Firms Betas - Equity, Debt betas My question is related to the following equation:
$$\beta_{asset}=\frac{Equity}{Equity+Debt}\beta_{equity}+\frac{Debt}{Equity+Debt}\beta_{debt} $$


Why can the $\beta_{asset}$ be seen as if the firm had been financed only with equity?
We can measure the $\beta_{equity}$ by CAPM, but how can we measure $\beta_{debt}$?


Any help would be appreciated
$
\newcommand{\Cov}{\text{Cov}}
$


  
  Why can the $\beta_{asset}$ be seen as if the firm had been financed only with equity?
  


The general idea with regard to this question stems from a discussion about how equity is a residual claim. To go through this step-by-step, recall that the value of a firm is given by the discounted present value of its cash flows. Equity is a claim to whatever is left over after other financial claim-holders have been paid (e.g., debt).  In this way, equity can be viewed as a call option and debt can be viewed as a put option. (See here and here.)

So, if the firm was financed only by equity, then equity would be the claim to 100% of the cash flow. Thus, $\beta_{\text{asset}}$ in that case can be seen as the if the firm was only financed with equity.


  
  We can measure the $\beta_{equity}$ by CAPM, but how can we measure $\beta_{debt}$?
  


You say that you can measure $\beta_{\text{equity}}$ by CAPM. However, you can measure any cash-flow by CAPM---even debt. Since $V = D + E$, keep in mind that
$$
\Cov(X + Y, Z) = \Cov(X,Z) + \Cov(Y,Z).
$$
The problem, I guess, is that you need to know the distribution of debts cash flows. This depends on the face value of total debt and the distribution of the underlying cash flows. This can get especially complicated if you're thinking about considering a dynamic setting. 

This question, I'm guessing, came from a discussion of the Modigliani-Miller theorem. Remember the idea behind Proposition 2. It's illustrated in the following graph: