Private Equity puzzles, solved step by step

About 20%, and 20.1% exactly. Bracket it with round rates you can compound in your head. 1.2 to the fifth is 2.49 and 1.25 to the fifth is 3.05, so 2.5x sits right at the bottom of the bracket, a hair above 20%. Straight division, 150% over five years, gives 30% and overstates the return badly because it ignores compounding.

How do you compound 1.2 five times in your head?

Do it by squaring, the way you would fold a sheet of paper. 1.2 squared is 1.44. Square again to get the fourth power: 1.44 times 1.44 is about 2.07. One more 1.2 gives about 2.49. Chaining small multiplications you trust is safer than reaching for a formula you half remember, and it is exactly what the interviewer wants to watch. The same chain at 1.25 runs 1.56, then 2.44, then 3.05.

Why is the answer not 30%?

Think of a savings account that pays interest on interest. If it pays 20% a year, the gain in year five is far larger than the gain in year one, because it is earned on a bigger balance. An IRR is a compound rate, so the five years of gains are not equal slices of the 150% total; the later years carry more of it. Dividing 150 by five assumes every year earns 30 on the original 100, which is simple interest. The compound rate needed is lower: 20.1%.

What table is worth memorising before a buyout interview?

A small grid of multiples by holding period. Once you know that 2x in five years is about 15%, 2.5x is about 20% and 3x is about 25%, most return questions become a lookup with a small adjustment. If you need more precision, interpolate between rungs: 2.5 is 0.01 of the way along a 0.56 gap from 2.49 to 3.05, which adds about a tenth of a point to 20%, giving 20.1%. State that this assumes a single cash flow in and a single cash flow out; interim dividends would change it.

360, or 90% of the sponsor's cost. Debt at 2x EBITDA of 120 is 240; at 5x it is 600. The 360 of new borrowing is paid out as a dividend, and 360 over the original 400 is 90%. The sponsor gets most of its money back in year three while keeping full ownership, and the business now carries two and a half times the debt it did.

What exactly is a dividend recap doing?

Think of a family that has paid down most of its home loan, then takes a fresh, larger loan against the same house and spends the cash. Nothing about the house changed; the family simply took equity out and left a bigger loan behind. A dividend recapA recapitalisation in which a company borrows new debt and pays the proceeds to its shareholders as a dividend. borrows against the company's current earnings and hands the cash to the owner, so the only new money is the increase in debt. Here that increase is 600 less 240, which is 360.

Has the sponsor made money, or just moved it?

Mostly moved it. Suppose the business is worth 10x EBITDA, which is 1,200. Before the recap the sponsor's equity is worth 1,200 less 240 of debt, which is 960. After it, the equity is worth 1,200 less 600, which is 600, plus 360 already in the bank: the same 960. A recap does not create value; it changes when the sponsor is paid and how much risk sits on the remaining equity. The returns arithmetic still improves, because cash returned in year three lifts the IRR and takes 90% of the cost off the table, which is why sponsors use the tool when credit markets are open.

Who carries the risk afterwards?

The company and its lenders. At an assumed 8% interest rate, interest rises from 19.2 to 48 a year, so EBITDA covers interest 2.5 times instead of 6.25. A 20% fall in EBITDA, to 96, now takes cover down to exactly 2x, a level at which many lenders start to worry, while the sponsor's money is already mostly home. Say the limitation too: lenders price this risk, covenants may cap the payout, and a board must be satisfied the company stays solvent after paying the dividend.

The LPs earn 1.56x net. The 850 invested at 2.0x returns 1,700. LP profit is 1,700 less the 1,000 they paid in, fees included, which is 700. Carry at 20% of 700 is 140, so LPs receive 1,560 on 1,000. Fees cost 0.30x and carry 0.14x, so a 2.0x gross fund delivers 1.56x.

Why do fees and carry need separate treatment?

Think of a farmer who hands a contractor 100 kilos of seed. The contractor keeps 15 kilos as a fee and sows 85, then takes a fifth of whatever extra grain the harvest brings. Fees shrink the base that gets invested, while carry takes a share of the profit after the LPs have their money back. They hit at different points, so they cannot simply be added as percentages, and trying to do it in one step is where most answers go wrong.

So run it in order. Fees of 150 mean only 850 reaches companies. At 2.0x, 850 becomes 1,700. LPs paid in 1,000 in total, fees included, so the profit is 700. Carry at 20% of 700 is 140. The LPs receive 1,700 less 140, which is 1,560, a net multiple of 1.56x.

How big is the gap between gross and net, really?

Measured as a multiple, fees and carry take 0.44x off 2.0x, just over a fifth. Measured as profit, the gap is far larger: the LPs keep 560 of what would have been 1,000 of gross profit had every rupee been invested at 2.0x, so fees and carry together take 44% of the gain. That framing is why LPs negotiate hard over the fee base and the fee step-down after the investment period. A fee that looks small as an annual percentage of committed capital takes a large share of the profit over a ten-year life.

What has the simple version left out?

Two things worth naming. Most funds pay carry only after LPs earn a hurdleA minimum return, often expressed as an annual rate, that LPs must receive before the GP earns carried interest. and then let the GP catch up; at a 2.0x outcome a full catch-up still leaves the GP with 20% of all profit, so the answer here holds. Timing is the bigger omission: net IRR falls further than net multiple, because fees are paid early and proceeds arrive late. Recycling of proceeds and fee offsets from portfolio companies would also change the figure.

