Cost-Volume-Profit & Break-Even

Financial Performance & Management Control - TUHH Institute of Management Accounting & Simulation, Hamburg · part of my Technology Management MBA · study notes for revision.

Chapter 8 sorted costs into fixed and variable buckets. This chapter puts those buckets to work. Cost-volume-profit analysis (CVP), which most people just call break-even analysis, is the planning tool that tells me how many units I must sell before a project stops losing money, how profit reacts when volume moves, and which cost structure I should choose when someone offers me a flat fee versus a percentage deal.

The maths is deliberately simple - one price, one variable cost per unit, one block of fixed costs. That simplicity is the point: it is a first cut that turns a vague business idea (an internet café in Kingston, a software booth at a convention) into a single number I can argue about. The second half of the chapter shows the same “only the costs that change matter” logic applied to two classic decisions: closing a loss-making division and accepting a below-cost special order.

1 · What CVP analysis looks at

CVP studies how three totals behave - total revenues, total costs and operating profit - when I push on one of three levers: the output level (how many units), the selling price, or the cost structure (variable and fixed costs). Because it links the levers to the totals with a formula, it can answer what-if questions instantly: what happens to revenues and costs if we produce more? What if we drop the price? What if the landlord swaps a fixed rent for a share of sales?

Move a leverunits sold, selling price, variable or fixed costs

→linear formula

Totals respondtotal revenues and total costs

→the difference

Operating profitzero at the break-even point

CVP in one line: change an input, read off the new revenues and costs, and see what is left as operating profit. The break-even point is the output where that leftover is exactly zero.

The course frames it as an important planning tool: before committing money, I want to know the volume at which revenues finally cover total costs - beyond that point every extra unit is profit.

2 · The assumptions - and the special case they describe

The model is only as honest as its simplifications. The standard CVP rests on five:

Total costs split into a fixed and a variable part - nothing in between.
Revenues and costs behave linearly - a straight line per unit, no volume discounts, no step costs.
A single product (or a constant sales mix) is analysed.
The time value of money is ignored - a euro next year counts like a euro today.
Production equals sales - no inventory building up or being run down.

Those assumptions describe a special case of CVP. The course is explicit that the general case needs more sophisticated maths:

Special case what this chapter solves

One revenue driver: output units
One cost driver: output units
Short-run decisions - a time span, typically under a year, in which fixed costs stay fixed inside the relevant range

General case needs heavier tools

Many revenue drivers (mix, channels, customer groups)
Many cost drivers (batches, set-ups, complexity)
Various time spans: short run, long run, whole product life cycles

Keep the relevant range idea from the last chapter in mind: fixed costs are only “fixed” between the volumes the business was set up for. Step outside that band and the whole line shifts.

3 · The vocabulary and the core formulas

The abbreviations show up in every exercise, so I want them automatic:

Symbol	Stands for	In plain words
USP	unit selling price	what one unit sells for
UVC	unit variable cost	what one extra unit costs to make and sell
UCM	unit contribution margin	USP minus UVC - what one unit contributes to fixed costs and profit
FC	fixed costs	the block that does not move with volume (direct or indirect)
Q	quantity	units sold (equals units made, by assumption)
TOP	target operating profit	the profit I want to plan for
NP	net profit	what is left after non-operating items and tax

Two accounting identities sit underneath. Total costs are variable plus fixed costs (each of which can be direct or indirect). Operating profit is total revenues from operations minus total costs from operations. Net profit takes operating profit, adds non-operating revenues, subtracts non-operating costs and then income tax - but in this chapter the non-operating items are assumed to be zero, so OP is the number we chase.

Operating profitOP = (USP × Q) - (UVC × Q) - FC

Unit contribution marginUCM = USP - UVC

Units for any target profitQ = (FC + TOP) / UCM

Break-even (TOP = 0)Q = FC / UCM

The contribution margin is revenues minus variable costs - the course’s one-line takeaway. It is not profit; it is what is available to cover fixed costs, and only the excess becomes profit.

4 · Finding the break-even point three ways

The break-even point is the output quantity at which total revenues equal total costs, so operating profit is zero. The course gives three routes to it - all land on the same answer, so pick whichever the question makes easiest.

The running example. A seller plans to offer a software package at a two-day computer convention. She can buy each package from a wholesaler for 120 with the right to return every unsold unit for a full 120 refund, so the purchase cost is purely variable. She sells at 200. She has already paid 2,000 for the booth. No other costs. How many units to break even?

