Cost, volume and price relationships
Introduction
Annual income twenty pounds, annual expenditure nineteen pounds, nineteen shillings and six pence result happiness. Annual income twenty pounds, annual expenditure twenty pounds, and six pence, result misery.
CHARLES DICKENS (1812–1870) (MR MICAWBER)
Introduction
The material covered to date has concerned itself with the company as a single overall unit. In this chapter we probe down into the detailed operations. Of course we find at this level that there are a multitude of separate products within the overall company. Each of these is a small business in itself, and each has its own associated revenue and cost.
The internal operating dynamics of these sub-units can be quite complicated. We use a cost/management accounting system to give information and help to make decisions at this level. We would like to be in a position where we can establish a total cost for a product, add on a profit margin and then sell at the target selling price. Rarely are we able to do this. The price we would like to charge is generally 10 per cent above that of our competitors. The manager is constantly being asked to get costs down or to sell at or below the total cost figure.
How is it possible to make profits in this scenario? There is no magic answer, but what we can do is to recognize that the main function of cost accounting is not to arrive at a selling price but to identify the areas where profit is possible. This means understanding how costs behave – in particular, how they respond to changes in the variables: price, volume and, above all, product mix.
It was mentioned earlier that the performance model which copes well with the big business scene is not good at this level of detail. However, it is at this level that profits are made and lost. Also, it is at this level that day-to-day decisions are made as to what price to quote for a job, whether to accept or reject a price on offer, whether and how to pay commission to salespeople and so on.
The monthly management accounts supplied to managers are used by them for decision-making. However, the way in which the information is presented is often not helpful for this purpose. For instance, costs may be classified in a way that does not allow their recalculation on the basis of an alternative course of action.
This chapter will illustrate a series of calculations around this subject that enable managers to tread their way safely through these difficulties. These are essential, interesting and powerful tools. In the following pages we will work through a number of examples that show their application.
Costing illustration
Figure 15.1 shows a budget for the company Lawnmowers Ltd. We will use the numbers to illustrate some fundamental facts about the behavior of costs. These numbers are kept very simple so that they will embed themselves in the mind very quickly. We will then do many calculations around them.
The crucial facts are:
- this company has only one product, which it sells for a unit price of $200
- it plans to manufacture and sell 1000 units in the coming year at a profit of $20 each
- the total cost of $180 consists of the elements shown in figure 15.1.
The company has received an offer from a very large retailer to purchase 200 extra units at the special price of $160. The sales manager is faced with the dilemma of whether the company can sell at a price that is below the calculated total cost and still bring benefit to the company. (Of course the marketing implications of selling at a special low price are very significant, but, for this exercise, they will be ignored in order to tease out the financial aspects.)
We instinctively know that the total cost as shown in the budget will change if the forecast number of units is changed. The manager could, therefore, recalculate the total cost per unit after allowing for the extra volume of 200 units. This would give a revised overall profit.
A decision could be made on the basis of this type of calculation, but it would not help much in determining the lowest price that could be charged in a really competitive situation. Neither would it be of very much use for producing a whole set of scenarios for sales of different volumes at different prices.
An alternative approach is illustrated in the following pages.
Figure 15.1 Summary budget of Lawnmowers Ltd
Cost classification
Of the many ways in which costs can be classified, possibly the most useful for decision-making purposes is on the basis of their response to volume changes. This well-known classification uses the following terms:
- Fixed costs: These are costs where the total expenditure does not change with the level of activity. For instance rent of a factory will not increase or decrease if volume of throughput goes up or down by 10 per cent.
- Variable costs: These vary directly with changes in output. The cost of materials consumed in the product will vary almost in direct proportion to changes in volume.
An interesting paradox is that a fully variable cost is always a fixed charge per unit irrespective of volume. On the other hand, fixed costs charged to each unit fall with volume increases.
