By definition, the bond price, *B*_{t} is:

where *T* is the maturity, *k*+ 1 is the number of coupon payments remaining, *C* is the half-year coupon payment rate, *A* is the face value of debt, and *r* is the yield to maturity for *k*+ 1 coupon payments remaining. We assume that *r*_{t} follows some unspecified stochastic process. By Ito's lemma we have:

- ((A1))

where Λ_{t} is the square of the diffusion coefficients of *r*_{t} process. If *r*_{t} is a multivariate process, then Λ_{t} should also include the covariance terms. Therefore, from equation (A1):

- ((A2))

where *D*_{t} is the bond's duration. We can rewrite equation (A2) as:

- ((A3))

Barring arbitrage, there exists some state price density process, Λ_{t}, such that:

In equilibrium, the risky bond return should satisfy:

- ((A4))

where cov _{t}(*dP*_{t}/*P*_{t}, *d*Λ_{t}/Λ_{t}) is the instantaneous conditional covariance, and *r*_{t}=−μ_{Λ,t}/Λ_{t} is the risk-free rate. We obtain the second equality above by using equation (A3). Following the discrete-time empirical literature, we further assume that the state price density is a linear function of both the market equity return and the long-term risk-free bond return (e.g., Scruggs (1998)). This implies that:

- ((A5))

where, γ_{i}, *i*= 1, 2, is the price of risk associated with the respective state variable, *dB*_{l,t}/*B*_{l,t} is the long-term bond return, and *dM*_{t}/*M*_{t} is the equity market return. In the empirical implementation, we will make the two adjustments. First, we only measure the proportional daily price change in *dB*_{t}/*B*_{t}. Second, we do not consider daily accrued interest. The latter condition means we consider only clean prices. In summary, bond price changes are only driven by long-term risk-free bond returns and equity returns. We also assume the conditional covariances are constant.^{11} This leads to the following specification for equation (1) in the text:

where *R*_{j,t} is the daily return for bond *j* that investors would bid given zero transaction costs, *Duration*_{j,t} is the bond's duration, and Δ S&P Index _{t} is the daily S&P equity return. Duration _{j,t}×Δ*R*_{l,t} is the proportional bond return of a long-term risk-free bond adjusted by the duration of the risky bond. The scaling of the market sensitivities by duration is consistent with Jarrow (1978).