Cost-volume-profit analysis

Cost-volume-profit (CVP) analysis answers the questions every business owner asks first: how many must we sell to cover our costs, how far can sales fall before we make a loss, and what must we sell to earn the profit we want? It works by splitting costs into fixed and variable and following one number, contribution, as volume changes. In PM, CVP appears in objective test questions on break-even, margin of safety and target profit, often with several products, and in questions that ask you to read a break-even or profit–volume chart or to discuss the limitations of the technique. The arithmetic is short; the marks are lost by mixing up units and revenue, and by forgetting that a multi-product answer only holds for one sales mix.

Contribution: the engine of CVP

Contribution per unit is selling price less variable cost per unit. Each unit sold adds its contribution towards the fixed costs; once the fixed costs are covered, each further unit's contribution is profit. So:

Profit = total contribution − fixed costs.

At the break-even point, total contribution exactly equals fixed costs and profit is nil. Everything else in this lesson follows from that sentence.

The single-product formulas

The C/S ratio (sometimes called the profit–volume or P/V ratio) tells you how many cents of every dollar of revenue are contribution. Use it whenever the question talks in revenue rather than units, or when there are several products with different prices.

The margin of safety measures risk: how far sales can fall below budget before the business makes a loss. A high margin of safety means a cushion; a low one means a small shortfall in sales wipes out profit. It can be given in units, in revenue or as a percentage, so give it in the form asked.

Several products: the weighted average C/S ratio

With more than one product, there is no single contribution per unit, so CVP uses an assumption: products are sold in a constant mix. Two methods then give the same answer.

1. Weighted average C/S ratio. Divide total budgeted contribution by total budgeted revenue. Break-even revenue is fixed costs ÷ that ratio. Split it between the products in proportion to their budgeted revenue.
2. Contribution per mix. Treat the mix (for example 16 units of X, 5 of Y and 10 of Z) as one bundle. Break-even bundles are fixed costs ÷ contribution per bundle, and each product's break-even units follow from the bundle.

The answer holds only for that mix. If the mix shifts towards products with a higher C/S ratio, the weighted average ratio rises and break-even revenue falls; a shift towards low C/S products raises it.

The break-even chart

The PM study guide asks you to interpret break-even and profit–volume charts, so know what each line and point means. On a break-even chart:

• the horizontal axis shows activity (units or revenue) and the vertical axis shows dollars;
• the fixed cost line is horizontal, at the level of fixed costs;
• the total cost line starts on the vertical axis at the fixed costs and rises by the variable cost per unit;
• the revenue line starts at the origin and rises by the selling price per unit, so it is steeper than the total cost line;
• break-even is where the revenue line crosses the total cost line; the gap to the right is profit, the gap to the left is loss;
• the margin of safety is the horizontal distance between budgeted activity and break-even.

A contribution break-even chart draws the variable cost line from the origin and adds fixed costs on top, so that the gap between revenue and variable cost, the contribution, can be seen at every level of activity.

The profit–volume chart

A profit–volume (P/V) chart plots profit directly. The horizontal axis shows activity, and the vertical axis shows profit above zero and loss below. For one product it is a single straight line: it starts at a loss equal to the fixed costs when sales are nil, rises by the contribution per unit (or by the C/S ratio when the axis shows revenue) and crosses zero at break-even.

For several products, the multi-product P/V chart uses revenue on the horizontal axis and plots the products one after another in order of C/S ratio, highest first. The result is a bowed line made of steeper and flatter segments: each segment's slope is that product's C/S ratio. A straight line joining the starting point to the final point shows the products sold in their budgeted mix, and it crosses zero at the weighted average break-even revenue. A product whose segment runs downwards has a negative contribution and is reducing profit.

Limitations of CVP

• It assumes selling price and variable cost per unit are constant, so revenue and costs are straight lines. In reality, discounts, overtime and learning effects bend them.
• It assumes fixed costs stay fixed. That is true only within a relevant range of activity; beyond it, fixed costs step up.
• It assumes every cost can be split cleanly into fixed and variable elements.
• It assumes one product, or several in a constant mix.
• It assumes production equals sales, so that inventory movements do not change profit.
• It is short-term and static, and it ignores uncertainty: a single break-even figure looks more certain than the estimates behind it.

Worked example

Example 1: one product. Lagos Lamps, a fictional manufacturer, sells a solar lamp for $50. Variable cost is $30 a lamp, fixed costs are $160,000 a year and budgeted sales are 12,000 lamps.

A useful cross-check: budgeted profit equals the margin of safety multiplied by the contribution per unit, 4,000 × $20 = $80,000, because the lamps sold beyond break-even are the only ones earning profit.

These are the figures a break-even chart and a P/V chart for Lagos Lamps would plot.

