Cost–Volume–Profit Analysis

For one product within a relevant range, let P be unit price, V unit variable cost, F total fixed cost and Q units sold:

Unit contribution margin is the selling price less unit variable cost. Each unit sold contributes that amount towards fixed cost; once fixed cost is covered, the remaining contribution becomes profit.

Northstar base case

• price: £600 per City bike;
• variable cost: £360 per bike;
• annual fixed cost: £480,000;
• forecast sales: 2,800 bikes.

Break-even marks the point at which the modelled profit is zero, rather than a profit target to aim for.

Derive the formula, then check the answer

Revenue is PQ and variable cost is VQ, so subtracting both cost types gives profit = (P − V)Q − F. To break even, set profit to zero; to earn a target T, solve (P − V)Q = F + T.

The units provide a check: pounds of fixed cost divided by pounds of contribution per bike gives bikes. Dividing by the contribution ratio instead gives pounds of sales. Northstar's contribution ratio is £240 / £600 = 40%, so break-even sales revenue is £480,000 / 40% = £1,200,000, the same as 2,000 bikes × £600.

The safety margin is a buffer against a fall in sales under unchanged assumptions. At the forecast, 800 bikes × £240 contribution = £192,000 profit. A change in price, cost or capacity changes that buffer; it is not a probability of avoiding loss. This chapter's target is modelled operating profit before financing and tax, not a promised cash surplus.

The 5% discount is not a 5% volume problem

If price falls 5% to £570 and variable cost remains £360:

• contribution falls from £240 to £210, a 12.5% decline;
• break-even rises to 2,286 units;
• target-profit volume rises from 2,500 to 2,858 whole bikes, a 14.3% increase.

The exact target is 600,000 / 210 = 2,857.14 bikes. Rounding to the nearest bike would miss the target: 2,857 × £210 − £480,000 = £119,970, whereas 2,858 bikes produce £120,180. Use upward rounding when the question asks for the minimum whole units needed to meet a threshold. Keep unrounded values for intermediate calculations.

A small price change can therefore have a much larger effect on profit. What matters is the contribution left after variable cost, not total revenue.

Explore the model

Try changing price, variable cost and fixed cost one at a time, then in combination. Compare the results to see which assumptions matter most to the decision.

Assumptions to expose

Basic CVP assumes:

• constant unit price and variable cost within the range;
• fixed total cost within the range;
• units produced equal units sold, or inventory effects are handled separately;
• a stable sales mix for multiple products;
• volume is the main activity driver.

Above 3,000 bikes, Northstar may need a second shift, which would increase fixed cost. The chart therefore stops at 3,000. Any target beyond that requires revised capacity and cost assumptions: correct arithmetic alone cannot make the existing cost model valid outside its relevant range.

Contribution must also be positive for the usual break-even formula to work with positive fixed costs. If price equals variable cost, every sale covers only its own variable cost; if price is lower, each extra sale increases the loss. A negative or undefined calculated break-even is not a usable sales target.

Multiple products

Use a weighted-average contribution only if the sales mix is explicit and reasonably stable. If demand shifts toward lower-contribution Cargo contracts, a single break-even total can conceal the mix effect.

Quick check

Variable cost rises to £390 with price and fixed cost unchanged. What is break-even?

Continue with the price-cut and capacity exercise. For another worked derivation, see OpenStax's break-even chapter.

Next: Budgeting and Variances