# Breakeven point

The break-even point (also benefit threshold ; English break-even point ) is in the economics of the point at which revenues and total costs of production (or product ) are equal and thus neither loss nor profit is earned.

## General

In simplified terms, the contribution margin of all products sold is identical to the fixed costs at the breakeven point . If the breakeven point is exceeded, profits arise, if it falls below it, losses arise accordingly. The breakeven point can be calculated for one product ( one-product consideration ) or several products (multiple -product consideration ).

In economic mathematics, the breakeven point, like the profit limit, is a zero of the profit function . In both places, revenues and costs are the same. However, the break-even point is the lower zero and the break-even point is the upper zero: profit is made when the break-even point is reached, and losses are written when the profit limit is reached.

The starting point of the break-even analysis are the following questions:

• How many products have to be produced and sold to cover the fixed costs? (One-product consideration)
• How much turnover does the products under review have to generate in order to cover the fixed costs? (Multiple product consideration)

The break-even analysis is an important tool for corporate planning. It helps to analyze the influence of changes in the cost structure and to determine the requirements for the sales volume .

## Break-even point analysis

The break-even point analysis is essential for a company in order to determine the sales volume (also called sales volume) at which full cost recovery occurs. This full-cost recovery will break-even point (BEP short) break or minimum sales called.

A break-even analysis can only be carried out if the costs are broken down into fixed and variable costs and the contribution margin (DB for short) is known. The BEP is an operational indicator that shows how much sales can decline if prices remain the same so that the total costs are just covered.

## Calculation of the break-even point in general

The question about the break-even point is: at what quantity is the profit equal to 0? ${\ displaystyle x}$${\ displaystyle G}$

${\ displaystyle G (x) = 0}$

Generally calculated from the revenues minus the costs . ${\ displaystyle G}$ ${\ displaystyle E}$ ${\ displaystyle K}$

${\ displaystyle G (x) = EK}$

The BEP can be found by equating both of the above equations. It turns out

${\ displaystyle 0 = EK \ quad \ Leftrightarrow \ quad E = K}$

The BEP is therefore the point where the revenue equals the total cost. By equating and inserting the individual straight lines of the respective functions, the formula shown above results. This formula can ultimately be converted to the minimum sales amount.

At the break-even point, the proceeds are equal to the costs

${\ displaystyle E = K \,}$

The function of the revenue ( ) is the unit price multiplied by the units sold or the number of units ${\ displaystyle E}$

${\ displaystyle E = p \ cdot x}$

The total costs consist of the fixed and the variable costs ${\ displaystyle K}$

${\ displaystyle K = K _ {\ mathrm {f}} + k_ {v} \ cdot x}$

When equating the formulas for the revenue, the point of intersection results, which is the BEP

${\ displaystyle x \ cdot p = x \ cdot k_ {v} + K _ {\ mathrm {f}}}$

Dissolved according to the minimum sales amount results ${\ displaystyle x}$

${\ displaystyle x = {\ frac {K_ {f}} {p-k_ {v}}}}$

The contribution margin per unit ( ) is equal to the unit price minus the variable costs per unit. ${\ displaystyle db}$

${\ displaystyle p-k_ {v} = db \,}$
${\ displaystyle x = {\ frac {K _ {\ mathrm {f}}} {db}}}$
• ${\ displaystyle p}$ price per unit
• ${\ displaystyle k_ {v}}$ variable cost per unit
• ${\ displaystyle K _ {\ mathrm {f}}}$ total fixed costs
• ${\ displaystyle x}$ Minimum sales amount.

For companies with more than one product, the minimum turnover is determined in terms of value.

${\ displaystyle x = 100 \ cdot {\ frac {K _ {\ mathrm {f}}} {d}}}$
• ${\ displaystyle x}$ minimum sales in terms of value
• ${\ displaystyle K _ {\ mathrm {f}}}$ total fixed costs
• ${\ displaystyle d}$the contribution margin as a percentage of sales${\ displaystyle DB}$

The break-even point is a tool for the entrepreneur. Therefore, there is a certain degree of freedom as to which costs and revenues (or positive or negative aspects) result from this point.

In the case of step-fixed costs , there may be several break-even points. That means that after a certain sales volume you reach the profit zone. However, due to the influence of the step-fixed costs, a loss zone is reached again. In practice, a linear course is usually used to simplify presentation and handling.

## Graphic representation: the break-even chart

The break-even chart graphically shows the relationship between revenue and costs over the unit quantity.

On the x-axis the amount is plotted on the y-axis of the revenue or cost, sometimes the profit.

## Use of break-even information

The question of the break-even point is an economic consideration to weigh up the negative or positive influencing factors. Negative factors represent, for example, various costs. A positive characteristic is the achievable income from a product or service.

This method can be used in a company not only for cost and revenue analyzes of production quantities, but also for other business issues.

Break-even analyzes, also known as profit or threshold analyzes, can be viewed as a decision-making tool that supports management. The relevant management process can be divided into the planning and control process phases.

In planning, goals are first defined and specified. After problems have been identified and structured, another key task is to identify alternatives. The alternatives are ranked using the results of a break-even analysis and the best alternative is selected. The control follows the planning phase. Here the selected alternative is enforced and executed. The monitoring of the implementation by controls is followed by the backup phase in which the adaptation measures are carried out.

Break-even analyzes are formal representations through mathematical calculations that represent a high reduction in complexity. This enables a problem to be presented simply to the decision maker. For example, in the case of a decision problem about the production of a certain product, the question is answered whether the expected sales volume is below or above the BEP. The break-even analysis converts the existing data into important key figures.

As the example above shows, the reduction in complexity is only possible if there is clarity about the objective in the background. The break-even analysis then provides information about the threshold values ​​that form the limit points of advantage.

a) Planning
With consistent and consistent planning based on break-even figures, the entire planning and control process can be based on this instrument. In the planning of company processes, break-even analyzes provide information about the target effect of possible alternative measures. They serve in particular to assess and compare the alternatives with regard to satisfaction targets or critical target lower limits. For the profit target, the break-even analysis results in the breakeven point, for other targets there are correspondingly different targets. The analysis forms the basis on which a planning system is established and is an indicator of critical situations.
b) Control
In the second part, the control, the break-even information has the character of a default value. This specification can be a minimum or limit value or a desired target value. In any case, a target-actual or target-actual control is in the foreground.

In the first case it is a control to be carried out after the process, in the second case an in-process control. The latter is a plan progress check. It brings about timely control, since adaptation measures can be initiated at an early stage. Prerequisites for this are well-founded forecasts from the planning system and an efficient control system.

## Interpretation of the break-even analysis

The key figure of the minimum turnover is primarily a danger signal, which indicates to the management that measures must be taken when approaching this point: measures such as increased sales efforts, lowering of fixed or variable costs or if these measures are not sufficient, even the shutdown of production. The determination of the BEP is intended to ensure that the company does not get into trouble, as it can recognize the danger early enough and take measures.

## Relationship with the payback period

The payback period can be calculated using the break-even point using the expected sales volume

${\ displaystyle {\ text {Amortization period}} [{\ text {in periods}}] = {\ text {Break-even-Point}} [{\ text {Quantity}}] / {\ text {Expected sales volume by period }}}$

This also answers the question of the time after which the break-even point is reached. When to get out “par” or how long it takes for an investment to pay off.

## One-product consideration

Graphic representation: The amount required to reach the breakeven point (coverage amount) corresponds to the x-value of the intersection of the cost curve with the revenue curve.

Let it be:

• ${\ displaystyle K (x) = k_ {v} \ cdot x + K _ {\ mathrm {f}}}$the cost function
• ${\ displaystyle E (x) = p \ cdot x}$the revenue function
${\ displaystyle K (x) = E (x) \ quad \ rightarrow \ quad {\ text {to}} \ x}$dissolve. That is the breakeven point.

In which:

• ${\ displaystyle k_ {v}}$: variable unit costs
• ${\ displaystyle K _ {\ mathrm {f}}}$: total fixed costs
• ${\ displaystyle p}$: Price per product unit
• ${\ displaystyle x}$: Production / sales volume of the product

This results in the following value for the breakeven point : ${\ displaystyle x_ {G}}$

${\ displaystyle x_ {G} = {\ frac {K_ {f}} {p-k_ {v}}} = {\ frac {K_ {f}} {db}}}$

So products have to be sold to cover all costs. The difference between the sales revenue ( price ) and the variable unit costs is also known as the contribution margin per unit of measure ( ). Geometrically, the breakeven point corresponds to the intersection of the cost function with the revenue function. ${\ displaystyle x_ {G}}$${\ displaystyle db}$

An example: (revenue - variable costs = contribution margin). The contribution margin is the portion that remains to cover the fixed costs. ${\ displaystyle E-K_ {v} = DB}$

### Cost example

Tanning salon fixed costs: € 5,000.00 / month net:

Post amount description
Revenue per tanning session (net) € 5.03 proceeds
Deduction for electricity / tanning (net) € 1.05 Variable costs, because increasing proportionally with the quantity!
Deduction of tube costs (net) € 0.35 Variable costs because increasing proportionally
= € 3.63 Remaining as contribution margin

Fixed costs: 5,000.00 € / 3.63 € = 1,377.41 So this would mean that a tanning salon, after deducting the variable costs, needs 1,378 tanning sessions (must always be rounded up!) Per month in order to meet the fixed (fixed costs) cover. The variable costs only arise when the service is consumed.

## More product consideration

Volume changes in the break-even diagram

When looking at several products, the breakeven point can no longer be specified by the amount of products sold, since the breakeven point can be reached through several different sales quantities of the individual product types . Therefore, the sales to be achieved are used here, which must be generated by the products.

The following formula then results for the breakeven point:

${\ displaystyle U_ {BEP} = {\ frac {K_ {f}} {\ frac {\ sum _ {j = 1} ^ {n} {(p_ {j} -k_ {j}) \ cdot x_ {j }}} {\ sum _ {j = 1} ^ {n} {p_ {j} \ cdot x_ {j}}}}} = {\ frac {K_ {f}} {\ frac {\ sum _ {j = 1} ^ {n} {db_ {j} \ cdot x_ {j}}} {\ sum _ {j = 1} ^ {n} {p_ {j} \ cdot x_ {j}}}}}}$

in which

${\ displaystyle U_ {BEP}}$: Sales that must be achieved in order to break even
${\ displaystyle n}$: Number of product types
${\ displaystyle p_ {j}}$: Selling price of product ${\ displaystyle j}$
${\ displaystyle k_ {j}}$: variable cost of product ${\ displaystyle j}$
${\ displaystyle x_ {j}}$: Production / sales volume of the product ${\ displaystyle j}$
${\ displaystyle db_ {j}}$: Contribution margin from product ${\ displaystyle j}$

## Colloquial

Colloquially, the breakeven point also refers to

• the monetary achievement of the breakeven point of a company (i.e. not a number of units, but a point in time)
• the price at which a securities account, taking into account the fixed costs, reaches the profit zone (break-even price or break-even price)

## Premises

The break-even point analysis is based on certain premises:

• Breakdown of costs into variable and fixed costs
• Production volume = sales volume, storage must be calculated additively
• constant sales prices over the course of the accounting period
• constant production program over the course of the accounting period
• Comparison of positive and negative effects