Investment Banking puzzles, solved step by step

About 71% for 30 people, and the odds pass 50% at 23. Work the chance that nobody matches: the second person avoids the first with 364/365, the third avoids both with 363/365, and so on. Multiply 30 of these and you get about 29.4%, so a shared birthday has a 70.6% chance. Repeating the product shows it first passes half at 23 people.

Why does the answer feel far too high?

Your instinct pictures yourself in the room: what are the odds someone shares my birthday? That is about 30 in 365, under 10%. But the question allows any two people to match. In a group of 30, every person can be compared with every other, and there are 30 x 29 / 2 = 435 such pairs. The number of pairs, not the number of people, drives the answer, and pairs grow roughly with the square of the group. Four hundred chances to hit a one-in-365 coincidence is a lot of chances.

How do you compute it without a list of 435 cases?

Count the opposite. Matching can happen in many overlapping ways, but nobody matching happens in exactly one: each new person must miss every birthday already taken. When at least one is messy, compute none and subtract from one.

For a quick estimate in your head, use pairs: the chance of no match is close to e raised to minus pairs over 365. With 253 pairs that is e to the minus 0.69, almost exactly a half, which is why 23 is the crossing point. The method assumes birthdays are spread evenly across the year; real birthdays bunch a little, which only nudges the odds higher. By 60 people the chance is 99.4%.

About 20.1% a year. Revenue multiplied by 2.5 over five years, so the annual rate is 2.5 to the power one fifth, minus one. A quick check: 1.2 to the fifth power is about 2.49, so the rate is just above 20%. Dividing the 150% total growth by five gives 30%, which overstates it, because 30% compounded for five years would take 100 to about 371.

Why is 30% wrong when 150% over five years looks like 30% a year?

A child's height or pocket money grows on what is already there. If your allowance rises 10% a year, year three's rise is 10% of a bigger allowance than year one's. Compound growth earns on its own past growth, so a constant rate produces bigger rupee steps every year, and dividing the total by the years overstates the rate. Thirty per cent a year would take 100 to 130, 169, 220, 286 and finally 371.

How do you get 20.1% without a calculator?

Guess and check from a round rate. Try 20%: 1.2 squared is 1.44, cubed is 1.73, then 2.07, then 2.49, just shy of 2.5, so the answer is a hair above 20%. A second route is the rule of 72: at 20% money doubles in about 3.6 years, and 2.5x in five years is a little more than one doubling plus a bit, consistent with roughly 20%.

Say the limitation. CAGR describes only the start and end points. Revenue could have jumped to 250 in year one and sat flat, or fallen and recovered; the CAGR is the same 20.1%. If the path matters, ask for the yearly figures.

About Rs 5.24. A call plus cash worth the strike's present value pays the larger of the share price and 100 at expiry, and so does a put plus the share. Identical payoffs must cost the same today: 10 + 100/1.05 = P + 100. The present value of 100 is 95.24, so the put is 5.24. No pricing model or volatility estimate is needed.

Why can you price the put without knowing volatility?

If two combo meals contain exactly the same food, a restaurant cannot charge different prices for long: everyone would buy the cheaper one. Options work the same way. Two portfolios that pay the same in every outcome must cost the same today, so if you know the price of one, the other follows without any view on where the share goes. Volatility matters for what a call is worth, but the call's price already contains it. The put inherits it through the parity relationship.

Build the two portfolios. A: buy the call and put aside 95.24 at 5%, which grows to exactly 100. B: buy the put and buy one share. If the share ends at 130, A exercises the call for 30 and adds its 100 cash, B holds a share worth 130 and lets the put expire: both have 130. If the share ends at 70, A lets the call expire and holds 100 cash, B exercises the put to sell the share for 100: both have 100.

What does the formula say, and what if the put traded at Rs 7?

If the put traded at 7, portfolio B would cost 107 against 105.24 for A, so you would sell B and buy A, pocketing 1.76 today with payoffs that cancel at expiry. Selling B means writing the put and short selling the share. Traders arbitraging that gap is what holds parity in place, which is why quoted put and call prices sit close to it in practice.

Say the assumptions: European options, so no early exercise; no dividend before expiry; and annual compounding. With continuous compounding the strike's present value is 100 x e to the minus 0.05, about 95.12, and the put comes out near 5.12. Give the convention, not just the number.

17 minutes. The 1 and 2 cross (2), the 1 returns (1), the 5 and 10 cross together (10), the 2 returns (2), and the 1 and 2 cross again (2). The obvious plan, the fastest person escorting each of the others, takes 19. The saving comes from pairing the two slowest so the 5 minute walk overlaps the 10 minute one.

Why does the escort plan feel right but lose two minutes?

