When I assume an one-factor Gaussian term structure model such as the Vasicek model

$dr_t = \kappa(\mu - r_t)dt + \sigma dW_t$

and specify a constant market price of risk of $dW$ to be $\lambda_t = \lambda_y$ a constant, use the usual change of measure

$d\widetilde{W}_t = dW_t + \lambda_y dt$

and solve for the drift term of the return of the zero coupon bond

$\frac{dP(t, \tau}{P(t, \tau)} = \mu_P(r_t, \tau) dt + \sigma_P(r_t, \tau) dW_t$

I get the bond risk premia to be

$\mu_P - r_t = -b(\tau)\sigma\lambda_y$

which is a CAPM-type equation.

However, if I apply CAPM to the zero-coupon bond, e.g. calculate the returns on the zero coupon bond and calculate its correlation $\rho_{r_B, r_M}$ with some global investible market $M$, I get the expected excess return on $\tau$-year zero coupon bond to be $\rho_{r_B, r_M} \sigma_{r_B} \lambda$ where $\lambda$ is the "global price of risk" calculated from the returns of the global investible market.

These two bond risk premia are both CAPM-type but I have trouble internalize them together. For example, in the Gaussian affine model case, there is not a term that looks like correlation, which appears in the CAPM solution. What is the connection between the bond risk premier from the Gaussian affine term structure model and the one from the good old CAPM?


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