Chapter 5

Modeling Risk Factors*

We now turn to an analysis of the distribution of risk factors used in financial risk management. A common practice is to use the volatility as a single measure of dispersion. More generally, risk managers need to consider the entire shape of the distribution as well as potential variation in time of this distribution.

The normal distribution is a useful starting point due to its attractive properties. Unfortunately, most financial time series are characterized by fatter tails than the normal distribution. In addition, there is ample empirical evidence that risk changes in a predictable fashion. This phenomenon, called volatility clustering, could also explain the appearance of fat tails. Extreme observations could be drawn from periods with high volatility. This could cause the appearance of fat tails when combining periods of low and high volatility.

Section 5.1 discusses the sampling of real data and the construction of returns. It shows how returns can be aggregated across time or, for a portfolio, across assets. Section 5.2 then describes the normal and lognormal distributions and explains why these choices are so popular, whereas Section 5.3 discusses distributions that have fatter tails than the normal distribution.

Section 5.4 then turns to time variation in risk. We describe the generalized autoregressive conditional heteroskedastic (GARCH) model and a special case, which is RiskMetrics' exponentially weighted moving average (EWMA). These ...