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Unformatted text preview: WEB APPENDIX 14A
Degree of Leverage
In our discussion of operating leverage in Chapter 14, we made no mention of
financial leverage; and when we discussed financial leverage, operating leverage
was assumed to be given. Actually, the two types of leverage are interrelated. For
example, a firm reducing its operating leverage would probably lead to an increase
in its optimal use of financial leverage. On the other hand, if the firm decided to
increase its operating leverage, its optimal capital structure would probably call for
less debt.
The theory of finance has not been developed to the point where we can
specify simultaneously the optimal levels of operating and financial leverage.
However, we can see how operating and financial leverage interact through an
analysis of the degree of leverage concept. Degree of Operating Leverage (DOL)
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:
ÁEBIT
Percentage change in EBIT
¼ EBIT
DOL ¼
ÁQ
Percentage change in sales
Q 14A1 In effect, the DOL is an index number that measures the effect of a change in sales
on operating income, or EBIT.
DOL can also be calculated using Equation 14A2, which is derived from
Equation 14A1:
DOLQ ¼ Degree of operating leverage at Point Q
QðP À VÞ
¼
QðP À VÞ À F 14A2 Or it can be based on dollar sales rather than units:
DOLS ¼ S À VC
S À VC À F 14A2a Here Q is the initial units of output, P is the average sales price per unit of output,
V is the variable cost per unit, F is fixed operating costs, S is initial sales in dollars,
and VC is total variable costs. Equation 14A2 is normally used to analyze a single
product such as IBM’s PC, whereas Equation 14A2a is used to evaluate an entire
firm with many types of products, where “quantity in units” and “sales price” are
not meaningful.
Equation 14A2 is developed from Equation 14A1 as follows. The change in
units of output is defined as DQ. In equation form, EBIT ¼ Q(P À V) À F, where Q
is units sold, P is the price per unit, V is the variable cost per unit, and F is the total
fixed costs. Since both price and fixed costs are constant, the change in EBIT is
DEBIT ¼ DQ(P À V). The initial EBIT is Q(P À V) À F, so the percentage change in
EBIT is shown as follows:
%ÁEBIT ¼ ÁQðP À VÞ
QðP À VÞ À F 14A1 14A2 Web Appendix 14A The percentage change in output is DQ/Q, so the ratio of the percentage change in
EBIT to the percentage change in output is shown as follows: 14A2 ÁQðP À VÞ
Q ðP À V Þ À F
ÁQðP À VÞ
Q
QðP À VÞ
DOL ¼
¼
¼
ÁQ
QðP À VÞ À F ÁQ
QðP À VÞ À F
Q Applying Equation 14A2a to data for an illustrative firm, Hastings Inc., at a sales
level of $200,000 as shown in Table 14A1, we find its degree of operating leverage
to be 2.0:
$200,000 À $120,000
DOL$200,000 ¼
$200,000 À $120,000 À $40,000
¼ $80,000
¼ 2:0
$40,000 Thus, an X% increase in sales will produce a 2X% increase in EBIT. For example, a
50% increase in sales, starting from sales of $200,000, will result in a 2(50%) ¼
100% increase in EBIT. This situation is confirmed by examining Section I of
Table 14A1, where we see that a 50% increase in sales, from $200,000 to $300,000,
causes EBIT to double. Note, however, that if sales decrease by 50%, EBIT will
decrease by 100%. This is again confirmed by Table 14A1, as EBIT decreases to $0
if sales decrease to $100,000.
Note also that the DOL is specific to the initial sales level; thus, if we evaluated
DOL from a sales base of $300,000, it would be different from the DOL at $200,000
of sales.
$300,000 À $180,000
DOL$300,000 ¼
$300,000 À $180,000 À $40,000
¼ $120,000
¼ 1:5
$80,000 In general, if a firm is operating at close to its breakeven point, the degree of
operating leverage will be high, but DOL declines the higher the base level of sales
is above breakeven sales. Looking back at the top section of Table 14A1, we see
that the company’s breakeven point (before consideration of financial leverage) is
at sales of $100,000. At that level, DOL is infinite.
DOL$100,000 ¼
¼ $100,000 À $60,000
$100,000 À $60,000 À $40,000
$40,000
¼ undefined but % infinity
0 When evaluated at higher sales levels, DOL progressively declines. Degree of Financial Leverage (DFL)
Operating leverage affects earnings before interest and taxes (EBIT), whereas
financial leverage affects earnings after interest and taxes, or the earnings available
to common stockholders. In terms of Table 14A1, operating leverage affects the top
section, whereas financial leverage affects the lower sections. Thus, if Hastings
decided to use more operating leverage, its fixed costs would be higher than
$40,000, its variable cost ratio would be lower than 60% of sales, and its EBIT would
be more sensitive to changes in sales. Financial leverage takes over where operating Web Appendix 14A Table 14A1 14A3 Hastings Inc.: EPS with Different Amounts of Financial Leverage (Thousands of
Dollars, except PerShare Figures) I. Calculation of EBIT, Total Assets $200,000
Probability of indicated sales
Sales
Fixed costs
Variable costs (60% of sales)
Total costs (except interest)
Earnings before interest and taxes (EBIT) 0.2
$100.0
40.0
60.0
$100.0
$ 0.0 0.0
0.0
$ 0.0
0.0
$ 0.0
$ 0.0 $ 40.0
0.0
$ 40.0
(16.0)
$ 24.0
$ 2.40
$ 2.40
$ 1.52
0.63 $ 80.0
0.0
$ 80.0
(32.0)
$ 48.0
$ 4.80 $ III. Situation If Debt/Assets (D/A) 50%
EBIT (from Section I)
Less interest (0.12 $100,000)
Earnings before taxes (EBT)
Taxes (40%; tax credit on losses)
Net income
Earnings per share (EPS) on 5,000 sharesa
Expected EPS
Standard deviation of EPS
Coefficient of variation 0.2
$300.0
40.0
180.0
$220.0
$ 80.0 $ II. Situation If Debt/Assets (D/A) 0%
EBIT (from Section I)
Less interest
Earnings before taxes (EBT)
Taxes (40%)
Net income
Earnings per share (EPS) on 10,000 sharesa
Expected EPS
Standard deviation of EPS
Coefficient of variation 0.6
$200.0
40.0
120.0
$160.0
$ 40.0 $ 40.0
12.0
$ 28.0
(11.2)
$ 16.8
$ 3.36
$ 3.36
$ 3.04
0.90 $ 80.0
12.0
$ 68.0
(27.2)
$ 40.8
$ 8.16 0.0
12.0
($ 12.0)
4.8
($ 7.2)
($ 1.44) a The EPS figures can also be obtained using the following formula in which the numerator amounts to an income statement at a given sales level
displayed horizontally.
EPS (Sales Fixed costs Variable costs Interest)(1 Tax rate)
______________________________________________________
Shares outstanding For example, with zero debt and Sales
EPSD/A 0 EPSD/A 0.5 $200,000, EPS is $2.40. (_____________________________________
$200,000 $40,000 $120,000 0)(0.6)
10,000 With 50% debt and Sales (EBIT I)(1 T )
_________________
Shares outstanding $2.40 $200,000, EPS is $3.36. (___________________________________________
$200,000 $40,000 $120,000 $12,000)(0.6)
5,000 $3.36 The sales level at which EPS will be equal under the two financing policies, or the indifference level of sales, S I, can be found by setting EPSD/A = 0
equal to EPSD/A = 0.5 and solving for SI.
EPSD/A 0 S (__________________________
SI $40,000 0.6SI 0)(0.6)
10,000 (________________________________
SI $40,000 0.6SI $12,000)(0.6)
5,000 EPSD/A = 0.5 $160,000 By substituting this value of sales into either equation, we can find EPS I, the earnings per share at this indifference point. In our example,
EPSI $1.44. 14A4 Web Appendix 14A leverage leaves off, further magnifying the effects on earnings per share of changes in the
level of sales. For this reason, operating leverage is sometimes referred to as firststage
leverage and financial leverage may be referred to as secondstage leverage.
The degree of financial leverage (DFL) is defined as the percentage change in
earnings per share that results from a given percentage change in earnings before
interest and taxes (EBIT), and it is calculated as follows:
DFL ¼
¼ 14A3 Percentage change in EPS
Percentage change in EBIT
EBIT
EBIT À I Equation 14A3 is developed as follows:
1. Recall that EBIT ¼ Q(P – V) – F.
2. Earnings per share are found as EPS ¼ [(EBIT – I)(1 – T)]/N, where I is interest
paid, T is the corporate tax rate, and N is the number of shares outstanding.
3. I is constant, so DI ¼ 0; hence, DEPS, the change in EPS, is
ÁEPS ¼ 4. ðÁEBIT À ÁIÞð1 À TÞ ÁEBITð1 À TÞ
¼
N
N The percentage change in EPS is the change in EPS divided by the original
EPS.
ÁEBITð1 À TÞ
ÁEBITð1 À TÞ
N
ÁEBIT
N
¼
¼
ðEBIT À IÞð1 À TÞ
N
ðEBIT À IÞð1 À TÞ
EBIT À I
N 5. The degree of financial leverage is the percentage change in EPS over the
percentage change in EBIT.
14A3 6.
ÁEBIT
ÁEBIT
EBIT
EBIT
¼
DFL ¼ EBIT À I ¼
ÁEBIT
EBIT À I ÁEBIT
EBIT À I
EBIT This equation must be modified if the firm has preferred stock outstanding.
Applying Equation 14A3 to data for Hastings at sales of $200,000 and an EBIT
of $40,000, the degree of financial leverage with a 50% debt ratio is 1.43:
DFLS ¼ $200,000, D ¼ 50% ¼ $40,000
¼ 1:43
$40,000 À $12,000 Therefore, a 100% increase in EBIT would result in a 1.43(100%) ¼ 143% increase
in earnings per share. This may be confirmed by referring to the lower section of
Table 14A1, where we see that a 100% increase in EBIT, from $40,000 to $80,000,
produces a 143% increase in EPS:
%EPS ¼ ÁEPS $8:16 À $3:36 $4:80
¼
¼ 1:43 ¼ 143%
¼
EPS0
$3:36
$3:36 If no debt were used, the degree of financial leverage would by definition be 1.0;
so a 100% increase in EBIT would produce exactly a 100% increase in EPS. This can
be confirmed from the data in Section II of Table 14A1. Combining Operating and Financial Leverage (DTL)
Thus far, we have seen:
1. That the greater the use of fixed operating costs as measured by the degree of
operating leverage, the more sensitive EBIT will be to changes in sales. Web Appendix 14A 2. That the greater the use of debt as measured by the degree of financial
leverage, the more sensitive EPS will be to changes in EBIT. Therefore, if a firm uses a considerable amount of operating and financial leverage, even small changes in sales will lead to wide fluctuations in EPS.
Equation 14A2 for the degree of operating leverage can be combined with
Equation 14A3 for the degree of financial leverage to produce the equation for the
degree of total leverage (DTL), which shows how a given change in sales will
affect earnings per share. Here are three equivalent equations for DTL:
DTL ¼ ðDOLÞðDFLÞ DTL ¼ 14A4 QðP À VÞ
QðP À VÞ À F À I 14A4a S À VC
S À VC À F À I 14A4b DTL ¼ Equation 14A4 is simply a definition, while Equations 14A4a and 14A4b are
developed as follows:
1. Recognize that EBIT ¼ Q(P À V) À F; and then rewrite Equation 14A3 as
follows:
DFL ¼ 2. EBIT
Q ðP À V Þ À F
S À VC À F
¼
¼
EBIT À I QðP À VÞ À F À I S À VC À F À I 14A3a The degree of total leverage is equal to the degree of operating leverage times
the degree of financial leverage, or Equation 14A2 times Equation 14A3a.
DTL ¼ ðDOLÞðDFLÞ 14A4 ¼ ðEquation 14A2ÞðEquation 14A3aÞ
Q ðP À V Þ
QðP À VÞ À F
¼
QðP À VÞ À F QðP À VÞ À F À I
¼ QðP À VÞ
Q ðP À V Þ À F À I ¼ S À VC
S À VC À F À I 14A4a
14A4b Applying Equation 14A4b to data for Hastings at sales of $200,000, we can
substitute data from Table 14A1 into Equation 14A4b to find the degree of total
leverage if the debt ratio is 50%.
$200,000, 50% ¼ $200,000 À $120,000
$200,000 À $120,000 À $40,000 À $12,000 ¼ DTL $80,000
¼ 2:86
$28,000 Equivalently, using Equation 14A4, we get the same result. DTL $200,000, 50% ¼ ð2:00Þð1:43Þ ¼ 2:86 14A5 14A6 Web Appendix 14A We can use the degree of total leverage (DTL) number to find the new earnings per
share (EPS1) for any given percentage increase in sales (%DSales), proceeding as
follows
EPS1 ¼ EPS0 þ EPS0 ½ðDTLÞð%ÁSalesÞ
14A5 ¼ EPS0 ½1:0 þ ðDTLÞð% ÁSalesÞ For example, a 50% (or 0.5) increase in sales, from $200,000 to $300,000, would
cause EPS0 ($3.36 as shown in Section III of Table 14A1) to increase to $8.16.