In practice, there are very few, if any, costs that are either totally fixed or totally variable across the whole range of possible outputs. However, it is useful for our example to make some simplifying assumptions. The fixed and variable costs for Lawnmowers Ltd are shown in figure 15.2.
The variable costs are:
- materials
- labour.
(In practice, while direct materials are almost always fully variable, direct labour is more than likely to be largely fixed.)
The fixed costs are assumed to be:
- administration
- selling.
(This assumption implies that these amounts will not change at all with any changes in volume.)
The variable and fixed costs per unit are $120 and $60 respectively. They sum to a total cost of $180. Note that management is faced with the question of whether or not to accept a large order at a price of $160 per unit.
Figure 15.2 Fixed and variable costs for Lawnmowers Ltd
Contribution
The division of costs into fixed and variable allows us to re-examine unit costs. Figure 15.3 shows two different approaches to cost reporting:
- total cost (traditional) approach
- contribution approach.
In the traditional breakdown the unit selling price of $200 is split into:
- variable cost $120
- fixed cost $60
- profit $20.
With the contribution approach, however, the selling price is split into just two components:
- variable cost of $120
- contribution of $80.
Contribution, therefore, replaces the figures of fixed cost of $60 and profit of $20. We remove the fixed cost charge from the product. The concept of contribution is important and useful. It is worthwhile spending some time examining it.
Cash flow and contribution
We have arrived at the contribution figure of $80 by adding back the fixed cost of $60 to the profit of $20. However, we can see that the same figure is arrived at when we deduct variable cost from the selling price, i.e., $200 – $120. This latter definition is the most useful one.
We will extract cash flow meaning from it. Each extra unit sold at the budget price results in a cash in-flow of $200. However, this extra unit also increases variable costs by $120. These are direct cash costs which will give rise to a corresponding cash out-flow. One extra unit sold gives rise to extra revenue of $200 and extra cash costs of $120. The difference of $80, which we call contribution, is the net cash in-flow resulting from an extra unit sold.
The $80 cash sacrificed from the loss of one unit is even more obvious. If the company suffers the loss of one unit it loses $200 revenue but it saves only $120. So the cash loss is $80.
Contribution can therefore be defined as the net cash flow from a single transaction. In other words, it is the cash gained from the sale of an extra unit or the cash lost from the sale of one unit less.
The relationship of contribution to profit will be pursued in the following pages.
Figure 15.3 Alternative costing approaches applied to Lawnmowers Ltd’s figures
Contribution and profit
Contribution flows into the business as cash from the sale of each unit. However, it is not free cash because it must first be assigned to the payment of fixed costs. But once these fixed costs have been paid in full, the contribution stream goes straight into profit.
Figure 15.4 illustrates the process. It shows two water tanks that represent volumes of fixed costs and profit respectively. The fixed cost tank has a capacity of $60,000 representing the total fixed costs of the company. Each unit sold pumps $80 into this tank. When a sufficient number of $80s have filled this tank with cash, then all the extra contribution overflows into the profit tank.
It is easy to see that the cash from the first 750 units (750 × $80 = $60,000) remains in the fixed cost tank, it is only the cash from units in excess of 750 that goes to profit. If the company sells its budgeted volume of 1,000 units, then the final 250 units will provide profits of $20,000 (250 × $80). So the budgeted profit figure is not built up at the stated rate of $20 per unit. Rather it accrues at the rate ot $80 per unit after fixed costs have been covered.
This important figure of $80 is referred to as contribution per unit (CPU). It is used a great deal in profit/volume calculations and you will find it to be a very useful tool of analysis.
Figure 15.4 also highlights a second important concept, which is total contribution. The total contribution in this plant is $80,000. In the upper part of the diagram it is shown as ‘units multiplied by CPU’. In the lower part, the same figure is produced from ‘fixed costs plus profit’. We will find ourselves making use of this formula in many situations.
