Statistical Arbitrage Trading

The primary outcome measure was the amount of fictitious
Swedish kronor (fSEK) as scored on the distributed form. This
score was used to determine “survivability” and the number of
traders able to earn money trading over the long run.
A trader who, at some point during simulation, reached a total
capital of zero fSEK or less, that is, lost all of his/her money, was
defined as bankrupt and did not “survive” trading the markets.
Consistently, traders increasing total capital to an amount greater
than the initial 10,000 fSEK were defined as winning traders, able
to trade profitably over the long run. Remaining participants, who
lost money (decreasing total capital to less than 10,000 fSEK) but
not all of it, were defined as losing traders.
There was a lot to gain (a remuneration of 200 SEK) for the traders
meeting the profit objective on level 2, but there was not much to
loose even if they lost all but 1.00 fSEK of their trading capital. As
long as they had just a little fictitious capital left, they were entitled
to keep their 100 SEK remuneration. This could cause traders to
totally abandon their position-sizing strategy, when time was
running out, in order to meet the objective. To have a profit
objective that must be met within a tight timeframe and not really
need to take the consequences if wrong is not a realistic scenario.
In order to reflect a more realistic image of the trader’s positionsizing
strategies, focus was shifted from the outcome of the very
last trade to a trade earlier in the sequence. The trade in focus was
set to the forty-fifth trade at each trader’s final level. A number
close enough to the final trade, yet far enough away from risking to
be strongly influenced by a “make or brake” bet. All the
participants who lost their total capital did so considerably prior to
the forty-fifth trade. They have therefore, most probably, not
unjustly been defined as bankrupt.


Statistical Trading Analysis

The effect of position-sizing was studied during simulated stock
trading, by measuring the performance of participants recruited
from Uppsala University. Information about the study was given in
a few classes among economics, law, and psychology majors. On
public bulletin boards within the departments, notes were posted
and anyone interested could sign up during September/October
2000. No prior knowledge of trading/investing was required.
Participants were notified by phone how they would participate
and that the remuneration could be lost if performing poorly.
A total of 62 students were randomly assigned into three groups:
20 + 20 were assigned to the three-hour lecture on positionsizing,
and 22 were assigned to the control group. The participants
were asked to state their knowledge of trading/investing in the
stock markets as no knowledge, some knowledge or active
Of the 40 participants assigned a lecture, 3 did not show up at the
lecture and 5 were not able to participate in the simulations (e.g.,
because of illness or non-compliance). Two forms of the
participants in the control group were suspected to have been
tampered with, and these two results were therefore not included
in any computations. Totally, 10 individuals (16%) were dropouts,
and the remaining 52 participants, 18 female and 34 male students
(see Table 2), were ranging in age from 19-56 years (mean age =
24.9 years).
Table 2. Distribution of gender and prior knowledge of
trading/investing (N = 52)
Gender/Knowledge Treatment Control
Female 8 10
Male 24 10
Active Traders 5 4
Some knowledge 8 6
No knowledge 19 10
The participants were randomly assigned into three groups. Two
experimental groups and one group acting as a control. The
experimental groups were given a three-hour lecture at separate
occasions and, in order to minimize the effect of the lecturer, by
separate lecturers. The curriculum was the same for both groups.
Both lecturers used the same content, including position-sizing, risk
management, and psychological biases, according to Appendix 1.

