Private Equity puzzles, solved step by step

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

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069About 10% of acquisition targets have a real accounting problem. A quality-of-earnings review flags 90% of the problem cases, but it also flags 15% of clean ones. The review flags your target. What is the chance it really has a problem?Probability and expected value in dealsHardMid-market buyout fundPortfolio operations team
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Given a flag, how likely is a real problem?
Show the worked solution
About 40%. Picture 1,000 targets: 100 have a problem and the review flags 90 of them. Of the 900 clean targets, it wrongly flags 15%, which is 135. So 225 targets are flagged and only 90 of those are real problems, 90 divided by 225. A good test on a rare problem still throws up more false alarms than true ones.
Why is the answer not 90%?
Think of a smoke alarm that goes off for almost every real fire, but also for burnt toast. In a house where fires are rare and toast is daily, most alarms are toast. The 90% is the chance of a flag when there is a problem; the question asks the reverse, the chance of a problem when there is a flag, and the two differ because clean targets are far more common. Counting people instead of using percentages makes the reversal visible.
Of 1,000 targets, the review flags 90 of the 100 real problems and 135 of the 900 clean ones, so only 90 of the 225 flags, 40%, are real problems.
The relationship
$P(\text{problem} \mid \text{flag}) = \frac{0.10 \times 0.90}{0.10 \times 0.90 + 0.90 \times 0.15} = \frac{0.09}{0.225} = 40\%$
0.10 the share of targets with a real problem
0.90 the chance the review flags a real problem
0.15 the chance the review flags a clean target
What it says in wordsThe chance a flag is real is the true flags divided by all flags, true and false.
What should a deal team do with a 40% flag?
Treat the flag as a reason to dig, not a reason to walk away. A flag lifts the chance of a problem from 10% to 40%, four times the starting level, but it still leaves the target more likely clean than not. A second, independent check changes the picture: if a forensic review with the same hit and false alarm rates also flags it, the chance rises to about 80%. The limit to state: that assumes the two reviews make independent mistakes, which two teams looking at the same ledgers may not.
Where candidates lose it
The common answer is 90%, mixing up the chance of a flag given a problem with the chance of a problem given a flag. Interviewers ask this precisely because the confusion is so natural.
The second slip is forgetting the false alarms entirely and answering 100% or near it. Write the four groups of 1,000 down, 90, 10, 135 and 765, before you compute anything.
What the interviewer asks next
A second independent review also flags it. What is the chance now?
How would the answer change if only 2% of targets had problems?
Which matters more to a fund, the false alarm rate or the miss rate, and why?
071A corridor has 100 closed lockers and 100 people. Person 1 opens every locker. Person 2 toggles every second locker, person 3 every third, and so on, until person 100 toggles only locker 100. How many lockers are open at the end, and which ones?Logic and brainteasersHardMid-market buyout fund
Try it first
How many lockers end up open?
Show the worked solution
Ten lockers are open: 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100, the perfect squares. Locker n is toggled once by each person whose number divides n, so its final state depends on how many divisors n has. Divisors come in pairs that multiply to n, which makes the count even, except for a perfect square, where one divisor pairs with itself. An odd count leaves the locker open.
What decides whether one locker ends open?
Think of a light switch flicked by everyone who walks past: if an odd number of people flick it, the light ends on. Locker n is flicked by person d exactly when d divides n, so the number of toggles is the number of divisors of n, and the locker ends open only if that number is odd. The whole puzzle reduces to one question: which numbers have an odd number of divisors? Do not simulate a hundred people; reason about one locker.
Every locker is toggled once for each divisor of its number, and only the ten perfect squares from 1 to 100 have an odd number of divisors, so they are the only lockers left open.
Why do only perfect squares have an odd number of divisors?
Pair each divisor with its partner: for 12, 1 goes with 12, 2 with 6, 3 with 4. Every divisor has a different partner, so the count is even and the locker closes. For a perfect square, one divisor is its own partner, 6 x 6 for 36, so it is counted once and the total turns odd. There are ten perfect squares from 1 to 100, so ten lockers stay open. The answer for any number of lockers N is the whole part of the square root of N.
The relationship
$\text{open lockers} = \lfloor \sqrt{N} \rfloor = \lfloor \sqrt{100} \rfloor = 10$
N the number of lockers, 100
floor of root N how many perfect squares lie between 1 and N
What it says in wordsThe open lockers are the perfect squares, and there are as many of them as the whole part of the square root of N.
Where candidates lose it