Picture sharing taxis to the airport. Putting the two people with the latest flights in the same cab costs nothing extra for either; putting each in a cab with someone in a hurry makes every ride as slow as its slowest passenger. A pair always moves at the slower pace, so the 5 minute walker is free if he travels with the 10, and costs a full five minutes if he travels with anyone else. The escort plan pays for the 10, the 5 and the 2 separately.

How do you know 17 cannot be beaten?

Count what every plan must pay. Five crossings are needed: three trips over and two back with the torch. The 10 must cross at least once, costing 10. If the 5 crosses separately from the 10, that is another 5 on top. Pairing them leaves two forward trips and two returns to do with the 1 and the 2, and the cheapest way to do that is 2 + 1 + 2 + 2. The trade is paying the 2 minute walker an extra return in order to save the 5 minute walker's crossing, and 2 is less than 5 minus 1.

The rule is worth stating because it is not always pairing. If the walkers took 1, 4, 5 and 10 minutes, twice 4 is 8, more than 1 plus 5, so escorting wins: 21 minutes against 23 for pairing. Showing the interviewer you know when your trick stops working is better than the trick.

About Rs 441 crore apart: Rs 1,913 crore with the add-back against Rs 1,471 crore without, a 30% gap. Treat the compensation as a real cost. It is paid in shares rather than cash, but existing holders bear it through dilution. Either deduct it from cash flow and use today's diluted share count, or add it back and count every future share it will create. Adding it back and ignoring the dilution counts the cost as zero.

If no cash leaves, why is stock compensation a cost?

Imagine a family business that pays its manager not in salary but by handing over a slice of the shop every year. No cash leaves the till, but each year the family owns less of the shop. Stock compensation is a real expense paid in ownership instead of money, and today's shareholders pay it through dilution. If the company had paid staff Rs 30 crore in cash, everyone would deduct it. Paying in shares changes who bears the cost, not whether there is one.

The two values follow from the perpetuity formula. With the add-back, 130 x 1.03 / (10% - 3%) is about Rs 1,913 crore. Treated as a cost, 100 x 1.03 / 7% is about Rs 1,471 crore. The gap, about Rs 441 crore, is exactly the present value of paying Rs 30 crore a year, growing at 3%, in shares.

Can the add-back version ever be right?

Yes, if you finish the job. Add the Rs 30 crore back, value the firm at Rs 1,913 crore, then recognise that the company will issue Rs 30 crore of new shares every year, growing at 3%, forever. Those future holders own a stream worth Rs 441 crore of the total, leaving Rs 1,471 crore for today's shareholders. Both consistent methods land on the same value for existing holders; the error is adding back the expense and then dividing by today's share count, which counts the cost nowhere. In practice, deducting it is simpler, because projecting every future grant into the share count is hard.

One honest limitation: existing options and restricted shares already granted are a separate matter, counted in today's diluted share count by the treasury method. The question here is future grants, which the cash flow treatment decides.

EBITDA is unchanged, net income falls by 50, cash is unchanged, and goodwill and equity both fall by 50. The impairment sits below EBITDA, so operating profit and pre-tax profit drop 50. Because it is not deductible, tax does not change and net income falls the full 50. The cash flow statement adds the non-cash charge back. On the balance sheet goodwill drops 50 and retained earnings drop 50.

What is an impairment actually admitting?

Suppose you paid Rs 10 lakh for a used car three years ago and a dealer now offers 6 lakh. Writing the car down in your notebook does not take money out of your wallet; the money left when you bought it. A goodwill impairment admits that an acquisition was overpaid for in the past; it does not spend any new cash. GoodwillThe part of an acquisition price above the fair value of the identifiable net assets bought, carried as an asset on the buyer balance sheet. is the premium paid above the target's net assets, and the write-down says part of that premium is no longer supported by the business's expected cash flows.

Why does net income fall the full 50 rather than 37.5?

Tax is what makes most non-cash charges reach cash. Depreciation, for example, usually cuts the tax bill, so a 50 charge at 25% lowers tax by 12.5. Goodwill impairment is generally not deductible, so the tax line does not move and the whole 50 hits net income. Confirm the rule in the relevant tax regime before relying on it. If the charge were deductible at 25%, net income would fall 37.5, cash would rise 12.5 from the lower tax bill, and the balance sheet would show cash up 12.5, goodwill down 50 and equity down 37.5.

Why does an analyst care if nothing happened to cash?

Because it tells you something about management's past decisions and future cash flows. An impairment says an acquisition is earning less than was paid for it, which usually reflects weaker expected cash flows. It also lowers book equity, which can push up leverage ratios measured on book values and occasionally trip a covenant. Most analysts strip it out of adjusted earnings as a one-off, but a string of them is a pattern worth asking about.

