This document is a seminar paper on the topics of operational and financial leverage. It provides definitions and explanations of these concepts from various academic textbooks and articles over time. While the concept of financial leverage is defined relatively uniformly, there are inconsistencies in how operational leverage is defined and measured. The paper aims to bring clarity to these definitions and measurements through analysis and examples.

1. Seminar paper
“Financial Management”
FAMA College
Ph. D. Halil Kukaj, associated professor
Arbnor Hoxhaj
252 Meriman Jakupi
20000, Prizren, Kosovo
+386(0)49 675-772
E-mail: arbnor.hoxhaj@gmail.com
File.nr. 002/13
Prishtina, 2014
This term paper was
written for one of my
seminars in 2014. It
is excellent both in
form and contents
and received the
mark 10 (excellent).
The term paper is
made available here
as an example of a
very good paper by
consent of its author,
whose intellectual
property it remains.

2. 2
Introduction to operational and financial leverage by Arbnor Hoxhaj
Copyright ©2014 by Arbnor Hoxhaj
All rights reserved. No part of this paper may be reproduced, in any form or by any means, without permission in
writing from the author.
FAMA College
Prizren, Kosovo

3. 3
Contents
Introduction..................................................................................................................................... 5
Operating leverage.......................................................................................................................... 7
What Counts: The Bottom Line.................................................................................................... 10
A Suggested New Way to Measure Operating Leverage ............................................................. 11
Financial Leverage........................................................................................................................ 12
Suggested New Way to Measure Financial Leverage .................................................................. 12
Interaction between Operating and Financial Leverage ............................................................... 13
Measuring Combined Leverage.................................................................................................... 14
Conclusions................................................................................................................................... 15
References..................................................................................................................................... 16
Appendix 1.................................................................................................................................... 18
Appendix 2.................................................................................................................................... 18

4. 4
List of key words
 Operating leverage
 Financial leverage
 Operating income
 Degree of operating leverage
 Percentage change in operating income
 Percentage change in earnings per share
 Fixed cost
 Variable cost
List of abbreviations
 EBIT The percentage change in operating income
 DOL The degree of operational leverage
 DFL The degree of financial leverage
 EPS The percentage change in earnings per share
List of tables
Table 1: Widget Works, Inc................................................................................................ 8
Table 2: Bridget Brothers.................................................................................................... 8
Table 3: The Impact on Financial Leverage of Increasing the Level of Output............... 13

5. 5
Introduction
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 a change would
affect risk and return.
A business that makes few sales, with each sale providing a very high gross margin, is
said to be highly leveraged. A business that makes many sales, with each sale
contributing a very slight margin, is said to be less leveraged. As the volume of sales in a
business increases, each new sale contributes less to fixed costs and more to profitability.
A business that has a higher proportion of fixed costs and a lower proportion of variable
costs is said to have used more operating leverage. Those businesses with lower fixed
costs and higher variable costs are said to employ less operating leverage.
Operating leverage is the name given to the impact on operating income of a change in
the level of output. Financial leverage is the name given to the impact on returns of a
change in the extent to which the firm’s assets are financed with borrowed money.
Despite the fact that both operating leverage and financial leverage are concepts that have
been discussed and analyzed for decades, there is substantial disparity in how they are
defined and measured by academics and practitioners.
In their 1969 college textbook, Weston and Brigham told some of today’s businessmen
and women that, "High fixed costs and low variable costs provide the greater percentage
change in profits both upward and downward”. [Weston, 86]
Today, in his 1995 textbook, Brigham says that, "If a high percentage of a firm’s costs
are fixed, and hence do not decline when demand decreases, this increases the company’s
business risk. This factor is called operating leverage." [Brigham, 425] "If a high
percentage of a firm’s total costs are fixed, the firm is said to have a high degree of
operating leverage." [Brigham, 426] "The degree 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 an index number which measures the
effect of a change in sales [number of units] on operating income, or EBIT." [Brigham,
440]
In their 1970 textbook, Grunewald and Nemmers told them that, "When fixed costs are
very large and variable costs consume only a small percentage of each dollar of revenue,
even a slight change in revenue will have a large effect on reported profits." [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 in sales...

6. 6
And the more important, proportionally, are fixed costs in the total cost structure, the
more marked is the effect on EBIT." [Cherry, 254]
In his 1971 textbook, Van Horne said that, "one of the most dramatic examples of
operating leverage is in the airline industry, where a large portion of total costs are fixed."
[Van Horne, 680]
Archer and D’Ambrosio in their 1972 textbook said that, "The higher the proportion of
fixed costs to total costs the higher the operating leverage of the firm..." [Archer, 421]
In their 1972 textbook, Schultz and Shultz, said that, "Since a fixed expense 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. These results in fact will be
subject to magnification, the degree of which depends on the relative size of fixed costs
vis-a-vis the potential range of sales volume. This entire subject 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 measures the 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 as the impact of a
change in revenue on profit or cash flow. It arises, they say, whenever a firm can increase
its revenues without a proportionate increase in operating expenses. Cash allocated to
increasing revenue, such as marketing and business development expenditures, are
quickly. "consumed by high fixed expenses." (This is certainly a different definition!)
In his 1997 article, Rushmore says that positive operating leverage occurs at the point at
which revenue exceeds the total amount of fixed costs.

