Private Equity puzzles, solved step by step

The split stack is cheaper, at a blended 8.33% against 8.5%. Two thirds of the debt costs 7% and one third costs 11%, so the blend is 7% plus a third of the 4-point gap. On EBITDA of 100, that is 37.5 of interest a year against 38.25, a gap of only 0.75. Sponsors often pay that for one lender, one set of documents and a surer, faster close.

How do you blend two rates quickly?

If you borrow Rs 2 lakh from a bank at 7% and Rs 1 lakh from a relative at 11%, you are not paying 9% on the whole: most of the money is the cheap kind. A blended rate is a weighted average, so start from the cheaper rate and add the gap times the share of debt that is expensive. Here the second lien is a third of the stack, so the blend is 7% plus a third of 4 points, 8.33%.

Why would a sponsor pay more for one lender?

Because the 0.75 a year buys simplicity that has real value. With two tranches there are two lender groups and an intercreditor agreementThe contract between two groups of lenders to the same company that sets who is paid first, who controls enforcement and what each may do without the other. between them, and every waiver or amendment means two negotiations. A unitranche puts one lender across the table, which usually means a faster close, fewer parties who can block a change, and more certainty in a competitive auction. The limit: compare all-in costs, because upfront fees and call protection can differ between the two and move the answer by more than 0.17 points.

About 60, released once. Revenue of 730 is 2 a day. At 90 days, customers are holding 180 of unpaid invoices; at 60 days, 120. Collecting 30 days faster brings in the 60 difference as cash, one time. After that, receivables simply stay at the lower level, so the cash flow benefit does not repeat each year.

What does a day of DSO actually hold?

Picture a tailor who lets regular customers pay at the end of the month. On any given day, a month's worth of stitched clothes is out there unpaid, and that money is not in the tailor's drawer. Days sales outstanding counts how many days of sales are sitting with customers, so each day of DSO is one day of revenue held as receivables instead of cash. Here one day is 730 over 365, which is 2.

Why does a buyout fund care that it happens only once?

Because a one-off release must not be valued like a recurring profit. The 60 can pay down debt or fund a dividend once, but it adds nothing to EBITDA and nothing to next year's cash flow. If a seller's numbers show a strong cash year driven by a receivables squeeze, a buyer strips it out before using that year to set the price. And as the company grows, receivables grow with it: 20% more revenue at 60 days lifts receivables to 144, absorbing 24 of cash.

About 9 crore tyres a year, on the assumptions stated. Take about 20 crore two-wheelers in use, each riding about 7,000 km a year on 2 tyres: that is 2.8 lakh crore tyre-kilometres. If a tyre lasts about 30,000 km, roughly 9.3 crore tyres wear out each year. Tyre life is the sensitive input: 20,000 km gives 14 crore and 40,000 km gives 7 crore.

Why start from the bikes on the road rather than new sales?

Think of how many toothbrushes a household buys. It depends on how many people live there and how often a brush wears out, not on how many babies were born this year. A replacement market is sized off the installed base, the stock in use, multiplied by how fast each unit wears out. New two-wheelers arrive with tyres already fitted at the factory, which is a separate market sold to manufacturers, so counting new sales here would mix two different customers.

How do you defend each assumption out loud?

Say where each number comes from and how you would check it. Registered vehicle counts overstate the bikes in use, because scrapped vehicles stay on the register, so cut the registered figure down and say you are doing so. Seven thousand km is about 20 km a day, a commute and errands. Then run the cross-check a rider can judge: at 7,000 km a year a 30,000 km tyre lasts about four years, which sounds right for a commuter bike. All three inputs are illustrations to be confirmed against current industry data, not facts.

Close by naming the input you would research first. Tyre life halves or doubles the answer across a plausible range, and rear tyres wear faster than front ones, so a sharper version splits the two. A sponsor looking at a tyre retailer would care about exactly that split, because the rear tyre is the more frequent purchase.

Levered free cash flow is 92.5. EBITDA is 250. Take off D&A of 40 and interest of 60 to get pre-tax profit of 150, tax of 37.5 and net income of 112.5. Add back the 40 of D&A because it is not cash, then subtract capex of 50 and the 10 of working capital the growth absorbs. What remains, 92.5, is cash the owners could take out.

Why does depreciation come off and then go back on?

