Private Equity puzzles, solved step by step

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

Showing 81–90 of 100

081At exit a business has EBITDA of 100, sells at 10x and carries net debt of 400. Which adds more to the equity: one extra turn of exit multiple, or 10% more EBITDA? Above what multiple does EBITDA growth win?Mental paper LBOsHardMid-market buyout fund
Try it first
Which adds more equity value at 10x?
Show the worked solution
At 10x they tie: each adds 100, taking equity from 600 to 700. One turn adds one year of EBITDA, 100. Ten per cent more EBITDA adds 10% of EBITDA times the multiple, 10 x 10, also 100. Above 10x the EBITDA growth adds more, below 10x the turn does. The crossover is one divided by the growth rate.
Why do the two levers tie at exactly 10x?
Enterprise value is EBITDA times the multiple, a product of two numbers, like a shop's takings being customers times average spend. A 10% rise in either factor lifts the product by 10%. One extra turn is a 10% rise in the multiple only when the multiple is 10, which is why the two levers tie there and nowhere else. At 8x, one turn is a 12.5% rise in the multiple; at 14x it is about 7%.
At 10x, one more turn and 10% more EBITDA each lift enterprise value from 1,000 to 1,100 and equity from 600 to 700; across multiples a turn always adds 100 while the EBITDA growth adds ten times the multiple, so the lines cross at 10x and growth wins above it.
The relationship
$\Delta EV_{\text{turn}} = E \qquad \Delta EV_{\text{growth}} = g\,E\,M \qquad \text{equal when } M = \tfrac{1}{g} = 10\times$
E exit EBITDA, here 100
M exit multiple
g EBITDA growth, here 10%
What it says in wordsA turn is worth one year of EBITDA; growth is worth the extra EBITDA times the multiple, so they match when the multiple equals one over the growth rate.
What does the equity holder actually feel, and which lever would you underwrite?
Net debt does not change in either case, so the full 100 lands on the equity: 600 to 700, a 16.7% lift. The arithmetic ties, but the two levers are not equally bankable: EBITDA growth is something the sponsor can plan and track, while the exit multiple is set by the market on the day of sale. That is why most buyout cases hold the exit multiple at or below entry and earn the return from EBITDA and debt paydown.
Give the general rule after the number: at 8x, 10% more EBITDA adds only 80 against 100 for a turn; at 14x it adds 140. One limitation is worth a sentence: higher EBITDA usually also brings extra cash that pays down debt, so in a full model growth gets a small bonus the turn does not.
Where candidates lose it
Candidates answer the multiple, because a turn sounds big, or the EBITDA, because growth sounds operational, and do not calculate. The question is built so that the arithmetic ties; guessing either way loses the point.
The second loss is stopping at the tie. The follow-up about the crossover is the real question: say one over the growth rate and give an example either side.
What the interviewer asks next
At what multiple does 20% more EBITDA tie with one more turn?
If net debt were 800 instead of 400, does the answer change in rupees or only in percentage terms?
Why might a sponsor still prefer to buy a business where multiple expansion is likely?
082Money compounds at 9% a year. Roughly how long does it take to double by the rule of 72, and how close is that to the exact answer?Compounding and time valueWarm upMid-market buyout fund
Try it first
Answer inside five seconds.
Show the worked solution
About 8 years by the rule of 72, and 8.04 years exactly. Divide 72 by the rate in per cent: 72 over 9 is 8. The exact answer is the log of 2 over the log of 1.09. The rule is near exact around 8% a year and drifts at very low or very high rates, where 69 or 70 works better.
Why does dividing 72 by the rate work?
Doubling needs the growth factor to reach 2, and the log of 2 is about 0.693. For small rates, the log of 1 plus r is close to r, so doubling time is about 69.3 divided by the rate in per cent. 72 is used instead of 69 because it divides cleanly by 2, 3, 4, 6, 8, 9 and 12, and because it corrects for the approximation at the rates people use most. It is a calculator in your head, like knowing that a dozen eggs at Rs 6 each is Rs 72.
At 9% a year money crosses twice its starting value at 8.04 years against the rule of 72's 8, and across rates the rule stays within about a tenth of a year from 8% to 12% but drifts to 0.33 years at 4% and 0.22 years at 24%.
The relationship
$t = \frac{\ln 2}{\ln(1.09)} = \frac{0.693}{0.0862} \approx 8.04 \qquad \text{rule: } \frac{72}{9} = 8$
t years to double
ln 2 natural log of 2, about 0.693
ln(1.09) natural log of the growth factor, about 0.0862
What it says in wordsExact doubling time is log 2 over log of one plus the rate; the rule of 72 approximates it with simple division.
Where does a buyout interviewer use this?
Everywhere returns are quoted. A deal that doubles the money in about four years has an IRR near 18%, and one that doubles in about three years is near 24%, because 72 over 4 is 18 and 72 over 3 is 24. That lets you check a quoted IRR against a quoted money multiple in seconds. At 24% the rule says 3 years and the truth is about 3.2, so for high-return deals you shade the rule slightly.
Where candidates lose it
The common slip is using simple interest and saying about 11 years, 100 divided by 9. Compounding means each year's interest itself earns interest, so doubling comes sooner.
The other is giving 8 and stopping when asked how exact it is. Know that the rule is closest around 8% and that the true figure here is just over 8 years.
What the interviewer asks next
How long does it take money to triple at 9%?
A deal returns 2x in 3 years. Roughly what IRR is that?
Why does the rule of 72 overstate doubling time at low rates and understate it at high rates?