7. 7
There seems to be more uniformity in the definition of financial leverage. "Financial
leverage," say Block and Hirt, reflects the amount of debt used in the capital structure of
the firm. Because debt carries a fixed obligation of interest payments, we have the
opportunity to greatly magnify our results at various levels of operations. [Block, 116]
According to Weston and Brigham back in 1969, the degree of financial leverage is
computed as the percentage change in earnings available to common stockholders
associated 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 given percentage
change in earnings before interest and taxes (EBIT), and it is calculated as follows:"
DFL = Percentage change in EPS divided by Percentage change in EBIT
This calculation produces an index number which if, for example, it is 1.43, this means
that a 100 percent increase in EBIT would result in a 143 percent increase in earnings per
share. (It makes no difference mathematically if return is calculated on a per share basis
or on total equity, as in the solution of the equation EPS cancels out.)
Clarity in regard to operating and financial leverage is important because these concepts
are important to businesses. As Conrad Lortie observes in an article, small and medium-
sized business often has difficulty using the highly sophisticated quantitative methods
large companies use. Fortunately, he observes, the simple break-even graph is simple and
easy to interpret; yet it can provide a significant amount of information. The algebra
necessary to compute operating and financial leverage, too, is not very complex.
Unfortunately, it comes in a several guises; not all equally easy to understand or equally
useful. [Brigham, 442]
Operating leverage
To make it readily apparent something that is wrong with the typical description 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 unit of $1.00. Bridget
Brothers, on the other hand, has fixed costs of $2,000 and variable costs per unit of $1.60.
Both firms’ selling price is $2.00 per unit. Shown in Tables 1 and 2 (below) are their
revenues and costs for the production of up to 25,000 units of output.

8. 8
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: Bridget 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 are familiar with descriptions of
operating leverage like those cited earlier would say that Widget Works, Inc. has the
higher degree of operating leverage because its fixed cost is absolutely and relatively
larger than Bridget Brothers’. Yet, computing operating leverage as Brigham does: the
percent change in operating profit (EBIT) divided by the percent change in the number of
units produced, indicates that both firms experience the same amount of operating
leverage when these firms increase their output from 5,000 to 10,000 units. (See below.)
DOL = [c (p -v)] / [q (p - v) -f] divided by c/q
DOL = operating leverage
Where:
 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:

9. 9
DOL = [q (p - v)] divided by [q (p - v) - f]
Therefore, when quantity increases from 5,000 to 10,000:
 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 Hirt’s method produces the same 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. have fixed costs of
$10,000 and variable costs per unit of $1.00, while Bridget Brothers has fixed costs of
only $100 and variable cost per unit of $1.99. Observe that now Widget Works’ fixed
costs are 100 times Bridget Brothers’, and that its variable costs are 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 by 20,000($2.00 - $1.99) - $100 = 2
The explanation for the equality of operating leverage in the two examples above when,
if the equation for figuring the degree of operating leverage did what is supposed to do:
reflect the difference in the relative importance of fixed cost, is that in both cases break-
even takes place at the same level of output, and each product sells for the same price.
Why this is true is explained in Appendix 1.
This is not the only situation in which operating leverage does to 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 Bridget Brothers:
OL w = [100($3.00 - $1.00)] / [100($3.00 - $1.00) - $100] = 2
OL b = [100($0.40 - $0.20)] / [100($0.40 - $.20)] - $10] = 2

10. 10
How to determine which sets of data produce the same DOL is shown in Appendix 2.
Fixed costs play no role in determining how rapidly profit rises after break-even. This is
determined by the ratio of variable cost per unit to price per unit.
It is true, of course, that if a business substitutes capital for labor; thereby raising its fixed
costs, it will simultaneously reduce a variable cost, labor cost, per unit. Some businesses
by their very nature, such as airlines, must employ a high ratio of capital to labor. If at the
maximum possible level of output fixed costs are a large percent of total costs, price per
unit will have to be high relative to variable cost per unit in order for the business to be
able to earn a profit. If a price much greater 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 on equity, it would seem that
the most appropriate method of computing operating leverage is to compute what EBIT
will be at various levels of output.
The change in the rate of return as a result of 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 is changed from q1 to q2 where q1 < q2.
In evaluating the wisdom of their investment in a corporation, its owners should use the
current market value of its stock, because this is what they would have available to invest
elsewhere if they liquidated the stock.
Businesses change the level of output in order increase the rate of return enjoyed by their
owners.
This can be done either by selling more units or avoiding producing 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 is certainly often true in the real
world for a small business.
Owners’ rate of return before interest and taxes (r) = EBIT divided by e or:
r = q (p - v) - f divided by e

11. 11
Where:
 e = equity
 r’s value after there is a change in level of output =
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 change in owners’ rate
of return resulting from a change in the level of output is not affected by interest expense.
A Suggested New Way to Measure Operating Leverage
Tables 1 and 2 make clear the fact that the difference in the bottom-line impact of
changing the level of output between Widget Works, Inc. and Bridget Brothers isn’t the
rate at which profit expands after break-even, as in both cases it doubles between 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, a simpler, more
meaningful way to the owner(s) of a business to measure operating leverage is to
compute the change in the following ratio resulting from a given increase or decrease in
the level of output from its current level.
m = p q - (qv + f) divided by p q
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 cost per unit to price per
unit; so, the greater is this ratio, the higher is operating leverage.

12. 12
Financial Leverage
Operating leverage refers to the fact that a lower ratio of variable cost per unit to price
per unit causes profit to vary more with a change in the level of output than it would if
this ratio was higher. Financial leverage refers to the fact that a higher ratio of debt to
equity causes profitability to vary more when earnings on assets changes than it would if
this ratio was lower. Obviously, the profits of a business with a high degree of both kinds
of leverage vary more, everything else remaining the same, than do those of businesses
with less operating and financial leverage. Greater variability of profits, of course, means
risk is higher. Therefore, in deciding what is the optimum level of leverage, what is an
acceptable risk/return tradeoff must be determined.
The degree of financial leverage (DFL) is sometimes measured 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 firm’s liabilities and 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 or less debt financing
or paying different interest rates for a given amount of debt financing.
Suggested New Way to Measure Financial Leverage
It is quite simple to compute what the impact on owners’ rate of return will be as a result
of borrowing a given percent of the money used to finance a firm’s 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 (%)

13. 13
Example: Assuming 70 percent of a firm’s assets are financed with debt costing 8
percent and return on assets is 12 percent, this equation indicates owners will 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 financial leverage obtained by 70
percent debt financing at a cost of 8 percent. If borrowing rose above 70 percent, this
figure would rise, that is, financial leverage would be greater. If financial leverage is
measured, instead, as an index number, an additional calculation is necessary to
determine what return on equity it produces.
To confirm that this equation is a valid way to measure the impact on the bottom 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 of output.
The return on assets and the return on equity =
r a = (q p - qv - f )/a and
r e = (d/e)[(q p - qv - f )/a) – r d] + (q p - 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 Financial Leverage
The interaction of operating and financial leverage is illustrated using data in Table 3.
Table 3: The Impact on Financial Leverage of Increasing the Level 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

14. 14
In the example shown in Table 3 (above), the interest rate is 5% ($50/$1,000). When the
level of output is 200, the return on assets ($2,000) is 7.5% (EBIT = $3 x 200 - $2 x 200 -
$50). When the output level is 400, the return on assets is 17.5% (EBIT = $3 x 400 - $2 x
400 - $50).
The return on equity ($1,000) when the level of output is 200 is 10% (EBT = $3 x 200 -
$2 x 200 - $50 - $50). When the level of output is 400, it is 30% (EBT = $3 x 400 - $2 x
400 -$50 - $50). Therefore, when the level of output is 200, owners’ rate of return is
increased from the 7.5% they would have earned if they had invested $2,000 to the 10%
they would earn by investing only $1,000 and borrowing another $1,000 at a cost of 5%.
That is, their return is increased by 33.3%--1.33 times more. When the level of output is
400, they experience an even higher degree of favorable financial leverage, earning 30%,
rather than 17.5% - 1.71 times more. (If the return on assets fell below 5%, the rate of
return they earn would be less than they would have earned if they had not borrowed any
money.)
The bottom-line impact of financial leverage can be measured 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 held constant in order to determine the
advantage of using creditors money, rather than owners' money. Therefore, if there is
no borrowing, equity (e) will be greater by the amount of foregone 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 by 1.33%, or that it
will rise from 7.5% to 10 percent? While, obviously, each can be used to determine the
other, why would the business person want to go through an intermediate step in order to
determine the bottom-line effect which, most assuredly, is something he or she will want
to know!
Measuring Combined Leverage
The combined impact of operating and financial leverage can be measured by an 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:

