Problem 2.1 This problem is based on the data contained in the script file utilities.asc.

Opening it in S-Plus, and running it as a script creates a matrix with two columns. Each row corresponds to a given day. The first column gives the log of the weekly return on an index based on Southern Electric stock value and capitalization, (we’ll call that variable X), and the second column gives, on the same day, the same quantity for Duke Energy (we’ll call that variable Y ), another large utility company.

1. Compute the means and the standard deviations of X and Y , and compute their correlation coefficients.

2. We first assume that X and Y are samples from a jointly Gaussian distribution with parameters computed in question 1. Compute the q-percentile with q = 2%of a the variables X + Y and X − Y .

3. Fit a generalized Pareto distribution (GPD) to X and Y separately, and fit a copula of the Gumbel family to the empirical copula of the data.

4. Generate a sample of size N (where N is the number of rows of the data matrix) from the joint distribution estimated in question 3.
4.1. Use this sample to compute the same statistics as in question 1 (i.e. means and standard deviations of the columns, as well as their correlation coefficients), and compare the results to the numerical values obtained in question 1.
4.2. Compute, still for this simulated sample, the two percentiles considered in question 2, compare the results, and comment.