 | Insights into Operating and Financial Leverage |
by Carole E. Scott
CaroleE. Scott is a Professor of Economics at the State University of West Georgia and Editor ofB>Quest.Ifyou do not like the background color, you can change it by highlighting the color youprefer in the scroll box below.
Should a business increase or reduce the number of units it is producing?Should it rely more or less heavily on borrowed money? The answer depends upon how achange would affect risk and return. Operating leverage is the name given to the impact onoperating income of a change in the level of output. Financial leverage is the name givento the impact on returns of a change in the extent to which the firms assets arefinanced with borrowed money. Despite the fact that both operating leverage and financialleverage are concepts that have been discussed and analyzed for decades, there issubstantial disparity in how they are defined and measured by academics and practitioners.
In their 1969 college textbook, Weston and Brigham told some oftodays businessmen and women that, "High fixed costs and low variable costsprovide the greater percentage change in profits both upward and downward." [Weston,86]
Today, in his 1995 textbook, Brigham says that, "If a high percentageof a firms costs are fixed, and hence do not decline when demand decreases, thisincreases he companys business risk. This factor is called operating leverage."[Brigham, 425] "If a high percentage of a firms total costs are fixed, the firmis said to have a high degree of operating leverage ." {Brigham, 426] "Thedegree of operating leverage (DOL) is defined as the percentage change in operating income(or EBIT) that results from a given percentage change in sales....In effect, the DOL is anindex number which measures the effect of a change in sales [number of units] on operatingincome, or EBIT." [Brigham, 440]
In their 1970 textbook, Grunewald and Nemmers told them that, "Whenfixed costs are very large and variable costs consume only a small percentage of eachdollar of revenue, even a slight change in revenue will have a large effect on reportedprofits." [Grunewald, 76]
In his 1970 textbook, Cherry said that, "Operating leverage, then,refers to the magnified effect on operating earnings (EBIT) of any given change insales...And the more important, proportionally, are fixed costs in the total coststructure, the more marked is the effect on EBIT." [Cherry, 254]
In his 1971 textbook, Van Horne said that, "one of the most dramaticexamples of operating leverage is in the airline industry, where a large portion of totalcosts are fixed." [Van Horne, 680]
Archer and DAmbrosio in their 1972 textbook said that, "Thehigher the proportion of fixed costs to total costs the higher the operating leverage ofthe firm..." [Archer, 421]
In their 1972 textbook, Schultz and Shultz, said that, "Since a fixedexpense is being compared to an amount which is a function of a fluctuating base (sales),profit-and-loss results will not bear a proportionate relationship to that base. Theseresults in fact will be subject to magnification, the degree of which depends on therelative size of fixed costs vis-a-vis the potential range of sales volume. This entiresubject is referred to as operating leverage." [Schultz, 86]
where:
q = quantity
p = price per unit
v = variable cost per unit
f = total fixed costs
Block and Hirt in their 1997 textbook say that operating leverage measuresthe effect of fixed costs on the firm, and that the degree of operating leverage (DOL)equals:
DOL = q(p - v) divided by q(p - v) - f
that is:
Degree of operating leverage =
Sales revenue less total variable cost divided by sales revenue less total cost
[Block, 116]
In their 1997 article, Buccino and McKinley define operating leverage asthe impact of a change in revenue on profit or cash flow. It arises, they say, whenever afirm can increase its revenues without a proportionate increase in operating expenses.Cash allocated to increasing revenue, such as marketing and business developmentexpenditures, are quickly. "consumed by high fixed expenses." (This is certainlya different definition!)
In his 1997 article, Rushmore says that positive operating leverage occursat the point at which revenue exceeds the total amount of fixed costs.
There seems to be more uniformity in the definition of financial leverage."Financial leverage," say Block and Hirt, reflects the amount of debt used inthe capital structure of the firm. Because debt carries a fixed obligation of interestpayments, we have the opportunity to greatly magnify our results at various levels ofoperations. [Block, 116]
According to Weston and Brigham back in 1969, the degree of financialleverage is computed as the percentage change in earnings available to common stockholdersassociated with a given percentage change in earnings before interest and taxes.
