Private Equity puzzles, solved step by step

Puzzles: 100
Traced to a firm: 22
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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?
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?
091Deal A returns 3x in 7 years. Deal B returns 2x in 3 years. Which has the higher IRR? Then show that if B's proceeds can only be reinvested to earn 1.5x over the remaining 4 years, both end at 3x after 7 years.Returns mathsHardLarge-cap buyout fundSecondaries and fund of funds
Try it first
Which deal has the higher IRR?
Show the worked solution
B has the higher IRR, about 26.0% against 17.0% for A. But if B's 2x can only earn 1.5x over the next four years, about 10.7% a year, it ends at 3x after seven years, exactly where A ends. A higher IRR is worth more only if the cash it returns early can be redeployed at a comparable rate.
How do you get the two IRRs in your head?
Use the rule of 72 and a couple of anchors. B doubles in three years, and 72 over 3 is 24, so B is in the mid twenties; exactly, 2 to the one-third is 1.26, so 26.0%. A triples in seven years. Tripling takes about 1.6 times as long as doubling, so a seven-year triple is like doubling in about 4.4 years, which the rule of 72 puts in the high teens. Exactly, 3 to the one-seventh is 1.170, so 17.0%.
Deal A compounds steadily at 17.0% to 3x in year 7, while deal B reaches 2x in year 3 at 26.0% but, reinvested at only 10.7% a year, also ends at 3x in year 7, so its higher IRR adds no wealth.
Why does the higher IRR not make B the better deal?
Think of two fixed deposits: one pays 26% but matures in three years, after which the best rate on offer is 10%; the other locks in 17% for seven. Over seven years you end up in the same place. IRR assumes the cash a deal returns can be reinvested at the same IRR, and when it cannot, the IRR overstates what the investor actually ends up with. B ties A when its proceeds earn 10.7% a year after year 3; any less and A wins over seven years, any more and B does.
The relationship
$2 \times 1.5 = 3.0\times \qquad 1.5^{1/4} - 1 = 10.7\% \qquad 3^{1/7} - 1 = 17.0\%$
2 B's money multiple at year 3
1.5 what the proceeds earn over the next four years
3^(1/7) the seven-year rate both deals deliver
What it says in wordsB's two stages multiply to the same 3x over seven years as A, so the seven-year rate is identical.
This is why fund investors look at both numbers. A fund with high IRRs from quick flips may return less money over a decade than one with lower IRRs and bigger multiples, if the investors cannot put the early cash back to work at a similar rate.
Where candidates lose it
Candidates pick A because 3x sounds better than 2x, confusing size with speed. Or they pick B on IRR alone and never ask what happens to the cash after year 3.
The point of the second half is reinvestment. Say the condition: B is better only if its proceeds can be redeployed above about 10.7% a year for the remaining four years.
What the interviewer asks next
What reinvestment multiple makes B end at 3.5x after seven years?
Why do secondaries buyers often care more about the money multiple than the IRR?
How can a fund raise its reported IRR without raising its money multiple?
096A buyout is financed with senior debt of 4x EBITDA at 8% and mezzanine of 1.5x at 13%. What is the blended cost of debt? If the lender's floor is 2.0x interest cover, how much all-senior debt at 8% could EBITDA of 100 carry?Leverage and capital structureHardPrivate credit
Try it first
What is the blended cost of the 5.5x of debt?
Show the worked solution
The blended cost is about 9.36%, and all-senior debt at 8% could reach 625, 6.25x, at 2.0x cover. Interest is 32 on the senior and 19.5 on the mezzanine, 51.5 on 550 of debt. A 2.0x floor on EBITDA of 100 allows 50 of interest, which buys 625 at 8%. The planned 5.5x mix costs 51.5, cover of 1.94x, just under the floor.
Why is the blended rate not the average of 8% and 13%?
If you borrow Rs 4 lakh from a bank at 8% and Rs 1.5 lakh from a relative at 13%, your average cost is pulled towards 8% because most of the money is cheap. A blended cost of debt is a size-weighted average: total interest divided by total debt. Here that is 51.5 divided by 550, about 9.36%, well below the simple average of 10.5%.
Drawn with width equal to size and height equal to rate, the 400 of senior debt at 8% and 150 of mezzanine at 13% cost 51.5 of interest, a blended 9.36%; at a 2.0x cover floor EBITDA of 100 carries 625 of all-senior debt but only about 534 at the blended rate, so the 5.5x mix sits just over the limit.
The relationship
$r_{blend} = \frac{400 \times 8\% + 150 \times 13\%}{550} = \frac{51.5}{550} = 9.36\% \qquad D_{max} = \frac{100 / 2.0}{8\%} = 625$
400, 150 senior and mezzanine debt
51.5 total interest
100 / 2.0 the most interest a 2.0x cover floor allows
What it says in wordsBlend the rates by size; then divide the interest the cover floor allows by the rate to find how much debt it supports.
Why does cover limit the debt before the leverage multiple does?
