This paper solves numerically the intertemporal consumption and portfolio choice problem of an infinitely lived investor who faces a time varying equity premium. There is now considerable evidence that the excess return on stocks over Treasury bills is predictable (see Campbell 1987, Campbell and Shiller 1988, Fama and French 1988, 1989, Hodrick 1992, or the textbook treatment in Campbell, Lo, and MacKin lay 1997, Chapter 7). Merton (1969, 1971), Samuelson (1969), and Giovannini and Weil (1989) have shown that time variation in investment opportunities affects portfolio choice unless investors have unit relative risk aversion. But the large literature on the equity premium puzzle finds that average excess stock returns are too high to be consistent with a representative0investor model with unit relative risk aversion (see Campbell 1996, Cecchetti, Lam, and Mark 1994, Cochrane and Hansen 1992, Hansen and Jagannathan 1991, Kocherlakota 1996, Mehra and Prescott 1985, or the textbook treatment in Campbell, Lo, and MacKinlay 1997, Chapter 8). Therefore, it is important to analyze optimal consumption and portfolio decisions when there is time variation in the investment opportunity set and investors have risk aversion different from one.
The problem, however, is not trivial analytically. Nonlinearities in both the Euler equations and the intertemporal budget constraint make it extremely hard to find exact analytical solutions. Recently a few special cases have been solved. In a continuous time model with a constant riskless interest rate and a single risky asset whose expected return follows a mean0reverting AR(1) process, for example, the model can be solved if long lived investors have power utility defined over terminal wealth (Kim and Omberg 1996), or if investors have power utility defined over consumption and the innovation to the expected asset return is perfectly correlated with the innovation to the unexpected return, making the asset market effectively complete (Wachter 1999), or if the investor has Epstein0Zin utility with intertemporal elasticity of substitution restricted to equal one (Campbell and Viceira 1999, Schroder and Skiadas 1999).
We use a numerical solution method that allows us to consider a somewhat more general discrete time model than any of these special cases. We assume a constant riskless interest rate and an AR(1) process for the risky asset return, but we do not assume perfect correlation between innovations to the expected and unexpected return, and we allow the investor to have general Epstein0Zin utility defined over consumption. We begin by discretizing the state0space and approximating the distribution for the innovations in the random variables using Gaussian quadrature. The solution algorithm assumes a portfolio allocation rule which is a p'th order polynomial in the state variable and uses a variant of the Newton Raphson algorithm to optimize over the coefficients of this polynomial. We use the Den Haan0Marcet (DHM) statistic (Den Haan and Marcet 1994) to choose the optimal value for p, and to evaluate the accuracy of the numerical solution. We find that the portfolio rule is approximately linear in the state variable while the log consumption wealth ratio is approximately quadratic. These approximations break0down as the state variable deviates substantially from its unconditional mean.
Our use of Epstein Zin preferences (Epstein and Zin (1989), Weil (1990)) allows us to consider different combinations of risk aversion and the elasticity of intertemporal substitution in consumption. We find that the importance of hedging demand for portfolio choice depends strongly on risk aversion, but hardly at all on the elasticity of intertemporal substitution.
We also consider a constrained version of the problem in which the investor is not allowed to borrow at the riskless interest rate or to short sell the risky asset. Such constraints are realistic, and they affect the form of the solution since the investors optimal plans take account of the possibility that the constraints may bind in the future, even if they are not binding today. We find that, in the region in which they are not binding, these constraints have a large impact on the consumption rule but little effect on the portfolio rule.
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PDF Stock Market Mean Reversion and the Optimal Equity Allocation of a Long-Lived Investor