A fundamental principle underlying the material of this chapter is that our objective in decision-making should be to ‘maximize contribution’. In following this principle the greatest profit will be achieved.
Figure 15.4 Contribution’s relationship to fixed costs and profit
Total contribution
The two definitions of total contribution mentioned can be combined in an important equation, as shown in figure 15.5.
Units × CPU = fixed cost + profit
This view of the cost dynamics of the company facilitates calculations to do with volume, cost and profits. For instance, we can start at units, and ascertain without difficulty what the profit would be at any level of unit sales. Alternatively we can work from the profit side and ask what unit sales are needed to deliver any desired level of profit. Finally we can combine price and volume changes and instantly translate them into profit.
For instance:
- What profit would result from an increase in sales of 10 per cent?
Answer $28,000 - How many units must be sold to provide a profit of $32,000?
Answer 1,150 units - Finally, a very important question, how many units must be sold to break even, i.e., make neither a profit nor a loss?
Answer 750 units.
This latter calculation gives the break-even point, a concept that is used widely in business. The formula for its calculation in terms of units is derived directly from the above equation. We simply set profit equal to zero and solve for the number of units. We will look at this concept some more in the following pages.
It is important to remember that we are still thinking in terms of physical units. The formulae and methods illustrated above work only in the case of a business that sells one type of unit. However, these limited formulae provide a useful stepping off point for the next stage. As soon as the fundamentals have been grasped, it is possible to move easily to the more general case where we can do calculations for a company that produces many different units.
Figure 15.5 Working out the effects of change using contribution per unit for Lawnmowers Ltd
Break-even (B/E)
The importance of the break-even level of output has been mentioned. It is the level at which the company makes neither a loss nor a profit, but just covers its fixed costs. It is, therefore, a most important cross-over point in a plant’s level of activity and managers pay much attention to it.
The break-even chart is a well-known illustration of the concept. It shows the relationship between three important components over the total possible range of outputs:
- fixed costs
- variable costs
- revenue.
The horizontal axis in figure 15.6 is used to represent activity, which can be expressed in different ways, e.g. percentage of capacity, machine hours, etc. For our purposes here, both the number of units produced and capacity percentages are used. The chart covers the full output range – from zero to the full plant capacity of 1,500 units.
The vertical axis is used to plot costs and revenue. Figures from Lawnmowers Ltd are used for each of the three components (fixed costs, variable costs and revenue). These are plotted in three separate charts in figure 15.6.
Chart A illustrates the fixed costs, represented by a horizontal line at the $60,000 point on the vertical axis. This line is absolutely horizontal because fixed costs do not go up or down with changes in the level of output.
Chart B illustrates the variable costs. At zero output there are no variable costs, but, with each unit sold, the total grows by $120. At 600 units variable costs amount to $72,000. The relationship with output is strictly linear, hence the straight line sloping upwards.
Chart C illustrates the revenue. Again, at zero output, total revenue is zero. For every unit sold, revenue increases by $200. At 600 units, total revenue amounts to $120 000 (600 × $200). The relationship between revenue and output is also linear, and this is reflected in the more rapidly rising total revenue line.
Figure 15.6 The elements of a break-even chart
Break-even (B/E) Chart
In figure 15.7, the three separate charts of figure 15.6 have been combined to show how the three sections come together. First we see the band of fixed costs at the base of the chart. Positioned above this is the variable costs wedge. Along the upper boundary of this wedge is the total cost line. Finally, the total revenue line is shown rising from the zero revenue at zero output up through the two areas of cost until it breaks through the total cost line about 50 per cent of capacity. This is the break-even point. To the left of it lies loss, to the right profit.
The break-even point here occurs at a level of 750 units which equates to revenue of $150,000. Remember from figure 15.5 that 750 units will result in a contribution of $60,000 (750 × $80), which just equals the fixed cost.
