Investment Banking puzzles, solved step by step

Roughly 2,000 to 2,500 taxis, inside a defensible range of about 1,000 to 4,000. Assume 300,000 people in Central at lunch, 2% of them taking a taxi trip that touches the district in the peak hour, and each trip spending 15 minutes inside it: 1,500 cabs occupied at any moment. Add half as many again cruising or queuing empty and you reach about 2,250.

Why count trips and minutes rather than cabs?

Picture a busy restaurant. If 60 diners arrive an hour and each stays half an hour, about 30 seats are taken at any moment, whatever is on the menu. That rule, known as Little's lawThe number of items in a system on average equals the rate they arrive multiplied by how long each one stays., turns a flow into a count. The taxis in Central at one moment equal the trips touching Central per hour multiplied by the hours each trip spends there, plus the cabs that are empty. You cannot see cabs from an interview room, but you can reason about people and journeys.

Where do the numbers come from, and which one moves the answer most?

Every number is an assumption you state, not a fact you recall. Central is a dense office district, so assume 250,000 office workers plus 50,000 visitors and shoppers at lunch. Most walk or take the train; assume 2% take a taxi trip that starts or ends in Central in the busiest hour, which is 6,000 trips. A cab picking up, crawling through lunchtime traffic and dropping off spends perhaps 15 minutes in the district per trip. The minutes-per-trip assumption moves the answer as much as the head count does, so say it slowly and invite challenge on it. Empty cabs cruising or waiting at ranks add about half again.

How do you sanity-check it from the supply side?

Come at it from the fleet. Suppose the city has about 18,000 licensed taxis, a number you would state as an assumption and check against the current licence count, and the busiest business district holds 10% to 15% of them at peak. That gives 1,800 to 2,700. Two methods landing in the same range is worth more than either number on its own. If they had disagreed by a factor of three, you would say which assumption you trust less and why.

Roughly Rs 2.3 lakh crore a year, on stated assumptions. Assume about 90 lakh weddings a year, from a population of about 140 crore. Split them into budget, mid-range and premium tiers, each with its own plate count and cost per plate. Budget weddings contribute about Rs 47,000 crore, mid-range Rs 81,000 crore and premium about Rs 1 lakh crore: 5% of weddings, 44% of the spend.

How many weddings happen a year?

Start from people, not from a remembered statistic. Assume a population of about 140 crore, a figure to confirm, and an average lifespan near 70 years, so roughly one seventieth of people, 2 crore, reach marrying age each year. Assume 90% marry: 1.8 crore people, or 90 lakh weddings. A count built from cohorts can be checked line by line, which is what makes it defensible in the room. One caveat is worth saying: a young population has larger cohorts than one seventieth, so this may undercount a little.

Why split by tier rather than use one average wedding?

Think of a school canteen and a five-star hotel both serving lunch. Averaging their prices and their customer counts separately, then multiplying, gives a number neither of them earns. Big weddings have both more guests and a higher cost per plate, so averaging guests and price separately understates the market. On the tiers below, the average-wedding shortcut, 435 plates at Rs 400, gives Rs 1.57 lakh crore, about 32% less than the tiered Rs 2.29 lakh crore. Plates here count every guest meal across all the functions of a wedding, and catering means food only, not venue or decoration.

Which assumption moves the answer most?

The premium tier. It is 5% of weddings and 44% of the value, so a change in its plate count or price moves the total more than any change in the budget tier. When one segment dominates the value, spend your sanity checks there: is 1,500 plates across several functions at Rs 1,500 a plate plausible for the top twentieth of weddings? Halve the premium tier's value and the total falls to about Rs 1.8 lakh crore, which is the honest low end of the range to quote.

Roughly 1.2 crore cups a day. Take about 2 crore people as a round assumption, 75% adults, 40% of them buying street tea on a given day and two cups each: 1.2 crore cups. Check it from the supply side: about one stall per 400 people gives 50,000 stalls, each selling about 252 cups a day, which is 1.26 crore. Two routes landing within 5% make the estimate credible.

How do you structure the estimate before any number?

Think of how you would guess the rotis a hostel mess makes in a day: count the students, how many eat in the mess, and how many rotis each one eats. A market sizing is a chain of multiplications, and the interviewer scores the chain more than the final figure. Say the chain first: people, then adults, then the share who buy street tea on a given day, then cups per buyer. Each link is one assumption you can defend or change, and round numbers keep the arithmetic in your head.

