Private Equity puzzles, solved step by step

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

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006A sponsor invests 300 of equity in a buyout. Four years later it sells, and the equity is worth 750. What are the MOIC and the IRR?Mental paper LBOsWarm upWarburg PincusSan Francisco · 2014
Try it first
Which IRR is closest?
Show the worked solution
The MOIC is 2.5x and the IRR is about 25.7%. MOIC is money out over money in: 750 divided by 300. The IRR is the yearly rate that compounds 300 into 750 over four years, the fourth root of 2.5. Take the square root twice: 2.5 becomes 1.581, then 1.257. Check: 1.25 to the fourth is 2.44, just short of 2.5, so the rate sits a little above 25%.
Why work out the multiple before the rate?
Picture a plant that grew from 30 centimetres to 75 in four years. Asking how many times bigger it got is one division; asking how fast it grew each year needs a root. Getting the multiple first strips the problem down to one number, 2.5, so the only hard step left is a fourth root. In a paper LBO the multiple is also what the interviewer will ask for first, because it is the number a sponsor quotes when it talks about a deal.
Compounding at 25.7% a year takes 300 to 377, 474, 596 and then 750, the same x1.257 step each year, which is why a 2.5x multiple over four years is a 25.7% IRR and not the 37.5% a straight line suggests.
How do you take a fourth root in your head?
A fourth root is a square root of a square root. The square root of 2.5 is about 1.58, because 1.58 squared is about 2.5, and the square root of 1.58 is about 1.257, because 1.26 squared is 1.59. Then check from the other side with a round rate you can compound: 1.25 squared is 1.5625, and squared again is 2.44. That is a little short of 2.5, so the IRR is a little above 25%, which matches 25.7%.
The relationship
$\text{MOIC} = \frac{750}{300} = 2.5 \qquad \text{IRR} = 2.5^{1/4} - 1 \approx 25.7\%$
MOIC multiple on invested capital: equity out divided by equity in
1/4 the fourth root, because the money is invested for four years
What it says in wordsThe multiple tells you how much the money grew; the root of the multiple tells you how fast it grew each year.
What would change the IRR without changing the multiple?
Time and interim cash. The same 2.5x earned over five years is about 20%, and over three years about 36%, so a sponsor always quotes both numbers together. Say also that this calculation assumes one cash flow in and one out. A dividend paid in year two, or a second equity cheque in year one, means the IRR has to be solved over several cash flows, and the shortcut gives only an approximation.
Where candidates lose it
The fast wrong answer is 37.5%: a 150% gain split evenly over four years. It is the most common error in paper LBOs because the multiple and the rate get mixed up under time pressure.
The other loss is a correct number with no check. A fourth root guessed at 26% sounds lucky; 1.25 to the fourth equals 2.44, so slightly above 25% sounds reasoned.
What the interviewer asks next
What IRR does 2.5x give over five years?
The sponsor also took a 100 dividend in year two. Is the IRR higher or lower, and roughly by how much?
What exit equity value gives a 30% IRR over four years?
Asked at Warburg Pincus, Private Equity, San Francisco, 2014 (Wall Street Oasis): The interviews consisted of case studies and technical questions (paper LBO, basic returns, IRR, etc.)
010In your head, and out loud: what are 48 x 52, 97 x 103 and 499 x 501?Mental mathsWarm upGeneral AtlanticNew York · 2026
Try it first
What is 97 x 103?
Show the worked solution
2,496, 9,991 and 249,999. Each pair sits evenly either side of a round number, so the product is that number squared less the gap squared. 48 x 52 is 50 squared less 2 squared, 2,500 less 4. 97 x 103 is 10,000 less 9. 499 x 501 is 250,000 less 1. Say the rule first, then the three answers come in seconds.
What do the three pairs have in common?
Look at where each pair sits. 48 and 52 are both 2 away from 50; 97 and 103 are both 3 away from 100; 499 and 501 are both 1 away from 500. Two numbers placed symmetrically around a round number multiply to that number squared, less the gap squared. In symbols, (a minus b) times (a plus b) is a squared minus b squared. The round number is easy to square, the small gap is easy to square, and one subtraction finishes the job.
