ECOM 2001 Term Project S Thomas Due Tuesday, 5th October 2021 16:00 AWST Introduction The aim of this project is to prepare, evaluate and analyse stock market data and to recommend an optimal portfo- lio consisting of two stocks. You have been assigned three stocks, all three must be included in the analysis which works towards your recommendation of a final optimal portfolio. The project requires a deep understanding of both the statistics and the mathematics components of this unit. It is recommended that you work on this on a weekly basis. Refer to the rubric at the end of this document to understand how this assessment will be graded. In particular, note that all figures need to be numbered and labelled, and you need to include all the steps to involved with arriving at each of your answers. Your final report should be a pdf document. An RMarkdown document to get you started is available on the unit Blackboard site. Show all of your coding by keeping echo = TRUE. Make sure to update your name and student ID in the YAML of the document. 1 Import Data Import the adjusted stock prices for the three stocks which you have been assigned. See the Markdown file for hints. 2 The Analysis 2.1 Plot prices over time Plot the prices of each asset over time separately. Succinctly describe in words the evolution of each asset over time. (limit: 100 words for each time series). 2.2 Calculate returns and plot returns over time Calculate the daily percentage returns of each asset using the following formula: rt = 100 ∗ ln ( Pt Pt−1 ) Where Pt is the asset price at time t. Then plot the returns for each asset over time. 1 2.3 Histogram of returns Create a histogram for each of the returns series (explain how you determined the number of bins to use). 2.4 Summary table of returns Report the descriptive statistics in a single table which includes the mean, median, variance, standard deviation, skewness and kurtosis for each series. What conclusions can you draw from these descriptive statistics? 2.5 Are average returns significantly different from zero? Under the assumption that the returns of each asset are drawn from an independently and identically distributed normal distribution, are the expected returns of each asset statistically different from zero at the 1% level of signif- icance? Provide details for all 5 steps to conduct a hypothesis test, including the equation for the test statistic. Calculate and report all the relevant values for your conclusion and be sure to provide an interpretation of the results. 2.6 Are average returns different from each other? Assume the returns of each asset are independent from each other. With this assumption, are the mean returns statistically different from each other at the 1% level of significance? Provide details for all 5 steps to conduct each of the hypothesis tests using what your have learned in the unit. Calculate and report all the relevant values for your conclusion and be sure to provide and interpretation of the results. (Hint: You need to discuss the equality of variances to determine which type of test to use.) 2.7 Correlations Calculate and present the correlation matrix of the returns. Discuss the direction and strength of the correlations. 2.8 Testing the significance of correlations Is the assumption of independence of stock returns realistic? Provide evidence (the hypothesis test including all 5 steps of the hypothesis test and the equation for the test statistic) and a rationale to support your conclusion. 2.9 Advising an investor Suppose that an investor has asked you to assist them in choosing two of these three stocks to include in their portfolio. The portfolio is defined by r = w1r1 + w2r2 Where r1 and r2 represent the returns from the first and second stock, respectively, and w1 and w2 represent the proportion of the investment placed in each stock. The entire investment is allocated between the two stocks, so w + 1 + w2 = 1. The investor favours the combination of stocks that provides the highest return, but dislikes risk. Thus the investor’s happiness is a function of the portfolio, r: h(r) = E(r)− Var(r) 2 WhereE(r) is the expected return of the portfolio, andVar(r) is the variance of the portfolio.1 Given your values forE(r1),E(r2),Var(r1),Var(r2) andCov(r1, r2)which portfolio would you recommend to the investor? What is the expected return to this portfolio? Provide evidence to support your answer, including all the steps undertaken to arrive at the result. (*Hint: review your notes from tutorial 6 on portfolio optimisation. A complete answer will include the optimal weights for each possible portfolio (pair of stocks) and the expected return for each of these portfolios.) 