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Investor Behaviour and Lottery Stocks
Grace Gong*
Macquarie Graduate School of Management
CMCRC
Danika Wright
Business School, The University of Sydney
Abstract
Two competing theories currently exist to predict investor behaviour conditioned on past investment
performance. Under prospect theory, investors are risk averse following gains and risk seeking following
losses. The ‘house-money’ effect, on the other hand, predicts investors are more risk-seeking following
gains. This study analyses investor behavior using brokerage data for Australian retail investors from 1
Feb 2010 to 28 Feb 2013. Specifically, we examine investment into and performance of lottery stocks as
well as risk-seeking conditioned on performance of existing investments. Consistent with past findings,
lottery stocks are shown to offer inferior returns and represent risk-seeking behaviour. At the portfolio
level, investment in lottery stocks results in significant underperformance. This result is not biased by
portfolio size or diversification, which has implications for behavioural finance research and portfolio
management. Our results indicate investors are more likely to invest in lottery stocks following past
portfolio gains, supporting the house-money effect. This result is robust over various holding periods and
alternative behavioural explanations as such as over-confidence.
Keywords: Lottery stocks, Behavioural finance, House money effect, Prospect theory
* Contact author: Grace Gong. Address for correspondence: CMCRC, Level 3, 55 Harrington Street, The Rocks, NSW, 2000. Email: [email protected]
I. Introduction and Literature Review
Rational, risk averse utility maximising behaviour underpins neoclassical finance theory, yet seemingly
countless market anomalies contradicting this expectation have been identified. Investor behavior is a
fundamental area of financial markets research, with a significant literature emerging in the past two
decades dedicated to examining this divergence between expected and observed behaviour. Decisions
involving financial risk affect all market participants, with direct impacts on market efficiency and, as a
consequence, portfolio optimisation, asset pricing, trading strategies, policy-making and regulation.
Risk averse investors are assumed to prefer lower risk investments to higher risk investments, all else
being equal. To take an investment that is risky, a risk averse investor requires a premium over a riskless
investment. As the risk of the investment or the degree of risk aversion of the investor increases, so too
does the required premium. Alternative risk preferences include risk-neutrality (an indifference to risk)
and risk-seeking (a preference for risk). In the neoclassical model, investors are assumed to be
unconditionally risk averse. However, the behavioural finance literature has presented models of investor
behavior in which risk preference is changing.
The prospect theory of Kahneman and Tversky (1979) predicts that investors are risk averse following
gains and risk seeking following losses. Specifically, the authors show that “a person who has not made
peace with his losses is likely to accept gambles that would be unacceptable to him otherwise” (1979:
287). According to prospect theory, investors will make riskier decisions after a loss on their existing
investment than after a gain.
In contrast to this, the house money effect predicts that past gains lead to subsequent risk seeking
behaviour. First proposed by Thaler and Johnson (1990), the house money effect is an example of mental
accounting, whereby individuals segregate “winnings” (that is, gains on a position) from the initial
investment position. These winning are then treated differently by the individual, and an individual who is
more disposed to the house money effect will take more risks with it.
While both of the above theories have their merits, the predictions they make on investor risk preference
conditioned on prior performance diverge. The aim of this study is to examine investor behavior
following past investment gains, and specifically whether this behaviour is consistent with the
expectations of prospect theory (that is, risk averse), or consistent with the house money effect (risk
seeking).
Brown et al (2006) provide empirical evidence in support of the house money effect. Using five years of
daily exchange data, the authors find that when existing gains and losses are measured by the same stock,
and risk seeking is represented by holding onto winning stocks, investors are more risk seeking after prior
gains. The evidence from Brown et al (2006) is strong; however, it does not distinguish between a change
in risk preference and a change in the degree of risk aversion. That is, the authors cannot identify whether
their results are the effect of a shift to risk seeking behavior, or are reflective of milder ‘less loss averse’
behavior.
Polkovnichenko (2005) observes, “investors not only want protection from risk but also want to have a
‘shot at riches’ ” (2005: 1469). As a result of this desire, investors either gamble in the stock market
(Kumar 2009), or “attempt to ‘get ahead’ by hoping to capture large but unlikely extreme gains, gains
which are only possible in a relatively undiversified portfolio” (Polkovnichenko, 2005:1469). In this
instance, risk seeking exists at the same time as loss aversion. This cognitive dissonance can be seen to
manifest as otherwise risk-averse individuals engaging in gambling behaviours.
The notion of gambling in investment decisions was tested in Kachelmeier and Shehata (1991). These
authors use experimental lottery sessions to examine risk preference and find evidence of significant risk
seeking for small probabilities in the presence of high monetary incentives. That is, payoffs with large,
positive (right-tail) skew.
Research into stock market gambling has recently received more attention. Barberis and Huang (2008)
observe that certain risk-seeking investors can synthetically create lottery-like positions in their portfolios
by taking large and undiversified positions in securities with positive skew, similar to the lottery-like
payoffs used by Kachelmeier and Shehata (1991).
Kumar (2009) formalises the general notion of ‘lottery like’ and defines lottery stocks as stocks with a
small probability of a high reward, but a negative expected payoff. Lottery stocks feature high variance
and positive skew. Specifically, lottery stocks are identified as the joint set of stocks that have high
idiosyncratic volatility, high idiosyncratic positive skew and low stock price. He further defines investors
as holding either ‘lottery preferred’ accounts or non-lottery preferred using the weighting of lottery stocks
in the portfolio. Kumar (2009) finds that lottery stocks underperform and investors who prefer lottery
stocks suffer from lower returns.
In a more recent study, Bali et al (2011) define lottery stocks based on the maximum daily return of each
security during the previous month. Those stocks with extreme returns in the highest decile are classified
as lottery stocks for the given month. Bali et al (2011) show that stocks ‘maxed out’ this way
underperform the broad market and non-lottery stocks.
The Bali et al (2011) extreme return criteria, albeit relatively simple, is not inconsistent with the approach
investors are found to take in making investment decisions in existing studies. Odean (1998) find that the
market under reacts to highly relevant and reliable information when it is abstract or statistic, and over
reacts to information that is extreme and salient. Maximum stock return is information that is salient and
has extreme nature. Grinblatt and Keloharju (2001) find further empirical evidence that investors rely on
relatively simple trading rules, showing that trading activity is affected following monthly high or low
records. Furthermore, Barber and Odean (2008) show that stocks that are considered “attention-grabbing”
by exhibiting extreme daily returns are attractive to individual investors.
In this study, we use lottery-stocks as a signal of risk-seeking investor behaviour. Using a large sample of
Australian retail investors portfolio and trading data, we are able to identify investors who are more risk
seeking than others. We compare returns on lottery stocks and the portfolios of investors with different
risk preferences. The two competing theories of investor behaviour following gains (prospect theory and
house money) are then examined.
This study adds to the body of literature that has previously studied risk preferences at an individual level.
Much of the prior literature in this space has focused on the endogeneity of an investor’s risk preference
and their characteristics. These investor characteristics include demographic features such as gender
(Sunden and Surette, 1998; Croson and Gneezy, 2009), age (Hallahan et al 2004), genes (Slutske et al,
2000; Zhong et al, 2009), ethnic background (Yao et al, 2005; Fan and Xiao, 2006), income and education
(Shaw, 1996; Grabal, 2000), marital status (Halek and Eisenhauer 2001), and religious beliefs (Kumar et
al, 2011). Other studies have linked exogenous conditions to risk preferences. For example, Kumar (2009)
links macroeconomic changes to investor risk attitudes, showing that gambling-like activity in the stock
market increases during economic downturns.
Consistent with the results in Kumar (2009) and Bali et al (2011), we find that lottery stocks
underperform and that the portfolios of investors who hold a larger portion of lottery stocks also
underperform. We find that risk preference is not consistent. Rather, investors exhibit significantly more
risk seeking behaviours following portfolio gains. This indicates support for the house money effect over
prospect theory among retail investors. This has significant implications for behavioural finance research,
and applications of finance theory in portfolio management and asset pricing.
The remainder of the paper is structured as follows. Section 2 discusses the methodology by which we
identify lottery stocks, risk preference, and ex-post investor behaviour following portfolio gains. Section 3
discusses the data available to this study and presents descriptive statistics for stocks identified as lottery
stocks and non-lottery stocks. Section 4 reports the results and section 5 discusses these findings and
concludes.
II. Methodology
2.1 Lottery Stocks
Lottery stocks are identified using three methods. A replication of the Kumar (2009) and Bali et al (2011)
methods, and then an extension to the Bali et al (2011) method which relaxes the original assumptions of
investor trading patterns.
2.1.1 Kumar Method: Lottery-like payoff
Following, Kumar (2009), lottery stocks are intially defined as exhibiting high idiosyncratic volatility,
high idiosyncratic skewness and low absolute price. Idiosyncratic volatility is obtained as the variance of
residual by fitting a four-factor (risk premium, SMB, HML, momentum) model using daily data. The four
factors are calculated per standard Fama-french method using end-of-day stock price, stock market cap,
stock book value per share and daily All Ordinaries index price. Idiosyncratic skewness is obtained as the
third moment of the standardized residual obtained by fitting a two-factor (market excess returns and the
squared excess market returns) model to the daily stock returns time series.
Each month all stocks are ranked independently by idiosyncratic volatility, idiosyncratic skewness and
average end of day stock price for the preceding 6 (-6, -1) months. Lottery stocks for each month are
defined as the stocks which jointly exhibit idiosyncratic volatility in the top 50%, idiosyncratic skewness
in the top 50%, and price is in the bottom 50% at the same time. Stocks which meet none of these criteria
– i.e., are jointly below the median idiosyncratic volatility and skewness and above the median average
price – are defined as ‘Non-lottery’, while stocks which meet some lottery criteria but not all are
classified as ‘Other’.