Why does a demand estimate need a supply check?

Each link in the demand chain is a guess, and errors multiply. A second route built from different assumptions, here stalls and cups per stall, tests the first: if both land close together, neither is likely to be badly wrong. One stall per 400 people gives 50,000 stalls. A busy stall open 14 hours and selling 18 cups an hour sells about 252 cups a day, so supply comes to about 1.26 crore cups, within 5% of demand.

What do you say about the weakest link?

Name it before the interviewer does. The share of adults buying street tea on a given day drives the answer one for one: 30% instead of 40% cuts the demand estimate to 90 lakh cups. Office workers, drivers and shopkeepers buy several cups a day; many households brew at home and buy none. Then say which number you would go and observe, the stalls along one busy street, because that is the input you can check. The population figure is a round assumption to confirm against the latest census estimate, not a fact.

About 2 crore a year. India has about 140 crore people in roughly 30 crore households. If about half of households own a two-wheeler, 15 crore are in use. A vehicle lasts about 10 years, so 1.5 crore are replaced each year, and a fleet growing 3% adds about 45 lakh more, close to 2 crore. A top-down check, 18 crore vehicles on the road over a 9-year life, lands at 2.0 crore too. The published industry figure is the thing to confirm, not to recall.

Where do you start, and what do you refuse to guess?

Think of guessing how many birthday cakes a town buys in a year. Nobody knows the number, but everybody can picture the pieces: people, birthdays per person, the share with a cake. Market sizing works from a number you can picture to the number you cannot, and the only figure taken on trust is the population. About 140 crore people at 4.5 per household is 31 crore households, call it 30. The one real judgement is the ownership rate, the penetrationThe share of possible buyers who already own the product. Half of households here.: higher in towns, lower in villages, so about half overall.

How does the bottom-up build go?

Thirty crore households at 50% is 15 crore two-wheelers in use. Sales are what it takes to keep that fleet going plus what it takes to grow it. Yearly sales of a durable are the fleet divided by its life, plus the fleet's growth, so the life assumption matters as much as the ownership rate. At a 10-year life, 1.5 crore are replaced each year; a fleet growing 3% a year needs another 45 lakh; together about 1.95 crore. Say each assumption as you use it, so the interviewer can push on the one they disagree with.

Why build a second route, and what if it disagrees?

Come at it from the fleet instead. About 13 two-wheelers per 100 people is 18 crore on the road, and dividing by a 9-year average life gives 2.0 crore a year, within 4% of the first route. A demand estimate is credible only once an independent route lands near it; agreement does not prove either number, but disagreement tells you which assumption to revisit. Stress the two judgements: 40% ownership and a 12-year life give 1.4 crore, 60% and 8 years give 2.8 crore. So the honest answer is a range of about 1.5 to 3 crore with 2 crore most likely, and the published sales figure from the industry body is the check to look up afterwards.

About Rs 82 crore a year, built as vehicles, stop rate, litres and price. Say 15,000 vehicles pass a day and 5% stop, which is 750 fills. At 30 litres a fill that is 22,500.0 litres, and at Rs 100 a litre about Rs 22.5 lakh a day. Times 365 gives roughly Rs 82 crore. The stop rate is the shakiest input: 3% gives Rs 49 crore and 8% gives Rs 131 crore.

How do you turn a vague question into a chain you can multiply?

Think of a chai stall outside a railway station. You would not guess its takings in one go; you would ask how many people walk past, how many stop, what they buy and what it costs. A petrol pump is the same chain. Revenue is vehicles passing, times the share that stop, times litres per fill, times the price per litre, and the whole job is to say a defensible number for each link out loud. A busy national highway might carry 15,000 vehicles a day past one spot. If one in twenty pulls in, that is 750 fills. A car takes 25 to 35 litres and a truck far more, so 30 litres is a fair blended figure once you note that most vehicles are cars. Use Rs 100 a litre as a round illustrative price and say that the live pump price should be checked, because it moves.

Which assumption should you shake, and which can you leave alone?

Not every link is equally uncertain. Shake the input you know least, because the answer moves one for one with every link and the widest range sits in the stop rate. Litres per fill is boxed in by the size of a fuel tank: 25 to 35 litres moves the annual figure only from Rs 68 crore to Rs 96 crore. The stop rate could honestly be 3% on a stretch with pumps every few kilometres or 8% at the only station for an hour, and that range takes the answer from Rs 49 crore to Rs 131 crore. Saying which input is soft is worth more to the interviewer than the point estimate.