Quant Strategies for Stocks

The trading and investing industry is growing bigger and bigger.
Not only in the U.S., but in Europe as well. More and more
companies are focusing on education of traders/investors. It is
probably not a question if, but when, this will unfold in Sweden.
The prospects from the companies offering training and education
are packed with promises of greater wealth and market success,
but few of the methods have been verified empirically. Trainingpackages,
addressing individuals, emphasizing “money
management”, sometimes called asset allocation or position-sizing,
have very little scientific support. However, there is evidence of
the importance of how assets are allocated. Asset allocation is even
more important than stock selection or timing. Brinson, Singer, and
Beebower (1991) found asset allocation policy to be of primary
importance, accounting for 91.5% of the differential return of the
pension funds.
The reason for conducting this study was to provide evidence for
the importance of position-sizing, that is, how much of one’s assets
that is allocated at each trade, on trading performance. With the
comment of “market wizard” Bruce Kovner (see p. 8) in fresh
memory, the following hypotheses are made:
(1) Participants going bankrupt (losing all their capital) will take
larger position sizes than those being able to maintain some or
all of their initial capital.
(2) Participants losing money, but not all of it, will take larger
position sizes than those being able to gain money over the
long run.
Further, is it possible to teach traders/investors not to lose all their
money and to make profits from trading the markets?
(3) Participants receiving lectures in position-sizing, risk
management, and psychological biases (treatment group) will
take smaller position sizes on an average than participants not
receiving such lectures (control group).
(4) Participants receiving lectures in position-sizing, risk
management, and psychological biases (treatment group) will
lose all their money to a less extent than participants not
receiving such lectures (control group).
(5) Participants receiving lectures in position-sizing, risk
management, and psychological biases (treatment group) will as
a group gain higher profits than participants not receiving such
lectures (control group).
Possible factors contributing to the way the participants decide on
position size, other than receiving a lecture, are gender and prior
knowledge of trading/investing. In an experimental study, Powell
and Ansic (1997) investigated differences in financial decisionmaking.
The results showed that females are less risk seeking than
males. Further, Myagkov and Plott (1997) found risk seeking to
diminish with experience, contrary to Prospect theory. In what
way will these factors affect the results of this study?
(6) Women will lose all their money to a less extent than men.
(7) Women will as a group gain higher profits than men.
(8) Participants with prior experience of trading/investing will lose
all their money to a less extent than participants with less
(9) Participants with prior experience of trading/investing will as a
group gain higher profits than participants with less experience.

