# What is operating leverage?

The (degree of) operating leverage is the elasticity of the operating income to a change in revenues; it is a measure of how sensitive the Operating Income is to changes in revenues.

The mathematical representation of this definition is given by the following formula:

$OL_p = \frac{\text{%} \Delta OI_{p,p+1}}{\text{%} \Delta R_{p,p+1}} = \frac{\frac{OI_{p+1} - OI_{p}}{OI_{p}}}{\frac{R_{p+1} - R_{p}}{R_{p}}}$

where $$OL_p$$ is the degree of operating leverage in period $$p$$, $$\Delta OI_{p,p+1} = \frac{OI_{p+1} - OI_{p}}{OI_{p}}$$ is the percentage change in operating income between period $$p$$ and $$p+1$$, and $$\text{%} \Delta R_{p,p+1} = \frac{R_{p+1} - R_{p}}{R_{p}}$$ is the percentage change in revenues between the same periods.

This formula indicates that the degree of operating leverage is the factor by which you multiply the percentage change in revenues between a period $$p$$ and a period $$p+1$$ to obtain the percentage change in Operating Income between these two periods:

$\text{%} \Delta R_{p,p+1} \times OL_p = \text{%} \Delta OI_{p,p+1}$

Therefore, the degree of operating leverage tells how responsive or sensitive the operating income is to any change in revenues: it literally magnifies the impact of a change in revenues on Operating Income. An operating leverage of 3 means that a 10% increase in revenues leads to a 3 x 10% = 30% increase in operating income; it also means that a 10% decrease in revenues leads to a 3 x 10% = 30% decrease in operating income. The greater the operating leverage, the more responsive the operating income to a change in revenues, the greater the operating risk. The higher the degree of operating leverage, the more volatile the Operating Income because the slightest change in sales will have greater repercussions on profit. It is therefore a measure of operating risks. A firm with a low operating leverage is less vulnerable to economic downturns; however, it also reaps less benefits from economic growth.

The preceding formula is very helpful to predict profit knowing just a change in revenues and a degree of operating leverage. However, since it requires two distinct observations in time, it is often less convenient than the next formula to compute the degree of operating leverage:

$OL_p = \frac{CM_p}{OI_p}$

where $$OL_p$$ is the degree of operating leverage in period $$p$$, $$CM_p$$ is the contribution margin in period $$p$$, and $$OI_p$$ is the operating income in the same period. This formula is thus very easy to use when you already have the contribution income statement. Moreover, it relates directly the operating leverage to the cost structure: for a given Operating Income, a more variable cost structure will have a lower contribution margin ($$\frac{CM_p}{OI_p}$$ is smaller) and fixed cost structures a higher contribution margin ($$\frac{CM_p}{OI_p}$$ is greater). You can think of the relationship quite literally as leverages:

Finally, the operating leverage is the inverse of the margin of safety ratio:

$OL_p = \frac{\text{1}}{MSR_p} = \frac{\text{100%}}{MSR_p}$

You can infer this formula from the very definition of the margin or safety ratio and the operating leverage. The margin of safety ratio is the percentage change in revenues necessary to bring the company at break-even, i.e. to cause 100% change in operating income. This inverse relationship is represented in the following figure: