"Frontmatter". In: Analysis of Financial Time Series

ECONOMETRIC APPROACH TO VAR 265The 5% quantile of a Student-t distribution with 5 degrees of freedom is −2.015 andthat of its standardized distribution is −2.015/ √ 5/3 =−1.5608. Therefore, the 5%quantile of the conditional distribution of r 9191 given F 9190 is0.000367 − 1.5608 √ 0.0003386 =−0.028354.The VaR for a long position of $10 million isVaR = $10,000,000 × 0.028352 = $283,520,which is essentially the same as that obtained under the normality assumption. The1% quantile of the conditional distribution is0.000367 − (3.3649/ √ 5/3) √ 0.0003386 =−0.0475943.The corresponding VaR is $475,943. Comparing with that of Case I, we see theheavy-tail effect of using a Student-t distribution with 5 degrees of freedom; itincreases the VaR when the tail probability becomes smaller.7.3.1 Multiple PeriodsSuppose that at time h we like to compute the k-horizon VaR of an asset whose logreturn is r t . The variable of interest is the k-period log return at the forecast originh (i.e., r h [k] =r h+1 +···+r h+k ). If the return r t follows the time series model inEqs. (7.5) and (7.6), then the conditional mean and variance of r h [k] given the informationset F h can be obtained by the forecasting methods discussed in Chapters 2and 3.Expected Return and Forecast ErrorThe conditional mean E(r h [k] |F h ) can be obtained by the forecasting method ofARMA models in Chapter 2. Specifically, we haveˆr h [k] =r h (1) +···+r h (k),where r h (l) is the l-step ahead forecast of the return at the forecast origin h. Theseforecasts can be computed recursively as discussed in subsection 2.6.4. Using theMA representationr t = µ + a t + ψ 1 a t−1 + ψ 2 a t−2 +···of the ARMA model in Eq. (7.5), we can write the l-step ahead forecast error at theforecast origin h ase h (l) = r h+l − r h (l) = a h+l + ψ 1 a h+l−1 +···+ψ l−1 a h+1 ;