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# Statistical Analysis of Stock Prices: Proof that Mean Reversion Exists & Whether Profit/Loss is Guaranteed

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### Statistical Analysis of Stock Prices: Proof that Mean Reversion Exists & Whether Profit/Loss is Guaranteed

1. 1. STATISTICAL ANALYSIS OF STOCK PRICES: PROOF THAT MEAN REVERSION EXISTS & WHETHER PROFIT/LOSS IS GUARANTEED Ahrisue Choi Sigma Xi Student Research Showcase Monday, March 10, 2014 1
2. 2. ABSTRACT By following the phenomenon of mean reversion and calculating the one month trailing moving average (MA) of a company's stock prices of a particular day, one has the potential to make great proﬁt. Because the costs of shares are always ﬂuctuating, there are ways to exploit these prices to earn money. If certain amount of shares are bought when their prices are below the average, less money will be spent than when they are bought at prices exceeding the average. Thus, by selling these shares when they cost more than the average price, a greater amount of money is accumulated. However, this is not always the case. As the results of this experiment will later show, one can generate proﬁt with the assumption that prices will mean revert, but it is not necessarily true that if stocks are bought at prices lower than the average and sold at prices higher than the average, then proﬁt will be made. Furthermore, this project proves that mean reversion exists and that it is a set trend in the stock market. Monday, March 10, 2014 2
3. 3. INTRODUCTION In the economic life, there exists such a system involving stocks. Stocks are purchased shares of ownership of a particular company or industry. These shares are used for one’s beneﬁt because it provides a certain percentage of proﬁt, depending on the company’s success. However, if gone wrong, much money can be lost. [1] Focusing on stock price mean reversion and the ﬂuctuation of stock prices, the manipulation of these circumstances to one’s advantage was investigated. Monday, March 10, 2014 3
4. 4. Literature Review ‣Mean Reversion: suggests that prices will eventually return to their average values [2] ‣One Month Trailing Moving Average (MA): the sum of all the prices within the month trailing the focus date divided by however many days the stock market was open within that month •“trailing”: looking back •“moving”: average value over a set period [3] Monday, March 10, 2014 4
5. 5. Historical Review [1/3] ‣The oldest stock exchange market in the world is the Amsterdam Stock Exchange, which was founded four centuries ago. •It was in this market that the ﬁrst shares were established. Companies started to participate in a system, where they were able to trade their shares. •By the 17th century, the importance of the stock exchange grew as the amount of bonds and shares increased. Eventually, the Amsterdam Stock Exchange Association was founded in the 19th century for the organization and regulation of traded shares. •Subsequently, the association launched the European Options Exchange in the 20th century, expanding the stock market. [4] •As the market grew, more proﬁt was gained and more failures were experienced. It was from this historical progression that patterns in the stock market were observed. Participants took advantage of these patterns and exploited their resources to their beneﬁts. Monday, March 10, 2014 5
6. 6. Historical Review [2/3] ‣According to Chakraborty and Kearns [5], even the slightest mean reversion, or any movement within the stock market that indicates that prices are mean reverting, will yield proﬁt, although it is a random occurrence. •They introduce the Ornstein-Uhlenbeck (OU) process, a physics concept, which is a mean-reverting process that has been studied deeply, and it acknowledges the randomness of mean reversion. •They provide a differential equation that describe how the process satisﬁes certain conditions, while allowing space and time variables to transform. •The stock market has trends that are established as time progresses, so when this concept is applied to economics, it helps direct market makers toward a path closer to proﬁt than failure. Monday, March 10, 2014 6
7. 7. Historical Review [3/3] ‣According to Poterba and Summers [6], there are mean reversion presence tendencies, meaning mean reversion is more present in some situations and less present in others. •They found that when return variance rises less, with respect to time, it implies more mean reversion. •Furthermore, there tends to be more mean reversion in less sophisticated equity (value of shares that a company issues) markets. •Thus, around the time their experiment was conducted, mean reversion was more apparently present in less sophisticated equity markets established in foreign countries than in those established in the United States. Monday, March 10, 2014 7
8. 8. Gaps in Research ‣If such a pattern exists, why are the participants of the stock market not always so successful? ‣If there is such a concrete trend observed in the system, should not the companies always make proﬁt without fail? ‣The stock market is also known for its instability. Although it is a predictable arrangement of shares, failures always happen by chance. So what happens if they do? ‣How can chance be accommodated for when assuming mean reversion will constantly be in motion? Monday, March 10, 2014 8
9. 9. METHODOLOGY ‣The one month trailing moving average (MA) of the focus date should be calculated. This calculation, along with mean reversion, will play a big role in determining when to buy or sell shares. ‣There is an investing strategy that can be used, based on mean reversion, to gain proﬁt. •If a stockholder owns a few shares of Apple Inc., he should sell his shares when their prices exceed the MA, so that the money earned is set in his bank account. This is because, according to mean reversion, the prices are bound to decrease back to their average costs. Likewise, when the shares’ prices go below their mean value, the stockholder should buy more shares, since they will most likely increase back to the average. •Figure 1 illustrates this exact theory of mean reversion. Whenever the costs of Apple’s shares deviated from their MA, they always managed to return to the average price. Monday, March 10, 2014 9
10. 10. RESULTS: Apple Inc. [1/2] ‣September 23, 2011: The MA was \$388.82. This is because the sum of the closed prices of each day between August 23 and September 23, inclusive, were divided by the number of days the stock market was open within that time frame. Monday, March 10, 2014 10
11. 11. RESULTS: Apple Inc. [2/2] ‣October 4, 2011: An accountant withdrew \$1,500.00 from his bank account. Noticing the \$372.50 cost of Apple’s shares that day, referencing Table 1, which is approximately 5.00% lower than the MA of that day, he bought three of Apple’s stocks for \$1,117.50. [7] He was left over with a \$382.50 change. ‣October 18, 2011: The prices of the accountant’s three shares rose to \$422.24, which is about 6.15% greater than the MA. To avoid losing money when the prices return, he sold his three shares, and received \$1,266.72. He made roughly a 13.35% proﬁt. ‣However, if he had sold the shares on October 7, he would have lost money. On that day, the closed prices of Apple’s shares were \$369.80, which is 5.29% lower than their one month trailing moving average and 0.72% lower than the closed price of October 4. Had he sold his shares on that day, he would have lost \$8.10 ((3) (372.50)−(3)(369.80)), and made a 0.72% loss. Monday, March 10, 2014 11
12. 12. RESULTS: Google Inc. ‣October 11, 2013: An investor buys ﬁve of Google Inc.’s stocks at \$871.99 (1.09% lower than the MA) each. ‣November 11, 2013: He sells all of his stocks at \$1,011.00 (2.24% higher than the MA) each. His proﬁt is \$695.05, which is 15.94% proﬁt. ‣This instance also demonstrates that purchasing a stock below the MA and selling it when it is above the MA will generate proﬁt. However, there are some instances in which this is not true. Monday, March 10, 2014 12
13. 13. RESULTS: Amazon.com Inc. ‣February 8, 2012: An entrepreneur purchases four stocks from Amazon.com Inc. at \$185.48 (0.12% lower than the MA) each. ‣He sells his four stocks on March 9, 2012 at \$184.32 (0.81% higher than the MA) each. ‣Although he purchases at a price lower than the MA and sells at a price higher than the MA, he does not make proﬁt. In fact, he loses \$4.64, which is a 0.63% loss. ‣This case demonstrates that even if one purchases a stock at a price lower than the MA and sells it at a price higher than the MA, he is not guaranteed proﬁt because the selling price has to exceed the purchasing price. Monday, March 10, 2014 13
14. 14. RESULTS: Tiffany & Co. Inc. ‣August 2, 2012: A businessman buys two of Tiffany & Co.’s stocks at \$53.58 (1.03% lower than the MA) each. ‣He sells his stocks on September 27, 2012 at \$61.90 (1.28% higher than the MA) each. ‣He buys and sells stocks at prices lower than the MA, and yet makes proﬁt. He makes \$16.64, which is a 15.53% proﬁt. ‣This situation shows that even if one purchases and sells a stock at a price lower than the MA, it is still possible to make proﬁt. This is because although the selling price is lower than the MA, it is still higher than the initial cost, so proﬁt will be made. Monday, March 10, 2014 14
15. 15. DISCUSSION [1/2] ‣Looking back at the completed experiments that apply different mathematical concepts, a valid conclusion can be drawn: By using particular conditions of stock prices to one’s beneﬁt, proﬁt can be made. ‣As shown by the ﬁrst two examples in the results, the MA and the phenomenon of mean reversion help determine when to buy or sell shares. •The MA of a particular date is the average of all the closed prices of one month prior to that day. •Mean reversion describes the case of ﬂuctuating prices eventually returning to their long-term mean. ‣When a stockholder acquires a certain amount of shares at a price lower than the MA, he is spending less money than when the price is higher than the mean. Consequently, by selling the shares at a price higher than the original cost and average, a higher percentage of merit is produced. Monday, March 10, 2014 15
