This paper develops a new multivariate modeling framework for intertemporal portfolio choice under a stochastic variance covariance matrix. We consider an incomplete market economy, in which stochastic volatilities and stochastic correlations follow a multivariate diffusion process. Different than in previous GARCH-type specifications, in this setting volatilities and correlations can be conditionally correlated with returns and optimal portfolio strategies include distinct hedging components against volatility and correlation risk. We solve for the optimal portfolio problem and provide simple closed-form solutions that allow us to study the volatility and correlation hedging demands in several realistic asset allocation settings. We document the importance of modeling the multivariate nature of second moments especially in the context of optimal asset allocation. We find that the model can accomodate dynamics that feature nonlinear mean reversion and a non-autonomous Markov process for the correlation. In this setting, the optimal significantly different from the one implied by more common models with constant correlations or single-factor stochastic volatility. In particular, the hedging demand estimated in our model is typically four to five times larger than in univariate stochastic volatility models and includes a substantial correlation hedging component, which tends to increase with the persistence of variance covariance shocks, the strength of leverage effects and the dimension of the investment opportunity set.
The importance of solving portfolio choice models taking into account the time-variation in volatilities and correlations is highlighted by Ball and Torous (2000), who study empirically the correlation process of a number of international stock market indices. They find that the estimated correlation structure is changing dynamically over time. Moreover, they note that the stochastic interrelation of these markets might follow from different reactions to changes in economic policy, or other fundamental changes, and conclude that ignoring the stochastic component of the correlation can easily imply erroneous portfolio choices and risk management. An important strand of the literature has explored the characteristics of this time-variation. Longin and Solnik (1995) reject the null hypothesis of constant international stock market correlations and find that these increase in periods of high volatility. Ledoit, Santa-Clara, and Wolf (2003) show that the level of correlation depends on the phase of the business cycle. Moreover, Erb, Harvey, and Viskanta (1994) find that international markets tend to be more correlated when countries are simultaneously in a recessionary state. Moskowitz (2003) documents that covariances across portfolio returns are highly correlated with NBER recessions and that average correlations are highly time-varying. Ang and Chen (2002) show that the correlation between US stocks and the aggregate US market is much higher during extreme downside movements than during upside movements. Longin and Solnik (2001) and Barndorff-Nielsen and Shephard (2004) find similar results. Another important strand of the literature has provided direct evidence that market integration and financial liberalization change the correlation of emerging markets’ stock returns with the global stock market index (Bekaert and Harvey, 1995, 2000). The implication is that economic policies changing the degree of market integration have structural effects on the comovement of financial markets.
Time-varying correlation can be an important source of economic risk with wide-ranging implications. Pastor and Veronesi (2005) explain the behavior of asset prices during technological revolutions by modeling the change in the nature of the risk associated with new technologies. Initially, this risk is mostly idiosyncratic, due to the small scale of production and the low probability of adoption. However, for the technologies that are ultimately adopted the risk gradually changes from idiosyncratic to systematic as the correlation between cash flow shocks to the new-economy technology and representative agent’s wealth increases. The behavior of correlations plays an important role also in Moskowitz (2003), who argues that some pricing anomalies such as momentum and size effect can be explained by stochastic correlations. Driessen, Maenhout, and Vilkov (2006) document that the implied volatility smile is flatter for individual stock options than for index options and attribute the difference to a priced correlation risk factor. The practical importance of modeling conveniently correlation in portfolio choice has become particularly evident during the 2007-2008 financial markets crisis. To illustrate this, we plot in Figure 1 the joint correlation dynamics between the S&P500, the Nikkey and the FTSE weekly index returns in the period April 2004 to April 2008.
The x?axis (y?axis) represents weekly sample correlations between S&P500 and FTSE (Nikkey) returns, computed using overlapping windows of four months. Thus, each point in the graph represents couples of realized sample correlations between these stock indices. The sample average of the correlation is low for all markets, but the correlation volatility is very large, ranging between -0.1 and 0.6 (-0.3 and 0.8) for the Nikkey correlations (FTSE correlations). Moreover, correlation volatilities tend to be higher (lower) conditional on more moderate (extreme) correlation values. The two correlations processes seem far from being independent: As correlation with the FTSE increases to reach the highest level of about 0.8 in March 2008, the correlation with the Nikkey also increases to reach its highest value of 0.6 in the same month. Interestingly, the co-movement of correlation changes during the dramatic market downturn between November 2007 and March 2008 is almost one to one and it is much higher than in the previous period. The fact that average correlations of stock returns tend to be higher in phases of market downturn, i.e. stocks returns exhibit correlation leverage effects, is well-established in the literature; see Roll (1988) and Ang and Chen (2002), among others. Figure 2 illustrates the correlation leverage effects for the S&P500 returns, with respect to the FTSE and the Nikkey correlation.
