Investment Banking puzzles, solved step by step

Light rope A at both ends and rope B at one end at the same moment. When A burns out, 30 minutes have passed: light B's other end, and B burns out 15 minutes later, at 45 minutes. Lighting a rope at both ends halves whatever burning time it has left, however unevenly it burns, so B's remaining 30 minutes take 15.

Why can you not just cut a rope into pieces?

Because uneven burning means length tells you nothing about time. Think of a candle with a thick base and a thin top: half its height might burn in ten minutes or in forty. The only thing you know for certain about each rope is its total burning time, 60 minutes, so every step must work with time, never with length. Cutting, folding or marking a rope uses length, which is why every answer built on them fails.

Why does lighting both ends halve the time?

Picture the rope as a row of short segments, each with its own burning time, adding up to 60 minutes. Two flames eat into the row from both ends at the same moment and meet somewhere. When they meet, each has burned for the same length of time, and between them they have consumed all 60 minutes of segments. Two flames sharing 60 minutes of burning finish in 30, wherever along the rope they happen to meet. The same holds for a rope with any amount of burning time left: lit at both ends, it lasts half that.

How does that build 45 minutes?

Light A at both ends and B at one end at time zero. A is gone at 30 minutes, which is your signal. B has burned for 30 minutes from one end, so exactly 30 minutes of burning time remain in it, wherever the flame has reached. Light B's other end at that moment and those 30 minutes burn in 15. B goes out at 30 plus 15, which is 45 minutes, and no step relied on length. Say the answer as three events with clock times, 0, 30 and 45, so the interviewer can follow each one.

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.

Enterprise value is Rs 1,320 crore. Start with equity value of Rs 1,000 crore. Add the claims that rank alongside or ahead of the ordinary shareholders on the consolidated business: debt of Rs 400 crore, preference shares of Rs 100 crore and minority interest of Rs 50 crore. Subtract what the company owns that is not part of that operating business: cash of Rs 150 crore and the Rs 80 crore associate stake. 1,000 plus 550 less 230 is 1,320.

Why are some items added and others subtracted?

Picture buying a shop. You pay the owner for her shares, but you also take on the bank loan, you owe the silent partner his share of the profits, and you inherit the cash in the till and the shop's small stake in the bakery next door. The price of the shop itself is what you paid, plus the loan and the partner's claim, minus the cash and the bakery stake, which you could sell on day one without touching the shop. Enterprise value is the value of the consolidated operating business, so you add every claim on that business and subtract every asset whose earnings sit outside it.

What is the test for each line?

Ask where the item's earnings appear. Minority interestThe share of a consolidated subsidiary owned by outside shareholders. Its profit is inside consolidated EBITDA but belongs to someone else. is added because consolidation puts 100% of the subsidiary's EBITDA into the group figure even though outsiders own part of it; leave it out and EV is too small for the EBITDA it is set against. Preference shares are added because their dividend is a claim ahead of the ordinary equity. The associateA company in which the group holds a significant but not controlling stake, usually 20% to 50%, shown as one line of share of profit below operating profit. is subtracted for the mirror reason: its profits arrive as a single line below EBITDA, so its value must come out of the numerator to keep the multiple consistent. The rule is consistency between numerator and denominator: EV must describe exactly the business whose EBITDA you divide it by.

What would you say the question leaves out?

Three things an interviewer may push on. Net debt here is Rs 250 crore, but a company needs some cash to run, so only the excess is truly surplus; most desks ignore that in an interview and net all of it, which is the assumption to state. Debt-like items such as unfunded pension deficits, lease liabilities and earn-outs also belong on the add side, and an interviewer who adds a convertible bond is checking whether you treat it as debt or as equity at the current share price. Give the Rs 1,320 crore, then name one debt-like item you would ask about, because the bridge in a live deal is longer than the one in the question. The multiple this feeds, EV/EBITDA, is only as clean as the bridge beneath it.

The lottery ticket has a beta of zero, and the half-and-half portfolio has a beta of 0.5. Beta is the covariance of an asset's return with the market's return divided by the market's variance. A lottery draw does not move with the index at all, so its covariance is zero, however wild its payoff. Betas combine in proportion to weights, so half an index fund at beta 1 and half tickets at beta 0 give 0.5 x 1 + 0.5 x 0 = 0.5.

