Typically, we value 1 dollar at time $T$ at $e^{-Tr}$, where $r$ is the risk-free rate.
Why wouldn't we do this for future cash flows in expected earnings for a corporation? Why do we discount at WACC instead to provide a valuation?
I understand that one should try to "beat" the WACC rate to be profitable, but I don't see why this should affect valuation.
A dollar to be received with certainty (for example, you have purchased a bill from the US government) at time $t$ will be valued at $e^{-rt}$.
If you are uncertain about whether you will receive the dollar or not, you should take this into account. A simple model is that the institution who is supposed to pay you will default with probability $p$, in which case the value of the dollar is
$$ V = p \times 0 + (1 - p) \times e^{-rt} = (1-p)e^{-rt} $$
For convenience, we often write this as the discounted value of a dollar with a risky discount rate $\lambda$, that is,
$$ e^{-\lambda t} = (1-p)e^{-rt} $$
which rearranges to
$$ \lambda = r + \frac{1}{t}\log\left( \frac{1}{1-p} \right) $$
when $p$ is small, this is approximately $\lambda \approx r + p/t$. The following two situations are equivalent -