Think of a delivery van bought last year. Each year the accounts charge a slice of its cost as depreciation, but no money leaves the business that year: the van was paid for already. Depreciation reduces taxable profit, which saves real tax, but it is not a cash payment, so it is subtracted to work out tax and then added back to reach cash. The cash cost of assets enters through capex instead, here 50, which is more than the 40 of depreciation because the business is growing.

What makes it levered, and why does a buyout investor want that version?

Levered means after interest: the lenders have been paid. Levered free cash flow is the cash left for the equity holders after the lenders, the taxman and the business's own reinvestment, which is exactly the cash that repays debt in a buyout. Check it a second way from EBITDA: 250 less 60 less 37.5 less 50 less 10 is 92.5. The limit worth saying: working capital is modelled here as 10% of growth, and a real business can swing by far more in a single year.

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.

The rule of 72 gives 36%; the exact IRR is about 41.4%. Eight times is three doublings, so the money doubles every two years, and 72 divided by 2 is 36. The exact rate is 8 to the power one sixth, which is the square root of 2, so the IRR is 1.414 minus 1, 41.4%. The rule runs low because it is tuned for rates near 8%.

How does the rule of 72 apply to a multiple bigger than 2?

Think of folding a sheet of paper: each fold doubles the thickness, so eight layers is three folds. Any money multiple that is a power of 2 is a count of doublings, and once you know the years per doubling, the rule of 72 gives the rate. Eight is 2 x 2 x 2, so six years hold three doublings of two years each. The rule says a doubling every two years needs 72 over 2, which is 36% a year.

How do you get the exact answer with a basic calculator?

You need the sixth root of 8. Split it: the sixth root is the square root of the cube root, and the cube root of 8 is 2, so the answer is the square root of 2. One press of the square root key gives 1.414, an IRR of 41.4%. Without a square root key, use trial and error: 1.4 squared is 1.96, and 1.96 cubed is about 7.53, a little short of 8, while 1.42 to the sixth is about 8.20, a little over. The answer sits between 40% and 42%.

Then say why the rule ran low. The rule of 72 is built around rates near 8%; the higher the rate, the bigger the number you should divide into. For a two-year doubling the exact rate implies a numerator of about 83, not 72. Checking the shortfall out loud, 36% for six years gives only 6.33x, shows the interviewer you know the rule's limit.

The real return is about 1.9% a year, and after 10 years the money buys about 1.21x what it did. Real growth is the money growth divided by the price growth: 1.07 / 1.05 = 1.0190. Over ten years money grows 1.97x while prices grow 1.63x, so purchasing power grows 1.21x. Subtracting 5% from 7% gives 2% and 1.22x, close but slightly high.

Why divide instead of subtract?

Think in plates of biryani. If a plate costs Rs 100 and you have Rs 100, you can buy one. A year later you have Rs 107 and a plate costs Rs 105, so you can buy 107 / 105 plates, about 1.019. Real return measures how many more things your money buys, which is a ratio of two growth factors, not a difference of two rates. The subtraction shortcut ignores that inflation also eats into the 7% you earned, which is why it comes out slightly high.

When does the shortcut stop being good enough?

At low rates the gap is small: here 1.90% against 2.00%, and 1.21x against 1.22x after a decade. The error grows with the size of the rates, so with 20% returns and 15% inflation the shortcut says 5% when the truth is about 4.3%. A fund comparing returns across countries with different inflation, or across decades, should divide. In an interview, give the shortcut first, then the exact figure, and say which you would use when.

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.

Rs 160 crore, or 16% of commitments. In years one to five the fee is 2% of Rs 1,000 crore, Rs 20 crore a year, Rs 100 crore in all. In years six to ten it is 1.5% of the Rs 800 crore invested, Rs 12 crore a year, Rs 60 crore in all. The early fees are charged on the full commitment, whether or not the money has been invested yet.

Why does the base change halfway through?

Think of a gym that charges full membership from the day you sign, even before you start going, and then a lower fee once you only use part of the facilities. During the investment period, fees are charged on what investors have promised, because the manager is busy finding deals; afterwards they usually drop to what is actually invested, because the job shifts to managing what was bought. The rates and bases differ from fund to fund, so read the agreement rather than assuming.

Why does an investor in the fund care about this number?

Because fees come out of the investors' money before any profit is shared. Rs 160 crore of fees on Rs 1,000 crore of commitments means the investments must earn back 16% before the investors are even whole on what they put in. The early fees bite hardest: if only Rs 200 crore were invested in year one, an illustrative figure, the Rs 20 crore fee would be 10% of the money actually at work. That is why investors care about the fee base as much as the rate.