0264 · 4.4.3Break-even analysis

Costs, revenue and break-even

0:00A business can sell plenty of products and still struggle to cover its costs. Break-even is the level of output at which total revenue equals total costs. At this point, the business makes neither a profit nor a loss. We will work out where that point comes from, read it on a chart, and use it to judge business decisions. Revenue is the income a business receives from selling goods or services.

0:22Profit is total revenue minus total costs. A loss is when total costs are greater than total revenue. These definitions link sales and costs to the output where profit begins. Fixed costs are costs that do not change with the level of output in the short run; variable costs change directly with the level of output. Rent is usually fixed over the period considered, while materials increase as more products are made.

0:47Total cost equals total fixed costs plus total variable costs. Break-even analysis brings these costs together with the income from sales. Every figure must refer to the same period. Monthly rent belongs with monthly sales and monthly material costs. We will assume that the business sells everything it produces during that month. This allows us to connect output directly with sales, although real businesses sometimes have unsold products. Kestrel Cases makes protective covers for musical instruments.

1:15Its monthly fixed costs are twelve thousand dollars, its selling price is ten dollars, and its variable cost is four dollars per unit. We assume that the price and variable cost per unit stay constant across the output shown. These conditions give us a simple model that we can use consistently.

Calculating break-even output

1:32Before any products are sold, the business still has twelve thousand dollars of fixed costs to cover. Each sale brings in ten dollars, but making that unit adds four dollars to its costs. The remaining six dollars helps cover the monthly fixed costs. This remaining amount gives us the key to understanding the calculation. Contribution per unit is selling price per unit minus variable cost per unit. It shows how much each sale provides towards fixed costs and, once those are covered, profit.

2:00For Kestrel Cases, ten dollars minus four dollars gives six dollars per unit. Each additional sale therefore closes the gap between revenue and total costs by six dollars. The business needs enough six-dollar contributions to cover twelve thousand dollars of fixed costs. Division tells us how many of those contributions are required. Break-even output equals fixed costs divided by contribution per unit. Twelve thousand dollars divided by six dollars per unit gives two thousand units.

2:28At two thousand units, the combined contribution exactly covers the fixed costs. Each sale has already provided enough revenue to cover its own variable cost. Together, the sales now cover every cost for the month. That is why the answer represents the point where the business makes neither a profit nor a loss. We can check the answer using the underlying figures. Two thousand units sold at ten dollars each produce twenty thousand dollars of revenue.

2:53Two thousand units at four dollars each create eight thousand dollars of total variable costs. Twelve thousand dollars plus eight thousand dollars gives twenty thousand dollars of total costs. Revenue and total costs are both twenty thousand dollars, so the calculation passes our check. The answer is two thousand units because we are finding a quantity of products. If division produces part of a unit, a business selling whole products needs the next whole unit to cover its costs.

3:20Always keep the unit of measurement beside your answer.

Reading a break-even chart

3:24A break-even chart presents these same relationships visually. The vertical axis measures costs and revenue in dollars, while the horizontal axis measures output in units. We will show the fixed costs, total costs and revenue as three separate lines. Their positions explain why low sales create a loss and higher sales create profit. The fixed costs line is horizontal at twelve thousand dollars. At zero output, the total costs line starts at that same value because the business still pays its fixed costs.

3:53As output increases, the line rises by four dollars for each extra unit. Its slope therefore represents the variable cost per unit. The revenue line begins at the origin because zero sales produce zero revenue. Each additional unit adds ten dollars, so this line rises more steeply than total costs. At two thousand units, the lines meet at twenty thousand dollars. Dashed guides from their intersection show the corresponding values on both axes.

4:19The blue guide marks one thousand units. Revenue at that output is ten thousand dollars, while total costs are sixteen thousand dollars. Sixteen thousand dollars minus ten thousand dollars gives a loss of six thousand dollars. The vertical gap between the lines tells us the size of that loss. At the green guide, output reaches three thousand units. Revenue is thirty thousand dollars and total costs are twenty-four thousand dollars.

4:44Thirty thousand dollars minus twenty-four thousand dollars gives profit of six thousand dollars. Here, the revenue line sits above total costs, so the vertical gap represents profit. To read profit or loss, find the chosen output and trace vertically to both lines. Read their dollar values from the side axis, then calculate the difference. Break-even output comes from the horizontal position of their intersection. The chart therefore answers different questions through either a vertical gap or a position along the bottom.