Cutting the 2 by 48 strip from a 50 by 50 square that is missing a 2 by 2 corner and laying it along the bottom gives a rectangle 48 wide and 52 tall, so 48 x 52 equals 2,500 less 4, which is 2,496.
Why is the product always a little less than the square?
Think of a square garden 50 metres on each side. If you make it 2 metres longer and 2 metres narrower, the strip you gain along one side is 2 by 48, but the strip you lose is 2 by 50. You lose slightly more than you gain, and the shortfall is exactly the small 2 by 2 corner, which is why the answer is always the square less the gap squared. That is the picture worth having in your head when the interviewer asks why the trick works.
The relationship
$(a-b)(a+b) = a^2 - b^2 \qquad 499 \times 501 = 500^2 - 1^2 = 249{,}999$
a the round number in the middle, here 500
b the distance from it to each factor, here 1
What it says in wordsSquare the middle, square the gap, subtract.
Why would a growth equity interviewer bother with this?
Because deal conversations are full of quick products: 48 months of a 52 lakh run rate, a multiple of 9.7x on 10.3 crore of EBITDA. The test is not whether you know the identity but whether you look for structure before you start grinding through digits. If the numbers are not symmetric, make them so: 47 x 55 is 51 squared less 4 squared, 2,601 less 16, which is 2,585. Say the method aloud so the interviewer hears the route, not just the result.
Where candidates lose it
The common loss is grinding through long multiplication out loud, which works but takes a minute and invites a slip in the carries. The interviewer is looking for the moment you spot the symmetry.
The second trap is getting 10,009 for 97 x 103 by adding the gap squared. The product is always below the square, because the lost strip is bigger than the gained one.
What the interviewer asks next
What is 51 x 49?
What is 46 x 54, and what is 46 x 55?
Square 105 in your head.
Asked at General Atlantic, Generalist, New York, 2026 (Wall Street Oasis): The first round was behavioral with mental math at the end.
012A company's enterprise value is 800. It has 500 of senior debt, 400 of subordinated notes and equity beneath both. In a restructuring, what does each class recover? What changes if enterprise value is 1,000 instead?Leverage and capital structureWarm upKKRNew York · 2025
Try it first
At an enterprise value of 800, what do the subordinated notes recover?
Show the worked solution
At 800, senior recovers 100%, the notes 75% and equity nothing. Value is paid strictly by priority. Senior takes its 500 first, leaving 300 for the 400 of notes, which is 75%. At 1,000, value covers all 900 of debt, so both classes recover in full and equity is left with 100. The 200 increase in value goes 100 to the notes and 100 to equity.
In what order does the value get paid out?
Think of a row of buckets under one tap, each bucket overflowing into the next. The first fills completely before a drop reaches the second. A capital structure works the same way: each class is paid in full before the class below it receives anything. With 800 of value, the senior bucket takes 500 and is full. The remaining 300 flows into the notes bucket, which holds 400, so it is 75% full. Nothing reaches equity.
At an enterprise value of 800 the senior debt of 500 is paid in full, the subordinated notes get the remaining 300 of their 400, a 75% recovery, and equity gets nothing; at 1,000 both debt classes are whole and equity keeps 100.
Which security does the interviewer care about most?
The one the value line cuts through. That tranche is the fulcrum securityThe most senior class of claims that is not repaid in full, and so typically ends up owning the restructured company., here the subordinated notes at an EV of 800. Every change in enterprise value between 500 and 900 lands entirely on the notes, so they are the class whose price moves with the valuation argument while senior sits at par and equity sits at zero. In a restructuring the fulcrum holders usually swap their claim for the new equity, which is why distressed investors spend their time on it.
What changes if enterprise value is 1,000?