About 360 halls, and the wedding calendar sets that figure, not annual demand. Fifty lakh people marrying once in about 70 years gives roughly 36,000 weddings a year. If half use a hall for 1.5 events each, that is 27,000 hall bookings. Spread evenly that needs only 74 halls, but if 40% fall on 30 peak dates, those dates alone need 360.

How do you get from a population to a number of weddings?

Start with something every listener can check: most people marry once, and a city's population turns over roughly once a lifetime. Take a lifetime of about 70 years. Then 50 lakh people divided by 70 gives about 71,000 people marrying each year, and two people make one wedding, so about 36,000 weddings. A rate built from one human fact, how often a person marries, is easier to defend than a recalled statistic, because the interviewer can follow every step. You can refine it later for a young city or for weddings held in hometowns elsewhere.

Not every wedding uses a banquet hall; some are at home, at temples, on open lawns or in hotels. Assume half use a hall, so 18,000. Many families book a hall for more than one function, a reception or a sangeet as well as the main day, so assume 1.5 hall events each: 27,000 bookings a year. Every one of these is an illustrative assumption to be tested, not a sourced figure.

Why is 27,000 divided by 365 the wrong answer?

Think of an umbrella shop. Averaged over the year, it sells a few a day, but it must hold stock for the first heavy rain, when everyone wants one at once. A banquet hall cannot store an empty Tuesday and sell it on an auspicious Saturday, so capacity has to match the peak, not the average. Weddings bunch into a season, and within the season onto a small number of favoured dates. If 40% of bookings land on 30 dates, those dates carry about 360 events each, and with one main event a hall per evening, the city needs about 360 halls.

What does this tell an investor about the business?

Utilisation is the economics. A city sized for its peak runs most halls far below capacity most of the year, so a hall's earnings depend on how it fills off-peak dates with corporate events, smaller parties and exhibitions. The ratio here, 360 halls against 74 if demand were even, says average utilisation is around 20% on wedding business alone. Say the weak assumptions out loud: the peak share and the hall share move the answer most, so they are what you would test with local operators.

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.

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.

Ask either guide: which road would the other guide say leads to the site? Then take the other road. If you asked the truth-teller, he truthfully reports the liar's wrong answer. If you asked the liar, he lies about the truth-teller's right answer, which is also the wrong road. Either way the answer is the wrong road, so the right road is the one not named. The question works because it passes through both guides, so exactly one lie is applied.

Why does a direct question fail?

Imagine asking two shopkeepers whether a price is fair, knowing one of them always exaggerates, but not which. One answer is honest and one is not, and with no way to tell them apart you have learned nothing you can act on. A direct question about the road gives the truth from one guide and a lie from the other, and since you do not know which guide you asked, the answer carries no usable information. Asking a guide whether he is the truth-teller is worse: both say yes, the honest one because it is true and the liar because it is false. The trick has to work the same way whichever guide you happen to be facing.

How does routing the question through the other guide fix it?

By making sure the answer passes through exactly one liar. Ask the truth-teller what the liar would say and you get an honest report of a lie; ask the liar what the truth-teller would say and you get a lie about the truth; both contain one lie, so both point to the wrong road. The table in the figure checks all four cases: truth-teller or liar, left road or right road correct, and in every cell the named road is the wrong one. Because the answer is wrong by construction, you do not need to know which guide you asked. You simply take the other road.

Why does a deal desk ask a puzzle with no numbers in it?

Because it tests whether you can design a question whose answer is useful regardless of what you do not know. The puzzle is a small model of diligence: a management team, a seller's banker and a reference customer each have a reason to shade the truth, and the skill is to ask questions whose answers you can act on without first knowing who is being straight with you. Asking a supplier how the target compares with its other customers, rather than asking management how the supplier feels about them, is the same move: route the question so the bias cancels. The limit is that the puzzle assumes a guide who lies perfectly and consistently; a real counterparty who is merely selective needs several questions, not one.

The multiple is 2.4x in both cases, but the recap deal's IRR is about 24.1% against 19.1%. Money back is 60 plus 180, which is 240 on 100. The single exit's IRR is 2.4 to the power one fifth, less one. The recap returns part of the money in year 2, and cash that comes back sooner lifts the IRR even though the total is unchanged.

Why does the same multiple give a different IRR?

Lend a friend Rs 100 and get Rs 240 back. If Rs 60 of it comes back after two years, you have that money in hand for three years while waiting for the rest. The multiple counts how much money comes back; the IRR counts how fast it comes back, so pulling cash forward raises the IRR without touching the multiple. The two deals below differ only in timing.

How do you get the recap IRR without a spreadsheet?

The single exit is a one-line calculation: 2.4 to the power 0.2 is about 1.191, so 19.1%. The recap needs a guess and a check, because there are two inflows. Try 24%: 60 divided by 1.24 squared is about 39.0, and 180 divided by 1.24 to the fifth is about 61.4. Together that is 100.4, just above the 100 invested, so the true rate is a touch higher. Guess, discount each cash flow, and nudge the rate until the present values add back to the cheque. The answer is 24.1%.

Say what the recap costs, because an interviewer will push. The dividend was paid with new debt, which is why the exit cheque is 180 and not 240. The company carried more leverage for three years, so the higher IRR came with higher risk of a covenant problem. That is the honest trade: an LP sees a better IRR and early cash back, and the business sees a thinner cushion.