Equation method. Write operating profit in full and set it to zero: 200 × Q - 120 × Q - 2,000 = 0. That gives 80 × Q = 2,000, so Q = 25 units.
Contribution-margin method. Compute UCM first: 200 - 120 = 80 per package. Then divide the fixed costs by it: 2,000 / 80 = 25 units. Same answer, one line.
Graph method. Plot units on the horizontal axis and money on the vertical. Because everything is linear, two points per line are enough. The total cost line starts at the fixed-cost point (0 units, 2,000) and its second point is FC plus units × UVC. The total revenue line starts at the origin (0, 0) and its second point is units × USP. Where the two lines cross is the break-even point - here at 25 units and 5,000 of revenue.

Total cost linestarts at FC, slope = UVC

→crosses

Break-even pointrevenues = total costs, OP = 0

←crosses

Total revenue linestarts at the origin, slope = USP

The graph method. Left of the intersection the cost line sits above the revenue line (a loss); right of it the revenue line is higher (a profit). The vertical gap between the lines at any volume is the operating profit or loss.

Target profit is the same trick. If the seller wants an operating profit of, say, 1,200 from the convention, she needs (2,000 + 1,200) / 80 = 40 units. The formula Q = (FC + TOP) / UCM treats the profit target as if it were one more fixed cost to be covered.

5 · Sensitivity analysis - alternative cost structures

CVP shines when the question is “which cost structure?”. Sensitivity analysis means changing one input at a time and watching the break-even and the profit move. The convention organiser offers the seller three booth deals:

Option 1: a 2,000 fixed fee.
Option 2: a 1,400 fixed fee plus 5% of revenue from the software sales.
Option 3: no fixed fee, but 20% of revenue.

A percentage of revenue is a variable cost per unit in disguise: 5% of a 200 price is 10 per unit, 20% is 40 per unit. So the options are really three different (FC, UVC) pairs:

Option 1all fixed

FC 2,000 · UVC 120UCM = 80

Break-even 25 units2,000 / 80

Option 2mixed

FC 1,400 · UVC 130UCM = 70

Break-even 20 units1,400 / 70

Option 3all variable

FC 0 · UVC 160UCM = 40

Break-even 0 unitsnever loses money

The three booth deals as CVP structures. Moving from left to right the fixed block shrinks, the break-even falls, and the downside risk vanishes - but so does the upside, because each unit now contributes less.

The trade-off is the whole lesson: lower fixed costs mean a lower break-even and less risk, but a thinner margin per unit once you are past it. My own working on where the options swap places: option 1 beats option 2 only above 60 units (80Q - 2,000 = 70Q - 1,400), option 2 beats option 3 above about 47 units (70Q - 1,400 = 40Q), and option 1 beats option 3 above 50 units. So a pessimistic seller wants option 3, a confident one option 1, and option 2 wins in a narrow middle band. The recommendation depends entirely on how many units she believes she will sell - which is exactly what sensitivity analysis is for.

6 · Operating leverage - how hard profit swings

Operating leverage describes the effect fixed costs have on operating profit when units sold (and therefore the contribution margin) change. At a given volume:

Degree of operating leverageDOL = contribution margin / operating profit

The course evaluates the three options at 40 units:

At 40 units	Option 1	Option 2	Option 3
Contribution margin per unit	80	70	40
Contribution margin (40 units)	3,200	2,800	1,600
Operating profit (40 units)	1,200	1,400	1,600
Degree of operating leverage	2.67	2.0	1.0

Read the leverage as a multiplier: with a DOL of 2.67, a 10% change in units moves operating profit by roughly 26.7% (from 40 to 44 units, OP goes from 1,200 to 1,520). With option 3 there are no fixed costs, so profit moves exactly in step with volume - a DOL of 1. High fixed costs mean high leverage: small changes in sales cause large changes in operating profit, in both directions. That is why the leverage tells me about the profit potential beyond break-even and, equally, about the pain below it.

high fixed costs = high leveragehigh leverage = big swingsno fixed costs = DOL of 1leverage is volume-specific

7 · Use and misuse of CVP

Where CVP is genuinely useful

Environments where costs really are a function of volume - mass production systems
Special projects with separable, incremental fixed costs: adding a new product, running a one-off marketing campaign
As an input into setting a selling price
To compare the effect of alternatives - the three booth deals, the three speaker fees below

Where it misleads

The relevant range limits the model - costs are linear only over a narrow band of volumes
Selling prices vary by customer or market group; one USP is a fiction
It can breed the mindset that fixed costs cannot be managed - they can, just not by volume
Treating it as the final decision model rather than a first cut

8 · Relevant-cost thinking - only the costs that change matter

The break-even logic extends naturally to yes/no decisions. The rule: when comparing two courses of action, look only at revenues and costs that differ between them. Allocated overheads that carry on regardless, and fixed costs already committed, are noise.

