Private Equity puzzles, solved step by step

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.

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 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.

About 2,000 shops on these assumptions. Of 2 crore people, assume 30% drink coffee, a cup a day each: 4.2 crore cups a week. If 10% are bought outside the home, that is 42 lakh a week, 6 lakh a day. At 300 cups a shop a day the metro supports about 2,000. The share bought outside is the branch to test.

What is the chain, and why build it before any number?

Shops exist to serve demand, so go from people to cups to shops. Picture a single street: how many of the people passing drink coffee, how often, and how many of those cups are bought rather than made at home. A sizing answer is a chain of stated assumptions, and the interviewer is testing the chain, not the final figure. Say the chain first, then fill it.

How do you sanity check the supply side?

Three hundred cups a day is the number to defend. A shop open twelve hours selling 25 cups an hour reaches it; a quiet neighbourhood outlet may sell half that and a busy office-district one double. Check the answer from the other end: about one shop for every 10,000 people, which you can compare with what you see on a busy street. If your answer implied a shop for every 500 people, a branch is wrong, and you would say which.

A growth equity interviewer often follows with how many of those shops a single chain could own, which turns the sizing into a market share question. Have the number of shops in your answer split into chains and independents if you can, even roughly.