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Analysis of Financial Time Series, Third Edition by RUEY S. TSAY

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7.2 RiskMetrics

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 Inline, where μt is the conditional mean and Inline is the conditional variance of rt. In addition, the method assumes that the two quantities evolve over time according to the simple model:

7.2 7.2

Therefore, the method assumes that the logarithm of the daily price, pt = ln(Pt), of the portfolio satisfies the difference equation ptpt−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) ...

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