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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:

enter image description here

Hillier et al.:

enter image description here

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    $\begingroup$ The indicator measured on the horizontal axis is different, $D/(D+E)$ vs. $D/E$ ? $\endgroup$
    – Giskard
    Aug 23, 2022 at 9:13
  • $\begingroup$ @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 $D/E$? $\endgroup$
    – Richard Hardy
    Aug 23, 2022 at 9:59
  • $\begingroup$ Related question: Modigliani & Miller with tax: how is this equation derived? $\endgroup$
    – Richard Hardy
    Aug 25, 2022 at 12:24

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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.)

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  • $\begingroup$ 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 $D/E \to \infty$. $\endgroup$
    – Giskard
    Aug 23, 2022 at 10:25
  • $\begingroup$ The second $D/E$ in the above comment should be $D/V$ (: Thus $D/V \to 100\%$ is equivalent to $D/E \to \infty$. $\endgroup$
    – Giskard
    Aug 23, 2022 at 13:17
  • $\begingroup$ 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? $\endgroup$
    – Richard Hardy
    Aug 25, 2022 at 12:20

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