16. 16. DISCUSSION [2/2] ‣However, this does not always stand true. As seen in the cases provided for Amazon.com Inc. and Tiffany & Co., mean reversion is simply an assumption. It is a random, stochastic phenomenon that suggests that prices will eventually rise and fall to their respective MA. One can use this assumption to his advantage. •If he buys a stock at a price lower than the MA, he can assume that the price will increase, anticipating proﬁt in the near future. He can sell his stock at a price higher than the MA, but that price can also be signiﬁcantly cheaper than the initial cost he purchased the stock at, resulting in loss. In a polar situation, he can sell his stock at a price lower than the MA, but still generate proﬁt if the price is higher than the original cost he payed. ‣Because mean reversion occurs by chance, proﬁt and failure are never certain, but it is a reoccurring trend throughout the stock market, so one can rely on it to his advantage. Monday, March 10, 2014 16
17. 17. CONCLUSION [1/3] ‣The demonstrated trials show the application of simple mathematical ideas in a more advanced world. •The term “average” is introduced at the elementary level. Using this statistical concept that was previously learned in adolescent years, the one month trailing moving average of a company’s stock prices on a speciﬁc date was calculated. •Likewise, the numerical term “fraction” was also used in the computations of change and proﬁt percentages. ‣By using basic math notions, it was possible for many patterns and inferences to be drawn. This illustrates the universal practicality of mathematics, whether it be the utilization of number theory, statistics, algebra, or more. Hence its category, applied mathematics. [8] Monday, March 10, 2014 17
18. 18. CONCLUSION [2/3] ‣The next experiment to be done would incorporate other factors of the stock market, such as: •supply and demand •loans •insurance policies •other aspects of the economic realm that affect the proﬁting conditions of a company ‣This experiment only focused on a signiﬁcantly simpliﬁed situation. •How can the ﬁndings of this experiment be applied to other incidences? •What if a company fails? •Moreover, how can the randomness of mean reversion be accommodated for when acknowledging mean reversion exists? •How does mean reversion manifest itself within the stock market? Or in other words, what factors set the trends of the market prices? Monday, March 10, 2014 18
19. 19. CONCLUSION [3/3] ‣Furthermore, this experiment focused on short periods of time. •How does mean reversion work over long periods? •How can one observe prices mean revert in a more general, larger ﬁeld of view? ‣Additionally, the fact that one can generate proﬁt by relying on mean reversion calls to mind another string of questions to inquire. •If it is really that simple to make proﬁt from the stock market with the described strategies of manipulation, why is the world not infested with billionaires? •Why is the whole world not participating in the stock market for the sole purpose of proﬁting? •Why are there people pursuing PhD’s to answer these questions? •A signiﬁcantly crucial question to address in the future is how can future prices of stocks be determined, using mean reversion? Monday, March 10, 2014 19
20. 20. REFERENCES 1Stocks Basics: What Are Stocks?. (n.d.). Investopedia. Retrieved December 27, 2011, from http://www.investopedia.com/university/stocks/stocks1.asp 2DeFusco, R. A., McLeavey, D. W., Pinto, J. E., & Runkle, D. E. (2007). Quantitative Investment Analysis (2nd ed.). Hoboken, NJ: John Wiley and Sons. 3Gietzen, A. (1995). Advanced Cycle Trading: Cutting Edge Techniques for Proﬁting from Market Tops and Bottoms. Burr Ridge, IL: McGraw-Hill Professional. 4Amsterdam Stock Exchange. (n.d.). NYSE Euronext. Retrieved October 20, 2013, from http://www.nyx.com/who-we-are/history/amsterdam 5Chakraborty, T., & Kearns, M. (2011). Market Making and Mean Reversion. Proceedings of the12th ACM conference on Electronic commerce, 12, 307-314. 6Poterba, J. M., & Summers, L. H. (1988). Mean Reversion in Stock Prices: Evidence and Implications. Journal of Financial Economics, 22, 27-59. 7Tracy, J. A. (2008). Accounting for Dummies (4th ed.). Hoboken, NJ: John Wiley & Sons. 8Lancaster, K. (n.d.). What is Applied Mathematics?. WSU. Retrieved December 27, 2011, from http://kirk.math.twsu.edu/appliedmath.html Monday, March 10, 2014 20
21. 21. ACKNOWLEDGEMENTS I would like to express my thanks to Mr. Alex Jung, who mentored me throughout this experiment. He offered me the guidance that I needed to complete this project. Monday, March 10, 2014 21