Dynamic Volatilities and Correlations
The evidence for dynamic patterns in portfolio risk is very strong, with a variety of dynamic patterns clearly documented across a range of asset classes. The influence of these dynamic features on accurate portfolio risk analysis can be substantial. This chapter reviews some of the empirical evidence and discusses analytical refinements to portfolio risk analysis models to account for these dynamic patterns. Section 9.1 deals with generalized autoregressive conditional heteroskedasticity (GARCH) models and section 9.2 with stochastic volatility (SV) models. Section 9.3 discusses time aggregation of risk forecasts in the presence of risk dynamics. Section 9.4 examines the issue of asymmetry in asset return correlations, called asymmetric dependence. Section 9.5 discusses options-implied volatility and its use in portfolio risk management. Section 9.6 looks at portfolio risk for long return horizons and its dependence on expected return dynamics. Section 9.7 looks at dynamics in cross-sectional volatility.
The research literature on GARCH is enormous. This section describes some key results that have particular relevance in portfolio risk analysis. Readers wanting a more detailed treatment of GARCH models can consult Christoffersen (2003), Engle (1995), or Taylor (2005).
In this section we assume that expected return is time constant and known, rather than estimated, in order to focus on the time variation in risk. In any case, most of the dynamic volatility effects discussed in this chapter are more powerful at high frequencies (with the exception of the long-horizon effects discussed in section 9.6). Recall from chapter 1 that for high-frequency portfolio returns the accurate estimation of expected returns is inconsequential to risk measurement. Consider the variance of log return for an asset or portfolio with constant per-period mean return µ and constant per-period variance σ2. Consider the relationship