15. 15
 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 firm’s financing from
lenders will increase owners’ rate of return from 7.5% to 30% (4.0 times 7.5% = 30%).
Conclusion
Operating leverage has often been misleadingly described. It’s magnitude is determined
by the ratio of variable cost per unit to price per unit, rather than by the relative size of
fixed costs.
Because business owners evaluate the success of the operation of their business on the
basis rate of the return earned on equity, measures of operating and financial leverage
that produce percent rates of return would appear to be more useful to them than those
that produce index numbers.
The following equation can be used to determine the current rate of return on equity
before taxes and the impact on it of a change in the level of output, amount of debt
financing, cost of debt financing, price, and costs.
It can be used to compute the impact of either operating or financial leverage or both of
them simultaneously. (If no debt financing is used, the term d/e would, of course, be
omitted from the equation).
re = (d/e)[(q p - qv - f )/a) – r d] + (q p - qv - f )/a
That is:
The return on equity = (the ratio of debt to equity) x (the return on assets minus the
cost of debt plus the return on assets)

16. 16
A ratio of two versions of this equation produces an index number that will, by placing in
the numerator the equation involving the higher level of output and/or debt to equity
ratio, measure the degree of operating or financial leverage, that is, a ratio of the rate of
return on equity after the level of output is increased or more debt is utilized to the rate of
return before these changes are made.
It is to the business community’s advantage for methods of financial analysis to be easy
to learn and apply. Adopting this equation appears to be a way to achieve this.
List of References
Allen, David, "How Do You Leverage?" Management Accounting -London (May 1994),
p. 14
Archer, Stephen H. and Charles A. D’Ambrosio, Basic Finance (1972).
Block, Stanley B. and Geoffrey A. Hirt, Foundations of Financial Management (1997).
Blazenko, George W., "Corporate Leverage and the Distribution of Equity Returns,"
Journal of Business & Accounting (October 1996), p. 1097-1120).
Brigham, Eugene F., Fundamentals of Financial Management (1995).
Buccino, Gerald P. and Kraig S. McKinley, "The Importance of Operating Leverage in a
Turnaround," Secured Lender (Sept./Oct. 1997), p. 64-6
Cherry, Richard T., Introduction to Business Finance (1970).
Darrat, Ali F. and Tarun K. Mukherjee, "Inter-Industry Differences and the Impact of
Operating and Financial Leverages on Equity Risk," Review of Financial Economics
(Spring 1995), p. 141-155.
Dugan, Michael T., Donald Minyard, and Keith A. Shriver, "A Re-examination of the
Operating Leverage-Financial Leverage Tradeoff," Quarterly Review of Economics &
Finance (Fall 1994), p. 327-334.
Ghosh, Dilip K. and Robert G. Sherman, "Leverage, Resource Allocation and Growth,"
Journal of Business Finance & Accounting (June 1993), p. 575-582.
Grunewald, Adolph E. and Erwin E. Nemmers, Basic Managerial Finance (1970).

17. 17
Huffman, Stephen P., "The Impact of Degrees of Operating and Financial Leverage on
the Systematic Risk of Common Stock: Another Look," Quarterly Journal of Business &
Economics (Winter 1989), p. 83-100.
Lang, Larry, Eli Ofek, and Rene M. Stulz, "Leverage, Investment, and Firm Growth,"
Journal of Financial Economics (January 1996), p. 3-29.
Li, Rong-Jen and Glenn V. Henderson, Jr., "Combined Leverage and Stock Risk,"
Quarterly Journal of Business & Finance (Winter 1991), p. 18-39.
Lortie, Conrad, "Using Operating Leverage to Increase Small Business Profits," CMA
Magazine (November 1989), p. 32-34.
Marston, Felicia and Susan Perry, "Implied Penalties for Financial Leverage: Theory
Versus Empirical Evidence," Quarterly Journal of Business & Economics (Spring 1996),
p. 77-97.
Mock, E. J., R. E. Schultz, R. G. Schultz, and D. H. Shuckett, Basic Financial
Management (1968).
Petersen, Mitchell A., "Cash Flow Variability and Firm’s Pension Choice: A Role for
Operating Leverage," Journal of Financial Economics (December 1994), p. 361-383.
Rushmore, Stephen, "The Ups and Downs of Operating Leverage," Lodging Hospitality
(January 1997), p. 9.
Schultz, Raymond G. and Robert E. Shultz, Basic Financial Management, (1972).
Shih, Michael S., "Determinants of Corporate Leverage: A Time-series Analysis Using
U.S. Tax Return Data," Accounting Research (Fall 1996), p. 487-504.
Staats, William F., "Operating Leverage: It Enhances Profitability When Things Go
Well," Credit Union Executive (Fall 1989), p. 40, 42.
Van Horne, Financial Management and Policy (1971).
Weston, J. Fred and Eugene F. Brigham, Managerial Finance (1969).

18. 18
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 f1 in 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