ANALYSIS OF EQUITY MARKETS: A SPEARMAN RANK CORRELATION COEFFICIENT APPROACH Item Type text; Electronic Thesis Authors CHEN, ZHAOREN Publisher The University of Arizona. Rights Copyright is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. Download date 28/06/2018 11:29:01 Link to Item http://hdl.handle.net/10150/612636
1 ANALYSIS OF EQUITY MARKETS: A SPEARMAN RANK CORRELATION COEFFICIENT APPROACH By ZHAOREN CHEN A Thesis Submitted to The Honors College In Partial Fulfillment of the Bachelor s degree With Honors in Finance THE UNIVERSITY OF ARIZONA MAY 2016 Approved by: Mr. Arvind Singh Department of Finance
2 Contents 1 Abstract................................................................3 2 Introduction.............................................................3 3 Background on Equity Markets............................................. 4 3.1 Market Index...................................................4 3.2 Sectors ETF in the S&P 500.......................................4 4 Assumptions............................................................ 5 5 Methods from the Paper (Source [3]).........................................5 6 Data Collection..........................................................6 7 Analysis................................................................8 8 Test................................................................... 9 9 Conclusion............................................................ 16 10 Reference............................................................. 17
3 1. Abstract In this honors thesis, I attempted to develop an investment strategy to analyze the U.S. stock market by studying the correlations between nine sectors defined by the Select Sector Standard & Poor s Depository Receipts (SPDR). These funds are a group of exchange-traded funds (ETFs) which are traded worldwide and can adequately represent the performance of the sectors. Through my method, I hope to identify the sector trends in the future by evaluating the sector ETF s most recent short-term period data. This analysis will provide an alternate investment method for investors to avoid short-term investment risk as well as explanations of why I think the approach may not be valid. 2. Introduction As an inspiration from Analysis of Equity Markets: A Graph Theory Approach [1], a paper that my math team completed for our course project, I followed the similar procedures and utilized nine sector ETFs data from January 1, 2008 to December 31, 2015 to compute the correlations between each two sectors. The reason I only analyzed nine sector ETFs instead of normal ten sector ETFs is that the S&P Telecommunications sector ETF does not have trading data before Jan 27, 2011 [2]. From what I learned in my finance courses, many approaches had been developed to evaluate the stock market, such as macroeconomic analysis, fundamental stock analysis, and technical stock analysis. Although people are able to access all the help from different valuation techniques that created for equities markets, it is still an incredibly difficult task to minimize the short-term risk for equity selection. Therefore, more and more people switch to ETF investing because of its low expense ratios, instant diversification, and high tax efficiency. In this honor thesis, by following the method of Shirokikh, Pastukhov et al. [3], which utilized Spearman rank correlation as a measurement of similarity between stocks, I explored an
4 efficient way to define the correlations between sector ETFs and tried to discover a new method to analyze the stock market. 3. Background on Equity Markets 3.1 Market Index A market index is an aggregate value produced by combining several stocks or other investment vehicles together [4]. It is intended to represent an entire stock market, and its value is useful for investors to track changes in market values over a long period of time. Some examples are S&P 500, Dow, and Nasdaq. In this honors thesis, I focus on the S&P 500 because it has been widely utilized in stock market valuation, and it is one of the most important leading indicators of the U.S. equities. 3.2 Sectors ETF in the S&P 500 A sector ETF is a marketable security that tracks a basket of stocks in the same sector as an index fund. The advantage of using sector ETFs as the building blocks of a portfolio is that they can provide for finer tuning of a portfolio versus a broad-based ETF [5]. Also, in helping investors save money, sector ETFs offer low turnover and broad diversification. In addition, when people are trading large volumes of sector ETFs, they can redeem them for the shares of stocks that the ETFs track. This arrangement is able to minimize the tax implications for the investor exchanging the sector ETFs because the investor can defer most taxes until the investment is sold [6]. However, compared to the broad-based ETFs of which they are a part, the trading cost tends to be more expensive for sectors ETFs because the bid-ask spread widens on the more thinly traded sector ETFs. Cxoadvisory website [2] provided me a total nine sectors ETFs to analyze, and some examples include in the S&P 500 are Energy, Consumer Discretionary, and Consumer Staples.
