# CVP and Break-Even Concept Card: [Product, business unit or course]

> A concept card for cost-volume-profit (CVP) analysis as it is taught in the managerial accounting core of a leading
> MBA programme: cost behaviour, contribution margin, break-even in units and revenue, target profit before and after
> tax, margin of safety, operating leverage, multi-product break-even with a sales mix, and the choice between cost
> structures. It is for MBA students preparing for class, a case or the final exam, and for managers who need a quick,
> correct view of how profit moves with volume. Each idea comes in plain language, then as a formula with notation,
> then as a fully computed example with the Excel formula next to every result.
> Source basis: the contribution-margin approach of the managerial accounting literature (Johnson and Kaplan 1987 for
> its history), the standard textbooks named in the reading list, and the operating-leverage concept, written up in
> original words. The companies and numbers are fictional and were computed in Python.
> Replace every [bracketed] value and every row marked *Example*.

<!--
Guidance: How to use this template
- Read "When to use it" and "The idea in plain language" first. If you cannot explain contribution margin in one
  sentence to a non-finance colleague, stop and reread that section.
- Rebuild Worked example A in a blank spreadsheet with the cell layout shown. Getting the same numbers with your own
  formulas is the point: cases and exams test speed and accuracy with these formulas.
- Then fill every [ ] in Worked example B and solve Exercise C before you look at the answer key.
- Answer the self-test from memory, check the answer key, log every miss in mba/mba-error-log, and repeat the
  self-test after two days and after a week (spacing).
- The fictional setting is Northfield Outdoor Co., an outdoor-gear retailer and brand with stores, e-commerce and a
  small own-brand production unit. Every number is invented.
- Related templates: mba/mba-managerial-accounting-guide, mba/mba-relevant-costs-card,
  mba/mba-activity-based-costing-card, mba/mba-pricing-card, mba/mba-demand-elasticity-card,
  mba/mba-capital-structure-card, mba/mba-npv-irr-card, mba/mba-formula-sheet, mba/pre-mba-accounting-primer,
  mba/pre-mba-excel-modeling-drills, mba/mba-error-log, finance/unit-economics-analysis, finance/annual-budget,
  finance/variance-analysis, academic/stats-linear-regression.
-->

## Document control

<!-- Guidance: Record which course, case or business question this card supports, and who checked your numbers. Update the version when you replace the Northfield data with your own. -->

| Field | Value |
|---|---|
| Course or project | [Managerial accounting core / case name / business unit plan] |
| Learner or analyst | [Your name] |
| Business question | [Example: how many units must the new pack line sell to reach the profit target?] |
| Data source | [Northfield Outdoor Co. (fictional) / your budget file and date] |
| Spreadsheet used | [Excel / Google Sheets / LibreOffice Calc] |
| Reviewed by | [Study partner, professor or controller] |
| Version | [0.1] |
| Last updated | YYYY-MM-DD |

## Contents

- [When to use it](#when-to-use-it)
- [The idea in plain language](#the-idea-in-plain-language)
- [Formulas and notation](#formulas-and-notation)
- [Assumptions and limits](#assumptions-and-limits)
- [How managers use it](#how-managers-use-it)
- [Worked example A (fully solved)](#worked-example-a-fully-solved-the-ridgeline-pack-line)
- [Multi-product break-even](#multi-product-break-even-with-a-sales-mix)
- [Worked example B (half solved)](#worked-example-b-half-solved-choosing-a-cost-structure)
- [Exercise C (unsolved)](#exercise-c-unsolved-a-two-product-rental-service)
- [Connections to other cards](#connections-to-other-cards)
- [Common mistakes and misconceptions](#common-mistakes-and-misconceptions)
- [Self-test questions](#self-test-questions)
- [My notes and next steps](#my-notes-and-next-steps)
- [Sources and further reading](#sources-and-further-reading)
- [Answer key](#answer-key)

## When to use it

<!-- Guidance: Start with the one-sentence rule. If your decision involves only some costs changing (make or buy, special order), switch to mba/mba-relevant-costs-card instead. -->

**One-sentence rule:** use CVP analysis when you need to know how operating profit responds to changes in volume,
price, variable cost or fixed cost over a range of activity where the cost structure is stable.

| Use when | Do not use when |
|---|---|
| Planning a new product, store, channel or service line and asking "how many must we sell?" | The decision changes only some costs and revenues (use relevant costing) |
| Setting a sales or profit target in a budget | Costs are mostly step costs that jump within the range you study |
| Testing what a price cut, a wage rise or a new lease does to profit | Volume moves far outside the relevant range (new plant, new shift) |
| Comparing a cost structure with high fixed costs with one that has high variable costs | Cash timing matters more than profit (use a cash-flow forecast and NPV) |
| Explaining to a board why profit fell much faster than sales | The product mix is unstable and cannot be predicted |

In a case discussion, CVP is usually the first ten minutes of quantitative work: it tells the class whether a plan is
plausible before anyone builds a full model. In recruiting, consulting and product interviews often include a
break-even question in disguise ("how many subscribers does this app need to cover its engineering team?").

