Private Equity puzzles, solved step by step

About Rs 120 crore a year. 20 MW at 70% use is 14 MW of IT load. A PUE of 1.4 means the site draws 1.4 times that for cooling and losses, 19.6 MW. Running all year, 19.6 MW times 8,760 hours is about 171.7 million kWh. At Rs 7 a kWh that is about Rs 1,202 million, or Rs 120 crore. The tariff is an assumption to check.

What does PUE add to the bill?

A home air conditioner does not just cool the room; it also uses power to run its own compressor and fan. Power usage effectivenessTotal power drawn by the whole facility divided by the power used by the IT equipment alone. 1.0 would mean no overhead at all. is total site power divided by IT power, so a PUE of 1.4 means every 1 MW of servers needs another 0.4 MW for cooling, power conversion and lighting. Here that is 5.6 MW on top of 14 MW of IT load, and the bill pays for all 19.6.

How do you keep the units straight out loud?

Convert once, early, and say it. One megawatt running for one hour is 1,000 kWh, so 19.6 MW for a year is 19.6 x 8,760 x 1,000, about 171.7 million kWh. Then price it: Rs 7 times 171.7 million is about Rs 1,202 million. A crore is ten million, so divide by ten: Rs 120 crore. Most wrong answers to this question are right in method and a factor of ten off in units.

Now the investor's view. Power is usually the largest running cost of a data centre, so PUE is an economic number, not a technical one: improving it from 1.4 to 1.3 saves about Rs 8.6 crore a year here. Many colocation contracts pass power through to tenants, which shifts who bears the tariff risk. Ask how the contracts treat power before you treat the bill as the owner's cost.

9%, 7% and 2%: here every turn of debt lowers the equity return, because debt costs more than the asset earns. On 100 of assets earning 9, debt of 40 costs 4.8 and leaves 4.2 on equity of 60, which is 7%. Debt of 70 costs 8.4 and leaves 0.6 on equity of 30, which is 2%. Leverage helps only when the asset earns more than the debt costs; when it earns less, leverage works in reverse.

How does leverage change the equity return?

Borrow at 12% to buy a flat that earns 9% in rent and you lose 3% a year on every borrowed rupee, and the more you borrow, the bigger the loss as a share of your own money. Equity keeps whatever the asset earns after the debt is paid, so each unit of debt adds the gap between the asset return and the debt cost, times the debt-to-equity ratio. With a 9% asset and 12% debt that gap is minus 3 points.

Why does a buyout investor worry about this?

A business bought at a high multiple has a low unlevered return: at 12x EBITDA, pre-tax cash yield on the purchase price is in single digits. If borrowing costs rise above that yield, the leverage that used to lift returns now pulls them down, and the deal depends entirely on growth and multiple expansion. This is called negative leverage, and it is why higher rates squeeze what sponsors can pay.

Say the limitations. The sums ignore tax: interest is usually deductible, which lowers the after-tax cost of debt and moves the break-even. And they treat the asset return as fixed. A growing business can out-earn expensive debt later even if it does not today, which is the bet some sponsors knowingly make.

The earnout is worth about 15.9 to you, so the price in value terms is about 115.9. A 40% chance of 50 is 20 on average, payable in two years, and 20 / 1.12^2 is 15.9. The headline price is 150, but you are paying about 116 in value. A seller who thinks the chance is 80% values the same earnout at 31.9, which is why earnouts close price gaps.

How do you value a payment that may not happen?

A shopkeeper who offers a Rs 500 voucher to anyone who spends Rs 5,000 next month books its cost by how many customers she expects to claim it, not by the face value. An earnout is a contingent payment, so you value it as chance times amount, discounted for the wait: 0.4 x 50 / 1.12^2 is about 15.9. That number, not 50, is what the earnout adds to your price.

Why does an earnout make a deal possible?

Because both sides can be right in their own terms. The seller, confident in the plan, values the package at about 132; the buyer, more cautious, values it at about 116; and the contract lets each side bet on its own view. If the target is hit, the buyer pays more but has a business that performed. If not, the buyer has paid 100 for a business that did not.

Say the risks. Earnouts are a common source of disputes: how EBITDA is defined, which costs the buyer loads onto the business after closing, and whether the seller's management stays to run it. A buyer controls the business that decides the payout, so the contract needs a tight EBITDA definition and agreed accounting policies.

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.

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.

About Rs 720 crore a year, of which 60 clinics could hold roughly 7.5%. Take 25% of 1.2 crore people seeing a dentist, 2 visits each and 80% of visits at private clinics: 48 lakh visits. At Rs 1,500 a visit that is Rs 720 crore. Sixty clinics at 20 visits a day for 300 days handle 3.6 lakh visits, about Rs 54 crore of revenue. Every input is an assumption to be tested.

Why build the market from visits rather than from spend per person?

Think of sizing the market for haircuts. Average spend per person hides the fact that some people go monthly and others never; counting the people who go, how often, and what one cut costs is easier to defend line by line. A visit-based tree gives you inputs you can each test separately, and it produces the same unit, visits, that a clinic's capacity is measured in. That second point is why it suits a rollup question: the market and the business are counted in the same currency.

How do you test the demand side against supply?

Turn the visits into clinics and see if the number feels right. Forty eight lakh visits over 300 working days is 16,000 visits a day; at 15 a day for a small single-dentist clinic, that is about 1,067 clinic-equivalents. If a local count found far more or far fewer clinics than that, one of the demand assumptions is wrong, most likely the share who visit or the fee. Say that you would check it with a street count in two or three neighbourhoods before trusting the total.