The line clears the debt. Senior is still paid 500, the notes are now paid their full 400, and the 100 left over belongs to equity. The 200 of extra value is split 100 to the notes and 100 to equity, and none of it reaches senior, which was already whole. That asymmetry is the lesson: a senior lender's upside is capped at par, while the junior classes capture the swing. Say the limitations too: real cases add administrative claims ahead of senior debt, and negotiated outcomes sometimes give junior classes a little value to secure their agreement.
Where candidates lose it
The usual slip is sharing value pro rata, giving every creditor 800 over 900, about 89%. That ignores priority, which is the whole point of having senior and subordinated claims.
The second miss is stopping at the recoveries. The follow-up about 1,000 is there to see whether you notice which tranche absorbs the change in value.
What the interviewer asks next
At what enterprise value does equity start to recover anything?
If the senior debt were secured on assets worth only 400, how would the answer change?
Why might a distressed fund buy the notes at 60 when they recover 75?
Asked at KKR, Distressed Debt, New York, 2025 (Wall Street Oasis): What are your weaknesses? A capital structure question with enterprise value.
059Quick-fire round, about ten seconds each: 17 x 23, 1.08 cubed, 7/8 as a percentage, and 45% of 360. Give each answer and the trick that gets you there.Mental mathsWarm upOaktree Capital ManagementLos Angeles · 2022
Try it first
Which trick turns 17 x 23 into a one-step sum?
Show the worked solution
391, about 1.26, 87.5% and 162. 17 x 23 is (20 - 3)(20 + 3), so 400 - 9. 1.08 cubed is about 1 + 3 x 0.08 + 3 x 0.0064, which is 1.259, close to the exact 1.2597. Seven eighths is one less one eighth, 100% - 12.5%. And 45% of 360 is half of 360 less a twentieth of it, 180 - 18.
Why name the trick rather than just give the number?
A shopkeeper who adds a bill in his head is not calculating faster than you; he has a handful of shortcuts he has used ten thousand times. Speed in a quick-fire round comes from recognising which shortcut a question is built for, so say the shortcut as you give the answer. It shows the interviewer the answer is reliable, and it protects you when a number comes out slightly off: the method is audible even if the last digit slips.
Each of the four answers comes from one named trick: 17 x 23 is 400 less 9, 1.08 cubed is about 1 plus 0.24 plus 0.019, seven eighths is 100% less 12.5%, and 45% of 360 is 180 less 18.
How do the two harder ones work?
For 1.08 cubed, expand (1 + x) cubed as 1 + 3x + 3x squared + x cubed with x = 0.08. When x is small, the first two terms carry almost everything and the third is a small correction: 1 + 0.24 + 0.0192 is 1.2592, within 0.001 of the exact 1.2597. That is three years of 8% growth, about 26%, which is why the trick earns its place on a returns desk. For 17 x 23, check the trick fits: it only works when the two numbers sit the same distance either side of a round number.
The relationship
$(a-b)(a+b) = a^2 - b^2 \qquad (1+x)^3 \approx 1 + 3x + 3x^2$
a the round number in the middle, here 20
b the distance either side, here 3
x the growth rate, here 0.08
What it says in wordsA product of two numbers equally spaced around a round number is that number squared less the gap squared; a small rate cubed is about one plus three times the rate.
Fractions and percentages are best memorised in eighths: 12.5%, 25%, 37.5% and so on up to 87.5%. Percentages of awkward numbers split into easy pieces: 10%, 5%, 50%. With those four habits, most of what turns up in a quick-fire round is one step.
Where candidates lose it
The trap is going silent and grinding long multiplication for 17 x 23 or 1.08 cubed. Ten seconds is not enough, and the interviewer hears nothing to rescue. Reach for the shortcut out loud.
The second slip is on 1.08 cubed: answering 1.24 by tripling 8% and forgetting the compounding term. Three years at 8% is about 26%, not 24%; the extra two points are the growth on the growth.
What the interviewer asks next
What is 1.1 to the power 5, to two decimals?
What is 48 x 52?
If a date falls on a Monday this year, what day is it next year?
Asked at Oaktree Capital Management, Generalist, Los Angeles, 2022 (Wall Street Oasis): Quick mental math questions are unexpected. Was asked around 6 of them.
098A PIK note of 200 accrues 12% a year, compounded annually, for 5 years, with nothing paid until maturity. How much is owed at maturity, and how much more is that than five years of the same 12% paid in cash?Credit and PIK mathsWarm upAres ManagementNew York · 2026
Try it first
Roughly how much is owed after five years?
Show the worked solution
About 352.5 is owed, about 32.5 more than a cash-pay note's 320. PIK interest is added to the balance each year: 224, 250.9, 281.0, 314.7, then 352.5. A cash-pay note returns 24 a year plus 200 at the end, 320 in all. The extra 32.5 is interest on interest, which is why a PIK claim grows faster than its coupon suggests.
What does paid in kind actually mean for the balance?
A PIKPayment in kind: interest that is not paid in cash but added to the amount owed, so the loan balance grows instead. note pays its interest with more debt. It is like a credit card where you pay nothing and the interest is added to the bill each month, so next month's interest is charged on the bigger bill. Because each year's interest joins the balance, the 12% is earned on a growing base, and the claim compounds. After one year the balance is 224, after two 250.9.
A 200 PIK note at 12% compounds to 224, 250.9, 281.0, 314.7 and 352.5 over five years, while a cash-pay note's principal plus interest received rises in a straight line to 320, so the PIK claim ends 32.5 higher.
The relationship
$200 \times 1.12^5 = 352.5 \qquad 200 + 5 \times 24 = 320 \qquad 352.5 - 320 = 32.5$
1.12^5 five years of 12% compounding, about 1.762
24 the yearly cash interest on 200 at 12%
What it says in wordsPIK compounds the balance; cash-pay adds the same interest each year without compounding, and the gap is the interest on interest.
Why does a private credit lender or a sponsor care?
The borrower keeps its cash today, which helps a company that is growing or stretched. The price is a claim that grows every year, so at exit the sponsor's equity sits behind a bigger debt than the original 200. For the lender, PIK means more risk: nothing is received for five years, and if the company fails the larger balance may not be recovered. Lenders usually charge a higher rate for PIK than for the same loan paid in cash.
Where candidates lose it
The common slip is 320: adding five years of simple interest. That treats PIK as if the interest were paid in cash, missing that it compounds on the balance.
The other is calling the 32.5 free money for the lender. It is compensation for waiting five years with nothing in hand, and for the risk that the balance is never paid.
What the interviewer asks next
What is owed after 5 years if the PIK compounds semi-annually at 6% a half-year?
If the cash-pay interest could be reinvested at 12%, how do the two compare?
How does a PIK toggle option change who bears the risk?
Asked at Ares Management, Generalist, New York, 2026 (Wall Street Oasis): Asked me basic behaviorals as well as some technicals around accounting as well as PIK interest.

MOIC	multiple on invested capital: equity out divided by equity in
1/4	the fourth root, because the money is invested for four years

a	the round number in the middle, here 500
b	the distance from it to each factor, here 1

a	the round number in the middle, here 20
b	the distance either side, here 3
x	the growth rate, here 0.08

1.12^5	five years of 12% compounding, about 1.762
24	the yearly cash interest on 200 at 12%

Try it first

Why work out the multiple before the rate?

How do you take a fourth root in your head?

What would change the IRR without changing the multiple?

Where candidates lose it

What the interviewer asks next

Try it first

What do the three pairs have in common?

Why is the product always a little less than the square?

Why would a growth equity interviewer bother with this?

Where candidates lose it

What the interviewer asks next

Try it first

In what order does the value get paid out?

Which security does the interviewer care about most?

What changes if enterprise value is 1,000?

Where candidates lose it

What the interviewer asks next

Try it first

Why name the trick rather than just give the number?

How do the two harder ones work?

Where candidates lose it

What the interviewer asks next

Try it first

What does paid in kind actually mean for the balance?

Why does a private credit lender or a sponsor care?

Where candidates lose it

What the interviewer asks next