According to Brigham in 1995, "The degree of financial leverage (DFL)is defined as the percentage change in earnings per share [EPS] that results from a givenpercentage change in earnings before interest and taxes (EBIT), and it is calculated asfollows:"
DFL = Percentage change in EPS divided by Percentage change in EBIT
[Brigham, 442]
This calculation produces an index number which if, for example, it is1.43, this means that a 100 percent increase in EBIT would result in a 143 percentincrease in earnings per share. (It makes no difference mathematically if return iscalculated on a per share basis or on total equity, as in the solution of the equation EPScancels out.)
Clarity in regard to operating and financial leverage is important becausethese concepts are important to businesses. As Conrad Lortie observes in an article, smalland medium-sized business often have difficulty using the highly sophisticatedquantitative methods large companies use. Fortunately, he observes, the simple break-evengraph is simple and easy to interpret; yet it can provide a significant amount ofinformation. The algebra necessary to compute operating and financial leverage, too, isnot very complex. Unfortunately, it comes in a several guises; not all equally easy tounderstand or equally useful.
Operating Leverage
To make it readily apparent something that is wrong with the typicaldescription of operating leverage, a very simple example is used in Tables 1 and 2.Assumed is that Widget Works, Inc. has fixed costs of $5,000 and variable costs per unitof $1.00. Bridget Brothers, on the other hand, has fixed costs of $2,000 and variablecosts per unit of $1.60. Both firms selling price is $2.00 per unit. Shown in Tables1 and 2 (below) are their revenues and costs for the production of up to 25,000 units ofoutput.
Table 1
Widget Works, Inc.
| Number of Units | EBIT | Total Variable Cost | Total Cost | Profit |
| 5,000 | $10,000 | $ 5,000 | $10,000 | $ 0 |
| 10,000 | 20,000 | 10,000 | 15,000 | 5,000 |
| 15,000 | 30,000 | 15,000 | 20,000 | 10,000 |
| 20,000 | 40,000 | 20,000 | 25,000 | 15,000 |
| 25,000 | 50,000 | 25,000 | 30,000 | 20,000 |
Table 2
Gidget Brothers
| Number of Units | EBIT | Total Variable Cost | Total Cost | Profit |
| 5,000 | $10,000 | $ 8,000 | $10,000 | $ 0 |
| 10,000 | 20,000 | 16,000 | 18,000 | 2,000 |
| 15,000 | 30,000 | 24,000 | 26,000 | 4,000 |
| 20,000 | 40,000 | 32,000 | 34,000 | 6,000 |
| 25,000 | 50,000 | 40,000 | 42,000 | 8,000 |
Someone looking at the data in Tables 1 and 2 who is familiar withdescriptions of operating leverage like those cited earlier would say that Widget Works,Inc. has the higher degree of operating leverage because its its fixed cost is absolutelyand relatively larger than Bridget Brothers. Yet, computing operating leverage asBrigham does: the percent change in operating profit (EBIT) divided by the percent changein the number of units produced, indicates that both firms experience the same amount ofoperating leverage when these firms increase their output from 5,000 to 10,000units. (See below.)
DOL = [c(p -v)] / [q(p - v) -f] divided by c/q
where:
DOL = operating leverage
p = price per unit
q = original quantity
c = change in quantity
v = variable cost per unit
f = total fixed costs
The above equation simplifies to:
DOL = [q(p - v)] divided by [q(p - v) - f]
therefore, when quantity increases from 5,000 to10,000:
Widget Works: DOL = $5,000/$5,000 divided by 5,000/10,000 = 2
Bridget Brothers: DOL = $5,000/$5,000 divided by 5,000/10,000 = 2
Block and Hirts method produces thesame results when operating leverage is computed at the 10,000 unit level of output.
Widget Works: DOL = 10,000($2 - $1) divided by 5,000/10,000 = 2
Bridget Brothers: DOL = 10,000($2.00 - $1.60) divided by 10,000($2.00 - $1.60) - $2,000 = 2
An even more extreme case is produced by letting Widget Works, Inc. havefixed costs of $10,000 and variable costs per unit of $1.00, while Bridget Brothers hasfixed costs of only $100 and variable cost per unit of $1.99. Observe that now WidgetWorks fixed costs are 100 times Bridget Brothers, and that its variable costsare just barely over one-half of Bridget Brothers.