## The idea in plain language

<!-- Guidance: Read this section aloud. The whole card rests on separating costs by behaviour (fixed versus variable), not by function (production versus selling). -->

Every cost behaves in one of a few ways as activity changes. A **variable cost** rises in proportion to volume: the
fabric, zips and buckles in each backpack, the card fee on each online order, the sales commission on each unit. A
**fixed cost** stays the same in total within a range of activity: store rent, salaried staff, the design team, the
insurance policy. Some costs are **mixed** (a fixed base plus a variable part, such as a utility bill) and some are
**step costs** (fixed within a band, then jumping, such as adding a supervisor for every 25 operators). For CVP you
split every cost into its fixed and variable parts; mba/mba-managerial-accounting-guide shows how to do that with the
high-low method and regression.

Once costs are split, each unit sold earns a **contribution margin**: its price minus its variable cost. That amount
"contributes" first to covering fixed costs, then to profit. The **break-even point** is the volume at which the total
contribution exactly covers fixed costs, so profit is zero. Every unit above break-even adds its full contribution
margin to profit; every unit below it adds that amount to the loss.

The **contribution margin ratio** expresses the same idea per dollar of sales: if the ratio is 40%, every extra USD 1
of revenue adds USD 0.40 to profit. The ratio lets you work in revenue when units are not meaningful (a store with
thousands of items) or when a product mix is held constant.

The **margin of safety** says how far sales could fall before the business makes a loss. **Operating leverage**
measures how strongly profit reacts to a change in sales. A business with high fixed costs and low variable costs has
high operating leverage: profit grows fast when sales grow and collapses fast when sales fall. The degree of operating
leverage at a given sales level equals one divided by the margin of safety percentage, which links risk (how close you
are to break-even) and sensitivity (how much profit swings).

CVP is a contribution-format view of the income statement. A financial-reporting income statement groups costs by
function (cost of goods sold, then selling and administrative expenses) and, under IFRS and US GAAP, includes fixed
production overhead in inventory. The contribution format groups costs by behaviour instead, which is why it is used
for internal decisions and not for external reporting.

## Formulas and notation

<!-- Guidance: Learn the first three formulas by heart; derive the rest from them in the exam rather than memorising them. Every formula is the same equation, OI = (P - V) Q - F, rearranged. -->

The core equation is operating income as a function of volume:

$$
OI = (P - V)\,Q - F = CM \times Q - F
$$

Setting $OI = 0$ gives break-even in units and in revenue:

$$
Q_{BE} = \frac{F}{P - V} = \frac{F}{CM}, \qquad R_{BE} = \frac{F}{CMR}, \qquad CMR = \frac{P - V}{P}
$$

Target profit before tax $T$, or after tax $T_{AT}$ at tax rate $t$:

$$
Q_T = \frac{F + T}{CM}, \qquad Q_{T_{AT}} = \frac{F + \dfrac{T_{AT}}{1 - t}}{CM}
$$

Margin of safety and degree of operating leverage at planned volume $Q$:

$$
MOS = Q - Q_{BE}, \qquad MOS\% = \frac{Q - Q_{BE}}{Q}, \qquad DOL = \frac{CM \times Q}{OI} = \frac{1}{MOS\%}
$$

$$
\%\Delta OI \approx DOL \times \%\Delta \text{Sales} \quad \text{(exact when price and unit costs stay fixed)}
$$

Multi-product break-even with a constant sales mix (unit shares $m_i$ summing to 1):

$$
\overline{CM} = \sum_i m_i\,(P_i - V_i), \qquad Q_{BE}^{\text{total}} = \frac{F}{\overline{CM}}, \qquad
Q_{BE,i} = m_i\,Q_{BE}^{\text{total}}
$$

