"Frontmatter". In: Analysis of Financial Time Series

284 VALUEATRISKFollowing similar procedures as those of long positions, we obtain the (1 − p)thquantile of the return r t as⎧⎨β n + α n { }1 − [−n ln(1 − p)]k nif k n ̸= 0VaR = k n⎩β n + α n ln[−n ln(1 − p)] if k n = 0,(7.28)where p is a small probability denoting the chance of loss for holding a short position.7.7 A NEW APPROACH BASED ON THE EXTREME VALUE THEORYThe aforementioned approach to VaR calculation using the extreme value theoryencounters some difficulties. First, the choice of subperiod length n is not clearlydefined. Second, the approach is unconditional and, hence, does not take into considerationeffects of other explanatory variables. To overcome these difficulties, amodern approach to extreme value theory has been proposed in the statistical literature;see Davison and Smith (1990) and Smith (1989). Instead of focusing onthe extremes (maximum or minimum), the new approach focuses on exceedances ofthe measurement over some high threshold and the times at which the exceedancesoccur. For instance, consider the daily log returns r t of IBM stock used in this chapterand a long position on the stock. Let η beaprespecified high threshold. We maychoose η =−2.5%. Suppose that the ith exceedance occurs at day t i (i.e., r ti ≤ η).Then the new approach focuses on the data (t i , r ti −η).Herer ti −η is the exceedanceover the threshold η and t i is the time at which the ith exceedance occurs. Similarly,for a short position, we may choose η = 2% and focus on the data (t i , r ti − η) forwhich r ti ≥ η.In practice, the occurrence times {t i } provide useful information about the intensityof the occurrence of important “rare events” (e.g., less than the threshold η fora long position). A cluster of t i indicates a period of large market declines. Theexceeding amount (or exceedance) r ti − η is also of importance as it provides theactual quantity of interest.Based on the prior introduction, the new approach does not require the choiceof a subperiod length n, but it requires the specification of threshold η. Differentchoices of the threshold η lead to different estimates of the shape parameter k(and hence the tail index −1/k). In the literature, some researchers believe that thechoice of η is a statistical problem as well as a financial one, and it cannot be determinedpurely based on the statistical theory. For example, different financial institutions(or investors) have different risk tolerances. As such, they may select differentthresholds even for an identical financial position. For the daily log returns of IBMstock considered in this chapter, the calculated VaR is not sensitive to the choiceof η.The choice of threshold η also depends on the observed log returns. For a stablereturn series, η =−2.5% may fare well for a long position. For a volatile return