LONG-TERM BONDS: Introduction 2

Merton (1969, 1971, 1973) studied the intertemporal portfolio choice problem with time-varying investment opportunities, introducing the important concept of intertemporal hedging demand for financial assets, but he did not obtain explicit solutions for portfolio weights. Recently a number of authors such as Balduzzi and Lynch (1997), Barberis (1998), Brandt (1998), and Brennan, Schwartz, and Lagnado (1996, 1997) have used numerical methods to solve particular long-run portfolio choice problems, while Kim and Omberg (1996) and Campbell and Viceira (1999) have derived some analytical results, but these papers generally concentrate on the choice between cash and equities rather than the demand for long-term bonds.

In this paper we study intertemporal portfolio choice in an environment with random real interest rates. We use an approximation technique developed in our earlier papers (Campbell 1993, Campbell and Viceira 1999) to replace the intractable portfolio choice problem with an approximate problem that can be solved using the method of undetermined coefficients. We use the approximate solution to understand the demand for long-term bonds.

We calibrate our model to historical data on the US term structure of interest rates, and report optimal portfolios for investors with a wide range of different attitudes towards risk and intertemporal substitution of consumption. In order to study the effects of inflation risk on optimal bond portfolios and investor welfare, we compare the solutions to our model when only indexed bonds are available with the solutions when only nominal, or both nominal and indexed bonds are available. We also allow for borrowing and short-sales constraints, and for the possibility of investment in equities.

We begin by specifying a simple two-factor model of the term structure of interest rates, augmented to fit equity as well as bond returns. The two factors are the log real interest rate and the log expected rate of inflation. Each factor follows a normal first-order autoregressive (AR(1)) process with constant variance. This implies that log bond yields are linear in the factors and the model is in the tractable “affine yield” class (Dai and Singleton 1997, Duffie and Kan 1996). The model for the real term structure is a discrete-time version of Vasicek (1977), while the model for the nominal term structure is a discrete-time version of Langetieg (1980). Closely related models are discussed in Campbell, Lo, and MacKinlay (1977), Chapter 11.

Next we consider the portfolio choice problem for an infinitely-lived investor who has only financial wealth and must choose consumption and optimal portfolio weights in each period. Because the investor is infinitely-lived, she does not value stability of wealth at any unique horizon; rather she cares about the long-run properties of her consumption path. payday loan online