Ebook Dynamic Consumption and Portfolio Choice with Stochastic Volatility in Incomplete Markets
There is strong empirical evidence that the conditional variance of asset returns, particularly stock market returns, is not constant over time. Bollerslev, Chou and Kroner (1992), Campbell, Lo and MacKinlay (1997, Chapter 12), Campbell, Lettau, Malkiel and Xu (2001) and others review the main findings of the ample econometric research on stock return volatility: Stock return volatility is serially correlated, and shocks to volatility are negatively correlated with unexpected stock returns. Changes in volatility are persistent (French, Schwert and Stambaugh 1987, Campbell and Hentschel 1992). Large negative stock returns tend to be associated with increases in volatility that persist over long periods of time. Stock return volatility appears to be correlated across markets over the world (Engle, Ito and Lin 1990, Ang and Bekaert 1999).
While there is an abundant literature exploring the pricing of assets when volatility is time varying, there is not much research exploring optimal dynamic portfolio choice with volatility risk. This situation is unfortunate, because Samuelson (1969) and Merton (1969, 1971, 1973) have shown that time variation in investment opportunities imply optimal portfolio strategies for multi-period investors that can be different from those of single-period, or myopic, investors. Multi-period investors value assets not only for their short-term risk-return characteristics, but also for their ability to hedge consumption against adverse shifts in future investment opportunities. Thus these investors have an extra demand for risky assets that reflects intertemporal hedging.
Intertemporal hedging is not only conceptually interesting; it is also empirically relevant. Recent research summarized in Campbell in Viceira (2002) has found that intertemporal hedging is quantitatively important in light of the observed predictable variation in both interest rates and equity premia in the US (Balduzzi and Lynch 1997, Barberis 2000, Brandt 1998, Brennan, Schwartz and Lagnado 1996, 1997, Campbell and Viceira 1999, 2001, Campbell, Chan and Viceira 2002).
This paper, as well as concurrent work by Liu (2000, 2001) that we discuss below, explores systematically optimal portfolio choice with volatility risk in a continuous time setting. We solve for the optimal consumption and portfolio choice of long horizon investors when there is predictable variation in stock market return volatility, and use these solutions to evaluate the importance of volatility risk for intertemporal hedging in the US stock market.
Investors have two assets available for investment, a riskless asset and a risky asset (“stocks”). We assume that the return on the riskless asset and that the expected return on stocks are both constant. We also assume that stock return precision, the reciprocal of volatility, follows a mean-reverting, square-root process. Volatility is instantaneously correlated with stock returns. We allow for this correlation to be imperfect. As a result we work in a setting where markets are not complete. Itô’s Lemma implies a process for stock return volatility that inherits the properties of the process for precision. We work with precision instead of volatility for mathematical convenience. Note that our assumptions about precision or volatility capture the main stylized empirical facts about stock market volatility.
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