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UNIT 5.3

KEY TOPICS

• Contribution• Break-even analysis• Margin of safety• Constructing break-even chartsHigher Level Extension

• Effects of changes in price and/or cost on the break-even quantity, profit and margin of safety

• Assumptions and limitations of break-even analysis• Target profit and revenues

Break-even Analysis

He who has no thirst has no business at the fountain.

Dutch proverb

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Break-even exists when a business makes neither a profit nor a loss. This occurs at the level of output where total costs equal total revenue, i.e. TC = TR. Break-even is often a key objective of new and unestablished firms. This suggests that businesses need to pay careful attention to their cash-flow

situation (see Unit 3.3) by monitoring and controlling the money coming into the business (revenues) and the money leaving the firm (costs). This Unit examines the quantitative methods used to calculate a firm’s break-even level of output.

The concept of contribution (see Unit 5.2) is crucial to the understanding of break-even analysis. Recall that unit contribution is the difference between a product’s price and its variable costs of production, i.e. Contribution = P – AVC. Any product that makes a positive contribution will help towards paying some of the fixed costs of the business. Therefore, contribution analysis suggests three broad ways in which profits can be improved:

• Increasing sales revenue, for example by using appropriate marketing strategies to attract more customers

• Reducing variable costs, for example by seeking cheaper production methods, helping to raise the contribution made from selling each unit of output

• Reducing fixed costs, for example by negotiating cheaper rents or limiting extravagant company expenses, thereby helping to reduce the break-even level of output.

Managers of all businesses are concerned with the difference between revenue and costs. A business can only survive in the long run if revenues exceed costs, i.e. if it is profitable. New firms in particular will want to determine the level of sales that must be generated in order for the business to earn a profit. Break-even analysis is a management tool that can be used to serve this purpose.

A business can be in any one of the following financial situations:

• Loss – when costs of production exceed the revenues of the business

• Break-even – when the revenues of the business equal the costs of production

• Profit – when revenues exceed costs of production.

Carrying out a break-even analysis can inform managers of two things:

• Whether it is financially worthwhile to produce or launch a particular good or service

• The level of profits that the business is likely to earn if things go according to plan.Consider the following a numerical example. A

jeans retailer has fixed costs of $2,500 per month. Variable costs are known to be $10 per pair of jeans, with each selling for $30. There are three ways to calculate or determine the break-even point (see Box 5.3a for break-even formulae).1. Interpretation from a break-even chart

In Figure 5.3a, the break-even level of output can be seen at the point where TC = TR, i.e. 125 pairs of jeans.

INTRODUCTION

CONTRIBUTION

BREAK-EVEN ANALYSIS

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Figure 5.3a Break-even chart

2. Using the TC = TR ruleThe break-even point can be calculated by com-paring total sales revenues with total costs. Recall from Unit 5.2 that total revenue is calcu-lated as Price Quantity sold and that total costs consist of both fixed and variable costs. Break-even can then be calculated as:P Q = TFC + TVC

30Q = 2,500 + 10Q20Q = 2,500Q = 125 units (or pairs of jeans)

3. Using the Unit Contribution rule: Break-even = Fixed Costs ÷ Unit ContributionThis is the quicker of the two quantitative methods of calculating break-even. Using the figures gives us:Unit Contribution = Price minus average variable costs = $30 minus $10 = $20Therefore, the break-even is $2,500 ÷ $20 = 125 pairs of jeans.

All three methods give the same answer – the business needs to sell 125 pairs of jeans each month for it to reach break-even. The chart also shows that any sales beyond the break-even level of output generate a surplus (or profit) whereas levels of sales below the break-even point means that the firm makes a loss for that month. The break-even quantity (BEQ) is the level of output where total costs equal total revenues. This information can be shown in a break-even chart (see Figure 5.3b).

Figure 5.3b Break-even chart – profit or loss

Costs and Revenues

($)

Output (pairs of jeans p/m)

TC

TR

2,500

The break-even point refers to the position on a break-even chart where the total cost line intersects the total revenue line, i.e. where TC = TR.

Break-even point

125

Box 5.3a Break-even formulae

• Unit Contribution: P – AVC• Break-even: TC = TR

or

• Profit (or loss): TR – TC

Fixed costsUnit contribution----------------------------------------------

Exam Tip!Break-even is a popular examination topic. Therefore, it is important to be able to accurately construct and interpret information shown in a break-even chart. For example, candidates often label the axes on a break-even graph inaccurately. Perhaps more importantly, candidates must be able to modify a break-even chart and analyse its implications for a business.