5 The idea of sectors ETFs is crucial in my project goal because it helps to reduce my analysis to seek which parts of the market are most connected instead of focusing on the relationships between each stock. 4. Assumptions Before analyzing the sector strength, the following two assumptions are necessary to be mentioned: The financial market is efficient, which means market information is available to all participants at any given time, and everything relevant to the value of a sector ETF is discounted and reflected in its share price. This assumption provides me the capability to use technical analysis to analyze each sector. Trends sometimes appear in share price moves and when once started, these trends tend to persist. There is a correlation between the performances of every two sectors. 5. Methods from the Paper (Source [3]) In the Shirokikh, Pastukhov et al. paper [3], the authors use the Spearman Rank correlation coefficient to measure of similarity between stocks, which is denoted rxy, and it is defined in the following way: Figure 1 [7]
6 The above correlation is a nonparametric measure of statistical dependence between two variables [8]. In my project, they are two sectors ETFs. The Spearman Rank correlation coefficient assesses how well the relationship between two ranked time series, and it can be described using a monotonic function. If there are no repeated data values, a perfect Spearman correlation takes on any value in the interval [ 1, 1], where a correlation of 1 is a 100% direct correlation, a correlation of 1 is a 100% inverse correlation, and a correlation of 0 represents no correlation [1]. After reading the Shirokikh, Pastukhov et al. paper [3], I decided to use the Spearman correlation since this methodology will provide me a solid groundwork on computing correlations. 6. Data Collection I extracted data from Cxoadvisory website for the nine sector ETFs data that appeared in the S&P 500 from January 1, 2008 to December 31, 2015. Then, by using the daily adjust close price, I computed daily percentage return for each sector ETFs. After subtracting the average daily percentage return of sector ETFs from daily percentage return sector ETFs, I received the difference between the average daily percentage return from 2008 to 2015 and the daily percentage return of sector ETFs. Later, I summed the square of the differences and finished calculating half part of the denominator in the Spearman Rank correlation coefficient formula (Figure 2).
7 Date Adj Close Percentage Average Adj - Average Square (Adj - Average) Sum 2015/12/31 77.83962-0.010006248 0.000618208-0.010624456 0.000112879 0.469917268 2015/12/30 78.626375-0.008041204-0.008659412 7.49854E-05 2015/12/29 79.263751 0.011565787 0.010947579 0.000119849 2015/12/28 78.357485 0.002548379 0.001930171 3.72556E-06 2015/12/24 78.158308-0.002668697-0.003286905 1.08037E-05 2015/12/23 78.367447 0.00510923 0.004491022 2.01693E-05 2015/12/22 77.969085 0.007334026 0.006715818 4.51022E-05 2015/12/21 77.401421 0.005043303 0.004425095 1.95815E-05 2015/12/18 77.013021-0.015719434-0.016337642 0.000266919 2015/12/17 78.242956-0.016456806-0.017075014 0.000291556 2015/12/16 79.55213 0.016732164 0.016113957 0.00025966 Figure 2 At the end, I followed the Spearman Rank correlation coefficient formula to calculate the correlations between each sector ETF and received a correlation graph (Figure 3), which provide me a more visual way to see how each sector ETF (the same as each sector) is interacting with each other. Sector/Correlation Energy Industrial Consumer Staples Utilities Financial Technology Consumer Discretionary Health Care Materials Energy 1.0000 0.7829 0.6340 0.6841 0.6520 0.7539 0.7019 0.6463 0.8470 Industrial 1.0000 0.7564 0.6563 0.7888 0.8544 0.8747 0.7561 0.8659 Consumer Staples 1.0000 0.7071 0.6616 0.7486 0.7647 0.7598 0.6718 Utilities 1.0000 0.5551 0.6651 0.6201 0.6551 0.6318 Financial 1.0000 0.7611 0.8095 0.6299 0.7116 Technology 1.0000 0.8514 0.7291 0.8028 Consumer Discretionary 1.0000 0.7321 0.7840 Health Care 1.0000 0.6981 Materials 1.0000 Figure 3
8 7. Analysis By following the Spearman Rank correlation coefficient formula in Figure 1, I found an alternative method to reduce short-term investment risk by reverse engineering. In order to demonstrate my approach, I need to find the average percentage return for the nine sectors ETFs (Figure 4). Sector ETF Average Percentage Return (1/1/2008-12/31/2015) Energy (XLE) 0.000139 Industrial (XLI) 0.000360 Consumer Staples (XLP) 0.000438 Utilities (XLU) 0.000244 Financial (XLF) 0.000291 Technology (XLK) 0.000410 Consumer Discretionary (XLY) 0.000618 Materials (XLB) 0.000268 Health Care (XLV) 0.000499 Figure 4 First, assuming I have two sector s (for example energy and finance) daily performance for one month (for example April 1 st, 2016 to April 30 th, 2016), which will be x1, x2 x30 and y1, y2 y30 in the formula. Then, applying the Spearman Rank correlation coefficient formula and using the average percentage return for the energy sector ETF in Figure 4 (0.000139), I am able to calculate the 30-days short-term average percentage return of ӯ for the finance sector. By comparing ӯ to 0.000139, I will have a clear understanding of the performance of the finance sector in the future. If ӯ is higher than 0.000139, it indicates an upward trend for the finance sector and a high potential for investment. Otherwise, it provides a downward trend for the sector performance as well as a negative sign for the investment.
9 8. Test In order to prove the validity of my short-term investment strategy, I planned to test it by extracting data from January 1, 2008 to December 31, 208 from the source 2 for all nine sectors. Then, I inputted 252 daily percentage returns for these nine sectors into Mathematica (Figure 5) and received total 18 different ӯs (Figure 6-13). Figure 5
10 ӯ computed from Financial ETF and Energy ETF Figure 6 ӯ computed from Financial ETF and Industrial ETF Figure 7
11 ӯ computed from Financial ETF and Consumer Staples ETF Figure 8
12 ӯ computed from Financial ETF and Utilities ETF Figure 9 ӯ computed from Financial ETF and Technology ETF Figure 10
13 ӯ computed from Financial ETF and Consumer Discretionary ETF Figure 11 ӯ computed from Financial ETF and Health Care ETF Figure 12
14 ӯ computed from Financial ETF and Materials ETF Figure 13 In Figure 6 and Figure 13, the values of both ӯs I received are complex numbers, which will not be concerned as valid ӯs in my project. Then, by averaging the other 12 ӯs, I received the difference between the average ӯ and the average financial ETF percentage return: 0.002324504 (Figure 14), which indicates an upward trend of financial ETF in 2008. yb1 yb2 yb computed from Financial ETF and Industrial ETF -0.0010935 0.00618861 yb computed from Financial ETF and Consumer Staples ETF -0.0191967 0.0241176 yb computed from Financial ETF and Utilities ETF -0.0154023 0.021113 yb computed from Financial ETF and Technology ETF -0.00767038 0.0135396 yb computed from Financial ETF and Consumer Discretionary ETF -0.0124391 0.0157576 yb computed from Financial ETF and Health Care ETF -0.00877678 0.0152491 Average yb1 and Average yb2-0.01076313 0.01599425 Average (yb1 and yb2) = A 0.00261556 Average Financial ETF Percentage Return (1/1/2008-12/31/2015) = B 0.00029106 A - B 0.0023245 Figure 14
