Consider, again, the daily log returns of Alcoa (AA) stock in Exercise 7.5. Focus now on the daily positive log returns. Answer the same questions as in Exercise 7.5. However, use threshold 3% in fitting the GPD model.

Exercise 7.5:

Consider the daily returns of Alcoa (AA) stock and the S&P 500 composite index (SPX) from 1998 to 2008. The simple returns and dates are in the file d-aaspx9808.txt. Transform the simple returns to \(\log\) returns and focus on the daily negative log returns of AA stock.

(a) Fit the generalized extreme value distribution to the negative AA \(\log\) returns, in percentages, with subperiods of 21 trading days. Write down the parameter estimates and their standard errors. Obtain a scatterplot and a QQ plot of the residuals.

(b) What is the return level of the prior fitted model when 24 subperiods of 21 days are used?

(c) Obtain a QQ plot (against exponential distribution) of the negative \(\log\) returns with threshold \(2.5 \%\) and a mean excess plot of the returns.

(d) Fit a generalize Pareto distribution to the negative log returns with threshold 3.5%. Write down the parameter estimates and their standard errors.

(e) Obtain (i) a plot of excess distribution, (ii) a plot of the tail of the underlying distribution, (iii) a scatterplot of residuals, and (iv) a QQ plot of the residuals for the fitted GPD.

(f) Based on the fitted GPD model, compute the VaR and expected shortfall for probabilities \(q=0.99\) and 0.999 .