# Derivation of DOL Formula

Degree of operating leverage is defined as percentage change in operating income that occurs in response to a percentage change in sales.

Following are the most common formulas used to calculate degree of operating leverage:

$${\text{DOL}}^\text{1}\ =\ \frac{\text{Q}\times(\text{P}\ -\ \text{V})}{\text{Q}\times(\text{P}\ -\ \text{V})\ -\ \text{FC}}$$

$${\text{DOL}}^\text{2}\ =\ \frac{\text{Contribution Margin}}{\text{Operating Income}}$$

$${\text{DOL}}^\text{3}\ =\frac{\text{Contribution Margin Ratio}}{\text{%\ Operating Margin}}$$

In the following paragraphs, we will derive the above-mentioned formulas using the basic definition of degree of operating leverage:

$$\text{DOL}\ =\ \frac{\text{%\ Change in Operating Income}}{\text{%\ Change in Sales}}$$

Percentage change in operating income equals change in operating income divided by initial operating income.

$$\text{%\ Change in Operating Income}\ =\frac{\Delta \text{OI}}{\text{OI}}$$

Similarly, percentage change in sales equals change in sales divided by initial sales.

$$\text{%\ Change in Sales}\ =\frac{\Delta \text{S}}{\text{S}}$$

The basic formula for DOL can be written as follows:

$$\text{DOL}\ =\frac{\Delta \text{OI}}{\text{OI}}÷\frac{\Delta \text{S}}{\text{S}}=\frac{\Delta \text{OI}}{\Delta \text{S}}\times \frac{\text{S}}{\text{OI}}$$

Where

∆OI is the change in operating income which equals change in quantity (∆Q) multiplied by contribution margin per unit (i.e. price P – variable cost per unit V). Since fixed cost is the same in both cases, it doesn’t factor in the formula:

$$\Delta \text{OI} = \Delta \text{Q}\times (\text{P} - \text{V})$$

∆S is the change in sales which equals change quantity (∆Q) multiplied by price P:

$$\Delta \text{S} = \Delta \text{Q}\times \text{P}$$

Sales S equals the production of quantity Q and price P:

$$\text{S}=\text{Q}\times \text{P}$$

Operating income OI equals contribution margin minus fixed costs:

$$\text{OI}\ =\ \text{Q}\times(\text{P}\ -\ \text{V})-\text{FC}$$

Substituting these equations in the formula for DOL and cancelling common expressions, we get another expression for first formula of degree of operating leverage DOL1:

$$\text{DOL}\ =\frac{\Delta \text{Q}\times (\text{P} - \text{V})}{\Delta \text{Q} \times \text{P}}\times \frac{\text{Q}\times \text{P}}{\text{Q}\times (\text{P} - \text{V}) - \text{FC}}$$

$$\text{DOL}\ =\ \frac{\text{Q}\times(\text{P}\ -\ \text{V})}{\text{Q}\times(\text{P}\ -\ \text{V})\ -\ \text{FC}}$$

The numerator in the above expression equals total contribution margin and the denominator equals total operating income. This gives us the second formula for degree of operating leverage DOL2:

$$\text{DOL}\ =\ \frac{\text{Contribution Margin}}{\text{Operating Income}}$$

Dividing and multiplying the right-hand side of the above equation with sales give us another equation for DOL:

$$DOL\ =\ \frac{Contribution\ Margin}{Operating\ Income}\times\frac{Sales}{Sales}=\frac{\frac{Contribution\ Margin}{Sales}}{\frac{Operating\ Income} {Sales}}$$

Contribution margin divided by sales equals contribution margin ratio and operating income divided by sales equals percentage operating income. This gives us the third formula for degree of operating leverage DOL3:

$${\text{DOL}}^\text{3}=\frac{\text{Contribution Margin Ratio}}{\text{%\ Operating Margin}}$$