15 In addition, if I only average the ӯs I computed between the Financial sector ETF with another sector ETF, I am able to discover that the average ӯ calculated from the Financial sector ETF and the Hearth Care sector ETF indicates the highest percentage return of the Financial sector (0.00323616). In addition, the average ӯ calculated from the Financial sector ETF and the Consumer Discretionary sector ETF points out the lowest percentage return of the Financial sector (0.00165925) (Figure 15). By taking the absolute value of the difference between the computed ӯ of the Financial sector ETF and other sector ETFs average percentage return (from 2008 to 2016), the difference between the Financial sector ETF and the Hearth Care sector ETF has the highest absolute value (0.002737128), and the difference between the Financial sector ETF and the Consumer Discretionary sector ETF has the lowest absolute value (0.001041042). Moreover, in Figure 3, it shows that Financial ETF and Hearth Care ETF have the lowest correlation (0.6299) among the correlations between the Financial sector ETF and other sector ETFs, and the Financial ETF and the Consumer Discretionary ETF have the highest correlation (0.8095) among the correlations between the Financial sector ETF and other sector ETFs. This observation demonstrates that when two sectors are highly correlated, the difference between their average percentage returns will be low, and vice versa. Yb Computer from Fin ETF Avg Pct Return (yb1 and yb2) = C Other Sector ETF Avg Pct Return = D C - D yb computed from Financial ETF and Industrial ETF 0.002547555 0.000360495 0.00218706 yb computed from Financial ETF and Consumer Staples ETF 0.00246045 0.000437553 0.002022897 yb computed from Financial ETF and Utilities ETF 0.00285535 0.000243743 0.002611607 yb computed from Financial ETF and Technology ETF 0.00293461 0.000410394 0.002524216 yb computed from Financial ETF and Consumer Discretionary ETF 0.00165925 0.000618208 0.001041042 yb computed from Financial ETF and Health Care ETF 0.00323616 0.000499032 0.002737128 Figure 15
16 9. Conclusion After researching and looking at the sector performance in 2008 [9], I realized that the annual percentage return on the financial sector declined tremendously during the recession period, which contradicted what I received from my approach. Therefore, it illustrates that my approach may not be valid and further demonstrates the limitation of pure technical analysis for the U.S. stock market. However, if I am able to define which ӯ to choose instead of taking the average of ӯs in each equation, or if I reduce the test period from twelve months to three months (from September 1, 2008 to December 31, 2008), my investment strategy may work.
17 References [1] Abrams, R., Alcala, J., Baldwin, D., Gonda, R., and Chen, Z. (2016, May). Analysis of Equity Markets: A Graph Theory Approach. 4. [2] (2016, January 5). Sector Performance by Calendar. Retrieved from https://www.cxoadvisory.com/4408/calendar-effects/sector-performance-by-calendarmonth/ [3] Shirokikh, O., Pastukhov G., Boginski V., and Butenko, S. (2013, January 8). Computational study of the US stock market evolution: a rank correlation-based network model. Computational Management Science Comput Manag Sci, 10, 81-103. [4] Market Index. Retrieved from http://www.investopedia.com/terms/m/marketindex.asp [5] Hawkins, K. An Introduction To Sector ETFs. Retrieved from http://www.investopedia.com/articles/exchangetradedfunds/08/sector-etfs.asp [6] McWhinney, J. The Benefits Of ETF Investing. Retrieved from http://www.investopedia.com/articles/mutualfund/05/060605.asp [7] Correlation. Retrieved from https://www.mathsisfun.com/data/correlation.html [8] Spearman's rank correlation coefficient. Retrieved from https://en.wikipedia.org/wiki/spearman%27s_rank_correlation_coefficient [9] SECTOR RETURNS by Year 2007 2016*. Retrieved from http://www.sectorspdr.com/sectorspdr/pdf/all%20funds%20documents/document%20re sources/10%20year%20sector%20returns