2.1.2 Bali et al Method: Extreme daily maximum return in previous calendar month
The second lottery stock definition used in this paper follows the method in Bali et al (2011). Under this
definition, lottery stocks are identified as extreme maximum return stocks. Specifically, stocks are ranked
by their maximum daily return (close-to-close) in the previous calendar month. Those stocks ranked in the
top decile are defined as lottery stocks under the Bali definition.
2.1.3 Bali et al Extension: Extreme daily maximum return over past rolling-month
The third lottery stock definition we consider in this paper extends the Bali et al method discussed above.
Underlying in part Bali et al’s extreme daily maximum definition is the availability heuristic – that recent
events stick in investors’ minds, and so recent extreme return stocks appear attractive to a lottery-seeking
investor. While intuitively appealing, this effect is not going to be consistent across the original approach
of Bali at al due to the use of past calendar months. That is, a lottery-stock could be identified one-day
after its extreme return (if the maximum was observed on the last day of the previous month) up to 60-
days after (if the maximum was observed on the first of the previous month).
This paper provides a more robust approach to identifying lottery stocks using extreme past returns using
a past month rolling window. For each trading day t, we find the maximum daily return for each stock
over the previous 20 trading days, and again rank stocks by their maximum return. Stocks ranking in the
top decile are defined as lottery stocks for day t.
2.2 Lottery Accounts
Those accounts which prefer to hold lottery stocks need to be identified. These ‘lottery-preferred’
accounts indicate a risk seeking preference. In our first approach, we adapt the Kumar (2009) method and
define lottery preference based on the lottery stocks holding weight. We improve on the Kumar (2009)
holding weight measure by using daily holding data instead of month-end holding data, capturing a more
precise holding weight and therefore more accurate lottery preference. Specifically, we get the weight of
an account’s lottery stocks for each day, lottery-stock preferred accounts are defined as those with an
average daily lottery stock holding weight during the sample period ranked in the top decile across all
accounts. The simple lottery stock weight score for an account i¸on day t is computed following
Equation 1:
∑ ∊
∑ 100% (1)
where is the set of lottery-type stocks defined by Kumar (2009) on day t, is the number of stocks in
the portfolio of investor i on day t, is the number of shares of stock j in the portfolio of investor i on
day t, and is the close price of stock j on day t.
We believe that it is necessary to take into consideration the portfolio size. This is because an investor
may happen to have a larger weight of lottery stocks simply because of her large portfolio size. Following
Kumar (2009), our second approach of defining lottery preferred accounts is based on size-adjusted
lottery stock holding. For each day, we get each account’s normalized holding weighted and expected
normalised lottery weight, with which we calculated each account’s size adjusted lottery weight.
Accounts whose average size-adjusted lottery stock holding weight during the entire sample period is
ranked in the top decile are defined as the lottery-prefer accounts. The size-adjusted lottery weight score
for account i on day t is given by Equation 2:
100% (2)
In the above equation, and are account i’s normalised and expected normalised lottery stock
holding on day t. and are given by Equations 3 and 4:
(3)
(4)
Where is as defined in Equation 1, is the minimum portfolio lottery-stock holding
weight across all accounts on day t, is the maximum portfolio lottery-stock holding weight of
all accounts on day t, is the total value of account i on day t calculated with holding volume and
the closing price of the stock., is the minimum portfolio size all accounts on day t, and
is the maximum portfolio size of all accounts on day t.
Both the simple and size-adjusted methods of defining lottery-prefer accounts give similar results in the
account performance examination.
2.3 Measuring Account Performance
To isolate the effect of preferring (or not preferring) lottery stocks on a portfolio’s performance, we
construct a hypothetical portfolio for each investor portfolio every day by replacing the non-lottery
component1 with the market portfolio. In other words, the hypothetical portfolios are a combination of a
well-diversified market portfolio and lottery-stocks, where the lottery stock weighting is equal to the
1 The Non-Lottery component of a portfolio is all stocks held in the portfolio which are not defined as lottery-stocks using the three methods outlined in Section 2.1.
observed lottery-stock weighting for each account. This method isolates the effect on portfolio
performance by investing/holding lottery stock at the given weight. Monthly four factor alphas are
calculated for all hypothetical accounts. These are then averaged to find the portfolio monthly alphas by
lottery preference classification. Any difference across accounts is attributable to the lottery stock
component, not the stock selection of the particular investor.
2.4 Presence of Another Bias, Portfolio Gain and Investor Risk Preference
Logit regression is estimated to investigate whether it is lottery-liking (risk-seeking behaviour) or over-
confidence that is driving the purchase of lottery stocks. The regression is given by Equation 5:
(5)
where takes the value of 1 when there is a purchase order of lottery stock on the day and 0 if not;
takes value of 1 if the account is a lottery preferred account, and 0 otherwise; stakes the value of 1
if an account is defined as an overconfident account, 0 otherwise.
In the regression above, the purchase order of a lottery stock signals a risk-seeking decision. LPA
represents an account (investor)’s overall risk preference. We apply two methods to identify an
overconfident account ( ). The first one adopts Kumar (2009) by defining an over-confident account
as one whose average monthly turnover is ranked in the top 10% and average risk adjusted monthly return
(four factor alpha) is ranked in the lowest 10%. The average monthly turnover is a parameter representing
an account’s confidence and the second criteria reflects the fact that the confidence that is not justified;
that is, the combination of these parameters represents overconfidence.
In calculating average monthly turnover, we adopt the method in Barber and Odean (2001). Barber and
Odean (2001) compute the turn over for any month as the sum of the half monthly purchase turnover and
half monthly sale turnover using the difference of two months’ holding as a proxy for the monthly trade
value. This study makes use of actual trade data to obtain a more precise estimate of turnover.
Specifically, we calculate each purchase turnover as purchase value divided by portfolio value after
purchase and each sale turnover as the sale value divided by portfolio value before sale.2 Using this
method, each account’s average monthly turnover during the sample period is obtained.
2 This gives us a purchase of an account with zero holding a turnover value of 1 instead of a missing value, and a sale of the whole position a turnover value of 1 instead of a missing value.
Turnover, however, may be a biased measure of trading frequency for lottery stocks given their typically
lower price than non-lottery stocks. If, for example, investors make trading decisions based on volume,
not value, then a lottery stock trade will be typically smaller in value and value-based turnover than a
comparable volume trade of non-lottery stocks,. Investors who trade a lot and focus on trading lottery
stocks will have a low turnover value. As a result, we also use a second method to identify an over-
confident account. While an account still need to be in the lowest 10% average monthly risk-adjusted
return group, the measure of trading frequency is defined as the average number of trades per month. In
this second measure, accounts whose average monthly trade numbers are ranked as the top 10 % are
considered as frequent trading account. Among these frequent trading accounts, those who also are ranked
as the lowest 10% in their risk adjusted performance are considered to be over-confident accounts.
Having obtained results from the regression above, which shows that it is risk-seeking, rather than over-
confidence that leads to a risk-seeking decision, we continue test whether the purchase of lottery stocks
(risk-seeking) is predicted by the accounts’ earlier performance. Given the two theories of risk-seeking
conditioned on prior outcome differ in the directions of the prior outcome, i.e., gains or losses, rather than
the actual amount, we design a logit regression when the variables are binary values. The logit model to
be estimated is given by Equation 6:
(6)
where takes the value of 1 when there is a purchase order of lottery stock on the day and 0 if not;
takes value of 1 if the account is having a positive return and 0 otherwise for 1 day before, 5 trading days
before and 20 trading days before respectively in 3 independent logistic regressions.
To further examine whether it is confidence gained from success of previous investment in lottery stocks,
rather than the changed risk preference, that is motivating the purchase of lottery stocks, we run the above
regressions again separating lottery-liking accounts and the other accounts.
So that there is a distinct difference between the risk preferences, we only keep purchase orders of lottery-
stocks and non-lottery stocks in the regression analysis. The same 9 regressions are done with all
purchases orders as a robustness test. These regression results are consistent with the reported results in
supporting house money effect, although not as strong in terms of statistical significance.
As another robustness test, we also use account abnormal returns relative to the market. The returns of All
Ordinaries Accumulation Index returns are used to represent the market performance. When account
abnormal returns are used, takes value of 1 if the account is having a positive abnormal return and 0
otherwise for 1 day before, 5 trading days before and 20 trading days before in the 9 regressions described
above. Once again, the results are similar to the results reported later in the paper providing support to
house money effect.
In this analysis, the submission of orders is used to represent the individual investors’ decision time for
trading rather than trades.3 This makes intuitive sense as we are seeking to determine the investor risk
preference at the time behaviour is recorded. It is especially the case for limit orders whose limit prices
are far away from market price and do not get executed within a short time from order submission, and
for thin-trading stocks. Given the sample investors are retail investors, cases where there are multiple
order submissions for the same stock on the same day are very rare. As such, even though the orders in
our dataset have time stamps, it is the day of the order submission that is used.
We further investigate the decision to trade lottery stocks by separate limit orders and market orders,
which reflects the investors’ determination to trade lottery stocks. When only market orders are used for
the above regression, we still have similar results.
III. Data
This study utilises a large and comprehensive database of retail investor daily holding and account
information, intra-day buy and sell orders and executed trades for the sample period 1 February 2010 to
28 February 2013. These data are sourced several databases collated by a major Australian retail
brokerage house, and linked using a unique investor account identifier which is present in all files.
Specifically, a daily portfolio holdings file contains date, security code (stock ticker used on Australian
Securities Exchange), and the size of the security held in the portfolio. The account information file
records the account holder’s date of birth, gender and address. The equity order file contains date and
time-stamped order submission details including size of the order, limit price in the case of a limit order,
and the portion of the order which is filled upon submission. Finally, the trade execution file holds
security code, trade price and date records for each account.
For this study, it is necessary to know stock price and company information, as well as market risk-free
benchmarks and market portfolio returns. As such, we supplement our trading and portfolio data with the
following. Daily stock prices and information on stock splits and dividends (size and date) are sourced
from Thompson Reuters Tick History (TRTH). Daily stock market-capitalisation and book value per
share are obtained from Bloomberg. The rate on the 90 day BAB rate is assumed to represent the risk-free
3 Lee and Radhakrishna (2000).
interest rates and are obtained from the RBA website,4 while the All Ordinaries Accumulation Index
proxies for the market portfolio and daily adjusted prices are obtained from Yahoo Finance.5
The following data filtering procedures are taken. Stock daily prices are dividend and split adjusted. Daily
returns that are greater than 100% or lower than -50% are removed. On investigation, these outliers are
found to be mainly caused by data error (such as inconsistencies in dollar or cent reporting between the
data sources). These observations account for less than 1% of the total sample. Only accounts that have
had stock holding for more than 30 days are considered. Finally, accounts with a total value less than
$100 has the day portfolio value to be set missing in the procedure to decide the minimum portfolio size
on market for portfolio size standardizing.
Table 1 reports the statistics of stocks by their lottery categories. Whereas Kumar (2009) is the only
method out of the three methods that defines lottery stocks by the stock’s characteristics, lottery stocks
defined by all methods share similar features.
<Insert Table 1>
Lottery stocks exhibit a low price, low book to market ratio and low market capitalization. Lottery stocks
are traded less frequently when we measure the frequency by number of days traded per month. However,
on the days they trade, their trading volume is higher than other stocks.
IV. Results
4.1 Lottery stock performance
Tables 2, 3 and 4 report the performance of stocks by lottery categories defined using three different
methods using the average monthly four-factor alpha over the sample period. Under all three definitions,
lottery stocks underperform.
<Insert Table 2>
Table 2 presents the results from the Kumar lottery definition. Using this rule, lottery stocks
underperform by 0.28% per month and 0.75% per month when compared to other stocks and non-lottery
4 Reserve Bank of Australia, Statistical Tables, Interest Rates and Yields – Money Market – Daily, accessed 17 March 2013, http://www.rba.gov.au/statistics/tables/index.html#interest_rates 5 Yahoo Finance, Historical Prices, accessed 17 March, 2013, http://au.finance.yahoo.com/q/hp?s=^AORD
stocks respectively. The difference between lottery stocks and non-lottery stocks is 9% per year, which is
significant not only statistically, but also economically.
<Insert Table 3>
<Insert Table 4>
Table 3 reports the performance of lottery stocks using the Bali et al (2011) definition, and Table 4
reports the performance of lottery stocks using this study’s rolling extreme observation method. There is a
very distinct pattern: the more lottery-like the stocks are, the worse returns. The difference between the
most lottery-like stocks and the second least lottery-like stocks using the Bali et al (2011) definition is as
much as 0.42% per month, or 5.04% annually. This again is an economically significant figure.
The results in Table 4 from the rolling extreme observation definition are not as strong. However,
whereas the most lottery-like stocks have a monthly return that is not significantly different from zero, 7
out of the 9 other groups have a monthly return above zero, and the generally pattern shows that more
lottery-like stocks have worse performance. The stock performance results under all three definitions are
consistent with existing findings that lottery stocks underperform.
4.2 Lottery preferred accounts
Table 5 reports the performance of retail investor accounts by their lottery preference category. Lottery-
preferred accounts are defined as accounts whose average lottery stock holding weight during the sample
period is in the top decile. When we define lottery–liking accounts by ranking of non-size-adjusted lottery
stock holding weight, lottery-liking accounts in our sample underperform other accounts by 0.28% per
month or 3.36% per year. When we define lottery –preferred accounts by ranking of size-adjusted lottery
stock holding weight, lottery-liking accounts underperform other accounts by 1.22% annually. The
differences are both statistically and economically significant.
<Insert Table 5>
In addition to measuring an account’s risk-seeking (lottery-liking) preference, we also investigate what is
the reason behind an account’s risk-seeking behavior. Table 6 shows the results from our logit regressions
designed to examine whether it is the lottery-liking (risk-seeking) feature of an account, or the
overconfidence feature of an account that influences the decision to buy a lottery stock. Panel A shows
the results when average monthly turnovers are used to identify over-confident accounts. The negative
intercept suggests that overall investors are not likely to purchase lottery stocks. However, an account’s
lottery-liking feature predicts are higher likelihood of purchasing lottery stocks as indicated by the
significantly positive parameter estimate. An over-confident account, however, appears to be less likely,
although only to a very small extent, to purchase account. Panel B shows the results when average
monthly trade numbers are used to identify over-confident accounts, which are consistent with the results
in Panel A. Results in Panel B rule out the concern that the method used to define over-confident accounts
may lead to mechanically created results. Whereas the findings on the relationship between over-
confidence and gambling in the stock market is different from what we have expected and different from
existing literature (Kumar, 2009; Barber and Odean, 2001), they nevertheless support the conclusion that
the effect of lottery-liking, or risk-seeking, predominates in predicting a risk-seeking decision when both
bias are in presence.
