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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.

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    $\begingroup$ An equivalent question is, "Why do equities have higher expected returns than U.S. government bonds?" If cash flows from equities were valued by discounting at the risk free rate, the expected return on equities and long-term government bonds would be the same. $\endgroup$ – Matthew Gunn Jul 28 '17 at 5:58
  • $\begingroup$ If equities have a higher return, and discounted by the same rate as bonds, then wouldn't the expected return on equities be higher? $\endgroup$ – user28961 Jul 28 '17 at 6:23
  • $\begingroup$ Let $x$ be a future cash flow and let $r_f$ be the risk free rate. If you discount at the risk free rate and say the price of $x$ is given by $p = \frac{1}{1+r_f} \operatorname{E}[x]$ then the expected return is $\operatorname{E}\left[\frac{x}{p} \right] = 1 + r_f$. $\endgroup$ – Matthew Gunn Jul 28 '17 at 6:26
  • $\begingroup$ Got it, thank you! $\endgroup$ – user28961 Jul 28 '17 at 6:28
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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 -

  1. Discounting an uncertain payment with a risk-free discount rate
  2. Discounting an assumed certain payment with a risky discount rate
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