Private Equity puzzles, solved step by step

Puzzles: 100
Traced to a firm: 22
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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.
051A sponsor buys a business at 12x EBITDA, funded 50% with debt, and repays none of the debt. It expects to sell at 9x. By how much must EBITDA grow before the sponsor simply gets its money back?Mental paper LBOsHardLarge-cap buyout fund
Try it first
Before you work it: how much EBITDA growth returns the equity, and no more?
Show the worked solution
EBITDA must grow by 33.3% just to hand the sponsor its money back. Take EBITDA of 100: the business costs 1,200, funded by 600 of debt and 600 of equity. With no paydown the debt is still 600 at exit, so equity is 600 only if the business is again worth 1,200. At 9x that needs EBITDA of 133.3, which is 12 over 9 minus 1.
Why is the answer not 25%?
Picture a flat bought for 12 years of rent and sold later when buyers will pay only 9 years of rent. To get your price back, the rent has to rise until 9 years of it equals the old 12. That is a rise of 12 over 9, a third, not a quarter. A 25% cut in the multiple needs a 33.3% rise in earnings to undo it, because the recovery is measured on the smaller multiple. It is the same base effect that makes a 10% fall need an 11.1% rise.
Entry and exit both show an enterprise value of 1,200: twelve turns of EBITDA 100 going in, nine turns of EBITDA 133.3 coming out, so EBITDA has to grow 33.3% before the 600 of equity is returned.
Why does the 50% debt not change the answer?
Write the equity at exit as exit value less debt. The debt was 600 going in and, with no repayment, is 600 coming out. Equity is returned in full only when the exit value equals the entry value, so the break-even growth is the same at 30% debt, 50% debt or no debt at all. Leverage changes how fast equity grows once you pass break-even, and how fast it is wiped out below it. It does not move the break-even itself when nothing is repaid.
The relationship
$g = \frac{M_{\text{entry}}}{M_{\text{exit}}} - 1 = \frac{12}{9} - 1 = 33.3\%$
g EBITDA growth needed to get the equity back
M_entry the purchase multiple, 12x
M_exit the exit multiple, 9x
What it says in wordsWith no debt repaid, the break-even EBITDA growth is the entry multiple divided by the exit multiple, less one.
What does this do to a real growth plan?
Say the plan grows EBITDA 50% to 150. Sold at 12x, the business would be worth 1,800 and the equity 1,200, double the cheque. Sold at 9x it is worth 1,350 and the equity 750, only 1.25x. The first third of the growth plan is spent refilling the hole left by the lower multiple. This is why sponsors who pay high multiples underwrite their exit at the entry multiple or lower, and why the interviewer asks the question.
Where candidates lose it
The fast wrong answer is 25%, read straight off the drop from 12x to 9x. It measures the cut on the old multiple, when the earnings have to climb back on the new one: 12 over 9 is 1.333, not 1.25.
The second trap is letting the 50% debt pull you into a leverage calculation. With no paydown the debt cancels out of the break-even, and saying so in one sentence shows the interviewer you saw it.
What the interviewer asks next
The company repays 200 of debt over the hold. What EBITDA growth now returns the equity?
EBITDA grows 50% and the exit is at 9x after five years. What is the IRR? (equity 750 on 600)
Why might a buyer at 9x in five years pay less than you did, even for a better business?
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?
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?

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

g	EBITDA growth needed to get the equity back
M_entry	the purchase multiple, 12x
M_exit	the exit multiple, 9x

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

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

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

Why is the answer not 25%?

Why does the 50% debt not change the answer?

What does this do to a real growth plan?

Where candidates lose it

What the interviewer asks next

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

Try it first

What do you set up before the years?

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

Where candidates lose it

What the interviewer asks next