Investment Banking puzzles, solved step by step

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.