## Friday, September 6, 2013

### Relating ROE with ROA and Leverage

This is just a quick derivation where I use some simplifying assumptions. I'm going to show that return on equity (ROE) is a function of return on assets (ROA) and leverage. This post was partly motivated by a discussion here.

Here are the assumptions:
\begin{align} ROE &= \frac{EBIT - I}{E} \\ ROA &= \frac{EBIT}{A} \\ A &= D + E \end{align}
I'm either going to ignore taxes or assume $EBIT$ has already been adjusted for taxes. $I$ is interest charges. $A$ is the total invested assets which are debt ($D$) and equity ($E$). So let's do some derivations.
\begin{align*} ROE &= \frac{EBIT}{E} - \frac{I}{E} \tag{from 1} \\ ROE &= \frac{A}{A}\left(\frac{EBIT}{E}\right) - \frac{I}{E} \tag{Multiply by "1"} \\ ROE &= \frac{A}{E}\frac{EBIT}{A} - \frac{I}{E} \tag{rearrange} \\ ROE &= \frac{A}{E}ROA - \frac{I}{E} \tag{substitute in 2} \end{align*}
This shows Return on Equity as a function of three variables. The first variable is ROA. This tells gives us information about the overall productivity of the assets.

The other two variables are leverage variables.

ROE will be higher when debt is used since $\frac{A}{E}$ is greater than 1 whenever debt is added.

ROE is lowered by another leverage term: $\frac{I}{E}$. We can actually represent $\frac{I}{E}$ as:
$$\frac{I}{E}= \frac{i}{\frac{A}{D} - 1}$$
where $i = I/D$ is the interest rate. So if the interest rate is higher, that will lower ROE. This term also plays a very significant role when leverage is high since $\frac{A}{D}$ will approach 1.

To illustrate this, I will use two tables in which I vary the interest rate between 4% and 10%, vary the leverage between 1 and 2 (0% debt to equity to 100% debt to equity) and vary the return on assets.

Using the simple model I gave above, it shows about what you'd expect, that higher/lower ROA increases/decreases ROE. Increasing leverage (A/E) increases ROE unless interest is too high, exceeding ROA.

This illustrates how high debt firms can turn out to have lower ROEs either by having lower ROAs or lower ROAs relative to their interest rate.

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