Private Equity puzzles, solved step by step

Half the cost base is fixed. Take revenue of 100, so EBITDA is 20 and costs are 80. Revenue grows to 110 and EBITDA to 26, so costs rose by only 4. A 4 rise on 10 of extra revenue means variable costs are 40% of revenue, which is 40. The remaining 40 of the 80 does not move, so fixed costs are 50% of costs.

Why does profit grow faster than revenue at all?

A tea stall pays the same rent whether it sells 100 cups or 110. The milk and sugar go up with each cup; the rent does not. When some costs are fixed, every extra unit of revenue only has to cover its own variable cost, and the rest drops straight to profit. The bigger the fixed slice, the more of each new rupee of revenue falls through, and the faster profit grows relative to sales. That gap between the two growth rates is the clue the question hands you.

So pick round numbers and let the gap tell you the split. Revenue of 100 at a 20% margin means EBITDA of 20 and costs of 80. Ten percent growth takes revenue to 110; thirty percent growth takes EBITDA to 26. Costs are therefore 110 less 26, which is 84, up 4. Those 4 are the variable costs riding on 10 of new revenue, so variable cost is 40% of revenue: 40 of the original 80.

Is there a formula you can say out loud to check it?

Yes. The ratio of profit growth to revenue growth is the degree of operating leverageHow many per cent profit changes for each one per cent change in revenue, equal to contribution divided by profit., here 30 over 10, which is 3. Operating leverage equals contribution divided by EBITDA, so contribution must be three times EBITDA: 60 on revenue of 100. Revenue of 100 less contribution of 60 leaves variable costs of 40, and the cost base of 80 less 40 leaves fixed costs of 40. Two routes landing on the same 50% is the check the interviewer wants to hear.

Why would an operating partner care about this number?

Because it cuts both ways. The same 3x leverage that turned 10% growth into 30% profit growth turns a 10% revenue fall into a 30% profit fall. Say the limitation: real costs are rarely cleanly fixed or variable. Staff can be cut with a lag, rent steps up when a site is added, and a one-year jump can include price rises that carry no variable cost at all, which would make the fixed share look larger than it is.

A keeps 85 and B keeps 60; leverage goes to 4.7x and 6.7x. A's variable costs are 350, or 70% of revenue, so at 450 they are 315 and EBITDA is 450 less 315 less 50, which is 85. B's variable costs are 100, or 20%, so at 450 they are 90 and EBITDA is 450 less 90 less 300, which is 60. Debt of 400 was 4.0x at both; it is now 4.7x at A and 6.7x at B.

Why does the same revenue fall hit the two businesses so differently?

Think of two auto drivers. One rents his vehicle by the day and pays a fixed 300 whatever happens; the other owns his and pays mostly for fuel. On a slow day the owner-driver still goes home with something, while the renter may go home with nothing. Fixed costs do not shrink when revenue shrinks, so the whole of a revenue fall lands on profit after only the variable costs have been saved. At A, 70 of every 100 of lost revenue was variable cost that disappears with it, so profit falls by 30 on 50 of lost revenue. At B only 20 of every 100 was variable, so profit falls by 40 on the same 50.

What are the numbers, and what do they do to the lenders?

A: revenue 450, variable costs 70% of that is 315, fixed 50, EBITDA 85, down 15%. B: revenue 450, variable 20% of that is 90, fixed 300, EBITDA 60, down 40%. Debt of 400 was 4.0x EBITDA at both companies, and after one bad year it is 4.7x at A and 6.7x at B, which is the difference between a conversation with the lender and a covenant breach. The operating leverageThe ratio of the percentage change in profit to the percentage change in revenue. High fixed costs mean high operating leverage. here is 1.5 at A and 4.0 at B: each 1% of revenue lost costs B 4% of EBITDA. Lenders feel it first because their claim is fixed and sits ahead of the equity; the equity feels it hardest because it is what is left.

How should a lender and a sponsor use this?

By sizing the debt to the cost structure, not to the EBITDA alone. Two businesses with identical EBITDA can carry very different debt safely, because the one with high fixed costs needs far less of a downturn to stop covering its interest. B breaks even at revenue of 375, a fall of only 25%, while A breaks even at 167. A lender to B wants lower leverage, a wider cushion in the covenant and a close look at whether any of the 300 can be made variable. The limits: the split of costs into fixed and variable is never clean, fixed costs do move over a long enough horizon, and the same leverage cuts the other way in an upswing, where B's EBITDA would rise 40% on a 10% revenue gain.

EBITDA rises from 96 to 117.1, a 22% increase, and the margin from 12% to 14.6%. Costs are 800 less 96, which is 704. A 3% cut saves 21.1, and with revenue fixed every rupee of it reaches EBITDA. Because costs are about 7.3 times EBITDA, a small cost change is a large profit change: 3% times 7.33 is 22%.

Why does a 3% cost cut grow EBITDA by 22%?

A family earning Rs 1 lakh a month and spending Rs 88,000 saves Rs 12,000. Trim spending by 3%, about Rs 2,640, and savings jump to about Rs 14,640, up 22%. Profit is the thin slice left after costs, so any change to the large cost base is magnified when measured against the small profit. The magnifier is the ratio of costs to EBITDA, here 704 over 96, about 7.3.

How does an operating partner use this, and where does it break?

Multiply by the exit multiple to see what it is worth. At 8x EBITDA, 21.1 of lasting savings adds about 169 of enterprise value, which is why cost programmes sit at the centre of so many buyout plans. The thinner the margin, the bigger the magnifier: at a 5% margin costs are 19 times EBITDA and a 3% cut lifts EBITDA by 57%.

Say the limitations. Not all of 704 can be cut: raw materials may be fixed by contract and wages by law, so the 3% often has to come from a much smaller part of the base. Cuts that hurt service can cost revenue later. And the same magnifier works in reverse: a 3% cost overrun would take 22% off EBITDA.

The blended margin falls from 40% to 36%, with no change in either product's economics. Before, 0.5 x 60% + 0.5 x 20% = 40%. After, 0.4 x 60% + 0.6 x 20% = 24% + 12% = 36%. Moving 10 points of the mix from a 60% product to a 20% product costs 10% of the 40-point gap between them, which is 4 points of blended margin.

How can margin fall when nothing got worse?

A cafe earns 70% on coffee and 25% on sandwiches. If more customers come at lunch and order sandwiches, the cafe's overall margin drops even though neither the coffee nor the sandwich got less profitable. A blended margin is a sales-weighted average, so shifting weight towards the lower-margin product pulls the average down by itself. Each 1 point of mix moved costs 1% of the margin gap, here 0.4 points.

Why does a buyout team care, and what should it check next?

When a target's margin falls, the first question is whether the products got worse or the mix moved. A mix-driven margin fall is a different problem from a pricing or cost problem, and it can come with rising profit. If the low-margin product is growing fast, total gross profit may still rise even as the percentage falls: a business at a 36% margin on revenue of 150 earns more than one at 40% on 100.

Say the limitation. Gross margin ignores the overheads each product uses. A low-margin product that needs little selling effort can be more attractive after overheads than it looks here, so ask for contribution by product before judging the mix.

About 60, released once. Revenue of 730 is 2 a day. At 90 days, customers are holding 180 of unpaid invoices; at 60 days, 120. Collecting 30 days faster brings in the 60 difference as cash, one time. After that, receivables simply stay at the lower level, so the cash flow benefit does not repeat each year.

What does a day of DSO actually hold?

Picture a tailor who lets regular customers pay at the end of the month. On any given day, a month's worth of stitched clothes is out there unpaid, and that money is not in the tailor's drawer. Days sales outstanding counts how many days of sales are sitting with customers, so each day of DSO is one day of revenue held as receivables instead of cash. Here one day is 730 over 365, which is 2.

Why does a buyout fund care that it happens only once?