What does the share tell a buyout investor?

It frames the thesis. At about 7.5%, the rollup would be the largest single brand in a fragmented market without needing to win most patients, which is the shape a rollup thesis wants. The limits matter as much: the fee per visit blends cheap check-ups with expensive implants, so a chain that skews to higher-value treatment could earn more from the same visits, and capacity is not demand. Sixty clinics only reach 3.6 lakh visits if patients actually come.

Company A's equity falls 33%; Company B's is wiped out. Both businesses fall from 800 to 600 of enterprise value, a loss of 200. The debt does not fall with them, so the loss lands entirely on the equity. A had 600 of equity and keeps 400. B had only 200 and keeps nothing: its lenders are now owed exactly what the business is worth.

Why does the equity take the whole fall?

Picture two people who each own a Rs 80 lakh flat. One owes Rs 20 lakh on it, the other Rs 60 lakh. If flat prices drop by Rs 20 lakh, both still owe the bank the same amount, so the first owner's stake falls from 60 to 40 and the second's from 20 to nothing. Debt is a fixed claim, so any change in the value of the business falls first and entirely on the equity. The more debt, the thinner the equity cushion that absorbs it.

What does a buyout investor take from this?

Leverage works in both directions with the same multiplier. At 6x debt on 8x value, the equity is a quarter of the business, so every 1% move in enterprise value is a 4% move in the equity. That is what makes high leverage attractive on the way up and fatal on the way down. The limit worth saying: in practice B's equity rarely goes straight to zero on paper. It may keep some option value while the debt has years to run, and the lenders would usually step in through a restructuring before the equity holders lost control.

About 40%. Picture 1,000 targets: 100 have a problem and the review flags 90 of them. Of the 900 clean targets, it wrongly flags 15%, which is 135. So 225 targets are flagged and only 90 of those are real problems, 90 divided by 225. A good test on a rare problem still throws up more false alarms than true ones.

Why is the answer not 90%?

Think of a smoke alarm that goes off for almost every real fire, but also for burnt toast. In a house where fires are rare and toast is daily, most alarms are toast. The 90% is the chance of a flag when there is a problem; the question asks the reverse, the chance of a problem when there is a flag, and the two differ because clean targets are far more common. Counting people instead of using percentages makes the reversal visible.

What should a deal team do with a 40% flag?

Treat the flag as a reason to dig, not a reason to walk away. A flag lifts the chance of a problem from 10% to 40%, four times the starting level, but it still leaves the target more likely clean than not. A second, independent check changes the picture: if a forensic review with the same hit and false alarm rates also flags it, the chance rises to about 80%. The limit to state: that assumes the two reviews make independent mistakes, which two teams looking at the same ledgers may not.

Ten lockers are open: 1, 4, 9, 16, 25, 36, 49, 64, 81 and 100, the perfect squares. Locker n is toggled once by each person whose number divides n, so its final state depends on how many divisors n has. Divisors come in pairs that multiply to n, which makes the count even, except for a perfect square, where one divisor pairs with itself. An odd count leaves the locker open.

What decides whether one locker ends open?

Think of a light switch flicked by everyone who walks past: if an odd number of people flick it, the light ends on. Locker n is flicked by person d exactly when d divides n, so the number of toggles is the number of divisors of n, and the locker ends open only if that number is odd. The whole puzzle reduces to one question: which numbers have an odd number of divisors? Do not simulate a hundred people; reason about one locker.

Why do only perfect squares have an odd number of divisors?

Pair each divisor with its partner: for 12, 1 goes with 12, 2 with 6, 3 with 4. Every divisor has a different partner, so the count is even and the locker closes. For a perfect square, one divisor is its own partner, 6 x 6 for 36, so it is counted once and the total turns odd. There are ten perfect squares from 1 to 100, so ten lockers stay open. The answer for any number of lockers N is the whole part of the square root of N.

About 16.3%. The lender pays 100, receives 12 a year for five years and gets the 100 back at the end. On its own that is exactly a 12% return. The warrants are 2% of equity worth 1,500, which is 30, received with the final payment, taking year 5 to 142. That one extra payment lifts the IRR by about 4.3 points.

What does the lender earn before the warrants?

Think of a fixed deposit at 12% that pays its interest every year and returns the principal at maturity: it earns exactly 12%. A loan bought at par that pays its full coupon in cash and is repaid at par earns exactly its coupon rate, so the first 12 points of the IRR need no calculation at all. The question is only about what the warrants add on top.

How do you estimate the warrant's effect without a calculator?

The 30 arrives in five years. At around 16%, five years of discounting roughly halves money, so the warrants are worth about 14 to 15 in today's terms. Spread that extra 14 or so of value across five years of a 100 loan and it adds roughly 4 points a year to the coupon, which puts the IRR near 16%. Then confirm by trial: at 16%, the flows of 12 a year and 142 at the end discount to slightly above 100, so the IRR is a touch above 16%, 16.3%.

Say the risk too. The warrant is worth 30 only if the equity really is worth 1,500 at exit. If equity were worth 500, the warrants would bring 10 and the IRR would fall to about 13.5%. That is the point of mezzanine: a contractual coupon plus a slice of the upside, priced for sitting behind the senior lenders.