Candidates start simulating the first few people out loud and run out of time, or guess 50 because half the lockers seem to flip each round. The interviewer wants the switch from simulating everyone to analysing one locker.
The second miss is reaching divisors but then guessing primes. A prime has exactly two divisors, 1 and itself, which is even, so every prime locker ends closed.
What the interviewer asks next
Which lockers are toggled exactly twice?
With 1,000 lockers, how many are open?
Person 1 is absent. Which lockers are open now?
074A mezzanine lender puts in 100, earns 12% cash interest a year, and also receives warrants for 2% of the company's equity. The loan is repaid at the end of year 5, when the equity is worth 1,500. What is the lender's IRR?Credit and PIK mathsHardPrivate creditSpecial situations
Try it first
Roughly where does the IRR land?
Show the worked solution
About 16.3%. The lender pays 100, receives 12 a year for five years and gets the 100 back at the end. On its own that is exactly a 12% return. The warrants are 2% of equity worth 1,500, which is 30, received with the final payment, taking year 5 to 142. That one extra payment lifts the IRR by about 4.3 points.
What does the lender earn before the warrants?
Think of a fixed deposit at 12% that pays its interest every year and returns the principal at maturity: it earns exactly 12%. A loan bought at par that pays its full coupon in cash and is repaid at par earns exactly its coupon rate, so the first 12 points of the IRR need no calculation at all. The question is only about what the warrants add on top.
The lender pays 100, receives 12 a year for four years, and in year 5 collects 12 of interest, 100 of principal and 30 of warrant value, a stream with an IRR of 16.3% against 12.0% for the coupon alone.
How do you estimate the warrant's effect without a calculator?
The 30 arrives in five years. At around 16%, five years of discounting roughly halves money, so the warrants are worth about 14 to 15 in today's terms. Spread that extra 14 or so of value across five years of a 100 loan and it adds roughly 4 points a year to the coupon, which puts the IRR near 16%. Then confirm by trial: at 16%, the flows of 12 a year and 142 at the end discount to slightly above 100, so the IRR is a touch above 16%, 16.3%.
The relationship
$100 = \sum_{t=1}^{4} \frac{12}{(1+r)^t} + \frac{142}{(1+r)^5} \quad\Rightarrow\quad r \approx 16.3\%$
12 cash interest each year, 12% of 100
142 year 5: interest 12, principal 100 and warrants 30
r the lender's IRR
What it says in wordsThe IRR is the rate at which the coupons, the repayment and the warrant value discount back to the 100 lent.
Say the risk too. The warrant is worth 30 only if the equity really is worth 1,500 at exit. If equity were worth 500, the warrants would bring 10 and the IRR would fall to about 13.5%. That is the point of mezzanine: a contractual coupon plus a slice of the upside, priced for sitting behind the senior lenders.
Where candidates lose it
The common slip is adding the warrants as if they paid every year: 30 on 100 is 30%, plus 12%, gives an absurd 42%. The warrant value arrives once, at the end, and is spread over five years in the IRR.
The other miss is ignoring the coupon's own return and computing only the warrant uplift, or treating the 12% as compounding PIK. Here the interest is paid in cash each year, so it does not accrue onto the principal.
What the interviewer asks next
What if the 12% were PIK, added to the loan each year instead of paid in cash?
Equity is worth 500 at exit instead. What is the IRR?
Why would a sponsor give away warrants rather than pay a higher coupon?
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?
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?
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?

0.10	the share of targets with a real problem
0.90	the chance the review flags a real problem
0.15	the chance the review flags a clean target

N	the number of lockers, 100
floor of root N	how many perfect squares lie between 1 and N

12	cash interest each year, 12% of 100
142	year 5: interest 12, principal 100 and warrants 30
r	the lender's IRR

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

Try it first

Why is the answer not 90%?

What should a deal team do with a 40% flag?

Where candidates lose it

What the interviewer asks next

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What decides whether one locker ends open?

Why do only perfect squares have an odd number of divisors?

Where candidates lose it

What the interviewer asks next

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What does the lender earn before the warrants?

How do you estimate the warrant's effect without a calculator?

Where candidates lose it

What the interviewer asks next

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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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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

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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

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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

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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

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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