083A company trades at 15x earnings and 8x EV/EBITDA. Net debt is 200 and net income is 40. What is EBITDA?Valuation riddlesWarm upMid-market buyout fund
Try it first
What do you need to find before EBITDA falls out?
Show the worked solution
EBITDA is 100. Net income of 40 at 15x gives equity value of 600. Add net debt of 200 to reach enterprise value of 800. The EV/EBITDA multiple of 8x then gives EBITDA of 800 divided by 8, which is 100. Two multiples and the bridge between equity and enterprise value pin down the missing line.
Why can you not go straight from net income to EBITDA?
Net income sits below interest, tax, depreciation and amortisation, and the question gives none of them. The two multiples work on different values, P/E on equity and EV/EBITDA on the whole business, so the route runs through value, not through the income statement. It is like knowing a flat's price per square foot and the loan on it: you get to the total value first and then back out what you need.
Net income of 40 at 15x gives equity value of 600, net debt of 200 bridges that to enterprise value of 800, and dividing by the 8x EV/EBITDA multiple gives EBITDA of 100.
The relationship
$\text{EBITDA} = \frac{15 \times 40 + 200}{8} = \frac{800}{8} = 100$
15 x 40 equity value from the P/E
200 net debt, added to reach enterprise value
8 the EV/EBITDA multiple
What it says in wordsTurn earnings into equity value, add net debt for enterprise value, and divide by the EBITDA multiple.
What follow-up can you get ahead of?
Interviewers often ask what the gap between EBITDA of 100 and net income of 40 is made of. Sixty of EBITDA goes on depreciation, interest and tax, and with net debt of 200 interest is only a modest slice, so depreciation or tax must be large. At an assumed 8% rate, interest would be 16, leaving 44 for depreciation and tax. Saying that shows you read the result, not only compute it.
State one assumption as you go: net debt here is all the claims that sit between equity and enterprise value. If there were minority interests or preference shares, they would be added too and EBITDA would come out higher.
Where candidates lose it
The usual slip is subtracting net debt instead of adding it, which gives EV of 400 and EBITDA of 50. Equity holders stand behind lenders, so the whole business is worth equity plus net debt.
The other is trying to rebuild EBITDA from net income by adding back guessed interest and tax. The multiples are there so you do not have to guess.
What the interviewer asks next
Net debt is minus 200, a net cash position. What is EBITDA now?
If D and A is 30 and interest is 16, what tax rate is implied?
What would make the P/E high and the EV/EBITDA low for the same company?
084A fund of 1,000 is drawn in full on day one and returns 1,800 in a single distribution after 5 years. The hurdle is 8% compounding, with a full catch-up and 20% carry. How much carry does the manager receive?Fund economics numeracyCoreSecondaries and fund of fundsLarge-cap buyout fund
Try it first
Before working the tiers: how much carry?
Show the worked solution
The manager receives 160 of carry. Investors first get 1,000 of capital and 469.3 of preferred return, 8% compounded for five years. The manager then takes 100% of the next 117.3 as catch-up. The last 213.3 splits 80/20. Manager's total: 117.3 plus 42.7, which is 160, exactly 20% of the 800 profit.
What order does the money flow in?
A distribution waterfallThe agreed order in which a fund pays out cash: capital back, then the preferred return, then catch-up, then the profit split. pays tier by tier, like a row of glasses being filled in turn: the next glass gets nothing until the one before is full. Capital comes back first, then the hurdle, then the catch-up, and only then the 80/20 split. Compounding 1,000 at 8% for five years gives 1,469.3, so the preferred return is 469.3.
Of the 1,800 returned, 1,000 repays capital and 469.3 pays the 8% compound hurdle to investors, the manager takes the next 117.3 as catch-up, and the last 213.3 splits 80/20, leaving the manager 160, exactly 20% of the 800 profit.
How big is the catch-up, and why does it end at 20% of all profit?
The catch-up runs until the manager holds 20% of everything distributed above capital. Investors already hold 469.3 of profit, so the manager needs c where c equals 20% of 469.3 plus c. That solves to a quarter of the preferred return, 117.3, after which every rupee splits 80/20 and the manager's share of total profit stays at exactly 20%. There was enough money to finish the catch-up, so the answer is simply 20% of 800.
The relationship
$c = \frac{0.20}{0.80} \times 469.3 = 117.3 \qquad \text{carry} = 117.3 + 0.20 \times 213.3 = 160$
469.3 preferred return, 1,000 x 1.08^5 less 1,000
117.3 catch-up, a quarter of the preferred return
213.3 what remains for the 80/20 split
What it says in wordsThe catch-up brings the manager level at 20% of profit so far; the split keeps it there.
Say the shortcut, then show the tiers as proof. If the fund had returned only 1,500, the catch-up would not complete: after the hurdle only 30.7 would be left, all of it to the manager, and carry would fall well short of 20% of the 500 profit.
Where candidates lose it
Candidates treat the hurdle as a deduction and pay carry only on profit above it: 20% of 330.7, about 66. A hurdle with a full catch-up is a gate; once the fund clears it with room to spare, the manager is made whole to 20% of all profit.
The other slip is using a simple 8% a year, 400, instead of compounding. That understates the hurdle by 69.3 and puts the catch-up in the wrong place.
What the interviewer asks next
What is the carry if the fund returns 1,500 instead of 1,800?
How does a 50% catch-up instead of a full one change the answer at 1,800?
Why do investors care whether the hurdle compounds or is simple?

