Private Equity puzzles, solved step by step

Puzzles: 100
Traced to a firm: 22
Topics: 12
Hard: 30

Showing 1–6 of 6 · filtered from 100Clear filters

004A portfolio company's revenue grows 10% and its EBITDA grows 30%. The EBITDA margin was 20% before the growth. Assuming costs are either fixed or move in line with revenue, what share of the cost base is fixed?Operating levers and margin mathsHardPortfolio operations teamMid-market buyout fund
Try it first
Your instinct: what share of costs is fixed?
Show the worked solution
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.
When revenue rises from 100 to 110, variable cost rises from 40 to 44, fixed cost stays at 40 and EBITDA rises from 20 to 26, a 30% jump, which shows that half of the 80 cost base is fixed.
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.
The relationship
$\text{DOL} = \frac{\%\Delta \text{EBITDA}}{\%\Delta \text{Revenue}} = \frac{R - V}{\text{EBITDA}} = \frac{100 - 40}{20} = 3$
R revenue, set to 100
V variable costs, which move in line with revenue
R - V contribution, what is left to pay fixed costs and earn profit
What it says in wordsProfit moves three times as fast as revenue because contribution is three times profit.
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.
Where candidates lose it
Candidates often reach for the margin and answer 20%, or try to solve with two unknowns in their head and lose the thread. Fix revenue at 100 first; the problem becomes subtraction.
The other loss is quoting the fixed share of revenue, 40%, rather than of the cost base, 50%. Repeat the question's denominator back before you answer.
What the interviewer asks next
If revenue now falls 10% from 110, what happens to EBITDA?
What would the EBITDA growth be if all costs were variable?
How would a price increase with no volume change distort this calculation?
033A business has revenue of 800 and a 12% EBITDA margin. The new owner cuts the cost base by 3% with revenue unchanged. What are the new EBITDA, the new margin and the EBITDA growth?Operating levers and margin mathsCorePortfolio operations teamIndian mid-market PE
Try it first
By how much does EBITDA grow?
Show the worked solution
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.
Cutting a cost base of 704 by 3% saves 21.1, which lifts EBITDA from 96 to 117.1 and the margin from 12% to 14.6%, a 22% rise in profit from a small change in costs.
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%.
The relationship
$g_{EBITDA} = c \times \frac{\text{Costs}}{\text{EBITDA}} = 3\% \times \frac{704}{96} = 22\%$
c the percentage cut to the cost base
Costs / EBITDA how many times larger the cost base is than profit
What it says in wordsEBITDA growth from a cost cut equals the cut times the ratio of costs to EBITDA.
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.
Where candidates lose it
The common loss is answering 3%, applying the cut to EBITDA instead of to costs. The interviewer is testing whether you see that costs are many times larger than profit.
The second loss is computing a 15% margin by adding 3 points to 12%. The new margin is 117.1 over 800, which is 14.6%; it rises by 2.6 points, not 3.
What the interviewer asks next
What revenue growth, with costs growing at the same rate, would deliver the same EBITDA as the 3% cut?
Which costs in a manufacturing business are hardest to cut, and why?
If half of the cost base is fixed, what does a 10% revenue fall do to EBITDA?
039A company sells two products with gross margins of 60% and 20%. Sales shift from 50/50 to 40/60, towards the low-margin product. What happens to the blended gross margin?Operating levers and margin mathsWarm upPortfolio operations team
Try it first
Where does the blended margin go?
Show the worked solution
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.
With product margins fixed at 60% and 20%, a 50/50 mix earns 30 plus 10 on revenue of 100, a 40% blend, while a 40/60 mix earns 24 plus 12, a 36% blend, so mix alone takes 4 points off the margin.
The relationship
$m = w_A m_A + w_B m_B = 0.4 \times 60\% + 0.6 \times 20\% = 36\%$
w_A, w_B each product's share of sales
m_A, m_B each product's gross margin
What it says in wordsThe blended margin is each product's margin weighted by its share of sales.
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.
Where candidates lose it
The common loss is saying the margin stays at 40% because neither product changed. The interviewer is checking whether you know that a blend is a weighted average and moves with its weights.
The second loss is reading a falling margin as bad news without asking about volume. Margin percentage and profit in rupees can move in opposite directions.
What the interviewer asks next
What mix would bring the blended margin down to 30%?
If revenue grows from 100 to 130 with the new mix, does gross profit rise or fall?
How would you separate price, cost and mix effects in a margin bridge?
062A portfolio company has revenue of 730 a year. The operating team cuts days sales outstanding from 90 to 60. How much cash does that release?Operating levers and margin mathsWarm upPortfolio operations teamIndian mid-market PE
Try it first
How much cash comes out of receivables?
Show the worked solution
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.
At 2 of sales a day, 90 days of receivables hold 180 and 60 days hold 120, so cutting DSO by 30 days releases 60 of cash, once, and later growth in revenue starts to absorb cash again.
The relationship
$\Delta\text{Cash} = \frac{\text{Revenue}}{365} \times \Delta\text{DSO} = \frac{730}{365} \times 30 = 60$
Revenue/365 sales per day, here 2
Delta DSO the cut in days of receivables, 90 to 60
What it says in wordsCash released equals one day of sales times the number of days cut.
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.
Where candidates lose it
Two slips are common. The first is answering 30, the change in days, without converting days into money at 2 a day.
The second is treating 60 as an annual saving and putting it into the free cash flow of every year. It is a one-time release from a lower balance; a fund that capitalises it as recurring overpays.
What the interviewer asks next
Payables days go from 30 to 45 on cost of sales of 365. How much cash is released?
Why might cutting DSO cost the company revenue?
How would you spot a seller who squeezed receivables just before a sale?
090A portfolio company raises its price 5% and loses 5% of its volume. Its contribution margin was 40% of price. Does total contribution rise or fall, and by how much?Operating levers and margin mathsCoreMid-market buyout fundPortfolio operations team
Try it first
What happens to total contribution?
Show the worked solution
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%.
Before the change 100 units contribute 40 each for 4,000; after a 5% price rise and a 5% volume loss, 95 units contribute 45 each for 4,275, a 6.9% rise, because contribution per unit grew 12.5% while volume fell only 5%.
The relationship
$\frac{45 \times 95}{40 \times 100} - 1 = \frac{4{,}275}{4{,}000} - 1 = +6.9\% \qquad \text{breakeven loss} = 1 - \frac{40}{45} = 11.1\%$
40, 45 contribution per unit before and after
100, 95 units sold before and after
What it says in wordsTotal contribution is units times contribution per unit; volume can fall until the higher margin no longer covers the lost units.
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.
Where candidates lose it
Candidates say the changes cancel, because plus 5% and minus 5% look symmetric, or they compute revenue, which falls 0.25%, and call it a loss. Neither looks at margin.
Work in contribution per unit. The trap only works on people who forget that variable cost does not rise with price.
What the interviewer asks next
What if the contribution margin were 80%, as in software? How much volume could you lose?
With fixed costs of 3,000, what happens to operating profit in percentage terms?
How would you test customer price sensitivity before a full rollout?
093A portfolio company has fixed costs of 300 and a contribution margin of 40%. The sponsor adds a sales team costing 60 a year. How much new revenue must the team bring in to pay for itself, and how much if the new sales carry only a 35% contribution margin?Operating levers and margin mathsCorePortfolio operations team
Try it first
How much new revenue pays for a 60 sales team at a 40% contribution margin?
Show the worked solution
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.
Contribution at 40% of revenue covers fixed costs of 300 at revenue of 750; adding 60 of fixed cost moves breakeven to 900, so the sales team must bring in 150 of revenue, or about 171 if its sales carry a 35% margin.
The relationship
$\Delta R = \frac{\Delta F}{m} = \frac{60}{0.40} = 150 \qquad \frac{60}{0.35} \approx 171$
Delta F the new fixed cost, 60
m contribution margin on the new sales
Delta R new revenue needed to break even on the hire
What it says in wordsDivide the new fixed cost by the margin the new sales earn.
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.
Where candidates lose it
The fast wrong answer is 60: matching the cost with the same amount of revenue, as if revenue were profit. Variable costs take 60% of every rupee before anything is left to pay the team.
The second miss is assuming new sales carry the existing margin. A sales team that buys growth with discounts can raise revenue and still fail to pay for itself.
What the interviewer asks next
How many months of ramp can the sponsor afford if the team reaches 300 of annual sales by year end?
The team's sales carry a 50% margin because they sell a premium line. What is the hurdle now?
How does a commission-only sales structure change this calculation?

R	revenue, set to 100
V	variable costs, which move in line with revenue
R - V	contribution, what is left to pay fixed costs and earn profit

c	the percentage cut to the cost base
Costs / EBITDA	how many times larger the cost base is than profit

w_A, w_B	each product's share of sales
m_A, m_B	each product's gross margin

Revenue/365	sales per day, here 2
Delta DSO	the cut in days of receivables, 90 to 60

Try it first

Why does profit grow faster than revenue at all?

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

Why would an operating partner care about this number?

Where candidates lose it

What the interviewer asks next

Try it first

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

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

Where candidates lose it

What the interviewer asks next

Try it first

How can margin fall when nothing got worse?

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

Where candidates lose it

What the interviewer asks next

Try it first

What does a day of DSO actually hold?

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

Where candidates lose it

What the interviewer asks next

Try it first

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

What would an operating partner check before raising prices?

Where candidates lose it

What the interviewer asks next

Try it first

Why is the answer not simply 60?

What would an operating partner ask before approving the hire?

Where candidates lose it

What the interviewer asks next