Due 30 September at 9:00AM AWST

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 the YAML of the document.

You are NOT ALLOWED to engage any AI-assistive platforms to complete this assessments, unless you are told

otherwise (in 2 questions 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 (4 points)

Plot the prices of each asset over time separately.

Succinctly describe in words the evolution of each asset over time. All axes and figures have to be properly labeled

and described. (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:

rt = 100 ∗ ln Pt

Pt−1

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

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2.3 Histogram of returns (6 points)

Create a histogram for each of the returns series.

You have to explain your choice of bins.

You will need to carefully label all axes and figures.

You are expected to: – Write a short paragraph to describe the trend of each time series; – Discuss the formula to

calculate the bins.

(Hint: Discuss the formula you use to calculate the bins)

2.4 Summary table of returns (5 points)

Report the descriptive statistics in a single table which includes the mean, median, variance, standard deviation, skewness and kurtosis for each series. All tables need to be correctly labeled.

What conclusions can you draw from these descriptive statistics?

2.5 Are average returns significantly different from zero? (6 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.

All steps have to be shown and this part has to be repeated for each hypothesis test. (1 points)

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) (3 points – one for each stock)

Part 3: If you would have done this question using Chat-GPT, what answer will you get? (hints: you will need to

describe how you promptthe question in Chat-GPT to guide the answer (1 point), would expect your answer to be

different or similar to your answer above and provide your rationale? (1 point))

2.6 Are average returns different from each other? (7 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.

All steps have to be shown and this part has to be repeated for each hypothesis test. (2 points)

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.) (3 points)

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. (1 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? (1

points)

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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. All steps have to be shown and this part has to be repeated

for each hypothesis test.

2.9 Advising an investor (12 points)

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

w1 + 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

Given your values forE(r1), E(r2), Var(r1), Var(r2) and Cov(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. You will need to

solve the optinmisation problem using pen and paper, and you need to typeset your answer. You can then scan as

picture to attach here as your answer. You can show the summary statistics using the coding learned in class, but

the optimisation problem has to be solved by hand.

You will need to get your instructors to validate your identity of your work by asking them to sign your work when

you complete it. Without their validation, you will automatically get a zero for this question.

Note: You will need to typeset your answer. Then, you need to put your name and student ID number on every

page (and side) of your work. You will have the instructor to validate your information by signing your answer

sheet. Then, you can scan the answer as picture(s) and embed it here as your answer.

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.

1Note that E(r) = w1E(r1) + w2E(r2), and Var(r) = w2

1Var(r1) + w2

2Var(r2) + 2w1w2Cov(r1, r2)

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iii. All your calculations and steps used to produce the results should be included. So include any mathematical 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.

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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 4 (0/4) A time series plot of the

prices of each stock is

missing.

(2/4) Plots are present but

with omitted details or

formatted poorly.

Explanations are unclear

and/or have spelling and

grammatical errors. Coding

is omitted.

(3/4) 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.

(4/4) 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 6 (0/6) A histograms of the

returns are missing.

(3/6) 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.

(5/6) 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.

(6/6) 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 5 (0/5) A summary table of the

returns to each stock is

missing.

(2.5/5) 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.

(4/5) 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.

(5/5) 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 6 (0/6) An appropriate

statistical test is missing.

(3/6) 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.

(5/6) 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.

(6/6) 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.

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Table 1: Rubric (continued)

Question (Maximum Score) Fail (<25) Pass (25) Meets Expectations (25-40) Above Expectations (40-50]

2.6 7 (0/7) No statistical tests for

differences between average

stock returns are present.

(3.5/7) 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.

(6/7) 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.

(7/7) 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.

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 present, all 5 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.

Total 50 – – – –

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