085Without a calculator: what is 12.5% of 1,368 plus one third of 2,469?Mental mathsWarm upMid-market buyout fund

Try it first

Pick the answer before you work it.

Show the worked solution

994. Treat 12.5% as one eighth and halve 1,368 three times: 684, 342, 171. For a third of 2,469, split it into 2,400 and 69: a third of each is 800 and 23, so 823. Then 171 plus 823 is 994. Common percentages are fractions in disguise, and fractions are faster in your head.

Why turn 12.5% into a fraction?

Multiplying by 0.125 in your head means three digits of decimals to keep track of. Halving is something you have done since school, like splitting a restaurant bill between two, then four, then eight friends. 12.5% is exactly one eighth, so three halvings give the answer with no decimals at all. 1,368 halves to 684, then 342, then 171.

Taking 12.5% of 1,368 is halving three times to 171, and a third of 2,469 is a third of 2,400 plus a third of 69, 800 plus 23, so 823; the two add to 994.

How do you divide by three cleanly?

Split the number into a part that divides easily and a small remainder. 2,469 is 2,400 plus 69, and a third of each is 800 and 23, so a third of the whole is 823 with no long division. Check it: 823 times 3 is 2,469. The same trick works for any divisor: pick the nearest round multiple, then handle the leftover.

Percentage	Fraction	How to do it
12.5%	1/8	halve three times
16.7%	1/6	halve, then take a third
33.3%	1/3	split into friendly pieces
37.5%	3/8	an eighth, times three
62.5%	5/8	half plus an eighth
87.5%	7/8	the whole less an eighth

Percentages that appear often in deal maths and the fractions they hide: each turns a decimal multiplication into halving or dividing by a small number.

Buyout interviews use questions like this as a warm-up, often before a paper LBO, to see whether you will cope with arithmetic out loud. Saying each step lets the interviewer follow you and makes a slip easy to catch.

Where candidates lose it

The trap is attacking 0.125 times 1,368 as a decimal multiplication and losing a digit along the way, or rounding 2,469 to 2,500 and landing near 1,004. Both answers are close, which is exactly why they feel safe.

Say the fraction first, one eighth, and the split, 2,400 plus 69. The interviewer hears a method and the arithmetic becomes easy.

What the interviewer asks next

What is 37.5% of 2,496?
What is 87.5% of 640 minus one sixth of 1,242?
Why does knowing these fractions help when checking an IRR quickly?

086A sponsor buys a business with EBITDA of 100 at 7x, using 4x debt at 10%. D&A is 20, capex is 15 and tax is 25% of EBIT less interest. All free cash flow repays debt. EBITDA stays flat and the exit is at 7x after 3 years. What is the money multiple?Mental paper LBOsHardMid-market buyout fund
Try it first
Roughly what money multiple do you expect?
Show the worked solution
About 1.38x, an IRR of roughly 11%. Year one cash flow is EBITDA 100 less interest 40, tax 10 and capex 15: 35. Interest falls as debt is repaid, so cash rises to about 38 and 40, and debt ends near 287. Exit at 700 leaves equity of about 413 on 300. Holding cash flat at 35 gives a quick 1.35x.
What do you set up before the years?
Entry price 700, debt 400, so equity 300. Then one year of cash. Think of a rented flat bought with a loan: the rent does not rise and the flat's price does not move, so the only way your stake grows is that rent left over after costs pays the loan down. With flat EBITDA and the same exit multiple, enterprise value is the same 700 at exit, so every rupee of equity gain is a rupee of debt repaid.
In year one EBITDA of 100 loses 40 to interest, 10 to tax and 15 to capex, leaving 35 to repay debt; with enterprise value fixed at 700, three years of repayment take debt from 400 to 286.9 and equity from 300 to 413.1, a 1.38x money multiple.
Why does the cash grow each year when EBITDA is flat?
Interest is charged on a shrinking balance. Every rupee repaid saves 10 paise of interest next year, 7.5 paise after tax, so cash flow climbs from 35 to 37.6 to 40.4 with no change in the business. That compounding is small over three years, which is why the quick answer of 35 a year, 105 in total and 1.35x, is close enough to say first. Then refine it if asked.
Year Opening debt Interest Tax Cash to repay debt Closing debt
1 400.0 40.0 10.0 35.0 365.0
2 365.0 36.5 10.9 37.6 327.4
3 327.4 32.7 11.8 40.4 286.9
Interest is 10% of the opening balance and tax is 25% of EBIT of 80 less interest; free cash is net income plus D&A of 20 less capex of 15, and it repays 113.1 of debt over three years.
Close with the judgement the interviewer wants. A 11% IRR from debt paydown alone is below most buyout targets, so this deal needs EBITDA growth or a cheaper entry to work. That sentence turns the arithmetic into a view.
Where candidates lose it
The usual slip is forgetting tax, or taxing EBITDA instead of EBIT less interest, which changes cash flow by several points a year. D&A matters only through tax: it is not cash, but it shields 5 of profit from tax each year.
The other is spending ages on the precise schedule. Say 35 a year and 1.35x first, then show that falling interest lifts it to about 1.38x.
What the interviewer asks next
EBITDA now grows 5% a year. Roughly what does the money multiple become?
What exit multiple would give a 20% IRR with flat EBITDA?
Would you rather have 5x debt at 11% or 4x at 10% here, and why?
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?
088A business has EBITDA of 100 and debt of 5x EBITDA at 10%. Capex is 20 and tax is ignored. What is interest cover, and how far can EBITDA fall before free cash flow after interest reaches zero?Leverage and capital structureCorePrivate creditMid-market buyout fund
Try it first
How far can EBITDA fall before free cash flow hits zero?
Show the worked solution
Interest cover is 2.0x, and free cash flow reaches zero at EBITDA of 70, a 30% fall. Debt of 500 at 10% costs 50. EBITDA of 100 less capex of 20 and interest of 50 leaves 30 of free cash. That 30 is the cushion. At EBITDA of 70 the cover ratio still reads 1.4x, which looks fine just as the cash runs out.
Why is cover of 2.0x not the cushion it looks like?
A household earning Rs 1 lakh a month with a Rs 50,000 loan payment has income twice its instalment. But if it also spends Rs 20,000 on things it cannot skip, school fees and rent, only Rs 30,000 is truly spare. Interest cover ignores capex, so the real cushion is free cash flow after interest, 30 here, not the gap between EBITDA and interest. Cover would suggest EBITDA can halve; the cash says it can fall 30%.
EBITDA of 100 pays capex of 20 and interest of 50 and leaves 30 of free cash; at EBITDA of 70, a 30% fall, free cash is zero while interest cover still reads 1.4x.
The relationship
$\text{cover} = \frac{100}{50} = 2.0\times \qquad \text{breakeven EBITDA} = 20 + 50 = 70$
100 EBITDA
50 interest, 10% on debt of 500
20 capex the business must spend
What it says in wordsCover compares EBITDA with interest; the breakeven adds the capex that must also be paid.
