# Operating Leverage Interpretation

Operating Leverage is the ratio of Contribution margin and operating income(proxy of profit).

So, Operating Leverage = [Sales-Variable Cost]/[Profit] = Quantity*(Price-AVC)/Profit

Many literature including investopedia, CFA curriculum suggests that if operating leverage is k, then if sales go up by a%, the profit will go up by k times a% = ak%

But, if I use the formula, Sales going up by a% implies, quantity going up by a% as price and AVC are fixed. Then, how come Profit will increase by ak%?

Operating leverage $$k$$: $$k=\frac{Quantity*(Price-AVC)}{Profit}$$ but profit (actually operating profit) is: $$Profit=Quantity*(Price-AVC)-FixOpCosts$$ If quantity sold increases by $$a\%$$ then our new profit is: $$Profit_{new}=Quantity*(Price-AVC)(1+a)-FixOpCosts$$ Percentage change in Profit is: $$Profit_{new}/Profit-1$$ Then you are basically asking if $$ka\stackrel{?}{=}Profit_{new}/Profit-1$$
It is just simple algebra to show that it holds $$ka\stackrel{?}{=}\frac{Profit_{new}}{Profit}-1$$ $$ka\stackrel{?}{=}\frac{Quantity*(Price-AVC)(1+a)-FixOpCosts}{Quantity*(Price-AVC)-FixOpCosts}-1$$ $$ka\stackrel{?}{=}\frac{Quantity*(Price-AVC)(1+a)-FixOpCosts}{Quantity*(Price-AVC)-FixOpCosts}-\frac{Quantity*(Price-AVC)-FixOpCosts}{Quantity*(Price-AVC)-FixOpCosts}$$
$$ka\stackrel{?}{=}\frac{Q(Price-AVC)a}{Q(Price-AVC)-FixOpCosts}$$ Given our definition of k we have: $$\frac{Quantity(Price-AVC)}{Profit}a\stackrel{?}{=}\frac{Q(Price-AVC)a}{Q(Price-AVC)-FixOpCosts}$$
Given our definition of $$Profit$$ we see that it in fact holds: $$ka{=}\frac{Profit_{new}}{Profit}-1$$