What Is a Break-Even Analysis? Three Firms Disagree

Three firms, three things a break-even analysis puts above the line

The denominator is uncontroversial. Contribution per unit is price minus variable cost per unit, and every published case computes it the same way. The numerator is where the cases diverge, and the divergence is not small.

A candidate who divides without settling this has produced a number whose meaning is unknown to everyone in the room. The fix costs one sentence, and it belongs in the section below.

Which costs are fixed, and over what horizon

None of the three cases prints a rule for the split. Each one simply assigns, and the assignment is the decision the formula hides. Bain names a cost to run the shop each year and a separate cost to open it, and treats the £1 cost per cup as the only thing that moves with volume. McKinsey calls $40 million "total fixed costs" that SuperSoda "needs to bear" to launch, and puts the $1.90 to produce and deliver a bottle on the variable side. Oliver Wyman never draws the line at all, because it never divides.

Whether a given cost is genuinely variable at the volume under discussion is a question about the client's operations, and no formula settles it. Staff hours are variable over a year and fixed over a week. Rent is fixed until the lease renews. A delivery van is a fixed cost until volume needs a second one, and then the cost steps.

So ask which horizon the client has in mind, and if the case does not say, state the horizon you assumed. The horizon decides the split, the split decides the numerator, and the numerator decides the answer. That chain is why three published cases can disagree while each one stays internally correct.

Bain's coffee shop, recomputed to 561 cups a day

The Coffee Shop Co. case hands over four figures. Price per coffee £3, cost to open the shop £245,610, cost to run the shop each year £163,740, cost per cup £1. The firm then prints its own working, which is rare enough to be worth noting. It sets revenue of 3Q against the sum of the two costs plus 1Q and puts the whole expression at zero, reduces it to "2Q - 409,350 = 0", and lands on "Q = 204,675 cups to breakeven".

Redo it in the order the page poses it. Contribution per cup is £3 minus £1, so £2. The two costs above the line sum to £163,740 plus £245,610, which is £409,350. Divide: £409,350 over £2 per cup gives 204,675 cups. The firm's figure holds exactly.

Now read what that number is. The opening cost is a one-time outlay and it has been charged in full against a single year of trading. So 204,675 cups is the volume at which the owner recovers her capital within twelve months of opening. That is a demanding test and a defensible one. It is also not what most readers assume a break-even volume means.

Per day, 204,675 cups over 365 days is 560.75, so call it roughly 561 cups a day. Saying that out loud is worth more than the annual figure, because a reader can picture 561 cups and cannot picture 204,675.

The opening cost multiplies the volume by two and a half

Charge only the year of trading and the arithmetic moves a long way. £163,740 over £2 per cup is 81,870 cups, which is 224.3 a day, so roughly 224 cups. The gap between the two volumes is 204,675 minus 81,870, which is 122,805 cups, and that is exactly £245,610 divided by £2, as it has to be.

The ratio is the number to carry. 409,350 over 163,740 is 2.5 exactly. Bain's convention multiplies the volume you need by two and a half on this case, and the multiplier is simply the ratio of total first-year outlay to annual operating cost. On a capital-heavy venture it can be far larger than two and a half.

Bain writes the caveat itself. Its better answer says a stronger response "would take into account how long it would take for her to recoup the cost to open the store", and adds that "we'd want to understand more about our friend's investment timeline". Read next to the published equation, that is a firm noting that its own break-even has already made an assumption about the timeline nobody stated.

McKinsey's Electro-Light, where no period is attached

The Electro-Light case gives a 16-ounce bottle, "one-eighth of a gallon", priced at $2 to retailers, costing "$1.90 to produce and deliver", against "$40 million in total fixed costs" that SuperSoda "needs to bear" to launch.

Contribution per bottle is $2.00 minus $1.90, so $0.10. Divide $40,000,000 by $0.10 and the break-even is 400,000,000 bottles. At an eighth of a gallon each, that is 400,000,000 over 8, or 50,000,000 gallons.

Notice what the question does not say. It asks what share of the market is needed "to break even" and never names a year. The $40 million is described as the cost of being able to launch, so it is a one-time burden, and 400 million bottles is therefore a cumulative volume. Sell it in one year and the venture repays its launch in one year. Sell it over four and the same volume means something much weaker. The case is silent, which means the candidate has to choose and say so.

One more thing the page is silent about. The size of the electrolyte category lives in an exhibit image we were not able to read, so the market share this volume represents cannot be computed from the text. We do not compute it. The question's own target of 12.5 percent implies a category of 400 million gallons if 50 million gallons is 12.5 percent of it, and that implication is ours rather than the firm's.

Contribution per unit, then the same margin as a ratio

Work the denominator on the two cases that divide before touching either numerator. At the coffee shop, £3 of price less £1 of variable cost leaves £2 of contribution a cup. At SuperSoda, $2.00 less $1.90 leaves $0.10 a bottle. Both are the absolute form, money per unit, and it is the only form this page has used so far.

Now divide each one by its own price. £2 over £3 is two thirds, so 66.7 percent of every pound the shop takes is still there to cover fixed cost. $0.10 over $2.00 is one twentieth, so 5 percent. The two cases run the same arithmetic on volume and look nothing alike on this line, and a reader carrying only the unit figures would never see it.

The ratio earns its place twice over. Units cancel out of it, so cups, bottles and gallons stop blocking the comparison between two cases. And it is the only input a break-even in money needs, which is the next section. Say which of the two you are quoting as you quote it. A contribution of £2 and a contribution of 66.7 percent describe the same cup, and a listener who hears the second and writes down the first has lost the price.

The same break-even, expressed in revenue

Divide fixed cost by the margin ratio and the answer comes back in money. Multiplying by the reciprocal of that ratio is the same operation, and it survives being done by hand.

