<p>The indicators measured on the horizontal axes are different, <span class="math-container">$D/(D+E)$</span> vs
<span class="math-container">$D/E$</span>. Taking this into consideration we get consistent functional forms for <span class="math-container">$R_E$</span>, but not for <span class="math-container">$R_D$</span>.</p>
<p>Let <span class="math-container">$D/(D+E) = x$</span>. Then <span class="math-container">$D/E =x/(1-x)$</span>.</p>
<p>Assume that there is a functional relationship <span class="math-container">$R_E = f_1(x_1)$</span> in the first graph, the lower index denoting the graph. There is also a seemingly linear relationship
<span class="math-container">$R_E = R_A + bx_2$</span> in the second graph. Since <span class="math-container">$x_2 = x_1/(1-x_1)$</span>, we have
<span class="math-container">$$
f_1(x_1) = R_E = R_A + b\frac{x_1}{1-x_1}.
$$</span>
This is more or less consistent with the <span class="math-container">$f_1$</span> depicted in the first graph.</p>
<p>Since <span class="math-container">$R_A = (1-x) \cdot R_E + x \cdot R_D$</span>, we have
<span class="math-container">$$
R_D = \frac{1}{x}R_A - \frac{1-x}{x} R_E. 
$$</span>
In terms of the first graph this means
<span class="math-container">$$
R_D = \frac{1}{x_1}R_A - \frac{1-x_1}{x_1} f_1(x_1) = \frac{1}{x_1}R_A - \frac{1-x_1}{x_1}R_A - b = R_A - b.   
$$</span>
This would be consistent with the depiction of <span class="math-container">$R_D$</span> in the second graph, but not in the first graph. (Perhaps <span class="math-container">$R_E$</span> is not as linear in the second graph as it seems.)</p>

CC BY-SA 4.0
2022-08-23T10:25:10.537Z
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 $D/E \to \infty$.
Giskard
CC BY-SA 4.0
2022-08-23T13:17:25.027Z
The second $D/E$ in the above comment should be $D/V$ (: Thus $D/V \to 100\%$ is equivalent to $D/E \to \infty$.
Giskard
CC BY-SA 4.0
2022-08-25T12:20:43.773Z
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](https://economics.stackexchange.com/questions/52465/modigliani-miller-with-tax-how-is-this-equation-derived) on which I am still very much lost... Could you perhaps take a look at it?
Richard Hardy
CC BY-SA 4.0
2022-08-23T10:16:32.607Z
2022-08-23T10:16:32.607Z
Giskard
<p>I am comparing two textbook's presentations of capital structure and Modigliani &amp; Miller propositions. The first one is Berk &amp; DeMarzo <a href="https://www.pearson.com/se/Nordics-Higher-Education/subject-catalogue/finance/corporate-finance-5e-berk-and-demarzo.html" rel="nofollow noreferrer">&quot;Corporate Finance&quot;</a> (5th global ed., 2019), the second one is Hillier et al. &quot;Fundamentals of Corporate Finance&quot; (3rd ed., 2017) (<a href="https://www.mheducation.co.uk/fundamentals-of-corporate-finance-4e-9781526848628-emea-group" rel="nofollow noreferrer">here</a> is a link to a slightly different edition).</p>
<p>They seem to be illustrating the same thing but quite differently. So I guess it is <strong>not</strong> the same thing after all. <strong>What is the difference</strong> between the two books' <strong>setups/assumptions</strong> that makes the graphs look so different?</p>
<p>(Also, does it make sense that in the bottom graph (Hillier et al.), <span class="math-container">$R_D$</span> stays constant irrespective of the level of <span class="math-container">$D/E$</span>? Should the lenders not require higher compensation if <span class="math-container">$D/E$</span> is higher?)</p>
<h3>Berk &amp; DeMarzo:</h3>
<p><a href="https://i.sstatic.net/einwu.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/einwu.png" alt="enter image description here" /></a></p>
<h3>Hillier et al.:</h3>
<p><a href="https://i.sstatic.net/2lsQc.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/2lsQc.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2022-08-23T09:13:20.420Z
The indicator measured on the horizontal axis is different, $D/(D+E)$ vs. $D/E$ ?
Giskard
CC BY-SA 4.0
2022-08-23T09:59:06.647Z
@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 $D/E$?
Richard Hardy
CC BY-SA 4.0
2022-08-25T12:24:07.317Z
Related question: [Modigliani & Miller with tax: how is this equation derived?](https://economics.stackexchange.com/questions/52465)
Richard Hardy
<finance><corporate-finance><capital-structure>
Understanding Modigliani & Miller: different graphs in different textbooks
Richard Hardy
# Understanding Modigliani & Miller: different graphs in different textbooks

(Asked by: username_0 on 2022-08-23)

I am comparing two textbook's presentations of capital structure and Modigliani & Miller propositions. The first one is Berk & DeMarzo ["Corporate Finance"](https://www.pearson.com/se/Nordics-Higher-Education/subject-catalogue/finance/corporate-finance-5e-berk-and-demarzo.html) (5th global ed., 2019), the second one is Hillier et al. "Fundamentals of Corporate Finance" (3rd ed., 2017) ([here](https://www.mheducation.co.uk/fundamentals-of-corporate-finance-4e-9781526848628-emea-group) 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](https://i.sstatic.net/einwu.png)](https://i.sstatic.net/einwu.png)

### Hillier et al.:

[![enter image description here](https://i.sstatic.net/2lsQc.png)](https://i.sstatic.net/2lsQc.png)

**Tags:** finance, corporate-finance, capital-structure

> **Comments:**
> **username_1**: The indicator measured on the horizontal axis is different, $D/(D+E)$ vs. $D/E$ ?  
> **username_0**: @username_1, good point! But does it make sense that in the bottom graph (Hillier et al.), $R_D$ stays constant irrespective of the level of $D/E$?  
> **username_0**: Related question: [Modigliani & Miller with tax: how is this equation derived?](https://economics.stackexchange.com/questions/52465)  

---

## 1 answers

### Answer by username_1 on 2022-08-23. Score: 2

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.)

> **Comments:**
> **username_1**: 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 $D/E \to \infty$.  
> **username_1**: The second $D/E$ in the above comment should be $D/V$ (: Thus $D/V \to 100\%$ is equivalent to $D/E \to \infty$.  
> **username_0**: 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](https://economics.stackexchange.com/questions/52465/modigliani-miller-with-tax-how-is-this-equation-derived) on which I am still very much lost... Could you perhaps take a look at it?  

---

