Private Equity puzzles, solved step by step

28%. The easy route is through the opposite event. The chance there is no restatement is 80% and the chance the customer stays is 90%; because the risks are independent, the chance of neither is 0.8 x 0.9, which is 72%. At least one problem is everything else, 100% less 72%, or 28%. Adding 20% and 10% gives 30% and double counts the 2% chance of both.

Why not just add the two chances?

Picture two friends who each might be late for dinner. If you add their chances of being late, the evening where both are late gets counted once for each of them. Adding probabilities works only when the events cannot happen together; when they can, the overlap is counted twice. Here both problems happen together 0.2 x 0.1, or 2%, of the time, so 20 plus 10 overshoots by exactly that: 30 less 2 is 28.

Why is one minus none the safest route?

Because the opposite of at least one is a single clean case: nothing goes wrong. At least one problem is everything except the case where neither happens, so you multiply the two chances of no problem and subtract from one. It scales without effort. With five independent risks of 10% each, the chance none happens is 0.9 to the fifth, about 59%, so at least one is about 41%, a number that adding would put at 50%.

Is independence a fair assumption in diligence?

Usually not, and saying so earns credit. A company that needs a revenue restatement may have weak controls or stretched customer relationships, so the two risks tend to move together, which raises the chance of both and lowers the chance of at least one below 28%. If the risks were perfectly linked, so the customer leaves only when revenue is also restated, the answer would fall to 20%. The 28% is the answer the question asks for; the correlation point is the judgement the interviewer is listening for.

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.

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.

Carry of 50 is paid, and a 10% LP that sells receives 45. The transfer is a sale at 500 against a cost of 250, so the old fund books a profit of 250 and the GP takes 20% of it, 50. The remaining 450 belongs to the old fund's LPs; a 10% holder gets 45 in cash. The GP earns that carry at a price it helped set, which is the conflict LPs examine.

What is a continuation vehicle, in plain terms?

Picture a shopkeeper who manages a shop for a group of owners and is paid a share of the profit when the shop is sold. Instead of selling to an outsider, he sets up a new group, with some new owners and some old ones, and sells the shop to that group. A continuation vehicleA new fund, managed by the same GP, that buys one or more assets from the GP’s older fund so they can be held for longer. is the same GP selling an asset from its old fund to a new fund it also manages, so existing LPs can take cash or roll into the new vehicle. To the old fund the transfer is a sale, so its carry is calculated as though the asset had been sold.

Why does the transfer price matter so much?

Because the GP is on both sides of it. A higher price raises the carry the GP collects today; a lower price makes the new vehicle, which the GP also manages and earns carry from, a cheaper purchase. At 450, carry falls to 40 and old LPs get 410; at 550, carry is 60 and they get 490. Each 10 of price moves carry by 2 and the selling LPs by 8. That is why these deals usually rely on a third-party lead buyer setting the price and a fairness opinion, so the number is tested by someone without the conflict.

What has the simple version left out?

Carry in most funds is calculated across the whole fund, not deal by deal. If the old fund has losses elsewhere, or has not yet returned LP capital and the hurdle, the crystallised carry could be lower or held back in escrow. Transaction costs, a discount to NAV and the GP rolling part of its carry into the new vehicle all change the cash figures. Name those three, then give 50 and 45 as the answer on the question's terms.

He proposes 98 for himself, 0, 1, 0 and 1. Solve from the end. With two left, partner 4 takes everything. With three, partner 3 buys partner 5 for one unit. With four, partner 2 buys partner 4 for one. With five, partner 1 needs two votes besides his own and buys the two partners who would get nothing in the next round, 3 and 5, for one unit each.

Why start from the end rather than the beginning?

Think of planning a train journey with a fixed arrival time: you work back from when you must arrive to when you must leave. Each partner's vote depends on what they would get if the current proposer were removed, so you can only value a vote once you know the next round, and the only round you can solve directly is the last one. With two partners left, partner 4 proposes 100 for himself and 0 for partner 5. His own vote is one of two, which is half, and half is not fewer than half, so it passes.

With three left, partner 3 needs two yes votes. Partner 5 gets nothing if partner 3 is removed, so one unit buys him: 99, 0, 1. With four left, partner 2 needs two votes. Partner 4 gets nothing in the three-partner round, so one unit buys him: 99, 0, 1, 0.

Which votes does the most senior partner buy, and why so cheaply?

With five partners he needs three yes votes: his own and two more. Look at the four-partner row. Partners 3 and 5 get nothing there, while partner 4 gets one unit and partner 2 gets 99. The cheapest votes always belong to whoever would be worst off in the next round, and here that is partners 3 and 5, so one unit each beats their alternative of zero. He keeps 100 less 2, which is 98. The answer feels unfair, and that is the point: in this game power comes from position in the sequence, not from any idea of a fair share.

Which assumptions does the answer rest on?

Three, and naming them is part of the answer. The tie rule, the tie-breaking preference and pure self-interest each change the split if you alter them. If a plan needed a strict majority, the proposer would need more votes. If an indifferent partner voted yes, the bought votes would cost zero instead of one, and the senior partner would keep 100. And real partners care about fairness and future rounds of carry, which is why actual carry allocations are set by negotiation and track record rather than by this logic.

