1. (1) Pick any 5 of your favorite real numbers from the interval (0,100]. A pmf (f (x)) has the following form based on your picks {x1, x2, x3, x4, x5}, where
f (x) = 0.2 x=x1, x2, x3, x4, x5;
0 elsewhere.
(i) Compute E(x) and E(x2). (ii) Use the results to find E[(x + 2021)2]. (iii) Draw the graph of the corresponding CDF of f (x). (iv) What is the probability that two students pick the exactly same five numbers?
(2) Let f (x) be the pdf of the continuous r.v X, where
f (x) = kx 0 ≤ x ≤ 1;
0 elsewhere.
(i) What is the value of k? (ii) Compute the mean and variance of x; (iii) Compute E( 2019 )
and E( 2020 ) − 2021
; (iv) Draw the graph of the corresponding CDF of f (x).
2. Suppose you don’t know the distribution of the r.v.,
(i) Given the MGF of r.v. x as Mx(t) = exp(−2020t + 2021t2), use the property of MGF to find the mean and variance of x.
(ii) Given the MGF of r.v. y as My(t) = (1 − 2t)−505, use the property of MGF to find the mean and variance of y.
3. Empirical Data exercise.
(1) Pick a data set consisting of FIVE stocks or indices or FX from Yahoo Finance or other source at daily level. Time periods: from Jan. 01, 2010 to Dec. 31, 2020. Use all five time series to do the following. (As a note: you are going to use this dataset for all assignments throughout the whole term for this course.)
(2) Histogram the distributions of the prices for each. Calculate the mean, variance, skew- ness and kurtosis your price series. Plot the ACF for each series.
(3) Histogram the distribution of the returns (use the following fomula). Calculate the mean, variance, skewness and kurtosis your return series. Plot the ACF for each series. Do you think they are normally distributed?
rt = 100[log(pt) − log(pt−1)]
(4) Histogram the distribution of the squared returns. Plot the ACF for each squared return series.
(5) Calculate the correlation between the return and trading volume for each.
(6) Write a code for computing the following conditional probabilities for each data series:
(i) Prob (stock price increases at t | stock price increases at t − 1)
(ii) Prob (stock price increases at t | stock price decreases at t − 1)
(iii) Prob (stock price decreases at t | stock price increases at t − 1)
(iv) Prob (stock price decreases at t | stock price decreases at t − 1)
(7) Write a code for computing the following conditional probabilities for each data series:
(i) Prob (stock price increases at t | trading volume increases at t − 1)
(ii) Prob (stock price increases at t | trading volume decreases at t − 1)
(iii) Prob (stock price decreases at t | trading volume increases at t − 1)
(iv) Prob (stock price decreases at t | trading volume decreases at t − 1)
4. Do the following Monte Carlo simulation:
(1) Assume st N (μ, σ2). Set μ to be equal to the first non-zero number of your student ID. And set σ2 to be equal to the last non-zero number of your student ID. Simulate T data points for st. (T is equal to the sample size of your empirical returns in 3(3)).
(2) Recall that the returns are constructed as follows:
rt = 100[log(pt) − log(pt−1)]
(3) Pick one out of your five time series from the previous question. Run a regression of your empirical return data from (2) on the data simulated in (1). In other words, rt = α + βst + ut. Report your results. Are the coefficients statistically significant? What about R2?
(4) Repeat (3) 1000 times and count the times when the coefficients are statistically signif- icant. (You don’t need to report all 1000 times regressions’ results.)
