Private Equity puzzles, solved step by step

Puzzles: 100
Traced to a firm: 22
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023Two businesses each earn EBITDA of 100 on revenue of 500. A has fixed costs of 50; B has fixed costs of 300. Revenue falls 10% at both. What is each one's EBITDA, and what happens to 4x leverage?Operating levers and margin mathsHardOaktree Capital ManagementLos Angeles · 2024
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After the 10% revenue fall, B's EBITDA is
Show the worked solution
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.
A 10% revenue fall from 500 to 450 cuts A's EBITDA from 100 to 85 because its fixed costs are only 50, but cuts B's from 100 to 60 because 300 of its costs do not move, so 400 of debt goes from 4.0x to 4.7x at A and 6.7x at B.
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.
The relationship
$\text{EBITDA}_1 = R_1(1 - v) - F \qquad B: 450 \times (1 - 0.20) - 300 = 60 \qquad \frac{400}{60} = 6.7x$
R_1 revenue after the fall, 450
v variable costs as a share of revenue, 0.70 at A and 0.20 at B
F fixed costs, 50 at A and 300 at B
What it says in wordsProfit after the fall is the new revenue less the costs that move with it, less the costs that do not.
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.
Where candidates lose it
The common slip is to cut EBITDA by 10% along with revenue, giving 90 at both companies. That treats every cost as variable and misses the whole point of the question.
The second loss is stopping at 85 and 60. The question says 4x leverage for a reason: convert both numbers into leverage and name which one has become a lender's problem, and say that it is the debt holders who feel operating leverage first because their claim does not shrink.
What the interviewer asks next
What revenue fall would take B to zero EBITDA?
Revenue rises 10% instead. What is each EBITDA, and which company would you rather own the equity of?
How would you check what share of a target's costs is really fixed during diligence?
Asked at Oaktree Capital Management, Credit, Los Angeles, 2024 (Wall Street Oasis): How does operating leverage affect debt vs. equity holders
034Roughly how many cinema tickets are sold in India in a year? Build it from the demand side, then check it from the supply side.Market sizing and estimationHardAdvent InternationalBoston · 2022
Try it first
Before building anything: which order of magnitude feels right?
Show the worked solution
Roughly 0.8 to 0.9 billion tickets a year, on round assumptions. Demand: 1,400 million people, a quarter of whom go in a year, about 2.5 times each, gives about 875 million. Supply: about 9,000 screens, 4 shows a day, 200 seats, 30% full, gives about 788 million. Two routes agreeing within about 11% is the check; every input should be tested against an industry report.
Why build it twice?
If you want to know how many people eat at a canteen each day, you can count students and guess how often they eat there, or count tables, sittings and how full they look at lunch. Either guess on its own could be badly off; if both land near the same number you can trust it. A sizing built from demand and checked from supply catches the assumption that is wildly off, which a single chain never reveals. Interviewers care more about the check than about the final digit.
Demand of 1,400 million people times 25% who go times 2.5 visits gives 875 million tickets, and supply of 9,000 screens times 1,460 shows times 200 seats times 30% occupancy gives 788 million, so two independent routes agree on roughly 0.8 to 0.9 billion.
How do you defend each assumption out loud?
Segment before you guess. A quarter of the population going at least once a year is an average of very different groups: urban young adults may go many times, while much of rural India has no screen nearby. Say which segment drives the answer and why your share and frequency are reasonable for it, then give one number. On supply, the interviewer will push on occupancy: 30% is plausible because weekday afternoon shows run thin while weekend evenings fill. The screen count is an assumption to confirm, not a fact to recite.
Input Assumption If you halve it
People who go in a year 25% 438m demand
Visits per goer 2.5 438m demand
Occupancy 30% 394m supply
Screens 9,000 394m supply
Every input moves the answer one for one, so the honest range is wide and worth saying out loud.
