# Levered beta with changing equity/debt ratios

I know how to calculate a bottom up levered beta for a privately held and not publicly traded company with Hamada (Proof of Hamada's Formula (Relationship between levered and unlevered beta)) and the help of http://people.stern.nyu.edu/adamodar/New_Home_Page/TenQs/TenQsBottomupBetas.htm.

$$\beta_{L}=\beta_U×(1+(1-t)\times D/E)$$

However, all the explanations I see in the literature assume that the debt ratio $\theta = D/E$ is a constant. Only sometimes it is casually mentioned that the betas need to be adjusted when D/E changes, but this is never formalised.

What happens if I do a valuation of a company that plans to pay off all debt so that for the future periods $t = 1,2,...$ : $\theta_1 = 0.5$, $\theta_2 = 0.25$, $\theta_3 = 0$. Do we have a time series of betas?

$$\beta_{L,t}=\beta_U×(1+(1-t)\times \theta_t)$$

...and do we need to discount future cash flows in each period by its own levered beta? In $t = 1,2$ the free cash flows will be much lower since the debt is paid off but for $t \geq3: \beta_{L,t}=\beta_U$. The value of the business should be much higher because free cash flows are hihger and the discount factor is smaller. Am I right?