What does a private credit lender do with this?
A lender tests the downside in cash, not in ratios: how far EBITDA can fall before the company must borrow more, cut capex, or miss a payment. Here the answer is 30%, and a lender would compare it with how far EBITDA fell in the sector's last downturn. If capex could be cut to 10 in a crisis, the cushion widens to a 40% fall, which is why lenders ask how much of capex is maintenance and how much is growth.
One limitation: with tax ignored the picture is generous. Tax would take a slice of the 30, and working capital swings in a downturn usually absorb cash too.
Where candidates lose it
Candidates say EBITDA can halve because cover is 2.0x. That forgets capex, which the business must pay whether or not EBITDA falls.
The second miss is giving the breakeven as an EBITDA level only. Say the percentage fall too, 30%, because that is the number a credit committee compares with history.
What the interviewer asks next
Add tax at 25% on EBITDA less D&A of 20 less interest. Where is the breakeven now?
If the rate rises to 12%, how much cushion is left?
Which covenant would you set for this loan, and at what level?
089A fund makes two deals of 100 each. Deal A returns 300 in year 2. Deal B returns nothing and is written off in year 4. Carry is 20% with no hurdle. How much carry is paid under a deal-by-deal waterfall, and under a whole-fund waterfall?Fund economics numeracyHardSecondaries and fund of funds
Try it first
Under deal-by-deal, how much carry does the manager hold at the end of year 2?
Show the worked solution
Both end at 20 of carry, but deal-by-deal pays 40 in year 2 and needs a 20 clawback in year 4. Deal-by-deal pays 20% of A's 200 profit as soon as A exits. When B is written off, fund profit is only 100, so the manager owes 20 back. Whole-fund returns all 200 of capital first, then pays 20% of the 100 profit: 20, with nothing to settle.
What is the difference between the two waterfalls?
Picture a salesperson paid commission on each sale as it closes, against one paid at year end on the year's net result. The first gets paid for the good deals before the bad ones show up. A deal-by-deal waterfall pays carry on each exit as it happens; a whole-fund waterfallA distribution order in which investors get back all contributed capital across the fund, plus any hurdle, before the manager receives any carry. pays carry only after investors have their capital back across the entire fund. The final entitlement is the same here; the timing is not.
Under deal-by-deal the manager takes 40 when deal A exits in year 2 and must hand back 20 when deal B is written off in year 4; under whole-fund the 300 first repays all 200 of capital, then pays the manager 20 of the 100 profit, with nothing to settle.
Why does the clawback matter so much to investors?
The clawbackA promise by the manager to return carry it was paid early if the fund as a whole ends up earning less than the carry assumed. is only as good as the manager's ability to pay. By year 4 the 40 has been distributed to individual partners and often taxed, so recovering 20 can mean chasing people, not a fund account. That is why investors negotiate escrows that hold back part of early carry, or prefer whole-fund terms altogether. The manager, meanwhile, has had an extra 20 for two years for free.
Waterfall Year 2 Year 4 Final carry
Deal by deal +40 -20 clawback 20
Whole fund +20 0 20
Both waterfalls end with the manager holding 20 of carry, 20% of the fund's 100 profit, but deal-by-deal pays 40 two years early and recovers 20 through a clawback.
One honest caveat: whole-fund here still pays in year 2, because the 300 from A is enough to return both deals' capital. Had A returned only 200, the fund would have made no profit and whole-fund would never have paid carry, while deal-by-deal would have paid 20 on A's 100 of profit and then had to claw all of it back.
Where candidates lose it
The common slip is saying both waterfalls pay 20 and stopping. The interviewer wants the timing and the clawback, because that is where investors lose money in practice.
The other is ignoring B under deal-by-deal and leaving the carry at 40. Under either structure the manager is finally entitled to 20% of the fund's profit, not of the winners.
What the interviewer asks next
Add an 8% hurdle. How does each waterfall change?
Why do most buyout funds outside the United States use the whole-fund model?
How would an escrow of 30% of carry have changed the clawback problem here?
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?

