Are investment game players reluctant to realize their losses?
A study on the disposition effect in an investment game
______________________________________________________________________________
Sijbrand N. van der Mast
Student number 1413643
University Supervisors:
Dr. R.M. Salomons
Prof. Dr. R.A.H. van der Meer
Amsterdam, April 2010
RijksUniversiteit Groningen
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Are investment game players reluctant to realize their losses?
A study on the disposition effect in an investment game
______________________________________________________________________________
Abstract
In this paper I investigate whether investment game players are prone to a behavioral bias known
as the disposition effect. This is the tendency to sell stocks that are held at a gain more often than
stocks that are held at a loss. I find that the average investment game player is not subject to this
behavioral bias. This makes it difficult to extrapolate investment game findings to the real world.
In addition, I find that because of the unique dataset (both experimental and field characteristics)
used in this paper, an „effort‟ issue arises. The „effort‟ issue can potentially bias test results based
on investment game data. Furthermore I find that investment behavior differs between men and
women, and that frequent traders and experienced and sophisticated traders are able to eliminate
the behavioral bias. I conclude that investment game data can in the present form not be used as
a proxy for real life investment behavior.
Sijbrand van der Mast
snvdmast@hotmail.com
Student number: 1413643
JEL Codes: G10, G11, G12,
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Acknowledgements
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Table of Contents
1.
I
NTRODUCTION5
2.
L
ITERATURER
EVIEW7
2.1 The fundamental features of the disposition effect 7
2.2 Field research on the disposition effect 9
2.3 Experimental Research on the disposition effect 13
2.4 Concluding remarks 14
3.
H
YPOTHESES16
4.
D
ATA17
4.1 The Challenge 17 4.2 The Rules 17 4.3 Data Selection 18 4.4 Investor characteristics 18 4.5 Alternative challenge 19 4.6 Market environment 19 4.7 Trading Activity 215.
M
ETHODOLOGY22
5.1 PGR-PLR Method 22 5.2 Cross-sectional research 23 5.3 Logit regression 245.4 Concluding methodological remarks 25
6. D
ATAI
MPACT26
6.1 Experimental vs. real world research 26
6.2 Non-random selection of participants 27
6.3 Heterogeneity among investors 28
6.4 The role of experience 28
6.5 Stakes and cognitive costs 29
6.6 Short-run effects versus long-run effects 29
6.7 The „Effort‟ issue 30
7.
R
ESULTS33
7.1 Game 1 33
7.2 Game 2 34
7.3 Robustness test over time 34
7.4 Investor characteristics and the disposition effect 35
7.5 Investor characteristics and the disposition effect after four months 37
8.
D
ISCUSSION38
8.1 At first glance 38
8.2 Explanations 39
8.3 Discussing game 2 41
8.3 To extrapolate or not to extrapolate 41
9.
C
ONCLUSION&
R
ECOMMENDATIONS42
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List of Tables and Figures
T
ABLEST
ABLE2.1
Overview of related empirical studies 14T
ABLE4.1
Descriptive data of games 1 & 2 17T
ABLE4.2
Demographic characteristics of all players in game 118
T
ABLE6.1
Trading record of OBAM u004623 30T
ABLE7.1
Aggregated and mean PGR & PLR for all game 1 & 2 players after six months 32T
ABLE7.2
Aggregated and mean PGR & PLR for all game 1 & 2 players after four months 34T
ABLE7.3
Univariate cross-sectional results after six months 38T
ABLE7.4
Univariate cross-sectional results after four months 36F
IGURESF
IGURE2.1
Prospect Theory Value Function (Kahneman & Tversky, 1979) 7F
IGURE4.1
MSCI World Index 2008 19F
IGURE4.2
Aggregated trading activity of all game 1 players 205
1.
I
NTRODUCTIONOver the last few decades research on behavioural finance has been popping up like daisies. This research has provided more and more proof that psychological factors affect the behaviour of financial practitioners. And hence affects financial markets.
“Modern finance theorists have turned finance into a science, but they forgot that it is a social science.”1 Even those who truly believe in the modern portfolio theory of Markowitz (1952) or the efficient market hypothesis of Fama (1970) can no longer deny the importance of behavioural finance in order to explain asset pricing. Markets are not perfect and neither are the people who act in them; the human being is predictably irrational. Behavioural finance which recognises the cognitive limitations of the human brain and the influence of psychological and emotional factors, has become more and more accepted by finance theorists. Market imperfections that could not be explained by the standard modern finance theories are now better understood because of it.
One of the most puzzling but robust findings in financial markets that cannot be explained by standard modern finance theories is the so-called disposition effect, a behavioural phenomenon which was first identified by Shefrin and Statman (1985). In a world where the purchase price of a stock should not matter for an investor‟s decision to sell it, investors are anxious to sell their winners, but reluctant to sell their losers. Investors hold on to a stock because they tend to believe that the stock will bounce back. This is against one‟s better judgement, as numerous studies have shown; betting on a recovery of the losing stock is a fruitless quest. Simply taking the pain, by selling a losing stock sooner is more rewarding for an investor. “The first loss is the best loss” is the market‟s way of saying that the investor was wrong. This reluctance to sell losers can be explained by any number of potential psychological errors; over-optimism, overconfidence and the self-attribution bias.
The disposition effect has been researched and observed in both financial and experimental markets throughout the world. In this paper I also find the existence of a disposition effect. As far as I am aware of, this paper will be the first to examine the disposition effect in an investment game environment. The data used in this study makes this paper on the disposition effect unique in comparison to previous studies. It is the first study that examines the disposition effect using data that has both experimental and field characteristics. “Experimental” because the players are subjects of a game and “field” because the game perfectly simulates actual financial markets. Because of the unique dataset there are a number of factors that might influence the investing behaviour of the investment game players. These potentially influential factors are described in the paper of Levitt & List (2007). Levitt and List explain the potential
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dangers of extrapolating experimental data into the field. Using the methodology of Odean (1998) and the theoretical framework of Levitt & List (2007), I will investigate whether investment investors are also prone to the disposition effect. The findings in this paper are biased by two of the five factors of Levitt and List.
In addition, the dataset from the trading game provides the unique opportunity to examine whether demographic differences interfere with the detection of the disposition effect. Are men more rational then women? Does wisdom come with age? Does either gender, age or the nationality of an investor interfere with the detection of the disposition effect? I will later show that investor characteristics can indeed have an influence on the trading behaviour of a player.
Because of an additional game that was hosted by game host WebsiteXpress. I have additional data from an investment game played during the same period as the initial trading game. The players of this game are all active in the financial market. This gives me the opportunity to test whether experienced and sophisticated players are able to eliminate the behavioural bias. The yielded result is in line with earlier research.
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2.
L
ITERATURER
EVIEWThe disposition effect is one of the most observed regularities in financial markets. According to Shefrin & Statman (1985), the disposition effect is the tendency to sell winners too early and hold losers too long. In order to clarify this study on the disposition effect, I have divided the next section into four parts. First I will discuss the literature that explains the fundamental features of the disposition effect. In the second part I will present the literature that researched the disposition effect using financial market data. Thirdly I will discuss the literature that investigated the disposition effect using experimental data. Fourthly and finally I will summarize the findings of the first three parts of this literature review. In addition I have provided a literature overview in table 2.1.
2.1 The fundamental features of the disposition effect
In order to explain the disposition effect, Shefrin and Statman combine certain features from Kahneman & Tversky‟s prospect theory with behavioural elements such as Thaler‟s mental accounting theory, regret (loss) aversion and self-control.
Markowitz (1952) and Kahneman & Tversky (1979) were the first researchers to deviate from the “conventional” expected utility theory where final wealth is seen as the carrier of value in the utility function. Instead Markowitz and Kahneman & Tversky stated that the proper carrier of value in the utility function is the change in wealth. Markowitz was the first to propose that utility be defined on gains and losses rather than on final asset positions. This is an important element in Kahneman & Tversky‟s so-called prospect theory, which is the main behavioral alternative to expected utility.
Kahneman & Tversky describe the prospect theory as how people frame and value (utility) a decision involving uncertainty. Nofsinger (2002) explains: First, investors frame the choices in terms of potential gains and losses relative to a specific reference point. Although investors can anchor on various reference prices, the purchase price appears to be the most important. Second, investors value the gains/losses according to an S-shaped function. Many biased decisions made by investors can be explained by the prospect theory‟s value (utility) function.
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currently held for a loss, but prefers the risk of continuing to own the stock hoping to sell it for a gain. Hope and fear are important emotions that drive investors to make irrational decisions.
Figure 2.1 Prospect Theory Value Function (Kahneman & Tversky, 1979)
In addition to the prospect theory behavioural elements such as mental accounting, regret aversion and self control contribute to the explanation of the disposition effect. Mental accounting was first used by Thaler (1980), in order to track gains and losses people separate each investment into an individual position and periodically re-examine this position, each position that has been taken can be seen as an opened mental account. Barber et al (2007) explain that due to mental accounting, investors focus on gains and losses from individual stock positions rather than focusing on portfolio returns or total wealth levels. Separating investments is however not in line with the traditional portfolio approach because it limits the investor‟s ability to minimize risk and maximize return on the entire portfolio.
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The self control problem is another possible explanation why investors are prone to the disposition effect. According to Thaler & Shefrin (1981) the self control problem is due to the fact that the human being has two types of individuals within, the planner and the doer. As Nofsinger (2002) explains, the doer wishes to consume now instead of later and procrastinates on unpleasant tasks. The planner wishes to save for later consumption and complete unpleasant tasks now. People are influenced by long-term rational concerns and by more short term emotions. This conflict between desire and willpower can influence the decision making process. In the view of Sherfin & Statman, investors ride losers to postpone regret, therefore not realizing any pain, and sell winners “too quickly” because they want to hasten the feeling of pride at having chosen correctly in the past. Glick‟s (1957) study on professional futures traders showed that the traders were aware that riding losers was not rational. The problem was the lack of self control to close the accounts at a loss, which would limit their losses in the long-run.
To be comprehensive about the literature concerning the disposition effect, two more papers must be mentioned. These papers focus on whether the prospect theory is actually able to explain the disposition effect. In their 2009 paper Barberis and Xiong investigate whether the prospect theory preferences can predict a disposition effect. By using simulated artificial data on how prospect theory investors would trade over time, they check, using the methodology of Odean (1998), whether the prospect theory predicts a disposition effect. Barberis and Xiong (2009) find that the realized gain/loss model of Odean indeed predicts a disposition effect. They conclude that utility from realized gains and losses may therefore be a useful way of thinking about certain aspects of individual investor trading.
Using their own econometrical model, Hans and Vlcek (2006) conclude that the prospect theory can indeed explain the disposition behavior. Alternative explanations for the disposition effect according to Hans and Vlcek could include mental accounting combined with backward looking optimization.
The above mentioned behavioral biases lead investors to sell their gains too soon and hold on to their losses too long, hence be prone to the disposition effect. By having knowledge of these behavioral biases and the discipline to operate in a rational manner to avoid these biases, one can overcome being prone to the disposition effect. Section 2.2 will show that in general investors cannot overcome the behavioral bias.
2.2 Field research on the disposition effect
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In 1998 Odean was the first to test the disposition effect using real brokerage data. By analyzing ten thousand accounts at a large US discount brokerage house in the period from 1993 until 1996. Using his own PGR-PLR methodology, Odean finds in all months with the exception of December that aggregated investors sell winners at twice the rate of losers. In the month of December, Odean documents that losers are sold at a higher rate than winners. This can be explained by year-end “tax purposes”.
When an investor postpones selling gains by continuing to hold on to a profitable investment, he also avoids paying taxes. When an investor sells a loss by getting rid of a losing investment, he recognizes a tax benefit. Realizing a losing investment gives a tax advantage, however realizing a profitable investment results in a tax disadvantage. But why are investors reluctant to sell their losses and hold onto their gains for all months but December? According to Shefrin and Statman (1985) this is because the end of the year is the deadline to realize losses for tax purposes. The month of December is used as a measure of self-control. Realizing losses is therefore postponed to the month of December, when investors require themselves to sell their losses before the year end tax date.
In all months other than December, Odean states, the disposition effect can not be explained through tax reasoning, rebalancing of the portfolio or transaction costs. The information hypothesis seems to give an obvious explanation for the disposition effect. The information hypothesis says that investors trade because they have private information. Investors may sell stocks with paper gains because they have information that these stocks will do poorly in the near future. On the other hand they hold on to stocks with paper losses because they have private information that they will recover. Odean finds, that over the next year, the average return of prior winners that investors sell is 3.4% higher than the average return of the prior losers. These results obviously reject the information hypothesis as an explanation for the presence of the disposition effect among investors. In his subsequent paper Odean (1999) uses the same dataset, but other accounts, and documents the same results. In order to explain the disposition effect, in both papers Odean refers to the “behavioral” prospect theory of Kahneman and Tversky.
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Because their methodology allows them to perform research at individual investor level, they also find that men and women have similar propensities to sell and that “less” sophisticated investors (in comparison with professional investors) – households, general government and non-profit institutions – are more prone to the disposition effect. Finally they find that foreign investors in Finland are not affected by the disposition effect. A possible explanation for the difference in behavior between foreign investors and that of local investors is that foreign investors assume higher risk levels for their foreign investments which makes them more risk averse and in expectance of higher returns, therefore selling losing stocks and holding onto winning stocks. The ex-ante risk aversion of local investors can be reduced by familiarity bias that is generally regarded as the cause for the home bias portfolio allocation.
Shapira and Venezia (2001) use the data of a major Israeli brokerage house during 1994 to investigate whether the disposition effect also holds for professional investors. In their paper instead of using the methodology of Odean, they replicate the methodology used by Shefrin and Statman. They find that both professional and individuals exhibit the disposition effect. For the professionals however the effect is less in comparison to the individual investor, confirming the findings of Grinblatt & Keloharju. Dhar and Zu (2006) also document that professional investors demonstrate a lower disposition effect, therefore strengthening the findings of earlier research.
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and therefore are more sophisticated. In addition the paper of Feng and Seasholes is important because it does not only use a different methodology than Odean (1998) but also criticizes his PGR-PGL methodology. Krause, Wei & Yang also establish new findings concerning research of the behavioral bias. They are the first to acknowledge the existence of a reverse disposition effect alongside a disposition effect. The variables that explain this result are most notably the length of an investment strategy and market conditions.
After his 1998 paper Odean received critique for his methodology. The most important shortcoming is that it is not possible to perform a cross-sectional analysis using his methodology. The critique led Odean to adjust his methodology, which he then used in collaboration with Barber, Lee and Liu on their 2006 paper. In the methodology section of this paper the critique will be elaborated in further detail. Using data from the Taiwan Stock Exchange in the period 1994-1999, Barber, Lee, Liu & Odean (2006) find that individuals, corporations and dealers are prone to the disposition effect, mutual funds and foreigners are not. Furthermore they find that the reluctance to realize losses is greater for men then women, and that “the willingness to sell losers increases following strong market returns”. In earlier research Barber & Odean (2001) were one of the first to use actual trading data to investigate whether gender has an impact on the behavior of an investor. They found that in male-dominated realms such as finance, men trade 45 percent more than woman. Therefore reducing their returns (due to transaction costs) with a larger amount than women. This confirms that both psychology and empirical behavioral finance reveal gender differences in investment behavior.
Frazzini (2006) set out to find whether the disposition effect induces under reaction to news, which consequently leads to return predictability of stock prices. In order for Frazzini to answer his hypothesis, using Odean‟s PGR-PLR methodology, he calculates whether investors are biased by the disposition effect. Noteworthy for this study is that he uses 22 years of data (1980-2002), and finds that investors are prone to the disposition effect consequently over time. This confirms that over time disposition effect is robust phenomenon. Additionally Frazzini links fund performance to the degree that funds sell their winners relative to their losers. Frazzini finds that the best-performing funds are less likely to sell a winning position than a losing position and the worst-performing fund visa versa. The final conclusion of Frazzini‟s paper is that the disposition effect can induce under reaction to news, which indeed leads to return predictability.
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phenomenon can be explained by the disposition effect, a behavioral finance theory which is driven by prospect theory and mental accounting. According to Grinblatt & Han both of these behavioral phenomena generate predictable equilibrium prices and therefore influence market prices and create momentum.
In the last three years little distinct findings have been done concerning the research of the disposition effect. The most recent studies have mostly confirmed previous findings. Worth mentioning is that Talsepp (2009) uses two different methods, the Cox proportional hazard model and the PGR-PLR methodology, both yielding the same results. This is a confirmation that there are multiple robust methodologies to research the disposition effect. Further details of performed research on the disposition effect can be found in the literature overview in table 2.1.
2.3 Experimental Research on the disposition effect
As the dataset used in this paper has experimental characteristics, in this section I will summarize three articles that researched the disposition effect using experimental data.
Weber and Camerer (1997) perform an experimental analysis in an artificial setup and find that the subjects in the experiment are susceptible to the disposition effect. They also find that when the shares that the subjects held were automatically sold, the disposition effect was greatly reduced. This validates that self control is an important variable to explaining the disposition effect.
In their paper Da Costa et al (2008) set an experiment to replicate results from the Weber & Camerer (1998) study and show that a subject‟s gender may interfere with the disposition effect‟s detection. For this research the paper of Da Costa (2008) is particular interesting. They find that women do not keep losing stocks and sell winners, thus are not prone to the disposition effect.
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2.4 Concluding remarksThe existing research to date is quite unambiguous in its results concerning the disposition effect in general. Most papers find that investors (market data) or subjects (experimental data) are affected by the behavioral bias. Papers that performed a more in-depth analysis and did cross-sectional research, commonly find that sophistication and experience eliminate or at least diminish the disposition effect. Investor wealth and trading frequency are other variables that seem to reduce the disposition effect among investors. Because of Hofstede (2001) we know that culture does influence the way people do business. But within the world of investing, little research has been done on the influence of nationality on investor behavior. Nationality does not seem to have an influence on whether investors are biased by the disposition effect or not. The disposition effect is a global phenomenon as it has been found in stock markets all over the world. It also seems to be a robust occurrence over time as empirical research has observed the disposition effect on stock markets since the eighties until now. There has been too little research done on the investor characteristics gender and age too be conclusive about the impact of these demographical characteristics. Croson & Gneezy (2009) reviewed economic literature on gender differences. They document a few fundamental differences between men and women that are relevant to this study. For reasons of emotion, overconfidence and framing, most lab and field studies find that women are more risk averse than men. Further field and lab studies find that women‟s preferences for competitive situations are lower than those of men. It is unclear whether gender affects the trading behavior of investors with regards to the disposition effect. Earlier research has found various results. It is also unclear whether age influences the behavior of an investor. Too little research has been done on age as a potential determinant of the disposition effect. Therefore it is difficult to draw conclusions concerning age and the disposition effect.
All different methodologies that I have encountered in the above mentioned papers yield similar results. A number of papers compared methodologies to see whether the results would be aligned and concluded that this was the case. These findings confirm that the methodology of Odean that is used in this paper is a robust way to test for the disposition effect.
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Table 2.1 Overview of related empirical studies
Authors Research goal Data source Data
period Methodology Country Asset class Robustness test over time Number of individual investors Results Odean (1998)
Testing DE, and analysis of DE with tax-motivated selling Brokerage account data (1987-1993) PGR-PLR method (Odean 1998)
USA Securities Yes 10000
Aggregated investors show DE expect in December
were selling is tax motivated Barber, Lee, Liu & Odean (2006) Full analysis of DE on the Taiwanese stock exchange Brokerage account data (1994-1999) PGR-PLR method (including daily data)
Taiwan Securities Yes 3971249
Individuals, corporations and dealers prone to DE, mutual funds and foreigners
are not Feng & Seasholes (2005) Reasearch on DE, does experience eliminate bias Brokerage account data (1999-2001) Survival analyisis (based on Logit) China Securities No 1511
Trading experience can reduce DE, experience is asymmetric on behavior in the face of losses vs. gains. Krause, Wei & Yang (2006) Researching DE based on average length trading strategy Brokerage account data (1999-2003) Dispo methodology (own measure) China Securities No 4619
Overall DE found, reverse DE found during falling markets, lower DE when experienced & buy strategy Griblatt & Keloharju (2001) DE among individuals & institutional investors Finnish Central Securities Depository (1995-1997) Logit Regressions
Finland Securities Yes 20000+
DE among individuals and institutional investors, not
for foreign investors, gender unrelated to propensity to sell Nofsinger et al (2005) Cross-sectional
research on DE account data Brokerage (1999-2002)
PGR-PLR method + Logit regressions
China Securities No 46969
Emerging market investors make cognitive errors, savvier investors are less
inclined to make these errors Shapira & Venezia (2000) DE among individual vs. professional managed PF‟s Brokerage
account data (1994) Scharblaum et al (1978) technique
Israel Securities No 4330
Both professional and independent investors exhibit DE, effect is stronger for independent
investors Dhar & Zhu (2006) DE across individual investors Brokerage account data (1991-1996) PGR-PLR method + Logit regressions
USA Securities Yes 50000+
Wealthier & professional investors exhibit lower DE,
also trading frequency reduces the DE Weber & Camerer (1997) Experimental research on DE Experimental data 14 periods Experimental design Germany - - 29
Tested subjects show a DE, when automatic sell system is implicated DE vanishes Da Costa, Mineto & Da Silva (2008) Experimental research on the relationship DE and gender Experimental data 14 periods Experimental design Brazil - - 106
Detection of the DE and evidence that gender may
interfere with the DE detection Benzion & Shavit (2009) Experimental analysis of DE Experimental data 20 rounds Experimental design, based on PGR-PLR method Israel - - 50
This study demonstrates that the DE can be a
product of trading conditions. Boolell,, Broihanne & Merli (2008) In depth analysis of DE in France Brokerage account data (1999-2006) PGR-PLR method
France Securities Yes 90244
DE among French investors, sophistication does not eliminate DE,
tax-selling does not eliminate DE, but is less Talsepp (2009) Analysing the DE using data from Estonian Stock market Nasdaq OMX Talin (2004-2008) Cox proportional hazard model + PGR/PLR Estonia Securities No 24153
Found DE overall only foreign traders exhibit reverse DE experience & sophistication decrease bias Choe & Eom (2006) Analysis of DE in the Korean index futures market Korean Index Futures Market (2003-2005) PGR-PLR method Korea Futures No 62570 Individuals stronger DE then institutions & foreigners, sophistication &
experience eliminate DE Frazzini (2006) DE and market under reaction to news among fund managers CDA Spectrum mutual fund holdings (1980-2002) PGR-PLR method
USA Securities Yes 29000
Statically strong tendency that mutual fund managers
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3.
H
YPOTHESESIn this paper I want to investigate whether investment game investors have the tendency to sell winners too early and hold losers too long. This is interesting to investigate because the data used in this research is unique, it has both experimental as real world data characteristics. Considering my main research objective and the literature written on the disposition effect I have stated the following hypotheses and expectations:
Hypothesis1 Players in an investment game are reluctant to sell losses and hold onto gains
At account level the proportion of gains realized is significantly larger than the proportion of losses realized
Hypothesis2 Demographic differences interfere with the detection of the disposition effect
Gender, age or the nationality of an investor interferes with the detection of the disposition effect
Hypothesis3 The experienced and sophisticated players of game 2 are not reluctant to sell losses and
hold onto gains
The experience and sophistication of the players in game 2 helps them to overcome behavioral biases
Dependent on whether I will reject the above mentioned hypotheses I will be able to answer the following question: Is it possible to generalize the behavioral biases of a investment game players to real world investors?
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4.
D
ATAThis paper differs to previous papers on the disposition effect because of its unique dataset. Earlier papers on the disposition effect use data that was obtained from real life brokerage houses or from short term experimental research. In this paper I use trading data that was acquired from an investment game, the Fortis OBAM challenge. The Fortis OBAM challenge is an online investment game that was created in order to promote the Fortis OBAM fund 2. The host of the Fortis OBAM challenge and provider of the data is WebsiteXpress. WebsiteXpress main product myWallet provides a new and rich user experience in the field of online portfolio management and serious investment gaming3. MyWallet's unique feature set allows it to being used for both educational and business purposes. WebsiteXpress has been hosting investment games for students at the University of Groningen since the early nineties. Since then they have broadened their horizon and host games at national and international level for financial and educational institutions.
4.1 The Challenge
The goal of the game was to make higher returns than all other participants and to see whether the participants were able to outperform the Fortis OBAM fund. The participant with the best performance could win a holiday and a master class on investing, regardless of his/her performance compared to that the Fortis OBAM fund. 3379 participants subscribed to take part in the Fortis OBAM investment game. The game took place over 130 trading days, from the 1st of April 2008 until the 30th of September 2008. Because the main benchmark for the participants was the Fortis OBAM fund, the players each received 1 billion euro's (the same amount of assets under management as in the Fortis OBAM fund) to invest in all equity markets throughout the world. In order to contend for the first prize, each player had to follow a few rules. These rules were aligned with the guidelines of the fund. A player who did not stick to these rules for a minimum of 110 of the 130 days was disqualified from the game.
