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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. (Hint: Discuss the formula you use to calculate the bins)
22.4 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.
What conclusions can you draw from these descriptive statistics?
## Your summary table here. Be sure to format the table appropriately.
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. (1 points)
Part 2: Calculate and report all the relevant values for your conclusion and be sure to provide an interpre
tation 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 prompt the question in Chat-GPT to guide the answer (1 point), would expect
your answer to be different or similar to your answer above (1 point))
## Hint: you can extract specific values from t.test objects using the $
## Eg. using t.test(x,y)$statistic will extract the value of the test statistic.
## Consult the help file for the other values generated by the t.test() function.
## The relevant values are: the t-test method, the estimated mean , the test statistic,
## whether the test is one or two tailed, the degrees of freedom, and the p-value.
## (You might wish to present this in a table)
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.(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)
3## Decide on which test is appropriate for testing differences in mean returns
## Hint: Include the results of your supporting test for the differences in variances
## (include all 5 hypothesis step tests and the equation for the test statistics,
## and a clear interpretation of the result).
## Hint: http://www.sthda.com/english/wiki/one-way-anova-test-in-r
## So this section has (at least) 2 significance tests.
2.7 Correlations (2 points)
Calculate and present the correlation matrix of the returns.
Discuss the direction and strength of the correlations.
## Include a formatted correlation matrix here
## Hint: http://www.sthda.com/english/wiki/correlation-matrix-a-quick-start-guide-to-analyze-format-and-visualize-a-correlation-matrix-using-r-software
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.
## Report the results of tests for statistical significance of the correlations here.
## Hint: http://www.sthda.com/english/wiki/correlation-matrix-a-quick-start-guide-to-analyze-format-and-visualize-a-correlation-matrix-using-r-software
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 for E(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?
1Note that E(r) = w1E(r1) + w2E(r2), and Var(r) = w
2
1
Var(r1) + w
2
2
Var(r2) + 2w1w2Cov(r1, r2)
4Provide 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.)
You can use this section to create a table of your results.