\begin{equation}
    \label{rtk}
    r_{t(k)}:=\text{exp} \left[ \text{log} (\pi_k) + x_t'M^{-1} \hat{\mu}_k - \frac{1}{2} \text{trace}(M^{-1} (\hat{\mu}_k \hat{\mu}_k'+\hat{R}_k))
    +  \frac{1}{\sigma^2} y_{t+1} x_t' \hat{\beta}_k - \frac{1}{2 \sigma^2} \text{trace}(x_tx_t' \ (\hat{\beta}_k \hat{\beta}_k'+\hat{Q}_k)) \right]
\end{equation}