For Widget Works at 20,000 units of output:
DOL = 20,000($2.00 - $1.00) divided by 20,000($2.00 - $1.00) - $10,000 = 2
For Bridget Brothers at 20,000 units of output:
DOL = 20,000($2.00 - $1.99) divided by20,000($2.00 - $1.99) - $100 = 2
The explanation for the equality of operating leverage in the two examplesabove when, if the equation for figuring the degree of operating leverage did what issupposed to do: reflect the difference in the relative importance of fixed cost, is thatin both cases break-even takes place at the same level of output, and each product sellsfor the same price. Why this is true is explained in Appendix 1.
This is not, however, the only situation in which operating leverage doesto distinguish between firms whose fixed costs relative size differs. For example,assume that Widget Works, Inc. has a selling price of $3; variable costs per unit of $1;and fixed costs of $100. Assume that Bridget Brothers has a selling price of $0.40;variable costs per unit of $0.20, and fixed costs of $10. At 100 units of output,leverage, respectively, is, where w stands for Widget Works and B stands for BridgetBrothers:
OLw = [100($3.00 - $1.00)] / [100($3.00- $1.00) - $100] = 2
OLb = [100($0.40 - $0.20)] / [100($0.40- $.20)] - $10] = 2
How to determine which sets of dataproduce the same DOL is shown in Appendix 2.
Fixed costs play no role in determining how rapidly profit rises afterbreak-even. This is determined by the ratio of variable cost per unit to price per unit.
It is true, of course, that if a businesses substitutes capital for labor;thereby raising its fixed costs, it will simultaneously reduce a variable cost, laborcost, per unit. Some businesses by their very nature, such as airlines, must employ a highratio of capital to labor. If at the maximum possible level of output fixed costs are alarge percent of total costs, price per unit will have to be high relative to variablecost per unit in order for the business to be able to earn a profit. If a price muchgreater than variable cost per unit cannot be obtained, the business will be liquidated.
What Counts: The Bottom Line
Since the "bottom-line" for a business is the rate of return onequity, it would seem that the most appropriate method of computing operating leverage isto compute what EBIT will be at various levels of output.
The change in the rate of return as a resultof increasing the level of output is:
r2 - r1 = (q2 - q1)(p - v) divided by e
where:
e is the value of equity,
and r2 is the return after output ischanged from q1 to q2 where q1 < q2
In evaluating the wisdom of their investment in a corporation, its ownersshould use the current market value of its stock, because this is what they would haveavailable to invest elsewhere if they liquidated the stock.
Businesses change the level of output in order increase the rate of returnenjoyed by their owners. This can be done either by selling more units or avoidingproducing units which cannot be sold without a rate-of-return-reducing reduction in price.Here it is assumed that changing the level of output will not affect price, which iscertainly often true in the real world for a small business.
Owners rate of return before interest andtaxes (r) = EBIT divided by e or:
r = q(p - v) - f divided by e
where: e = equity
rs value after there is a change in level ofoutput =
q1(p - v) - f (+) or (-) (q2 - q1)(p - v) divided by e
let i = interest expense ($)
then:
r2 - r1 = [q2(p - v) - f - i - q1(p - v) - f - i] divided by e
this simplifies to:
(q2 - q1)(p - v) divided by e
The simplified version of equation of the equation reveals that the changein owners rate of return resulting from a change in the level of output is notaffected by interest expense.
A Suggested New Way to Measure OperatingLeverage
Tables 1 and 2 make clear the fact that the difference in the bottom-lineimpact of changing the level of output between Widget Works, Inc. and Bridget Brothersisnt the rate at which profit expands after break-even, as in both cases it doublesbetween 10,000 and 20,000 units; rises by fifty percent between 20,000 and 30,000; etc.,instead it is Widget Works higher ratio of profit to total revenue. Therefore, asimpler, more meaningful way to the owner(s) of a business to measure operating leverageis to compute the change in the following ratio resulting from a given increase ordecrease in the level of output from its current level.
m = pq - (qv + f) divided by pq
where:
m = profit margin before interest and taxes,that is, EBIT/sales revenue
The value of this ratio is greater the lower is the ratio of variable costper unit to price per unit; so, the greater is this ratio, the higher is operatingleverage.
