LONG-TERM BONDS: A Two-Factor Model

Specification of the model

Our focus in this paper is the microeconomic problem of portfolio choice for an individual investor facing exogenous bond returns. In order to generate empirically reasonable and theoretically well-specified bond returns, however, we start by writing down a general equilibrium bond pricing model. We consider a discrete-time, two-factor homoskedastic model that allows for non-zero correlation between innovations in the short-term real interest rate and innovations in expected inflation.

The real part of the model is determined by the stochastic discount factor (SDF) Mt+1 that prices all assets in the economy. In a representative-agent framework the SDF can be related to the marginal utility of a representative investor, but here we simply use it as a device to generate a complete set of bond prices. We assume that Mt+i has the following lognormal structure, a discrete-time version of Vasicek (1977):

where 7t£+1 is the log inflation rate and zt is the one-period-ahead conditional expectation of the inflation rate.

The system is subject to four normally distributed, white noise shocks ex,t+ъ £m,t+1, £z,t+1, and Sirj+i that determine the innovations to the log SDF, the log inflation rate, and their conditional means. These shocks are cross-sectionally uncorrelated, with variances a\, and a\. It is important to note that zt+i, the expected inflation rate, is affected by both a pure expected-inflation shock £Ztt+i and the shocks to the expected and unexpected log SDF eX)M and That is, innovations to expected inflation can be correlated with innovations in the log SDF, and hence with innovations in the short-term real interest rate. These correlations mean that nominal interest rates need not move one-for-one with expected inflation—that is, the Fisher hypothesis need not hold—and nominal bond prices can include an inflation risk premium as well as a real term premium.

We have written the model with a self-contained real sector (1) and a nominal sector (2) that is affected by shocks to the real sector. But this is merely a matter of modelling convenience. Our model is a reduced form rather than a structural model, so it captures correlations among shocks to real and nominal interest rates but does not have anything to say about the true underlying sources of these shocks.