Costs and Revenues

($)

Output (pairs of jeans p/m)

TC

TR

2,500 Break-even point

Break-even quantity

Profit

Loss

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The margin of safety measures the difference between a firm’s sales volume and the quantity needed to break-even, i.e. it shows how much demand (for a product) exceeds the break-even quantity. The larger the positive difference between sales output and BEQ, the safer the firm will be in terms of earning profits, especially if there are adverse changes in the marketplace. A positive margin of safety (or safety margin) means that the firm makes a profit, whereas a negative safety margin means the firm makes a loss. The margin of safety is calculated using the formula:

Safety margin = Level of demand less Break-even quantity

For example, if the demand for the jeans retailer in the above example is 200 pairs per month, then the safety margin is 75 units (i.e. 200 minus 125). This means that the business can sell 75 pairs of jeans less than its current level and still not make a loss (see Figure 5.3c). Hence, the smaller the safety margin the more vulnerable a business becomes to changes in the market. Many businesses prefer to express the safety margin as a percentage of the BEQ as this puts the figure into context and allows better comparisons to be made. In this case, the safety margin is 60 per cent higher than the break-even level of output. Therefore, the safety margin can reveal the degree of risk involved in a business decision.

Figure 5.3c Margin of safety

Question 5.3.1

a Use the unit contribution method to calculate the break-even quantity for a firm that has:• TFC = $200,000• Average Variable Costs = $5• Price = $30 [2 marks]

b Calculate the value of sales at the break-even quantity. [2 marks]

THE MARGIN OF SAFETY

Exam Tip!Far too often, candidates will express the margin of safety as a monetary value. This clearly shows a lack of understanding and application of the concept. The margin of safety is calculated and shown on the x-axis of a break-even chart, i.e. the unit of measurement is the volume of output and not the value of that output.

Costs and Revenues

($)

Output (pairs of jeans p/m)

TC

TR

2,500

Break-even quantity

Profit

Loss

125 200Level of demand

Margin of safety

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To construct an accurate break-even chart, use the following rules:

• Although not necessarily required to show break-even, it is conventional to draw and label the Total Fixed Costs (TFC) line.

• The Total Costs (TC) line is drawn and labelled. Recall that even when there is no output, fixed costs still have to be paid, therefore the TC line starts at the same level as total fixed costs.

• The Total Revenue (TR) line is drawn and labelled. When there is no output, the revenue is zero, so the TR line starts at the origin.

• The x-axis is labelled as ‘Output’ and measured in the appropriate units, per time period.

• The y-axis is labelled as ‘Costs, Revenues and Profits’ which are all expressed in terms of a currency.

• A title, put into the context of the business, should also be added.

Consider the following worked example as a point of illustration. Suppose that Katia Jewellery Ltd. sells hand-made earrings at an average price of $20 with variable costs averaging $8 per unit. The fixed costs are $4,500. In order to plot the break-even chart, it is necessary to first calculate:

• the BEQ. Using the unit contribution method, the BEQ is calculated as (4,500 / (20 – 8) = 375 units.

• the value of costs and revenue at the BEQ. Since the value of TC and TR are the same at the BEQ, it does not matter which component is worked out. For example, TC = 4,500 + (8 375) = $7,500. Equally, if we calculate the TR, the figure would be $20 375 = $7,500.Using this information, it is now possible to plot

the break-even chart. We know that the x-axis must go to beyond 375 units and that the y-axis must go beyond $7,500 (see Figure 5.3d on page 562).

Question 5.3.2

Calculating the margin of safety

Tread-it is a manufacturer of hiking shoes. Play-it produces wooden toys for children. Cost and revenue data for both businesses are shown in the table below.

a Calculate the missing figures for i, ii and iii in the table above. [3 marks]b Comment on which firm has the better margin of safety. [4 marks]

Tread-it Play-it

Break-even quantity 250 500

Output 500 i

Margin of safety (units) ii 300

Margin of safety (%) 100 iii

CONSTRUCTING BREAK-EVEN CHARTS

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Figure 5.3d Break-even chart for Katia Jewellery Ltd.

Exam Tip!If the BEQ is a decimal number, such as 185.33, then the value has to be rounded up (to 186 in this case) because firms cannot sell a third of a product to a customer!

Exam Tip!Before drawing a break-even chart, it is important to first work out the value of the costs and revenues at the break-even quantity. This will help to determine the scale needed to plot the figures on the y-axis. By working out the BEQ beforehand, it is also easier to determine the scale needed for the x-axis.