Indifference point between two cost structures 1 and 2 (same price):

$$
Q^{*} = \frac{F_2 - F_1}{V_1 - V_2}
$$

| Symbol | Meaning | Unit |
|---|---|---|
| $P$ | Selling price per unit | USD per unit |
| $V$ | Variable cost per unit (production, selling, delivery: every cost that moves with units) | USD per unit |
| $CM$ | Contribution margin per unit, $P - V$ | USD per unit |
| $CMR$ | Contribution margin ratio, $CM / P$ | % |
| $F$ | Total fixed costs for the period within the relevant range | USD |
| $Q$ | Units sold in the period | units |
| $OI$ | Operating income (profit before interest and tax) | USD |
| $T$, $T_{AT}$, $t$ | Target profit before tax, after tax, tax rate | USD, USD, % |
| $MOS$ | Margin of safety | units, USD or % |
| $DOL$ | Degree of operating leverage at a given volume | multiple |
| $m_i$, $\overline{CM}$ | Unit sales-mix share of product i, weighted average CM | %, USD per unit |
| $Q^{*}$ | Volume at which two cost structures give the same profit | units |

```mermaid
flowchart TD
  A["Business question about volume, price or cost"] --> B{"Do all costs and revenues move with the decision?"}
  B -->|"No, only some change"| R["Use relevant costing (mba/mba-relevant-costs-card)"]
  B -->|"Yes, a volume or structure question"| C["Split every cost into fixed and variable"]
  C --> D{"One product or a stable mix?"}
  D -->|"One product"| E["CM per unit and CMR"]
  D -->|"Several products, stable mix"| F["Weighted average CM from the mix"]
  D -->|"Mix unstable"| G["Scenario model per mix, report a range"]
  E --> H["Break-even, target volume, margin of safety, DOL"]
  F --> H
  G --> H
  H --> I["Sensitivity: price, unit cost, fixed cost, mix"]
  I --> J["Recommendation with the risk stated"]
```

## Assumptions and limits

<!-- Guidance: In a case write-up, state which assumptions you made and which one is most likely to fail. That sentence often earns more credit than the arithmetic. -->

CVP is a linear model. It is useful precisely because it is simple, and it is wrong in predictable ways when its
assumptions fail.

| Assumption | Why it matters | When it fails and what to do |
|---|---|---|
| Costs can be split cleanly into fixed and variable | Break-even depends directly on the split | Mixed costs: estimate the split with regression; step costs: model each band separately |
| Linear revenue: price is constant at every volume | A higher volume target may need a price cut | Model price as a function of volume (mba/mba-demand-elasticity-card) |
| Linear costs within a **relevant range** | Fixed costs are fixed only within capacity | Mark the range on the chart; outside it, add the new fixed cost |
| Sales mix is constant (multi-product) | A mix shift moves break-even even if total units stay the same | Run the analysis for two or three mixes |
| Units produced equal units sold | Otherwise absorption-costing profit differs from contribution-format profit | Use variable costing for CVP; see the guide for the reconciliation |
| One time period, no time value of money | Long-lived investments need discounting | Use NPV for multi-year decisions (mba/mba-npv-irr-card) |
| Certainty | Real volumes are uncertain | Add a scenario table or a simulation; report the probability of a loss where possible |

Two limits deserve special attention. First, the **cost classification depends on the time horizon and the decision**:
staff salaries are fixed for a month but variable over three years. Second, CVP gives **operating** break-even. A
company can be above operating break-even and still lose money after interest, or run out of cash because growth ties
up working capital. The cash break-even (fixed costs minus non-cash items such as depreciation, divided by CM) is lower
than the accounting break-even, and the break-even that also covers a required return on invested capital is higher.

## How managers use it

<!-- Guidance: Pick the uses that match your role or case and delete the rest when you adapt the card. -->

- **Launch decisions.** A product manager compares the break-even volume with a realistic market-size estimate. If
  break-even needs 30% of the reachable market in year one, the plan is unrealistic, however good the margin looks.
- **Pricing.** A price cut lowers CM per unit, so the volume needed to keep profit flat rises more than proportionally.
  The rule: with fixed costs unchanged, the volume increase needed to keep profit flat equals the price cut per
  unit divided by the new CM per unit (computed in self-test 6). Pair this with elasticity estimates from mba/mba-pricing-card.
- **Budget targets.** Controllers back-solve the sales target from the profit target, then test whether the sales team
  can reach it.
- **Cost structure choices.** Outsourcing, automation, leasing versus owning, and commission versus salary all trade
  fixed for variable costs. CVP shows the volume at which each choice wins and how much risk each one carries.
- **Risk communication.** Margin of safety and DOL turn "we are exposed" into a number a board can discuss: with a DOL
  of 3, a 10% sales shortfall cuts operating income by 30%.
- **Unit economics.** Start-ups and subscription businesses run the same logic per customer (contribution per
  customer, customers needed to cover fixed cost); see finance/unit-economics-analysis.

In spreadsheets, managers build a one-page CVP model with inputs at the top, then use **Goal Seek** (Data, What-If
Analysis) to back-solve the volume for a profit target and a **Data Table** to show profit over a grid of price and
volume. Keep inputs, calculations and outputs in separate blocks, as in mba/pre-mba-excel-modeling-drills.