About 25 lakh litres a day. One crore people at four per household is 25 lakh households. A household uses about 0.8 litres a day: 0.25 for tea, 0.35 for drinking and 0.2 for curd and cooking. That is 20 lakh litres at home. Tea stalls, restaurants and sweet shops add perhaps a quarter more, about 5 lakh litres, for roughly 25 lakh in total, or 250 ml a person.

Why build the estimate around a household?

Think about how milk actually enters a home: one or two packets delivered each morning, bought for the whole family. A household is the unit that buys milk and the unit whose uses you can picture, so an assumption made at household level is one the interviewer can check against their own kitchen. One crore people at four per household gives 25 lakh households. The household size is an assumption to state, not a statistic to recite; a city with more single migrants would have smaller households.

Then list the uses. Tea for four people, two cups each, takes about a quarter of a litre. A child's glass or two of drinking milk adds about 0.35. Curd, cooking and the odd sweet take another 0.2. That is 0.8 litres per household per day, and 25 lakh households make 20 lakh litres.

What does the household count miss?

Milk drunk outside the home. Tea stalls, restaurants, hotels and sweet shops are large buyers, and leaving them out understates demand in a city where many people take their tea at a stall. Adding a quarter to household use gives 5 lakh litres more and a total of about 25 lakh. Then do the sanity check: 25 lakh litres over 1 crore people is 250 ml a person a day, roughly one glass. If that sounds wrong for the city you have in mind, adjust the assumption you trust least.

How would a fund use this number?

As a frame, not a forecast. For a dairy deal the question is what share of that daily volume the target already sells and what share it could plausibly win, which needs the packaged versus loose milk split more than the total. A dairy selling 2 lakh litres a day here would hold about 8% of the market. Say the weakest assumption out loud, probably the out-of-home share, and say how you would test it: by counting the tea stalls on a few streets.

No, the IRR is unchanged at about 26%. 2x in three years means the money doubles every three years. 4x in six years is two doublings in six years, the same pace. The cube root of 2 is 1.26, and the sixth root of 4 is the same number, so both IRRs are 26.0%. The multiple doubled; the annual rate did not move.

Why does a bigger multiple not mean a better IRR?

Think of two runners: one covers 2 km in 10 minutes, the other 4 km in 20 minutes. The second went further, but neither ran faster. IRR measures pace, and the multiple measures distance, so doubling the multiple over double the time leaves the pace exactly where it was. Three years to double, then another three years to double again, is the same doubling time throughout.

How do you show it in two lines?

Write both as roots. The first IRR is 2 to the power of one third, less one. The second is 4 to the power of one sixth, less one. Since 4 is 2 squared, 4 to the one sixth is 2 to the two sixths, which is 2 to the one third: the same number, 1.26. So both are 26.0%. If the slip had produced only 3x by year six, the IRR would fall to 20.1%, a real cost of the delay.

So is the slipped exit just as good?

Not necessarily, and the judgement is the point of the question. Whether 4x in six years is as good depends on what the money could have done in years four to six: if the fund could have redeployed it at 26%, the two are equal; if not, the larger multiple is better. For the GP, 4x means more carry in absolute terms. For the LPs, the delay holds capital in an older fund for longer. And three extra years of exposure carry three more years of risk that the second doubling does not happen.

About 30% of the gross wealth. At 15%, 1 rupee grows to 16.37 in 20 years; at 13% it grows to 11.52. The investor keeps 11.52 over 16.37, about 70%, so fees take about 30%. Each year the fee removes 1 less 1.13/1.15, about 1.7%, of the wealth, and twenty years of that compounds to a loss of nearly a third.

Why is the fee's share so much bigger than 2%?

Think of a water tank with a small leak. A leak of a cupful an hour sounds trivial, but over a day it empties a good part of the tank, and every cup lost is water that would otherwise have been there. A fee taken every year removes not just that year's money but everything that money would have earned in all the years after, so the loss compounds just like the return. The fee is 2 points of a 15% return, but its effect on the end wealth is far larger.

How do you get to 30% without a calculator?

Work with the ratio, not the two big numbers. Each year the investor keeps 1.13 over 1.15 of what the gross fund keeps, about 0.983, so over twenty years the investor keeps 0.983 to the twentieth. Use the shortcut that (1 minus x) to the n is roughly e to the minus nx: 20 times 0.0174 is 0.35, and e to the minus 0.35 is about 0.70. So the investor keeps about 70% and the fees take about 30%. Checking against the full numbers, 11.52 over 16.37 is 0.704.

Why does an LP care about this arithmetic?

Because fees look small as annual rates and large as outcomes. The longer capital stays invested, the larger the share of wealth that a fixed annual fee takes, so long holding periods and multi-layer structures, such as a fund of funds charging on top of underlying funds, deserve the closest look. Say the limitation: real private equity fees are charged on committed or invested capital rather than as a clean drag on returns, and carry is a separate deduction, so this is the shape of the effect rather than an exact fund calculation.