EPS1 ¼ $3:36½1:0 þ ð2:86Þð0:5Þ
¼ $3:36ð2:43Þ
¼ $8:16 This figure agrees with the one for EPS shown in Table 14A1.
The degree of leverage concept is useful primarily for the insights it provides
regarding the joint effects of operating and financial leverage on earnings per
share. The concept can be used to show the management of a business, for
example, that a decision to automate a plant and to finance the new equipment
with debt would result in a situation wherein a 10% decline in sales would produce a 50% decline in earnings, whereas with a different operating and financial
leverage package, a 10% sales decline would cause earnings to decline by only
20%. Having the alternatives stated in this manner gives decision makers a better
idea of the ramifications of alternative actions.1
1
The degree of leverage concept is also useful for investors. If firms in an industry are ranked by degree of
total leverage, an investor who is optimistic about prospects for the industry might favor those firms with high
leverage and vice versa if industry sales are expected to decline. However, it is very difficult to separate fixed
from variable costs. Accounting statements simply do not make this breakdown, so an analyst must make the
separation in a judgmental manner. Note that costs are really fixed, variable, and “semivariable,” for if times get
tough enough, firms will sell off depreciable assets and thus reduce depreciation charges (a fixed cost), lay off
“permanent” employees, reduce salaries of the remaining personnel, and so forth. For this reason, the degree of
leverage concept is generally more useful for thinking about the general nature of the relationship than for
developing precise numbers and any numbers developed should be thought of as approximations rather than as
exact specifications. QUESTIONS
14A1
14A2
14A3 What effect would an increase in a firm’s operating leverage have on its use of debt in its
optimal capital structure? if it decreased the firm’s operating leverage? Explain.
If a firm decided to begin flexibly leasing (shortterm leases that are easily renewed) many
fixed assets rather than replacing them when they became old, what effect would this have
on its operating leverage? its financial leverage? its total leverage?
How are the degrees of operating, financial, and total leverage related? PROBLEMS
14A1 DEGREE OF OPERATING LEVERAGE Grant Grocers has sales of $1,000,000. The company’s
fixed costs total $250,000, and its variable costs are 60% of sales. What is the company’s
degree of operating leverage? If sales increased 20%, what would be the percentage increase
in EBIT? Web Appendix 14A 14A2 14A3 14A7 DEGREE OF FINANCIAL LEVERAGE Arthur Johnson Inc.’s operating income is $500,000, the
company’s interest expense is $200,000, and its tax rate is 40%. What is the company’s
degree of financial leverage? If the company could double its operating income, what
would be the percentage increase in net income?
DEGREE OF LEVERAGE A company currently has $2 million in sales. Its variable costs
equal 70% of its sales, its fixed costs are $100,000, and its annual interest expense is $50,000.
a.
b. If this company’s operating income (EBIT) rises by 10%, how much will its net income
increase? c. 14A4 What is the company’s degree of operating leverage? If the company’s sales increase 10%, how much will the company’s net income
increase? OPERATING LEVERAGE EFFECTS The Whitman Corporation will begin operations next
year to produce a single product at a price of $12 per unit. Whitman has a choice of two
methods of production: Method A, with variable costs of $6.75 per unit and fixed operating
costs of $675,000, and Method B, with variable costs of $8.25 per unit and fixed operating
costs of $401,250. To support operations under either production method, the firm requires
$2,250,000 in assets and it has established a debt ratio of 40%. The cost of debt is rd ¼ 10%.
The tax rate is irrelevant for the problem, and fixed operating costs do not include interest.
a. b. c.
d. The sales forecast for the coming year is 200,000 units. Under which method would
EBIT be more adversely affected if sales did not reach the expected levels? (Hint:
Compare DOLs under the two production methods.)
Given the firm’s present debt, which method would produce a greater percentage
increase in earnings per share for a given increase in EBIT? (Hint: Compare DFLs under
the two methods.)
Calculate DTL under each method; then evaluate the firm’s risk under each method.
Is there some debt ratio under Method A that would produce the same DTLA as the
DTLB you calculated in Part c? (Hint: Let DTLA ¼ DTLB ¼ 2.90 as calculated in Part c,
solve for I, and then determine the amount of debt that is consistent with this level
of I. Conceivably, debt could be negative, which implies holding liquid assets rather
than borrowing.) ...
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 Fall '06
 Tapley
 Finance, Leverage

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