A line is plotted representing the budgeted sales of 1,000 units at a unit price of $200. This puts the company a little way to the right of the break-even point. The surplus output over break even is sometimes referred to as the ‘margin of safety’. This margin of safety here is 25 per cent, which means that sales can fall by this percentage before the break-even point is reached.
A number of simplifying assumptions are incorporated into this chart. It has been assumed that (1) fixed costs are totally fixed for all levels of output, (2) that unit variable costs do not change irrespective of numbers produced and (3) that unit revenue is the same for all levels of sales. These assumptions may be valid over a fairly narrow range on each side of existing levels of activity. At very low or very high levels, however, the assumptions break down and the chart ceases to be accurate.
Notwithstanding these limitations, the break-even chart is a useful tool for presenting information, for explaining the dynamics of a production unit, for pointing out the essential features of the volume, cost and revenue system and for setting minimum sales targets.
This chart can also be used to identify companies with different cost structures and therefore different levels of what is called ‘operating leverage’. Operating leverage identifies the change in profit that results from a small change in sales, e.g. if a 10 percent change in sales produces a 40 per cent change in profit, then we say that the company has a high operating leverage of 4 times.
Figure 15.7 Break-even chart for Lawnmowers Ltd
Figure 15.8 shows charts for two types of company. The first (A) has a high operating leverage. With high fixed costs and low unit variable costs it has a total cost line that starts high on the chart but has a relatively flat trajectory. The total revenue line cuts the total cost line at quite a wide angle, which means that profit increases rapidly to the right of the break-even point. Unfortunately, to the left of this point, the negative gap also widens rapidly, meaning that the company gets into heavy losses very quickly as output falls away.
The second (B) has a low operating leverage. This company has low fixed costs but high unit variable costs. The total cost line starts low but climbs steeply with increased volume. The total revenue line produces a much narrower angle at the crossover point, indicating that both profits and losses grow more slowly either right or left of the break-even point.
In Company A, we can understand there is intense pressure to achieve volume sales in order to move out of the heavy loss situation to the left of the break-even point. And when the break-even point has been reached, there is the prize of a rapidly increasing profit figure to provide the incentive to achieve even greater volume. When total production capacity exceeds total demand, fierce price competition can erupt between firms of this type as they compete for volume. The companies with this cost profile are those with heavy fixed assets, such as steel and transport companies.
The second situation, with low operating leverage, is frequently found to exist in companies with smaller, flexible production systems and low fixed assets. These companies tend to have less risk, but also less reward, attached to them.
Contribution to sales percentage (CPS)
As noted earlier, ‘contribution per unit’ (CPU) is limited in its usefulness because it can be applied only to a one-product company. A more general ratio is therefore needed for the normal multi-product company. The following approach deals with this problem.
Figure 15.8 Operating leverage
Figure 15.9 shows the familiar diagram of unit selling price, variable costs and contribution. However one extra calculation has been added. The contribution figure of $80 is expressed as a percentage of the selling price of $200 to give the answer 40 per cent.
This is a very important percentage and a very important ratio. It has several different names in financial literature, such as, profit to volume (PV) ratio. As we have used the term ‘contribution per unit’ (CPU) when dealing with units, we will for consistency adopt the term ‘contribution percentage of sales’ (CPS) for this value.
The power of this ratio is that it can be used to convert any revenue figure into its corresponding contribution. Therefore it can be applied to the output of a total company or any sub-section of a company such as a particular product or market. It can be used to analyze separate individual products, groups of similar products or the total mix of products for a multi-unit business. When it is used to convert sales into contribution, the profit figure can be derived simply by deducting the fixed costs.
Total contribution – alternate views
It is important at this point to distinguish between unit contribution and the total contribution of the whole plant.
Unit contribution is simply the selling price less the variable costs figure for one unit.
Similarly total contribution is total revenue less total variable costs.
However it can also be derived from three separate sources (see figure 15.10):
- units × CPU
- revenue × CPS
- fixed costs + profit.
