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This simple condition results in the break-even-volume: <br> | This simple condition results in the break-even-volume: <br> | ||
: x* = K<sup>f</sup>/(p – k<sup>v</sup>). <br> | : x* = K<sup>f</sup>/(p – k<sup>v</sup>). <br> | ||
=== Standard break-even-analysis using contribution margins === | === Standard break-even-analysis using contribution margins === | ||
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The break-even point corresponds to the condition D(x) = K<sup>f</sup>:<br> | The break-even point corresponds to the condition D(x) = K<sup>f</sup>:<br> | ||
: x* = K<sup>f</sup>/d.<br> | : x* = K<sup>f</sup>/d.<br> | ||
=== Graphical break-even-analysis === | === Graphical break-even-analysis === | ||
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* The model assumes a '''simple production structure''' with one product. Convergent production is possible if the production coefficients are fixed. Divergent or [[Multi-product break-even-analysis|multi-product cases]] require more challenging approaches. | * The model assumes a '''simple production structure''' with one product. Convergent production is possible if the production coefficients are fixed. Divergent or [[Multi-product break-even-analysis|multi-product cases]] require more challenging approaches. | ||
* Assumptions concerning costs and sales revenues: | * Assumptions concerning costs and sales revenues: | ||
** Cost and sales revenues depend only on the same single factor, the volume or activity which can be measured by a continuous variable. Other factors are assumed to be constant, irrelevant of covered by the volume, too. | ** Cost and sales revenues depend only on the same single factor, the volume, number of units or activity which can be measured by a continuous variable. Other factors are assumed to be constant, irrelevant of covered by the volume, too. | ||
** Cost and sales revenues are '''linear''' but different. Non-linear or step-wise linear cost and sales revenue functions resulting, among others, from intensity effects, discrete capacity increases or volume discounts are more difficult to handle. | ** Cost and sales revenues are '''linear''' but different. Non-linear or step-wise linear cost and sales revenue functions resulting, among others, from intensity effects, discrete capacity increases or volume discounts are more difficult to handle. | ||
** It is possible to separate costs into a fixed and a variable part. Without this '''cost separation,''' the break-even-volume would be zero (if p ≥ k) or there wouldn’t be any break-even point at all (if p < k). | ** It is possible to separate costs into a fixed and a variable part. Without this '''cost separation,''' the break-even-volume would be zero (if p ≥ k) or there wouldn’t be any break-even point at all (if p < k). | ||
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These assumptions limit the use of break-even-analyses to a very narrow range of cases. For other cases, Schweitzer/Troßmann (1998) describe variants and extensions of the standard break-even model. <br> | These assumptions limit the use of break-even-analyses to a very narrow range of cases. For other cases, Schweitzer/Troßmann (1998) describe variants and extensions of the standard break-even model. <br> | ||
* '''Variants of break-even-analysis''' include (see Schweitzer/Troßmann 1998, pp. 55 ff.): | * '''Variants of break-even-analysis''' include (see Schweitzer/Troßmann 1998, pp. 55 ff.): | ||
** Separate break-even-points for different levels of financial goals. It is straightforward to calculate increasing break-even-points corresponding to a zero EBITDA, a zero EBIT, zero earnings or | ** Separate break-even-points for different levels of financial goals. It is straightforward to calculate increasing break-even-points corresponding to a zero EBITDA, a zero EBIT, zero earnings or other '''target earnings.''' | ||
** Cost-volume-profit-analyses (in the narrow sense of the term) look at the reaction of the break-even-point on variations of unit prices, unit variable costs, or the fixed costs. A typical application is configuration of a product through '''value engineering.''' | ** Cost-volume-profit-analyses (in the narrow sense of the term) look at the reaction of the break-even-point on variations of unit prices, unit variable costs, or the fixed costs. A typical application is the configuration of a product through '''value engineering.''' | ||
** Differences between production and sales volumes can be modeled through the introduction of inventories. | ** Differences between production and sales volumes can be modeled through the introduction of inventories. | ||
** Instead of comparing sales and costs of one project or technique, costs for different projects or techniques to manufacture the same product can be analyzed. Typical applications include decisions about rationalization, the choice among alternative plants, or technological innovations. The result is a recommendation whether to shift from one technique to the next, and a break-even-point for this shift.<br> | ** Instead of comparing sales and costs of one project or technique, costs for different projects or techniques to manufacture the same product can be analyzed. Typical applications include decisions about rationalization, the choice among alternative plants, or technological innovations. The result is a recommendation whether to shift from one technique to the next, and a break-even-point for this '''technology shift.''' <br> | ||
A common characteristic of these variants is that they calculate the break-even-points in the same ways as in the standard model. In general, finding the relevant information or simplifying it to fit to the assumptions of standard break-even analyses is more difficult than its use.<br> | A common characteristic of these variants is that they calculate the break-even-points in the same ways as in the standard model. In general, finding the relevant information or simplifying it to fit to the assumptions of standard break-even analyses is more difficult than its use.<br> | ||
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* Extensions of break-even-analyses<br> | * Extensions of break-even-analyses<br> | ||
: Extensions of break-even-analyses override the limits or the simplifications of the standard model. They require different computational approaches to determine the break-even-points. The most important extensions deal with (see Schweitzer/Troßmann 1998, pp. 122 ff.): | : Extensions of break-even-analyses override the limits or the simplifications of the standard model. They require different computational approaches to determine the break-even-points. The most important extensions deal with (see Schweitzer/Troßmann 1998, pp. 122 ff.): | ||
:* non-linear production structures, especially for [[Multi-product break-even-analysis|multi-product cases]] | :* non-linear production structures, especially for divergent or regrouping [[Multi-product break-even-analysis|multi-product cases]] | ||
:* non-linear cost and sales functions | :* non-linear cost and sales functions | ||
:* more than one factor | :* more than one factor, especially the [[Multi-product break-even-analysis|multi-product cases]] or the combination of output units with process conditions (temperature, ...) | ||
:* dynamic production processes, especially multi-period processes and inventories | :* dynamic production processes, especially multi-period processes and inventories | ||
:* multiple goals.<br> | :* multiple goals.<br> | ||
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== Dealing with risk in break-even-analyses == | == Dealing with risk in break-even-analyses == | ||
Break-even-analysis itself is a technique designed to deal with one form of '''uncertainty | Break-even-analysis itself is a technique designed to deal with one form of '''uncertainty:''' the uncertain outcome of the production or sales volume. The other input data of the break-even model (unit prices, unit variable costs, fixed costs, …) are deterministic. However, if additional information is available, further insights are possible. The additional information can consist in: | ||
* the '''expected value''' x<sup>e</sup> of sales volume: this information allows to calculate the following '''margin of safety''' S (cp. Tucker 1963):<br> | * the '''expected value''' x<sup>e</sup> of sales volume: this information allows to calculate the following '''margin of safety''' S (cp. Tucker 1963):<br> | ||
: S = (x<sup>e</sup> – x*)/x<sup>e</sup> <br> | : S = (x<sup>e</sup> – x*)/x<sup>e</sup> <br> | ||