Previous Research and Theory

Why does not everybody minimize losses and maximize profits?
Traditionally, economic theory is based on the idea that market
operators are rational and therefore make rational decisions.
Feelings and biases do not influence the operators’ judgement, only
relevant information effects their behavior. Decision-makers
decide on basis of the probability of each alternative outcome and
select the alternative giving the maximum return. This view is not
supported without exception (Sebora & Cornwall, 1995; Bell, Raifa
& Tversky, 1988).
As stated earlier, our choices are influenced by how a situation is
framed. A problem is positively framed when the options at hand
generally have a perceived probability to result in a positive
outcome. Negative framing occurs when the perceived probability
weighs over into a negative outcome scenario. In one of Kahneman
and Tversky’s (1979) experiments, the participants were to choose
one of two scenarios, a 80% possibility to win $ 4,000 and the
20% risk of not winning anything as opposed to a 100% possibility
of winning $ 3,000. Although the riskier choice had a higher
expected value ($ 4,000 x 0.8 = $ 3,200), 80% of the participants
chose the safe $ 3,000. When participants had to choose between
a 80% possibility to loose $ 4,000 and the 20% risk of not losing
anything as one scenario, and a 100% possibility of losing $ 3,000
as the other scenario, 92% of the participants picked the gambling
This framing effect, as described in Kahneman and Tversky’s (1979)
Prospect theory, occurs because individuals over-weight losses
when they are described as definitive, as opposed to situations
where they are described as possible. This is done even though a
rational economical evaluation of the two situations lead to
identical expected value. People tend to fear losses more than they
value gains. A $ 1 loss is more painful than the pleasure of a $ 1 gain
(Kahneman & Tversky, 1991). Describing a loss as certain, and
therefore more painful, will inflict investors trying to avoid such a
loss. As a consequence, they will take a greater risk and gamble in a
losing situation, holding on to the position in hope that prices will
recover. In a winning situation the circumstances are reversed.
Investors will become risk averse and quickly take profits, not
letting profits run. This goes for the professional investment
managers as well (Olsen, 1997), and this is not only a tendency in
the Western world (Sharp & Salter, 1997).
Costs, that is, losses, made at an earlier time may predispose
decision-makers to take risks. They are more risk seeking than
they would be if they had not made the earlier loss (Zeelenberg &
van Dijk, 1997). This effect is referred to as “the sunken cost
effect” and results in organizations and individuals “throwing good
money after bad” in order to make up for the loss (Ghosh, 1995).
The loss already incurred makes the context equivalent of a
negative frame, but with an increased commitment, for example,
buying more shares makes a recovery possible, although uncertain.
Nothing new under the sun, especially in the markets:
. . . I did precisely the wrong thing. The cotton showed
me a loss and I kept it. The wheat showed me a profit
and I sold it out . . . . Of all speculative blunders there
are few greater than trying to average a loosing game.
Always sell what shows You a loss and keep what
shows You a profit. That was so obviously the wise
thing to do and was so well known to me that even
now I marvel at myself for doing the reverse. (Lefèvre,
1923/1993, p. 154)
Investors and traders, shifting in risk tolerance according to
positively and negatively framed situations, show no risk aversion,
but an aversion against losses. Loss aversion applies when one is
avoiding a loss even if it means accepting a higher risk (Tversky &
Kahneman, 1986). The preference for risky actions to avoid an
impending loss over less risky options just to minimize the loss and
“bite the bullet” can be explained by “loss aversion” (Thaler &
Johnson, 1990).
Weber and Camerer (1998) describe selling assets that have gained
value and keeping assets that have lost value as “Disposition effect”
in a recent experimental study. The disposition effect is based on
two characteristics of prospect theory, namely the tendency of
individuals to value gains and losses relatively a reference point and
further, the tendency to be risk-seeking in situations where a loss
might occur and risk averse in situations where a certain gain is
possible. Weber and Camerer’s study showed that participants did
sell their winners and kept their losers.
Being poor Bayesians, is that our lot, or is this disposition effect
possibly alterable? Is it conceivable adopting the behavior of the
“market wizards” or at least avoiding the most flagrant mistakes? Is
it determined by chance if one is behaving like Nick Leeson, trading
Baring’s Bank into bankruptcy, or like Michael Marcus, who went
bankrupt in the beginning of his career and later turned $30,000
into $80 millions?

