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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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