4.2 The Rules
The players of the Fortis OBAM challenge had to abide by the following rules: Max 10% of AUM is allowed to be held as a cash position
Min 20% and Max 40% of AUM is allowed to be invested in the European region Min 40% and Max 60% of AUM is allowed to be invested in the North America region Min 10% and Max 30% of AUM is allowed to be invested in the other regions
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These rules will have no influence on whether a player is prone to the disposition effect or not. They will only force the player to diversify his portfolio, this will not have an influence on his selling behavior. 4.3 Data Selection
As mentioned, 3379 participants signed up for the game. The only requirement a participant had in order to register was that a he/she had to be older than eighteen years. Many participants that registered for the game did not start playing the game or did not follow the rules. In the real world of investing where real money is at stake, in general an investor will always care about his/her portfolio. Because the game lacked any true up- or downside risk, many registered players did not take the game and their portfolio seriously. Including data from these so-called junk portfolios would have biased the results. Therefore I only used data from portfolios of players that followed the rules and actively participated in the game. In order to obtain this information, I filtered out all junk portfolios and all players that were disqualified during the game were removed, hence I only used portfolios of players that followed the rules of the game. After filtering out all junk portfolios, 640 of the 3379 participants remained in the dataset. Moreover, following Odean (1998), only active accounts (accounts with at least one transaction) were used in the final dataset of this paper. The dataset diminishes from 640 to 564 accounts when I exclude all inactive accounts. The final number of accounts used in this study is therefore 564.
I have obtained the complete transaction history of all active traders participating in the investment game through 564 accounts during the whole game period. The trade data include the identity and investor characteristics of each trader (gender, age and nationality), the date of each transaction done and each position held, a stock identifier, order type (buy, sell, part sell or hold), transaction price, and the number of shares. The descriptive data statistics of the Fortis OBAM challenge are shown in table 4.1.
Table 4.1 Descriptive data of games 1 & 2
4.4 Investor characteristics
When subscribing for the game, each participant had to fill in his name, gender and age-group. The fourth investor characteristic is trading activity, which is split up into frequent and infrequent traders. The number of frequent and infrequent traders is obtained by using the methodology of Chen et al (2005). In their paper the top ten percent of all investors which exhibit the greatest turnover are the frequent traders,
Aggregated over all players (game 1) Aggregated over all players (game 2)
Number of accounts 564 55
Number of daily positions 3,488,953 395,583
Number of stock names 2,001 1,094
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the other ninety percent are the infrequent traders. With this additional information we have opportunity to see whether certain investor characteristics interfere with the detection of the disposition effect. The investor characteristics are subdivided in table 4.2.
Table 4.2 Demographic characteristics of all players in game 1
4.5 Alternative challenge
Over the same game period a separate challenge was hosted for a group of people that worked for a major Dutch financial institution. This challenge was completely aligned with the rules and requirements of the Fortis OBAM investment challenge, except for the fact that their game period was a week shorter. The data from this additional group only contains trading information, it does not provide any investor information. In addition to the research on the Fortis OBAM challenge I will also investigate whether this group of financial advisers is anxious to sell winners, but reluctant to sell losers. Because the subjects in this additional dataset are active as professionals in the financial markets, I assume there will be a difference in investment behavior between this group and the participants of the Fortis OBAM challenge. This assumption is based on findings of previous papers that studied the disposition effect. An ever-recurring finding in the research on the disposition effect is that experience and sophistication eliminate the behavioral bias of the disposition effect. Because the participants of the additional game are all active in financial markets I assume that they have experience and are sophisticated. Therefore I expect to find a difference in results between game 1 and 2. The descriptive data of game 1 and 2 is shown in table 4.1. 4.6 Market environment
Krause, Wei & Young (2009) and Barber et al (2007) both mention that the market environment could potentially influence the investment behavior of investors, consequently influencing the detection of the disposition effect. Therefore it is necessary to analyze the market environment during the game period. Because the players could trade in equity markets throughout the world, I use the MSCI world index from the 1st of April 2008 until the 30th of September as reference. The choice of using the MSCI world index is supported by the fact that the Fortis OBAM fund uses the MSCI world index as the benchmark for their fund performance.
Men Women 18-26 26-36 36+ N/A Dutch German Other Frequent Infrequent
Number of players 513 51 202 145 150 67 301 235 28 56 508
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Figure 4.1 MSCI World Index 2008
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4.7 Trading ActivityFigure 4.2 shows the aggregated trading activity. The trading activity is the sum of all buy/sell transactions aggregated over all selected players. In figure 4.2 the trading activity is split up into the six months that the game was played in.
Figure 4.2 Aggregated trading activity of all game 1 players
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5.
M
ETHODOLOGYIn the real world of investing, investors are subject to the disposition effect. In this paper I want to investigate whether investors of an investment game are also prone to the behavioural phenomenon of selling winners too soon and holding losers too long. In addition this paper will study whether there are differences between certain demographic groups in connection the disposition effect. The null hypothesis is that an investor is just as likely to realize a loss as a gain. In order to test the hypotheses I will use the same methodology that Barber, Lee, Liu & Odean use in their 2007 paper. The methodology of Barber, Lee, Liu & Odean 2007 paper is based on the methodology of Odean‟s 1998 paper with a few important adjustments.
5.1 PGR-PLR Method
As in Odean (2007) I analyze each investor‟s trading record in chronological order. I construct a portfolio of individual stocks for which the purchase date and prices are known. The counting of the paper gains and losses begins on the day following of the purchase of the security. Each stock that was in that portfolio at the beginning of the day but was not sold is considered to be a paper (unrealized) gain or loss. I compare the stock‟s daily closing price to the average purchase price of the stock and categorize unrealized positions as gains or losses. For stocks sold, the selling price for the stock is then compared to its average purchase price to determine whether that the stock was sold for a gain or a loss. The sale is realized as a gain if the sales price exceeds the average purchase price, and as a realized loss if the sales price is less than the average purchase price.