The score is 25 today, and at 25% growth the margin must reach 15%. The Rule of 40 adds revenue growth and profit margin: 35 plus minus 10 is 25, well short of 40. If growth slows to 25%, the margin has to be 40 less 25, or 15%. That is a swing of 25 points of margin, from minus 10% to plus 15%, just to hold the bar.

What is the Rule of 40 actually measuring?

Think of a young restaurant that can either open new branches fast and lose money doing it, or open slowly and keep every branch profitable. Either can be healthy; losing money while also growing slowly is not. The Rule of 40 adds growth and margin so that a software company can trade one for the other, and treats a combined score of 40 or more as a sign of a healthy balance. Growth of 35% and a margin of minus 10% score 25, which says the company is spending more on growth than the growth is earning.

Why does slowing growth make the margin job so large?

Because the score falls point for point. Losing 10 points of growth drops the score from 25 to 15 if nothing else moves, so the margin has to cover both the existing 15-point shortfall and the 10 points just lost. That is 25 points of margin. In practice it means cutting sales and marketing spend that was buying the growth, which is often exactly what happens when a buyout fund takes over a software business: growth slows by choice and margin is rebuilt.

What should you say about the rule's limits?

That it is a heuristic, not a law. Different investors use different margin definitions, EBITDA, operating or free cash flow, and the same company can pass on one and fail on another. The rule also treats a point of growth and a point of margin as equal, which is untrue for a small company where growth compounds into a much larger future business. Use it to frame the trade-off, then look at what is driving each side: retention, sales efficiency and gross margin.

About 1,133, so the winner overpays by about 133 on average. Each estimate is right on average, but the auction picks the highest of five. Five draws spread evenly between 800 and 1,200 sit on average at the sixths of the range, and the top one averages 800 plus five sixths of 400, about 1,133. To avoid overpaying, each bidder should bid below its own estimate, and by more as the number of bidders grows.

If every estimate is unbiased, how can the winner be wrong?

Picture five friends guessing the number of sweets in a jar. On average they are right, but the one who guesses highest is almost certainly too high. Each estimate is unbiased on its own, but the auction does not pick a random estimate; it picks the highest, and the highest of several noisy guesses is biased upwards. That is the {term('winner’s curse', 'The tendency for the winner of an auction, chosen because its estimate was highest, to have overestimated the value of what it bought.')}: winning is itself evidence that you overestimated.

Why does the top estimate average five sixths of the range?

Five points dropped at random on a line split it, on average, into six equal gaps. So the expected sorted estimates sit at one sixth, two sixths and so on up to five sixths of the way from 800 to 1,200: 867, 933, 1,000, 1,067 and 1,133. The top one is 400 x 5/6, or 333, above 800, which is 1,133. With n bidders the top estimate averages n over n plus 1 of the range, so ten bidders push it to about 1,164.

What does a disciplined bidder do with this?

Shade the bid. A bidder that wants to break even when it wins must bid as though its estimate is the highest of five, which in this setup means bidding about 133 below its estimate, and more in a crowded auction. That is the case for walking away from processes with many bidders and for building value on what the buyer can change rather than on a higher estimate of the same cash flows. Say the limits: real bidders have different information and different synergies, so not all of the gap is error, and a bidder with genuine private information is less exposed.

1.0x with no paydown and 1.75x with 300 repaid. The purchase price is 1,000, with 600 of debt and 400 of equity. With flat EBITDA and the same multiple, the exit value is again 1,000. If debt is still 600, equity is still 400: 1.0x. If 300 has been repaid, debt is 300 and equity is 700, which is 1.75x, an IRR of about 12% over five years.

Where does the return come from if nothing grows?

Think of buying a flat for 1 crore with a 60 lakh home loan, then renting it out and using the rent to pay down 30 lakh of the loan over five years. If the flat is still worth 1 crore, your share has gone from 40 lakh to 70 lakh, though the flat itself is unchanged. Debt paydown moves value from the lenders to the owner: the business is worth the same, but a larger slice of it belongs to the sponsor. The company's own cash flow is doing the repaying, so the sponsor's cheque never changes.

What are the numbers, step by step?

Entry: 10 x 100 is 1,000. Sixty per cent debt is 600, so equity is 400. Exit with nothing repaid: 1,000 less 600 is 400 of equity, 1.0x, a zero return over five years. Exit with 300 repaid: 1,000 less 300 is 700 of equity, and 700 over 400 is 1.75x. Over five years, 1.75x is about 11.8% a year: 1.12 to the fifth is 1.76, so a shade under 12%.

Is 300 of paydown realistic, and what is left out?

It means 60 a year of free cash flow after interest and tax on a business with EBITDA of 100, which is possible for a capital-light company and unlikely for a capital-hungry one. Paydown is the most reliable of the three return levers, alongside EBITDA growth and multiple change, because it depends on cash the business already generates rather than on the future. Say what the simple version ignores: transaction fees at entry and exit, any cash left on the balance sheet, and the risk that interest rates or a downturn absorb the cash meant for repayment.