Discount Rate of Variable Cash Flows - Investments and Real Options

Joachim Kuczynski, 26 March 2025

In this post I want do derive how you can calculate the discount rate of contribution cash flows.

The value of the asset is the value of fixed cash flows plus the value of contribution cash flows. The value of fixed cash flows is negative. To avoid later discussions about the sign I will write all expressions with its absolute value $V_f=-\left| V_f \right|$ . With that we obtain $V_a = V_f+V_v=-\left| V_f \right| + V_v$ .

According to CAPM the return rate of the variable cash flows $r_v$ is:

$r_v=r_f+\beta_v\left( r_m-r_f \right)$

$r_a=r_f+\beta_a\left( r_m-r_f \right)$

$r_v=r_a-r_f-\beta_a\left( r_m-r_f \right) + r_f+ \beta_v \left( r_m-r_f \right)$

$r_v=r_a- \left( r_m-r_f \right) \left( \beta_a -\beta_v \right)$

Next we have a look at the betas. The beta of a sum is the weighted average of the component’s betas. That means:

$\beta_a=\beta_f\frac{-\left| V_f \right|}{V_a}+\beta_v\frac{V_v}{V_a}=\beta_v\frac{V_v}{V_a}$

$\beta_a - \beta_v = \beta_a \left( 1-\frac{V_a}{V_v} \right) =\beta_a \frac{ \left| V_f \right| }{ \left| V_f \right| + V_a}$

Whith that we can write $r_v$ as:

$r_v=r_a- \left( r_m-r_f \right) \beta_a \frac{ \left| V_f \right| }{ \left| V_f \right| + V_a}$

Substituting $\left( r_m-r_f \right) \beta_a$ by $\left( r_a-r_f \right)$ leads to the final result:

$r_v= r_a- \left( r_a-r_f \right) \frac{ \left| V_f \right| }{ \left| V_f \right| + V_a}$

Next we want to have a look at some special cases. At first we assume to have no fixed cash flows. That means that the variable cash flows and the asset cash flows are the same. Hence also the asset discount rate has to be the same as variable cash flows discount rate. Setting $V_f = 0$ in the previous formula gives:

$r_v\left( V_f = 0 \right) = r_a$

Obviously this seems to be correct.

$r_v\left( V_a = 0 \right) = r_f$