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𝑃𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛 𝑜𝑓 𝑔𝑎𝑖𝑛𝑠 𝑟𝑒𝑎𝑙𝑖𝑧𝑒𝑑 𝑃𝐺𝑅 = 𝑅𝑒𝑎𝑙𝑖𝑧𝑒𝑑 𝑔𝑎𝑖𝑛𝑠
𝑅𝑒𝑎𝑙𝑖𝑧𝑒𝑑 𝑔𝑎𝑖𝑛𝑠 + 𝑃𝑎𝑝𝑒𝑟 𝑔𝑎𝑖𝑛𝑠
𝑃𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛 𝑜𝑓 𝑙𝑜𝑠𝑠𝑒𝑠 𝑟𝑒𝑎𝑙𝑖𝑧𝑒𝑑 𝑃𝐿𝑅 = 𝑅𝑒𝑎𝑙𝑖𝑧𝑒𝑑 𝑙𝑜𝑠𝑠𝑒𝑠
𝑅𝑒𝑎𝑙𝑖𝑧𝑒𝑑 𝑙𝑜𝑠𝑠𝑒𝑠 + 𝑃𝑎𝑝𝑒𝑟 𝑙𝑜𝑠𝑠𝑒𝑠
Following Barber, Lee, Liu & Odean, in order to ascertain statistical significance I will calculate the mean difference between the proportion of gains realized and the proportion of losses realized that are the outcome of aggregating and averaging the PGR-PLR among all investors. A significant difference between the proportion of gains realized and the proportion of losses realized indicates that an investor is reluctant to realize losses and is anxious to sell winners.
In his 1998 paper, Odean aggregates all gains and losses of all investors and calculates one PGR and one PLR. Barber, Lee, Liu & Odean (2007) recognizes that when aggregating all observations of all investors, the results might be biased due to independence issues. Barber et al state that this lack of independence between the observations will inflate the test statistics, though it will not bias the observed proportions. Instead of aggregating all observations into one PGR-PGL, Barber et al perform an additional test. They calculate the difference between PGR and the PLR for each account, and then all PGR-PLR are averaged among all accounts. They assume that the proportions of gains and losses realized in each account are independent of those realized in other accounts. Therefore cancelling out the independence issues. 5.2 Cross-sectional research
Though it is the most popular method to quantify the disposition effect, the PGR-PLR method is criticized by other papers. Feng & Seafoles (2005) reveal some potential drawbacks when using the PGR-PLR method of Odean (1998). When calculating the disposition effect on average for all investors, the measures of Odean seem to work well. The negative aspect of this method however is that it is problematic for cross-sectional research, as the method does not work well when performing research at individual account level. As mentioned above, in their 2007 paper Odean et al acknowledge that the method that Odean uses in his 1998 paper has issues when trying to research at cross-sectional level. Therefore they adjust the methodology of Odean 1998. Using the adjusted methodology of Odean (2007) it is possible to make direct comparisons of PGR and PLR across investors holding differing portfolio sizes.
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difference across investors for the entire investor group and for particular groups of investors (males, females, certain nationalities and certain age groups). Similar to Odean, statistical significance is based on the mean difference and the cross-sectional standard deviation of the difference.
As a robustness check I will perform a multivariate regression. A number of studies have used multivariate analysis in order to find cross-sectional differences between investors regarding the disposition effect. Grinblatt & Keloharju (2001) use a logit regression and Feng & Seasholes (2005) use the survival method (based on the method of Grinblatt & Keloharju). According to Feng & Seasholes (2005) both the logit methodology and survival analysis allow the econometrician to test for the disposition effect while at the same time controlling for factors that might be correlated with the propensity to trade. In this study I will test cross-sectional differences using the logit regression method of Grinblatt and Keloharju (2001).
5.3 Logit regression
As mentioned, the logit regression is a useful method to test the relationship between cross-sectional differences and the disposition effect. Unlike the PGR-PLR as dependent in a regression the dependent variable of the logit regression is not mechanically linked to the right hand side variables. The reason for this is that the logit regression uses a dummy variable instead of a ratio. The dependent dummy variable that is used by Grinblatt & Keloharju (2001) is equal to one if PGR is greater than PLR and zero if PGR is smaller of equal to zero (= 1 𝑖𝑓 𝑃𝐺𝑅 > 𝑃𝐿𝑅). The logit regression that will try to disentangle the multivariate effects of investor‟s personal characteristics on the disposition effect is presented below:
Dependent Variable = 𝛼 + 𝛽1 Investor Age + 𝛽2 Nationality Dummy +𝛽3 Gender Dummy + 𝛽4 Frequent Trading Dummy
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5.4 Concluding methodological remarks26
6. D
ATAI
MPACTBefore I start describing the results it is important to understand that the dataset used in this study is unique in comparison with earlier research on the disposition effect. The data I use in this paper can be seen as data from an experimental game with real world characteristics.
By using literature from
studies that explain the difference between the real world and the experimental world, I will
illustrate that certain data characteristics of this paper can influence the results of this research.
Possibly making it difficult to extrapolate the findings of this paper to the field.
6.1 Experimental vs. real world research
The reason laboratory experiments are carried out is that the interpretive data that are gained from these experiments give insights that can be extrapolated to the real world. Is it possible to generalize findings across all studies? Levitt & List (2006) explain that the basic strategy underlying laboratory experiments in the physical sciences and in economics is similar, but the fact that humans are the object of study in the latter raises special questions about the ability to extrapolate experimental findings beyond the lab, questions that do not arise in the physical sciences. Due to the way the human brain functions, certain factors may have an influence on human behavior and therefore outcomes between the lab en and the real world may systematically vary.
Based on decades of research in psychology and recent findings from experimental economics, Levitt & List (2007) reveal 5 factors that influence human behavior in the lab. According to Levitt & List these factors only hold if moral and wealth-maximizing actions are not competing objectives. There must be no inherent conflict between the moral choice and the wealth-maximizing choice within the game.
1) non-random selection of participants 2) group differences
3) the role of experience 4) stakes and cognitive costs
5) short-run effects versus long-run effects
These five issues may potentially distort the experimental results, and therefore make it difficult to extrapolate the findings to the real world.
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investment behavior does not influence real market prices. The game is not set up as an experiment but can be compared to an experiment. There are a variety of factors that may influence the investment game. Therefore the behavior of an investor, in this game (or lab), may systematically differ from the behavior of a real investor. I believe that there are no competing objectives between moral and wealth maximizing actions. There are no moral issues that will restrain a player to not do his best in order to win the investment game. In the next part I will address each of the five factors of Levitt & List in relation to the investment game. I will explain which potential influence these factors can have on the behavior of the players. Because the dataset in this paper is unique, it will be necessary to adjust and customize some of the factors. Contrary to real world investors, the investors that participate in the investment game (or lab experiment) might not be prone to the disposition effect due to the influence of the factors described by Levitt & List.
