Investment Banking puzzles, solved step by step
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041Your pay is 60% fixed salary and 40% bonus, and the bonus moves twice as much as the market. What is your personal beta, and what does it say about the market risk you carry before you invest a rupee of savings?Rothschild & CoNew York · 2026
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What is your personal beta?
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Your personal beta is 0.8. Salary does not move with the market, a beta of 0; the bonus moves twice as much, a beta of 2. Weighted by share of pay, 0.6 x 0 plus 0.4 x 2 gives 0.8. If the market falls 20%, the bonus falls 40% and total pay falls 16%. Your earnings already carry most of the market risk of an index fund before you invest a rupee of savings.
How do you get a beta for a person?
Think of a household with two earners, a schoolteacher on a fixed salary and a salesperson paid on commission. In a downturn the teacher's pay holds while the commission falls sharply, and the household's income falls by a blend of the two. The beta of a mix is the weighted average of the betas of its parts, so a pay packet's beta is each component's beta weighted by its share of pay. Salary has a beta of 0 and the bonus a beta of 2, so the blend is 0.8.
Salary is 60% of pay with a beta of 0 and the bonus 40% with a beta of 2, which blends to 0.8; on pay of Rs 25 lakh, a 20% market fall cuts the bonus to Rs 6 lakh and total pay to Rs 21 lakh, a 16% fall. What does 0.8 look like in rupees?
Take pay of Rs 25 lakh: Rs 15 lakh of salary and Rs 10 lakh of bonus. If the market falls 20%, the bonus falls 40% to Rs 6 lakh, and total pay falls from Rs 25 lakh to Rs 21 lakh, a 16% drop, which is 0.8 times the market's fall. The salary cushions the swing, but the bonus is exactly the part that shrinks in the years the market is weakest.
The relationship0.6, 0.4 each component's share of total pay 0, 2 how much each component moves when the market moves What it says in wordsWeight each part of pay by its share and add up how much each moves with the market.Why does it matter for your savings?
Your future earnings are an asset, often the largest one you own early in a career, and finance calls it human capitalThe value today of the earnings a person expects to make over the rest of a working life.. If that asset already has a beta of 0.8, your total wealth carries market risk before your savings add any, and the worst years for markets are the years your bonus, and perhaps your job, are most at risk. This is a way of counting risk, not a recommendation for anyone's savings: the point the interviewer wants is that risk is measured across everything you own, including your pay.
Where candidates lose it
Candidates answer 2, the bonus beta, or 0.4, the bonus weight, because they reach for one number instead of a weighted average. The beta of a mix always blends by weight.
The second loss is stopping at the number. The question asks what it says, and the answer is that a banker's pay is already a leveraged bet on markets, which is why the follow-up is about savings.
What the interviewer asks next
- What is the personal beta of a civil servant on a fixed salary, and what does it suggest?
- Your bonus rises to 60% of pay. What is your personal beta now?
- How would you put a value on your future earnings as an asset?
Asked at Rothschild & Co, Generalist, New York, 2026 (Wall Street Oasis):
How would you value your favorite animal? What is your personal beta?
042A company has revenue of Rs 730 crore and collects from customers in 45 days. Next year revenue grows 20% and collection slows to 60 days. How much cash does the receivables line absorb?Bulge bracket IBMiddle market IB
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How much cash does the receivables line absorb next year?
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About Rs 54 crore. Receivables today are Rs 730 crore times 45 over 365, which is Rs 90 crore. Next year revenue is Rs 876 crore and collection takes 60 days, so receivables are Rs 144 crore. The increase of Rs 54 crore is cash the business has earned but not collected. Rs 18 crore of it comes from growth and Rs 36 crore from slower collection, twice as much.
Why does growth eat cash at all?
Think of a tailor who lets regular customers pay a month and a half after collecting their clothes. If the business grows, more money is out with customers at any moment, even though nobody is paying late. Receivables are revenue earned but not yet collected, so when revenue grows at the same collection period the balance grows with it, and the increase is cash the business has to fund. Here 20% more revenue at 45 days lifts receivables from Rs 90 crore to Rs 108 crore.
