Investment Banking puzzles, solved step by step

Put one white ball alone in bowl A and the other 99 balls in bowl B. Bowl A gives white every time. Bowl B gives white 49 times in 99. Each bowl is picked half the time, so the chance of white is 0.5 x 1 + 0.5 x 49/99, about 74.7%, against 50% for an even split.

Why does the number of balls in a bowl not matter?

Think of two queues at a ticket counter, one with a single person and one with ninety nine. If a coin decides which queue gets served, the lone person gets served half the time. The bowls work the same way. The bowl is chosen with even odds, so a bowl holding one ball carries exactly as much weight as a bowl holding ninety nine. That is the lever: you can make one half of the draw certain at the cost of a single white ball.

How do you prove no other split beats it?

Bowl A can never beat 100%, and one ball is the cheapest way to get there. Every extra white ball you move into bowl A is a white ball taken out of bowl B, which lowers bowl B's odds without raising bowl A's. Adding black balls to bowl A only hurts it. So the best bowl A is one white ball, and bowl B's best case is whatever is left: 49 white out of 99.

Say the 74.7% figure out loud, then say the general rule: with n white and n black balls, the answer is one half plus one half of (n minus 1) over (2n minus 1), which creeps towards 75% as n grows. Interviewers like hearing that you see the limit.

488 small cubes are painted and 512 are not. The unpainted ones form a hidden core with one layer peeled off every side, so the core is 8 x 8 x 8, which is 512. The whole cube is 1,000, so 1,000 minus 512 leaves 488 with at least one painted face.

Why count the cubes you cannot see?

Picture a mango with its skin peeled. Measuring the skin piece by piece is fiddly; measuring the fruit left inside and subtracting from the whole is easy. The unpainted cubes form one clean block, so counting them and subtracting is faster and harder to get wrong than adding up the painted surface. Peeling one layer from both sides of each dimension takes 10 down to 8.

How do you check 488 the other way?

Count the surface directly, taking care not to double count. There are 8 corners, 12 edges each carrying 8 cubes once the corners are removed, and 6 faces each carrying an 8 x 8 centre. That gives 8 plus 96 plus 384, which is 488. Two methods agreeing is exactly what the interviewer wants to hear.

Day 29. The coverage doubles every day, so it halves when you step back one day. Full on day 30 means half on day 29, a quarter on day 28 and one thirty-second on day 25. Most of the growth in a doubling process arrives right at the end.

Why is day 15 wrong?

Day 15 would be right if the pond gained the same area every day, like a tank filling from a tap. Doubling is different: it is money compounding at 100% a day. Under doubling, each step back in time halves the amount, so half full is always exactly one step before full. On day 15 the pond is only one part in about 32,768 covered.

Why would a banking interviewer ask this?

It is a test of whether you think in growth rates or in straight lines. The same instinct that answers day 15 also overestimates how much of a compounding investment's value arrives early. Say that link in one sentence after you answer: it shows you know why the question is on the table.

The tax rate is 50%. P/E of 20 on equity of 200 gives net income of 10. Interest of 20 at 5% means debt of 400, so enterprise value is 600 and EBITDA is 60. Take off depreciation of 20 and interest of 20 to reach pre-tax profit of 20. Tax is 20 minus 10, which is 10, so the rate is 50%.

Where do you start when every number looks equally useful?

Start from what the answer is made of. A tax rate is tax divided by pre-tax profit, so the job is to find the two profit lines either side of the tax charge. Everything in the question is a road to one of those two numbers. Say that out loud first: it tells the interviewer you have a plan before you touch the arithmetic.

Net income is one step: equity value over the P/E, 200 over 20, is 10. Pre-tax profit takes longer. The enterprise valueThe value of the whole business to all its funders: equity value plus debt, less cash. multiple is on EBITDA, so you need EV, which needs debt, which the interest line hands you: 20 of interest at 5% means 400 of debt.

What assumption are you making, and should you say it?

