# Understanding Modigliani & Miller: different graphs in different textbooks

I am comparing two textbook's presentations of capital structure and Modigliani & Miller propositions. The first one is Berk & DeMarzo "Corporate Finance" (5th global ed., 2019), the second one is Hillier et al. "Fundamentals of Corporate Finance" (3rd ed., 2017) (here is a link to a slightly different edition).

They seem to be illustrating the same thing but quite differently. So I guess it is not the same thing after all. What is the difference between the two books' setups/assumptions that makes the graphs look so different?

(Also, does it make sense that in the bottom graph (Hillier et al.), $$R_D$$ stays constant irrespective of the level of $$D/E$$? Should the lenders not require higher compensation if $$D/E$$ is higher?)

### Berk & DeMarzo: ### Hillier et al.: • The indicator measured on the horizontal axis is different, $D/(D+E)$ vs. $D/E$ ? Aug 23, 2022 at 9:13
• @Giskard, good point! But does it make sense that in the bottom graph (Hillier et al.), $R_D$ stays constant irrespective of the level of $D/E$? Aug 23, 2022 at 9:59
• Related question: Modigliani & Miller with tax: how is this equation derived? Aug 25, 2022 at 12:24

The indicators measured on the horizontal axes are different, $$D/(D+E)$$ vs $$D/E$$. Taking this into consideration we get consistent functional forms for $$R_E$$, but not for $$R_D$$.
Let $$D/(D+E) = x$$. Then $$D/E =x/(1-x)$$.
Assume that there is a functional relationship $$R_E = f_1(x_1)$$ in the first graph, the lower index denoting the graph. There is also a seemingly linear relationship $$R_E = R_A + bx_2$$ in the second graph. Since $$x_2 = x_1/(1-x_1)$$, we have $$f_1(x_1) = R_E = R_A + b\frac{x_1}{1-x_1}.$$ This is more or less consistent with the $$f_1$$ depicted in the first graph.
Since $$R_A = (1-x) \cdot R_E + x \cdot R_D$$, we have $$R_D = \frac{1}{x}R_A - \frac{1-x}{x} R_E.$$ In terms of the first graph this means $$R_D = \frac{1}{x_1}R_A - \frac{1-x_1}{x_1} f_1(x_1) = \frac{1}{x_1}R_A - \frac{1-x_1}{x_1}R_A - b = R_A - b.$$ This would be consistent with the depiction of $$R_D$$ in the second graph, but not in the first graph. (Perhaps $$R_E$$ is not as linear in the second graph as it seems.)
• It seems likely that the model tries to make the assumption $R_D = R_A - b$, but since this is not possible at $D/V = 100\%$, some "adjustments" are necessary at this end. The problem only arises asymptotically when $D/E$ is measured, since $D/E \to 100\%$ is equivalent $D/E \to \infty$. Aug 23, 2022 at 10:25
• The second $D/E$ in the above comment should be $D/V$ (: Thus $D/V \to 100\%$ is equivalent to $D/E \to \infty$. Aug 23, 2022 at 13:17
• I get your point about the asymptotics, but does it make sense that in the bottom graph (Hillier et al.), $R_D$ stays constant irrespective of the level of $D/E$? Should the lenders not require higher compensation if $D/E$ is higher? I have also posted another, related question on which I am still very much lost... Could you perhaps take a look at it? Aug 25, 2022 at 12:20