Receivables rise from Rs 90 crore to Rs 144 crore: Rs 18 crore of the increase comes from 20% growth at the old 45 days and Rs 36 crore from collecting in 60 days instead, so slower collection absorbs twice as much cash as growth. How do you split the Rs 54 crore?
Take the two changes one at a time. Growth first: Rs 876 crore at the old 45 days is Rs 108 crore, so growth absorbs Rs 18 crore. Then slower collection: going from 45 to 60 days on Rs 876 crore adds 15 days of sales, Rs 36 crore. Fifteen extra days of collection on the larger business costs twice as much cash as 20% growth. The line to watch is days sales outstandingReceivables divided by revenue, times 365: the average number of days customers take to pay..
The relationship876 x 45/365 next year's receivables if collection had not slowed 90 today's receivables, 730 x 45/365 876 x 15/365 the extra 15 days of sales tied up by slower collection What it says in wordsThe change in receivables is the growth effect at the old collection period plus the slower collection on the new revenue.Does the order of the split matter?
A little. Slower collection first, on today's revenue, costs Rs 30 crore, and growth then adds Rs 24 crore; the Rs 6 crore difference is growth and slower collection compounding each other. The total is Rs 54 crore whichever way you split it, and slower collection is the bigger cause either way. Say which order you used. Then say what it means: a business growing fast while collecting slowly can report rising profit and falling cash at the same time, which is the pattern a lender looks for first.
Where candidates lose it
Candidates answer Rs 144 crore, the new balance, rather than Rs 54 crore, the change. Cash is absorbed by the increase in receivables, not by their level.
The second loss is blaming it all on growth. Splitting the change shows that slower collection costs twice as much, and that is the insight the question is built to draw out.
What the interviewer asks next
- Collection improves to 30 days instead. How much cash does the receivables line release?
- Payables also stretch from 30 to 45 days of costs. How does that change the answer?
- Why might a company's collection period lengthen as it grows?
043Comparable companies have an unlevered beta of 0.8. Your target will run at a debt-to-equity ratio of 50% with a 25% tax rate. The risk-free rate is 7% and the equity risk premium 6%. What is the target's cost of equity?Moelis & CompanyLos Angeles · 2026
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What cost of equity do you get?
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About 13.6%. The peers' 0.8 is an unlevered beta, the risk of the business with no debt. The target carries debt equal to 50% of its equity, so relever it: 0.8 times (1 plus 75% times 0.5) is 1.1. CAPM then gives 7% plus 1.1 times 6%, which is 13.6%. Using the unlevered 0.8 directly would understate the cost at 11.8%.
Why can you not use the peer beta as it is?
Think of two people with the same job and salary, one with no loan and one with a large home loan. A pay cut hurts the borrower more, because the loan payment does not shrink. Debt magnifies the swings that reach shareholders, so the same business carries a higher equity beta the more it borrows. Peers' betas are first stripped of their own debt to give an unlevered betaThe beta a business would have with no debt: the risk of its operations alone., then relevered at the target's own mix of debt and equity.
The peers' unlevered beta of 0.8 relevers to 1.1 at a 50% debt-to-equity ratio and a 25% tax rate, giving a cost of equity of 13.6%, while the three common slips give 11.8%, 13.0% and 14.2%. How does the relevering work?
Use the standard formula, which assumes the debt itself carries no market risk. Levered beta equals unlevered beta times one plus (one minus the tax rate) times debt over equity: 0.8 times (1 + 0.75 x 0.5), which is 0.8 times 1.375, or 1.1. The tax term is there because interest is tax deductible, so the tax saving softens a quarter of the debt's burden on shareholders. Then CAPM: 7% plus 1.1 times 6% is 13.6%.
The relationshipbeta_U the peers' unlevered beta, 0.8 t the tax rate, 25% D/E the target's debt divided by its equity, 50% What it says in wordsAdd back the target's own debt to the business risk, then price that risk with CAPM.Which slips change the answer?
Three, each worth naming. Skipping the relevering gives 11.8%, using debt over total capital instead of debt over equity gives 13.0%, and ignoring tax gives 14.2%. A spread of 2.4 points in the cost of equity moves a DCF a long way, so the interviewer wants to hear which ratio you are plugging in. State the limit too: the rates here are illustrations, and a live model takes the current government bond yield and the premium the bank uses.
Where candidates lose it
The fast loss is plugging 0.8 straight into CAPM. The peers' beta has been cleaned of debt, so using it unadjusted values the target as if it had none, and the cost of equity comes out too low.
The quieter loss is the ratio. The formula takes debt over equity, 50%, not debt over total capital, one third. Say the ratio out loud before you multiply.
What the interviewer asks next
- The target's debt rises to 100% of equity. What is the cost of equity now?
- What is the WACC if the target's debt costs 10% before tax?
- Why do you unlever each peer before averaging, rather than averaging the levered betas?
Asked at Moelis & Company, Investment Banking, Los Angeles, 2026 (Wall Street Oasis):
high-level technical questions aimed at accounting (NWC) and valuation topics (EV, DCF, UFCF, Cost of Equity)
044A company finds Rs 10 crore of cash lying on the street. What happens to its enterprise value and its equity value?UBSAnonymous interview candidate in · 2024
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What happens to enterprise value?
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Enterprise value stays the same and equity value rises by Rs 10 crore. Shareholders own the extra cash, so their equity is worth Rs 10 crore more. Enterprise value is equity plus debt minus cash: equity is up 10 and cash is up 10, so EV does not move. EV measures the operating business, and picking money up off the street does not change what that business earns. Ignoring tax on the windfall keeps the numbers clean.
What is enterprise value actually measuring?
Think of buying a shop whose till holds Rs 10,000. You would pay for the business plus the cash in the till, and the cash is worth exactly its face value to you, no more and no less. Enterprise value prices the operating business on its own, so cash, which is worth its face value to whoever holds it, is taken out of the bridge. Equity value is what shareholders own, and they own both the business and the cash.
Finding Rs 10 crore lifts equity value from Rs 500 crore to Rs 510 crore and cash from Rs 50 crore to Rs 60 crore, and because cash is subtracted in the bridge, enterprise value stays at Rs 750 crore. How does the bridge move?
Take a company with equity worth Rs 500 crore, debt of Rs 300 crore and cash of Rs 50 crore, so enterprise value is Rs 750 crore. Find Rs 10 crore: cash becomes Rs 60 crore and equity Rs 510 crore, and EV is 510 plus 300 minus 60, still Rs 750 crore. Nothing in the operating business changed, so multiples of EBITDAEarnings before interest, tax, depreciation and amortisation: a rough measure of the cash profit the operating business produces. or revenue do not change either.
The relationshipE equity value, what the shareholders own D debt C cash, subtracted because it is not part of the operating business What it says in wordsEquity and cash rise by the same amount, so their effects on enterprise value cancel.What assumptions should you say out loud?
Two. First, tax: the windfall is probably taxable income, so at a 25% rate equity and cash each rise by Rs 7.5 crore rather than Rs 10 crore. Whatever amount sticks, cash and equity move together and enterprise value stays put. Second, the cash sits idle. If the company used it to repay debt, EV would still not change, but the split between lenders and shareholders would. Naming both shows the interviewer you know which line each event touches.
Where candidates lose it
The common wrong answer is that enterprise value rises by Rs 10 crore because the company is worth more. The owners are richer, but the operating business is not, and EV only measures the business.
The opposite slip is saying EV falls because cash is subtracted. That forgets equity rises by the same amount. Walk the bridge line by line and the two moves cancel.
What the interviewer asks next
- The company uses the Rs 10 crore to repay debt. What happens to EV and equity value?
- The company issues Rs 100 crore of new shares for cash. What happens to EV?
- Why might a buyer pay less than face value for cash trapped in a foreign subsidiary?
Asked at UBS, Investment Banking, Anonymous interview candidate in, 2024 (Wall Street Oasis):
explain the assumptions behind it - if pick up 10 bucks, what happens to EV
045Twelve bankers meet at an offsite and every pair shakes hands exactly once. How many handshakes are there?Bulge bracket IBMiddle market IB
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Answer fast.
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66 handshakes. Each of the 12 bankers shakes hands with the other 11, which gives 12 x 11 = 132, but every handshake has two people in it and so has been counted twice. Half of 132 is 66. The same number falls out of adding 11 + 10 + 9 and so on down to 1, or from 12 choose 2.
Why do you halve the count?