Costs, Revenues and Profits ($)

Output (units of jewellery p/m)

TC

TR

4,500

BEQ375

TFC

7,500

Question 5.3.3

Lisa Chan’s Day-Care Centre

Lisa Chan runs a children’s day-care centre. The main clients are working parents, who pay a fixed $20 per child for the whole day. Children at the centre learn through play and are engaged in activities such as art, music, dance and physical education. The business is open for an average of 22 days each month. The firm’s expected costs and revenue for the next year are as follows:

a Calculate the sum of the fixed costs. [2 marks]b Calculate the break-even quantity per month. [2 marks]c Assuming the business works at 80% of its capacity, calculate the margin of safety. [2 marks]d Construct a fully labelled break even chart for Lisa Chan’s Day-Care Centre. [5 marks]e Identify the break-even point, the break-even output and the safety margin on your chart. [3 marks]f Examine the strengths and weaknesses of using break-even analysis for a business such as Lisa Chan’s

Day-Care Centre. [6 marks]

Capacity 25 children per day

Demand 80% of capacity

Price $20 per child per day

Materials $4 per child

Rent $600 per month

Salaries $1,000 per month

Administration $100 per month

Power $140 per month

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One criticism of break-even analysis (BEA) is that the model is static, i.e. it represents only a snapshot position of the business. Manual calculations of changes in break-even can also be onerous.

However, computer software packages can show managers the effects of changes in costs and revenues on the BEQ. This makes the BEA model a little more flexible.

Although break-even analysis can be a useful decision-making tool, there are several limitations. For example, the assumptions of the model are hardly ever met by any real business:

• BEA assumes that all cost functions are linear. In reality, the cost lines are unlikely to be linear because economies of scale (and hence lower average costs of production) could be enjoyed by operating on a larger scale. Fixed costs might also change, perhaps due to an increase in rent. This would lead to a ‘stepped’ fixed cost line, rather than a horizontal line.

• It also assumes the sales revenue function is linear. In reality, customers would demand discounts for larger orders, thereby distorting the sales revenue line. Indeed, demand theory states that to sell more, a business might need to reduce its prices. Also, a linear sales revenue function totally ignores price discrimination (see Unit 4.4) used by some businesses (when a firm is able to charge different prices to different

groups of customers, such as adults and children).

• BEA assumes that only one product is produced by the business. Despite this, the model can be used to make predictions that are more realistic rather than relying on simple guesswork. For example, a restaurateur can use past data and experience to estimate the average cost of a meal, the average number of customers and the average price paid for each meal. This data can therefore help a multi-product firm to work out, albeit inaccurately perhaps, its break-even level of sales.

• It assumes that the business will sell all of its output. However, in reality most businesses will have some unsold stock, which do not generate cash but cost the firm money (in terms of storage and insurance costs, for example). Furthermore, unsold stock might need to be sold at a discount (thereby reducing the profits) of the business to make space for new incoming stocks.

HIGHER LEVEL EXTENSION: CHANGES IN BREAK-EVEN

Question 5.3.4

Phoebe's Dance Studio Ltd.

Phoebe’s Dance Studio Ltd. has overhead costs of $3,000 per month, variable costs of $5 per unit and an average selling price of $20. There are typically 500 customers each month but the firm’s maximum capacity is 600 clients.1 Calculate the break-even quantity for Phoebe’s Dance Studio Ltd. [2 marks]

2 Calculate the margin of safety for the business. [2 marks]

3 Plot the break-even chart for Phoebe’s Dance Studio Ltd. [5 marks]

Suppose in the subsequent time period that rents increase, thereby raising the firm’s overhead costs to $4,000. In addition, average selling price has fallen to $17 and this has increased demand to 520 clients per month.4 Calculate the new break-even quantity and comment on your findings. [3 marks]

5 Illustrate the new break-even level of output on your original chart. [3 marks]

Phoebe’s Art Studio Ltd.Phoebe’s Art Studio Ltd. has overhead costs of $3,000 per month, variable costs of $5 per unit and an average selling price of $20. There are typically 500 customers each month but the firm’s maximum capacity is 600 clients.

a Calculate the break-even quantity for Phoebe’s Art Studio Ltd. [2 marks]b Calculate the margin of safety for the business. [2 marks]c Construct a break-even chart for Phoebe’s Art Studio Ltd. [5 marks]

Suppose in the subsequent period that rents increase, thereby raising the firm’s overhead costs to $4,000. In addition, average selling price has been reduced to $17 and this has increased demand to 520 clients per month.

d Calculate the new break-even quantity and comment on your findings. [3 marks]e Illustrate the new break-even level of output on your original chart. [3 marks]f Explain whether the change in price was a sensible decision. [4 marks]

HIGHER LEVEL EXTENSION: LIMITATIONS OF BREAK-EVEN ANALYSIS

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Other limitations of break-even analysis include:• BEA is a static model so might not be very

useful in a dynamic business environment. For example, it ignores the possibility that production costs can and do change at short notice, such as fluctuating exchange rates which affect the costs and revenues of exporting firms. The use of dedicated computer software, such as spreadsheets, can help to update data easier, but each set of break-even calculations will only be valid for one point in time.

