a) Choose a social/economic behavior/topic that you wish to examine;

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1 The aim of this assignment is to enable you to discover, analyze and critique the process by which economic models are created. As a result, you will work in pairs to complete the following: a) Choose a social/economic behavior/topic that you wish to examine; b) Then identify the main variables you wish to utilize to explain this topic. Provide reasons. c) Then create a mathematical model of your choice to explain articulate the relationship between the variables that you have chosen in b. d) Articulate the assumptions that you made in the process of creating this model. Do you simplify reality to make this model? Why? e) Articulate the aspects that you left out and explain your reasons. f) What are the shortcomings of your model in explaining the phenomena of your choice? /50 Some examples are provided on my site under the Resource section: Models Assignment Examples. You may use any form of software to build this model. 1

2 Models Assignment Modeling the Stock Market Introduction The stock market is generally considered unpredictable, a lottery, with studies going so far as to pit professional money managers against various animals, including cats, monkeys, and even an octopus, with results conclusively proving that the experts have no advantage over the animals. However, these experts often chose stocks based on either overly complex formulas, or simple human intuition and thus, the question: are there any simplistic formulas that can maximize revenue in an entirely nondeterministic system based on previous data? Procedure A computer program was created to help model and visually represent a very simplistic stock market. This market consisted of only 4 stocks, which followed the following rules: Their price was a random number between $7 and $13 Each day (one round of trading), the price either went up or down by some value between $0 and $1 The price was further modified based on how many stocks were available on the market of that type, up to a 50% increase The following formulae define a stock s price (P) in terms of the previous day s value (D): P = 7 + rand() % 7 P = D + rand() * 1000 / 1000 * MktVol/MktCap Furthermore, three distinct computer agents were created to compete in this market, each according to some simple formula, in order to see if any performed better than might be statistically expected. Each agent was charged with valuating all of the stocks, and based on their valuation, purchase or sell the stock. The formula for deciding how much of a stock to buy was represented by : ΔStocks = Factor * (Valuation ActualPrice) 3 Where factor had a constant value of 5, Actual Price was the price at which that stock was being sold at on that day, and Valuation was the dollar value that the agent assigned to that stock when it was asked to. ΔStocks represents the change in the agent s portfolio of that stock on that day (limited by the agent s available cash). 2

3 The agents all started with $1000 in cash, and followed these general archetypes: 1. Very Passive: This was an attempt to model a couch potato investor, who only participated in the market once a month (every 30 days). Their valuation of a stock was the average of the stock today, and 30 days ago, added to the value of the stock 15 days ago, according to the following formula: Bid = [Price(Today) + Price(Today -30)]/2 + Price(Today-15) 2. Semi-Passive: This agent tried to model a fairly interested person in stocks, but for whatever reason could only participate on a semi-regular basis: once a week (every 7 days). Their valuation of a stock was simply the average price of the stock in its entire history. Bid = Average(Price(Start to Today)) 3. Aggressive: This agent represented a day trader, who was active in the market every single day. They followed a slightly more intelligent formula, given the constraints of this model, whereby they valued a stock based on its 5 day performance and current price. Bid = Average(Price(Today-5 to Today)) + Price(Today) Clearly, the aggressive agent was predisposed (as is often the case in real life, too) to assume that stocks will tend to go up in value. Choice of Variables The most significant variable in this model is the portfolio (also referred to as cash) value of the agents. It was chosen because in the real world, the end goal of all participants in the stock market is to maximize the function ValueOfPortfolio(Cash, Time); such that cash and time are at global minima while ValueOfPortfolio is at a maximum. Individual stock prices are more important variables, as they have a direct impact on the value of an agent s portfolio, but the impact is not instantaneous, rather, it has some inherent lag time. Some other variables were also considered, such as the volume of a stock having an impact on its price, as a way to represent some level of control that buyers exert over the market, though not enough to direct prices in any particular direction. Assumptions Now, with a clear understanding of how the various elements of the market interact, it is critical to consider assumptions undertaken in the model. 1. All agents act at the same time, and all trades happen simultaneously. There can only be one trade per stock per agent per day. 2. All agents have perfectly accurate information about the previous values of the stock (and its market volume) but have no information about future prices. 3

4 3. All agents begin with exactly $1000, and they have no other access to money unless they sell some or all of their holdings. In some cases, agents end up going negative with cash (ie. borrowing), though this is emergent behaviour, and actually discouraged (however, it generally seems to work, more on this later). 4. There exist no fees of any type on trading, and trading can happen in non-integer quantities. 5. The simulation runs for 900 days, though it can optionally be continued for an indefinite amount of time. 6. All agents decide on the quantity to purchase based on the exact same cubic function. 7. Stock prices are completely random within a certain variation of the previous value. 8. Stocks that go bankrupt (value less than $0) are not rejected from the market, and can sometimes be hidden gems for agents that hold on to them. 9. The value of the dollar remains consistent throughout the simulation (ie. no inflation, deflation, etc. is considered). Simplifications This model is certainly several levels of abstraction higher than reality, though it is also fairly representative. Reality must be simplified because it is far too complex, and more importantly, too interconnected for a model to properly explain it. Models are best described as functions, where for some given input they return some output (in a completely deterministic fashion). Once we consider a mental model obsolete, it is scrapped and replaced by an upgraded version. Reality itself, however, is a fully intradependent structure, whereby any one thing relies on one or more other things, sometimes in a mutual arrangement. Each of the assumptions made were a direct result of some natural incongruity between reality and a model. 1. In reality, agents in a stock market do NOT act at the same time, which results in very curious problems. For example, the business of High Frequency Trading (HFT) is worth billions of dollars, because one agent can perform a transaction slightly faster, a matter of individual milliseconds, allowing them to earn a guaranteed profit by acting as a pseudo-middleman. In this model, either the first buyer always has their first choice of which stocks to purchase, or else, the later agents can buy and/or sell based on the previous agents actions, similar to HFT. This was a necessary simplification for the sake of equity, and because this is a very simplistic model, even going so far as to ignore transaction fees. Furthermore, all agents start at the same time (everyone participates in the market on the first day) to ensure the maximum equity between agents, and thus not needing a control group. (Duplicate agents will always have exactly identical portfolios, given they start at on the same day). 2. This was another necessary assumption, but is very justifiable. Although insider trading sometimes happens, pedestrian stock traders would have access to essentially identical information as these agents, but in even greater detail. The distinction is that all of these agents 4

5 have identical information at all instants in time, which of course, is not the case in reality. However, these small temporal differences have minimal effect on the stock trading behaviour of pedestrian stock traders, compared to something like HFT. 3. This assumption was also made to minimize the need for a control group. It was also a sum of money that was large enough to create noticeable returns, without being so large as to have a significant impact on the market (for example, if one agent were to buy all 10,000 units of a given stock). Borrowing is likewise heavily discouraged in this model, as it is in real life, as the stock market is unpredictable, and beside the possibility the debt will go unpaid, no one will actually lend money to be used on the stock market. In essence, wise people (who have no inside information) will never leverage or borrow in order to compete on the stock market. Thus, this assumption makes the model more realistic. 4. This simplification is necessary because it significantly simplifies the valuation of the stocks. Alternatively, one could consider the price of the stock to already include two sets of the fees, one for purchasing and one for selling. Having the fees included adds nothing to the model from a mathematical point of view, and a very minimal amount from a realistic point of view, at the cost of significantly greater complexity (ie. formulas accounting for buying and selling, ensuring profit despite these costs, etc.) days was chosen as it represents just slightly less than two and a half years. From a mathematical point of view, it has no bearing on the model at all; it was simply used for the display. It also provides enough time for somewhat significant changes in the prices of the various stocks without becoming overwhelming. 6. This assumption was made to simplify the process buyers go through in choosing stocks even though two buyers may agree on a valuation of a stock (ie. the value they think the stock actually holds, not its market price), they may not agree on how much of it to purchase. This simplifies that, and assumes that all humans would purchase following a similar model. The cubic function was chosen because it represents exceptionally greater purchases for greater profits, and has the advantage of also having negative values within its domain (unlike a quadratic). A linear function was also considered, but ultimately discarded, as humans would (generally) significantly favour higher returns (ie. greater difference between valuation and actual price) than lower returns. 7. This was another deliberate choice, and one of the most significant in the context of this assignment, as it best represents the stock market from the point of view of the agents, and thus, stock traders. The values of stocks in the stock market can be said to follow the Efficient Market Hypothesis, in which prices reflect all actionable information, and thus, from the point of view of the consumer, the fluctuations are random. Of course, stocks represent a company, and thus changes in actual stocks on the market are wholly dependent on the valuation of the underlying company long term. Over the short term, however, all stocks fluctuate as buyers and sellers jockey for ownership. It is this relationship which is being modeled. This also had a very interesting consequence just like in the real stock market, winners can often emerge for no real reason, and maintain that lead if they only have average luck from that point on. This assumption is the foundation for this model, and does not allow for events such as mergers or signing high-value deals into the model, despite these occurring commonly in the real world. 5