## Worked example A (fully solved): the Ridgeline pack line

<!-- Guidance: Rebuild this in a spreadsheet with the cell addresses shown before you read the solution. The Excel formula column is what you would type; the result column is what you should see. -->

**Situation (fictional).** Northfield Outdoor Co. plans to launch an own-brand 40-litre hiking backpack, the Ridgeline
40L, through its stores and website. Marketing proposes a price of USD 120. Sourcing and fulfilment give a variable cost
of USD 72 per pack (manufacturing by a contract supplier, freight, duty, card fees and the store commission). The line
needs USD 1,440,000 a year of fixed cost: a design and product team, a marketing launch budget and dedicated warehouse
space. The plan is to sell 45,000 packs in the first year. The tax rate is 25%. The CFO wants an after-tax profit of at
least USD 360,000 from the line.

**Spreadsheet layout.** B1 = price 120, B2 = variable cost 72, B3 = fixed cost 1,440,000, B4 = planned units 45,000,
B5 = tax rate 25%, B6 = after-tax target 360,000.

| Step | Quantity | Excel formula | Result |
|---|---|---|---|
| 1 | CM per unit | `=B1-B2` | USD 48.00 |
| 2 | CM ratio | `=(B1-B2)/B1` | 40.0% |
| 3 | Break-even units | `=B3/(B1-B2)` | 30,000 packs |
| 4 | Break-even revenue | `=B3/((B1-B2)/B1)` | USD 3,600,000 |
| 5 | Pre-tax target | `=B6/(1-B5)` | USD 480,000 |
| 6 | Units for the target | `=(B3+B6/(1-B5))/(B1-B2)` | 40,000 packs |
| 7 | Revenue for the target | `=(B3+B6/(1-B5))/((B1-B2)/B1)` | USD 4,800,000 |
| 8 | Planned revenue | `=B1*B4` | USD 5,400,000 |
| 9 | Planned total CM | `=(B1-B2)*B4` | USD 2,160,000 |
| 10 | Planned operating income | `=(B1-B2)*B4-B3` | USD 720,000 |
| 11 | Planned after-tax income | `=((B1-B2)*B4-B3)*(1-B5)` | USD 540,000 |
| 12 | Margin of safety (units) | `=B4-B3/(B1-B2)` | 15,000 packs |
| 13 | Margin of safety (%) | `=(B4-B3/(B1-B2))/B4` | 33.3% |
| 14 | DOL at plan | `=(B1-B2)*B4/((B1-B2)*B4-B3)` | 3.00 |
| 15 | OI if sales are 10% above plan | `=(B1-B2)*B4*1.1-B3` | USD 936,000 (+30.0%) |

**Reading the result.** The line breaks even at 30,000 packs, two thirds of the plan. It reaches the CFO's after-tax
target at 40,000 packs, so the plan of 45,000 clears the target with 5,000 packs to spare. The margin of safety is
15,000 packs or 33.3% of planned sales: sales could fall by a third before the line loses money. The DOL of 3.00 is the
reciprocal of that margin of safety (1 / 0.333 = 3.00), and step 15 confirms it: a 10% rise in sales lifts operating
income by 30%, from USD 720,000 to USD 936,000. The same lever works in reverse.

**Sensitivity.** Each line changes one input and keeps the others at plan (45,000 packs).

| Scenario *(Example)* | CM per unit | Break-even units | OI at 45,000 packs | Excel formula for break-even |
|---|---|---|---|---|
| Plan | USD 48.00 | 30,000 | USD 720,000 | `=B3/(B1-B2)` |
| Price 5% lower (USD 114) | USD 42.00 | 34,286 | USD 450,000 | `=ROUNDUP(B3/(114-B2),0)` |
| Variable cost 5% higher (USD 75.60) | USD 44.40 | 32,433 | USD 558,000 | `=ROUNDUP(B3/(B1-75.6),0)` |
| Fixed cost 10% higher (USD 1,584,000) | USD 48.00 | 33,000 | USD 576,000 | `=B3*1.1/(B1-B2)` |

A 5% price cut costs USD 270,000 of operating income, more than the other two shocks, because every dollar of price
falls straight through to contribution while the volume stays the same. Round break-even volumes **up** to whole
units: 34,285.7 packs means the 34,286th pack is the first profitable one.

**Recommendation sentence.** "The Ridgeline 40L breaks even at 30,000 packs and meets the after-tax target of
USD 360,000 at 40,000 packs; the 45,000-pack plan gives a 33% margin of safety. The main risk is price: a 5% discount
cuts operating income by 38%, so promotional pricing needs a volume plan behind it."

