Investment Banking puzzles, solved step by step

About Rs 50. Prices rising 6% a year compound to 1.06 to the power of 12, which is 2.01, so a basket costing Rs 100 today costs about Rs 201 in year 12. Rs 100 then buys what Rs 49.70 buys now. The rule of 72 gets there in one line: 72 divided by 6 is 12 years for prices to double.

Why does the answer land so close to half?

Think of a plate of biryani that costs Rs 100 today. If its price rises 6% every year, each rise is taken on the previous year's higher price, so the rises themselves grow: Rs 6 in the first year, close to Rs 12 by the twelfth. Inflation compounds like interest, so over 12 years at 6% prices roughly double, and the same Rs 100 note buys roughly half as much. To state a future rupee in today's money, you divide by the growth in prices, 1.06 to the 12th.

How do you get 1.06 to the 12th without a calculator?

Square your way up. 1.06 squared is 1.1236; squared again gives 1.2625 for four years; squared again gives 1.594 for eight years. Twelve years is eight plus four, so multiply: 1.594 x 1.262 is 2.012. Repeated squaring turns eleven multiplications into four, and every step can be said out loud. Then Rs 100 divided by 2.012 is Rs 49.70, and the rule of 72 has already told you to expect about 2.

What are the two wrong answers, and why are they tempting?

The straight-line answer, Rs 28, takes Rs 6 off every year as if prices rose by the same rupee amount each time. The subtler one, Rs 48, shrinks the money by 6% a year, 0.94 to the 12th. Inflation raises prices by 6%, which cuts buying power by 6 divided by 106, about 5.7% a year, not by a full 6%. The gap between Rs 48 and Rs 50 is small here and widens with the rate and the years.

Why a deal team cares: a return in rupees is not a return in buying power. Money growing at an assumed 12% a year while prices rise 6% grows in real terms by 1.12 divided by 1.06, minus one, which is 5.66% a year, a little under the 6% you get by simple subtraction. Over a long holding period that gap compounds too.

Enterprise value is unchanged, equity value rises by Rs 100 crore and EV/EBITDA stays the same. Equity value rises because Rs 100 crore of new shares now exist, and debt falls by the same Rs 100 crore. Enterprise value is equity plus net debt, so the two moves cancel. EBITDA sits above the interest line and is untouched, so the multiple cannot change either.

Why does enterprise value not move?

Think of a house worth Rs 1 crore with a Rs 40 lakh mortgage. If you pay down Rs 10 lakh of the loan with your savings, your stake rises to Rs 70 lakh and the loan falls to Rs 30 lakh, but the house is worth exactly what it was. Enterprise value measures the business, and swapping one kind of claim for another changes how that value is split, not how big it is. At first order, issuing shares to repay debt is exactly that swap.

Then what does change?

Everything below the interest line. Take EBITDA of Rs 100 crore, D&A of Rs 20 crore, Rs 400 crore of debt at 10% and a 25% tax rate, with 60 crore shares at Rs 10. Interest falls from Rs 40 crore to Rs 30 crore, so net income rises from Rs 30 crore to Rs 37.5 crore, and with equity now worth Rs 700 crore, P/E falls from 20.0x to 18.7x. Equity multiples such as P/E move with capital structure; enterprise multiples such as EV/EBITDA do not, which is why bankers compare companies on enterprise multiples.

EPS also rises, from Rs 0.50 to Rs 0.54. The reason is a comparison of two costs: the new shares were sold at an earnings yieldNet income divided by equity value, the inverse of the P/E. It is what each rupee of equity earns. of 5%, and the debt they retired cost 7.5% after tax. Replacing a 7.5% cost with a 5% cost lifts earnings per share.

What could make enterprise value move after all?

Three second-order effects, worth one sentence each. Fees on the share issue leak value out to advisers. Less debt means a smaller interest tax shield, which in Modigliani and Miller's analysis with taxes lowers value slightly. Against that, lower leverage reduces the risk and cost of financial distress. In an interview, give the first-order answer, unchanged, and then name the effects that could nudge it.

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.

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.

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.

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.

No, the price stays at Rs 100. Paying out Rs 10 forever at a 10% cost of equity is worth 10 over 10%, Rs 100. Retaining half and reinvesting at 10% gives growth of 50% times 10%, which is 5%, on a Rs 5 dividend: 5 over (10% minus 5%) is also Rs 100. Reinvesting at exactly the cost of capital swaps cash today for cash later at a fair rate, so value does not change.

Why does growth not add value here?

Imagine lending your bonus to a friend at exactly the rate your bank pays. You will have more money later, but you are no richer today, because the bank would have paid you the same. Retained earnings create value only if the company reinvests them at more than shareholders could earn elsewhere at the same risk, which is the cost of equity. At exactly 10% the company is just the bank: the growth is real, but it is paid for rupee for rupee by the dividend given up.

How do the two dividend streams compare?

The flat stream pays Rs 10 every year. The growing one starts at Rs 5 and rises 5% a year, so it overtakes Rs 10 only after about 14 years. Discounted at 10%, the early shortfall and the later surplus cancel exactly, and both streams are worth Rs 100. The tool is the Gordon growth modelA valuation of a stream that grows at a constant rate forever: the next payment divided by the discount rate minus the growth rate.: price equals next year's dividend over the cost of equity minus growth, and growth equals the share retained times the return on the money reinvested.

When would the decision change the price?

Change the return on the reinvested money. At 15%, growth is 7.5% and the price doubles to Rs 200; at 6%, growth is 3% and the price falls to about Rs 71. The same retention policy creates or destroys value depending only on whether the return beats 10%. That is the sentence the interviewer is waiting for: growth is not good in itself, profitable growth is.

Two weighings. Put three bars on each side and leave two off. If the scale balances, the light bar is one of the two left off, and weighing them against each other finds it. If one side is lighter, the light bar is among those three: weigh one against one, and if they balance it is the third. A weighing has three outcomes, so two weighings separate up to 3 x 3 = 9 bars.

Why is halving the wrong instinct?

Think of guessing a number with questions that can be answered higher, lower or spot on. Each answer splits the possibilities three ways, not two. A balance scale gives three outcomes, left lighter, right lighter or level, so each weighing should split the suspects into three groups, not two. Halving, four against four, uses only two of those outcomes and needs 3 weighings for eight bars.

Why can it not be done in one weighing?

One weighing has only three possible results, and any of the eight bars could be the light one. Three outcomes cannot point to eight different answers, so one weighing is never enough, and two weighings, with 3 x 3 = 9 outcome paths, are the fewest that can cover eight bars. That counting argument is the proof the interviewer wants: it shows two is a floor, not just a method that happened to work.

What does the puzzle teach beyond the scale?

The same counting sits behind any search. The fewest questions you need is set by how many answers each question can give, not by how clever the questions are. With nine bars, two weighings still suffice; with ten, you need a third. Saying where the method breaks shows the interviewer you understand why it works, which is the point of asking it.