"Frontmatter". In: Analysis of Financial Time Series

RISKMETRICS 259we use log returns r t in data analysis. The VaR calculated from the quantile of thedistribution of r t+1 given information available at time t is therefore in percentage.The dollar amount of VaR is then the cash value of the financial position times theVaR of the log return series.Remark: VaR is a prediction concerning possible loss of a portfolio in a giventime horizon. It should be computed using the predictive distribution of future returnsof the financial position. For example, the VaR for a 1-day horizon of a portfoliousing daily returns r t should be calculated using the predictive distribution of r t+1given information available at time t. From a statistical viewpoint, predictive distributiontakes into account the parameter uncertainty in a properly specified model.However, predictive distribution is hard to obtain, and most of the available methodsfor VaR calculation ignore the effects of parameter uncertainty.7.2 RISKMETRICSJ.P. Morgan developed the RiskMetrics TM methodology to VaR calculation; seeLongerstaey and More (1995). In its simple form, RiskMetrics assumes that thecontinuously compounded daily return of a portfolio follows a conditional normaldistribution. Denote the daily log return by r t and the information set available attime t −1byF t−1 . RiskMetrics assumes that r t | F t−1 ∼ N(µ t ,σt 2),whereµt is theconditional mean and σt2 is the conditional variance of r t . In addition, the methodassumes that the two quantities evolve over time according to the simple model:µ t = 0, σt 2 = ασt−1 2 + (1 − α)r t−1 2 , 1 >α>0. (7.2)Therefore, the method assumes that the logarithm of the daily price, p t = ln(P t ),of the portfolio satisfies the difference equation p t − p t−1 = a t ,wherea t = σ t ɛ tis an IGARCH(1, 1) process without a drift. The value of α is often in the interval(0.9, 1).A nice property of such a special random-walk IGARCH model is that the conditionaldistribution of a multiperiod return is easily available. Specifically, for ak-period horizon, the log return from time t + 1 to time t + k (inclusive) is r t [k] =r t+1 + ··· +r t+k−1 + r t+k . We use the square bracket [k] to denote a k-horizonreturn. Under the special IGARCH(1,1) model in Eq. (7.2), the conditional distributionr t [k] |F t is normal with mean zero and variance σt 2 [k], whereσ2t [k] can becomputed using the forecasting method discussed in Chapter 3. Using the independenceassumption of ɛ t and model (7.2), we haveσ 2t [k] =Var(r t[k] |F t ) =k∑Var(a t+i | F t ),where Var(a t+i | F t ) = E(σ 2t+i| F t ) can be obtained recursively. Using r t−1 =a t−1 = σ t−1 ɛ t−1 , we can rewrite the volatility equation of the IGARCH(1, 1) modeli=1