• As with all financial and numerical predictions, the principle of garbage in, garbage out (GIGO) applies. Unrealistic and outdated data input will generate dubious results. Hence, the accuracy of any BEA largely depends on the validity of the original data used to generate the calculations and on the skills and experiences of the managers in estimating costs, output and revenues.

• Other quantitative and qualitative factors that can alter the costs, revenues and profits of the business are ignored. For example, BEA ignores the impacts of staff working under increased pressures to maximise output, such as demotivation and declining productivity. The reaction of competitors, the availability of spare capacity and access to finance are also ignored in BEA.

• BEA is really only suitable to single-product firms. For firms with more than one type of product, overheads have to be split between the products in a rather subjective way (see Unit 5.2). Although there are software programmes that can help managers to calculate multiproduct break-even, these do not truly represent the break-even for each product.

Theory of Knowledge

How do models such as break-even analysis support or hinder our search for knowledge?

Question 5.3.5

Airbus A380Back in 2005, the Airbus A380 – the world’s largest commercial aircraft – was estimated to break-even on 270 aircraft orders. However, a year later the company announced that the break-even quantity had been revised to 420 aircraft following production delays and soaring costs that had plagued the European aircraft giant. At that time, Airbus had only sold 159 A380s, with the first plane delivered two years behind schedule. Airbus expects to sell more than 750 A380 planes over the life of the project.

a Calculate the revised margin of safety for the production of the A380. [2 marks]b Comment on how the change in the margin of safety will affect Airbus. [4 marks]c Explain how the delays and soaring production costs might affect the profits of Airbus. [4 marks]d Discuss the value of break-even analysis as a management tool for businesses such as Airbus. [7 marks]

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Continuing from the earlier example of Katia Jewellery Ltd., it is possible to use a break-even chart to work out the level of sales needed to earn a certain amount of profit. For example, in order for

the firm to earn a target profit of $3,000, it can be seen from the graph that the firm needs to sell 625 units of jewellery, per time period.

Figure 5.3e Break-even chart for Katia Jewellery Ltd.

Target profit can also be worked out manually, without the use of a break-even chart. Instead, the following steps are taken to work out the level of output needed for Katia Jewellery Ltd. to make $3,000 profit:

Profit = TR minus TC3,000 = 20Q – 4,500 – 8Q7,500 = 12QHence, Q = 625 unitsThe target revenue can then be worked out as TR = $20 625 = $12,500The total costs at 625 units of output = $4,500 + ($8 625) = $9,500Therefore the target profit is $12,500 – $9,500 = $3,000Whilst profit can be seen from a break-even

chart, it is typically calculated by working out the difference between total revenues and total costs. For example, suppose that Katia Jewellery Ltd. has sales of 700 units per month. Profit can be worked out using the TR – TC rule:

TR = 20 700 = $14,000TC = 4,500 + (8 700) = $10,100Profit = $3,900

However, contribution can also be used to work out profits at each level of output.

Profit = Total Contribution – TFC

Total contribution = (20 – 8) 700 = $8,400TFC = $4,500Profit = $3,900In reality, actual profits (or losses) are likely to

be different from those predicted in a BEA because there are so many factors that can affect the profit (or loss) of a business. These factors include:

• The difference between short run and long run profits. It may be necessary to lower prices (and hence break-even occurs at a higher level of output) in order to attract customers to a firm's products. In the long term, prices can be increased once a loyal customer base has been established.

• The level of demand is subject to change – Factors that affect demand, such as changes in income or fashion, will alter the BEQ and hence the value of profits.

• Profit also depends of the level of risk involved - Whilst low-risk projects generally lead to a quicker BEQ, the monetary value of profits is likely to be low. High-risk projects, such as the Airbus A380 (see Question 5.3.5) have the potential of returning huge amounts of profits.

HIGHER LEVEL EXTENSION: TARGET PROFIT AND REVENUE

Costs, Revenues and Profits ($)

Output (units of jewellery p/m)

TC

TR

4,500

BEQ375

TFC

7,5009,500

12,500

625

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• Innovation and the introduction of new ideas. For example, Dell computers, Dyson vacuum cleaners and Apple iPods have generated sales and profits far in excess of their original forecasts.