We can take the latter two and derive the identity:
Revenue × CPS = Fixed costs + Profit
Figure 15.9 Contribution to sales percentage
Figure 15.10 Alternative views of total contribution
Given the four variables in this equation, if we know any three of them we can find the remaining one.
Answer: ($35,000 × 22% – $3,800) = $3,900
Further examples are given in the next section.
Problem solving using CPS
We use the identity between the expressions:
- Revenue × CPS and
- Fixed costs + Profit
to solve problems in figure 15.11.
The problems illustrated here are the same as in figure 15.5, except that ‘contribution to sales percentage’ (CPS) has been used in place of ‘contribution per unit’ (CPU). Results are expressed in money terms rather than in units which is generally more useful. By converting units @ $200 sales price we see that the answers achieved by both approaches are identical.
As an extra example, consider the following:
- Budgeted sales for Lawnmowers Ltd are $200,000.
- At a CPS of 40 per cent the budgeted contribution is $80,000.
- If the revenue figure were to fall to $180,000, the resulting contribution would be $72,000 ($180,000 × 40 per cent).
- The profit from this reduced level of sales would be $12,000.
- ($72,000 contribution – $60,000 fixed cost = $12,000 profit).
Figure 15.11 Working out the effects of change using the contribution to sales percentage for Lawnmowers Ltd
Weighted CPS
We now know how to convert sales revenue into contribution by applying the ‘contribution to sales percentage’. However if a company’s sales increase by 10 per cent will contribution increase by the same amount? The answer is ‘yes’ if the CPS remains constant. However we can ask how constant is this factor likely to be? What drives it?
The selling prices of products are determined not by cost but by the market-place. Most companies do not have the luxury of fixing prices to give secure margins, they must accept or reject, within a range, the generally accepted price. For some products, the prices will be more satisfactory than for others.
A single product can have more than one price as discounts are given to bulk purchasers and so on. Therefore, the ‘contribution to sales percentage’ will vary widely for different products and in different markets. The overall contribution to sales percentage is a weighted average of the individual CPSs of the separate products and markets.
Figure 15.12 shows an example of a product that is sold though two different distribution channels. First it is sold direct to the public at a price of $500. It is also sold through agents at a discount of 20 per cent. The variable costs figure in both situations is $350.
The contribution for sales through the direct channel is $150 per unit, to yield a CPS of 30 per cent. Sales through agents give a contribution of only $50, to yield a CPS of 12.5 per cent.
As shown in Box 3, with revenues budgeted at $300,000 for direct sales and at $100,000 for agent sales the average CPS for all sales amounts to 25.6 per cent.
In Box 4 it is shown what happens when sales to agents increase substantially. The average CPS falls to 19 per cent. A change in the mix of sales has brought on a dramatic fall in the weighted CPS. The result in this exaggerated example is that a significant increase in sales volume has not produced a corresponding increase in profit.
Many companies work on the basis of average values. This can be dangerous. The average is determined by the mix of high and low contribution products and this mix must be managed. This point is illustrated more fully in the next section.
Figure 15.12 Overall contribution to sales percentage as a weighted average for a mix of sales
Product mix
In figure 15.13, data is given about a company that sells three products (A, B and C) through two distribution networks (direct and through agents). There are, therefore, six product-market segments as shown in the matrix in the top right-hand corner. The contribution to sales percentage for each segment is shown in the matrix in the upper right-hand corner.
Budgeted and actual sales of the product market segments are shown. Beneath these, the contributions from the two sets of data have been calculated.
As shown by the arrows, the sales value in each segment is multiplied by its corresponding CPS to arrive at its contribution. The contributions from the individual segments are accumulated to give the total contribution.
Fixed costs are $75,000 for both budgeted and actual results, and these are deducted to give the final budgeted and actual profit. A budgeted profit of $16,000 has turned into an actual loss of $1,000.
