J. P. Morgan developed the RiskMetrics methodology to VaR calculation; see Longerstaey and More (1995). In its simple form, RiskMetrics assumes that the continuously compounded daily return of a portfolio follows a conditional normal distribution. Denote the daily log return by rt and the information set available at time t − 1 by Ft−1. RiskMetrics assumes that , where μt is the conditional mean and is the conditional variance of rt. In addition, the method assumes that the two quantities evolve over time according to the simple model:
Therefore, the method assumes that the logarithm of the daily price, pt = ln(Pt), of the portfolio satisfies the difference equation pt − pt−1 = at, where at = σtϵt is an IGARCH(1,1) process without drift. The value of α is often in the interval (0.9, 1) with a typical value of 0.94.
A nice property of such a special random-walk IGARCH model is that the conditional distribution of a multiperiod return is easily available. Specifically, for a k-period horizon, the log return from time t + 1 to time t + k (inclusive) is rt[k] = rt+1 + ⋯ + rt+k−1 + rt+k. We use the square bracket [k] to denote a k-horizon return. Under the special IGARCH(1,1) ...