Introduction

The aim of this project is to prepare, evaluate and analyse stock market data and to recommend an optimal portfolio 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.

YOU MUST USE THE STOCKS ASSIGNED TO YOU. Any deviation from the assigned stocks will results in a grade of zero.

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 theYAML of the document.

You are NOT ALLOWED to engage any AI-assistive platforms to complete this assessments, unless you are told otherwise (in 1 question below).

1    Import Data (2 points)

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 (3 points)

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 (4 points)

Calculate the daily percentage returns of each asset using the following formula:

Where Pt is the asset price at time t. Then plot the returns for each asset over time.

2.3    Histogram of returns (4 points)

Create a histogram for each of the returns series.

You have to explain your choice of bins. (Hint: Discuss the formula you use to calculate the bins)

2.4    Summary table of returns (4 points)

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? (5 points)

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 significance?

Part 1: Provide details for all 5 steps to conduct a hypothesis test, including the equation for the test statistic. Part 2: Calculate and report all the relevant values for your conclusion and be sure to provide an interpretation of the results. (Hint: you will need to repeat the test for expected returns of each asset)

2.6 Are average returns different from each other? (6 points)

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.)

If you have a chance to engage Chat-GPT, how would you approach this question? That is, you need to clearly lay out ALL STEPS that you would ask the question to Chat-GPT. (0.5 points)

Now, compare your answer to Chat-GPT, why do you think your answer is different or similar? Please attach a picture of the screenshot of the answer you have got from Chat-GPT. What do you learn from this exercise? (0.5 points)

2.7    Correlations (2 points)

Calculate and present the correlation matrix of the returns.

Discuss the direction and strength of the correlations.

2.8    Testing the significance of correlations (2 points)

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 (12 points)

Note: You need to show all steps in this questions in RStudio to be able to get full marks.

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 = w1 r1 + w2 r2

Where r1 andr2 represent the returns from the first and second stock, respectively, and w 1 and w 2 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)

Where E(r) is the expected return of the portfolio, and Var(r) is the variance of the portfolio.1

Ginivest(ven)r(o)?uWrvhaat(lu)eisst(f)e e(r E)x(pe(r1))c,te(E)r2e)t,ur(V)t(r)rt1h)i,sp(V)o(a)o2l)io(a)?nd Cov(r1 , r2 ) which portfolio would you recommend to the

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 (2 points)

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. i. Keep the sections as they are in this document

ii. Ensure that all Figures and Tables are numbered, and have appropriate captions.

iii. 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.

Rubric

The submission is worth 50 Points in total and will be worth 50% of your final grade.

Table 1: Rubric