"Frontmatter". In: Analysis of Financial Time Series

EXTREME VALUE TO VAR 281VaR =−2.583 − 0.945 {1 − [−63 ln(1 − 0.01)] −0.335}−0.335=−3.04969.Thus, for daily log returns of the stock, the 1% quantile is −3.04969. If one holds along position on the stock worth $10 million, then the estimated VaR with probability1% is $10, 000, 000 × 0.0304969 = $304,969. If the probability is 0.05, then thecorresponding VaR is $166,641.If we chose n = 21 (i.e., approximately 1 month), then ˆα n = 0.823, ˆβ n =−1.902,and ˆk n =−0.197. The 1% quantile of the extreme value distribution isVaR =−1.902 − 0.823−0.197 {1 −[−21 ln(1 − 0.01)]−0.197 }=−3.40013.Therefore, for a long position of $10,000,000, the corresponding 1-day horizon VaRis $340,013 at the 1% risk level. If the probability is 0.05, then the correspondingVaR is $184,127. In this particular case, the choice of n = 21 gives higher VaRvalues.It is somewhat surprising to see that the VaR values obtained in Example 7.6using the extreme value theory are smaller than those of Example 7.3 that uses aGARCH(1, 1) model. In fact, the VaR values of Example 7.6 are even smaller thanthose based on the empirical quantile in Example 7.5. This is due in part to thechoice of probability 0.05. If one chooses probability 0.001 = 0.1% and considersthe same financial position, then we have VaR = $546,641 for the Gaussian AR(2)-GARCH(1, 1) model and VaR = $666,590 for the extreme value theory with n =21. Furthermore, the VaR obtained here via the traditional extreme value theory maynot be adequate because the independent assumption of daily log returns is oftenrejected by statistical testings. Finally, the use of subperiod minima overlooks thefact of volatility clustering in the daily log returns. The new approach of extremevalue theory discussed in the next section overcomes these weaknesses.Remark: As shown by the results of Example 7.6, the VaR calculation based onthe traditional extreme value theory depends on the choice of n, which is the lengthof subperiods. For the limiting extreme value distribution to hold, one would preferalargen. But a larger n means a smaller g when the sample size T is fixed, whereg is the effective sample size used in estimating the three parameters α n ,β n ,andk n .Therefore, some compromise between the choices of n and g is needed. A properchoice may depend on the returns of the asset under study. We recommend that oneshould check the stability of the resulting VaR in applying the traditional extremevalue theory.7.6.1 DiscussionWe have applied various methods of VaR calculation to the daily log returns of IBMstock for a long position of $10 million. Consider the VaR of the position for the