Why is a wildly risky asset given a beta of zero?

Think of two shops on one street. One sells umbrellas, whose takings rise and fall with the weather; the other sells birthday cards, whose takings have nothing to do with it. The card shop's sales swing plenty, but not with the rain. Beta measures only the part of an asset's movement that follows the market, so an asset whose payoff is decided by a draw of numbered balls has a beta of zero no matter how large its swings. On a Rs 10 ticket with an assumed one in a crore chance of a Rs 5 crore prize, the standard deviation of the return is about 1,600 times the stake and the beta is still zero. Volatility is total riskThe full spread of an asset's returns, usually measured by standard deviation, including both the part linked to the market and the part unique to the asset.; beta is only its market-linked part.

How does the portfolio beta follow?

Covariance is linear in the weights, so the beta of a portfolio is the weighted average of the betas inside it. An index fund tracking the market has a beta of one by construction. Half at beta 1 and half at beta 0 gives 0.5, the same beta as a portfolio that is half the index and half cash. That comparison is the point: in the capital asset pricing model the lottery half and a cash half demand the same return, the risk-free rate, because neither adds market risk; the lottery's enormous extra risk is diversifiable in theory and simply unrewarded.

What does the security market line say about the blend?

Plot required return against beta. With an assumed 6% risk-free rate and 6% equity premium, the line runs from 6% at beta 0 to 12% at beta 1, and the blend at beta 0.5 would be required to earn 9%. Its expected return is nothing like that: if a ticket pays out an assumed half of its price in prizes, its expected return is -50%, and the blend's is 0.5 x 12% plus 0.5 x -50%, which is -19%. Beta tells you where an asset sits along the line, and the lottery ticket sits about 56 points below it: a zero-beta asset with an appalling expected return. That is the limitation to state: beta says nothing about expected return outside the model's assumptions, and nothing about risk the market does not share.

Why a banker asks it: the same logic decides what goes into a cost of equity. A project whose cash flows do not move with the economy, a regulated pipeline or a contracted toll road, carries a low beta and a low cost of equity even if its engineering risk is large, because that risk is specific and diversifiable. Candidates who confuse volatility with beta tend to over-discount exactly those assets, and to under-discount a cyclical business that happens to look calm.

About Rs 27.3 lakh, the value of selling it. Keeping the elephant forever costs Rs 20 lakh a year, minus Rs 2 crore in today's money at 10%. Selling means one more year of upkeep and Rs 50 lakh at the end, a net Rs 30 lakh worth Rs 27.3 lakh today. Donating is worth zero. A gift is worth its best use, and the free exit means it can never be worth less than zero.

Why does the elephant not have one value?

Think of being handed an old car that needs Rs 30,000 of repairs before anyone will buy it. The car is not worth its sticker price; it is worth what you can do with it, net of what doing that costs. An asset with a carrying cost has no value of its own: each option for it has a value, and the asset is worth the best of them. So the first move is to list the options out loud, keep it, sell it or give it away, and then price each one at the same 10% rate.

How do you price each option?

Keeping is a perpetuityA payment that repeats every year forever. Its value today is the yearly amount divided by the discount rate. of costs: Rs 20 lakh a year divided by 10% is minus Rs 200 lakh, or minus Rs 2 crore. Selling means paying one more year of upkeep while the paperwork clears, then collecting Rs 50 lakh. Net of upkeep, the sale delivers Rs 30 lakh at the end of year one, and Rs 30 lakh a year away is worth Rs 27.3 lakh today. Donating costs nothing and brings nothing. The working assumes upkeep is paid at the year end; if it is paid upfront, the sale is worth Rs 25.5 lakh instead, and the choice does not change.

When would the gift be worth nothing?

Change one number. If the best sale price were Rs 15 lakh, the sale would net minus Rs 5 lakh at the year end, minus Rs 4.5 lakh today, and donating would win at zero. The free exit puts a floor under the gift: it can be worth nothing, but never less than nothing, as long as someone will take it away for free. Remove the sanctuary and the floor goes too. That is why bankers ask about exit options before valuing a loss-making plant, a lease that cannot be broken or a subsidiary with heavy fixed costs.

