Practice Exam (S1-2025)
To prepare yourself for the exam:
1. Read the seminar slides (important).
2. Revise the tutorial questions (important).
3. Revise the workshop questions (important).
4. The Final Exam will be based on the seminar slides, tutorial questions and workshop questions.
5. Be able to read EViews output and interpret the results.
Question 1
1. An investor is interested in allocating his wealth across the following two securities: Apple (AAPL) and Microsoft (MSFT).
(a) Using the Eviews output in Figure 1(a) on AAPL (R_AAPL) and MSFT (R_MSFT) returns respectively, what percentage of the investorís wealth would you advise him to allocate to each security?
(b) Verify your answer to Q1(a) by making use of the Eviews output in Figure 1(b) and the optimal weights formulae derived from the risk minimisation problem applied to the portfolio consisting of AAPL and MSFT.
2. Consider the following Fama-French four factor model
rt - r ft = β0+ β1(rmt - rft) + β2SMBt + β3HMLt + β4MOMt + ηt ; (1)
where rt denote monthly returns on the Fama-French durables (DUR) portfolio (includes consumer durables - Cars, TVs, Furniture, Household Appliances), (r mt - r ft) is the excess market return (the market factor - MKT-RF), rft is the risk free interest rate, SMB is the size factor, HML is the value factor, and MOM is the momentum factor. OLS output from estimating (1) over the period January 2005 to February 2023 is shown in Figure 2(a).
(a) Show the analytical decomposition of total risk into systematic and idiosyncratic risks derived from (1), if we assume that β0 = β2 = β3 = β4 = 0 and r ft is time invariant.
(b) Using the Eviews output in Figure 2(a), report what fraction of total risk implied by (1) is systematic and what fraction is idiosyncratic.
3. Using the Eviews output of regression (1) in Figure 2(a):
(a) Interpret the coefficient estimate of the SMB factor. Test whether this coefficient is statistically significant at 1% significance level. Briefly comment on your conclusion.
(b) Test whether the one factor CAPM model is preferred over (1) using the information provided in Figures 2(a) and 2(b). What conclusion do you draw?
4. In the regression results of Figure 2(a), HAC standard errors have been used. What two properties of the error term does this adjustment correct for? Write the relevant auxiliary regressions, and null and alternative hypotheses in each case. [Hint: do not write all steps of the testing procedures].
Figure 1 (a)
Figure 1 (b)
Figure 2 (a)
Figure 2 (b)
Statistical tables are provided after the end of each question.
Question 2
Consider the daily price of Apple (AAPL) from 2 January 2020 to 30 April 2025.
1. Let pt denote the log price of AAPL. Assume that we model pt as:
pt = μ + pt-1 + εt (2)
where p0 = 0 and εt ~ i.i.d. (0; σ2). Explain the di§erence between setting μ = 0 versus μ ≠ 0 in (2). How is model (2) named in each instance?
2. The Augmented Dickey-Fuller (ADF) test results applied to p t are provided in Figure 3(a). Write the null and alternative hypotheses that are implied by this test. What conclusion do you draw about the pt process?
3. Let Ft-1 denote the information set available at time t - 1. We compute the time series of log returns on Apple, rt = pt - pt-1 . Suppose rt is stationary and follows an AR(1) process:
rt = 0.1 + 0.5rt-1 + ut (3)
where ut ~ i.i.d. (0; 4) with V (ut jFt-1) = V (ut) . Provide numerical answers to the following questions where applicable.
(a) What is the conditional mean of rt?
(b) Using your answer above, what is the unconditional mean of rt?
(c) What is V (rt jFt-1)?
(d) Using your answers above, what is V (rt)?
(e) Compare your answers to parts (c) and (d). Are they as you would expect? Briefly explain.
4. Using Figure 3(b), write down the fitted model for rt. Table 1 presents the observed return rt ; the residuals ^(u)t obtained from Figure 3(b), and the fitted conditional variance, also obtained from Figure 3(b). Assuming that rt follows a normal distribution, compute the 95% prediction interval for the one-step-ahead forecast (1 May 2025) of rt.
5. Using Figure 3(b), calculate the 1-day-ahead 5% VaR on the $1m investment in AAPL stock.
Figure 3 (a)
Figure 3 (b)
Table 1
rt (%) ^(u)t Fitted conditional variance
|
29 April 2025
|
0.734
|
0.6189
|
5.2336
|
|
30 April 2025
|
0.309
|
0.2660
|
4.6638
|
Statistical tables are provided after the end of each question.
Question 3
Nicky is interested in investing in Volkswagen security. In order to understand the evolution of this security over time, she models its daily log returns, denoted by rV;t, over the period 4 January 2000 to 24 April 2023.
1. She uses a constant mean equation and conducts an ARCH(5) test on the residuals of this model, ^(v)t. The results are shown in Figure 4.
(a) Provide all steps in the testing procedure. What conclusion do you draw?
(b) Why do you think Nicky conducted the ARCH test on residuals, ^(v)t, rather than the White test? Brieáy explain the di§erence between the two tests. [Hint: do not write all steps of the testing procedure].
2. Nicky applies a GARCH(1,1) model to ^(v)t. Results are shown in Figure 5(a).
(a) Neatly write the theoretical model that is implied by the results of Figure 5(a) and verify that the parameter estimates comply with the conditions of this model. Why are these conditions needed?
(b) Derive the unconditional skewness of the errors vt from the theoretical model in Q3.2(a). What conclusion do you draw?
(c) Figure 5(b) provides the empirical distributional properties of the standardised residuals from the GARCH(1,1) model. Is the empirical skewness in line with your theoretical findings in Q3.2(b)? Briefly explain.
3. Nicky further applies a GJR-GARCH(1,1,1) model to the Volkswagen returns equation, rV;t.
(a) Neatly write the theoretical model corresponding to GJR-GARCH(1,1,1) and briefly ex- plain how this model differs from the GARCH(1,1) theoretical model.
(b) Which parameters from the GJR-GARCH(1,1,1) model determine the curvature of its news impact curve (NIC)? Provide the numerical values of the NIC curvature using the information in Figure 6.
Figure 4
Figure 5 (a)
Figure 5 (b)
Figure 6
Statistical tables are provided after the end of each question.
Question 4
An investor runs the following Fama-French four factor model over the period January 2009 to December 2019:
rM;t - r ft = β 0+ β 1(r mt - r ft) + β 2SMBt + β 3HMLt + β 4MOM t + ut ; (4)
where rM;t denote monthly returns on the Microsoft security, (r mt - r ft) is the excess market return (the market factor - MKT-RF), rft is the risk free interest rate, SMB is the size factor, HML is the value factor, and MOM is the momentum factor.
1. Set up a test on the residuals in (4), ^(u)t, to verify whether the Efficient Market Hypothesis holds.
2. Based on the results of Figure 7 what conclusion do you draw for the selected time period?
3. Which are the three forms of EMH? How do they differ from each other? Briefly explain.
4. Enumerate other tests that can be used to verify the validity or not of EMH. [Hint: do not write all steps of the testing procedures].
Figure 7
Statistical tables are provided after the end of each question.