A growth investor would go one step further: tickets times an average price gives box office revenue, and food and drink add a second stream on the same visits. Say the limitation: streaming changes how often people go, so a figure built on old habits may overstate tomorrow's market.
Where candidates lose it
The common loss is a single long chain with one unrealistic link, usually that most Indians go to the cinema several times a year, giving a number in the billions with no check. The interviewer then asks how many screens that implies, and the answer falls apart.
The second loss is reciting a screen count or a ticket figure from memory as fact. Present every input as an assumption, explain why it is reasonable, and offer the second route as the check.
What the interviewer asks next
What average ticket price would you assume, and what does that make the box office?
How would you split the answer between multiplex and single-screen cinemas?
Which single assumption would you research first, and where?
Asked at Advent International, Private Equity, Boston, 2022 (Wall Street Oasis): Market sizing questions, e.g.: how many books were sold this year in the US?
044Work backwards: EBITDA grows from 100 to 140 over 5 years and the business exits at 9x. Entry debt is 5x EBITDA and 200 of it is repaid by exit. What is the highest entry multiple that still returns 3x the equity?Mental paper LBOsHardNeuberger BermanNew York · 2022
Try it first
Where does the maximum entry multiple land?
Show the worked solution
About 8.2x. Exit EV is 140 x 9 = 1,260. Debt falls from 500 to 300, so exit equity is 960. A 3x return allows entry equity of 960 / 3 = 320. Add the 500 of entry debt and the most you can pay is an EV of 820, which is 8.2x entry EBITDA. Pay more and the 3x target is missed; fees and costs would lower the ceiling further.
Why work backwards instead of guessing an entry price?
If you want to arrive at a wedding by 7 and the drive takes two hours with half an hour of traffic, you leave at 4:30. You fix the end point and walk back. A target multiple fixes the exit equity you need relative to the cheque, so the maximum price is found by running the LBO in reverse: exit value, less exit debt, divided by the target, plus entry debt. Every number you need is in the question.
Exit EV of 1,260 less 300 of remaining debt leaves exit equity of 960, which supports entry equity of 320 at a 3x target; adding 500 of entry debt sets the maximum entry EV at 820, or 8.2x EBITDA of 100.
The relationship
$EV_{0}^{max} = \frac{E_5 \times m_{exit} - D_5}{\text{target}} + D_0 = \frac{1260 - 300}{3} + 500 = 820$
E_5 x m_exit exit EV, 140 x 9
D_5 debt left at exit, 300
target the money multiple required, 3x
D_0 entry debt, 5 x 100 = 500
What it says in wordsThe most you can pay is the exit equity divided by the target multiple, plus the debt you borrow at entry.
What does 8.2x tell the investment committee?
It is a ceiling, not an offer. Entering at 8.2x and exiting at 9x means the plan relies on a little multiple expansion as well as 40% EBITDA growth and 200 of debt paydown, so the committee will test each of those three. 3x over five years is an IRR of about 25%: 3 to the power one fifth is about 1.246.
Say the limitations. Transaction fees and financing costs come out of the equity at entry, so the true ceiling on the headline price is lower. A real model would also hold some cash at exit and might pay interest in kind. And if the lenders will not provide 5x at an 8.2x price, the equity cheque grows and the ceiling falls.
Where candidates lose it
The common loss is forgetting to add the entry debt back: candidates divide exit equity by 3, get 320, and say 3.2x. That is the equity cheque over EBITDA, not the enterprise multiple.
The second loss is using entry debt instead of exit debt when computing exit equity, which ignores the 200 repaid and makes the ceiling look lower than it is. Track debt at both ends.
What the interviewer asks next
If the exit multiple is 8x instead of 9x, what is the new ceiling?
How much does the ceiling fall if 20 of fees are paid out of the equity at entry?
What IRR does 3x in five years imply, and what multiple would 25% need?
Asked at Neuberger Berman, Private Equity, New York, 2022 (Wall Street Oasis): Interviews 4-5 were very technical again and also included multiple paperback LBOs and other, more advanced technicals.
087Two identical office buildings. A is let for 10 more years to a strong tenant at a net income of Rs 8 crore a year. B earns Rs 10 crore, but its lease ends in 2 years; the market rent is Rs 8 crore and re-letting takes a year. At a 10% discount rate and a 7.5% exit cap rate, which is worth more?Valuation riddlesHardApollo Global ManagementWilliamsport · 2022
