Private Equity puzzles, solved step by step

EBITDA must grow by 33.3% just to hand the sponsor its money back. Take EBITDA of 100: the business costs 1,200, funded by 600 of debt and 600 of equity. With no paydown the debt is still 600 at exit, so equity is 600 only if the business is again worth 1,200. At 9x that needs EBITDA of 133.3, which is 12 over 9 minus 1.

Why is the answer not 25%?

Picture a flat bought for 12 years of rent and sold later when buyers will pay only 9 years of rent. To get your price back, the rent has to rise until 9 years of it equals the old 12. That is a rise of 12 over 9, a third, not a quarter. A 25% cut in the multiple needs a 33.3% rise in earnings to undo it, because the recovery is measured on the smaller multiple. It is the same base effect that makes a 10% fall need an 11.1% rise.

Why does the 50% debt not change the answer?

Write the equity at exit as exit value less debt. The debt was 600 going in and, with no repayment, is 600 coming out. Equity is returned in full only when the exit value equals the entry value, so the break-even growth is the same at 30% debt, 50% debt or no debt at all. Leverage changes how fast equity grows once you pass break-even, and how fast it is wiped out below it. It does not move the break-even itself when nothing is repaid.

What does this do to a real growth plan?

Say the plan grows EBITDA 50% to 150. Sold at 12x, the business would be worth 1,800 and the equity 1,200, double the cheque. Sold at 9x it is worth 1,350 and the equity 750, only 1.25x. The first third of the growth plan is spent refilling the hole left by the lower multiple. This is why sponsors who pay high multiples underwrite their exit at the entry multiple or lower, and why the interviewer asks the question.

About 2.25x. Three repayments take debt from 600 to 300, and three years of 10% growth take EBITDA from 100 to 133.1. Leverage is 300 over 133.1, or 2.25x, down from 6.0x. Halving the debt alone would have left 3.0x; growth removes most of another turn, and that matters to a lender reading the covenant.

Why does leverage fall faster than the debt?

Think of a family loan measured against the family's income. Paying down the loan helps, and so does a pay rise, because the bank looks at how many years of income the loan represents. Leverage is debt divided by EBITDA, so it falls when the top shrinks and when the bottom grows, and here both are happening at once. Track both lines year by year rather than jumping to the end.

How much of the drop came from growth?

Split the move multiplicatively, which is exact for a ratio. Halving the debt divides leverage by 2.0; growing EBITDA by 33.1% divides it by 1.331. Six divided by 2.0 and then by 1.331 is 2.25x. Measured that way, repayment does about 71% of the work and growth about 29%, so growth is worth roughly three quarters of a turn here. Say the limit too: the 100 a year of repayment is an assumption, and in a real deal the cash for it comes from the same EBITDA that is growing.

Yes: the study is worth Rs 7.5 crore in expectation and costs Rs 2 crore, a net gain of Rs 5.5 crore. Without it, the fund carries a 25% chance of losing Rs 30 crore, an expected loss of Rs 7.5 crore. With it, the fund walks away in the bad case and pays only the fee. The study stays worth buying as long as the chance of a problem is above 2 over 30, about 6.7%.

What exactly is the fund buying?

Think of paying a mechanic to inspect a second-hand car before you buy it. The inspection is worth nothing if the car is sound and a great deal if the gearbox is about to fail. A diligence study is worth the loss it can prevent, multiplied by the chance that the loss is really there. It is not worth the loss itself, and it is not worth zero because the problem is unlikely. Draw both choices as branches and the answer is visible.

What assumptions is that answer resting on?

Two, and you should say both. First, the study is perfect: it always finds the problem when it is there and never raises a false alarm. If the study catches the problem only 60% of the time, its value falls to 0.25 x 0.6 x 30, Rs 4.5 crore, still above the fee. Second, walking away costs nothing beyond the fee. In a live auction, pausing for a study can lose the deal to a faster bidder, and that cost belongs on the yes branch.