• Luck! Every business needs a little bit of luck to succeed. External factors such as changes in exchange rates, unemployment, national income and interest rates can have a direct impact (positive or negative) on the profitability of businesses.

Despite its limitations, BEA is a useful strategic planning tool for presenting cost and revenue data to aid decisions, such as:

• Product portfolio management – BEA helps to assess the expected BEQ prior to the launch of a new product, thereby helping firms to manage their product portfolio. Although the analysis works best for single-product firms, allocating overheads (see Unit 5.2) can help to alleviate this issue.

• Risk assessment – Calculating the margin of safety helps managers to gauge the level of risk involved in a particular project. A predicted safety margin of 250 per cent might justify the go-ahead for a project, whereas a negative safety

margin can prevent a loss being made had the firm invested in the plan.

• Make-or-buy decisions – As the name suggests, these decisions refer to a firm’s choice of whether to produce a product itself or to buy it from a supplier. BEA shows the relative benefits of either decision.

• Special order decisions – Special orders are atypical and/or one-off orders for which a business will charge a price that differs from the norm. For example, some customers might demand speedier delivery times or changes to the product specification, thereby raising production costs. BEA helps to assess whether the change in profits – by accepting the special order – is worthwhile e (see Box 5.3b).

Question 5.3.6

RT’s HotdogsRhys Thomas runs a hotdog stall outside a busy shopping mall. His expected costs and revenues for the next few months are shown below:

a Construct a break-even chart for Rhys Thomas, showing the monthly break-even quantity. [5 marks]b Assume that the average daily sales volume increases to 70% of capacity and that rents rise by 50%.

Show the effect of these changes on the break-even chart and comment on your findings. [5 marks]

Capacity 200 hotdogs per day

Sales volume 110 hotdogs per day

Unit price $2.50

Ingredients and materials $0.80 per hotdog

Rent $200 per month

Salary $500 per month

Other overhead costs $320 per month

BREAK-EVEN ANALYSIS AND BUSINESS STRATEGY

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As with all quantitative tools, it is important to consider the analysis in the context of the business. It is also essential to remember that break-even analysis should be used with caution, bearing in mind the assumptions of the model. Finally, the analysis should be used alongside other tools such as SWOT analysis (see Unit 1.6) and investment appraisal (see Unit 3.2) to form more comprehensive and coherent decision making.

1 Explain the meaning of ‘break-even’.2 What is ‘contribution’ and why is it important

to understanding break-even analysis?3 What are the three generic ways that profits can

be increased?

4 State the two formulae for calculating break-even.

5 Outline the difference between the ‘break-even point’ and the ‘break-even quantity’.

6 What is the ‘margin of safety’ and how is it calculated?

Higher Level Extension

7 State three assumptions made when carrying out a break-even analysis.

8 Outline three limitations of using break-even analysis.

9 What is a ‘make-or-buy decision’?10 What are ‘special order decisions’?

Break-even chart is the name given to the graph that shows a firm’s costs, revenues and profits (or losses) at various levels of output.Break-even point refers to the position on a break-even chart where the total cost line intersects the total revenue line, i.e. where TC = TR.Break-even quantity (BEQ) refers to the level of output that generates neither any profit nor loss. It is shown on the x-axis on a break-even chart.Contribution per unit (or unit contribution) is the difference between the selling price of a product and its variable costs of production. The surplus goes towards paying fixed costs.Margin of safety (or safety margin) is the difference between a firm’s level of demand and its break-even quantity. A positive safety margin means the firm can decrease output (sales) by that amount without making a loss. A negative safety margin means that the firm is making a loss.Profit is the positive difference between a product’s revenue and its costs at each level of output. On a break-even chart, profit can be seen to the right of the break-even quantity.Special order decisions are unique and/or unusual orders for which a customer will pay a price that differs from the norm.

Box 5.3b Special order decisions – a numerical example

Suppose a charity wishes to buy 10 computers at a price of $500 each instead of the usual $800 price tag. Average variable costs of the computers are $200 and fixed costs are allocated at $3,000. Assuming the business has spare capacity, should it take on this special order?Break-even analysis shows that the BEQ is 10 computers, i.e. $3,000 ÷ ($500 – $200).There are three possible outcomes:• Based on financial grounds, the firm only breaks

even. For some managers, this will not be worthwhile, especially if production costs turn out to be higher than expected.

• Based on qualitative factors, such as corporate social responsibility, the business may take on the order since they feel that the charity is supporting a worthwhile cause.

• Based on contribution analysis, the fixed costs exist with or without this special order. So, by taking on the special order, there is a contribution of $300 per computer. Hence, some managers would accept the special order.

R EVIEW QUESTIONS 5.3

TERMS