ETF5952 Quantitative Methods for Risk Analasis
Assignment 2
Assignment guidelines:
• This assignment covers topics from workshops 1 to 7.
• The assignment has two parts. Part I is similar to the format of the final exam. Part II requires
the use of R for data analysis and computation.
• This assignment will be graded out of 100 points, which will then be scaled to a mark out of 15
when calculating your final grade.
• This is a group assignment.
– Each group member will also be required to complete an anonymous peer evaluation
survey. You will be asked to rate your group member’s participation and effort. These
surveys will be used to adjust your assignment marks. Further details on the survey will be
released via Moodle in due course.
Submission guidelines:
• You must submit two PDF files corresponding to the two parts of the assignment:
– Part 1: Submit your answers (either typed or scanned handwritten responses) as Part1.pdf
– Part 2: Submit your typed analysis and results as Part2.pdf
• Upload both PDF files through Moodle using the link titled “Assignment 2 Online Submission.”
No email submission is accepted.
• Your work must be clear and legible. If it cannot be read, or read only with difficulty, your
assignment will not be marked and you will receive a zero mark for this assignment.
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Part I
Exam-format exercises (70 Points)
Please note that N(a, b) denotes a normal distribution with mean a and variance b, and t(a, b, c)
denotes a t distribution with mean a, variance b, and degree of freedom c.
Multiple Choices
Instruction: Choose one best answer to each question. You do not need to provide reasoning
for your choice.
1. (5 points) Which of the following distribution has the fattest tail (highest kurtosis)?
(a) N(0, 1) (b) t(1, 2, 4) (c) t(2, 4, 20) (d)N(2, 4)
2. (5 points) Which of the following statements is consistent with volatility clustering?
(a) Variance is constant over time.
(b) Positive and negative shocks to the price have the equal impact on volatility.
(c) Large changes in prices tend to be followed by large changes.
(d) Only positive shocks to the price increase volatility.
3. (5 points) Suppose that a series of financial returns {rt} follows an ARCH(1) model with rt =
σtzt, which of the following equation is not consistent with the definition of σ2t ?
(a) σ2t = Et−1(r2t )
(b) E(σ2t ) = E(r2t )
(c) Et−1(σ2t ) = σ2t
(d) σ2t = var(rt)
4. (5 points) Which of the following statements is consistent with the conditional variance in a
GARCH model?
(a) It is unrelated to previous shocks/residuals.
(b) It is a function of past squared shocks/residuals.
(c) It depends on the sign of the past shocks/residuals.
(d) It is the variance of the past shocks/residuals.
5. (5 points) Which of the following is not consistent with the definition of unconditional Value-at-
Risk (VaR)?
(a) The 5% VaR is the absolute value of the 5th quantile of the return’s unconditional distri-
bution.
(b) If returns are i.i.d., given a sample of 200 observations, we can compute the 1% VaR using
the 2nd smallest value in the sample.
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(c) The 5% VaR is the value such that the area to the right of this value and below the return’s
unconditional pdf function is equal to 0.05.
(d) The 1% VaR is always larger in magnitude than the 5% VaR.
6. (5 points) What does Expected Shortfall (ES) measure?
(a) The standard deviation of the portfolio’s returns
(b) The expected loss given that the loss exceeds Value-at-Risk (VaR)
(c) The minimum expected return over a given period
(d) The total exposure of a portfolio
Short Answers
Instruction: Answer the following questions and support your answers with brief explanations or
calculations if required. Label each answer with the corresponding question number.
Question A
A financial anslyst is assessing the risk associated with the stock of NVIDIA Corp. They have fitted the
ARCH(2) model to the daily returns of NVIDIA between 1 Jan 2021 and 10 Apr 2025 (in percentages)
and obtained the following output in R:
7. (5 points) Write down the fitted ARCH(2) model using the R output above.
8. (5 points) Derive the formula for computing the unconditional volatility of return based on the
ARCH(2) model.
9. (5 points) Given the ARCH(2) estimates, compute the unconditional volatility of the return on
11 Apr 2025 (round to 1 decimal place).
10. (5 points) Given the ARCH(2) estimates and the returns in Table 1, compute the one-step-ahead
conditional volatility of the return on 11 Apr 2025 (round to 1 decimal place).
Table 1: Returns on selected days (in percentages)
Date 7 Apr 8 Apr 9 Apr 10 Apr
Index T − 3 T − 2 T − 1 T
Return 3.4 -1.4 17.2 -6.1
11. (5 points) Compare the volatility from the last two questions and discuss why the values are
different.
12. (5 points) Without doing any computation, discuss whether varT (rT+5) based on the ARCH(2)
model will be larger, smaller, or equal to σ2T+1.
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Question B
The financial analyst has fitted a GARCH(1,1) model to the same dataset used in Question A and
produced the following diagnostic plots based on the model:
• the autocorrelation function (ACF) of the squared standardized residuals;
• the Q-Q plot of the standardized residuals.
1 3 5 7 9 11 14 17 20 23 26 29
ACF of Squared Standardized Residuals
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norm − QQ Plot
Theoretical Quantiles
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Figure 1: Estimation results
13. Evaluate whether the GARCH(1,1) model appears to be a good fit for the data based on both
diagnostic plots. Justify your answer using evidence from the plots.
14. Based on your analysis of the diagnostic plots from the GARCH(1,1) model: what changes, if
any, would you recommend the analyst to make to improve the model?
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Part II
Compute-based exercises (30 points)
General Instructions:
• Use R to perform all data analysis, visualizations, and computations required for this assignment.
• Do not submit your R code. Only submit a clean, well-organized report as a PDF file containing:
– Your final answers
– Any relevant tables or figures (exported from R)
– Clear written explanations and interpretations of your results
• Limit your submission to 1 page.
Question C
The Invesco QQQ ETF (ticker: QQQ) tracks the Nasdaq-100 index which consists of 100 of the largest
non-financial companies listed on the Nasdaq stock exchange. The performance of QQQ reflects the
performance of major technology and growth-oriented companies in the U.S. In this question, you will
estimate and compare the volatility of QQQ returns during the presidential terms of Donald Trump
(first term) and Joe Biden. Use log daily returns in percentages to answer the following questions.
• (10 points) Fit a GARCH(1,1) model to the returns during the first Donald Trump presidency
between Jan 21, 2017 to Jan 20, 2021 and report the estimated parameters (round to 2
decimal place). Compute the 5% one-day conditional Value-at-Risk during the Trump presidency
and present the VaR in a plot.
• (10 points) Fit a GARCH(1,1) model to the returns during the Joe Biden presidency between
Jan 21, 2021 to Jan 20, 2025 and report the estimated parameters (round to 2 decimal place).
Compute the 5% one-day conditional Value-at-Risk during the Biden presidency and present the
VaR in a plot.
• (10 points) Compare the model estimates in the two periods and discuss your findings. Focus
on the follow aspects: the unconditional volatility of return, the estimated GARCH parameters,
the “memory” of volatility implied by the estimated GARCH parameters, and the range of the
5% one-day VaR.
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