About Losses and Gains

If a trader has a long position, then prices need to rise before
he/she is gaining any profit. If the trader is short, then prices must
decline to make him/her a profit. The market moves regardless of
the position of one particular trader. In order to make money
trading, one must be positioned on the right side when prices are
moving. There is an indefinite number of methods used around the
globe, to increase probability of getting positioned on the right
side. However, nobody knows for certain whether the market will
rise or decline.
When the market moves against the trader’s position and he/she
decides it is time to close the trade, the price movement multiplied
by position size determines the size of the loss. Accordingly, the
risk can be estimated as the drop from entry point to exit point,
that is, the difference between actual buying price and
predetermined selling price multiplied by the number of shares
sold. Following this reasoning, the potential profit that one can
receive depends on price rise and position size. Mastering these
two concepts “Cut your losses short” and “Let your profits run”
seems to be the common denominator making the “market
wizards” so successful, rather than having a high percentage of
winning trades, being able to pick the “right” stock or ignoring a
losing trade.
When a loss is realized there is an obvious mathematical rule
regarding drawdowns affecting recovery that sometimes is
overlooked. If losing 1,000 SEK out of a total of 10,000 SEK (a 10%
loss), then to get even, there is a need of a 11.1% increase on the
remaining 9,000 SEK. The larger the loss, the greater profit must
be obtained to recover (see Table 1). A 30% loss requires a profit
on remaining capital of 43%. That is more than twice as much as
the broad “Generalindex” of Stockholm Stock Exchange rose
during the eleven years from the beginning of 1990 to the end of
2000 (an average of approx. 16% annually; “Generalindex” rose
from 1231 points at 1989-12-29 to 4735 points at 2000-12-29, a
total increase of 285%). Taking losses bigger than that requires
extraordinary profit compared to index, and still that is just to
Table 1. Drawdown effects
Size of draw-down on
initial capital
Percent gain to recover
5% 5,3%
10% 11,1%
15% 17,6%
20% 25,0%
25% 33,3%
30% 42,9%
40% 66,7%
50% 100 %
60% 150 %
70% 233 %
80% 400 %
90% 1000 %
The importance of cutting losses short is obvious. If the trader is
unable to survive in the markets on a near term basis, then he/she
will not be around when opportunities arise to make money on the
long term. Again, the price movement multiplied by position size
determines the size of the loss. The greater the number of shares,
that is, the position size, the greater the loss. The quotations
below, from Schwager (1993), reflect a top trader’s view on
managing risk.
Risk management is the most important thing to be
well understood. Undertrade, undertrade, undertrade
is my second piece of advice. Whatever you think your
position ought to be, cut it at least in half [Bruce
Kovner] (Schwager, 1993, p. 82)
To sell an asset that is losing money is definitely a measure being
questioned. Weekly, there are experts and analysts participating in
talk shows and news broadcasts on national television, reassuring
small savers to ” . . . just sit tight”, ” . . . if you sell now, you’ll sell at
the bottom” and ” . . . speculating in stocks is a long term
business”. However, it can be somewhat arduous to maintain this
strategy when there have been down moves from peak to trough
of 75% on a stock, by itself representing 40% of the major index
when trading at all time high (Ericsson B, 2000-03-06 — 2001-03-
17, Stockholm Stock Exchange, Sweden). Especially hard, when
fund managers and “the big money” have been selling the stock the
entire journey down. This can be seen as the number of
shareholders more than doubled, from 272,000 to 586,400, during
the decline throughout the year (Sundin & Sundqvist, 2001).
Nevertheless, for those trading the stock markets there is little
advise to follow but to buy. The short selling recommendations are
very few, as are the recommendations to sell in order to take
profits. According to U.S. statistics: of 28,000 recommendations by
brokerage-house analysts, 99% of those recommendations on U.S.
companies were “strong buy”, “buy” or “hold”. Only 1% of the
time, analysts recommended “sell” (Thomson Financial/First Call
Corp., 2001.) The “dot-com” companies’ rise and fall, another
trying example for the long term buy-and-hold strategist, seems to
validate a more than 70-year-old biography quotation: “The big
money in booms is always made first by the public- on paper. And
it remains on paper” (Lefèvre, 1923/1993, p. 265).
Further, when trading options and futures, which are time limited
by their nature, there is no choice. A paper loss will become real,
since there is someone else, the counterpart, who will close the
trade for you. To close a trade with an unrealized loss or not is a
topic by itself, but will not be addressed any further in this paper.

