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Analysis of Financial Time Series, Third Edition by RUEY S. TSAY

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7.6 Extreme Value Approach to VaR

In this section, we discuss an approach to VaR calculation using the extreme value theory. The approach is similar to that of Longin (1999a,b), who proposed an eight-step procedure for the same purpose. We divide the discussion into two parts. The first part is concerned with parameter estimation using the method discussed in the previous subsections. The second part focuses on VaR calculation by relating the probabilities of interest associated with different time intervals.

Part I

Assume that there are T observations of an asset return available in the sample period. We partition the sample period into g nonoverlapping subperiods of length n such that T = ng. If T = ng + m with 1 ≤ m < n, then we delete the first m observations from the sample. The extreme value theory discussed in the previous section enables us to obtain estimates of the location, scale, and shape parameters βn, αn, and ξn for the subperiod maxima {rn, i}. Plugging the maximum-likelihood estimates into the CDF in Eq. (7.16) with x = (r − βn)/αn, we can obtain the quantile of a given probability of the generalized extreme value distribution. Let p* be a small upper tail probability that indicates the potential loss and Inline be the (1 − p*)th quantile of the subperiod maxima under the limiting generalized extreme value distribution. Then we have

where it is understood that for ξ ...

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