"Frontmatter". In: Analysis of Financial Time Series

288 VALUEATRISKTable 7.3. Estimation Results of a Two-Dimensional Homogeneous Poisson Modelfor the Daily Negative Log Returns of IBM Stock From July 3, 1962 to December31, 1998. The Baseline Time Interval is 252 (i.e., One Year). The Numbers inParentheses Are Standard Errors, Where “Thr.” and “Exc.” Stand For Thresholdand the Number of Exceedings.Thr. Exc. Shape Par. k Log(Scale) ln(α) Location β(a) Original log returns3.0% 175 −0.30697(0.09015) 0.30699(0.12380) 4.69204(0.19058)2.5% 310 −0.26418(0.06501) 0.31529(0.11277) 4.74062(0.18041)2.0% 554 −0.18751(0.04394) 0.27655(0.09867) 4.81003(0.17209)(b) Removing the sample mean3.0% 184 −0.30516(0.08824) 0.30807(0.12395) 4.73804(0.19151)2.5% 334 −0.28179(0.06737) 0.31968(0.12065) 4.76808(0.18533)2.0% 590 −0.19260(0.04357) 0.27917(0.09913) 4.84859(0.17255)Example 7.7. Consider again the daily log returns of IBM stock from July 3,1962 to December 31, 1998. There are 9190 daily returns. Table 7.3 gives someestimation results of the parameters k,α,β for three choices of the threshold whenthe negative series {−r t } is used. We use the negative series {−r t }, instead of {r t },because we focus on holding a long financial position. The table also shows the numberof exceeding times for a given threshold. It is seen that the chance of dropping2.5% or more in a day for IBM stock occurred with probability 310/9190 ≈ 3.4%.Because the sample mean of IBM stock returns is not zero, we also consider thecase when the sample mean is removed from the original daily log returns. From thetable, removing the sample mean has little impact on the parameter estimates. Theseparameter estimates are used next to calculate VaR, keeping in mind that in a realapplication one needs to check carefully the adequacy of a fitted Poisson model. Wediscuss methods of model checking in the next subsection.7.7.3 VaR Calculation Based on the New ApproachAs shown in Eq. (7.30), the two-dimensional Poisson process model used, whichemploys the intensity measure in Eq. (7.31), has the same parameters as those of theextreme value distribution in Eq. (7.27). Therefore, one can use the same formula asthat of the Eq. (7.28) to calculate VaR of the new approach. More specifically, for agiven upper tail probability p, the (1 − p)th quantile of the log return r t is{ {β +αk1 − [−T ln(1 − p)]k } if k ̸= 0VaR =β + α ln[−T ln(1 − p)] if k = 0,(7.34)where T is the baseline time interval used in estimation. Typically, T = 252 in theUnited States for the approximate number of trading days in a year.