**Break-even chart.** Revenue and total cost cross at 30,000 packs and USD 3.6m. The vertical distance between the two
lines to the right of break-even is operating income; to the left it is the loss.

```vega-lite
{
  "$schema": "https://vega.github.io/schema/vega-lite/v5.json",
  "title": "Ridgeline 40L break-even chart (Example, fictional data)",
  "width": 420,
  "height": 260,
  "data": {"values": [
    {"units": 0, "line": "Revenue", "usd": 0}, {"units": 0, "line": "Total cost", "usd": 1440000}, {"units": 0, "line": "Fixed cost", "usd": 1440000},
    {"units": 10000, "line": "Revenue", "usd": 1200000}, {"units": 10000, "line": "Total cost", "usd": 2160000}, {"units": 10000, "line": "Fixed cost", "usd": 1440000},
    {"units": 20000, "line": "Revenue", "usd": 2400000}, {"units": 20000, "line": "Total cost", "usd": 2880000}, {"units": 20000, "line": "Fixed cost", "usd": 1440000},
    {"units": 30000, "line": "Revenue", "usd": 3600000}, {"units": 30000, "line": "Total cost", "usd": 3600000}, {"units": 30000, "line": "Fixed cost", "usd": 1440000},
    {"units": 40000, "line": "Revenue", "usd": 4800000}, {"units": 40000, "line": "Total cost", "usd": 4320000}, {"units": 40000, "line": "Fixed cost", "usd": 1440000},
    {"units": 50000, "line": "Revenue", "usd": 6000000}, {"units": 50000, "line": "Total cost", "usd": 5040000}, {"units": 50000, "line": "Fixed cost", "usd": 1440000},
    {"units": 60000, "line": "Revenue", "usd": 7200000}, {"units": 60000, "line": "Total cost", "usd": 5760000}, {"units": 60000, "line": "Fixed cost", "usd": 1440000}
  ]},
  "layer": [
    {
      "mark": {"type": "line", "point": true},
      "encoding": {
        "x": {"field": "units", "type": "quantitative", "title": "Packs sold per year"},
        "y": {"field": "usd", "type": "quantitative", "title": "USD", "axis": {"format": "~s"}},
        "color": {"field": "line", "type": "nominal", "title": "Line"}
      }
    },
    {
      "data": {"values": [{"units": 30000, "label": "Break-even 30,000 packs"}, {"units": 45000, "label": "Plan 45,000 packs"}]},
      "mark": {"type": "rule", "strokeDash": [4, 4]},
      "encoding": {"x": {"field": "units", "type": "quantitative"}, "tooltip": {"field": "label", "type": "nominal"}}
    }
  ]
}
```

## Multi-product break-even with a sales mix

<!-- Guidance: Use units-based mix when the products are sold in a natural bundle or ratio; use revenue-based mix (weighted CMR) when units are not comparable. Both give the same answer if the mix is consistent. -->

Northfield's trail category sells three own-brand products with shared fixed costs of USD 2,700,000 a year (the
category team, a catalogue, a dedicated distribution zone). The expected **unit** mix is 50% daypacks, 20% tents and
30% jackets.

| Product *(Example)* | Price | Variable cost | CM per unit | Unit mix | Weighted CM | Excel formula |
|---|---|---|---|---|---|---|
| Daypack | USD 60 | USD 33 | USD 27 | 50% | USD 13.50 | `=(B2-C2)*E2` |
| Tent | USD 300 | USD 180 | USD 120 | 20% | USD 24.00 | `=(B3-C3)*E3` |
| Jacket | USD 150 | USD 90 | USD 60 | 30% | USD 18.00 | `=(B4-C4)*E4` |
| **Weighted average** | USD 135 | | | 100% | **USD 55.50** | `=SUMPRODUCT(B2:B4-C2:C4,E2:E4)` |

| Result | Excel formula | Value |
|---|---|---|
| Weighted CM ratio | `=SUMPRODUCT(B2:B4-C2:C4,E2:E4)/SUMPRODUCT(B2:B4,E2:E4)` | 41.11% |
| Break-even, total units | `=2700000/F5` | 48,649 units (48,648.65) |
| of which daypacks | `=48648.65*E2` | 24,324 |
| of which tents | `=48648.65*E3` | 9,730 |
| of which jackets | `=48648.65*E4` | 14,595 |
| Break-even revenue | `=2700000*B5/F5` | USD 6,567,568 |

At break-even, revenue splits USD 1,459,459 daypacks, USD 2,918,919 tents and USD 2,189,189 jackets (revenue shares
22.2%, 44.4% and 33.3%), and contribution covers the USD 2.7m of fixed cost exactly. At a planned 60,000 units in the
same mix, operating income is USD 3,330,000 minus USD 2,700,000, or USD 630,000; the margin of safety is 18.9% and the
DOL is 5.29, much higher than for the Ridgeline line, because the category runs closer to break-even.