6 The main reason for omitting these kinds of events is because from the point of view of the average end user, these events are identical to the current implementation of stock price variations, but with greater profits and/or losses. In other words, they do not add meaningful depth to the model, but rather further increase the dependence on luck in order to achieve success. 8. This assumption was made in large part for the sake of code simplification. However, it is not unjustifiable. Companies can occasionally reach penny stock levels and climb back up. Because stocks represent the values of companies, and company values have a limited rate at which they can change, for good or bad, it is reasonable to say that this model removes penny stocks too quickly (as all price changes are fixed at a maximum change of $1/day). Therefore, agents that own those stocks but chose not to sell them retain them, and sometimes the stock re-emerges, and the agent is able to buy low and sell high. Of course, this behaviour is considered highly risky and relatively few respectable stock brokers do this in real life, but since this market is quite limited, this interaction is on the whole more beneficial to consider than to eliminate. 9. This is a critical assumption despite not being explicitly modeled. Rather than accounting for inflation/deflation and having to consider all of those extra factors, this model simply works in real values relative to the starting point of the simulation. In that sense, it can also be thought of as modeling an economy without inflation or deflation, though that deviates further from reality. Ignored Factors Some factors that are integral to the function of markets were excluded from the model for various reasons, mostly related to them being only tangentially relevant to the model itself. For example, all forms of derivatives were left out, as that only complicated the model, but added nothing. In a sense, the agents already act on the principles of derivatives, by purchasing or selling stocks according to their own valuation of the stocks, just as one might do when buying derivatives. Another important example is the nonexistence of the commodities market oil, gold, silver, gas, and so on. Although these are important to markets, they are somewhat unimportant in a heavily abstracted sense, as is the case in this model. One purpose of the commodities market is that if one believes that a certain resource will grow in value, but does not believe any companies that utilize that resource will also grow in value, or will not grow as much as the resource itself, one can opt to buy the resource itself. In a sense, this is similar to the derivatives market, though in a somewhat opposite sense. Rather than purchasing the end product, the derivative, one invests in the input product, the stock, or here, the commodity. Similar to commodities, competition between companies was also ignored. No companies were considered to be substitute goods to one another, as may be the case in real markets. However, this abstraction can be justified by saying that the stocks in this model actually represent indices, and each index is a different sector or industry, with no overlap between them. This also somewhat explains the random variations, as although it may be feasible to have insider knowledge about one or two companies, it is a lot harder to be able to predict the aggregate value of all companies in a sector. This also encourages diversification in every stock, a behaviour the agents follow agents are not allowed to 6

7 purchase a quantity of an individual stock at a cost greater than one quarter of their current cash. This market model also assumes that all agents are completely independent actors, and have no influence over one another outside the market, as should be the case in real life (though is not). Pump and dumps, as well as other schemes are not possible in the scope of the model all agents have exactly identical information. Agents also cannot team up, nor engage in corporate take overs. There are certainly other critical functions of markets that were ignored, but between the omissions herein discussed and the various assumptions that were already explained, this model is very capable of standing on its own. Shortcomings/Flaws This model s greatest flaw is that it does not permit for any interaction between agents. Although they can indirectly influence stock prices, by purchasing or selling and thereby changing the modifier on the stock price s variance, they cannot force prices up or down. To an extent, however, this is also one of the model s strengths, in that casual investors in the stock market also have no impact on a stock s price, despite various economic theories claiming they do have an impact. The model also makes several assumptions about the buying patterns of humans that they are consistent, that they are logical, and even that they do not take loans. In no human is that the case the agents are simply fundamentally inhuman. The agents follow specific algorithms for evaluating the stocks, but even still are different from humans following an identical algorithm; humans are controlled in equal parts by logic and emotion, whereas the agents have no emotion. A human, with the exact same formula for buying and selling, is far more influenceable by other, completely irrelevant factors. After a good day, people are more likely to invest; a bad one may cause people to become overly protective of their money. The agents demonstrate perfect discipline in executing the formulas, but people rarely are so algorithmic, even if the formula is an internal part of them (ie. they are not following it because someone told them to, but because they genuinely believe it to be the best possible formula). Another arguable flaw is that none of the agents can actually guarantee a profit, nor do they always have the same relative values at the end of the simulation. Each agent can end with either the greatest profit, or the greatest loss the results are completely unpredictable, and fairly non-reproducible. But then, that is also the case in real life. The flaw, then, is that the model fails at proving that there exists a simple algorithm for stock valuation that beats the market, ie. beats random chance. Furthermore, there is an element of knowledge that is present in actual stock trading which is not present in this simulation. There is a notable distinction between insider knowledge and industrial or applicable knowledge, in that applicable knowledge is technically available to everyone; it is simply knowledge about how an industry works. For example, one could strategically invest in Apple ~6 months after they released the latest version of the iphone knowing that in another 6 months, a new iphone will be released, inciting consumption, and thereby raising the value of Apple s stock. Another example might be a company that specializes in watering lawns will see a decrease in sales after a rain storm rain storms are a predictable event, and because they have significant impacts on this imaginary company s stock, the company s stock becomes in part predictable as well. Of course, there is likely a point beyond which a company is too big for these predictions to work. 7