# Does WACC not depend on the cost of debt?

According to chapter 17 of Ross's Corporate Finance (Brazilian translation of 2nd edition),

$$r_{WACC} = \frac{S}{S+B}r_S + \frac{B}{S+B}r_B(1 - T)$$

and

$$r_S = r_0 + \frac{B}{S}(1 - T)(r_0 - r_B)$$

where $$S$$ is equity, $$B$$ is debt, $$T$$ is tax rate, $$r_0$$ is the unlevered cost of equity, $$r_S$$ is the levered cost of equity, and $$r_B$$ is the cost of debt.

By replacing $$r_S$$ in the first formula and simplifying, I get

$$r_{WACC} = \Bigg{(}\frac{S + B(1 - T)}{S + B}\Bigg{)}r_0$$

which would mean the weighted average cost of capital does not depend on the cost of debt, $$r_B$$. This formula yields the same WACC as the one in the book, and I checked on some other examples as well.

Did I get this right? If so, is this because the higher tax shield compensates the additional risk from higher interest payments? If not, what am I doing wrong?