#### The Previous Derivation

The previous derivation rested on the following assumptions/definitions:

\[\begin{align}

ROE &= \frac{EBIT - I}{E} \\

ROA &= \frac{EBIT}{A} \\

A &= D + E

\end{align}\]

In addition to these, I defined the interest rate and now I'll be adding one additional term for "leverage":

\[\begin{align}

i &= \frac{I}{D} \\

L &= \frac{A}{E}

\end{align}\]

One of the expressions that I derived was the following (which is where I'll be starting from today) is:

$$ROE = \frac{A}{E}ROA - \frac{I}{E}$$

#### A New Expression

I'm not sure why I didn't notice this. It looks like I implicitly derived this when using the "ROE Tables" that I constructed.

I can divide the top and bottom of $\frac{I}{E}$ by $D$ to obtain:

$$ROE = \frac{A}{E} ROA - \frac{I/D}{E/D}$$

From definition (3) it can be shown that:

$$\frac{E}{D} = \frac{1}{A/E-1}$$

Hence, our new expression (along with the substititions from (4) and (5)) is:

$$ROE = L (ROA - i) + i$$

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