The coffee shop first, carrying the full first-year numerator. Two thirds inverts to one and a half, so £409,350 multiplied by 1.5 is £614,025 of revenue. Check it against the volume already computed: 204,675 cups at £3 is £614,025. Charge only the year of trading and £163,740 multiplied by 1.5 is £245,610, which is 81,870 cups at £3.

That second figure equals the published cost to open the shop, to the pound, and the coincidence is pure arithmetic: £245,610 happens to be exactly one and a half times £163,740, and one and a half is what this margin ratio inverts to. Read nothing into it.

Electro-Light next. Five percent inverts to twenty, so $40 million multiplied by 20 is $800 million of revenue, and 400 million bottles at $2 is $800 million.

The reciprocal is the sentence a client recognizes. Every pound of fixed cost at the coffee shop needs one pound fifty of revenue standing behind it, and every dollar of SuperSoda's launch cost needs twenty. The same fixed burden is 13.3 times heavier to cover at the bottling margin, because 20 over 1.5 is 13.3.

Judging the volume against the market you were handed

Bain's recommendation reads: "Because there are 7,000,000 cups sold and the break-even point is 204,675 cups, which is ~3% of the market share, breaking even is achievable." Check it. 204,675 over 7,000,000 is 2.92 percent, and the tilde in front of the 3 declares the rounding, so the claim is sound. The operating-only volume is 81,870 over 7,000,000, which is 1.17 percent.

The text of the case, though, gives 100,000 residents drinking one cup a day. That is 100,000 times 365, or 36,500,000 cups a year. Against that base, 204,675 cups is 0.56 percent. The 7,000,000 figure is roughly 19 percent of 36,500,000, and the steps that narrow one to the other sit in an image we could not read. A candidate working from the printed text alone will not reach the share the page announces. Both readings are arithmetically fine on their own base, and we leave them side by side rather than deciding which one the firm meant.

The transferable habit here is a sentence rather than a figure. State the base before the percentage. Two point nine percent of the addressable market and just over half a percent of total consumption are the same volume described against different denominators, and only one of them is the market the owner can actually reach.

What a small margin cut does to the volume you need

This is where commentary goes wrong most often, so hold the two percentages apart. Suppose the coffee shop drops its price to £2.75. Contribution falls from £2.00 to £1.75. Keeping the full first-year numerator, the break-even becomes £409,350 over £1.75, which is 233,914 cups, so roughly 233,900.

Now the three percentages, each measured on its own base. The price fell by £0.25 on £3.00, which is 8.3 percent. The contribution fell by £0.25 on £2.00, which is 12.5 percent. The volume you need rose from 204,675 to 233,914, which is 14.3 percent. Three different numbers from one price change, and quoting the wrong one is how a correct calculation becomes a wrong recommendation.

The same cut read in the revenue form. At £2.75 the ratio is £1.75 over £2.75, which is 63.6 percent, and the break-even revenue becomes £643,264, because 233,914 cups at £2.75 is £643,264. The volume needed rose 14.3 percent while each cup now sells for 8.3 percent less, and 1.143 times 0.917 is 1.048, so the revenue needed rose 4.8 percent. Quote the form along with the base.

The rule behind it is worth memorizing, because it makes the third number arrive without arithmetic. If contribution falls by a fraction f, required volume rises by f divided by (1 minus f). Here f is 0.125, so 0.125 over 0.875 is 0.143. A margin cut always costs more volume than the cut itself, and the gap widens fast as f grows.

How we score a break-even answer, and whose scale this is

No firm publishes a marking scheme for quantitative work. The hundred points below are ours, written by us, and read out of src/lib/grading/rubrics.ts. They are what our own grader measures. No consulting firm has seen them, and none of them describes its own weighting anywhere we have read.

The four moves before you divide

Run them in this order and the division at the end is uncontroversial.

1. Compute contribution per unit and say it aloud. Price minus variable cost, one number, before anything else enters.
2. List the costs you propose to recover, and name the period. One year of trading, the capital, or both.
3. Divide, then state the volume twice. Per year for the record, per day for the interviewer.
4. Test it against a base you can defend. Name the denominator before the percentage.

Then ask which costs the client wants recovered, and over what period. Phrase it as a choice rather than a request for help: "I will treat the opening cost as recovered inside year one, which makes this a capital-payback volume. If you would rather see the operating break-even I will drop it and the number falls by roughly three fifths." On the coffee shop, dropping it moves 204,675 to 81,870, and 81,870 is 40 percent of 204,675, so the fall is 60 percent. That is the conversation the three firms' conventions are silently having, and saying it out loud puts you on the right side of it.

Where the answer is a decision about capital over time rather than a volume, break-even stops being the right tool. Discounted cash flow handles the case where timing changes the sign of the answer, and return and payback handles the case where the client has named a time criterion, as Oliver Wyman's published case does when it requires a five-year payback and never computes a break-even at all.

Drilling the division

Work both numerators on every case you meet. Compute the operating break-even, then the one that recovers the capital, then say which one you are recommending on and why. The habit takes twenty seconds and it carries most of the Interpretation dimension on its own.

Then check your own direction of error. Most people are consistent about which costs they forget. Some always drop a one-time outlay out of the numerator and produce a volume that looks easy. Others sweep depreciation and allocated overhead in and produce a volume nobody could reach, then recommend against a venture that was fine. Finding out which of the two you are is worth more than another twenty break-even questions, because the fix is a checklist rather than practice.

The weights above and the reasoning behind them sit on how an answer is assessed. The math drill reports the four dimensions separately after each attempt and names the one to fix first, so a Units problem stops being mistaken for an Answer problem. Nothing here is behind a sign-up, and the arithmetic above is reproducible from the three firm pages linked in the text.