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 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.

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.

Break-even is Rs 5,000 a passenger; a 10% margin on revenue needs about Rs 5,556. A 2-hour flight costs 2 x Rs 3.6 lakh, Rs 7.2 lakh. An 80% load factor fills 144 of 180 seats, so each passenger must cover Rs 7.2 lakh / 144, Rs 5,000. For cost to be 90% of revenue, divide by 0.9, not multiply by 1.1, which gives about Rs 5,556.

Why divide by passengers and not by seats?

Think of a shared taxi to the airport that costs Rs 1,000 whether four people ride or two. If only two turn up, each pays Rs 500, not Rs 250. A flight costs the same to operate with empty seats, so the fare has to be set on the seats that are actually sold. That is why the load factor sits at the centre of airline economics: at 80%, every paying passenger carries a quarter of an empty seat.

What would you ask the interviewer for next?

In this format you can ask for more data, and asking well scores points. The two inputs that move the answer most are the load factor and what is inside the cost per hour, because fuel and aircraft leases dominate it. At a 70% load factor the break-even rises to about Rs 5,714. You might also ask about ancillary revenue, bags and seats, which lets the base fare sit lower. The interviewer's follow-up, why airlines have often earned poor returns, sits on the same logic: high fixed cost per flight and thin margins mean small swings in load factor or fuel swing profit hard.

About 11.65%. In rupees the investment grows by a factor of 1.15 a year. Converting back, each dollar now costs 1.03 times as many rupees, so the dollar value grows by 1.15 divided by 1.03, which is 1.1165. Over five years that compounds to 2.01x in rupees but only 1.74x in dollars. Subtracting 3 from 15 gives 12%, a close but slightly low shortcut.

Why divide by the currency move rather than subtract it?

Picture a relative abroad who sends you dollars to invest in a rupee deposit. The deposit grows, but when you convert the money back, each dollar costs more rupees than before. The dollar investor's growth is the rupee growth divided by the rise in the rupee price of a dollar, because both are growth factors applied to the same money. It is the same arithmetic as turning a nominal return into a real one: a ratio, with subtraction as the rough version.

Does it matter how the depreciation is quoted?

Yes, and saying so earns credit. If instead the rupee loses 3% of its dollar value each year, the factor is 0.97 rather than 1 over 1.03, and the dollar IRR is 1.15 x 0.97 minus 1, 11.55%. The two conventions differ by only a tenth of a point here, but state which one you are using before you compute. The bigger point for a foreign fund is the size of the drag: 3 points a year off a 15% rupee return is a fifth of the return, every year, before any fees.

Equity earns 17% when the asset earns 12%, and only 3% when it earns 5%. Per Rs 100 of asset, lenders take a fixed 3.5. At 12% equity keeps 8.5 on its 50; at 5% it keeps 1.5 on its 50. Leverage adds the spread between the asset return and the cost of debt, times debt over equity, in both directions.

Why does borrowing push the equity return above the asset return?

Think of buying a flat for Rs 100 lakh with Rs 50 lakh of your own money and a Rs 50 lakh loan at 7%. If the flat earns 12 lakh a year in rent and value, the bank takes 3.5 lakh and everything else is yours. Lenders get a fixed amount whatever the asset earns, so any return above the cost of debt on the borrowed half lands on the equity. The borrowed 50 earns 12% and costs 7%; that 5 point spread on 50 is 2.5, added to the 6 the equity's own half earns.

What happens when the asset earns less than the debt costs?

The same formula runs backwards. At 5%, the spread is minus 2 points, and with one rupee of debt per rupee of equity the equity return falls to 5 minus 2, which is 3%. Leverage is not a return booster; it is a multiplier on the spread, and the spread can be negative. Below a 3.5% asset return equity earns nothing at all, and below zero it loses money faster than the asset does.

In the room, give both numbers and then the rule in one sentence. A private credit interviewer is checking that you see the downside as clearly as the upside, because their job is to sit on the other side of that fixed 3.5.

About Rs 21,250 crore a year on these assumptions. Take roughly 25 crore school students, assume 85% attend schools that require uniforms, two sets a year and an average of Rs 500 a set: 25 x 0.85 x 2 x 500. Then say which input is weakest. Price is, because government and private sets differ several times over.

How do you structure the estimate before any number?

Say the chain out loud first: students, times the share who must wear a uniform, times sets a year, times price. It is how a shopkeeper would size the season: how many children in the area, how many of their schools insist on a uniform, how many sets each family buys, what they pay. The structure earns most of the marks, because the interviewer can disagree with a number and still follow your answer.

Then put a number on each branch and label it an assumption. Enrolment is the one input with a public source, the government's school statistics; use about 25 crore and say you would confirm the current figure. The uniform share, sets a year and price are judgement calls: 85%, two sets and Rs 500 a set are reasonable starting points, not facts.

Which branch do you name as the biggest uncertainty, and why?

Swing each input across a range you could defend, holding the others. The uniform share from 75% to 95% moves the pool by about Rs 5,000 crore. Sets a year from 1.5 to 2.5 moves it by about Rs 10,625 crore. Price from Rs 300 to Rs 800 moves it by Rs 21,250 crore, so price is where you would spend your first hour of real work. The reason is the mix: many state schools supply uniforms cheaply in bulk, while private schools sell branded sets at a premium.

A strong close splits the pool into those two segments and says which one a buyout fund would care about: the private school segment, where the price is set by the school and the supplier relationship is sticky.

Down 1%. Start from 100. A 20% rise takes it to 120, a 25% fall takes away 30 to leave 90, and a 10% rise adds 9 to reach 99. In one line, 1.2 x 0.75 x 1.1 = 0.99. Percentage changes multiply; adding them gives plus 5%, which never happened.

Why can you not just add the percentages?

A shop marks a Rs 100 kurta up 20% to Rs 120, then runs a 25% sale, then nudges the price up 10%. The sale takes 25% of 120, which is 30, not 25. Every percentage is measured against the level it starts from, so each change has a different base and the changes multiply rather than add. Adding them treats every change as if it were applied to the original 100.

How do you do 1.2 x 0.75 x 1.1 fast?

Pick the pair that cancels nicely. 1.2 times 0.75 is exactly 0.9, because three quarters of 1.2 is 0.9, and 0.9 times 1.1 is 0.99. Saying the intermediate 0.9 out loud lets the interviewer follow you and catch a slip early. The order does not matter: multiplication gives the same answer whichever change comes first.

In private equity this is the same arithmetic as a multiple that expands 20% at one point and contracts later, or a business that grows, shrinks and recovers. Revenue up 20% and back down 20% does not return to the start: it ends 4% lower.