Source: Chapter 5, Exercises 5-1 through 5-13, Problems 5-19, 5-22, Appendix 5A
Tags: CVP analysis, contribution margin, break-even point, target profit, CM ratio, operating leverage, variable costs, fixed costs, high-low method, cost-volume-profit, managerial accounting
Difficulty: Intermediate | Prerequisites: Understanding of variable vs. fixed cost behaviour (Chapter 2), contribution format income statement basics.
Cost-volume-profit (CVP) analysis is one of the most practical tools in managerial accounting. It tells managers how changes in selling prices, costs, and volume affect profit, and it underpins almost every short-term business decision: pricing, product mix, advertising spend, automation trade-offs. If you missed earlier chapters, you need a solid grasp of the distinction between variable and fixed costs and the contribution format income statement before this material will click. CVP sits at the intersection of cost behaviour and profit planning, and the formulas here will reappear in budgeting, relevant-cost analysis, and performance evaluation later in the course.
CVP analysis uses the contribution margin (selling price minus variable cost per unit) to work out how many units a company must sell to break even or hit a target profit. The contribution margin ratio lets you do the same analysis in dollar terms. Operating leverage tells you how sensitive profit is to changes in volume, and the high-low method helps estimate mixed costs so you can slot them into the CVP framework.
Contribution margin (CM)
Selling price per unit minus variable expenses per unit. In simple terms, it is the amount each unit sold contributes toward covering fixed costs and then generating profit.
Contribution margin ratio (CM ratio)
Contribution margin divided by sales, expressed as a percentage. Think of it as the fraction of every sales dollar that is available to cover fixed costs and profit.
Break-even point
The level of sales (in units or dollars) at which total revenue equals total costs and net operating income is zero. In simple terms, it is the point where the company neither makes money nor loses money.
Variable expenses
Costs that change in direct proportion to changes in activity level (e.g. direct materials, sales commissions). Per-unit variable costs stay constant; total variable costs rise and fall with volume.
Fixed expenses
Costs that remain constant in total regardless of changes in the activity level within the relevant range (e.g. rent, salaries, insurance). Per-unit fixed costs change as volume changes, but the total stays the same.
Net operating income (NOI)
Sales minus variable expenses minus fixed expenses. This is the profit figure CVP analysis is built to predict.
Target profit analysis
An extension of break-even analysis that calculates the sales volume needed to achieve a specific profit target rather than zero profit.
Degree of operating leverage (DOL)
Contribution margin divided by net operating income. Think of it as a multiplier: it tells you the percentage change in profit for a given percentage change in sales.
High-low method
A technique for separating a mixed cost into its variable and fixed components by comparing the highest and lowest activity levels. In simple terms, you draw a line through the two extreme data points and use the slope as the variable rate.
Mixed cost (semi-variable cost)
A cost that contains both a variable element and a fixed element (e.g. a utility bill with a base charge plus a per-unit usage charge).
Sales mix
The relative proportion in which a company's products are sold. In a multiproduct firm, the overall CM ratio depends on which products make up the mix.
Contribution format income statement
An income statement organised by cost behaviour (variable vs. fixed) rather than by function. Sales minus variable expenses equals contribution margin; contribution margin minus fixed expenses equals net operating income.
The contribution format groups costs by behaviour, not function.
Structure: Sales – Variable expenses = Contribution margin – Fixed expenses = Net operating income.
Per-unit figures stay constant for sales price, variable expense, and contribution margin. Total figures scale with volume. Fixed expenses stay the same in total.
When volume changes and nothing else does, the change in net operating income equals the change in units multiplied by the unit contribution margin.
Example (Exercise 5-1): Whirly Corporation sells 10,000 units at $35 each, with variable expenses of $20 per unit. CM per unit = $15. If volume rises by 100 units, profit rises by 100 × $15 = $1,500 (from $15,000 to $16,500). If volume drops by 100 units, profit drops by $1,500 (to $13,500).
At 9,000 units, the company hits exactly $0 net operating income, which is its break-even point.
Fixed costs do not change when volume changes, so only the contribution margin matters.
CM ratio = Contribution margin / Sales. It can be computed per unit (CM per unit / Selling price per unit) or in total.
Example (Exercise 5-4): Holiday Creations has sales of $200,000, variable expenses of $120,000, and CM of $80,000. CM ratio = $80,000 / $200,000 = 40%.
