ECMT5001: Mid-semester Examination (2021s2)
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ECMT5001: Mid-semester Examination (2021s2)
1. Let Jt be the indicator that a big movement occurs in the stock market on day t (i.e., Jt = 1 if there is a big move; Jt = 0 otherwise). The joint probability distribution of Jt 
1 and Jt is given in the table below:
|
|
Jt = 1 |
Jt = 0 |
|
Jt |
0:3 |
0:2 |
|
Jt |
0:2 |
0:3 |
(a) [2 marks] What is the probability that there are at least one day with big market
movement out of two consecutive days?
(b) [3 marks] Given that there is a big market movement today, what is the probability
that there was a big market movement yesterday?
(c) [3 marks] Are Jt 
1 and Jt independent? Explain.
(d) [2 marks] Compute E(Jt).
(e) [3 marks] Compute Var(Jt).
(f) [4 marks] Compute Corr(Jt 
1 ;Jt).
2. There are two major blue-chip stocks in the stock market: Apple (AAPL) and Berry (BRRY). Let A and B denote, respectively, the annual returns of a share of AAPL and BRRY. It is known that A has mean 0.1 and standard deviation 0.4, and B has mean 0.05 and standard deviation 0.2.
Carol decided to form a portfolio by investing in 50% of AAPL and 50% of BRRY.
(a) [3 marks] Find the expected annual return of Carolís portfolio.
(b) [5 marks] It is also known that the correlation between A and B is -0.25. Find the
standard deviation of the annual return of Carolís portfolio.
3. The stocks in the technology sector performed quite well in 2020. Their annual returns follow a normal distribution with mean 0.2 and standard deviation 0.5. Little Bob formed
a mutual fund by randomly picking 25 stocks with equal weights from the technology sector.
(a) [5 marks] What is the probability that the annual return of little Bobís mutual fund
was negative in 2020?
(b) [6 marks] Little Bobís mutual fund achieved an average annual return of 0.35 in
2020. Test at the 5% signiÖcance level that little Bobís mutual fund outperformed the technology sector of the stock market with a higher average annual return. Show all your steps. A complete response should include:
i. setting up the null and alternative hypotheses;
ii. deÖning an appropriate test statistic;
iii. stating the distribution of your test statistic under the null hypothesis;
iv. computing the test statistic based on the sampled data;
v. making a decision using a correct method (e.g., critical value approach or p-value approach); and
vi. drawing a conclusion.
(c) [4 marks] Simon formed his own mutual fund by randomly picking 100 stocks with equal weights from the technology sector. He also achieved an average annual return of 0.35 in 2020. Do you think Simonís mutual fund outperforms the technology sector of the stock market at the 5% signiÖcance level? Explain your answer.
2022-03-25