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A P P E N D I X 1 3 A D E G R E E O F L E V E R A G E 13A-
DEGREE OF LEVERAGE
13A-
In our discussion of operating leverage in Chapter 13, 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 exam- ple, if a firm reduced its operating leverage, this would probably lead to an increase in its optimal use of financial leverage. On the other hand, if it 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 actually specify simultaneously the optimal levels of operating and financial leverage. How- ever, we can see how operating and financial leverage interact through an analysis of the degree of leverage concept.
D E G R E E O F O P E R AT I N G L E V E R A G E ( D O L )
The degree of operating leverage (DOL) , is defined as the percentage change in oper- ating income (or EBIT) that results from a given percentage change in sales:
(13A-1)
In effect, the DOL is an index number which measures the effect of a change in sales on operating income, or EBIT. DOL can also be calculated by using Equation 13A-2, which is derived from Equation 13A-1:
(13A-2)
or, based on dollar sales rather than units,
(13A-2a)
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 13A-2 is normally used to analyze a single product, such as IBM’s PC, whereas Equation 13A-2a is used to evaluate an entire firm with many types of products, where “quantity in units” and “sales price” are not meaningful.
DOLS
S VC
S VC F
Q(P V)
Q(P V) F
DOLQ Degree of operating leverage at Point Q
DOL
Percentage change in EBIT Percentage change in sales
EBIT
EBIT
Q
Q
13A
13A-2 A P P E N D I X 1 3 A D E G R E E O F L E V E R A G E
Equation 13A-2 is developed from Equation 13A-1 as follows. The change in units of output is defined as Q. In equation form, EBIT Q(P V) 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 EBIT Q(P V). The initial EBIT is Q(P V) F, so the percentage change in EBIT is
The percentage change in output is Q/Q, so the ratio of the percentage change in EBIT to the percentage change in output is
(13A-2)
Applying Equation 13A-2a to data for an illustrative firm, Hastings Inc., at a sales level of $200,000 as shown in Table 13A-1, we find its degree of operating leverage to be 2.0:
Thus, an X percent increase in sales will produce a 2X percent increase in EBIT. For example, a 50 percent increase in sales, starting from sales of $200,000, will result in a 2(50%) 100% increase in EBIT. This situation is confirmed by examining Sec- tion I of Table 13A-1, where we see that a 50 percent increase in sales, from $200, to $300,000, causes EBIT to double. Note, however, that if sales decrease by 50 per- cent, then EBIT will decrease by 100 percent; this is again confirmed by Table 13A-1, 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, of sales:
In general, if a firm is operating at close to its breakeven point, the degree of op- erating 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 13A-1, 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:
When evaluated at higher sales levels, DOL progressively declines.
undefined but infinity.
DOL$100,000
DOL$300,000
DOL$200,000
DOL
Q(P V)
Q(P V) F
Q
Q
a
Q(P V)
Q(P V) F
b a
Q
(Q)
b
Q(P V)
Q(P V) F
%EBIT
Q(P V)
Q(P V) F
13A-4 A P P E N D I X 1 3 A D E G R E E O F L E V E R A G E
D E G R E E O F F I N A N C I A L L E V E R A G E (DFL)
Operating leverage affects earnings before interest and taxes (EBIT), whereas finan- cial leverage affects earnings after interest and taxes, or the earnings available to com- mon stockholders. In terms of Table 13A-1, operating leverage affects the top sec- tion, 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 vari- able cost ratio would be lower than 60 percent of sales, and its EBIT would be more sensitive to changes in sales. Financial leverage takes over where operating 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 first-stage leverage and fi- nancial leverage as second-stage leverage. The degree of financial leverage (DFL) is defined as the percentage change in earn- ings per share that results from a given percentage change in earnings before inter- est and taxes (EBIT), and it is calculated as follows:
(13A-3)
Equation 13A-3 is developed as follows:
1. Recall that EBIT Q(P V) F. 2. Earnings per share are found as EPS [(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 I 0; hence, EPS, the change in EPS, is
4. The percentage change in EPS is the change in EPS divided by the original EPS: 5. The degree of financial leverage is the percentage change in EPS over the per- centage change in EBIT:
. (13A-3)
