4.1. CHAPTER OVERVIEW

We now turn to the third and final part of the stepwise distribution modeling (SDM) approach, namely accounting for conditional non-normality in portfolio returns. In Chapter 1, we saw that asset returns are not normally distributed unconditionally. If we construct a simple histogram of past returns on the S&P 500 index, then it will not conform to the density of the normal distribution: The tails of the histogram are fatter than the normal, and the histogram is more peaked around zero. From a risk management perspective, the fat tails, which are driven by relatively few but very extreme observations, are of most interest. These extreme observations can by symptoms of liquidity risk or event risk as defined in Chapter 1.

One motivation for the time-varying variance models discussed in Chapter 2 is that they are capable of accounting for the unconditional non-normality of the data. For example, a GARCH(1,1) model with normally distributed innovations will imply an unconditionally non-normal distribution, so if one drew a histogram of returns from the GARCH model, they would have fat tails.

Simple normal GARCH models by definition do not capture conditional non-normality in the returns. Returns are conditionally normal if the standardized returns (i.e., returns divided by their time-varying standard deviation) are normally distributed. Unfortunately, histograms from standardized returns typically do not conform to the normal density. Figure 4.1 illustrates this point. The top panel shows the histogram of the raw returns superimposed on the normal distribution, and the bottom...