Think of counting the phone calls in a group where everyone rang everyone else once. If each person reports the calls they took part in, every call shows up in two reports. Each handshake involves two people, so counting from each person's side counts every handshake exactly twice, and the true number is half. Twelve people times eleven others is 132 ends of handshakes, which is 66 handshakes.
Twelve people on a circle with one line for every pair make 66 lines: each person sits at the end of 11 lines, giving 132 line ends, and every line has two ends, so the count halves to 66. How do you check 66 another way?
Let people arrive one at a time. The second person shakes 1 hand, the third 2, and so on to the twelfth, who shakes 11, and adding 1 through 11 gives 66. Pair the terms to add them quickly: 1 and 11, 2 and 10, and so on make five pairs of 12 plus a 6 in the middle, 66. Two routes to the same number is the check interviewers like to hear.
The relationship12 x 11 each person times the others they meet, counting every handshake from both sides 2 the two people in every handshake What it says in wordsPairs from a group of n are n times (n minus 1), halved.Where does this count show up on a desk?
Anywhere pairs matter. The number of pairs grows roughly with the square of the group: double the group to 24 and the handshakes rise to 276, more than four times as many. That is why a 12-stock portfolio has 66 pairwise correlationsMeasures of how closely two things move together, from minus 1 to plus 1. to estimate, and why a deal with many parties needs far more conversations than its headcount suggests.
Where candidates lose it
The fast wrong answer is 132, from 12 times 11 without halving. It treats one handshake between two people as two separate events.
The other slip is 144 or 78, from letting people shake their own hand. Say each person shakes hands with the other 11, and both errors disappear.
What the interviewer asks next
- How many people must be in the room for there to be 105 handshakes?
- At a dinner of six couples, everyone clinks glasses with everyone except their own partner. How many clinks?
- How many pairwise correlations does a 30-stock portfolio have?
046Fund A returns +30% and then -10%. Fund B returns +10% and then +10%. Starting with Rs 100 in each, which ends higher?Middle market IBPrivate equity
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Which ends higher?
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Fund B ends higher, at Rs 121 against Rs 117. Fund A goes to 130 and then loses 10% of 130, ending at 117. Fund B compounds 10% twice to 121. Both average 10% a year, but A's compound growth rate is only about 8.2% because its returns swing. Volatility drags on compound returns: roughly half the variance comes off the average each year.
Why does the same average give different endings?
Think of a salary that rises 30% one year and is cut 10% the next, against one that rises 10% twice. The cut lands on the higher salary, so it takes away more rupees than the same percentage would have earlier. Returns multiply rather than add, so a loss after a gain is taken from a bigger base, and an uneven path ends below a smooth one with the same average. Rs 100 grows to 130 and then gives back Rs 13; the steady fund never gives anything back.
Fund A rises to 130 and falls to 117 while Fund B climbs to 110 and then 121; both average 10% a year, but A compounds at only 8.2% because of its swing, so B ends Rs 4 higher. How big is the drag, and can you estimate it in your head?
The growth rate that matters is the geometric averageThe constant yearly return that would turn the starting amount into the ending amount over the same period.. Fund A's is the square root of 1.17 minus 1, about 8.2%, against B's 10%, even though both arithmetic averages are 10%. A quick estimate: subtract half the variance. A's returns sit 20 points either side of 10%, so half of 0.2 squared is 2 points, and 10% minus 2% is about 8%, close to the exact 8.2%.
The relationshipr-bar the arithmetic average return, 10% sigma how far returns swing around the average, 20 points for Fund A What it says in wordsCompound growth is roughly the average return minus half the variance.Why would an interviewer care?
Because fund reports often quote average returns, and investors live on compound ones. Two funds with the same average return can leave an investor with very different money, and the more volatile one leaves less. The same arithmetic explains why a fund that rises 50% and then falls 50% is down 25%, and why leverage that doubles volatility can lower long-run growth even while it raises the average.
Where candidates lose it
The trap is answering that they end level, because both funds average 10% a year. Averaging percentages assumes they add, and returns multiply.
The second loss is getting 117 and 121 without saying why. Name volatility drag and give the half-the-variance estimate; that turns a calculation into an insight.
What the interviewer asks next
- A fund rises 50% and then falls 50%. Where does it end?