6.2 Non-random selection of participants
When one tries to imitate the real world it is important that the actors in the experimental world are systematically similar to those of the real world. According to Levitt & List the attempts to directly generalize experimental results otherwise might be frustrated. For the experiment to be optimal, subjects that participate in the experiment should have rules of behavior that they have learned in the outside world. The bias that might occur in an experiment vs. the real world is called the selection-effect.
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6.3 Heterogeneity among investorsA lot of research has been done in order to find out whether all individual investors behave in the same way. According to psychologists, cognitive biases differ among different groups of people. The demographic issues of gender, age and nationality could have an impact on the results of this research. Over the years experimental and field research has been done on the influence of investor characteristics with regard to investor behavior. Both experimental and field research yield some interesting results. The disposition effect is documented as global phenomenon as the human being is prone to the disposition effect regardless of its nationality. Therefore I believe that it will not influence the experimental results of this research. Research on the disposition effect concerning the determinants age and gender is not as straightforward in its results. Because of the limited amount of research, it is unclear whether age influences the investment behavior of an investor. Earlier research has found evidence that gender has an impact on the general behavior of an investor. Considering the disposition effect, however, research does not find substantial differences in behavior between males and females.
Demographic differences between the real world and the experimental world can bias test results. In this case one can not extrapolate the results found in an experiment to the real world. Because of the limited amount of research on investor characteristics, it is unclear whether differences in investor characteristics between the OBAM investment game (experiment) and the field (due to selection effects) could bias the results. In this paper I perform cross-sectional tests, using nationality, age and gender as determinants. The tests in this paper reveal that investor characteristics influence the results (section 7). In addition the cross-sectional tests will make this research more robust.
6.4 The role of experience
Everyone is familiar with the phrase “Once bitten, twice shy.” This means that it is important to acknowledge mistakes that you make and learn from them in order not to make the same mistake again. Levitt & List state that due to the lack of experience and expertise the participants might not provide accurate guidance as to how decision making occurs in the field. One of the most important facets of experience is that one becomes familiar with certain situations, and therefore knows how to react in the correct manner once ending up in the same situation. Due to experience, rules of thumb and heuristics are developed. Numerous papers 4 have researched whether investor experience can eliminate the disposition effect. The findings of these papers show that investor experience can in a game diminish or even eliminate the behavioral bias.
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Experience thus influences the decision making process. Therefore if the level of experience of the experimental subjects differs to that of field subjects the results could be biased. The problem with this factor is that the level of experience is difficult to quantify because we have little background information of the subjects. This makes it difficult to measure whether the level of financial experience of the players influences the results. Because I assume that the players of game 2 are experienced, I will compare the results of game 1 and game 2. This might give some insight to the role of experience.
6.5 Stakes and cognitive costs
Theory suggests that the likelihood of the wealth-maximizing choice being made is an increasing function of the gap between the stakes of the game and the costs of effort (Levitt & List, 2007). Past research has confirmed the cognitive-effort theory. Levitt and List believe that a difference in stakes between the field and the experimental world could possibly lead to a difference in effort between investors (field) and subjects (experiment). This might ultimately make it impossible to extrapolate the experimental findings to the field.
In the OBAM investment game there is no a clear incentive or reward structure built into the game. In the real world of investing in general there are two types of investors, the institutional investor and the private investor. Both investor types are driven by a monetary incentive. In addition the institutional investor has carrier risk as an incentive to perform well. The incentives and stakes for the real world investor are clear. The only incentive a game player has, is that the winner of the OBAM game is rewarded with a holiday and a master class on investing. All other investors, regardless of their rank or performance don‟t win anything. More important is the fact that an investor participating in the OBAM investment game has nothing at stake. A participant of the game has no downside and little upside. This can influence the amount of effort a player puts into the game and also the risk behavior of a player. Because the investor in the investment game has nothing to lose, his perception of risk is totally different in comparison to the real world investor who does have downside risk. Both the risk and effort issue influence the wealth-maximizing choice of the investment game player. The question arises whether this will influence the investment behavior of a player. By using trading data of a OBAM game player, at the end of this section I will show that a combination of the factors stakes and cognitive effort and the length of the game, influences the results of this research.
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By short- and long-run effects Levitt & List (2007) mean the time period in which people have to make a decision. Psychology literature finds that the decision making behavior of a human being is different when a decision has to be made in the short-run (hot) compared to decision making in the long-run (cold). When a human being has to make a decision in the short-run he is led by emotion and visceral effects. When he has more time to make a decision it is easier to suppress the emotions he feels in the hot-state and rationalize more. Because the investment game simulates the real world of investment perfectly for six months, I believe that the hot/cold approach will be similar for the investment game and the real world. I do believe that the length of the game can have an other influence on the behavior of an investor. Because of the unique dataset of this paper I will adjust the factor of Levitt & List. Therefore from now on I will talk about the impact of the length of the game instead of short-run vs. long-run effects.
In contrast with the typical experiment and the real world of investing, the game period in this study is unique. The game length is too long to be compared with an experiment and too short to be compared with the real world. In both the experimental and field environment previous papers have found that investors are prone to the disposition effect. In both cases the subject‟s level of „effort‟ is not an issue. In the experimental environment the participation period is short and subjects can effortlessly participate during the entire period. In the field, the length of the investment period for investors is indefinable, stakes drive an investor to actively participate in financial markets. Therefore effort is not a questionable issue for real world investors. The time frame of the OBAM investment game is 6 months, this can potentially influence the behavior of the participants. While the number of days remaining until the finishing point of the game gets less and less, an investor might come to realize that he is not in contention for winning the game. This can lead to three different scenarios. Alternatively the investor maintains his game plan and keeps playing on, not changing his behavior. The investor stops putting effort into the game, thus stops making decisions, which may possibly bias the results. Or third, in a final attempt to make returns the investor might adjust his trading strategy. For example a trader might start selling stocks in order to be able to buy more volatile stocks, by changing his risk attitude he will hope to make excessive returns just before the trading deadline. This of course can influence the results of this research on the disposition effect. For the subjects in this study the combination of the six month game period and there only being a first prize at stake, raises an „effort‟ issue. In 5.7 I will show to what extent stakes and cognitive costs and the length of the game can influence the results.
6.7 The ‘Effort’ issue
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the following paragraph I will illustrate how „effort‟ has an influence on the results. To clarify the „effort‟ issue I must try to quantify effort.