<Insert Table 6>
Our last examination is to test the existing theories that predict risk seeking behavior based on prior
outcomes in the investment context. Under the house money effect it is expected that when investors
make money, they become more risk seeking. Evidence in support of the house money effect will come
from an observation of favorable account performance leading to the purchasing of lottery stocks.
Table 7 reports the logit regression results designed to test this hypothesis when account actual returns are
used. Panel A shows the results when the regressions are done across all investors. The dependent
variable takes the value of 1 when there is a submission of an order to buy a lottery stock on the day, 0
otherwise. The explanatory binary variables in the three regressions are the account performance 20
trading days before, 5 trading days before and 1 trading day before respectively. The explanatory variable
takes the value of 1 if the corresponding periodical performance is positive, and 0 if not.
These results support the conjecture that while generally investors are risk averse, as shown by the
negative intercept value, a positive account performance predicts more risk seeking investment decision.
The variables are significant at the 0.01 level of significance. Results reported in Table 7 use both limit
and market purchase orders of lottery stocks and non-lottery stocks. Results from using purchase orders of
all stocks (which include lottery stocks, non-lottery stocks and other stocks) are consistent with the results
reported. 6 We have considered that the negative intercept, which is very robust, may have been caused by
the mere fact that there are fewer lottery stocks. Further investigations rule out this conjecture. Of all the
6 Furthermore. the results are nominally the same when using abnormal returns and only market orders for the regressions.
stocks in the sample, lottery stocks and non-lottery stocks are about 22% and 23% respectively, with the
remaining being ‘other stocks’ category which takes up about 50% in terms of numbers of stocks. When
only lottery stock and non-lottery stock purchase orders are examined, where there are about the same
number of lottery stocks made as non-lottery stocks, the purchase orders of lottery stocks are only 25% of
all purchases orders. When all purchase orders are examined, the purchase orders of lottery stocks are
only about 10% of all purchase orders, although the number of lottery stocks is nearly one quarter of all
stocks in that sample.
<Insert Table 7>
Panel B in Table 7 follows the same procedure as used in Panel A, however we separate ‘lottery-liking’
investors and investors who do not prefer lottery stocks. The results are in line with a priori expectations.
Lottery-prefer accounts have a ‘built-in’ tendency to buy lottery stocks. In addition to a positive intercept
estimate for this group, we find there are more lottery stock purchase orders for this risk-seeking investor
group. This tendency, which is a significant effect in the unconditional estimates, becomes even stronger
when the prior outcome is ‘winning money.’ This finding supports the house money effect, as investors
are increasing their risk-seeking following prior gains. The results are statistically significant at 0.1%
level over all periods during which prior performance is measured.
This house-money effect is not confined to the lottery-preferring investors. Investors who do not have
lottery preference are, by definition, less inclined to buy lottery-stocks. Regressions for this group have
significantly negative intercept values and there are much fewer lottery stock purchase orders than non-
lottery stocks,. However, our results show that positive past performance increases the likelihood of
lottery stock purchases for this group of risk-averse investors as well. In fact, the greater values of
estimates for non lottery-prefer accounts suggests that for this group of risk-averse investors, house
money effect plays a big role than it does for lottery-prefer investors.
These regression results provide strong evidence for the house money effect, unconditional on investor
risk preference. By showing an increased likelihood of lottery stock purchase following gains in both
lottery-preferring and non-lottery preferring accounts, this finding can not be attributed to overconfidence
or the specific past performance of lottery stocks.
V. Conclusion
This study provides strong evidence that it is the heuristic of risk-seeking itself, rather than over-
confidence that drives the investment into lottery-like securities. The findings presented in this paper
indicate that, regardless of an investor’s prior degree of risk-aversion, past investment outcomes affect the
investor’s next investment decision. This contradicts the assumptions of rational investor behaviour that
form the neoclassical finance framework, but is consistent with the predictions of behavioural finance
theories.
Two competing theories attempt to predict the ex-post decisions of investors following past investment
gains. Prospect theory (Kahneman and Tversky, 1979) shows that investors are risk averse following
gains, and risk-seeking following losses. Under the house money effect, on the other hand, investors are
expected to act in a risk-seeking way following gains (Thaler and Johnson, 1990).
The results presented in this paper indicate that investors are more likely to make risk-seeking decisions
following past investment gains, supporting the house money effect. These findings have significant
implications for behavioural finance research and our understanding of market efficiency.
Investor risk preferences, particularly risk-seeking investment decisions, and the impact and cause of such
behaviour are examined in this study. Using a large proprietary database of portfolio holdings, order
submissions and trade information as well as investor characteristics, this study is able to identify
investors with a risk-seeking preference by their investment in lottery stocks.