- What steady yearly return matches Fund A over the two years?
- Fund C returns +40% and then -20%. How does it compare with A and B?
047A deal is agreed at a fixed exchange ratio of 0.5 acquirer shares for each target share, when the acquirer trades at Rs 200. Before closing, the acquirer's shares fall 10%. What do target holders now receive, and how would a fixed-price structure have differed?Elite boutique IBPrivate equity
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Under the fixed ratio, what is a target share now worth?
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Under the fixed ratio, target holders receive Rs 90 of value per share instead of Rs 100. They still get 0.5 acquirer shares, but each is now worth Rs 180. Under a fixed price they would still receive Rs 100, and the acquirer would issue 100 over 180, about 0.556 shares per target share, 11.1% more than planned. A fixed ratio shares the price risk with the target; a fixed price keeps it with the acquirer and its shareholders.
What does each structure actually fix?
Think of agreeing to swap your car for 50 grams of gold, rather than for Rs 5 lakh paid in gold. If the gold price falls before the swap, the first deal still hands you 50 grams, now worth less; the second hands you more grams so that you still get Rs 5 lakh. A fixed exchange ratioThe number of acquirer shares paid for each target share in a stock deal. fixes the number of shares the target receives; a fixed price fixes their rupee value, and the share count moves instead. Whichever number floats carries the risk of the acquirer's share price between signing and closing.
After the acquirer falls from Rs 200 to Rs 180, a fixed ratio still pays 0.5 shares, now worth Rs 90 per target share, while a fixed price still pays Rs 100 by issuing 0.556 shares per target share, 11.1% more than planned. How do the two structures play out across the whole deal?
Assume the target has 10 crore shares and the acquirer 50 crore. Fixed ratio: target holders receive Rs 900 crore of value instead of Rs 1,000 crore, and own 5 crore of 55 crore shares, 9.1% of the combined company. Fixed price: they still receive Rs 1,000 crore, so the acquirer issues 5.56 crore shares, 11.1% more than planned, and the target's holders own 10.0%. The acquirer's existing shareholders give up more of the company to pay the same rupees.
Per target share unless stated Fixed ratio Fixed price Acquirer shares received 0.500 0.556 Value at Rs 180 Rs 90 Rs 100 New acquirer shares, crore 5.00 5.56 Target holders' share of the combined company 9.1% 10.0% With 10 crore target shares and 50 crore acquirer shares, the fixed ratio leaves target holders with Rs 90 a share and 9.1% of the combined company, while the fixed price gives them Rs 100 a share and 10.0%. Which side wants which, and how do bankers split the difference?
A target worried that the acquirer's shares will fall wants a fixed price; an acquirer worried about issuing too many shares wants a fixed ratio. A collar is the usual compromise: in one common form the ratio is fixed while the acquirer's price stays inside a band and the value is fixed outside it, so each side carries the risk only up to a point. Interviewers ask this to see whether you think about who carries risk between signing and closing, not just the headline value.
Where candidates lose it
Candidates say target holders still get Rs 100, because that was the headline price. A fixed ratio never promised rupees, only shares, and the shares are now worth less.
The second miss comes under the fixed price: saying nothing changes. The acquirer must issue 11.1% more shares, so its own shareholders absorb the fall through extra dilution.
What the interviewer asks next
- The acquirer's shares rise 10% instead. Which structure would the target have preferred?
- Design a collar that protects the target against falls of more than 10%.
- Why might an acquirer with a volatile share price prefer to pay cash?
048You have 1,000 bottles of wine and exactly one is poisoned. A single sip makes a taster ill after exactly 24 hours. With 10 testers and one day, how do you find the poisoned bottle?Consulting style brainteasersSales and trading
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How can 10 testers cover 1,000 bottles in one round?
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Number the bottles 1 to 1,000, write each number in 10-digit binary, and have tester k sip from every bottle whose kth digit is a 1. After 24 hours, write a 1 for each tester who is ill and a 0 for each who is not. That 10-digit binary number is the poisoned bottle. It works because 10 yes-or-no results can tell apart 2 to the 10th, 1,024 cases, which is more than 1,000.
Why does giving each tester a separate batch fail?