22 times. The minute hand goes round 12 times in 12 hours and the hour hand once, so the minute hand gains 11 laps and passes the hour hand 11 times. The overlaps come every 12/11 hours, about 65 minutes 27 seconds apart, and the one you expect between 11 and 12 is the one at 12:00 itself. Two 12-hour cycles make 22, counting midnight once.

Why is it not once an hour?

Think of two runners on a track, one doing 12 laps while the other does 1. The faster runner passes the slower one only 11 times, because the slower runner's single lap is subtracted from the faster one's total. Overlaps count the laps the minute hand gains on the hour hand, and it gains 12 minus 1, which is 11, every 12 hours. The hour hand keeps creeping forward, so each catch-up takes a little longer than an hour.

When exactly do the hands meet?

The minute hand moves 6 degrees a minute and the hour hand half a degree, so the minute hand gains 5.5 degrees a minute. A full lap of 360 degrees takes 360 over 5.5, which is 65 5/11 minutes, so the meetings come at 12:00, about 1:05:27, about 2:10:55 and so on. The eleventh gap lands exactly back on 12:00, which is why there is no meeting between 11 and 12: the hands meet at about 10:54:33 and next at 12:00.

What do you say about the edges?

State how you count midnight. Counting from one midnight up to but not including the next gives 22; counting both midnights gives 23. That one sentence shows you saw the edge case rather than stumbling into it. The same lap method answers the follow-ups in seconds: the hands point in exactly opposite directions 22 times a day as well, and form a right angle 44 times, because a right angle happens twice per lap gained.

Dilutive, by about 5.3%. At Rs 45 a share, Rs 1,200 crore needs 26.7 crore new shares, taking the count to 126.7 crore. Earnings rise to Rs 360 crore, so EPS falls from Rs 3.00 to about Rs 2.84. The shortcut: in an all stock deal, buying at a higher P/E than your own always dilutes, because each rupee of earnings bought costs more than each rupee handed over. It takes Rs 20 crore of after-tax synergies to break even.

What is the shortcut that answers it in ten seconds?

Picture swapping a shop for a share in a friend's shop. If your shop is priced at 15 times its yearly profit and your friend's at 20 times, every rupee of your profit you hand over buys only three quarters of a rupee of theirs. In an all stock deal the acquirer gives up its own earnings yield and receives the target's, so a target on a higher P/E means dilution. Here the acquirer's earnings yieldEarnings divided by price, the inverse of the P/E. A P/E of 15 is an earnings yield of 6.7%. is 1 over 15, 6.7%, and the target's is 1 over 20, 5.0%.

How do the full numbers work?

New shares: Rs 1,200 crore divided by Rs 45 is 26.7 crore, so the count rises from 100 to 126.7 crore. Earnings rise from Rs 300 crore to Rs 360 crore. When the share count grows faster than earnings, EPS falls: Rs 360 crore over 126.7 crore shares is Rs 2.84, down 5.3% from Rs 3.00.

What would make the deal break even?

Turn it round. The enlarged share count needs Rs 3.00 of earnings per share, so Rs 3.00 times 126.7 crore, which is Rs 380 crore. The deal needs Rs 20 crore a year of after-tax synergies, a third of the target's own earnings, before EPS is back where it started. Saying that number tells the interviewer how big the problem is, which matters more than the sign. Close with the limit: one year of EPS is not value, and a dilutive deal can still be worth doing if the target grows much faster than the acquirer.

2/11, about 18.2%. Of the 36 equally likely outcomes, 11 contain at least one 4: six with the first die showing 4, six with the second, less the double 4 counted twice. Of those 11, only 4 and 3 and 3 and 4 sum to 7. The new information shrinks the world to 11 outcomes, and 2 of them qualify.

What does being told something actually change?

