Betty Battle manufactures a product which has a selling price of $20 and a variable cost of $10 per unit. The company incurs annual fixed costs of $29,000. Annual sales demand is 9,000 units.
New production methods are under consideration, which would cause a $1,000 increase in fixed costs and a reduction in variable cost to $9 per unit. The new production methods would result in a superior product and would enable sales to be increased to 9,750 units per annum at a price of $21 each.
If the change in production methods were to take place, the break-even output level would be:
A 400 units higher B 400 units lower C 100 units higher D 100 units lower
(The answer is at the end of the chapter)
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9 Break-even graphs, contribution graphs and profit/volume charts
9.1 Break-even graphs
Section overview
• The break-even point can also be determined graphically using a break-even graph or a contribution break-even graph. These graphs show approximate levels of profit or loss at different sales volume levels within a limited range.
• The profit/volume (P/V) chart is a variation of the break-even graph which illustrates the relationship of costs and profits to sales and the safety margin. It shows clearly the effect on profit and break-even point of any change in selling price, variable cost, fixed cost and/or sales demand.
A break-even graph has the following axes:
• A horizontal axis showing the sales/output (in value or units)
• A vertical axis showing $ for sales revenues and costs The following lines are drawn on the break-even graph:
(a) The sales line
(i) Starts at the origin
(ii) Ends at the point signifying expected sales (b) The fixed costs line
(i) Runs parallel to the horizontal axis
(ii) Meets the vertical axis at a point which represents total fixed costs (c) The total costs line
(i) Starts where the fixed costs line meets the vertical axis
(ii) Ends at the point which represents anticipated sales on the horizontal axis and total costs of anticipated sales on the vertical axis
The break-even point is the intersection of the sales line and the total costs line.
The distance between the break-even point and the expected (or budgeted) sales, in units, indicates the safety margin.
Worked Example: A break-even graph
The budgeted annual output of a factory is 120 000 units. The fixed overheads amount to $40 000 and the variable costs are 50c per unit. The sales price is $1 per unit.
Construct a break-even graph showing the current break-even point and profit earned up to the present maximum capacity.
Solution
We begin by calculating the profit at the budgeted annual output.
Sales (120 000 units) 120 000 $
Variable costs 60 000
Contribution 60 000
Fixed costs 40 000
Profit 20 000
Break-even graph (1) is shown on the following page.
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The graph is drawn as follows:
(a) The vertical axis represents money (costs and revenue) and the horizontal axis represents the level of activity (production and sales).
(b) The fixed costs are represented by a straight line parallel to the horizontal axis (in our example, at $40,000).
(c) The variable costs are added 'on top of' fixed costs, to give total costs. It is assumed that fixed costs are the same in total and variable costs are the same per unit at all levels of output.
The line of costs is therefore a straight line and only two points need to be plotted and joined up.
Perhaps the two most convenient points to plot are total costs at zero output, and total costs at the budgeted output.
• At zero output, costs are equal to the amount of fixed costs only, $40,000, since there are no variable costs.
• At the budgeted output of 120,000 units, costs are $100,000.
Fixed costs 40 000 $
Variable costs 120 000 × 50c 60 000
Total costs 100 000
(d) The sales line is also drawn by plotting two points and joining them up.
(i) At zero sales, revenue is nil.
(ii) At the budgeted output and sales of 120,000 units, revenue is $120,000.
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The break-even point is where total costs are matched exactly by total revenue. From the graph, this can be seen to occur at output and sales of 80 000 units, when revenue and costs are both $80 000. This break-even point can be proved mathematically as:
=
Re quired contribution ( fixed cos ts) Contribution per unit =
unit per 50c
000
$40 = 80 000 units
The safety margin can be seen on the graph as the difference between the budgeted level of activity and the break-even level.
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9.2 The value of break-even graphs
Break-even graphs are used as follows:
• To plan the production of a company's products
• To market a company's products
• To give a visual display of break-even calculations
Worked Example: Variations in the use of break-even graphs
Break-even graphs can be used to show variations in the possible sales price, variable costs or fixed costs. Suppose that a company sells a product which has a variable cost of $2 per unit. Fixed costs are
$15,000. It has been estimated that if the sales price is set at $4.40 per unit, the expected sales volume would be 7,500 units; whereas if the sales price is lower, at $4 per unit, the expected sales volume would be 10,000 units.
Draw a break-even graph to show the budgeted profit, the break-even point and the safety margin at each of the possible sales prices.
Solution
Workings
Sales price $4.40 per unit Sales price $4 per unit
$ $
Fixed costs 15 000 15 000
Variable costs (7 500 × $2.00) 15 000 (10 000 × $2.00) 20 000
Total costs 30 000 35 000
Budgeted revenue (7 500 × $4.40) 33 000 (10 000 × $4.00) 40 000
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(a) Break-even point A is the break-even point at a sales price of $4.40 per unit, which is 6 250 units or $27 500 in costs and revenues.
(check: Re quired contribution to break - even
Contribution per unit $2.40per unit 000
$15 = 6 250 units)
The safety margin (A) is 7 500 units – 6 250 units = 1 250 units or 16.7% of expected sales.
(b) Break-even point B is the break-even point at a sales price of $4 per unit which is 7 500 units or
$30 000 in costs and revenues.
(check: Re quired contribution to break - even
Contribution per unit $2per unit 000
$15 = 7 500 units)
The safety margin (B) = 10 000 units − 7 500 units = 2 500 units or 25% of expected sales.
Since a price of $4 per unit gives a higher expected profit and a wider safety margin, this price will probably be preferred even though the break-even point is higher than at a sales price of $4.40 per unit.
9.3 Contribution (or contribution break-even) graphs
As an alternative to drawing the fixed cost line first, it is possible to start with variable costs. This is known as a contribution graph. An example is shown below using the data in Paragraph 9.2.
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One of the advantages of the contribution graph is that is shows clearly the contribution for different levels of production (indicated here at 120,000 units, the budgeted level of output) as the 'wedge' shape between the sales revenue line and the variable costs line. At the break-even point, the contribution equals fixed costs exactly. At levels of output above the break-even point, the contribution is larger, and not only covers fixed costs, but also leaves a profit. Below the break-even point, the loss is the amount by which contribution fails to cover fixed costs.
9.4 The profit/volume (P/V) chart
The profit/volume (P/V) chart is a variation of the break-even graph which illustrates the relationship of costs and profits to sales and the safety margin.
A P/V chart is constructed as follows (look at the graph in the example that follows as you read the explanation).
4: Cost behaviour and CVP analysis 121 (a) 'P' is on the y axis and actually comprises not only 'profit' but contribution to profit (in monetary
terms), extending above and below the x axis with a zero point at the intersection of the two axes, and the negative section below the x axis representing fixed costs. This means that at zero
production, the company is incurring a loss equal to the fixed costs.
(b) 'V' is on the x axis and comprises either volume of sales or value of sales (revenue).
(c) The profit-volume line is a straight line drawn with its starting point (at zero production) at the intercept on the y axis representing the level of fixed costs, and with a gradient of contribution/unit (or the P/V ratio if sales value is used rather than units). The P/V line will cut the x axis at the break- even point of sales volume. Any point on the P/V line above the x axis represents the profit to the company (as measured on the vertical axis) for that particular level of sales.
Worked Example: P/V chart
Let us draw a P/V chart for our example (Paragraph 9.1). At sales of 120 000 units, total contribution will be 120 000 × $(1 – 0.5) = $60 000 and total profit will be $20 000.
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9.5 The advantage of the P/V chart
The P/V chart shows clearly the effect on profit and break-even point of any changes in selling price, variable cost, fixed cost and/or sales demand.
If the budgeted selling price of the product in our example is increased to $1.20, with the result that demand drops to 105 000 units despite additional fixed advertising costs of $10 000. At sales of 105 000 units, contribution will be 105 000 × $(1.20 – 0.50) = $73 500 and total profit will be $23 500 (fixed costs being $50 000).
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x 20
Breakeven point 1 BREAKEVEN
LOSS PROFIT
Profit/loss
$’000 P/V chart (2)
Sales volume '000 (units) 10
10 20
30 40 50
Breakeven point 2
105 120
30
x
x
x
The diagram shows that if the selling price is increased, the break-even point occurs at a lower level of sales revenue (71 429 units instead of 80 000 units), although this is not a particularly large increase when viewed in the context of the projected sales volume. It is also possible to see that for sales above 50 000 units, the profit achieved will be higher (and the loss achieved lower) if the price is $1.20. For sales volumes below 50 000 units the first option will yield lower losses.
The P/V chart is the clearest way of presenting such information; two conventional break-even graphs on one set of axes would be very confusing.
Changes in the variable cost per unit or in fixed costs at certain activity levels can also be easily
incorporated into a P/V chart. The profit or loss at each point where the cost structure changes should be calculated and plotted on the graph so that the profit/volume line becomes a series of straight lines.
For example, suppose that in our example, at sales levels in excess of 120 000 units the variable cost per unit increases to $0.60 (perhaps because of overtime premiums that are incurred when production exceeds a certain level). At sales of 130 000 units, contribution would therefore be 130 000 × $(1 – 0.60) = $52 000 and total profit would be $12 000.
20
BREAK-EVEN
LOSS
Fixed costs PROFIT
Profit/loss
$ 000 P/V chart (3)
Sales volume 000 (units) 10
10 20 30 40
Break-even point
120 130
x x
x
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