6. This equation must be modified if the firm has preferred stock outstanding.
DFL
EBIT
EBITI
EBIT
EBIT
a
EBIT
EBITI
b a
EBIT
EBIT
b
EBIT
EBITI
EBIT(1 T)
N
(EBIT I)(1 T)
N
c
EBIT(1 T)
N
d c
N
(EBIT I)(1 T)
d
EBIT
EBIT I
EPS
(EBITI)(1T)
N
EBIT(1T)
N
EBIT
EBITI
DFL
Percentage change in EPS Percentage change in EBIT
13A-
Applying Equation 13A-3 to data for Hastings at sales of $200,000 and an EBIT of $40,000, the degree of financial leverage with a 50 percent debt ratio is
Therefore, a 100 percent increase in EBIT would result in a 1.43(100%) 143 per- cent increase in earnings per share. This may be confirmed by referring to the lower section of Table 13A-1, where we see that a 100 percent increase in EBIT, from $40,000 to $80,000, produces a 143 percent increase in EPS:
If no debt were used, the degree of financial leverage would by definition be 1.0, so a 100 percent increase in EBIT would produce exactly a 100 percent increase in EPS. This can be confirmed from the data in Section II of Table 13A-1.
C O M B I N I N G O P E R AT I N G A N D F I N A N C I A L L E V E R A G E (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, and 2. That the greater the use of debt as measured by the degree of financial lever- age, the more sensitive EPS will be to changes in EBIT.
Therefore, if a firm uses a considerable amount of both operating and financial lever- age, then even small changes in sales will lead to wide fluctuations in EPS. Equation 13A-2 for the degree of operating leverage can be combined with Equa- tion 13A-3 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:
(13A-4)
(13A-4a)
(13A-4b)
Equation 13A-4 is simply a definition, while Equations 13A-4a and 13A-4b are developed as follows:
1. Recognize that EBIT Q(P V) F, and then rewrite Equation 13A-3 as follows:
DFL (13A-3a)
EBIT
EBIT I
Q(P V) F
Q(P V) F I
S VC F
S VC F I
DTL
S VC
S VC F I
DTL
Q(P V)
Q(P V) F I
DTL (DOL)(DFL).
%EPS
EPS
EPS
DFLS$200,000, D50%
A P P E N D I X 1 3 A D E G R E E O F L E V E R A G E
13A-
Having the alternatives stated in this manner gives decision makers a better idea of the ramifications of alternative actions. 1
P R O B L E M S
Grant Grocers has sales of $1,000,000. The company’s fixed costs total $250,000, and its vari- able costs are 60 percent of sales. What is the company’s degree of operating leverage? If sales increased 20 percent, what would be the percentage increase in EBIT? Arthur Johnson Inc.’s operating income is $500,000, the company’s interest expense is $200,000, and its tax rate is 40 percent. What is the company’s degree of financial leverage? If the company were able to double its operating income, what would be the percentage increase in net income? A company currently has $2 million in sales. Its variable costs equal 70 percent of its sales, its fixed costs are $100,000, and its annual interest expense is $50,000. a. What is the company’s degree of operating leverage? b. If this company’s operating income (EBIT) rises by 10 percent, how much will its net in- come increase? c. If the company’s sales increase 10 percent, how much will the company’s net income in- crease? 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 percent. The cost of debit is k (^) d 10 percent. The tax rate is irrelevant for the problem, and fixed operating costs do not include interest. a. 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.) b. Given the firm’s present debt, which method would produce the greater percentage increase in earnings per share for a given increase in EBIT? (Hint: Compare DFLs under the two methods.) c. Calculate DTL under each method, and then evaluate the firm’s risk under each method. d. Is there some debt ratio under Method A that would produce the same DTL (^) A as the DTL (^) B that you calculated in part c? (Hint: Let DTL (^) A DTLB 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.)
13A- Operating leverage effects
13A- Degree of leverage
13A- Degree of financial leverage
13A- Degree of operating leverage
A P P E N D I X 1 3 A D E G R E E O F L E V E R A G E
(^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 ana- lyst must make the separation in a judgmental manner. Note that costs are really fixed, variable, and “semi- variable,” 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 on. 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.