**Mix matters.** If the mix shifts to 60% daypacks, 10% tents and 30% jackets, the weighted CM falls to USD 46.20 and
break-even rises to 58,442 units, about 20% more, even though prices and costs did not change. When a sales team is
paid on units or revenue rather than contribution, it tends to push the easiest product to sell, which is often the
lowest-margin one. CVP makes that cost visible.

## Worked example B (half solved): choosing a cost structure

<!-- Guidance: Fill every [ ] yourself before you open the answer key. Write the Excel formula as well as the number. -->

**Situation (fictional).** Northfield's small production unit will make a premium tent priced at USD 300. It can set up
the line in two ways:

- **Manual:** variable cost USD 180 per tent, fixed cost USD 600,000 a year.
- **Automated cutting and sealing:** variable cost USD 150 per tent, fixed cost USD 1,200,000 a year (equipment lease
  and maintenance contract).

Expected demand is 25,000 tents a year, but the range of outcomes is wide.

| Step | Quantity | Manual | Automated | Excel formula (manual) |
|---|---|---|---|---|
| 1 | CM per unit | USD 120 | [ ] | `=300-180` |
| 2 | Break-even units | 5,000 | [ ] | `=600000/(300-180)` |
| 3 | OI at 25,000 tents | USD 2,400,000 | [ ] | `=(300-180)*25000-600000` |
| 4 | DOL at 25,000 tents | 1.25 | [ ] | `=(300-180)*25000/((300-180)*25000-600000)` |
| 5 | OI at 15,000 tents | [ ] | USD 1,050,000 | [ ] |

6. Compute the indifference point $Q^{*}$ at which both structures give the same operating income, and that operating
   income. [ ]
7. Above $Q^{*}$ which structure gives more profit, and why? [ ]
8. Write a two-sentence recommendation that names the demand risk. [ ]

## Exercise C (unsolved): a two-product rental service

<!-- Guidance: Solve on paper first, then in a spreadsheet. Show the Excel formula for each answer. -->

Northfield runs a winter rental service at three mountain stores. A ski rental day earns USD 45 with variable cost
USD 12 (cleaning, waxing, wear). A snowshoe rental day earns USD 25 with variable cost USD 7. Rentals split 70% skis and
30% snowshoes by rental days. Fixed costs for the season are USD 198,000 (seasonal staff, insurance, fleet
depreciation). The manager wants an operating profit of USD 66,000.

1. Compute the weighted average CM per rental day.
2. Compute break-even in total rental days, split by product, and break-even revenue.
3. Compute the rental days needed for the profit target, split by product.
4. The manager forecasts 9,000 rental days. Compute operating income, the margin of safety in percent and the DOL.
5. Name two reasons the 70/30 mix might not hold, and say which way each would move break-even.

## Connections to other cards

<!-- Guidance: Use this list to plan your revision order: CVP first, then relevant costs, then ABC, then the guide's mini-case. -->

| Card or template | Connection |
|---|---|
| mba/mba-managerial-accounting-guide | Cost estimation (high-low, regression), variable versus absorption costing, budgets and variances that start from the same cost split |
| mba/mba-relevant-costs-card | Uses the same contribution logic for one-off decisions; adds opportunity costs and constraints |
| mba/mba-activity-based-costing-card | Shows that many "fixed" overheads are driven by batches, products and customers, which changes the CVP view of cost |
| mba/mba-pricing-card and mba/mba-demand-elasticity-card | Replace the constant-price assumption with a demand curve |
| mba/mba-capital-structure-card | Financial leverage stacks on operating leverage: total leverage = DOL x DFL |
| mba/mba-npv-irr-card | Multi-year launches need discounted cash flows, not single-period break-even |
| finance/unit-economics-analysis | CVP per customer for subscription and marketplace businesses |
| finance/annual-budget | Profit targets and volume plans in the budget come from CVP |

## Common mistakes and misconceptions

<!-- Guidance: Before an exam, read this list and mark the two mistakes you are most likely to make. Check for those two on every answer. -->

1. **Splitting costs by function instead of behaviour.** Cost of goods sold is not "variable" and selling and admin
   expenses are not "fixed": a sales commission is a variable selling cost, factory rent is a fixed production cost.
2. **Using a unit cost that includes allocated fixed cost as if it were variable.** A "full cost of USD 104 per pack"
   includes fixed cost spread over a planned volume; it does not stay at USD 104 if volume changes.
3. **Forgetting the tax gross-up.** An after-tax target must be divided by (1 - t) before it is added to fixed costs.
4. **Rounding break-even down.** 34,285.7 units means 34,286 units; fractional units are not sold.
5. **Treating DOL as a constant.** DOL is specific to a volume level: it falls as sales rise above break-even and is
   infinite at break-even.
6. **Ignoring the relevant range.** Doubling volume may need a second shift or a new warehouse, which adds fixed cost.
7. **Assuming the mix is fixed.** Multi-product break-even is valid only for the mix used; report a range of mixes.
8. **Averaging contribution margin ratios without weights.** The weighted CMR uses revenue weights, not a simple average
   of the product ratios.
9. **Confusing break-even with success.** Break-even covers operating costs, not interest, taxes, capital cost or the
   owner's opportunity cost.
10. **Reading CM per unit as the ranking rule under a constraint.** When capacity is limited, rank by contribution per
    unit of the constrained resource (mba/mba-relevant-costs-card).

## Self-test questions

<!-- Guidance: Answer from memory, then check the answer key. Repeat after two days and after a week. -->

1. Define contribution margin per unit and contribution margin ratio, and say when you would use each.
2. Why is the DOL at break-even undefined (infinite), and what does that tell a manager?
3. A shop has fixed costs of USD 500,000 and a CM ratio of 25%. What is break-even revenue? Give the Excel formula.
4. A firm's DOL is 4.0 at its current sales level. Sales fall by 5%. What happens to operating income?
5. CM per unit is USD 40, fixed costs are USD 800,000 and the pre-tax profit target is USD 200,000. How many units are
   needed?
6. In Worked example A, by how much must volume rise to keep operating income at USD 720,000 after the 5% price cut?
7. Explain in two sentences why a sales mix shift towards daypacks raised Northfield's trail-category break-even.
8. A colleague says: "Our cash break-even is lower than our accounting break-even, so we are safe." What is right and
   what is missing?
9. Give two ways in which outsourcing production changes the CVP profile of a business.
10. Which CVP assumption is most likely to fail for a new product launch, and how would you handle it in a case?

## My notes and next steps

<!-- Guidance: Write what you will do next, not what you read. One concrete action per line, with a date. -->

| Date | Note or action *(Example rows)* |
|---|---|
| YYYY-MM-DD | *Example:* Rebuilt Worked example A in Excel, all 15 results match; added a two-way Data Table for price and volume |
| YYYY-MM-DD | *Example:* Missed self-test 6 (forgot to divide by the new CM); logged in mba/mba-error-log |
| YYYY-MM-DD | [Next: read mba/mba-relevant-costs-card and do its Worked example A] |

## Sources and further reading

<!-- Guidance: Cite the primary and textbook sources in your own write-ups; never paste textbook problems into shared notes. -->

- Johnson, H. T., and Kaplan, R. S. (1987). *Relevance Lost: The Rise and Fall of Management Accounting*. Harvard
  Business School Press. (History of contribution and cost-behaviour analysis in management accounting.)
- Cooper, R., and Kaplan, R. S. (1988). Measure costs right: make the right decisions. *Harvard Business Review*,
  66(5). (Why fixed overhead is often driven by activities, a limit of simple CVP.)
- Textbooks, by name only: Datar, S. M., and Rajan, M. V., *Horngren's Cost Accounting: A Managerial Emphasis*
  (Pearson); Garrison, R. H., Noreen, E. W., and Brewer, P. C., *Managerial Accounting* (McGraw Hill); Zimmerman, J. L.,
  *Accounting for Decision Making and Control* (McGraw Hill).
- Open resource: OpenStax, *Principles of Accounting, Volume 2: Managerial Accounting* (CC BY-NC-SA 4.0),
  https://openstax.org/details/books/principles-managerial-accounting, chapter on cost-volume-profit analysis.
- Spreadsheet tools: Microsoft Support pages for Goal Seek and Data Tables (What-If Analysis).