Picture 10 friends each tasting their own 100 bottles. One friend falls ill and you know the poison is among their 100, but not which one, and the day is over. When testers taste separate batches, each result answers one question about one batch, so 10 testers can single out at most 10 groups, not 1,000 bottles. The fix is to let testers overlap, so that every bottle is sipped by its own unique combination of testers.
Writing each bottle number in 10-digit binary assigns one digit to each tester, so bottle 357, which is 0101100101, is sipped by testers 2, 4, 5, 8 and 10, and if exactly those five fall ill their place values add back to 357. How does the binary code name the bottle?
Give each tester a place value: tester 1 is worth 512, tester 2 is worth 256, and so on down to tester 10, worth 1. Bottle 357 equals 256 + 64 + 32 + 4 + 1, so testers 2, 4, 5, 8 and 10 sip from it. If exactly those five fall ill, adding their place values gives back 357, and no other bottle has the same set of tasters. Every bottle number is a different set of testers, so the pattern of illness points to one bottle only.
The relationship2^10 the number of different ill-or-well patterns 10 testers can show 0101100101 bottle 357 in binary, one digit per tester What it says in wordsTen yes-or-no results give enough patterns to label every bottle uniquely.What is the general lesson?
Each tester is one yes-or-no question asked of every bottle at once. Ten yes-or-no answers can tell apart 1,024 possibilities, so the right design makes each answer carry one binary digit of the bottle's number. The same idea finds one faulty item among many with very few tests, and it is why 2,000 bottles need only one more tester. With two days, each tester has three outcomes, ill on day one, ill on day two, or never, and 7 testers would be enough.
Where candidates lose it
Most candidates split the bottles into 10 batches of 100, find the right batch, and run out of time. Separate batches waste the testers, because each one answers only a single question about a single batch.
The second loss is saying binary without showing the decoding. Work one bottle through, such as 357, so the interviewer sees the pattern of illness turn back into a number.
What the interviewer asks next
- How many testers would you need for 2,000 bottles?
- With two poisoned bottles, why does this scheme break down?
- You have two days instead of one. How few testers can you manage with?
049A bank trades at 2.0x book value, earns a 16% return on equity and has a 12% cost of equity. Using a dividend discount model, what long-run growth rate is the market pricing in?Rothschild & CoRiyadh · 2026
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What growth is priced in?
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About 8% a year, forever. In a dividend discount model, a bank that earns its ROE on book and grows at g must retain g over ROE of its earnings and pays out the rest, which makes its fair price to book (ROE minus g) over (cost of equity minus g). Setting (16% minus g) over (12% minus g) equal to 2.0 gives g of 8%, with 50% of earnings paid out. Whether 8% forever is believable is the real question.
Where does the price-to-book formula come from?
Think of a fruit tree. The fruit you eat is the dividend; the fruit you replant grows the orchard. A bank earning its ROE on book value can grow only by retaining earnings, so growth equals ROE times the share retained, and the rest is paid out as dividends. Put that dividend into the dividend discount modelA valuation that prices a share as the present value of its future dividends; with constant growth it becomes the next dividend divided by the cost of equity minus growth. and divide by book value, and price to book becomes (ROE minus g) over (r minus g).
With a 16% return on equity and a 12% cost of equity, the justified price to book rises from 1.33x at zero growth to 2.0x at 8% growth and 3.0x at 10%, so a 2.0x price implies the market expects 8% growth forever. How do you solve for the growth?
Set the formula equal to the price. (16% minus g) over (12% minus g) equals 2.0, so 16 minus g equals 24 minus 2g, and g is 8%. Check it with rupees: on book of Rs 100 the bank earns Rs 16, keeps Rs 8 to grow book by 8%, and pays Rs 8. Rs 8 over (12% minus 8%) is Rs 200, twice book.
The relationshipROE return on equity, 16% r cost of equity, 12% g the growth rate the price implies What it says in wordsThe justified price to book depends on how far ROE and the cost of equity each sit above growth.Is 8% growth forever reasonable?
That is the judgement the interviewer is waiting for. Look at the curve: near 8% the price is very sensitive to growth, so a small shortfall, 7% instead of 8%, would justify only 1.8x book. Compare 8% with the nominal growth the economy could sustain over decades, a figure to look up rather than assume. And note what the formula says about returns: if the bank earned only its 12% cost of equity, it would be worth exactly book value, whatever its growth.
