Investment Banking puzzles, solved step by step

The price falls by about 3.5%. Modified duration says how many per cent the price moves for a one-percentage-point move in yield, in the opposite direction. A 50 basis point rise is half a point, so 7 x 0.5 = 3.5% down. On a Rs 100 crore holding that is a loss of about Rs 3.5 crore. The bend in the price curve makes the true fall slightly smaller.

Why do bond prices fall when yields rise?

A bond pays fixed cheques. If new bonds start paying more, nobody will pay the old price for the old, smaller cheques, so the old bond's price drops until its yield matches the market. Think of a flat let at a fixed rent when rents nearby jump: a buyer now pays less for it. Price and yield move in opposite directions, and duration measures how hard the price reacts. A longer bond reacts more, because more of its cash arrives far in the future, where a change in the discount rate bites hardest.

How accurate is the duration answer?

Duration draws a straight line, the tangent, along a price curve that actually bends. For small moves the line and the curve are almost the same, and for large moves the curve sits above the line, so the true loss is smaller than duration says. Take a zero-coupon bond yielding 6% with 7.42 years to run, which has a modified duration of exactly 7. A 50 basis point rise cuts its price by 3.43%, against 3.5% from the shortcut. A 300 basis point rise cuts it by 18.7%, against 21.0% from the line.

Add one sentence that marks you out. The bend is convexityHow much the price curve of a bond bends, which is the change in duration as yields move., and for an ordinary bond it works in the holder's favour both ways: the price falls less than duration predicts when yields rise and gains more than predicted when they fall. For a 50 basis point move the effect is a few hundredths of a per cent, so 3.5% is the answer to give.

A 25% premium. Each target share is exchanged for 0.5 acquirer shares worth Rs 200 each, so the offer is worth Rs 100 a share. The target trades at Rs 80 undisturbed, so holders get Rs 20 more, and 20 over 80 is 25%. The exchange ratio is not itself the premium: a straight swap at market prices would be 80 over 200, or 0.4.

How do you turn an exchange ratio into rupees?

Picture a swap where someone offers you half a gold coin for your silver one. To judge the offer you price both coins first. Multiply the exchange ratio by the acquirer's share price to get what each target share is being offered: 0.5 x Rs 200 = Rs 100. Then compare that with the undisturbed priceThe target share price before news or rumours of the deal moved it., Rs 80. The premium is the gap over the undisturbed price, Rs 20 on Rs 80, which is 25%.

How do you check it a second way?

Work out the ratio that would be a straight swap at today's prices: Rs 80 over Rs 200, or 0.4 acquirer shares per target share. An offer of 0.5 is 25% more shares than 0.4, the same premium seen through share counts instead of rupees. Two routes agreeing is what an interviewer wants to hear before you commit to a number.

What can happen to the premium before the deal closes?

In a fixed exchange ratio deal, the target holders' payout moves with the acquirer's share price. If the acquirer falls 10% to Rs 180, the offer is worth Rs 90 and the premium shrinks to 12.5%; if the acquirer rises, the premium grows. That is why target boards push for collars or cash, and why the premium quoted on announcement day is only a snapshot.

Take the 10% revenue rise: it adds Rs 0.4 crore of EBITDA. Revenue rises by Rs 1 crore, and with variable COGS at 60%, Rs 0.4 crore of it reaches EBITDA, a 20% jump. A 5% cut in COGS saves 5% of Rs 6 crore, Rs 0.3 crore. A 5% rise in EBITDA is 5% of the smallest number, Rs 0.1 crore. Translate every option into rupees of EBITDA before choosing.

Why can't you compare the three percentages directly?

Ten per cent off a car and ten per cent off a cup of coffee are not the same saving. Each percentage is taken on a different base, so convert all three into one unit, rupees of EBITDA, before you choose. Revenue is the biggest base at Rs 10 crore, COGS is Rs 6 crore and EBITDA is the smallest at Rs 2 crore. A small percentage of a big number can beat a big percentage of a small one, and here it does.

What should you ask before you answer?

Two readings change the numbers, and asking about them is part of the answer. First, is the revenue rise from volume or from price? Selling 10% more units drags 60% of the new revenue away as COGS, but a 10% price rise costs nothing extra, so all Rs 1 crore reaches EBITDA. Second, does a 5% fall in COGS mean 5% of COGS, or COGS falling from 60% to 55% of revenue? The first saves Rs 0.3 crore; the second saves Rs 0.5 crore and would beat a volume-led revenue rise.

Close with what a banker adds beyond the arithmetic. Operating leverageHow much profit swings for a given change in sales, because fixed costs do not move with volume. is why the revenue rise is so strong: with Rs 2 crore of fixed costs, a 10% rise in sales lifts EBITDA by 20%. The same leverage works in reverse, so a 10% fall in volume would cut EBITDA by 20% as well.

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 83.2%. Go through the back door. Each deal fails to close with probability 0.7, and because the deals are independent the chance that all five fail is 0.7 to the fifth power, about 16.8%. At least one closing is the only other possibility, so it is 1 minus 16.8%, which is 83.2%. Adding five lots of 30% to get 150% is the trap.

Why can you not add the five chances?

If five friends each have a 30% chance of turning up to dinner, you would not say there is a 150% chance someone comes. Adding counts the evenings when two or three friends arrive several times over. Probabilities of separate events add only when the events cannot both happen, and here two deals can close in the same quarter. Five times 30% is instead the expected number of deals that close, 1.5, which is a useful number but a different question.

Why is the complement the fast route?

At least one means one, or two, or three, or four, or five deals closing, and working each case out is slow. The opposite of at least one is none, and none is a single clean case: every deal fails. Independence lets you multiply the five 0.7s, and the number shrinks fast: 70%, 49%, 34.3%, 24.0%, then 16.8%. Subtract from 100% and you have the answer in one line. Whenever a question says at least one, reach for the complement before anything else.

What does the independence assumption hide?

The question tells you the deals are independent, and you should say that you are leaning on it. In a real quarter the deals share a market: a rate shock or a credit squeeze that stalls one is likely to stall the others, so the true chance of at least one closing is lower than 83.2%. If you want to show range, add that the chance of exactly one closing is 5 x 0.3 x 0.7 to the fourth, about 36.0%, so most of the 83.2% is a single close rather than a flood.

A always tells the truth and B always lies. Suppose A is a liar. Then A's statement is false, so neither of them is a liar, which means A tells the truth: a contradiction. So A is truthful, the statement is true, at least one of the two is a liar, and since A is not, B is. B never needed to speak.

Why does assuming A is a liar fall apart?

Picture someone who says, with a straight face, "I am lying right now." If that is a lie, they are telling the truth, and if it is the truth, they are lying; the sentence eats itself. A's statement is a milder cousin. A liar can never truthfully announce that there is a liar among us, because the announcement would then be true. So test the liar case first: if A lies, the statement is false, which means neither person is a liar, which means A is truthful. The assumption contradicts itself and is thrown out.

How does B get classified without saying a word?

Once A is truthful, A's sentence becomes a fact about the pair: at least one of them lies. A has been cleared, so the only person left who can be the liar is B, and B's silence changes nothing because the information came from A. The question is built this way to see whether you notice that evidence about a person can come from someone else. Say the two steps in order, A first and then B, and name the contradiction out loud rather than skipping to the answer.

Why do interviewers like this shape of puzzle?

It rewards the method you need for any logic test: assume a case, chase it to a contradiction, and move on. The same discipline is how you check a model: assume the circular reference resolves, trace it, and if it cannot, the assumption was wrong. Candidates who fail it usually try to argue from B's silence, which carries no information, instead of from A's sentence, which carries all of it.

The bet is exactly fair: its expected value is zero. A six comes up one time in six, so the win is worth 1/6 x Rs 10,000 = Rs 1,667. The other five faces each cost Rs 2,000, worth 5/6 x Rs 2,000 = Rs 1,667. They cancel. Whether to take it is then a question about risk: five rolls in six you hand over Rs 2,000, and the swing on one roll is about Rs 4,472 either way.

How do you weigh a big rare win against a small frequent loss?

Imagine playing the game six hundred times. You would expect a six about a hundred times, collecting Rs 10 lakh, and the other five hundred rolls would cost Rs 10 lakh. Expected value is each outcome multiplied by its probability, added up, and here the two sides come to Rs 1,667 each, so they cancel exactly. Say the arithmetic as fractions of 6 so the interviewer can follow: 10,000 over 6 against 5 x 2,000 over 6, and 10,000 equals 10,000.

If the maths is a tie, what decides it?

Risk appetite, and the question is really about whether you can say so. A fair bet with an uneven shape is not neutral to everyone: you lose on five rolls in six, and the spread of outcomes, about Rs 4,472 on a single roll, is large next to the stake. Someone for whom Rs 2,000 is lunch money might play for fun; someone for whom it is a week's groceries should not, because the most likely result of a single roll is a loss. Interviewers want the number, then a sentence that separates expected value from the experience of one roll.

How would you make the bet worth taking?

Move one number and watch the sign. Break-even on the win is 5 x Rs 2,000 = Rs 10,000, so any win above Rs 10,000 makes it positive, and at Rs 12,000 the expected value is Rs 333 a roll. Alternatively, ask to play many times: over 600 rolls the expected result is still zero, but the swing shrinks relative to the stake, which is why a casino is happy with a tiny edge and a huge number of hands. Offering that framing shows you understand why desks care about both edge and variance.

24 trailing zeros. Each zero at the end is a factor of 10, which is a 2 paired with a 5. Among the numbers 1 to 100 there are 97 factors of 2 but far fewer 5s, so the 5s decide it. The 20 multiples of 5 contribute one 5 each, and the four multiples of 25 contribute a second, giving 20 + 4 = 24. There is no multiple of 125 below 100, so there is no third layer.

Why is the answer about fives and not about tens?

Think of making pairs of shoes from a pile of lefts and rights. If you have 97 left shoes and 24 right shoes, you can make 24 pairs; the extra lefts do not help. A trailing zero is a pair of one 2 and one 5 multiplied together, and in 1 to 100 the 2s vastly outnumber the 5s, so the number of 5s sets the number of zeros. Counting multiples of 10 misses the zeros built from a 5 in one number and a 2 in another, such as 5 x 2 = 10 or 15 x 4 = 60.

Where do the extra four come from?

A number like 50 is 2 x 5 x 5: it carries two fives, not one. Every multiple of 25 hides a second five, and 25, 50, 75 and 100 are the four of them below 100, so the count is 20 from the multiples of 5 plus 4 from the multiples of 25. The next layer would be multiples of 125, which carry a third five, but 125 is beyond 100, so the count stops at 24. The pattern continues for larger factorials: divide by 5, then 25, then 125, and add the whole-number parts.

How do you check it without writing out 100 factorial?

Test the method on a factorial you can write. 10 factorial is 3,628,800, with two trailing zeros, and the rule gives 10 over 5 = 2, plus 10 over 25 = 0, which matches. Showing the rule on a small case you can verify is the fastest way to earn trust for the big case you cannot. Then say the general version: the number of trailing zeros of n factorial is the whole-number sum of n over 5, n over 25, n over 125 and so on, and for 1,000 factorial that is 200 + 40 + 8 + 1 = 249.

The company spending 20% is cheaper, and EV over EBITDA less capex shows it. Give both Rs 100 crore of EBITDA, so each is worth Rs 800 crore. The first keeps Rs 80 crore after capex and trades at 10x that cash figure; the second keeps Rs 40 crore and trades at 20x. EV/EBITDA cannot see capital intensity, so when capex differs this much, compare value against cash earnings instead.

Why does the same 8x mean different things?

Two taxi drivers each earn Rs 1 lakh a month before costs. One drives a new car that needs a service a year; the other drives an old one that eats Rs 60,000 a month in repairs. You would not pay the same for both businesses. EBITDA is earnings before the spending needed to keep the assets working, so two companies with equal EBITDA can hand their owners very different amounts of cash. At 8x, both are valued at Rs 800 crore on Rs 100 crore of EBITDA. The first spends Rs 20 crore of that on capex, the second Rs 60 crore, and that difference never appears in the headline multiple.

Which multiple fixes it?

Take the capex off before dividing. EV over EBITDA less capex puts the two companies on cash they can actually keep, and the gap opens up to 10x against 20x. EV over EBIT does a rougher version of the same job, because depreciation is yesterday's capex spread over time, but it lags when a company is spending more than it depreciates. The cleanest version is EV over unlevered free cash flow, which also takes tax and working capital off. Name the limitation too: capex can be lumpy, so use a normal year or an average, and growth capex that adds capacity deserves a different reading from maintenance capex that merely keeps the lights on.

When would the market be right to pay 8x for both?

Say the case against your own answer. If company B's Rs 60 crore of capex is building capacity that will lift EBITDA sharply, the market may be paying for growth that company A will never deliver. The question has no growth figures, so the honest answer is that on today's cash the 20% company is cheaper, and that the interviewer would need to tell you about growth and the split between maintenance and growth capex before you could defend an equal multiple.

About 42% of the gross ending wealth. Rs 1 compounding at 10% for 30 years becomes Rs 17.45. With a 2% fee the money compounds at 8% and becomes Rs 10.06. The gap is Rs 7.39, and 7.39 over 17.45 is 42%. A fee that sounds like a fifth of the return takes more than two fifths of the final pot, because every rupee of fee also forfeits the growth it would have earned.

Why does a 2% fee take far more than 2%?

Imagine a farmer who hands over two in every hundred sacks of grain each harvest. The loss is not two sacks; it is two sacks plus everything those sacks would have grown into had they been sown. A fee is taken from a growing pot, and each slice removed stops compounding, so the cost builds year on year. After one year the fee has cost about 2% of the pot. After 10 years it has cost 17%, after 20 years 31%, and after 30 years 42%. The number is still climbing at year 30; it only stops when the money is withdrawn.

How do you get the number in your head?

Use the rule of 72 for the two growth rates. At 10% money doubles roughly every 7.2 years, so in 30 years it doubles a little over four times, about 17x. At 8% it doubles every 9 years, so just over three times, about 10x. The ratio of the two end points, 10 over 17, is roughly 0.58, so the fee has taken about 42% of the gross pot. The exact figures are 17.45x and 10.06x, and the exact share is 42.3%, but the rule of 72 gets you within a point in ten seconds.

What does an interviewer want after the number?

The link to the desk. Carried interest, management fees and transaction costs all work the same way: a small annual drag compounds into a large share of the terminal value, which is why net-of-fee returns, not gross, are what an investor should compare. Say the limitation too. The 10% gross return is an illustration, not a forecast, and a fee is worth paying if the manager's gross return beats the alternative by more than the fee. The question is about arithmetic, not about whether any fund is worth its fee.