In comparing budgeted with actual figures, note the following:
- total revenue of $300,000 is unchanged
- selling prices are unchanged
- cost per unit figures are unchanged
- fixed costs are unchanged
- budgeted profit is converted into an actual loss.
The sole reason for this adverse turnaround is to be found in the mix of products.
The bulk of sales has moved from high to low contribution segments. For example, in the bottom right-hand last segment, which has the lowest CPS value (product C sold through agents), sales have increased from a budgeted $20,000 to an actual $60,000. This increase has been offset by reductions in the better segments. Similar other adverse movements can be identified. The overall product mix has been diluted and the company has suffered grievously.
Product mix is a potent driver of profit. It must be identified, understood by all and positively managed to achieve maximum potential.
A step beyond contribution to sales percentage
Product mix expressed in terms of CPS by product/market segment is, as noted previously, a powerful tool to enable management to achieve the best from the market-place. It is difficult to exaggerate the importance to a company’s financial health of directing sales away from low to high contribution areas.
However this is not the end of the story. This technique alone will not always succeed in maximizing contribution. There are situations in which the promotion of sales of products with high CPS is not to the advantage of the company.
This is often true when a company is working close to maximum capacity and when different products are competing for this capacity. In such situations, additional output of one product means a reduction in the output of another. So a gain from one product may be more than offset by loss from the other.
Consider a manufacturing situation where a high-value machine is used to manufacture two products, each of which sells for $5,000 (see figure 15.14). Product A has a CPS of 30 per cent while that of product B is 20 per cent. The absolute contribution from product A is $1,500 and from product B is $1,000.
From all that has been said so far, it would seem that management should always favour product A over product B. However, it is possible for product B to be more profitable than product A. How can this be?
When a company is operating at, or close to, maximum capacity, a further calculation beyond CPS is necessary. We extract a value that we call ‘contribution per unit of capacity’ or ‘contribution per unit of the limiting factor’. It can be a great identifier of sources of profit to a company, and so it will be explored in the next section.
Figure 15.14 Selling price to cost relationships of products A and B
Contribution per unit of capacity
Central to figure 15.15 is an illustration of machine capacity. We will assume that the machine can (after allowing for downtime, maintenance, cleaning and so on) deliver 2,000 productive hours per annum. We further assume that these productive hours can be more than fully utilized by existing products. Is it possible to increase the profitability of this machine?
We start our analysis by looking at what proportion of this capacity is used by each product, and we then relate this usage of capacity to the total contribution of each product.
Product A
One unit of product A takes 10 hours of machine capacity. The total contribution of this unit is $1,500. Therefore, the contribution per hour (CPH) of this product is $150. The capacity of the machine, remember, is 2,000 hours per annum, so, at $150 per hour, the maximum contribution attainable from the machine for this product in one year is $300,000.
Product B
One unit of product B takes four hours of machine capacity. The total contribution of this unit is $1,000. Therefore, the CPH of this product is $250. At $250 per hour, the maximum contribution attainable from the machine for this product in one year is $500,000.
Product B is by far the more profitable product in this situation, even though it has a lower CPS.
The determining conditions for these circumstances to exist is that capacity is limited, and there is competition between products for this capacity. In this situation contribution will be maximized – and, therefore, so will profit – when activity is directed into those products that have, not the highest contribution to sales percentage, but the highest contribution per hour value.
We have discussed here a specific instance of the more general case referred to as ‘contribution per unit of the limiting factor’. The limiting factor is the constraint that puts a ceiling on output. Machine hours are the most common limitation, but there can be others, such as raw materials, or working capital.
Errors are frequently made in pricing products without reference to this concept.
Figure 15.15 Contribution per machine hour of products A and B
Summary
It is important to:
- Classify costs to distinguish fixed from variable
- Identify contribution by product group
- Actively direct sales from low to high contribution areas
- Where capacity hours are a constraint, identify and direct sales into products with high CPH values.