How far sales can fall; what a price, VC or FC change does to break-even

◈ 10 cards

Compute margin of safety in dollars, units and as a ratio, and restate break-even and operating income after a change in price, variable cost or fixed cost — one at a time, with the rest held.

The cushion

Northlake Nordic Centre Inc. expects 20,000 skier-days and breaks even at 15,000. The margin of safety is the distance between the two — how far sales can fall before the season loses money — and it is quoted three ways:

$Margin of safety = Actual (or expected) sales - Break-even sales$

in dollars: 600,000 − 450,000 = 150,000
in units: 20,000 − 15,000 = 5,000 skier-days
as a ratio: 150,000 ÷ 600,000 = 0.25 — a quarter of the expected season could vanish before a loss

The ratio's denominator is actual sales, not break-even, and the subtraction runs actual minus break-even — a negative margin of safety means the business is already below break-even. A bad-snow year that costs Northlake 4,000 skier-days still leaves 1,000 of cushion; one that costs 6,000 does not.

One variable at a time

The owner has three levers, and the paper tests each separately, with everything else held constant — say so in the answer, because the marker wants the assumption stated.

(a) Raise the price 10 %, to 33. Variable cost stays 9, so CM per skier-day rises to 24 and the ratio to 24 ÷ 33 = 0.727. Break-even falls: 315,000 ÷ 24 = 13,125 skier-days. If 20,000 skier-days are still sold, operating income is 24 × 20,000 − 315,000 = 165,000 — up 57.1 % on a 10 % price rise, because the whole 3 goes to contribution.

(b) Variable cost rises 1.50, to 10.50 (a fuel price rise). CM per skier-day falls to 19.50, the ratio to 19.50 ÷ 30 = 0.65. Break-even rises: 315,000 ÷ 19.50 = 16,153.8 → 16,154 skier-days. At 20,000 skier-days, operating income is 19.50 × 20,000 − 315,000 = 75,000.

(c) Fixed costs rise 21,000, to 336,000 (a rental-fleet lease). CM per skier-day is still 21 and the ratio is still 0.70 — a fixed cost touches neither. Break-even rises: 336,000 ÷ 21 = 16,000. At 20,000 skier-days, operating income is 420,000 − 336,000 = 84,000.

The direction table

Change	CM per unit	CM ratio	Break-even	OI at 20,000
Price ↑	↑	↑	↓	↑
Variable cost ↑	↓	↓	↑	↓
Fixed cost ↑	—	—	↑	↓

The row to remember is the last: a fixed-cost change moves break-even and income but leaves the ratio alone, so a case that reports a changed CM ratio is telling you the price or the variable cost moved. And never compare two scenarios' incomes without rebuilding both statements; a percentage read off one line will mislead.

The caveat the owner needs

Scenario (a) held volume at 20,000. A price rise may cut skier-days. The useful number is the volume at which the new price gives the same 105,000 as today: (105,000 + 315,000) ÷ 24 = 17,500 skier-days. The owner can raise the price to 33 and lose up to 2,500 skier-days before being worse off — that is the sentence a recommendation carries (L10.7).

One lever at a time, everything else held. The fixed-cost row leaves the ratio at 0.70 — the tell that separates it from the other two.