In order to quantify „effort‟, I analyze the trading activity among all game players. As can be seen in figure 4.2, the trading activity declines every month by quite a substantial amount. The obvious explanation for this decline is that players stop believing that they can win the game. Because the participants of the game are anonymous, giving up is not paired with the public humiliation of failure, making quitting easier. Assuming that players actually stop being active during the game, this can bias the results considerably. When players decide to throw in the towel, they do not sell their portfolio like real world investors would do, but they just stop playing and therefore implicitly still participate. When using the methodology of Barber et al (2007), unrealized gains and losses are counted, regardless of whether a player quits. These unrealized gains and losses are still counted and used in the final calculation to see whether participants of the investment game are prone to the disposition effect. The next example should shed more light on this important issue:
Consider the example of a player who participated in the OBAM investment game in figure 6.1. In the first three months the player was an active player, in the last three months of the game he did not execute one trade.
Table 6.1 Trading record of OBAM u004623
I assume that player OBAM u004623 quit playing the game, and did not halt trading in the last three months as an investment strategy. In order to illustrate the impact of a quitted player I will calculate the disposition effect (𝐷𝐸 = 𝑃𝐺𝑅 − 𝑃𝐿𝑅) after three months, at the moment the player quitted, and after six months, when the game finished. After three months the player realizes his losses 2.89 times faster than his gains but after six months the player realizes his gains 0.74 times faster than his losses. This means that after three months the investor is prone to a reverse disposition effect, while after six months he is prone to a disposition effect. The difference in results after three and six months can be explained by the fact that the quitted player is still unconsciously participating in the game. Although he is not trading anymore, the number of unrealized gains and losses are still being counted and this has a great influence on the PGR and the PLR ratios. In the last three months of the game the players experienced a strong bear market, turning gains into losses (table 5.1). The significant influence that quitting has on the PGR and
OBAMu004623 Apr May Jun Jul Aug Sep
Number of RG 16 45 37 0 0 0
Number of RL 26 37 24 0 0 0
Number of UG 195 415 245 137 121 71
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PLR ratios is strengthened by the bear market. If the player had quit and there would have been a strong bull market, then after six months the results would have shown a stronger reverse disposition effect would be the result. When player OBAM u004623 quit the game, the results were biased substantially. In the end this can lead to false interpretation of the results.
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7.
R
ESULTS7.1 Game 1
In Table 7.1, I present the total number of realized gains, realized losses, unrealized gains and unrealized losses for all players. Each field in panel A is summed across all players and over the whole six month game period. These values make it possible to calculate the proportion of gains realized (PGR) and the proportion of losses realized (PLR) in table 7.1, panel B. First, consider the results of the players of the Fortis OBAM challenge (game 1). In panel B it can be seen that PGR is realized at a daily rate of 1.7 percent, while losses are realized at a daily rate of 1.35 percent. In aggregate, the difference between PGR and PLR is 0.36 percent. PGR and PLR differ at a 1 percent significance level, giving an indication that the aggregate investor is reluctant to realize losses. However, according to Odean (1998) calculating the disposition effect by aggregating investors brings heterogeneity issues. Therefore in order to formally test the disposition effect among the players, we must separately calculate the PGR and PLR per player and then average these ratios across investors. These results are presented in panel C of table 7.1. For the average player, the proportion of gains realized is 1.86 percent, with on the other hand 1.63 percent for losses realized. Therefore the difference between PGR and PLR is 0.23 percent, this difference is reliably positive at a 5 percent significance level. This means that on average, the players of the Fortis OBAM challenge seem prone to the disposition effect. Which confirms the results of panel B.
Table 7.1 Aggregated and mean PGR & PLR for all game 1 & 2 players after six months
Significant at a ten (*), five (**) and one (***) percent significance level
Players game 1 Players game 2
Number of Investors 564 55
Panel A: Total number of RG, RL, UG and UL
Realized gains 21,597 1,465
Realized losses 26,250 1,975
Unrealized gains 1,245,126 134,846
Unrealized losses 1,924,975 204,019
Panel B: PGR and PLR based on aggregated values of RG, RL, UG and UL
Percentage gains realized 1.70% 1.07%
Percentage losses realized 1.35% 0.96%
PGR-PLR 0.36% 0.12%
t-statistic 25*** 3.29***
Panel C: Mean PGR and PLR across investors
Percentage gains realized 1.86% 1.10%
Percentage losses realized 1.63% 1.22%
PGR-PLR 0.23% -0.12%
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7.2 Game 2For this paper I also analyzed data from a game that was played by people that work for a large Dutch financial institution. The results of the players of this additional game can be found in the second column of table 7.1, under game 2. Panel B provides the aggregated values for PGR and PLR. With a PGR of 1.07 percent and a lower PLR of 0.96 percent it seems that the players with a financial background are reluctant to sell their losses, this is in line with previous literature on the disposition effect. Though when we separately calculate the PGR and PLR for each player and average the ratios across all players, thus calculating the disposition effect in the formal manner, the test yields different results. The results of this test can be found in Panel C of table 7.1 and show that there is no significant difference between PGR and PLR. This provides support that on average the players that participated in the additional game are not prone to the behavioral bias of selling their winners at a higher rate then their losers. The difference in result between panel B and C is most likely due to the fact that relative to the positions they hold, large investors tend to trade more actively than small investors.
7.3 Robustness test over time
In order to shed more light on the „effort‟ issue (mentioned in section 6) and to see whether the results are consistent and thus robust over time, a robustness test is performed. In prior papers on the disposition effect the dataset is split up into two periods and the PGR and PLR ratios are calculated over each period. There is however a huge difference between this paper and the other papers which study the disposition effect. In comparison to other papers which have multiple years of data I only have six months of data. The shorter data period could give some potential problems when trying to perform a robustness test. The literature overview in table 2.1 shows that the papers with the longest data period are the ones that perform a robustness test. This is due to the combination of the way the disposition effect is calculated and the manner the data is selected. Following Odean (1998), only active accounts (accounts with at least one transaction) are used in the final dataset of this paper. With the assumption of a minimum of one transaction per investor, the vast majority of investors will have a PGR that is in equal to the PLR. However when the dataset is split up for one of the two periods many traders who did not trade frequently will have PGR and PLR of zero, this does not represent their true investment behavior. Without the sale of a loss and the sale of a gain the ratios will be zero, hence PGR-PLR is zero. When a short time period is used to analyze the disposition effect, the less frequent traders will bias the results when I split up the dataset in order to perform a robustness check.