The implied enterprise value rises 32%, from 500 to 660. A 20% premium on equity of 800 is 160, taking the offer to 960. The 300 of cash is worth 300 to everyone, so the whole 160 is a premium on the operating business. With net cash, the premium on EV is always larger than the headline premium on equity; with net debt, it is smaller.

Why can the premium not land on the cash?

Picture buying a shop that has Rs 3 lakh sitting in its till. You may pay extra for the shop's location and customers, but nobody pays Rs 3.6 lakh for Rs 3 lakh of cash: the cash is worth its face value to every buyer. The equity price is cash plus the business, and since the cash is fixed at 300, the full 160 of premium has to sit on the business. That turns a 20% premium into a 32% premium on the part the buyer is really valuing.

Why would a buyout investor care which premium you quote?

Because the sponsor underwrites the business, not the cash. If the business earned EBITDA of 50, the market price would be 10x and the offer is 13.2x. A 20% headline premium sounds modest, but on a cash-rich target it can mean paying a third more for the operations. The limit runs the other way too: on a company with large net debt, a 20% equity premium is a much smaller premium on EV, so headline premiums are never comparable across capital structures without this step.

The company must repay 320, which is the whole equity cheque. The business costs 800, funded by 480 of debt and 320 of equity. With flat EBITDA and the same multiple it is still worth 800 at exit, so equity of 640 needs debt to fall to 160. That is 80 a year for four years, equal to all of EBITDA, so in practice this deal cannot reach 2x without growth.

Why does the repayment equal the equity cheque?

Think of a house bought for Rs 80 lakh with a Rs 48 lakh loan and Rs 32 lakh of your own money. If the house is still worth Rs 80 lakh years later, the only way your share doubles to Rs 64 lakh is for the loan to shrink to Rs 16 lakh. When the enterprise value does not move, equity gains exactly what the debt loses, so doubling the equity means repaying an amount equal to the original equity. That holds at any leverage, which makes it a fast check.

Is that repayment realistic?

Now test the answer against the business. Repaying 80 a year means every rupee of EBITDA goes to the lenders as principal. Interest on 480 of debt at 8% is about 38 in year one alone, before any tax or capex, so the cash to repay 80 a year simply does not exist. The honest conclusion is the useful one: a flat business bought at 10x with 6x debt cannot reach 2x from paydown. It needs EBITDA growth, a higher exit multiple, or a lower entry price.

Choose levered free cash flow, and Company A is the cheaper one. Both cost 9.0x EBITDA, but A keeps 90 after capex and B only 50, so the same 900 buys cash at 10.0x in A and 18.0x in B. After 25% tax the gap holds at 13.3x against 24.0x. Levered free cash flow is the one number that has already paid for capex, working capital, interest and tax: it is what an owner can actually take out.

What does EBITDA leave out that matters here?

Picture two taxis, each bringing in Rs 1 lakh a month after fuel and the driver. One is new and needs little upkeep; the other is old and eats half its takings in repairs. Before repairs they look identical, and nobody would pay the same for both. EBITDA is earnings before the business pays for the assets it uses up, so two companies with equal EBITDA can leave their owners very different amounts of cash. Company B spends half its EBITDA just to keep running.

Why levered free cash flow rather than EBIT?

EBIT does subtract depreciation, which here happens to equal capex, so EBIT would rank the two correctly. But depreciation is an accounting estimate of past spending, not the cash going out this year, and it ignores working capital, interest and tax. Levered free cash flow is the only one of the three that is cash in the owner's hand after every claim ahead of the owner has been paid. Assume no debt and depreciation equal to capex: A's free cash flow is 100 less tax of 22.5 less capex of 10, which is 67.5; B's is 37.5.

What is the limit of your answer?

