# LONG-TERM BONDS: A Two-Factor Model 3

The variance term on the left hand side of (9) is a Jensen’s Inequality correction that appears because we are working in logs. The conditional covariance of the excess bond return with the log SDF determines the risk premium. In our homoskedastic model the conditional covariance is constant through time but dependent on the bond maturity; thus the expectations hypothesis of the term structure holds for indexed bonds. It is important to realize that constant risk premia do not imply constant investment opportunities because real interest rates are stochastic in our model read more.

Since Bn-1 > 0, the Jensen 7s-inequality-corrected risk premium is negative if Pmx > 0> znd positive otherwise. With positive f3mx) long-term indexed bonds pay off when the marginal utility of consumption for a representative investor is high, that is, when wealth is most desirable. In equilibrium, these bonds must have a negative risk premium. With negative on the other hand, long-term indexed bonds pay off when the marginal utility of consumption for a representative investor is low, and so in equilibrium they have a positive risk premium.

Equations (8) and (9) imply that the Sharpe ratio for indexed bonds is ~/Зтх Since both Mt+i and IIi+i are jointly lognormal and homoskedastic, Mf+l is also log-normal. The log nominal return on a one-period nominal bond is = — log Et [МД. J, which implies that r* f+1 is a linear combination of the expected log real SDF and expected inflation given in the Appendix. Since nominal bond prices are driven by shocks to both real interest rates and inflation, they have a two-factor structure rather than the single-factor structure of indexed bond prices. Inflation affects the excess return on an n-period nominal bond over the one-period nominal interest rate, so risk premia in the nominal term structure include compensation for inflation risk. Like all other risk premia in the model, however, the risk premia on nominal bonds are constant over time; thus the expectations hypothesis holds for nominal as well as for real bonds.

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