Add the target to fixed costs; gross up an after-tax target first

A target is a fixed cost with a name

Break-even asks how many skier-days cover 315,000. Northlake Nordic Centre Inc.'s owner asks a harder one: how many skier-days give an operating income of 147,000? The contribution margin must now cover the fixed costs and the target, so the target goes into the numerator beside the fixed costs:

$$\text{Units for a target}=\frac{\text{Fixed costs}+\text{Target operating income}}{\text{CM per unit}}=\frac{315,000+147,000}{21}=22,000\text{ skier-days}$$

Which is the +10 % season of L9.6, arrived at from the other end. The target is added to fixed costs, not to revenue; it is a sum the contribution has to reach, and every skier-day still contributes only 21 toward it. In dollars: 22,000 × 30 = 660,000, or (315,000 + 147,000) ÷ 0.70 = 660,000.

The bank wants net income, after tax

The loan covenant is written on net income — after tax — and the bank wants 100,800. Northlake pays income tax at 20 %. The profit equation produces operating income, which is taxed; so the after-tax figure must first be translated into the pre-tax figure that yields it. If tax takes 20 %, net income is 80 % of pre-tax income:

$$\text{Pre-tax target}=\frac{\text{After-tax target}}{1-t}=\frac{100,800}{0.80}=126,000$$

The gross-up is a division. Multiplying — 100,800 × 0.80 = 80,640 — goes the wrong way and lands short of the target. Now the pre-tax 126,000 is an ordinary target:

$$\frac{315,000+126,000}{21}=21,000\text{ skier-days}\Rightarrow 21,000\times 30=630,000\text{ of sales}$$

And the proof, which is where the tax line finally appears:

Sales                    21,000 × 30     630,000
Less: variable costs     21,000 × 9      189,000
Contribution margin      21,000 × 21     441,000
Less: fixed costs                        315,000
Operating income                         126,000
Income tax at 20 %                        25,200
Net income                               100,800

Tax is 20 % of operating income — never of fixed costs, never of sales. The statement lands on exactly the 100,800 the bank asked for, which is the check that the gross-up was done right.

Reading the case

When a case gives a tax rate and says "target profit" or "net income", the target is after tax and must be grossed up. When it says "operating income" or gives no tax rate, use the target as it stands. Cedar Ridge Golf Club Ltd. wants net income of 78,000 at a 25 % rate: pre-tax 78,000 ÷ 0.75 = 104,000; (335,000 + 104,000) ÷ 52 = 8,442.3 → 8,443 rounds, rounded up as at break-even. A target in dollars for a ratio-only business runs the same way: pre-tax target, plus fixed costs, divided by the CM ratio.