Completing charts and margin of safety

5:14Completing a chart means using the figures to supply its missing parts. If Kestrel Cases has a missing total costs line, begin at twelve thousand dollars on the vertical axis. Plot twenty thousand dollars at two thousand units and join those points with a straight line. Label the line clearly so its meaning is visible. For a missing revenue line, start at the origin and use the selling price to find another point.

5:36Two thousand sales at ten dollars each give the point at twenty thousand dollars. After completing the line, mark where it meets total costs. Use the chart’s scale carefully, especially when each square represents several hundred units or dollars. Margin of safety is the actual number of sales minus the break-even number of sales, in units. It measures how far sales can fall before the business reaches break-even. Kestrel Cases actually sells three thousand units during the month.

6:03Three thousand units minus two thousand units gives a margin of safety of one thousand units. The bracket along the chart’s output axis shows that one-thousand-unit cushion. Sales could fall by that amount and the business would still cover its total costs. A further fall would place it in the loss area. A larger positive margin provides a greater buffer against falling sales, assuming prices and costs remain unchanged. We can also verify the profit at actual sales through a full calculation.

6:30Three thousand units multiplied by four dollars gives twelve thousand dollars of total variable costs. Total cost equals total fixed costs plus total variable costs. Twelve thousand dollars plus twelve thousand dollars gives twenty-four thousand dollars of total costs. Profit equals total revenue minus total costs. Three thousand units multiplied by ten dollars gives thirty thousand dollars of revenue. Thirty thousand dollars minus twenty-four thousand dollars gives six thousand dollars of profit.

6:59The arithmetic agrees with the vertical gap at the green guide on our chart.

Effects of changing costs and price

7:03Break-even analysis helps managers examine changes in selling price, fixed costs and variable cost per unit. We will change each factor separately from the original Kestrel Cases figures. This keeps the cause of each result clear. After every comparison, we return to the original conditions before changing a different factor. Suppose the selling price rises from ten dollars to twelve dollars per unit. Variable cost remains four dollars per unit, and fixed costs remain twelve thousand dollars.

7:31Twelve dollars minus four dollars gives contribution of eight dollars per unit. Twelve thousand dollars divided by eight dollars per unit gives break-even output of fifteen hundred units. Each sale now contributes more towards the same fixed costs, so fewer sales are needed to cover them. On the chart, the revenue line becomes steeper while the cost lines remain unchanged. The intersection moves left to fifteen hundred units, where revenue and total costs both equal eighteen thousand dollars.

7:59If actual sales remain three thousand units, three thousand minus fifteen hundred gives a margin of safety of fifteen hundred units. Customers’ responses remain important to that decision. A higher price can reduce the number of sales if customers choose other suppliers. Managers should compare the new break-even output with realistic expected sales. The lower threshold is useful evidence, while the forecast of customer demand determines whether the business can actually reach it.

8:26Return to the original price and variable cost, then raise fixed costs to fifteen thousand dollars. Contribution remains six dollars per unit because those two unit figures are unchanged. Fifteen thousand dollars divided by six dollars per unit gives break-even output of twenty-five hundred units. More sales are needed because the business must cover a larger fixed amount. The fixed costs line moves upwards to fifteen thousand dollars. Total costs also move upwards by the same amount at every output, while their slope stays unchanged.

8:56Revenue keeps its original position, so the intersection moves right to twenty-five hundred units and twenty-five thousand dollars. At unchanged actual sales, three thousand minus twenty-five hundred gives a narrower margin of safety of five hundred units. A manager could use this comparison before renting larger premises. The higher fixed costs increase the sales needed to cover all costs. Extra space might also allow the business to serve more customers.

9:22The decision depends on whether realistic additional sales justify the higher threshold and the reduced cushion at existing output. Finally, restore the original fixed costs and raise variable cost to five dollars per unit. The selling price remains ten dollars. Ten dollars minus five dollars gives contribution of five dollars per unit. Twelve thousand dollars divided by five dollars per unit gives break-even output of twenty-four hundred units. Every sale now provides less towards fixed costs, so the business needs more sales to cover them.

9:52Total costs still begin at twelve thousand dollars, but their line becomes steeper. The fixed costs and revenue lines remain unchanged. Break-even moves right to twenty-four hundred units, where both lines reach twenty-four thousand dollars. Three thousand minus twenty-four hundred leaves a smaller margin of safety of six hundred units. For the business, this result highlights the effect of more expensive materials. Managers might seek another supplier to protect contribution per unit.

10:20A cheaper input must still meet customers’ expectations, because lower sales could offset the cost saving. Break-even analysis clarifies the financial relationship, while other information helps managers judge the practical choice. Changes in the opposite direction reverse these relationships, with other factors held constant. A lower selling price reduces contribution and raises break-even output. Reducing fixed costs lowers the amount that sales must cover. A lower variable cost per unit increases contribution, so fewer sales are required to break even.

Limitations and exam practice

10:51The model has five important limitations: unsold output, changing fixed costs, straight-line assumptions, a single product and forecast figures. Each limitation concerns a difference between the simplified chart and business conditions. Managers gain a clear comparison from the model. Its usefulness depends on how closely those assumptions match the decision. Inventory is the goods and materials a business holds, for use in production or for sale. The simple model assumes all output is sold with no inventory left.

11:21If some completed products remain unsold, the expected revenue has yet to be earned. Using production alone would then give managers an inaccurate picture of sales income. Fixed costs may change when output reaches a higher level. A business might need another building or an additional salaried supervisor. The horizontal fixed costs line remains suitable only for the output range where those costs stay constant. Managers should revise the chart when expanding capacity changes the amount that sales must cover.

11:50Straight lines assume a constant selling price and variable cost per unit. Suppliers may offer discounts on larger orders, so unit costs can fall. A business may also reduce its selling price to attract additional customers. These changes alter the slopes, so one straight-line chart can oversimplify the relationship at higher output. The simple chart covers one product with one selling price and one variable cost per unit. Businesses selling several products have different contributions from each.

12:17A change in the mixture sold can therefore change the overall result. One product’s break-even output gives a limited view of a business with a varied product range. Forecast figures are estimates of future conditions. Actual demand, material prices or rent may differ from the amounts managers expect. A precise calculated answer still depends on the accuracy of those estimates. Managers should update the figures as conditions change and combine the chart with information about customers and capacity.

12:45In examination questions, keep the meanings of the chart’s lines and distances clear. Revenue measures income from sales, while profit is the amount remaining after total costs. Contribution first helps cover fixed costs and then provides profit once those costs are covered. Margin of safety is measured along the output axis in units. Paper One Short Answer and Data Response and Paper Two Case Study can connect these ideas to business information.

13:11You may calculate an output, complete a chart or explain the effect of a change. Calculate means work out from given facts, figures or information. Explain means set out purposes or reasons, say why and how, supported with relevant evidence. Willow Repairs services household equipment. Its monthly fixed costs are eighty-four hundred dollars, its charge per job is forty dollars, and its variable cost is sixteen dollars per job. Actual monthly sales are five hundred jobs, and every job completed is sold.

13:41An exam-style question asks you to calculate break-even output and margin of safety. Contribution per unit equals selling price per unit minus variable cost per unit. Forty dollars minus sixteen dollars gives twenty-four dollars per job. Break-even output equals fixed costs divided by contribution per unit. Eighty-four hundred dollars divided by twenty-four dollars per job gives three hundred and fifty jobs. Margin of safety equals the actual number of sales minus the break-even number of sales, in units.

14:11Five hundred jobs minus three hundred and fifty jobs gives one hundred and fifty jobs. Sales can fall by that amount before the business reaches break-even. Showing the working and the units makes the reasoning clear. Suppose a further question asks you to explain the effect of higher rent on its break-even output. Higher rent increases Willow Repairs’ fixed costs. With the charge and variable cost per job unchanged, contribution remains the same.

14:36More jobs are required to cover the increased fixed costs, so break-even output rises. Break-even is the level of output at which total revenue equals total costs. Contribution per unit is selling price per unit minus variable cost per unit. Dividing fixed costs by that contribution finds the number of sales needed to cover every cost. On the chart, this quantity appears where revenue meets total costs. Margin of safety is the actual number of sales minus the break-even number of sales, in units.

15:06Its bracket shows the sales cushion, while the vertical gap between revenue and total costs shows profit or loss. Changes in price or costs alter those relationships. Use the calculations to understand the decision, then check whether the assumptions fit the business.