2.10 The impact of financial events on returns Two significant financial events have occurred in recent history. On September 15, 2008 Lehman Brothers declared bankruptcy and a Global Financial Crisis started. On March 11, 2020 the WHO declared COVID-19 a pandemic. Use linear regression to determine if a. Any of the stocks in your data exhibit positive returns over time. b. Either of the two events had a significant impact on returns. Report the regression output for each stock and interpret the results to address these two questions. How would you interpret this information in the context of your chosen portfolio? Submission 1. Submit the pdf output of your completed project to the Turnitin.com link on the BlackBoard site for our unit. ii. Keep the sections as they are in this document iii. Ensure that all Figures and Tables are numbered, and have appropriate captions. iv. All your calculations and steps used to produce the results should be included. So include any math- ematical calculations and set echo=TRUE in all of your code chunk headers, including those used to generate figures. 2. Additional details • All results (numbers) should be accurate to 3 decimal places. • Proof-read your report - do not include spelling or grammatical errors. 1Note thatE(r) = w1E(r1) + w2E(r2), andVar(r) = w21Var(r1) + w22Var(r2) + 2w1w2Cov(r1, r2) 3 Rubric The submission is worth 50 Points in total and will be worth 50% of your final grade. Table 1: Rubric Question (Maximum Score) Fail (<25) Pass (25) Meets Expectations (25-40) Above Expectations (40-50] 1 2 (0/2) The data are not imported into R, or the incorrect stock symbols were imported. (1/2) The data were imported but the code or assigned symbols are not clear. (2/2) The assigned stocks are correctly imported and identified in the report title. 2.1 3 (0/3) A time series plot of the prices of each stock is missing. (1.5/3) Plots are present but with omitted details or formatted poorly. Explanations are unclear and/or have spelling and grammatical errors. Coding is omitted. (2/3) Plots are clear, but missing components such as captions or numbers. Explanations are present but may have spelling, grammatical or other minor errors. Coding is present but inadequately commented. (3/3) Plot axes are labelled and the plot has an appropriate caption and Figure number. Explanations are clear, concise and free of spelling and grammatical errors. Plot coding is clear and commented. 2.2 4 (0/4) The calculation of returns is not present, no time series plot of the returns of each stock is included. (2/4) Calculation of the returns is present, but may include errors. A time series plot is included by is missing details or poorly formatted. Coding is not commented. (3/4) Calculation of returns is present and correct. A time series plot of the returns is present but details such as a caption and Figure number are missing. Code may be partially commented. (4/4) Returns are correctly calculated and coding is clearly commented. A well-formatted time series plot of the returns to each stock is present, axes are labelled, the figure has a caption and a Figure number. 2.3 4 (0/4) A histograms of the returns are missing. (2/4) A histogram of the returns is present for each stock, but is missing key details such as axis labels, caption or figure number. The selection of the number of bins to include is not discussed. Code may be uncommented. (3/4) A histogram of returns is present for each stock, but some details may be missing. The selection of the number of bins to include is discussed, but may be poorly motivated or contain spelling or grammatical errors. Code is partially commented. (4/4) Histograms of the returns to each stock are present, clearly formatted and include figure captions and numbers. A discussion of the method used to select the number of bins in the histogram is included. Code is commented. No spelling or grammatical errors are present. 2.4 4 (0/4) A summary table of the returns to each stock is missing. (2/4) A summary table of the returns to each stock is included but omits key summary statistics (or included summary statistics which were not requested), poorly formatted and/or missing a table caption and number. Code is absent or uncommented. No conclusiosn or erroneous conclusions are drawn. (3/4) A summary table of the returns to each stock is included and is formatted appropriately. Not more than 1 summary statistic is absent and no additional summary measures are included. Coding is commented but is missing some details. An explanation is provided but may have minor errors and/or spelling and grammatical errors. (4/4) A well formatted summary table is presented which includes the requested summary statistics for each stock.The table has an appropriate caption and Table number. Code is well commented. Explanations are clear, concise and without spelling or grammatical errors. 2.5 5 (0/5) An appropriate statistical test is missing. (2.5/5) Appropriate statistical tests are present for each stock but the results are poorly formatted. The steps involved in the hypothesis test procedure are present with errors. The equation of the test statistic contains errors. The results are partially interpreted. Coding is not commented. (4/5) Appropriate statistical tests are present for all three stocks and the results are appropriately formatted. All five steps of the hypothesis testing procedure are present. The equation for the test statistic is present. Inference is drawn. Code is at least partially commented. (5/5) An appropriate statistical test is present for each stock and the results appropriately formatted. All five steps of the hypothesis testing procedure are present and correct. The equation for the test statistic is present and correct. Inference is correct and well presented. Coding is clearly commented. 4 Table 1: Rubric (continued) Question (Maximum Score) Fail (<25) Pass (25) Meets Expectations (25-40) Above Expectations (40-50] 2.6 6 (0/6) No statistical tests for differences between average stock returns are present. (3/6) Statistical tests for differences between average stock returns are presented but missing key details. The testing procedure is partially explained and equations for test statistics may be present but contain errors. Results are poorly formatted. Coding is incompletely commented. (5/6) Appropriate statistical tests are presented and explained but poorly formatted. Hypothesis testing procedures are included and coding is at least partially commented. Inferences are drawn but may contain minor errors. (6/6) Appropriate statistical tests are presented and well formatted. All five steps of the hypothesis testing procedures are clearly explained. Inference drawn from the test(s) is correct and informs any follow up tests. Code is fully commented. 2.7 2 (0/2) Correlations between stocks are missing. (1/2) Correlations are present but formatted poorly. The direction and strength of correlations are discussed but may contain errors in interpretation. (1.5/2) A correlation matrix is present and appropriately formatted. Direction and strength of correlation is discussed but may contain minor errors in interpretation, spelling or grammar. Code is present and at least partially commented. (2/2) A well formatted correlation matrix is presented and discussed. All five steps of the hypothesis testing procedure are included as well as an equation for the test statistic. Interpretations are accurate and free of spelling and grammatical errors. Code is well commented. 2.8 2 (0/2) No test for the significance of correlations is present. (1/2) A test for the significance of correlation is present but poorly formatted and or incompletely explained (steps to the hypothesis testing procedure and an equation for the test statistic are incomplete). Inference may be incorrect. (1.5/2) Tests for the significance of correlations is present and appropriately formatted. All five steps of the hypothesis testing procedure are present as well as the equation for the test statistic. Inference drawn may contain minor error including spelling and/or grammatical errors. (2/2) A complete set of tests for significance of correlations between stock returns is presetn, all steps to the hypothesis testing procedure are documented and include the test statistic. The results are well formatted, code is well commented and inferences drawn are correct and free of spelling and grammatical errors. 2.9 12 (0/12) No response is present. (6/12) Some attempt to explain the mathematical process for arriving at the optimal portfolio is presented but may be poorly formatted or contain errors. A reccomendation is made, but may be accompanied by a weak rationale. Results of all portfolios are present but may be poorly formatted. Coding may be present but incompletely commented. (8/12) The mathmatical process for arriving at the optimal portfolio is provided but may contain minor errors or lack appropriate supporting tests. Results for all portfolios are presented. A recommendation is provided and supported by a rationale. The answer may contain spelling and grammatical errors. Code is partially commented. (12/12) A complete explanation of the process for arriving at the optimal portfolio is explained, and the result is supported by appropriate follow up tests. Coding is well commented and intelligently applied. Results are presented in a well formatted table that includes the optimal proportion to invest in each stock, as well as the expected return for each possible portfolio. A recommendation for the optimal portfolio is given and supported by a clear rational. No spelling or grammatical errors are present. 2.10 6 (0/6) No response is present. (3/6) A regression analysis is presented but poorly formatted. An explanation is given but may contain errors in interpretation. Coding is a least partially commented. (5/6) An appropriate regression analysis if presented and adequately formatted. The results are interpreted to answer the posed questions, but may contain minor errors. Implications for the recommended portfolio are given but may lack clarity. Coding is present and at least partially commented. (6/6) An appropriate regression analysis is presented and well formatted. Results are accurately interpreted to answer the posed questions. Implications for the recommended portfolio are discussed. Coding is well commented and the response is free of spelling and grammatical errors. Total 50 - - - - 5