Financial Leverage
Operating leverage refers to the fact that a lower ratio of variable costper unit to price per unit causes profit to vary more with a change in the level of outputthan it would if this ratio was higher. Financial leverage refers to the fact that ahigher ratio of debt to equity causes profitability to vary more when earnings on assetschanges than it would if this ratio was lower. Obviously, the profits of a business with ahigh degree of both kinds of leverage vary more, everything else remaining the same, thando those of businesses with less operating and financial leverage. Greater variability ofprofits, of course, means risk is higher. Therefore, in deciding what is the optimum levelof leverage, what is an acceptable risk/return tradeoff must be determined.
The degree of financial leverage (DFL) is sometimesmeasured in the following manner:
DFL = [q(p - v) - f - i] / e divided by [q(p -v) - f] / [e + d]
where: d is the value of a firms liabilitiesand equity plus liabilities = assets = e + d
that is:
DFL = rate of return on equity when borrowed money
is used divided by rate of return on assets
By assuming various levels of debt financing at various interest rates,equation 13 can be used to judge the impact at various levels of output of using more orless debt financing or paying different interest rates for a given amount of debtfinancing.
Suggested New Way to Measure FinancialLeverage
It is quite simple to compute what the impact on owners rate ofreturn will be as a result of borrowing a given percent of the money used to finance afirms assets:
re = (d/e)(ra - rd) + ra
where:
d = debt (either as $ or %)
e = equity (either as $ or %)
rd = interest rate on debt (%)
ra = return on assets (%)
Example: Assuming 70 percent of a firms assets are financed withdebt costing 8 percent and return on assets is 12 percent, this equation indicates ownerswill earn 21.33 percent:
2.33 (.12 - .08) + .12 = 2.33(.04) + .12 = .093 +.12 = .213
Owners return rises by 9.33 percent as a result of the financialleverage obtained by 70 percent debt financing at a cost of 8 percent. If borrowing roseabove 70 percent, this figure would rise, that is, financial leverage would be greater. Iffinancial leverage is measured, instead, as an index number, an additional calculation isnecessary to determine what return on equity it produces.
To confirm that this equation is a valid way to measure the impact on thebottom line of financial leverage, assume that assets = $100,000. This means that EBIT =$12,000. Interest cost is $5,600 (.08 x $70,000). So EBT is $6,400 ($12,000 - $5,600).$6,400 is a 21.3 percent return on equity of $30,000.
The return on assets would, of course, vary with the assumed level ofoutput.
The return on assets and the return onequity =
| ra = (qp - qv - f )/a and |
| re = (d/e)[(qp - qv - f )/a) - rd] + (qp - qv - f )/a |
that is:
The return on equity = (the ratio of debt to equity) times
(the return on assets minus the cost of debt plus the return on assets)
Interaction Between Operating and FinancialLeverage
The interaction of operating and financial leverage is illustrated usingdata in Table 3.
Table 3
The Impacton Financial Leverage ofIncreasing theLevel of Output
| Financial Leverage | Level of Output | Equity | Debt | Interest Expense | Price Per Unit | Variable Cost Per Unit | Total Fixed Cost |
| 1.33 | 200 | $1,000 | $1,000 | $50 | $3 | $2 | $50 |
| 1.71 | 400 | 1,000 | 1,000 | 50 | 3 | 2 | 50 |
In the example shown in Table 3 (above), the interest rate is 5 percent($50/$1,000). When the level of output is 200, the return on assets ($2,000) is 7.5percent (EBIT = $3 x 200 - $2 x 200 - $50). When the output level is 400, the return onassets is 17.5 percent (EBIT = $3 x 400 - $2 x 400 - $50).
The return on equity ($1,000) when the level of output is 200 is 10percent (EBT = $3 x 200 - $2 x 200 - $50 - $50). When the level of output is 400, it is 30percent (EBT = $3 x 400 - $2 x 400 -$50 - $50). Therefore, when the level of output is200, owners rate of return is increased from the 7.5 percent they would have earnedif they had invested $2,000 to the 10 percent they would earn by investing only $1,000 andborrowing another $1,000 at a cost of 5 percent. That is, their return is increased by33.3 percent--1.33 times more. When the level of output is 400, they experience an evenhigher degree of favorable financial leverage, earning 30 percent, rather than 17.5percent--1.71 times more. (If the return on assets fell below 5 percent, the rate ofreturn they earn would be less than they would have earned if they had not borrowed anymoney.)