About 4.8% a year, forever. Write the terminal value both ways. The multiple says 10 x EBITDA. The Gordon formula says 0.5 x EBITDA x (1 + g) / (0.10 - g). Setting them equal, EBITDA cancels: 1.0 - 10g = 0.5 + 0.5g, so g = 0.5 / 10.5 = 4.76%. Every exit multiple implies a growth rate, and the check is whether that rate is believable for this business.

Why does a multiple contain a growth rate at all?

If a buyer wants a 5% return and pays 25 times this year's rent for a flat, they are counting on the rent rising: with flat rent, a 5% return supports a price of only 20 times. A multiple is a shorthand price for a stream of future cash flows, so every multiple implies some combination of growth and discount rate. In a DCF the exit multiple sits beside an explicit WACC, which means the growth assumption can be solved for exactly.

Set the two terminal values equal. The multiple gives 10 x EBITDA. The perpetuity growth method gives next year's free cash flow over (r minus g), and next year's cash flow is 0.5 x EBITDA x (1 + g). EBITDA cancels from both sides, leaving one equation in g, which solves to 4.76%.

How do you judge whether 4.8% is believable?

Perpetual growth means growth forever, so it should not exceed the long-run nominal growth of the economy whose currency the cash flows are in, or the business eventually becomes larger than that economy. Whether 4.8% passes depends on that currency's inflation and real growth, so state the benchmark you are using rather than a number from memory. The useful habit is running the check both ways: an exit multiple that implies an unbelievable growth rate is a sign the multiple came from a peer set that does not match this business.

Say the convention too. If you apply the multiple to final-year EBITDA but take next year's cash flow without the (1 + g) uplift, the answer becomes 10% minus 0.5/10, which is 5.0%. A small difference, but naming the convention shows you know where it comes from.

5,898 dollars. The fee is 1.5% of Rs 5,00,000, which is Rs 7,500, leaving Rs 4,92,500 to convert. Divided by 83.50 that is about 5,898.2. A quick route: 83.5 x 6,000 is 5,01,000, so Rs 5 lakh buys about 5,988 dollars before the fee, and taking 1.5% off 5,988 gives the same 5,898.

What is the question really testing?

These questions sit in the timed numerical section of an online assessment, and the arithmetic is easy. What they test is whether you apply each adjustment to the right base under time pressure. Think of a restaurant bill with a 10% service charge: it is 10% of the food, not of the food plus tax, and getting the base wrong gives a number that looks almost right. The test designers put the almost-right numbers in the options, so the base you apply the fee to decides whether you pick the correct one.

Does it matter whether you take the fee before or after converting?

Not for a percentage fee. Taking 1.5% off the rupees and then converting, or converting and then taking 1.5% off the dollars, multiplies by the same two numbers in a different order: 5,988 x 0.985 is also 5,898. Order matters only for a flat fee or a fee charged on a different base; a percentage fee on the same amount commutes with the conversion. Knowing this lets you pick whichever order makes the mental maths easier.

Where does 5,900 come from? Dividing by 1.015 answers a different question: how much could you convert if the 1.5% were charged on top of the converted amount. That base is smaller than Rs 5 lakh, so the fee comes out as about Rs 7,389 instead of Rs 7,500. On a test with options two dollars apart, that is a wrong answer.

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.

75 km. The trains close the 100 km gap at a combined 100 km/h, so they meet after exactly one hour. The bird flies without stopping for that hour at 75 km/h, so it covers 75 km. You never need to track the individual legs, though they do add up: 60 km, then 12, then 2.4, each a fifth of the one before, summing to 75.

What question should you ask first?

If a friend walks a dog in the park for an hour and the dog runs at 10 km/h the whole time, the dog covers 10 km however it zig-zags between trees. Nobody measures each dash. Distance is speed times time, so find how long the bird flies and the shape of its path stops mattering. The trains settle the time: they approach each other at 50 plus 50, which is 100 km/h, and 100 km apart means one hour.

If the interviewer insists on the legs, how do you sum them?

On the first leg the bird and the oncoming train close at 75 plus 50, which is 125 km/h, so they meet after 100 / 125 = 0.8 hours; the bird has flown 60 km. In that time the trains have closed 80 km, leaving 20 km between them. The next leg is the same problem a fifth the size: 12 km, then 2.4, and so on. Each leg is a fixed fraction of the last, which makes the total a geometric series that sums to 60 / (1 - 1/5) = 75 km.

Why ask this in a banking interview? It is a test of whether you look for the simpler quantity before diving into the mechanics, which is what a good analyst does with a messy model: find the number that settles the question and compute that. Answer the short way, then offer the series as a check if there is time.