The division-closure trap

A company has two divisions and is thinking of shutting the Northern sales office. Divisional fixed costs would disappear with the division; allocated corporate costs would continue.

	Northern	Southern	Total
Sales	550,000	500,000	1,050,000
Variable costs	275,000	200,000	475,000
Divisional fixed costs	180,000	150,000	330,000
Contribution to corporate costs	95,000	150,000	245,000
Allocated corporate costs	170,000	135,000	305,000
Operating profit	-75,000	15,000	-60,000

Northern “loses” 75,000 on paper, so closing it looks tempting. But after closure the full 305,000 of corporate costs still has to be paid, now by Southern alone: 150,000 contribution minus 305,000 corporate costs = -155,000. The group result worsens from -60,000 to -155,000, because Northern was quietly contributing 95,000 towards corporate costs. Do not close it. A segment with positive contribution above its own avoidable fixed costs is helping, however ugly its allocated-cost line looks.

The one-off special order

A canner sells 1,000 crates of peaches a month at 10 each. Costs are classed as manufacturing or marketing, each with a fixed and a variable part:

Monthly	Fixed	Variable	Per crate (at 1,000 crates)
Manufacturing	2,000	4,000	4.00 variable + 2.00 fixed = 6.00 absorption cost
Marketing	1,600	1,400	1.40 variable

So the income statement reads: sales 10,000, total variable costs 5,400, contribution margin 4,600, fixed costs 3,600, operating profit 1,000. The normal mark-up is 4,000 / 6,000 = 66.7% on absorption cost, or 4,600 / 5,400 = 85.2% on total variable cost.

A new customer wants 200 crates at 5.50, needs no extra marketing, but requires 200 of special packaging. The customer is winding down in six weeks, so it is a genuine one-off. Management wants to refuse because 5.50 is below the 6.00 absorption cost. Is that right?

Relevant costs are the ones this order creates: 200 × 4 = 800 of variable manufacturing plus 200 of packaging = 1,000, i.e. 5.00 per crate.
Any price above 5.00 adds contribution; at 5.50 the order adds 200 × 0.50 = 100 to operating profit.
The 6.00 absorption figure smuggles in 2.00 per crate of fixed manufacturing cost that is paid anyway - irrelevant here.
Capacity is 1,500 crates and the relevant range for fixed manufacturing cost is 500 to 1,500; the order lifts output from 1,000 to 1,200, so fixed costs really do stay fixed.

Accept. But if the customer then decides to stay in business, the strictly short-run lens stops being appropriate: other customers may learn about the 5.50 price and demand it, and a permanent low-price customer is a pricing decision, not a spare-capacity decision.

Worked example

(a) Caribbean Internet Café - is the venture feasible?

A returning MBA graduate plans an internet café in New Kingston, Jamaica (all figures in Jamaican dollars). He has 500,000 of savings; the telecom company offers to invest a further 500,000 for a 50% share and lend 1,250,000 at 10%. Equipment costs 1,426,000. The group solution builds the CVP model from the case data:

Fixed cost item (per year)	JA$	Where it comes from
Student staff wages	374,400	2 on duty × 90 hours a week × 40 an hour × 52 weeks
Manager	480,000	40,000 a month
Lease	360,000	30,000 a month
Telephone and utilities	180,000	15,000 a month
Internet link	120,000	10,000 a month
Insurance	120,000	10,000 a month
Advertising	120,000	10,000 a month
Miscellaneous admin and maintenance	600,000	50,000 a month
Interest on the loan	125,000	10% of 1,250,000
Depreciation of equipment	475,333	1,426,000 straight-line over 3 years
Total fixed costs	2,954,733

On top come one-off start-up costs of 147,000 (utility deposit 7,000, pre-opening advertising 20,000, legal, licences and decorating 120,000).