Try it first
Which building is worth more?
Show the worked solution
A is worth more, by about Rs 2.5 crore. From year 4 the buildings earn the same 8, so only the first three years differ. B earns 2 more in years 1 and 2, worth 3.47 today, and 8 less in year 3, worth 6.01. Valued on three years plus a sale at a 7.5% cap, A is about 100.0 and B about 97.5.
If the buildings are identical, what is actually different?
Two identical flats on the same floor can sell at different prices if one has a tenant paying above market on a lease about to end and the other a reliable tenant at market. Buildings are valued on the certainty and timing of their income, not their bricks. A has eight crore a year from a strong tenant for ten years. B has ten crore for two years, then a vacancy, then whatever the market pays, which is eight.
Building B earns 2 more than A in years 1 and 2 but nothing in year 3, and from year 4 the two are identical; the differences are worth +1.82, +1.65 and -6.01 today, so B is worth about Rs 2.54 crore less than A.
How do you value them so the comparison is fair?
Use the same method for both: three years of cash, then a sale at the end of year 3 at a 7.5% cap rateNet operating income divided by property value. A building earning 8 a year at a 7.5% cap rate is worth 8 divided by 0.075, about 106.7. on the 8 a year both will earn from then, which is 106.7. Discount at 10%. A comes to 100.0 and B to 97.5, and the gap of 2.54 is just the present value of the three years in which they differ.
The relationship
$V_B - V_A = \frac{+2}{1.1} + \frac{+2}{1.1^2} + \frac{-8}{1.1^3} = 1.82 + 1.65 - 6.01 = -2.54$
+2 B's extra rent over A in years 1 and 2
-8 the year 3 void, when B earns nothing and A earns 8
1.1 one plus the 10% discount rate
What it says in wordsOnly the years in which the buildings differ matter, and the void in year 3 outweighs two years of higher rent.
Say the limitation. The absolute values shift with the exit year you pick, because a 10% discount rate against a 7.5% cap implies rents that grow, while this question holds rent flat. The gap does not shift, because after year 3 the buildings are the same. In practice B's discount is also larger: re-letting costs agents' fees and incentives, and the market rent of 8 is a forecast while A's is a contract.
Where candidates lose it
The classic error is capping B's in-place rent: 10 divided by 7.5% is 133.3, a quarter more than A at 106.7. That pays full value for rent that disappears in two years and ignores the empty year.
The second is calling them equal because the bricks are the same. The interviewer chose identical buildings to strip out everything except the lease, so the lease is the answer.
What the interviewer asks next
How long can B's void last before the gap reaches Rs 10 crore?
B's tenant offers to renew at 9 for five years. What is that worth?
Why might a buyer still prefer B despite the lower value?
Asked at Apollo Global Management, Generalist, Williamsport, 2022 (Wall Street Oasis): Comparing two identical buildings, how would you value them?

R_1	revenue after the fall, 450
v	variable costs as a share of revenue, 0.70 at A and 0.20 at B
F	fixed costs, 50 at A and 300 at B

Input	Assumption	If you halve it
People who go in a year	25%	438m demand
Visits per goer	2.5	438m demand
Occupancy	30%	394m supply
Screens	9,000	394m supply

E_5 x m_exit	exit EV, 140 x 9
D_5	debt left at exit, 300
target	the money multiple required, 3x
D_0	entry debt, 5 x 100 = 500

+2	B's extra rent over A in years 1 and 2
-8	the year 3 void, when B earns nothing and A earns 8
1.1	one plus the 10% discount rate

Try it first

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

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

How should a lender and a sponsor use this?

Where candidates lose it

What the interviewer asks next

Try it first

Why build it twice?

How do you defend each assumption out loud?

Where candidates lose it

What the interviewer asks next

Try it first

Why work backwards instead of guessing an entry price?

What does 8.2x tell the investment committee?

Where candidates lose it

What the interviewer asks next

Try it first

If the buildings are identical, what is actually different?

How do you value them so the comparison is fair?

Where candidates lose it

What the interviewer asks next