What does the equation $COV_{\frac{1}{2}} r_t; x_t = COV(\frac{1}{2} r_t; x_t) + COV(\frac{1}{2} r_{t1}; x_t)$ demonstrate?
This equation shows that the overall beta of $r_t$ with respect to $x_t$ is influenced by both the contemporaneous and lagged betas.