LONG-TERM BONDS: A Two-Factor Model 2

Campbell, Lo and MacKinlay (1997) note that £m>t+i only affects the average level of the real term structure and not its average slope or time-series behavior. Accordingly, we can either drop it or identify its variance with an additional restriction. We follow the second approach and introduce equities in the model. We assume that the unexpected log excess return on equities is affected by shocks to both the expected and unexpected log SDF:

The variance term on the left hand side of (4) is a Jensen’s Inequality correction that appears because we are working in logs, and the terms on the right hand side relate the risk premium on equities to the covariance of equity returns with innovations in the SDF. This specification implies that the equity premium, like all other risk premia in the model, is constant over time. Thus it ignores the time-variation in the equity premium that is the subject of our earlier paper on long-rim portfolio choice (Campbell and Viceira 1999).

Pricing indexed bonds

Our model can price both indexed bonds and nominal bonds. In this section we show how to price indexed bonds, defined as zero-coupon bonds paying one unit of consumption at maturity and free of default risk.

Characterizing the stochastic discount factor is equivalent to characterizing the return on the one-period indexed bond, since rlit+i = — logEt[Mf+i]. Because Mt+i is lognormal, we have that real payday loans