ROE, ROA and Leverage (Update)

Sometimes I don't see alternative ways of writing expressions until after the fact. So this will just be a brief modification of a previous post: Relating ROE with ROA and Leverage.

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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