Revenue per visit 248

Computer time 48 - only 40% of visitors pay the 120 hourly rate
Food 60
Drinks 140

Variable cost per visit 104

Internet 24 - 40% of visitors × 60 per computer-hour paid to the provider
Food 30
Drinks 50

Unit contribution margin 144

248 - 104 = 144 per visit
Every visit pays 144 towards the fixed block

Break-even visits

Year one: (2,954,733 + 147,000) / 144 ≈ 21,540 visits, about 64 customers a day
Year two (no start-up costs): 2,954,733 / 144 ≈ 20,519 visits, about 61 a day

A neat side-calculation from the group: the first customer effectively costs about 3.1 million (all fixed costs plus start-up plus 104) - a vivid way to feel the fixed block.

Scenarios. The market study agrees on a target segment of 20,000 people and offers three usage estimates:

Scenario	Share of segment	Visits per person	Visits per year	Year-1 operating profit	Year-2 operating profit
Best	50%	5	50,000	about +4.10m (4,098,267)	about +4.25m
Average	40%	3	24,000	about +0.35m (354,267)	about +0.50m
Worst	30%	2	12,000	about -1.37m (-1,373,733)	about -1.23m

The average case only just clears break-even (24,000 visits against 21,540 needed); the worst case is a heavy loss. Averaging the three outcomes with equal weights gives a “probable” operating profit of roughly 1.03m in year one, of which the founder’s 50% share is about 0.51m - but the case gives no probabilities, so that average is a guess. The group’s conclusion: feasibility depends on how risk-averse the founder is, plus qualitative factors (see the case corner).

(b) The motivational speaker - which fee schedule?

A law association wants a speaker for an all-day seminar and is offered three fee arrangements: Schedule 1 a flat 8,000; Schedule 2 2,000 fixed plus 20 per person; Schedule 3 50 per person. Attendees each pay 200.

Attendees	Schedule 1 (8,000 flat)	Schedule 2 (2,000 + 20 each)	Schedule 3 (50 each)	Cheapest
50	8,000 total · 160 per head	3,000 · 60	2,500 · 50	Schedule 3
200	8,000 · 40	6,000 · 30	10,000 · 50	Schedule 2
500	8,000 · 16	12,000 · 24	25,000 · 50	Schedule 1

The pattern is the booth story again. The all-fixed schedule has a falling unit cost (160 → 40 → 16) and wins when the room is full; the all-variable schedule has a constant 50 per head and wins when attendance is small and uncertain; the mixed schedule wins in between. My own crossover arithmetic: Schedule 2 overtakes Schedule 3 above about 67 attendees, and Schedule 1 overtakes Schedule 2 above 300. Because each attendee pays 200, every schedule is covered at every size shown - the choice is about how much risk the association wants to carry if turnout disappoints.

Case corner

Caribbean Internet Café. The case is a full feasibility study run through CVP. The break-even of roughly 21,500 visits a year (about 64 a day on a 50-seat floor) is the anchor; the three market scenarios then show that the venture is comfortably profitable if half the segment visits five times a year, marginal in the realistic case, and loss-making if only 30% turn up twice. The 50% partner changes the picture: the telecom company puts in 500,000 of equity and 1,250,000 of debt, so the founder shares both the profit and the risk, and the partner also earns 125,000 of interest a year plus the internet charges regardless of the café’s result. The qualitative issues the group flagged sit outside the model: an IT and internet bubble (the case is set in 1996 to 1997), the state of Jamaica’s telecom access (scarce phone lines in parts of Kingston, high provider rates, low private usage expected for at least three years), and whether Jamaica’s economy is on the rise. Also worth noting: no competitor offered internet access, and coffee was not a traditional drink - both an opportunity and a demand risk.

The division-closure trap. Whenever a segment shows a loss after allocated corporate costs, ask what would actually disappear if it were closed. If its contribution exceeds its own avoidable fixed costs, closing it makes the company poorer - the allocated costs simply land on whoever is left.

Key terms

Term	What it means in plain words
CVP analysis	Planning tool showing how revenues, costs and operating profit respond to changes in volume, price or cost structure
Break-even point	The output where total revenues equal total costs and operating profit is zero
Unit selling price (USP)	The price one unit sells for
Unit variable cost (UVC)	The cost of making and selling one more unit
Unit contribution margin (UCM)	USP minus UVC - what each unit gives towards fixed costs and profit
Contribution margin	Total revenues minus total variable costs
Fixed costs (FC)	Costs that do not change with volume inside the relevant range
Relevant range	The band of volumes within which the fixed and variable cost assumptions hold
Target operating profit (TOP)	The planned profit; units needed = (FC + TOP) / UCM
Operating vs net profit	Operating profit is revenues minus costs from operations; net profit also deducts non-operating items and tax
Sensitivity analysis	Changing one input at a time to see how break-even and profit move
Operating leverage	Contribution margin divided by operating profit at a given volume - how strongly profit reacts to volume
Relevant cost	A cost that differs between the alternatives being compared
Absorption cost	Full unit cost including a share of fixed manufacturing cost - a poor guide for one-off orders

