Consider the daily \(\log\) returns of the GE stock of Exercise 7.1. Obtain estimates \(\hat{\theta}_{b}^{(1)}\) and \(\hat{\theta}_{r}^{(3)}\) of the extremal index of

(a) the positive return series and

(b) the negative return series, using block sizes \(k=5\) and 10 and threshold \(2.5 \%\).

Exercise 7.1:

Consider the daily returns of GE stock from January 2, 1998, to December 31, 2008. The data can be obtained from CRSP or the file d-ge9808 . txt. Convert the simple returns into log returns. Suppose that you hold a long position on the stock valued at \(\$ 1\) million. Use the tail probability 0.01 . Compute the value at risk of your position for 1-day horizon and 15-day horizon using the following methods:

(a) The RiskMetrics method.

(b) A Gaussian ARMA-GARCH model.

(c) An ARMA-GARCH model with a Student- \(t\) distribution. You should also estimate the degrees of freedom.

(d) The traditional extreme value theory with subperiod length \(n=21\).