Private Equity puzzles, solved step by step

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.

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.

About 1.52x. The buyer pays 85 for the NAV today and 20 more when the unfunded commitment is drawn, 105 in all. It receives 130 when the NAV realises at 1.3x and 30 when the drawn 20 returns 1.5x, 160 in all. 160 over 105 is 1.52x, roughly 15% a year if the money went out at the start and came back at year three. At full NAV the same deal would be 1.33x.

Why does the unfunded commitment belong in the sum at all?

Think of taking over a friend's gym membership halfway through. You pay him something for the months already paid up, but you also inherit the remaining monthly instalments, and in return you get every month of the membership that is left. Buying an LP interest transfers both the assets already in the fund and the obligation to fund the rest of the commitment, so the cheque you write today is only part of what the position costs. Here the buyer pays 85 now and 20 later, and the unfunded commitmentThe part of an LP commitment that the fund has not yet called. The buyer of the interest takes over the obligation to pay it when called. is money the buyer must find whether or not it wants to.

What are the numbers, line by line?

Money out: 85% of 100 is 85, plus 20 when it is drawn, so 105. Money in: the existing NAV of 100 grows 1.3x to 130, and the 20 of new capital earns 1.5x, which is 30, so 160. 160 over 105 is 1.52x, and the discount is what lifts it, because at full NAV the cost would be 120 and the multiple only 1.33x. Note that the 15-point discount is 15% of NAV but only 12.5% of the 120 of total exposure. Quoting the discount on NAV alone makes a secondary look cheaper than it is when a lot of the commitment is still to be drawn.

What does the simple version leave out?

Timing, mostly. The 20 is drawn after the purchase and the 160 arrives in pieces, so the 15% figure is only a placeholder for an IRR that depends on when each call and distribution lands. A later draw raises the IRR because less money is out for less time, which is one reason secondary buyers like interests with a large unfunded tail: the headline discount applies to a small NAV and the rest of the exposure is priced at cost. Say that out loud, then say what else is missing: the GP's management fee on the commitment, transfer costs and the fact that 1.3x and 1.5x are assumptions, not knowledge.

Reinvest 25% and the P/E is 15x; at a 10% return on capital it falls to 10x. To grow 5% a year on a 20% return, the business reinvests 5 over 20, a quarter of profit, and pays out 75 of every 100. 75 over 10% less 5% is 1,500, fifteen times profit. At a 10% return it must reinvest half, pays out 50, and 50 over 5% is 1,000, ten times profit, the same value it would have with no growth at all.

Why does growth cost anything in the first place?

Picture a tiffin service that wants 5% more customers next year. It needs more tiffin boxes and a bigger kitchen, and that comes out of this year's profit before the owner can take anything home. Growth has to be funded, and the amount of profit that must be ploughed back is the growth rate divided by the return the business earns on new capital. A business earning 20% on capital needs to reinvest 5 over 20, a quarter of profit, to add 5% a year. One earning 10% needs 5 over 10, half its profit, for the same growth. The owner of the second business gives up twice as much to get the same thing.

How does the reinvestment rate turn into a multiple?

Value is the cash the owner actually receives, growing at 5%, discounted at 10%. With 20% returns the owner receives 75 of every 100, so value is 75 over 5%, which is 1,500 and a P/E of 15x; with 10% returns the owner receives only 50, and 50 over 5% is 1,000, a P/E of 10x. Compare that with no growth at all: the owner receives the whole 100, value is 100 over 10%, which is also 1,000 and 10x. So growth at a 10% return on capital has moved the multiple from 10x to exactly 10x. The 5% growth consumed exactly as much as it created, because each rupee reinvested earned precisely the rate the owner demanded.

What is the lesson for a buyer, and where does the formula break?

That a growth story is only worth paying for when the growth is earned at a return above the cost of capital. Below 10% in this example, growth destroys value: the business would reinvest more than it earns on the reinvestment, and the multiple would fall below the no-growth 10x. That is why a buyout investor asks what return the incremental capital earns before asking how fast revenue grows. The limits are real: the formula assumes the growth and the return last forever, that profit is a fair proxy for cash, and that the business can keep finding projects at the same return, which it usually cannot as it grows.

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.

EBITDA can fall only 10%, from 100 to 90, before breach; after the drop to 85, about 110 of debt must be repaid. Breach comes at 450 divided by 5.0, which is 90. At 85 leverage is 5.29x, already in breach. For EBITDA to be able to fall 20% from 85, to 68, debt must be no more than 5.0 x 68 = 340. That is a repayment of 110.

Why measure headroom in EBITDA rather than in turns of leverage?

Think of a lift rated for 500 kg carrying 450 kg. The spare 50 kg sounds like a tenth of the rating, and it is: one more passenger and the alarm goes. A covenant works the same way, except that the load is fixed and the rating shrinks. Debt does not move between test dates, EBITDA does, so the only honest measure of headroom is how far EBITDA can fall before the ratio hits the limit. Here that point is 450 divided by 5.0, which is 90, a fall of 10% from today.

Half a turn of spare leverage, 4.5x against 5.0x, feels like comfort. It is not much. A business that misses budget by one bad quarter can lose 10% of its trailing EBITDA. The trailing twelve monthsThe covenant usually tests EBITDA over the last four quarters, so one weak quarter feeds straight into the ratio. test makes this worse, because a weak quarter enters the ratio in full and stays there for a year.

How do you size the repayment that restores a 20% cushion?

Define the cushion the same way you measured headroom: the share by which EBITDA could fall before breach. A 20% cushion on EBITDA of 85 means the covenant must still hold at 85 x 0.8 = 68, so debt can be at most 5.0 x 68 = 340. Debt is 450, so 110 has to be repaid. Leverage after the repayment is 340 / 85 = 4.0x.

Say where the money comes from, because the interviewer will ask. A business that just lost 15% of its EBITDA rarely has 110 of spare cash, so this usually means an equity cureA sponsor injection of new equity, allowed by many loan agreements, used to repay debt or to count towards EBITDA for a covenant test. The terms vary by document., an asset sale, or a negotiated reset with lenders. Which of these the loan agreement allows is a question for the document, not for arithmetic.

The EBITDA margin is 20%. Both multiples share the same enterprise value on top. Divide EV/Sales by EV/EBITDA and EV cancels, leaving EBITDA divided by sales: 2 over 10, which is 20%. Check it with any EV you like: at 1,000, sales are 500 and EBITDA is 100, and 100 is 20% of 500.

Why does the enterprise value not matter?

A cricket bat costs as much as 10 balls, and as much as 2 sets of pads. How many balls is a set of pads worth? Five, and you never needed the price of the bat. When two ratios share the same top line, dividing one by the other cancels it and leaves the ratio of the two bottom lines. Here the shared top line is enterprise value, and the bottom lines are sales and EBITDA.

How do you check the direction of the division?

A margin must be smaller than 100%, and EBITDA is a slice of sales, so EBITDA has to be the smaller number. The higher multiple sits on the smaller number, so the margin is the low multiple over the high multiple, never the other way. Dividing 10 by 2 gives 5, which would be a 500% margin and is impossible. Dividing 2 by 10 gives 0.2.

A buyout investor uses this the other way round all the time. If comparable companies trade at 2x sales and the target earns a 10% margin, then 2x sales is 20x EBITDA for this target, which is expensive. Sales multiples hide margin differences; converting to EBITDA puts them back. The limitation: EBITDA multiples carry their own blind spots, such as heavy capital spending that EBITDA leaves out.