The bottom-line impact of financial leverage can bemeasured in the following way:
rd - re = [q(p - v) - f - i] / e - [q(p - v) - f]/ ( e + d)
where:
rd = return with borrowing and re= return without borrowing
NOTE: The amount of assets being financed is heldconstant in order to determine the advantage of using creditors money, rather than owners'money. Tberefore, if there is no borrowing, equity (e) will be greater by the amount offoregone debt (d). The larger amount of equity is measured above as (e + d).
Which is more useful? Knowing that your rate of return will increase by1.33 percent, or that it will rise from 7.5 percent to 10 percent? While, obviously, eachcan be used to determine the other, why would the business person want to go through anintermediate step in order to determine the bottom-line effect which, most assuredly, issomething he or she will want to know!
Measuring Combined Leverage
The combined impact of operating and financial leverage can be measured byan index number in the following manner:
OFL = [q2(p -v) - f - i] / e divided by [q1(p-v) - f] /( e + d)
Solving this equation where:
| p = $ 2 | v = $ 1 | f = $ 50 | i = $ 50 |
| e = $1,000 | d = $1,000 | q1 = 200 | q2 = 400 |
means: DFL = .30/.075= 4.0
Adding 200 additional units of output and obtaining half the firmsfinancing from lenders will increase owners rate of return from 7.5 percent to 30percent (4.0 times 7.5 percent = 30 percent.).
Conclusions
Operating leverage has often been misleadingly described. Itsmagnitude is determined by the ratio of variable cost per unit to price per unit, ratherthan by the relative size of fixed costs.
Because business owners evaluate the success of the operation of theirbusiness on the basis rate of the return earned on equity, measures of operating andfinancial leverage that produce percent rates of return would appear to be more useful tothem than those that produce index numbers.
The following equation can be used to determine the current rate of returnon equity before taxes and the impact on it of a change in the level of output, amount ofdebt financing, cost of debt financing, price, and costs. It can be used to compute theimpact of either operating or financial leverage or both of them simultaneously. (If nodebt financing is used, the term d/e would, of course, be omitted from the equation.)
re = (d/e)[(qp - qv - f )/a) - rd] + (qp - qv - f )/a
That is:
The return on equity = (the ratio of debt to equity) times (the return on assets minus the cost of debt plus the return on assets) |
A ratio of two versions of this equation produces an index number thatwill, by placing in the numerator the equation involving the higher level of output and/ordebt to equity ratio, measure the degree of operating or financial leverage, that is, aratio of the rate of return on equity after the level of output is increased or more debtis utilized to the rate of return before these changes are made.
It is to the business communitys advantage for methods of financialanalysis to be easy to learn and apply. Adopting this equation appears to be a way toachieve this.
References
Allen, David, "How Do You Leverage?" ManagementAccounting-London (May 1994), p. 14.
Archer, Stephen H. and Charles A. DAmbrosio, BasicFinance (1972).
Block, Stanley B. and Geoffrey A. Hirt, Foundations ofFinancial Management (1997).
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Brigham, Eugene F., Fundamentals of FinancialManagement (1995).
Buccino, Gerald P. and Kraig S. McKinley, "TheImportance of Operating Leverage in a Turnaround," Secured Lender(Sept./Oct. 1997), p. 64-68.
Cherry, Richard T., Introduction to Business Finance(1970).
Darrat, Ali F. and Tarun K. Mukherjee, "Inter-IndustryDifferences and the Impact of Operating and Financial Leverages on Equity Risk," Reviewof Financial Economics (Spring 1995), p. 141-155.
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Appendix 1
B = f/(1 - v/p)
where: B = break-even level of sales
If B1 = B2, that is, f1 / (1 - v1/p) = f2 / (1 - v2/p)
and p1 = p2.
then: [q(p - v1)]/ [q(p - v1)- f1] will equal [q(p - v2)] / [q(p - v2)- f2]
This is because, simplifying the above:
f1(1 - v2/p) = f2(1- v1/p) or
f1 = [f2(p - v1)]divided by [p - v2]
and substituting this value of f1in the equation for the operating leverage produces:
DOL1 = [q(p - v2)]divided by [q(p - v2) - f2]
Appendix 2
This type of result shown is obtained by setting:
[q(p1 - v1)] / [q(p1- v1) - f1] = [q(p2 - v2)] / [q(p2- v2) - f2]
where: f1> f2 and v2/p2 > v1/p1