Say two things. First, one year of free cash flow can be lumpy: a big capex year or a working capital swing can make a good business look expensive. With a single period of data, levered free cash flow is the most honest number, but it is also the noisiest. Second, levered cash flow belongs to the equity, so you compare it with equity value, not enterprise value. In this example there is no debt, so the two are the same; with debt you would divide by equity value instead.

1,000 / 7 is 142.857 and 22 / 0.35 is about 62.86. Seven goes into 1,000 142 times with 6 left, and six sevenths is 0.857142, repeating. For the second, multiply both numbers by 100 to get 2,200 / 35, divide both by 5 to get 440 / 7, and that is 62 with 6 left: 62 and six sevenths, about 62.86. Both answers end in the same sevenths.

Why is one seventh worth memorising?

Think of a clock face with six numbers on it instead of twelve. Every seventh, whatever it is, lands on the same circle of six digits, 1, 4, 2, 8, 5, 7, and only the starting point changes. Once you know 1/7 = 0.142857, every other seventh is a rotation of the same six digits: 2/7 starts at the 2, 3/7 at the 4, and 6/7 at the 8. That turns both of these divisions into one fact you already hold.

How do you show the long division out loud?

Seven into 1,000 is 142, because 7 x 142 is 994, leaving 6. Bring down a zero: 60, seven goes 8 times, 4 left. 40 gives 5, with 5 left. 50 gives 7, with 1 left. The moment a remainder you have seen before comes back, the digits start repeating, so you can stop dividing and write the cycle. Three decimals is 142.857; the next digit is a 1, so no rounding is needed.

Interviewers ask these at desks that live in spreadsheets because quick division is how you sanity check a multiple or a margin in a meeting. A 22 crore profit on a 0.35 crore unit is about 63 units. Saying the rescaling step out loud matters more than speed: it shows a method you would trust on any number.

The IRR falls, from 18.9% to about 17.1%, even though the money multiple rises from 2.0x to 2.2x. IRR is an average yearly growth rate, and a year that adds 10% is below the 18.9% the deal has been earning. To keep the IRR flat, the fifth year must grow equity by 18.9% itself, to about 2.38x. The hurdle for holding on is the IRR already achieved.

Why can a bigger multiple mean a lower IRR?

Think of a batsman averaging 19 runs a match over four matches. If he scores 10 in the fifth, his total rises but his average falls. IRR is the average yearly growth of the equity, so any year that grows it by less than the current IRR lowers the average, even while the total keeps rising. The money multiple is the total; the IRR is the average. Here the deal has averaged 18.9% a year and the fifth year offers 10%.

How would a fund actually weigh the choice?

Both numbers matter to an investor, and they point opposite ways. The extra year adds 0.2x of money but costs about 1.8 points of IRR, so the right call depends on what the cash would earn if it were returned now. If the investors could put it to work at more than 10% elsewhere, selling at year 4 is better; if not, the extra 0.2x is real money. Say this trade-off out loud, then say the limit: the 10% is a forecast, while the 2.0x offer may be on the table today.

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.

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.

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.

Two weighings. Put three bars on each pan and three aside. If one pan rises, the light bar is among those three; if they balance, it is among the three aside. Then take that group of three and weigh one bar against one: the pan that rises holds it, and a balance means it is the bar left out. Each weighing has three outcomes, so two weighings tell nine bars apart.

Why split into three groups and not two?

Think of a balance scale as a question with three possible answers: left is lighter, right is lighter, or they balance. Asking a yes or no question wastes one of those answers. A weighing that splits the suspects into three equal groups uses every outcome, so each weighing cuts the suspects to a third rather than a half. Nine bars become three after one weighing and one after two.

How do you prove that one weighing is not enough?

Count the answers a single weighing can give: three. There are nine bars, so nine different possible answers to the question which bar is light. One weighing can distinguish at most three cases, two weighings at most nine, so with nine bars two is both enough and the minimum. The same count gives the general rule: n weighings can find one light bar among up to 3 to the power n bars, so three weighings handle 27.