Test yourself

A seller buys packages at 120 with full return rights, sells them at 200 and has paid 2,000 for a booth. Find the break-even quantity using the contribution-margin method, and state the break-even revenue.
Under the same deal, how many units must she sell to earn an operating profit of 1,200? And under Option 2 (1,400 fixed plus 5% of revenue)?
At 40 units under Option 1, compute the degree of operating leverage and use it to estimate the operating profit if sales rise by 10%.
Northern division shows an operating loss of 75,000 after allocated corporate costs of 170,000. Explain in numbers why closing it would hurt the company.
The canner is offered 200 crates at 5.50 when its absorption cost is 6.00 per crate. Should it accept? Give the relevant cost per crate and the change in operating profit.
The internet café has fixed costs of 2,954,733, start-up costs of 147,000 and a UCM of 144 per visit. What is the break-even in year one, and why does it fall in year two?

UCM = 200 - 120 = 80; Q = 2,000 / 80 = 25 units; break-even revenue = 25 × 200 = 5,000.
Option 1: (2,000 + 1,200) / 80 = 40 units. Option 2: UVC becomes 130, UCM 70, so (1,400 + 1,200) / 70 ≈ 37.1, i.e. 38 whole units.
CM = 40 × 80 = 3,200; OP = 3,200 - 2,000 = 1,200; DOL = 3,200 / 1,200 = 2.67. A 10% rise in units lifts OP by about 26.7%, to roughly 1,520 (check: 44 × 80 - 2,000 = 1,520).
Northern contributes 550,000 - 275,000 - 180,000 = 95,000 towards corporate costs. Those 305,000 of corporate costs continue after closure, so Southern alone shows 150,000 - 305,000 = -155,000 against a combined -60,000 today. Keep it open.
Yes. Relevant cost = 4.00 variable manufacturing + 1.00 packaging (200 / 200) = 5.00 per crate; at 5.50 the order adds 200 × 0.50 = 100 to operating profit. The 2.00 fixed share in the 6.00 absorption cost is irrelevant, and 1,200 crates stays inside the 500 to 1,500 relevant range.
Year one: (2,954,733 + 147,000) / 144 ≈ 21,540 visits (about 64 a day). Year two the one-off start-up costs are gone, so 2,954,733 / 144 ≈ 20,519 visits.

Revision summary

The one-minute recap

CVP (break-even) analysis shows how total revenues, total costs and operating profit react to changes in volume, price and variable or fixed costs, and answers what-if questions. A planning tool, a first cut.
Assumptions: costs split into fixed and variable, linear behaviour, single product, no time value of money, production equals sales. That is the special case (one revenue driver, one cost driver, short run inside the relevant range); the general case needs more.
UCM = USP - UVC; contribution margin = revenues minus variable costs. OP = USP × Q - UVC × Q - FC.
Break-even three ways: equation (set OP to zero), contribution margin (Q = FC / UCM), graph (cost line from FC, revenue line from the origin, intersection). Target profit: Q = (FC + TOP) / UCM. Software booth: UCM 80, break-even 25 units.
Sensitivity analysis: 2,000 fixed → BE 25; 1,400 + 5% → UCM 70, BE 20; 20% only → UCM 40, BE 0. Less fixed cost = lower break-even, less risk, thinner margin.
Operating leverage = CM / OP at a volume: at 40 units 2.67 / 2.0 / 1.0. High fixed costs = profit swings harder with volume.
Use it where costs follow volume, for separable projects, pricing input and comparing alternatives; beware the relevant range, varying prices, the “fixed costs cannot be managed” mindset, and treating it as the final word.
Relevant costs: only what changes matters. Do not close a division that covers its own fixed costs (Northern contributes 95,000; closing turns -60,000 into -155,000). Accept a one-off order above incremental cost (5.00, not the 6.00 absorption cost) - unless the customer becomes permanent.
Caribbean Internet Café: fixed about 2.95m plus 147,000 start-up, UCM 144, break-even about 21,540 visits (64 a day); best +4.1m, average +0.35m, worst -1.37m; the answer depends on risk appetite and qualitative factors.

Next: Costing & Pricing → - from what a product costs to what it should sell for.