You are assuming the company holds no cash, so enterprise value is simply equity plus debt. Say the assumption before you use it, because with any cash on the balance sheet EV falls, EBITDA falls, and the tax rate changes. A 50% rate is also high for most countries, which is worth one sentence: the interviewer built round numbers, not a real company.

About 11.1%. A price of 100 falls 10% to 90. Getting back to 100 needs a rise of 10 on a base of 90, which is 10 divided by 90, or 11.1%. The general rule: to recover a loss of L, you need L divided by (1 minus L), which grows much faster than the loss itself.

Why is the recovery bigger than the fall?

A 10% discount on a 100 rupee shirt takes it to 90. If the shop then marks it up by 10%, the tag reads 99, not 100, because the markup is taken on 90. Percentages are always measured against a base, and after a fall the base is smaller, so the same rupee gain is a bigger percentage.

What do you add to sound like a banker rather than a calculator?

Give the number, then the pattern. Large drawdowns are hard to climb out of: a fund that loses 50% has to double just to break even. That one sentence shows you see why lenders and investors care about downside, which is the real reason the question is asked.

Fill the 3-litre bottle, pour it into the 4, fill the 3 again and pour until the 4 is full: 2 litres are left in the 3-litre bottle. That is four moves. Because 3 and 4 share no common factor, fills and pours can build every whole number of litres, so any total from 1 to 7 is possible across the two bottles.

What does each move actually measure?

Think of paying for something with only Rs 3 and Rs 4 coins. You can pay Rs 2 by handing over two Rs 3 coins and taking a Rs 4 coin back as change. The bottles work the same way. Every fill adds a full bottle and every pour into a full bottle sets one bottle's worth aside, so the volumes you can make are sums and differences of 3s and 4s. Two litres is 3 plus 3 minus 4: you draw two small bottles from the tap, the big bottle swallows 4 of those 6 litres, and 2 are left.

Is there a second route, and which volumes are out of reach?

Start from the big bottle and it still works, only slower. Fill the 4, pour into the 3 to leave 1, empty the 3, move the 1 litre across, fill the 4 again and top up the 3, which takes 2 litres and leaves 2 in the big bottle. That is six moves, 4 plus 4 minus 3 minus 3. Saying both routes and choosing the shorter one shows the interviewer you searched the problem rather than remembered it. Write each state as a pair, small bottle first, so neither of you loses track.

For the second half of the question, use BezoutA result in number theory: whole-number combinations of two numbers produce exactly the multiples of their highest common factor, and nothing else.'s rule: two bottle sizes can measure exactly the multiples of their highest common factor. The highest common factor of 3 and 4 is 1, so every whole number of litres can be built, limited only by capacity: up to 4 in one bottle and 7 across both. Contrast a 4 and a 6: their common factor is 2, so 3 litres is impossible however long you pour.

About Rs 92.4 per Rs 100 of face value. Discount the five Rs 8 coupons and the Rs 100 repayment at 10%: the coupons are worth Rs 30.33 today and the repayment Rs 62.09. It sits below par because it pays Rs 2 a year less than a new bond, and a buyer wants those five years of shortfall, worth Rs 7.58 today, taken off the price.

How do you price it from first principles?

Imagine you let a flat on a five-year lease at Rs 8,000 a month, and every similar flat now fetches Rs 10,000. Someone buying your lease will pay less for it, by roughly the rent you miss over the remaining term. A bond is the same lease on money. A bond's price is every future payment discounted at the yield the market now demands, so when that yield rises above the coupon, the price has to fall below par. At 10%, each Rs 8 coupon and the final Rs 100 are worth less today the further away they sit.

Is there a shortcut that skips most of the discounting?

Yes, and it is the one to use out loud. Price equals par minus the yearly coupon shortfall times the annuity factor: 100 minus 2 x 3.79 is 92.42. The annuity factorThe present value of Rs 1 a year for a set number of years. At 10% for five years it is about 3.79. is the only number that needs any work, and at 10% for five years it is close to 3.8. The same rule runs the other way: a bond paying above the market yield trades above par by the extra coupon times the annuity factor.

