest 2ϕt−1 (st−1 ) − ψt−1 (kt−1 ) < 0 can be is −1, since ϕt−1 and ψt−1 are probabilities. In this case the

inequality becomes

2L − H > 0 (E.19)

Finally, we compute the equilibrium gain or loss of asset i, according to the above definition. If xi,t0 ≥ 0,

Pt−1
gi,t ≡ Di,t−1 − pi,t0 = j=t0 vi,j − λi,t0 (st0 −1 , kt0 −1 ) · ωi,t0
(E.20)
Pt−1
= j=t0 vi,j − λi,t0 (st0 −1 , kt0 −1 ) · (βi,t0 (st0 −1 , kt0 −1 ) · θi,t0 + zi,t0 ).

If xi,t0 < 0,

Pt−1
gi,t ≡ pi,t0 − Di,t−1 = λi,t0 (st0 −1 , kt0 −1 ) · ωi,t0 − j=t0 vi,j
(E.21)
Pt−1
= λi,t0 (st0 −1 , kt0 −1 ) · (βi,t0 (st0 −1 , kt0 −1 ) · θi,t0 + zi,t0 ) − j=t0 vi,j .

41