## Answer key

<!-- Guidance: Check only after you have written your own answers. For each miss, write why your first answer was wrong. -->

**Worked example B.**

| Step | Manual | Automated | Excel formula (automated) |
|---|---|---|---|
| 1 | USD 120 | USD 150 | `=300-150` |
| 2 | 5,000 | 8,000 | `=1200000/(300-150)` |
| 3 | USD 2,400,000 | USD 2,550,000 | `=(300-150)*25000-1200000` |
| 4 | 1.25 | 1.47 | `=(300-150)*25000/((300-150)*25000-1200000)` |
| 5 | USD 1,200,000 (`=(300-180)*15000-600000`) | USD 1,050,000 | `=(300-150)*15000-1200000` |

6. $Q^{*} = (1{,}200{,}000 - 600{,}000)/(180 - 150) = 20{,}000$ tents (`=(1200000-600000)/(180-150)`), where both
   structures earn USD 1,800,000.
7. Above 20,000 tents the automated line earns more, because each extra tent adds USD 150 instead of USD 120; below
   it, the manual line wins because it carries USD 600,000 less fixed cost.
8. "At the expected 25,000 tents, automation earns USD 150,000 more (USD 2.55m versus USD 2.40m), but its break-even is
   8,000 tents instead of 5,000 and its profit falls faster if demand disappoints. Choose automation only if we are
   confident demand stays above 20,000 tents; otherwise start manual and automate once volume is proven."

**Exercise C.** (1) Weighted CM = 0.7 x 33 + 0.3 x 18 = USD 28.50 (`=SUMPRODUCT({33,18},{0.7,0.3})`).
(2) Break-even = 198,000 / 28.50 = 6,947.4, so 6,948 rental days: 4,864 ski and 2,085 snowshoe days (each rounded up
from 4,863.2 and 2,084.2); break-even revenue USD 270,947 (`=198000/28.5*(0.7*45+0.3*25)`). (3) (198,000 + 66,000) / 28.50
= 9,263.2, so 9,264 rental days: about 6,485 ski and 2,779 snowshoe days (rounded up from 6,484.2 and 2,778.9) (`=(198000+66000)/28.5`). (4) OI = 9,000 x 28.50 -
198,000 = USD 58,500; margin of safety (9,000 - 6,947.4) / 9,000 = 22.8%; DOL = 256,500 / 58,500 = 4.38. (5) Examples:
a snow-poor season shifts demand to snowshoes (lower CM, break-even rises); a ski-school partnership adds ski days
(higher CM, break-even falls).

**Self-test.**

1. CM per unit = P - V, used when units are meaningful (one product, a capacity plan). CMR = CM / P, used when working
   in revenue (many items, a store, a fixed mix).
2. At break-even OI = 0, so DOL = CM / 0 is undefined. Close to break-even, a tiny change in sales causes a very large
   percentage change in profit: the business is at its most fragile.
3. 500,000 / 0.25 = USD 2,000,000 (`=500000/0.25`).
4. Operating income falls by about 4 x 5% = 20% (exact when price and unit costs are unchanged).
5. (800,000 + 200,000) / 40 = 25,000 units (`=(800000+200000)/40`).
6. New CM is USD 42, so the volume for OI of USD 720,000 is (1,440,000 + 720,000) / 42 = 51,428.6, so 51,429 packs,
   an increase of 14.3% over 45,000. Shortcut: price cut per unit / new CM per unit = 6 / 42 = 14.3%
   (`=6/42`).
7. Daypacks have the lowest CM per unit (USD 27). More daypacks and fewer tents (USD 120) lower the weighted CM from
   USD 55.50 to USD 46.20, so more units are needed to cover the same fixed cost.
8. Right: cash break-even excludes non-cash fixed costs such as depreciation, so it is lower. Missing: debt service,
   capital expenditure, working capital needs and the return owed to investors; accounting losses eventually become
   cash needs when assets must be replaced.
9. It turns fixed production cost into variable purchase cost, which lowers break-even and DOL; and it usually lowers
   CM per unit, which lowers the profit gained from extra volume.
10. Constant price and volume certainty fail most often. Handle it with a scenario table (low, base, high), the
    margin of safety, and the indifference point between alternatives, and state which assumption drives the answer.

## Revision history

<!-- Guidance: Add a row each time you change the data, the analysis or the conclusion. -->

| Version | Date | Author | Change summary |
|---|---|---|---|
| 0.1 | YYYY-MM-DD | [Your name] | First draft from the template |
| 0.2 | YYYY-MM-DD | [Your name] | *Example:* Replaced Northfield data with the course case, added a mix scenario table |

## Standard and licence note

Based on the cost-volume-profit and contribution-margin analysis found in the management accounting literature listed
above (Johnson and Kaplan 1987; Cooper and Kaplan 1988) and in the standard managerial accounting textbooks named as
further reading. The text, the worked examples, the exercises and the questions are original; no textbook text or
problem is copied or paraphrased, and Northfield Outdoor Co. is a fictional company. This card is independent and not
affiliated with or endorsed by any business school, publisher or professional body. Check your course's notation and
your spreadsheet's current function documentation before you rely on it.
Template licence: MIT.