ECMT6006/ECON4916 Semester 1, 2025
Assignment 3
Due: 11:59PM Sunday 1 June 2025
Academic Honesty
Academic honesty is a core value of the University, and all students are required to act honestly,
ethically and with integrity. The consequences of engaging in plagiarism and academic dishonesty,
along with the process by which they are determined and applied, are set out in the Academic
Honesty in Coursework Policy 2015. Under the same policy, as the unit coordinator, I must report
any suspected plagiarism or academic dishonesty.
Instructions
Please read the following instructions carefully before you start the assignment.
• This is a group assignment worth 10% of your final grade. Each group may have
a maximum of two members. Before submitting, ensure that your Assignment 3
Group is properly registered on Canvas.
• Individual submissions are also accepted. If you are completing the assignment on your
own, you do not need to sign up in a group on Canvas.
• You may either handwrite your answers (ensuring they are clear and legible) or type
them. Regardless of the format, all answers must be compiled into a single pdf file
and submitted via file upload on Canvas. You may submit up to two times; only the
latest submission will be graded. Submissions are limited to 15 pages, including any
appendix. Submissions exceeding this limit may incur a penalty.
• The assignment consists of 2 questions (with sub-questions). Please attempt all ques-
tions. The total mark of this assignment is 25 points. The breakdown of the points
is shown next to each question.
• For the analytical questions, please show all derivations and working. Answers without
intermediate steps will be considered as incomplete.
• For the empirical questions, you may use any statistical software. Ensure that you
present all required results, including figures, and provide interpretations where re-
quested. If you use MATLAB live script, you may export your work into a document
that include code, output, and written explanations. If you use separate code files,
please include them in an appendix at the end of your PDF submission.
• In accordance with the University late policy, a late submission will incur a penalty of
5% of the total mark per calendar day. I will not accept any submissions later
than June 6, to ensure that the suggested solution is published several days before
the final exam on June 11.
• Regarding the use of AI tools: I will not impose unenforceable restrictions. You are
permitted to use AI to assist with this assignment; however, you must clearly declare
any such use and briefly explain how it contributed to your work.
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Questions
Q1 (17pt) The data file APPL ORCL.xlsx contains the daily percentage (arithmetic) net
returns of APPL (Apple Inc.) and ORCL (Oracle Corp.) from January 2013 to December
2019. Let r1t and r2t denote the daily returns of APPL and ORCL, respectively.
An investor is evaluating different portfolio strategies involving these two risky assets and
a risk-free asset. For simplicity, we assume the risk-free rate (the return of the risk-free asset)
rf is constant at 5% per annum, which corresponds to approximately 0.02% per trading day.
The investor has a quadratic utility function1 given by
u(Rt) = Rt − 0.5 cR2t ,
where Rt is the portfolio return at time t, and c = 0.02 is the risk aversion parameter.
(i) Create a single figure with two subplots displaying the daily percentage returns of
AAPL and ORCL. Compute and report the summary statistics for both return series—
including maximum, minimum, mean, standard deviation, skewness, excess kurtosis—
as well as the sample correlation between the two series. Provide a brief interpretation
of your findings. (2pt)
(ii) Construct an equal-weighted portfolio with the portfolio return defined as
REQt = 0.5 r1t + 0.5 r2t.
Compute and report the summary statistics for REQt over the sample period, including
those listed in part (i). Briefly compare the distribution of the portfolio returns with
those of the individual asset returns. (2pt)
(iii) What is the (unconditional) Sharp ratio of the equal-weighted portfolio in the sample
period? (1pt)
(iv) Explain the RiskMetrics method for estimating the conditional covariance matrix Σt
of the two stock returns. (2 pt)
(v) Estimate the conditional covariance matrix Σt of the two stock returns using the Risk-
Metrics method. Plot the estimated individual conditional volatilities (i.e., the square
roots of the diagonal elements of Σt) and the conditional correlation over the sample
period. (4pt)
(vi) Assuming the conditional mean of (r1t, r2t)
′ is a constant vector µ, the optimal dynamic
portfolio weight vector that maximizes the conditional expected quadratic utility is
given by2
wt = c
−1(Σt + µµ′)−1µ.
1This was discussed in Q2 of the Week 9 tutorial.
2This was derived in Q2 (iv) of the Week 9 tutorial.
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Construct the optimal portfolio weights using µˆ, the sample mean of returns, and
Σˆt, the RiskMetrics estimates of the conditional covariance matrix. Plot the dynamic
portfolio weights on APPL and ORCL in a single figure. (3pt)
(vii) Including a risk free asset to ensure that the portfolio weights sum up to one, the return
on the dynamically weighted portfolio with weights (w1t, w2t) from part (vi) is given
by3
RRMt = w1t r1t + w2t r2t + (1− w1t − w2t) rf ,
where a negative weight denotes a short position (assuming no short-sale or borrowing
restrictions). Please complete the following table and comment on how the investor
should make the portfolio allocation decision based on the results. (3pt)
Dynamically Weighted Equally Weighted
Average APPL weight 0.5
Average ORCL weight 0.5
Average rf weight 0
Average portfolio return
Portfolio return std. dev.
Portfolio return skewness
Portfolio return ex. kurtosis
Sharp ratio
Average utility (×100)
Q2 (8pt) The files VaRtest.csv and EStest.csv contain the VaR and ES forecasts for
daily APPL returns over the test window from January 2016 to December 2021, generated
using various models. The forecasting procedure was discussed in the Week 11 lecture. Focus
on the 1%-VaR and ES forecasts produced by the Historical Simulation (HS) and Filtered
Historical Simulation (FHS) methods, and answer the following questions.
(i) Explain the key differences between the HS and FHS methods for forecasting return
VaR and ES. (2pt)
(ii) Evaluate the two 1%-ES forecasts from the HS and FHS methods by regressing a proxy
for ES on a constant, the ES forecast, and the lagged hit variable. Report the fitted
regression equation and the p-values for testing the the null of optimality. (3pt)
(iii) Compare the VaR and ES forecasts jointly using the Diebold-Mariano test and the
“FZ” loss function. Report the resulting t statistic and provide an overall conclusion
on which model performed best in forecasting VaR and ES. (3pt)
3This formula is equivalent to optimizing the allocation between two risky assets, since RRMt − rf =
w1t (r1t − rf ) + w2t (r2t − rf ).
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