E	exit EBITDA, here 100
M	exit multiple
g	EBITDA growth, here 10%

t	years to double
ln 2	natural log of 2, about 0.693
ln(1.09)	natural log of the growth factor, about 0.0862

15 x 40	equity value from the P/E
200	net debt, added to reach enterprise value
8	the EV/EBITDA multiple

469.3	preferred return, 1,000 x 1.08^5 less 1,000
117.3	catch-up, a quarter of the preferred return
213.3	what remains for the 80/20 split

Try it first

Why do the two levers tie at exactly 10x?

What does the equity holder actually feel, and which lever would you underwrite?

Where candidates lose it

What the interviewer asks next

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Why does dividing 72 by the rate work?

Where does a buyout interviewer use this?

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Why can you not go straight from net income to EBITDA?

What follow-up can you get ahead of?

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What order does the money flow in?

How big is the catch-up, and why does it end at 20% of all profit?

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Why turn 12.5% into a fraction?

How do you divide by three cleanly?

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What do you set up before the years?

Why does the cash grow each year when EBITDA is flat?

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If the buildings are identical, what is actually different?

How do you value them so the comparison is fair?

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Why is cover of 2.0x not the cushion it looks like?

What does a private credit lender do with this?

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What is the difference between the two waterfalls?

Why does the clawback matter so much to investors?

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

What would an operating partner check before raising prices?

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