Lottery stocks have characteristics similar to a common speculative lottery, such as high volatility and
skew, a low probability of a high positive return, and are cheap to purchase. This lottery-like payoff
would not be acceptable to a risk-averse investor, and instead is representative of risk-seeking behaviour.
We find that across a range of different definitions for lottery stocks, such assets significantly
underperform on a risk-adjusted basis. Furthermore, risk-seeking investors who tend to overweight their
portfolios to these lottery stocks suffer from inferior investment returns.
These findings have significant implications for future research in behavioural finance, portfolio
management and the operations of capital markets. It is shown that investors are more likely to increase
their risk-seeking following past gains, which contradicts in part the predictions of prospect theory.
Possible research questions could examine adjustments to prospect theory, or search for conditions under
which it holds. This study focuses on endogenous investor trading characteristics, though it would be of
interest to see if a wider set of investor traits (such as age and gender) or exogenous macro factors affect
the results.
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Table 1: Sample Statistics
This table reports statistics for stocks by their lottery categories that have been traded/held by individual investors from a large retail brokerage house during the sample period 1 February 2010 to 28 February 2013. Panel A presents statistics using the Kumar (2009) lottery stock definition. Panel B presents statistics using the Bali et al (2011) lottery stock definition. Panel C presents these statistics using the extension to the Bali et al (2011) method.
Daily Trade Volume Stock Price ($) Trade Value ($)Days Traded Per
Month
Book to Market
Ratio
Market
Capitalisation
($ million)
Non Lottery 1,619,161 6.82 8,514,134 19 2.9 3,745
Other 1,081,723 1.53 1,983,615 16 1.5 2,144
Lottery 1,539,917 0.08 91,005 15 0.11 103
Least Lottery 986,668 8.22 4,750,667 8 4.25 6,449
2 1,842,193 10.46 14,081,887 18 4.15 7,595
3 1,500,406 4.86 7,708,489 18 2.88 3,690
4 1,337,661 2.68 3,998,417 18 2.05 2,320
5 1,198,994 1.45 1,809,481 18 1.4 1,018
6 1,114,663 0.80 905,036 17 0.87 637
7 1,140,226 0.48 544,049 17 0.72 428
8 1,288,761 0.33 346,066 17 0.49 325
9 1,402,050 0.27 237,342 16 0.44 345
Most Lottery 2,252,276 0.22 307,070 15 0.37 210
Least Lottery 986,668 8.22 4,750,667 8 4.25 6,449
2 1,842,193 10.46 14,081,887 18 4.15 7,595
3 1,500,406 4.86 7,708,489 18 2.88 3,690
4 1,337,661 2.68 3,998,417 18 2.05 2,320
5 1,198,994 1.45 1,809,481 18 1.4 1,018
6 1,114,663 0.80 905,036 17 0.87 637
7 1,140,226 0.48 544,049 17 0.72 428
8 1,288,761 0.33 346,066 17 0.49 325
9 1,402,050 0.27 237,342 16 0.44 345
Most Lottery 2,252,276 0.22 307,070 15 0.37 210
Panel C: Rolling‐ Month Bali method
Panel B: Bali definition
Panel A: Kumar definition
Table 2: Stock Performance using Kumar Definition
This table shows sample stocks’ average monthly weighted four factor alphas in different category. Stocks in this sample include all the stocks that have been traded/held by individual investors from a large retail brokerage house during the sample period (1 February 2010 to 28 February 2013). Lottery stocks and non-lottery stocks are defined per Kumar (2009). Stocks are ranked by their idiosyncratic volatility, idiosyncratic sknewness and price in the previous month. Each of the three rankings is independent. Stocks in the joint set of highest 50% by ranking of idiosyncratic volatility, highest 50% by ranking of idiosyncratic skewness and lowest 50% by ranking of price are defined as lottery stocks. Stocks in the joint set of lowest 50% by ranking of idiosyncratic volatility, lowest 50% by ranking of idiosyncratic skewness and highest 50% by ranking of price are defined as non-lottery stocks. Stocks that are neither lottery stocks nor non-lottery stocks are classified as ‘other’. Lottery stocks underperform non-lottery stocks and other stocks during the sample period.
Stock Category Average Monthly Weighted Alpha - Stocks t-statistics
Non-Lottery Stocks 0.76% 25.05
Others 0.29% 12.97
Lottery Stocks 0.01% 9.14
Table 3: Stock Performance by Bali Definition
This table shows stocks’ monthly weighted four factor alpha by their lottery-like ranking. Stocks in this sample include all the stocks that have been traded/held by individual investors from a large retail brokerage house during the sample period (1 February 2010 to 28 February 2013). The ranking follows Bali (2011): each month stocks are ranked by the previous calendar month’s maximum daily return, the decile of stocks whose maximum daily returns in the previous month are highest is defined as the most lottery-like stocks. For nine out of the ten groups, except for the least lottery-like group, the more lottery-like the stocks are, the worse they perform.
