Staying ahead on downside risk 147
exist portfolios that would be chosen by all nonsatiable risk averse agents over
the mean – variance solution.
6.2.2 Modeling EVaR dynamically
Another notable advantage of expectiles is their mathematical tractability,
as argued in Newey and Powell (1987). Dynamic expectile models have been
derived in Taylor (2008) , Kuan et al. (2009), and De Rossi and Harvey (2009).
An early example appeared in Granger and Sin (2000). Computationally, simple
univariate models compare favorably to GARCH or dynamic quantile models.
The analysis presented in this chapter is based on the dynamic model of De
Rossi and Harvey (2009). Intuitively, their estimator produces a curve rather
than a single value μ , so that it can adapt to changes in the distribution over
time. The two parameters needed for the estimation are ω and q.
The former can be interpreted as a prudentiality level: the lower ω , the more
risk aversion. Figure 6.1 shows a typical expectile plot for alternative values of ω.
By decreasing ω , one focuses on values that are further out in the lower tail,
i.e., more severe losses.
By increasing q , we can make the model more flexible in adapting to the
observed data. The case q 0 corresponds to the constant expectile (estimated
by the sample expectile). As Figure 6.2 shows, larger values of q produce esti-
mated curves that follow more and more closely the observations.
0.040.02Return0.00–0.02–0.041996 1998 2000 2002 2004 2006 2008Figure 6.1 Time-varying expectiles. The solid black lines represent estimated dynamic
expectiles for ω 0.05, 0.25, 0.5 (mean), 0.75, and 0.95.