**Cost-volume-profit analysis looks primarily at the effects of differing levels of activity on the financial results of a business**

In any business, or, indeed, in life in general, hindsight is a beautiful thing. If only we could look into a crystal ball and find out exactly how many customers were going to buy our product, we would be able to make perfect business decisions and maximise profits.

Take a restaurant, for example. If the owners knew exactly how many customers would come in each evening and the number and type of meals that they would order, they could ensure that staffing levels were exactly accurate and no waste occurred in the kitchen. The reality is, of course, that decisions such as staffing and food purchases have to be made on the basis of estimates, with these estimates being based on past experience.

While management accounting information can’t really help much with the crystal ball, it can be of use in providing the answers to questions about the consequences of different courses of action. One of the most important decisions that need to be made before any business even starts is ‘how much do we need to sell in order to break-even?’ By ‘break-even’ we mean simply covering all our costs without making a profit.

This type of analysis is known as ‘cost-volume-profit analysis’ (CVP analysis) and the purpose of this article is to cover some of the straight forward calculations and graphs required for this part of the Performance Management syllabus, while also considering the assumptions which underlie any such analysis.

### The objective of CVP analysis

CVP analysis looks primarily at the effects of differing levels of activity on the financial results of a business. The reason for the particular focus on sales volume is because, in the short-run, sales price, and the cost of materials and labour, are usually known with a degree of accuracy. Sales volume, however, is not usually so predictable and therefore, in the short-run, profitability often hinges upon it. For example, Company A may know that the sales price for product X in a particular year is going to be in the region of $50 and its variable costs are approximately $30.

It can, therefore, say with some degree of certainty that the contribution per unit (sales price less variable costs) is $20. Company A may also have fixed costs of $200,000 per annum, which again, are fairly easy to predict. However, when we ask the question, ‘Will the company make a profit in that year?’ the answer is ‘We don’t know’. We don’t know because we don’t know the sales volume for the year. However, we can work out how many sales the business needs to achieve in order to make a profit and this is where CVP analysis begins.

**Methods for calculating the break-even point**

The break-even point is when total revenues and total costs are equal, that is, there is no profit but also no loss made. There are three methods for ascertaining this break-even point:

**(1) The equation method**

A little bit of simple maths can help us answer numerous different cost‑volume-profit questions.

We know that total revenues are found by multiplying unit selling price (USP) by quantity sold (Q). Also, total costs are made up firstly of total fixed costs (FC) and secondly by variable costs (VC). Total variable costs are found by multiplying unit variable cost (UVC) by total quantity (Q). Any excess of total revenue over total costs will give rise to profit (P). By putting this information into a simple equation, we come up with a method of answering CVP type questions. This is done below continuing with the example of Company A above.

Total revenue – total variable costs – total fixed costs = Profit

(USP x Q) – (UVC x Q) – FC = P

(50Q) – (30Q) – 200,000 = P

Note: total fixed costs are used rather than unit fixed costs since unit fixed costs will vary depending on the level of output.

It would, therefore, be inappropriate to use a unit fixed cost since this would vary depending on output. Sales price and variable costs, on the other hand, are assumed to remain constant for all levels of output in the short-run, and, therefore, unit costs are appropriate.

Continuing with our equation, we now set P to zero in order to find out how many items we need to sell in order to make no profit, i.e. to break even:

(50Q) – (30Q) – 200,000 = 0

20Q – 200,000 = 0

20Q = 200,000

Q = 10,000 units.

The equation has given us our answer. If Company A sells less than 10,000 units, it will make a loss. If it sells exactly 10,000 units it will break-even, and if it sells more than 10,000 units, it will make a profit.

**(2) The contribution margin method**

This second approach uses a little bit of algebra to rewrite our equation above, concentrating on the use of the ‘contribution margin’. The contribution margin is equal to total revenue less total variable costs. Alternatively, the unit contribution margin (UCM) is the unit selling price (USP) less the unit variable cost (UVC). Hence, the formula from our mathematical method above is manipulated in the following way:

(USP x Q) – (UVC x Q) – FC = P

(USP – UVC) x Q = FC + P

UCM x Q = FC + P

Q = __FC + P__

UCM

So, if P = 0 (because we want to find the break-even point), then we would simply take our fixed costs and divide them by our unit contribution margin. We often see the unit contribution margin referred to as the ‘contribution per unit’.

Applying this approach to Company A again:

UCM = 20, FC = 200,000 and P = 0.

Q =__ FC __

UCM

Q = __200,000__

20

Therefore, Q = 10,000 units

The contribution margin method uses a little bit of algebra to rewrite our equation above, concentrating on the use of the ‘contribution margin’.

**(3) The graphical method**

With the graphical method, the total costs and total revenue lines are plotted on a graph; $ is shown on the y axis and units are shown on the x axis. The point where the total cost and revenue lines intersect is the break-even point. The amount of profit or loss at different output levels is represented by the distance between the total cost and total revenue lines. **Figure 1** shows a typical break-even chart for Company A. The gap between the fixed costs and the total costs line represents variable costs.

Alternatively, a contribution graph could be drawn. While this is not specifically covered by the Performance Management syllabus, it is still useful to see it. This is very similar to a break-even chart; the only difference being that instead of showing a fixed cost line, a variable cost line is shown instead.

Hence, it is the difference between the variable cost line and the total cost line that represents fixed costs. The advantage of this is that it emphasises contribution as it is represented by the gap between the total revenue and the variable cost lines. This is shown for Company A in **Figure 2**.

Finally, a profit–volume graph could be drawn, which emphasises the impact of volume changes on profit (**Figure 3**). This is key to the Performance Management syllabus and is discussed in more detail later in this article.

### Ascertaining the sales volume required to achieve a target profit

As well as ascertaining the break-even point, there are other routine calculations that it is just as important to understand. For example, a business may want to know how many items it must sell in order to attain a target profit.

**Example 1**

Company A wants to achieve a target profit of $300,000. The sales volume necessary in order to achieve this profit can be ascertained using any of the three methods outlined above. If the equation method is used, the profit of $300,000 is put into the equation rather than the profit of $0: