Break-even and cost-volume-profit analysis

How many units must a business sell to cover its costs? How much profit will an extra thousand sales bring? What happens to profit if rent goes up, or if the price is cut by 10 %? Cost-volume-profit (CVP) analysis answers these questions by modeling how costs and revenue change with the volume of sales. It is one of the most practical tools in business mathematics, used in business plans, pricing decisions, budgeting and loan applications. This lesson builds the model from the distinction between fixed and variable costs and applies it to break-even points, target profits, risk and sales mix.

Fixed and variable costs

A variable cost changes in proportion to the number of units produced or sold: ingredients, packaging, sales commissions, credit card fees, direct materials, piece-rate labor. A fixed cost stays the same over a range of activity, regardless of volume: rent, salaried wages, insurance, loan interest, depreciation, basic utilities, software subscriptions. Some costs are mixed, with a fixed base plus a variable part, such as a phone plan with a monthly charge plus usage, and are split into their fixed and variable pieces for analysis.

The word 'fixed' applies only within a relevant range. A café's rent is fixed whether it sells 2,000 or 5,000 coffees a month, but if it needs a second location to sell 20,000, its fixed costs jump. CVP results are reliable only within the range where the cost assumptions hold.

Contribution margin

The contribution margin per unit is the selling price minus the variable cost per unit. It is the amount each unit sold contributes, first toward covering fixed costs and then, once they are covered, toward profit. The contribution margin ratio is the contribution margin as a percentage of the price.

With price P, variable cost per unit VC, fixed costs FC and volume Q, the profit equation is:

The break-even point

The break-even point is the volume at which profit is zero: total contribution exactly covers fixed costs. Below it the business loses money; above it, every additional unit adds its contribution margin to profit.

Target profit

To find the volume needed for a target operating profit, treat the profit as an extra fixed cost that the contribution margin must cover.

If the target is an after-tax profit, first convert it to the pre-tax profit that leaves that amount after tax: pre-tax profit = after-tax profit ÷ (1 − tax rate).

A US C corporation pays a flat federal corporate income tax of 21 % on its taxable income, plus state corporate income tax in most states, at the time of writing (October 2026). Many small businesses are instead organized as sole proprietorships, partnerships, S corporations or LLCs, whose profit passes through to the owners and is taxed on their personal returns, along with self-employment tax.

Margin of safety and operating leverage

The margin of safety is how far sales can fall before the business starts losing money. It is usually expressed in units, in dollars and as a percentage of expected sales. A larger margin of safety means a less risky operation.

Operating leverage measures how sensitive profit is to changes in sales. A business with high fixed costs and low variable costs has high operating leverage: once fixed costs are covered, profit climbs quickly with sales, but it also falls quickly when sales drop. The degree of operating leverage (DOL) at a given volume is the total contribution margin divided by operating profit; a 1 % change in sales then changes profit by DOL %.

What-if analysis

Because the CVP model is so simple, it is easy to test the effect of changes: a price rise, a new supplier, a rent increase, a new salaried employee. Change the inputs and recompute. Spreadsheets make this particularly easy: put price, variable cost and fixed costs in input cells, and every result updates instantly.

Several products: the sales mix

Most businesses sell more than one product, each with its own contribution margin. If the sales mix (the proportion of each product in total sales) is stable, the business can be analyzed with a weighted average contribution margin.

Limits of the model

CVP analysis assumes that price and variable cost per unit stay constant, that fixed costs stay fixed within the relevant range, that everything produced is sold and that the sales mix is stable. Real businesses face volume discounts, overtime, price cuts to move more product and changing customer preferences. The model is best used as a first approximation and a way to see which assumptions matter most, not as a precise forecast.

# Cost-volume-profit model for the cafe
P, VC, FC = 6.00, 2.40, 9000
CM = P - VC
print(f"CM {CM:.2f}, CMR {CM / P:.0%}, break-even {FC / CM:,.0f} cups = ${FC / (CM / P):,.0f}")
print(f"Cups for $5,400 profit: {(FC + 5400) / CM:,.0f}")
Q = 3500
profit = Q * CM - FC
print(f"At {Q} cups: profit {profit:,.0f}, margin of safety {(Q - FC / CM) / Q:.1%}, DOL {Q * CM / profit:.2f}")
mix = {"coffee": (0.5, 3.60), "pastry": (0.3, 2.40), "sandwich": (0.2, 4.50)}
wcm = sum(share * cm for share, cm in mix.values())
print(f"Weighted CM {wcm:.2f}, break-even {FC / wcm:,.1f} items")
for q in (2000, 2500, 3000, 3500, 4000):
    print(q, "cups ->", round(q * CM - FC))

CM 3.60, CMR 60%, break-even 2,500 cups = $15,000
Cups for $5,400 profit: 4,000
At 3500 cups: profit 3,600, margin of safety 28.6%, DOL 3.50
Weighted CM 3.42, break-even 2,631.6 items
2000 cups -> -1800
2500 cups -> 0
3000 cups -> 1800
3500 cups -> 3600
4000 cups -> 5400