Because a one-off release must not be valued like a recurring profit. The 60 can pay down debt or fund a dividend once, but it adds nothing to EBITDA and nothing to next year's cash flow. If a seller's numbers show a strong cash year driven by a receivables squeeze, a buyer strips it out before using that year to set the price. And as the company grows, receivables grow with it: 20% more revenue at 60 days lifts receivables to 144, absorbing 24 of cash.

Levered free cash flow is 92.5. EBITDA is 250. Take off D&A of 40 and interest of 60 to get pre-tax profit of 150, tax of 37.5 and net income of 112.5. Add back the 40 of D&A because it is not cash, then subtract capex of 50 and the 10 of working capital the growth absorbs. What remains, 92.5, is cash the owners could take out.

Why does depreciation come off and then go back on?

Think of a delivery van bought last year. Each year the accounts charge a slice of its cost as depreciation, but no money leaves the business that year: the van was paid for already. Depreciation reduces taxable profit, which saves real tax, but it is not a cash payment, so it is subtracted to work out tax and then added back to reach cash. The cash cost of assets enters through capex instead, here 50, which is more than the 40 of depreciation because the business is growing.

What makes it levered, and why does a buyout investor want that version?

Levered means after interest: the lenders have been paid. Levered free cash flow is the cash left for the equity holders after the lenders, the taxman and the business's own reinvestment, which is exactly the cash that repays debt in a buyout. Check it a second way from EBITDA: 250 less 60 less 37.5 less 50 less 10 is 92.5. The limit worth saying: working capital is modelled here as 10% of growth, and a real business can swing by far more in a single year.

Contribution rises about 6.9%. On a price of 100 with variable cost of 60, each unit contributes 40. A 5% price rise adds 5 straight to that margin, making 45, a 12.5% rise. Selling 95 units at 45 gives 4,275 against 4,000 before. Volume would have to fall 11.1% before the price rise stopped paying.

Why does a 5% price rise beat a 5% volume loss?

A tea stall sells a cup for Rs 20 that costs Rs 12 to make, keeping Rs 8. Raise the price by Rs 1 and the stall keeps Rs 9: a 5% price rise is a 12.5% rise in what it earns per cup. A price increase falls entirely on the margin because variable cost does not change, so the lower the margin, the bigger the percentage boost. Here the margin is 40% of price, so a 5% price rise lifts contribution per unit by 5 divided by 40, 12.5%.

What would an operating partner check before raising prices?

The arithmetic says volume could fall 11.1% before this price rise lost money, so the real question is how customers respond. A business whose customers cannot easily switch can usually hold more than 89% of volume after a 5% rise; one in a crowded market with lookalike products may lose far more. Operating teams test price on one region or product line first. Revenue here actually falls 0.25%, which is why a revenue-only dashboard would wrongly call the move a failure.

The team needs 150 of new revenue at a 40% margin, and about 171 at 35%. A fixed cost is paid for out of contribution, and each rupee of revenue brings only 40 paise of it: 60 divided by 0.40 is 150. If the team wins sales by discounting and the margin falls to 35%, the hurdle rises to 60 divided by 0.35, about 171. Breakeven revenue moves from 750 to 900.

Why is the answer not simply 60?

A shop hires a helper for Rs 6,000 a month. If every Rs 100 of sales costs Rs 60 in stock, the helper must bring in Rs 15,000 of extra sales, not Rs 6,000, because only Rs 40 of each Rs 100 is left to pay wages. New fixed costs are paid out of contribution, so the revenue needed is the cost divided by the contribution margin. At 40%, 60 of cost needs 150 of revenue.

What would an operating partner ask before approving the hire?

Breakeven is the floor, not the case for the hire: the team should bring in well above 150 within a reasonable ramp, at a margin close to the existing 40%. Sales teams often win volume with discounts, which is exactly what drops the margin to 35% and lifts the hurdle to 171. The partner would also ask how long the ramp takes, because a team that needs a year to sell anything costs 60 before it earns a rupee.