Lottery-Like Ranking Average Monthly Weighted Alpha - Stocks t-statistics
Least 0.06% 5.68
1 0.43% 16.30
2 0.25% 15.03
3 0.16% 10.62
4 0.07% 10.53
5 0.04% 7.71
6 0.02% 3.74
7 0.02% 8.76
8 0.01% 5.23
Most 0.01% 3.82
Table 4: Stock Performance by Bali Extension Method
This table shows stocks’ average monthly value weighted return by their rolling lottery-like ranking. Stocks in this sample include all the stocks that have been traded/held by individual investors from a large retail brokerage house during the sample period (1 February 2010 to 28 February 2013). The ranking is an extension of Bali (2011): each day stocks are ranked by the previous 20 trading days’ maximum daily return, the decile of stocks whose maximum daily returns in the previous month are highest is defined as the most lottery-like stocks. There is a general pattern that the more lottery-like stocks have worse monthly returns.
Lottery-like Ranking Average Monthly Value Weighted Return - Stocks t-statistics Least 0.40% 2.82
1 -0.03% -0.17 2 0.35% 1.29 3 0.53% 4.29 4 0.43% 4.63
5 0.27% 4.27 6 0.36% 6.39
7 0.25% 7.08 8 0.27% 7.77
Most 0.14% 0.71
Table 5: Account Performance
This table reports the account performance difference between Non Lottery-liking accounts and Lottery-liking accounts. We define lottery-liking accounts as those whose (size-adjusted) average lottery stock holding weight during the sample period is ranked as the top decile. Statistical significance is denoted as * at
the 5% level, ** at the 1% level, and *** at the 0.1% level. The lottery weight for account i on day t is computed as: ∑ ∊
∑ 100%. The size-
adjusted lottery weight score SALW for account i on day t is given by 100%.
Average Monthly Four Factor Alpha
Test of Difference Between Means
Pooled t-test Sattherthwaite t
Panel A: Account type defined without size adjustment
Non Lottery-Liking Accounts 1.10% ***
Lottery-Liking Account 0.82% *** 9.32 *** 3.54 ***
Panel B: Account type defined with size adjustment
Non Lottery-Liking Accounts 1.08% ***
Lottery-Liking Account 0.97% *** 9.32 *** 3.54 ***
Table 6: Logit Regression – Risk Seeking or Over-confidence
This table reports the results of estimations the logit regression designed to test whether the purchase of lottery stocks (risk-seeking) is predicted by the accounts risk-seeking bias or by the accounts over-confidence bias: where takes the value of 1 when there is a purchase order of lottery stock on the day and 0 if not; takes value of 1 if the account is one that is define as lottery prefer account and 0 if not; takes the value of 1 if an account is an over-confident account and 0 if not. Panel A reports the parameter estimates for the regression when account average monthly turnover is used to define an over-confident account. Panel B reports the parameter estimates for the regression when account average monthly trade number is used to define an over-confident account. Lottery-like stocks and Lottery Prefer Accounts here are defined using Kumar (2009) definition. Statistical significance is denoted as * at the 5% level, ** at the 1% level, and *** at the 0.1% level.
Panel A: Defining Over-confident Account by Average Monthly Turnover
-1.883*** 2.686*** -0.167***
Panel B: Defining Over-confident Account by Average Monthly Trade Number
-1.885*** 2.683*** -0.055***
Table 7: Logit Regression: Prediction of Purchasing Lottery-stocks by Account Performance
This table reports the results of estimations the logit regression designed to test whether the purchase of lottery stocks (risk-seeking) is predicted by the accounts performance: where takes the value of 1 when there is a purchase order of lottery stock on the day and 0 if not; takes value of 1 if the account is having a positive return for 1 day before, 5 trading days before and 20 trading days before respectively in 3 independent logistic regressions. Stepwise procedure is used to only keep the explanatory variable with statistical significance greater than the 0.05 level. Panel B presents the results when the same regressions are run separating lottery-liking accounts from other accounts. Lottery-like stocks and Lottery Prefer Accounts here are defined using Kumar (2009) definition. In this analysis, only the purchase of lottery-stocks and non-lottery stocks are kept for regression. Statistical significance is denoted as * at the 5% level, ** at the 1% level, and *** at the 0.1% level.
Panel B: By Account Type
No of Trading Days Before Purchasing Lottery-stocks Account Types
20 Lottery_Liking 0.782*** 0.039***
Non Lottery-Liking -1.962*** 0.180***
5 Lottery_Liking 0.742*** 0.115***
Non Lottery-Liking -1.981*** 0.203***
1 Lottery_Liking 0.756*** 0.082***
Non Lottery-Liking -1.986*** 0.201***
Panel A: Across All Accounts
No of Trading Days Before Purchasing Lottery-stocks 20 -1.185*** 0.217***
5 -1.201*** 0.170***
1 -1.186*** 0.130***