Exercise Solution 14.9

1. b. Given the assumption that ^tL is conditionally normal with ^t–1E(^tL) = 0, and knowing that the the .99-quantile of a normal distribution occurs 2.326 standard deviations above the mean, we can infer from the value-at-risk values the conditional standard deviations of the ^tL assumed by those value-at-risk values—and hence the conditional normal distributions. With [14.9], recalculate the quantiles ^tu at which losses ^tl occur. With [14.10], we convert these to values ^tn. See spreadsheet.

3. d.  e. We arrange the ^tn in ascending order to form the n_j. We calculate values  as describe in Section 14.4.2. Plotting the against the n_j, we obtain (see the same spreadsheet)

The points fall more-or-less near a line passing through the origin, but the fit is poor. Based on just the graph, it cannot be determined whether the poor fit reflects a shortcoming of the VaR measure or merely the limited data.

6. We calculate the correlation between the n_j and as 0.985. Based on Exhibit 14.6, we reject the VaR measure at the .05 significance level.