Quantitative Trading

Buying and selling stocks and derivatives have increased
enormously over the last decade. An occupation, earlier restricted
to a few well-situated capital owners, has now become almost a
national movement, involving a majority of the Swedes. There are
reports estimating 80% of the Swedes, 16 years of age and above,
to be shareholders, directly in the markets or indirectly by pension
funds (Modig, 2001).
The stock market is a popular subject of discussion at work, at
home, and in the tabloids. Media are reporting of people gaining
huge amounts in the markets, but also giving hindsight descriptions
of how one could have made millions, or more recently, how much
capital that was lost in the latest decline. During the last quarter of
1999 and first quarter of 2000, when stock market indices around
the Western world soared to new highs, there seemed to be one
question on everyone’s mind; what stock should I buy to get the
best profit? However, since March 2000, during the decline, the
focus has somewhat changed to how one should avoid getting
ruined. Why do some people succeed in the markets, while others
are going bankrupt? Some possible clues can be found when
reviewing the psychological research that has been made within
the domain of behavioral finance.
When participants of the markets are studied in real life, they seem
to present a number of shortcomings, one of them can be
characterized as overconfidence (Scott, Stumpp & Xu, 1999).
Camerer and Lovallo (1999) found that overconfidence presented
by business managers leads to excessive business entry. When the
results were based on the participants’ abilities, individuals tended
to overestimate their relative success and enter more frequently.
This was not because of irrational information processing or
neglecting the competition they were up against. They were just
overconfident about their relative skill. Studies made by Kahneman
and Tversky (1973) show that humans have a tendency to
overestimate the probability of one’s forecasts. Among other
reasons, such as a prolonged bull-market, huge financial resources
and numerous media reports of rising markets and big gains, an
overconfidence effect could be a contributing factor to the great
number of “new” and inexperienced investors entering the stock
and derivatives markets.
Investors adjust their expectations slowly (Daniel, Hirshleifer, &
Subrahmanyam, 1998), and as a possible effect, they did not see
when the bull-market turned into a bear-market, leading to holding
on to their positions longer then expected.
Further, when we as humans make decisions under uncertainty,
our choices are influenced by the way we describe, “frame”, the
situation rather than the absolute value of the result. When we
perceive the situation as a loosing scenario, a negative framing, we
tend to be risk seeking. Consequently, if a scenario is perceived as
positive we will become risk-averse (Kahneman & Tversky, 1979).
This could have caused investors to take greater risks during the
big decline than they otherwise would judge as reasonable.
Altogether, these human foibles make investing or trading in the
stock markets a difficult task. How could one possibly become a
successful market player?
One of the recipes of success, at least according to non-academic
literature, is to control one’s risk and utilize proper “money
management”. The definition of money management is not
perfectly clear and according to trading coach Van K. Tharp, it is
not “risk control” per se, “diversification” or “how one makes
trading decisions” as sometimes stated (Tharp, 1998). Risk control
and maximization of profits is rather a result of implementing
money management strategies. Tharp emphasizes that money
management or position-sizing (this term will be used in the
following) answers the question: “How much?” or “How many?”
(Tharp, 1997). In the meaning of “how much of available capital is
to be put at risk?” or “how many contracts or shares are to be
bought?” In this paper the following definition of money
management will be used: Money management determines how
much of available capital is to be allocated in a specific market
position, that is, the number of shares bought or percentage of
total capital spent.
Author/trader Jack Schwager has published two bestsellers, Market
Wizards (1993) and New Market Wizards (1994) where
approximately forty exceptionally successful traders were
interviewed. These traders were chosen on the basis of
consistently high annual return or extraordinary growth:
I was looking for people who had attained “incredible”
achievements in the market, such as a 12-yr average
annual return of 45% with only a 5% max draw-down
(Gil Blake), or turning $30,000 into $80 million
(Marcus) etc. The methodologies used was[sic] NOT a
precondition but rather an information item I sought
out. (Schwager, 2000).
There was not one particular method of analyzing the markets, nor
one sole buy-sell strategy that could account for how these traders
could be so prosperous. However, one common trait in their
approach towards the markets was their ability to manage risk.
Earlier during their careers some of them had, completely or
almost, lost their trading capital. The one thing separating these
future “market wizards” from other losers was their ability to learn
from their mistakes by analyzing the risks they had taken, to
develop and launch strategies for never letting themselves get
stuck in the loss-trap again.


Non-academic literature on stock and futures trading emphasizes the importance of “money management”, here defined as “how much of available capital is to be allocated in a specific market position”, also called position size. The effect of position size was experimentally studied by letting two groups trade fictitious capital through a series of trades, with only one variable available for manipulation by the participants, that is, how much of available capital to be put at risk in each and every trade. The treatment group had received a three-hour lecture in position sizing, risk management, and psychological biases, whereas the control group did not. The results showed that participants in the treatment group lost all their money to a lesser extent (p < .01) than those in the control group. However, the treatment group did not gain significantly higher profits than the control group. Traders being
able to gain money over the long run were taking smaller positions than losing and bankrupt traders were (p < .0001). By receiving a theoretical education, without any practical training, the risk for a trader of going bankrupt when trading simulated stocks was decreased to a tenth.