Two limits are worth one sentence each. The yield must come from bonds of the same maturity and credit quality, which is what similar bonds means; in today's market you would read it off comparable issues. Real bonds also pay coupons twice a year in many markets and trade with accrued interest, which shifts the answer by small amounts. For an interview answer, annual discounting is the expected level.

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.

Net income falls Rs 60, cash falls Rs 10, debt rises Rs 50 and retained earnings fall Rs 60. The full Rs 100 is expensed, so pre-tax income drops 100 and, after 40% tax, net income drops 60. The cash flow statement adds back the Rs 50 of PIK as a non-cash expense. On the balance sheet the PIK accrues to the debt, and assets of minus 10 equal liabilities and equity of plus 50 minus 60.

What does paid in kind mean in plain terms?

Suppose you owe a friend Rs 100 of interest, pay Rs 50 in cash and write an IOU for the other Rs 50. Your wallet is Rs 50 lighter, and you now owe Rs 50 more than before. Paid in kindInterest settled by adding it to the loan principal instead of paying cash. Often shortened to PIK. interest works exactly like that IOU. PIK interest is a real expense paid with more debt instead of cash, so it reduces profit while adding to the loan. That one sentence tells you where every number in the walk will land.

Why does cash fall by only Rs 10?

Because tax is worked out on the whole expense, not just the cash part. Assume both halves of the interest are tax deductible, which is the usual set-up for this question; real deductibility rules vary by country and are often capped, so say the assumption. The Rs 100 expense saves Rs 40 of tax, and the only cash going out is the Rs 50 of cash interest, so cash falls Rs 50 minus Rs 40, which is Rs 10. On the cash flow statement the same Rs 10 appears as net income of minus 60 plus the Rs 50 PIK add-back.

How do you prove the balance sheet balances?

Say it in one breath. Assets fall 10 through cash. Liabilities rise 50 because the PIK interest is added to the principal of the debt. Equity falls 60 through retained earnings. Minus 10 on the asset side equals plus 50 minus 60 on the other side, so the balance sheet balances. Saying the check out loud is what turns a recited answer into a reasoned one, and it catches the slip of forgetting where the PIK went.

EV/Revenue is 2.0x and price to sales is 1.8x. Take Rs 100 of revenue. A 20% margin gives EBITDA of Rs 20, and at 10x the enterprise value is Rs 200, two times revenue. Net debt equals one year of EBITDA, Rs 20, so equity value is Rs 200 minus Rs 20, which is Rs 180, or 1.8 times revenue.

How do multiples convert into one another?

Think of a corner shop priced at 10 years of profit, where profit is a fifth of takings. Priced against takings, it costs two years' worth: 10 times a fifth. An enterprise value multiple converts into another by the ratio of their denominators, so EV/Revenue equals EV/EBITDA times the EBITDA margin. Working per Rs 100 of revenue keeps every step visible: EBITDA Rs 20, enterprise value Rs 200.

Why is price to sales lower than EV to revenue here?

Price to sales is an equity multiple: the market value of the shares over revenue. To reach it from enterprise value you cross the equity bridgeThe steps from enterprise value to equity value: subtract debt and similar claims, add cash.. Equity value is enterprise value less net debt, so an equity multiple sits below the enterprise multiple when a company has net debt, and above it when it holds net cash. Net debt of one year's EBITDA is Rs 20 per Rs 100 of revenue, which takes Rs 200 down to Rs 180, or 1.8x.

The limitation to state is why bankers prefer one family over the other. Revenue belongs to every funder of the business, so comparing it with enterprise value is consistent; comparing it with equity value alone mixes a whole-business line with one funder's slice. Two companies with identical operations but different debt will show the same EV/Revenue and different price to sales ratios.