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The ability to foresee stock market trends and the potential performance of specific financial instruments is key to a stock trader’s success. Yet studies show that both traders and financial analysts often make mistakes. **Lyudmila Egorova**, Junior Research Fellow of the HSE International Laboratory of Decision Choice and Analysis, applied mathematical methods to calculate which strategies can help brokers make a profit and avoid bankruptcy.

Having analysed the transactions in 66,500 brokerage accounts over five years, international researchers found that with a 17.9% yield on the market portfolio, traders' average returns stood at 16.4%, and even less, 11.4%, for active traders.

In another example, a review of financial analysts' forecasts revealed that less than 50% of such forecasts proved accurate (a similar study in Russia showed a 56.8% accuracy rate). Even specialist investment funds employing numerous analysts and using sophisticated forecasting instruments do not always show optimum performance.

Classical finance theory assumes, first, that traders always make rational, utility-maximising decisions, and second, that they are aware of the laws governing the distribution of future prices of financial instruments. However, this assumption is not always correct. "There is ample evidence of traders' deviation from rational behavior and their inability to make the right investment decisions," explains **Egorova** in her dissertation "Behavioural Models of Stock Market Participants."

In addition, rational agent models do not always hold true in a crisis when some traders try to grab the opportunity to earn big money by taking advantage of stock market shocks. In such cases, mathematical models can be used to clarify expectations, suggest decision-making guidelines, and project potential results based on the chosen course of action.

The purpose of Egorova's research was to create a mathematical model describing the behavior of stock exchange participants in both a stable economic situation and in a crisis. She also attempted to create a software package to analyse traders' behaviour. She examined individual stock exchange participants (traders) focusing specifically on the behavioural aspects of their activities. She used the theory of decision-making under uncertainty, probability theory and simulation modeling.

Her study of stock traders' behaviour covered two types of strategies, namely short-term strategies of daily stock transactions and long-term investment strategies.

She found that traders' short-term strategies relied mainly on their knowledge of mathematics and their ability to use market analysis tools. According to Egorova, "there is a need for new mathematical models allowing traders – who sometimes tend to act irrationally – to assess realistically their chances of hitting their investment targets."

In other words, traders need user-friendly market analysis methods with the following characteristics:

- simple and convenient for those traders who do not have advanced training in mathematics;
- capable of assessing the accuracy of a trader’s projections concerning the potential future value of various financial instruments; and
- capable of quantifying the long-term prospects of specific financial strategies in accordance with current economic theories and the validity of certain hypotheses made by ‘lucky’ traders (whose luck is sometimes incorrectly attributed to unconventional theories).

During phase one, Egorova modeled trader behavior in a stable economy, when traders are generally expected to make predictions with a degree of certainty. The author used the Bernoulli distribution proceeding from the assumption that the proportion of correct predictions of the value of securities reflects the trader's ability to make accurate projections for the future. More than 151 tests were required to estimate the probability of accurate projections rounded to the first decimal place, and 16,401 if rounded to the second decimal place. According to Egorova, in order to run a sufficient number of tests, a special program was required for simulating an individual's actions based on their chosen strategy.

Next, Egorova considered two of the most common situations:

- the trader has certain assumptions on how the prices of financial instruments may change (i.e. they can predict the price range);
- the trader can only project the direction of price change (i.e. whether prices will go up or down).

The first option is more common among those traders who tend towards technical analysis. They determine, using price charts, the levels of resistance and support, based on analysis e.g. of price patterns or Fibonacci levels.

The second option is more common and can be used by those not keen on technical analysis.

In times of crisis, the market environment changes too fast, making any model designed for a stable economy inapplicable. Most traders suffer losses in a crisis, while many small and medium-sized players go bankrupt. For those traders who do not have access to insider information, a stock market crisis can be Taleb's 'black swan', i.e. a rare and unpredictable event of enormous impact. The study's author suggests using Taleb's 'black swan' theory to analyse traders' decisions in a crisis.

In her model using some elements of the queuing theory, a trader is presented as a system which serves two streams of applications distributed according to Poisson's distribution of rare events, and makes its choices using decision trees.

Egorova's models describe traders' decision-making in a crisis, including options with rewards and declining rewards and a learning model. She has tested her models using data from international stock market indices and data on the stock of S&P 500's largest companies.

She found that a trader could make gains just by predicting regularly occurring events slightly more than half the time, regardless of whether or not the trader could foresee crises. Thus, investors can make a gain by expecting crises, yet it is not necessary for trading profitably.

Based on her analysis, Egorova developed three programs for evaluating the financial performance of traders using three different strategies:

- 'conventional traders' independently making decisions and predictions regarding possible changes in the prices of financial instruments;
- 'followers' who repeat the actions of their chosen leader with a delay of one cycle; and
- 'black swan seekers' who are not very good at predicting price changes in quiet periods, yet rarely make mistakes in a crisis (Taleb's strategy).

In each case, Egorova examined three characteristics: average wealth on the final date; the proportion of agents whose end-date wealth exceeded the starting day wealth, and the proportion of bankrupts; she conducted 100 to 150 experiments in each case.

She found that most 'conventional traders' who make decisions by tossing a coin and do not trade with leverage will neither go bankrupt nor make a substantial gain after ten years of trading – most likely, such traders will make a small loss estimated at 9,500 conventional units for the initial amount of 10,000. If an agent trades without leverage, they will avoid bankruptcy with a probability of 0.99 after ten years just by making slightly more than half correct decisions; the likelihood of making a profit in this case is assessed at 0.7.

Generally, the 'follower' strategy tends to produce the same results as tossing a coin, with a lower likelihood of bankruptcy compared to 'conventional' speculators and an expected ten-year yield approaching zero.

In contrast, ‘black swan seekers’ experience a similar 10-year yield to conventional traders, but the former's likelihood of bankruptcy is much higher, since 'black swans' are rare events which can rarely provide a solid foundation for long-term trading success. On average, conventional agents, even though they may suffer losses in a crisis, still tend to be better off than black swan seekers. The latter can be exposed to a higher risk of bankruptcy – in the course of Egorova's experiment, 239 agents from this group went bankrupt, versus 12 conventional traders.

Author:
Гринкевич Владислав Владимирович,
November 05, 2015