Where candidates lose it
Candidates reach for P/B equals ROE over cost of equity, get 1.33x, and cannot reconcile it with 2.0x. That shortcut is the no-growth case; the gap between 1.33x and 2.0x is exactly the growth the market is paying for.
The second loss is solving for 8% and stopping. The point of an implied number is to judge it: say whether 8% a year forever is plausible for a bank, and what happens to the price if it is not.
What the interviewer asks next
- What payout ratio is consistent with this valuation?
- If the market cut the bank to 1.5x book, what growth would that imply?
- Why is a bank earning exactly its cost of equity worth book value whatever its growth?
Asked at Rothschild & Co, Generalist, Riyadh, 2026 (Wall Street Oasis):
DDM model mechanics and A/P & A/R increase
050A screening test flags accounting fraud with 95% accuracy both ways: it flags 95% of companies that commit fraud and clears 95% of companies that do not. In the sample, 2% of companies commit fraud. A company is flagged. What is the probability it is actually committing fraud?Bulge bracket IBSales and trading
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Instinct first: how likely is a flagged company to be a fraud?
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About 28%. Take 1,000 companies. Twenty commit fraud and the test flags 19 of them. The other 980 are clean, and the test wrongly flags 5% of them, which is 49. So 68 companies are flagged and only 19 are frauds: 19 over 68 is 27.9%. The test is accurate, but fraud is rare, so a small error rate on the large clean group produces more false flags than true ones.
Why is the answer not 95%?
Think of a smoke alarm that sounds for 95% of real fires and stays quiet 95% of the time there is no fire. In a building where fires are rare, most alarms you hear are burnt toast, because the quiet days vastly outnumber the fires and 5% of them still ring. A test's accuracy tells you how it behaves when the truth is known; the question asks the reverse, how likely the truth is given the test, and the base rateHow common the thing is before any test is run. Here, 2% of companies commit fraud. sits between the two. With 2% frauds, a 5% error on the 98% of clean companies creates 49 false flags for every 19 true ones.
Of 1,000 companies, the test flags 19 of the 20 frauds and 49 of the 980 clean companies, so 68 are flagged and only 19 of them are frauds, which makes a flag 28% likely to be right rather than 95%. How do you work it with whole numbers?
Start with 1,000 companies and walk them down the tree. Frauds: 2% of 1,000 is 20, and 95% of those, 19, are flagged. Clean: 980, and 5% of those, 49, are wrongly flagged. 19 true flags and 49 false flags make 68 flagged companies, so the chance that a flagged company is a fraud is 19 over 68, about 28%. Working in counts rather than percentages is faster in the room and harder to get wrong, and it shows the interviewer where every number came from.
The relationship0.02 x 0.95 the share of all companies that are frauds and get flagged 0.98 x 0.05 the share that are clean and get flagged anyway What it says in wordsDivide the true flags by all the flags, true and false together.What moves the number, and what does that mean for a screen?
Two things. The base rate: if 10% of the sample committed fraud, the same test would make a flag 68% reliable. The false-flag rate: cutting it from 5% to 1% lifts the answer to 66% even at a 2% base rate. A screen for a rare event is a filter that earns a closer look, not a verdict, and the second look does the real work: run an independent test of the same accuracy on the 68 flagged companies and a double flag is 88% likely to be a fraud. That is how diligence red flags, transaction monitoring and credit early-warning lists are meant to be read. The working assumes the two tests err independently; if they look at the same ratios, the second adds less than the arithmetic suggests.
Where candidates lose it
Candidates answer 95%, reading the test's accuracy as the answer. The question asks the reverse conditional, and when the event is rare the two numbers can sit far apart.
The second loss is reaching for the formula in symbols and getting lost in it. Say 1,000 companies, 20 frauds, 19 caught, 49 false flags, 19 over 68; the counts carry you through and the interviewer can follow every step.
What the interviewer asks next
- The company is also flagged by a second, independent test with the same accuracy. Now what is the probability?
- How rare does fraud have to be before a flag is more likely wrong than right?
- The test clears a company instead. What is the chance it is actually clean?
