Investment Banking puzzles, solved step by step

42. The gaps between terms are 4, 6, 8 and 10, rising by 2 each time, so the next gap is 12 and 30 + 12 = 42. The rule underneath is that the nth term is n x (n + 1): 1 x 2, 2 x 3, 3 x 4, 4 x 5, 5 x 6, and next 6 x 7, which is 42.

What should you do first with any number series?

Imagine a taxi meter that charges a little more for each extra kilometre than it did for the one before. The fares look irregular, but the jumps between them tell the story. Write the differences between terms underneath the series before guessing, and if they are not constant, take differences again. Here the first differences are 4, 6, 8 and 10, and the second differences are a steady 2, which tells you the rule is a quadratic: something times something.

How do you check the answer, not just find it?

A second route that lands on the same number is your proof. Each term splits into two neighbouring whole numbers: 2 = 1 x 2, 6 = 2 x 3, 12 = 3 x 4, 20 = 4 x 5, 30 = 5 x 6. Two methods agreeing, differences and a formula, turn a guess into an answer. The formula also lets you jump ahead: the 10th term is 10 x 11 = 110 without writing out the terms in between.

Why does a bank put this in an online test?

Numerical and logical screens are timed tightly and filter large numbers of applicants before anyone reads a CV. The skill being tested is spotting structure fast, the same skill you use when a line in a model grows by a changing amount each year. A revenue line whose yearly increase itself rises by a fixed amount has exactly this shape, and seeing it saves you from projecting the last increase forward as if it were fixed.

About 2.67%. A P/E of 15 means each Rs 100 of share price buys Rs 6.67 of yearly earnings, an earnings yield of 1 over 15. The company pays out 40% of that, Rs 2.67, as dividend. Dividend yield is dividend over price, Rs 2.67 over Rs 100. In one line: the payout ratio divided by the P/E, 0.40 over 15.

How do you turn a P/E into something you can multiply?

Flip it. A P/E of 15 says you pay 15 rupees for every rupee of yearly profit, so every rupee of price buys one fifteenth of a rupee of profit. Inverting the P/E gives the earnings yieldEarnings per share divided by the share price: the inverse of the P/E., 1 over 15 or about 6.67%, and every other per-share ratio then falls out by multiplication. Picking a round share price of Rs 100 makes it concrete: earnings per share are Rs 6.67.

Why is the dividend only part of the shareholder's return?

Think of a shopkeeper who takes home 40% of each year's profit and leaves 60% in the shop to buy more stock. The retained Rs 4.00 is not lost to shareholders; it is reinvested, and if it earns a decent return it should lift future earnings and dividends. That is why a low dividend yield on its own says little about whether a share is cheap. A company paying out everything would show a 6.67% yield and little room to grow.

Say the general rule after the number, because the follow-up usually changes one input. If the P/E doubles to 30 with the same payout, the yield halves to 1.33%; if the payout doubles to 80% at the same P/E, the yield doubles to 5.33%. Having the one-line formula ready means you answer those in a breath.

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.

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.

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.

2,496 and 9,991. Both pairs sit the same distance either side of a round number, so use the difference of squares: (a minus b)(a plus b) is a squared minus b squared. 48 x 52 is 50 squared less 2 squared, 2,500 minus 4, 2,496. 97 x 103 is 100 squared less 3 squared, 10,000 minus 9, 9,991. Say the middle number and the gap out loud, then subtract.

Why is the product always a little less than the middle number squared?

Take a square courtyard 50 paces on each side, 2,500 square paces. Cut a strip 2 paces wide off one side and lay it along the bottom. The courtyard is now 48 by 52, but the strip you moved was only 48 long, so a 2 by 2 corner is left bare. Two numbers either side of a round one multiply to the round number squared minus the gap squared, because the strip you move never quite fills the corner. 48 x 52 is 2,500 less 4, which is 2,496; the bare corner is the 4.

How do you spot when the trick applies?

Add the two numbers and halve: if the result is round, the trick is on. 48 plus 52 is 100, half is 50, gap 2. 97 plus 103 is 200, half is 100, gap 3. Any pair that straddles a round number is a difference of squares: 73 x 67 is 70 squared less 3 squared, 4,900 minus 9, 4,891. If the pair does not straddle a round number, fall back on splitting one factor: 48 x 53 is 48 x 52 plus 48, 2,496 plus 48, 2,544. The reflex to look for the structure first is the skill the question tests.

Why does a banking interview bother with this?

Because the same reflex speeds up every number you say in a meeting. A 4% discount on a Rs 1,040 crore valuation, or a share price of Rs 97 times 103 crore shares, is a difference of squares in disguise, and getting it in two seconds without a calculator buys you credibility. Say your working out loud as you go: middle, gap, square, subtract. The interviewer is listening for the method as much as the answer, and a candidate who mumbles the right number silently earns less than one who narrates the route.

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.

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.

Division C grew fastest, at 25%, but Division A contributed most, adding Rs 42 crore of the Rs 111 crore increase. Growth rates are 42/420 = 10%, 39/260 = 15% and 30/120 = 25%. Rupee increases are 42, 39 and 30. A's slower rate on a much bigger base adds more. Total revenue grew 13.9%, and A supplied 37.8% of that growth.

Why are these two different questions?

A child's pocket money rising from 100 to 150 is 50% growth; a parent's salary rising from 1 lakh to 1.1 lakh is 10%. The household budget still cares far more about the salary. A growth rate measures change relative to each part's own size; contribution measures change in rupees, which is what adds up to the total. Online tests ask both in one question because candidates who rush answer the same division twice.

How do you do it fast under a timer?

Compute the increases first, because they are subtractions: 42, 39, 30. Then compute rates only where needed, using easy fractions: 42 on 420 is a tenth, 30 on 120 is a quarter, 39 on 260 sits between them at 15%. Write the two added columns next to the table rather than holding the numbers in your head, because the next question in the set often reuses them.

That weighting is the third trap a test can set. The simple average of 10%, 15% and 25% is 16.7%, which is wrong because it gives the small division the same say as the large one. Total revenue went from 800 to 911, a rise of 13.9%. If an option shows 16.7%, it is there to catch exactly that shortcut.