Imagine a teacher tells you that your friend is one of the 11 students who scored above 90. Your friend's chance of topping the class is now measured against those 11, not the whole class of 36. Conditional probability throws away every outcome the new information rules out and counts only inside what is left. Here the information rules out the 25 outcomes with no 4 anywhere, and 11 remain.

Why is 1/6 tempting and wrong?

1/6 comes from fixing one die at 4 and asking for a 3 on the other, which is the right answer to 'the first die shows a 4'. 'At least one shows a 4' is a weaker statement: it lets the 4 sit on either die, which creates 11 outcomes rather than 6. If you counted the double 4 twice you would have 12 outcomes and get 2 in 12, exactly 1/6. Counted once, as it must be, there are 11, and the answer edges up to 2/11.

How do you check it quickly?

Use the complement for the denominator. The chance of no 4 on either die is 5/6 times 5/6, which is 25/36, so at least one 4 is 11/36. Two cells sum to 7 and contain a 4, so 2/36 over 11/36 is 2/11. Two routes to the same 11 is what the interviewer wants to hear before you commit to the answer.

The 12.5% loan compounded annually is cheaper. 12% a year compounded monthly means 1% a month, and 1.01 to the twelfth is 1.1268, an effective rate of 12.68% a year. That is about 0.18 points more than 12.50%. On a Rs 100 crore loan the monthly loan costs about Rs 18 lakh a year more. Compare effective annual rates, never quoted rates with different compounding.

Why is 12% compounded monthly not really 12%?

Think of a savings account that credits interest every month. In February you earn interest on January's interest as well as on your deposit, so by December the year's earnings come to a little more than twelve single months. A rate compounded monthly is charged as one twelfth each month, and each month's interest then earns interest of its own, so the true yearly cost is higher than the quote. The true yearly cost is the effective annual rateThe rate that, charged once a year, costs the same as the quoted rate with its compounding. It is the only fair basis for comparing loans., and it is the only number worth comparing.

How do you work out 1.01 to the twelfth in your head?

Expand it. Twelve months of 1% give 12% of simple interest. Every pair of months adds a little interest on interest, and there are 66 pairs, so add 66 times 0.01 squared, 0.66%. The first two terms give 12.66%, the next adds a sliver more, and the total is 12.68%, which beats 12.50% by 0.18 points. Saying the shortcut out loud shows you can reason about compounding without a calculator.

What else should you say before you finish?

Turn the comparison round as a check. The monthly quote that would match 12.5% effective is 12 times (1.125 to the power of one twelfth, minus 1), about 11.84%. Any lender quoting monthly compounding has to quote below 11.84% to beat the 12.5% annual loan. Then state the limit: fees, prepayment penalties and the timing of repayments can matter more than 0.18 points, so the effective rate is where a comparison starts, not where it ends.

50 teams. Choosing any 3 of 8 analysts gives 8 x 7 x 6 / 6 = 56 teams. The forbidden ones contain both analysts who refuse to work together, and the third seat can go to any of the other six, so there are 6 forbidden teams. 56 minus 6 leaves 50. Count everything, then subtract the cases the rule forbids.

Why count the teams you do not want?

Think of seating guests at a dinner when two of them have fallen out. Listing every acceptable table is slow; counting all possible tables and removing those that seat the two together is quick. When a rule forbids a small set of cases, count everything and subtract the forbidden ones, because the forbidden set is usually the easy one to count. Here a forbidden team is the two feuding analysts plus a third person, and there are only 6 choices for the third.

How do you get 56 without a formula sheet?

Fill the team one seat at a time: 8 choices, then 7, then 6, which is 336 ordered picks. Each team of three turns up in 3 x 2 x 1 = 6 different orders, so divide by 6 to get 56 distinct teams. That is the combinationA selection where order does not matter. The number of ways to choose k from n is written n choose k. count, 8 choose 3. Subtract the 6 forbidden teams and 50 remain.

How do you check 50 a second way?

Count the allowed teams directly, split by how many of the feuding pair they include. Teams with neither of the two are 6 choose 3, which is 20; teams with exactly one are 2 times 6 choose 2, which is 30; together 50. Two methods agreeing is the answer the interviewer remembers, and the second method is the one you will need when a follow-up adds a second rule.