A lender's cover test caps interest, not turns of debt. The cheaper the debt, the more of it fits under the same interest ceiling, so 8% senior debt supports 6.25x while debt at the blended 9.36% supports only about 5.34x. That is the surprise in this question: replacing the mezzanine with more senior debt would let the sponsor borrow more, not less, if a senior lender would go that far. In practice senior lenders also cap leverage directly, which is why mezzanine exists at all.
Close with the practical point. The planned structure fails a 2.0x floor by a whisker, 1.94x. A credit interviewer wants you to notice that and suggest a fix: trim the mezzanine to about 1.4x, or negotiate the cover test on a measure that adds back something, and say which you would try first.
Where candidates lose it
The common slip is averaging the two rates, 10.5%, which overweights the small expensive tranche. Weight by size every time.
The second is assuming more debt always means more interest cover pressure in the same proportion. Cover depends on the rate as well as the amount, so a cheaper tranche changes the capacity even at the same leverage.
What the interviewer asks next
How much mezzanine can stay in if senior is fixed at 4x and cover must be 2.0x?
Rates rise by 2 points on the senior debt only. What is cover now?
Why would a sponsor accept 13% mezzanine at all if senior debt is cheaper?
100A sponsor invests 100, sells half its stake for 150 in year 3 and the rest for 200 in year 5. The money multiple is 3.5x. Estimate the IRR by trial, and explain why it beats receiving the full 3.5x in year 5.Returns mathsHardLarge-cap buyout fundSecondaries and fund of funds
Try it first
Which is closest to the IRR?
Show the worked solution
About 36.9%, against 28.5% if the full 3.5x came back in year 5. Try 30%: the 150 and 200 are worth about 122 today, too much. Try 42%: about 87, too little. The rate that values them at exactly 100 is about 37%. The early 150 is discounted for three years, not five, which pulls the IRR up for the same money multiple.
How do you find an IRR by trial?
The IRR is the discount rate at which the money coming back is worth exactly what went in. Pick a rate, discount each inflow, and see whether the total is above or below 100. At 30%, 150 over 1.3 cubed is 68.3 and 200 over 1.3 to the fifth is 53.9, together 122.1, so 30% is too low; at 42% they come to 87.0, too high. Interpolating between the two lands near 37%, and the exact answer is 36.9%.
The sponsor puts in 100 and gets 150 back in year 3 and 200 in year 5; the value of those flows falls as the discount rate rises, from +22.1 at 30% to -13.0 at 42%, crossing zero at an IRR of 36.9%, well above the 28.5% of a single year-5 exit.
The relationship
$-100 + \frac{150}{(1+r)^3} + \frac{200}{(1+r)^5} = 0 \quad\Rightarrow\quad r \approx 36.9\%$
r the IRR
150 proceeds from selling half the stake in year 3
200 proceeds from the rest in year 5
What it says in wordsThe IRR is the rate that makes the discounted inflows exactly equal to the 100 invested.
Why does the partial exit beat 3.5x in year 5?
Rs 150 received in year 3 is worth more than Rs 150 received in year 5, just as a bonus paid this year is worth more than the same bonus promised later. IRR measures speed, so pulling part of the money forward lifts it even though the total returned, 3.5x, is the same. A single year-5 exit of 350 is 3.5 to the one-fifth, 28.5%. That gap is why sponsors like early dividends and partial sales, and why investors check the money multiple alongside the IRR.
Where candidates lose it
The common slip is treating 3.5x as if it all arrived in year 5 and answering 28.5%. The timing of the 150 is the whole question.
The other is guessing without a bracket. Name a rate that is too low and one that is too high, then interpolate out loud; an answer within a couple of points, reached that way, is what the interviewer is looking for.
What the interviewer asks next
If the 150 had come in year 2 instead of year 3, roughly what is the IRR?
Why might an investor prefer the single 3.5x exit despite the lower IRR?
How does a dividend recap in year 1 change the IRR and the money multiple?

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

Waterfall	Year 2	Year 4	Final carry
Deal by deal	+40	-20 clawback	20
Whole fund	+20	0	20

2	B's money multiple at year 3
1.5	what the proceeds earn over the next four years
3^(1/7)	the seven-year rate both deals deliver

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

Try it first

What is the difference between the two waterfalls?

Why does the clawback matter so much to investors?

Where candidates lose it

What the interviewer asks next

Try it first

How do you get the two IRRs in your head?

Why does the higher IRR not make B the better deal?

Where candidates lose it

What the interviewer asks next

Try it first

Why is the blended rate not the average of 8% and 13%?

Why does cover limit the debt before the leverage multiple does?

Where candidates lose it

What the interviewer asks next

Try it first

How do you find an IRR by trial?

Why does the partial exit beat 3.5x in year 5?

Where candidates lose it

What the interviewer asks next