The CM ratio is useful for predicting the effect of dollar changes in sales on profit. Multiply the increase in sales dollars by the CM ratio to get the increase in contribution margin (and therefore in profit, assuming fixed costs do not change).
Example: A $1,000 increase in sales at a 40% CM ratio adds $400 to contribution margin.
These factors rarely change one at a time. CVP analysis handles combined changes by rebuilding the contribution format income statement under the new assumptions.
Example (Exercise 5-5, Part 1): Hermann Corporation has a CM of $27 per unit (selling price $90 – variable expenses $63), fixed expenses of $30,000, and sells 2,000 units. If advertising (a fixed cost) rises by $5,000, volume rises by 100 units, and total sales increase by $9,000, the incremental CM from the extra sales is $9,000 × 30% CM ratio = $2,700, minus the $5,000 advertising cost, for a net decrease of $2,300.
Example (Exercise 5-5, Part 2): If variable expenses rise by $2 per unit (CM drops from $27 to $25) but unit sales increase by 10%, the new total CM = 2,200 units × $25 = $55,000 versus the old $54,000 (2,000 × $27). Net increase = $1,000, so the higher-quality components should be used.
Miller Company (20,000 units, $15 selling price, $9 variable, $6 CM, $70,000 fixed, $50,000 NOI):
15% volume increase: 23,000 units × $6 CM = $138,000 – $70,000 = $68,000 NOI.
Price drops $1.50, volume rises 25%: 25,000 units × $4.50 CM = $112,500 – $70,000 = $42,500 NOI.
Price rises $1.50, fixed costs rise $20,000, volume drops 5%: 19,000 units × $7.50 CM = $142,500 – $90,000 = $52,500 NOI.
Price rises 12%, variable expenses rise $0.60, volume drops 10%: 18,000 units × $7.20 CM = $129,600 – $70,000 = $59,600 NOI.
Formula: Break-even units = Fixed expenses / Unit contribution margin.
Example (Exercise 5-6): Mauro Products sells baskets at $15 with variable expenses of $12. CM per unit = $3. Fixed expenses = $4,200. Break-even = $4,200 / $3 = 1,400 baskets.
If fixed expenses increase by $600 (to $4,800), the new break-even = $4,800 / $3 = 1,600 baskets.
Formula: Break-even dollars = Fixed expenses / CM ratio. Alternatively, multiply break-even units by the selling price.
Example: 1,400 baskets × $15 = $21,000. With increased fixed expenses: 1,600 × $15 = $24,000.
Or: $4,200 / 0.20 CM ratio = $21,000.
The break-even formula is extended by adding the target profit to fixed expenses in the numerator.
Formula: Units for target profit = (Fixed expenses + Target profit) / Unit CM.
Example (Exercise 5-7): Lin Corporation sells at $120, variable expense $80, CM $40, fixed expenses $50,000.
For a $10,000 target profit: ($50,000 + $10,000) / $40 = 1,500 units.
For a $15,000 target profit: ($50,000 + $15,000) / $40 = 1,625 units. Dollar sales = 1,625 × $120 = $195,000.
Minden Company sells at $70, variable expense $40, CM $30, fixed expenses $540,000, volume 15,000 units.
Current NOI = (15,000 × $30) – $540,000 = a loss of $90,000.
Break-even at current price = $540,000 / $30 = 18,000 units (or $1,260,000 in sales).
A marketing study suggests every $2 price reduction adds 5,000 units of volume. The profit-maximising price is $58 (CM = $18, volume = 45,000 units, total CM = $810,000, NOI = $270,000).
At $58, the break-even shifts to $540,000 / $18 = 30,000 units ($1,740,000 in sales). The break-even is higher in units because the CM per unit is lower, even though total profit is higher.
PEM sells laptop batteries at $30, variable expenses $21 per unit, CM $9, CM ratio 30%, fixed expenses $180,000, selling 19,500 units, NOI = loss of $4,500.
Break-even: $180,000 / $9 = 20,000 units, or $180,000 / 0.30 = $600,000 in sales.
Scenario: $16,000 more advertising + $80,000 more sales → incremental CM = $80,000 × 0.30 = $24,000, less $16,000 = $8,000 increase. Loss turns to $3,500 profit.
Scenario: 10% price cut + double volume + $60,000 more fixed costs → sales 39,000 units at $27, variable $21, CM $6. Total CM $234,000, fixed $240,000, loss of $6,000.
Target profit with new packaging ($0.75 more variable cost per unit): CM = $30 – $21.75 = $8.25. Units for $9,750 target = ($180,000 + $9,750) / $8.25 = 23,000 units.
Automation (variable drops $3, fixed rises $72,000): new CM = $12, CM ratio = 40%. Break-even = $252,000 / $12 = 21,000 units ($630,000). At 26,000 units, automated NOI = $60,000 vs. non-automated $54,000. Automation is better above approximately 24,000 units.
DOL = Contribution margin / Net operating income.
Example (Exercise 5-9): Engberg Company has CM of $48,000 and NOI of $10,000. DOL = 4.8.
Interpretation: a 5% increase in sales produces a 5% × 4.8 = 24% increase in NOI.
Verification: sales rise to $84,000, variable expenses to $33,600, CM to $50,400, less $38,000 fixed = $12,400 NOI. Change = $2,400 / $10,000 = 24%.
Companies with higher fixed costs relative to variable costs have higher operating leverage, meaning their profits are more sensitive to changes in volume.
When a company sells more than one product, the overall CM ratio is a weighted average based on the sales mix.
Example: Olongapo Sports sells Flight Dynamic ($150,000 sales, 80% CM ratio) and Sure Shot ($250,000 sales, 36% CM ratio). Total sales = $400,000. Total CM = $120,000 + $90,000 = $210,000. Overall CM ratio = $210,000 / $400,000 = 52.5%.
Break-even in dollars = Fixed expenses / Overall CM ratio = $183,750 / 0.525 = $350,000.
If sales increase by $100,000 at the same mix, the additional CM = $100,000 × 52.5% = $52,500, all of which flows to profit (assuming no change in fixed expenses).
The break-even calculation assumes the sales mix stays constant. If the mix shifts toward lower-margin products, the overall CM ratio falls and the break-even point rises.
Purpose: to split a mixed cost into its variable rate and fixed component.
Steps:
Identify the periods with the highest and lowest activity levels.
Variable rate = (Cost at high activity – Cost at low activity) / (High activity – Low activity).
Fixed cost = Total cost at either activity level – (Variable rate × Activity level).
Example (Exercise 5A-1, Cheyenne Hotel):
High activity: August, 2,406 occupancy-days, $5,148 electrical cost.
Low activity: October, 124 occupancy-days, $1,588.
Variable rate = ($5,148 – $1,588) / (2,406 – 124) = $3,560 / 2,282 = $1.56 per occupancy-day.
Fixed cost = $5,148 – ($1.56 × 2,406) = $5,148 – $3,753 = $1,395 per month.
Example (Archer Company shipping):
High: June, 8 units shipped, $2,700. Low: July, 2 units, $1,200.
Variable rate = $1,500 / 6 = $250 per unit shipped.
Fixed cost = $2,700 – ($250 × 8) = $700 per month.
Cost formula: Y = $700 + $250X.
\text{Contribution Margin per Unit} = \text{Selling Price} - \text{Variable Expenses per Unit}\text{CM Ratio} = \frac{\text{Contribution Margin}}{\text{Sales}}\text{Break-even (units)} = \frac{\text{Fixed Expenses}}{\text{Unit CM}}\text{Break-even (dollars)} = \frac{\text{Fixed Expenses}}{\text{CM Ratio}}\text{Units for Target Profit} = \frac{\text{Fixed Expenses} + \text{Target Profit}}{\text{Unit CM}}\text{Degree of Operating Leverage} = \frac{\text{Contribution Margin}}{\text{Net Operating Income}}\text{Variable Rate (High-Low)} = \frac{\text{Cost}_{\text{high}} - \text{Cost}_{\text{low}}}{\text{Activity}_{\text{high}} - \text{Activity}_{\text{low}}}\text{Mixed Cost Formula: } Y = a + bX \quad (a = \text{fixed}, \; b = \text{variable rate}, \; X = \text{activity})CVP analysis is how a restaurant owner works out how many covers per evening they need to break even, or how a SaaS company decides whether to lower its subscription price to gain volume. Any business that has fixed overheads and a per-unit margin uses this logic, whether they call it CVP or not. The high-low method is a quick first pass for estimating mixed costs when you do not have the time or data for regression analysis, common in settings like hotel utilities, fleet maintenance, and call-centre staffing.
Students often think that lowering the selling price will always reduce profit. It depends: if the volume increase is large enough, total contribution margin can rise even though CM per unit falls. Problem 5-19 shows the optimal price at $58, not the original $70.
Students confuse the CM ratio with the gross margin percentage. The CM ratio uses variable costs only; gross margin uses cost of goods sold (which may include fixed manufacturing overhead under absorption costing).
Students sometimes apply the CM ratio to a change in units rather than a change in dollars. The CM ratio works with dollar changes in sales; for unit changes, use the unit contribution margin.
Students forget that the break-even point shifts when the sales mix changes in a multiproduct company. The overall CM ratio is mix-dependent, so a new mix means a new break-even.
⚠️ Break-even and target profit calculations are near-certain exam questions. Know both the unit formula and the dollar formula.
⚠️ Expect a scenario question where multiple variables change at once (price, volume, costs). Rebuild the full contribution format income statement, do not try to shortcut.
⚠️ The degree of operating leverage is a common short-answer or multiple-choice item. Remember: DOL × % change in sales = % change in NOI.
⚠️ The high-low method is tested as a standalone calculation. Be careful to pick the highest and lowest activity levels, not the highest and lowest costs (they are usually but not always the same).
⚠️ Multiproduct break-even questions test whether you can compute a weighted-average CM ratio. The sales mix assumption is always worth stating in your answer.
True or false: If a company's unit contribution margin is $8 and fixed costs are $40,000, the break-even point is 5,000 units. (True)
Fill in the blank: The CM ratio is calculated as ______ divided by ______. (Contribution margin divided by Sales)
True or false: The degree of operating leverage changes as the company's sales level changes. (True, because NOI changes while fixed costs stay the same.)
True or false: In a multiproduct company, the break-even point is unaffected by changes in the sales mix. (False. A shift in mix changes the overall CM ratio and therefore the break-even point.)
Fill in the blank: Using the high-low method, the variable cost rate equals the change in ______ divided by the change in ______. (Cost divided by Activity)
Q: A company sells a product for $50. Variable expenses are $30 per unit. Fixed expenses total $60,000 per month. What is the break-even point in units and in dollar sales?
A: CM per unit = $20. Break-even units = $60,000 / $20 = 3,000 units. Break-even dollars = 3,000 × $50 = $150,000 (or $60,000 / 0.40 CM ratio = $150,000).
Q: Using the data above, how many units must be sold to earn a target profit of $20,000?
A: ($60,000 + $20,000) / $20 = 4,000 units.
Q: A company has a contribution margin of $100,000 and net operating income of $25,000. What is its degree of operating leverage? If sales increase by 10%, what is the expected percentage increase in NOI?
A: DOL = $100,000 / $25,000 = 4.0. Expected increase in NOI = 10% × 4.0 = 40%.
Q: A hotel's utility costs were $3,000 at 500 occupancy-days (low) and $5,400 at 1,300 occupancy-days (high). Using the high-low method, what is the variable rate per occupancy-day and the fixed cost per month?
A: Variable rate = ($5,400 – $3,000) / (1,300 – 500) = $2,400 / 800 = $3.00 per occupancy-day. Fixed cost = $5,400 – ($3.00 × 1,300) = $5,400 – $3,900 = $1,500.
Q: A company sells two products. Product A has sales of $80,000 and a CM ratio of 60%. Product B has sales of $120,000 and a CM ratio of 25%. Fixed expenses are $50,000. What is the company's break-even point in dollar sales?
A: Total CM = ($80,000 × 0.60) + ($120,000 × 0.25) = $48,000 + $30,000 = $78,000. Overall CM ratio = $78,000 / $200,000 = 39%. Break-even = $50,000 / 0.39 = $128,205 (rounded).
This connects to Chapter 6 (Variable and Absorption Costing) because CVP analysis relies on the contribution format, which is a variable-costing framework. When companies report under absorption costing, the CVP relationships can seem to break down, and reconciling the two methods requires understanding how fixed manufacturing overhead moves in and out of inventory.
CVP concepts also feed directly into budgeting and profit planning (later chapters), where managers use break-even and target profit analysis to set volume targets and evaluate spending decisions.
The high-low method connects to Chapter 2's cost behaviour material and to regression analysis, which is a more precise alternative for separating mixed costs.
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