On the break-even chart, the revenue and total cost lines meet at 8,000 lamps and $400,000. On the P/V chart, the line starts at a loss of $160,000 and rises $20 for every lamp, reaching zero at 8,000 lamps.

Example 2: three products. Enugu Electricals, also fictional, sells three products in a constant mix. Fixed costs are $150,000.

Weighted average C/S ratio = $300,000 ÷ $800,000 = 37.5%. Break-even revenue = $150,000 ÷ 0.375 = $400,000, half of budgeted revenue, so the margin of safety is 50%. Revenue for a profit of $90,000 would be ($150,000 + $90,000) ÷ 0.375 = $640,000, in the same mix.

Splitting break-even revenue in the budgeted revenue mix (50% X, 25% Y, 25% Z) gives X $200,000 (4,000 units), Y $100,000 (1,250 units) and Z $100,000 (2,500 units). Check: contribution is (4,000 × $20) + (1,250 × $40) + (2,500 × $8) = $80,000 + $50,000 + $20,000 = $150,000, exactly the fixed costs. The bundle method agrees: the unit mix is 16 X : 5 Y : 10 Z, contribution per bundle is $320 + $200 + $80 = $600, and $150,000 ÷ $600 = 250 bundles, which is 4,000 X, 1,250 Y and 2,500 Z.

The multi-product P/V chart takes the products in order of C/S ratio: Y (50%), then X (40%), then Z (20%).

The bowed line is steepest for Y and flattest for Z. The straight line from (0, −$150,000) to ($800,000, $150,000) has a slope of 37.5% and crosses zero at $400,000, the break-even revenue in the budgeted mix.

Common mistakes

• Dividing fixed costs by the selling price, or by the C/S ratio when the question asks for units.
• Using a simple average of the products' C/S ratios instead of the weighted average, total contribution ÷ total revenue.
• Forgetting that a multi-product break-even point assumes the budgeted mix, and so is wrong as soon as the mix changes.
• Adding the target profit to contribution instead of to fixed costs.
• Giving the margin of safety in units when the question asks for a percentage or for revenue.
• Including absorbed fixed overhead in the "variable cost" per unit.

Exam technique

Before calculating, write down three things: contribution per unit, fixed costs and whether the answer is wanted in units or revenue. Most CVP objective test questions are then one line of arithmetic, and the wrong options are the other lines. In multi-product questions, lay out a small table with a column per product and a total column, as in Example 2, so the weighted C/S ratio comes from the totals. For a question on a chart, name the line or point, say what it represents and, if asked to interpret, explain what it tells management about risk, such as how close budgeted sales are to break-even.

Study with AI

I am studying ACCA Performance Management (PM), cost-volume-profit analysis. Give me
six short questions, one at a time: break-even units, break-even revenue with the C/S
ratio, margin of safety, target profit, a three-product break-even with a constant
mix, and one question asking me to describe a point on a multi-product profit-volume
chart. Wait for my answer each time, then show the full working. Use $.

AI assistants can mix up units and revenue or use a simple average C/S ratio: check every answer against these notes before you rely on it.

Lecturer's note: Stop memorising eight formulas and remember one: profit equals contribution less fixed costs. Break-even is the point where profit is nil, a target profit is just a different profit figure, and the margin of safety is the volume earning that profit. When there are several products, the only new idea is that "contribution per unit" has to be replaced by an average, and an average only exists for a stated mix.

Key points

• Contribution per unit is price less variable cost; profit is total contribution less fixed costs.
• Break-even units = fixed costs ÷ contribution per unit; break-even revenue = fixed costs ÷ C/S ratio; for a target profit, add the profit to the fixed costs.
• The margin of safety is budgeted sales less break-even sales, and shows how far sales can fall before a loss.
• With several products, use the weighted average C/S ratio (total contribution ÷ total revenue); the answer holds only for that mix.
• Break-even and P/V charts show the same relationships in pictures; CVP assumes linear costs and revenues, fixed costs within a relevant range, a constant mix and no inventory changes.

Quick check

1. A product sells for $25 with a variable cost of $15. Fixed costs are $60,000 and budgeted sales are 9,000 units. What are the break-even point and the margin of safety as a percentage?
2. A business has a C/S ratio of 30% and fixed costs of $90,000. What revenue is needed for a profit of $30,000?
3. Two products are sold in a constant mix: 60% of revenue comes from a product with a C/S ratio of 50%, and 40% from a product with a C/S ratio of 25%. Fixed costs are $160,000. What is the break-even revenue?

Answers

1. Contribution $10 a unit; break-even $60,000 ÷ $10 = 6,000 units; margin of safety (9,000 − 6,000) ÷ 9,000 = 33.3%.
2. ($90,000 + $30,000) ÷ 0.30 = $400,000.
3. Weighted C/S ratio (60% × 50%) + (40% × 25%) = 30% + 10% = 40%; break-even revenue $160,000 ÷ 0.40 = $400,000.