Extreme Value Theory

I begin by applying extreme value theory (EVT) to model fat-tailed return distributions, specifically, the distribution of losses exceeding a prespecified lower threshold. Extreme value theory has received much attention in the insurance industry to predict rare events such as floods, earthquakes, and other natural disasters and has a large devoted literature. See for example, the text by Embrechts, Kluppelberg, and Mikosch (1997). Our interest centers not only on applying extreme value theory to predict rare market events but, more importantly, monitoring changes in the parameters of the extreme value distributions themselves, which may signal a fundamental shift in downside risks. There is also a deep and rich literature on applying EVT to financial markets. See, for example, Cotter (2006), Longin and Solnik (2001), LeBaron, Blake, and Samanta (2004), Malevergne, Pisarenko, and Sornette (2006).

There is a related literature on extreme outcomes, specifically, the distribution of order statistics (see, for example, Mood et al. 1974 for an introduction, and the related quantile regression theory developed by Koenker and Basset [1978], which models specific extreme quantiles, for example, the fifth percentile of the distribution of returns). EVT, on the other hand, models the likelihood function for the tail density and not a quantile, which in our case, is the set of minimum returns found in the left-hand tail. The challenge with EVT is to estimate the tail parameters from a finite sample of data using either maximum likelihood (as in this chapter) or Monte Carlo simulation (Longin and Solnik 2001). Thus, we will model downside risks, where downside is taken to be those observed returns that fall below a specific loss threshold—in our case, the fifth percentile. Our choice of the empirical quantile that determines the threshold is subjective but sensible (Ledford and Tawn 1996).

Graphs such as those shown in Figure 13.1, illustrate this process.

Figure 13.1 Pre- and Post-Lehman Tail Densities

Let's define the distribution of interest as the extra-threshold returns—what we call the fail event criteria (losses beyond the defined threshold). More formally, we model the likelihood function for these returns using the generalized extreme value (GEV) probability density and estimate its parameters using the method of maximum likelihood. We assume intervals are independent with exponential distribution. The generalized extreme value distribution is a three-parameter family: scale (σ), a measure of the dispersion of return events, location (μ), which shows the average position of the extreme within the distribution, and shape (κ), a measure of skewness. In general, these loss distributions will be skewed to the left but their specific shapes will depend on the values of their specific shape (κ > 0), and scale (σ), parameters. Denoting daily returns by r, the density function for the GEV distribution is defined as (Embrechts, Kluppelberg, and Mikosch [1997]):

Malevergne, Pisarenko, and Sornette (2006) show that estimates of the GEV are inefficient in the absence of the independence of returns and that this inefficiency confounds the precision of point estimates of the distribution's parameters. They show that the generalized Pareto (GPD) performs somewhat better but suffers likewise in the presence of strong dependence in returns. We show results from both distributions with the GPD density given by:

This is a three-parameter distribution as well, but where θ is the threshold value. In both cases—GEV and GPD—we bootstrap the parameter estimates to provide standard errors. A discussion of these results follows with 13.1, 13.2, and 13.3.

Table 13.1 Daily Total Return Series Descriptions.

Table 13.2 GEV Parameter Estimates and Confidence Intervals.

Table 13.3 GPD Parameter Estimates and Confidence Intervals.

Let's now bring this together through an examination of the recent volatile market events. The five asset classes analyzed are U.S. equities, non- U.S. equities, U.S. fixed income, high yield bonds, and REITs. They're all described in Table 13.1.

We compare results across two periods. The pre-Lehman period is estimated from February 1, 1996, through August 31, 2008, and the post-Lehman period has the same start date but is estimated through December 2008. These periods serve to demonstrate the impact of the turbulent markets witnessed in the fall of 2008. Table 13.1 shows a large post-Lehman shift in the GEV parameters (location, shape, and scale) and this, in turn, clearly reveals the increased likelihood of downside risks.

The parameter estimates for virtually all of these return distributions appear to have degraded into higher risk states; average tail losses are higher (μ has declined), dispersion has increased (σ) and losses have become more skewed (κ). Nevertheless, a higher risk state does not necessarily follow from these changed parameter values for the reasons cited in Malevergne et al. (2006), that is, in the presence of low parameter precision, these values may be statistical artifacts from the same distribution. We therefore bootstrapped 1,000 samples from both the pre-Lehman and post-Lehman histories (pre-Lehman is a subset of the post-Lehman set). We estimated the parameters of the likelihood function for each sample and then located the 2.5 percent and 97.5 percent percentiles along with the median across the bootstrapped samples. These results are presented in Table 13.2 for the GEV and in Table 13.3 for the GPD.

From Table 13.2, we see that the median value for the location parameter, post-Lehman, lies outside the 95 percent band constructed from the pre-Lehman returns data for all markets but fixed income. The same conclusion holds for the scale parameter, but only for non-U.S. equity and REITs regarding the shape parameter. Table 13.3 echoes these results for the threshold parameters but shows somewhat weaker results for scale and shape. In all cases, equity and REITs show the strongest divergence from the pre-Lehman downside risks.

While these may not constitute formal tests, they do support the empirical evidence of increased incidences of extreme losses and at the very least are useful monitoring devices. Failure to recognize shifts in these risk parameters would result in an underestimation of the probability of larger losses occurring in the future. This cost, in percent, is approximated in Figure 13.1 by the area between the pre and post distribution functions; if we believe that these distributions have changed, then for REITs, this cost amounts to an 8.44 percent underestimate of the mass to the left of the fifth percentile while that for high yield would have been 7.5 percent. Furthermore, losses would be underestimated by 7.25 percent for non-U.S. equity and 6.67 percent for U.S. equity. REITs, especially, experienced a large parametric shift in risk with expected downside loss increasing by roughly 20 percent, (µ changes from –1.97 percent to –2.32 percent) and with volatility nearly doubling (σ changes from 0.41 percent to 0.67 percent).