What does a banker add after the number?

Say what the number is not. Revenue is not profit: a dealer keeps a margin of a few rupees a litre, so Rs 80 crore of sales is a small fraction of that in gross margin, and the rest passes through to the oil company and the exchequer. Then offer one sanity check: Rs 80 crore a year is about Rs 22 lakh a day, which at Rs 3,000 a fill is roughly 750 customers, about one every two minutes across an 18-hour day. If that sounds like a busy forecourt, the estimate hangs together.

About 2.7 lakh ATMs, on these assumptions. Take 100 crore adults, 80% with a bank account and half of those using ATMs: 40 crore users. Two withdrawals a month each is 80 crore a month, or 2.67 crore a day. If one machine handles about 100 withdrawals a day, you need roughly 2.67 lakh machines. Every input is an assumption to say out loud, not a fact.

Where does population stop being the useful number?

Think of a petrol pump. Nobody decides how many pumps a town needs by counting residents; they count cars, how often each one fills up, and how many fills one nozzle can manage in a day. ATMs work the same way. Population only tells you demand; capacity per machine is what turns demand into a number of machines. So the estimate has two halves: build daily withdrawals from people, then divide by what one machine does.

The demand half, with round numbers. Start with about 100 crore adults, rounded so the arithmetic stays easy. Say 80% hold a bank account with a debit card, and half of those actually take cash from a machine in a typical month: 40 crore users. Two withdrawals a month each gives 80 crore withdrawals, which over 30 days is about 2.67 crore a day.

How many withdrawals can one machine really handle?

A withdrawal takes about two minutes from card in to cash out, so a machine busy all 24 hours could do 720 a day. It never is. Demand bunches around mornings, lunch, evenings and salary days, and sits near zero at three in the morning. Size the machine for its realistic daily use, not its theoretical maximum, because idle night hours are capacity nobody can use. About 100 a day, a little over three hours of real use, is a fair working figure, and the one you should defend if pushed.

How do you sense-check 2.7 lakh?

Turn the answer into a ratio a person can judge. 2.67 lakh machines for 100 crore adults is about 27 ATMs per lakh adults, or one machine for every 3,750 adults. In a city that feels like a machine every few streets; in a village it feels sparse, which matches deployment leaning toward cities. RBI publishes the actual installed count, so say you would check your figure against the latest data. The second check is direction: digital payments cut cash withdrawals, so the honest answer drifts down over time, not up. If withdrawals fall from two a month to 1.5, the requirement falls to about 2.0 lakh.

About Rs 3.1 crore a year, on these assumptions. Take 5 shows a day and 200 seats. Weekday shows run about 25% full over 261 days and weekend shows about 55% over 104 days, giving roughly 122,450 tickets. At an average of Rs 250 a ticket that is about Rs 3.06 crore. Occupancy, not seat count, is the input that swings the answer.

How do you structure it so the interviewer can follow?

Think of a bus route: revenue is trips a day times seats times how full the bus is times the fare. A screen is the same machine with a different product. Revenue is capacity times utilisation times price, and the estimate is only as good as the utilisation guess. Lay the chain out first, then fill each slot with a number and a one-line reason.

Capacity: about 5 shows a day, since a film of two and a half hours plus cleaning takes roughly three hours a slot from late morning to past midnight, and about 200 seats in a mid-sized auditorium. Utilisation: a Tuesday afternoon show might be a fifth full while a Saturday night show is close to sold out, so split the year. With 104 weekend days at 55% and 261 weekdays at 25%, the blended occupancy is 33.5%. Price: Rs 250 as an average across morning discounts and premium evening seats.

Why is occupancy the swing input and not seats?

Each input moves revenue in proportion, so what matters is how wrong you could plausibly be. Seats are visible on a booking app's seat map and you will not be more than 10% off. Occupancy could reasonably be anywhere from 20% to 45% blended, and every 10 points is about Rs 0.91 crore, close to 30% of the answer. When you present an estimate, name the input you are least sure of and show what a fair range for it does to the total.

State the boundary of what you sized. This is gross ticket revenue before taxes, and the operator shares part of it with the film's distributor. Food, drinks and on-screen advertising sit outside it and can add a large share on top, so say which number you were asked for.