Hedge Funds puzzles, solved step by step

Puzzles: 100
Traced to a firm: 38
Topics: 14
Hard: 30

Showing 81–90 of 100

081Which is more likely: at least one six in four rolls of a single die, or at least one double six in twenty-four rolls of a pair of dice?Expected value and dice gamesCoreQuant and systematic fundsProp and quant trading firms
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Before any arithmetic, which do you back?
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The single six in four rolls: 51.8% against 49.1%. Use the complement for each. Four rolls with no six happen (5/6)^4 = 48.2% of the time, so at least one six is 51.8%. Twenty-four rolls of two dice with no double six happen (35/36)^24 = 50.9% of the time, so at least one is 49.1%. Only the first is better than even money.
Why does the proportional argument give the same answer for both?
The old gamblers' rule, in the problem usually credited to the Chevalier de Méré, went like this: a six comes up one time in six and you get four tries, so 4/6; a double six comes up one time in thirty-six and you get twenty-four tries, so 24/36, also 4/6. Think of phoning a friend four times: four tries do not give four times the chance of getting through, because once they pick up, the later calls add nothing. Adding the chance per try counts the runs with two or more hits more than once, so it overstates the chance of at least one hit, and the overstatement grows with the number of tries.
At least one six in four rolls comes up 51.8% of the time and at least one double six in twenty-four rolls only 49.1%, on opposite sides of even money, while the proportional rule wrongly puts both at 66.7%.
How does the complement settle it?
Ask how likely it is that nothing happens. Four rolls with no six: (5/6)^4 = 625/1,296 = 48.2%, so at least one six is 51.8%. Twenty-four rolls with no double six: (35/36)^24 = 50.9%, so at least one is 49.1%. The rare event with many tries falls short, because its misses compound over six times as many rolls.
The relationship
$1 - \left(\tfrac{5}{6}\right)^{4} = 0.518 \qquad 1 - \left(\tfrac{35}{36}\right)^{24} = 0.491$
5/6 the chance one roll of a die is not a six
35/36 the chance one roll of two dice is not a double six
4, 24 the number of tries in each bet
What it says in wordsThe chance of at least one hit is one minus the chance that every single try misses.
Why is the gap so small, and what is it worth as a bet?
The two probabilities are only 2.6 points apart, which is why the question needed a careful calculation to settle and why it still tests method rather than intuition. As an even-money bet, the first earns 3.5 paise per rupee staked on average: 51.8% of winning a rupee less 48.2% of losing one. The second loses 1.7 paise per rupee. A small edge repeated many times is the whole business of a casino, and of many trading strategies. Give the second bet one more roll and it tips over: 1 - (35/36)^25 = 50.6%.
Where candidates lose it
The trap is the proportional argument: 4 x 1/6 = 24 x 1/36 = 2/3, so the bets are equal. It is the reasoning the question was built to catch, and any answer that makes the chance grow in a straight line with the tries fails the moment the tries pass six.
The second loss is getting 51.8% for the first bet and assuming the second is also above half. Compute both; the point of the question is that they fall on opposite sides of 50%.
What the interviewer asks next
How many rolls of two dice do you need before a double six is more likely than not?
You are offered even money on the second bet. What is your expected result per Rs 100 staked?
Why is 1 - (1 - p)^n close to 1 - e^(-np) when p is small, and what does that give for the second bet?
082A car drives 60 miles at an average speed of 30 mph. How fast must it drive the 60 miles back to average 60 mph over the whole round trip?Logic and brainteasersWarm upMan GroupLondon · 2016
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Answer inside ten seconds.
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It cannot be done at any finite speed. Averaging 60 mph over the 120-mile round trip means finishing in 2 hours. The outward 60 miles at 30 mph already took 2 hours, so the return leg would have to take no time at all. Driving back at 90 mph, the tempting answer, gives an average of only 45 mph.
Why is 90 mph the wrong instinct?
Averaging 30 and 90 to get 60 treats the two speeds as if they counted equally. They do not, because the car spends far longer at the slow speed. Think of a student who scores 30% on a three-hour paper and 90% on a ten-minute quiz: nobody would call that a 60% performance. Average speed is total distance over total time, so the slow leg carries more weight because it takes up more of the clock.
A 60 mph average over 120 miles allows 2 hours, and the outward leg at 30 mph uses all of them, so a return at 90 mph ends 40 minutes late for a 45 mph average and even 300 mph ends 12 minutes late for 54.5 mph.
How do you prove it cannot be done?
Work in time, not speed. At 60 mph, 120 miles takes exactly 2 hours, and the first leg has already spent those 2 hours. Any return speed, however fast, adds some time, which pushes the average below 60 mph. At 90 mph the return takes 40 minutes and the average is 45 mph; at 300 mph it takes 12 minutes and the average is 54.5 mph. The average creeps towards 60 but never reaches it.
The relationship
$\bar v = \frac{120}{2 + 60/v} < 60 \quad \text{for every finite } v$
v the speed on the return leg, in mph
2 hours already spent on the outward leg
60/v hours the return leg takes
What it says in wordsThe average is the whole distance over the whole time, and the whole time is always more than the 2 hours a 60 mph average allows.
Where does the same mistake show up on a desk?
It appears whenever numbers are averaged without the right weights. A position that falls 50% and then rises 50% does not break even, because the second move works on a smaller base; the average that matters is the one weighted the way the thing actually compounds. Speeds over equal distances call for the harmonic mean, which sits below the simple average whenever the numbers differ. It is the same reason that putting a fixed rupee amount into a fund each month buys units at an average cost below the average price over those months.
Where candidates lose it
90 mph is the whole trap, and it comes from averaging the speeds instead of dividing distance by time. The interviewer asks it quickly precisely so that the symmetric answer comes out first.
The second loss is saying impossible without the reason. Give the time budget in one line: 2 hours allowed, 2 hours already used.
What the interviewer asks next
What return speed gives a round-trip average of 45 mph?
The car drives the first 60 miles at 40 mph instead. What speed back gives 60 mph overall?
Why is the average cost of buying a fixed rupee amount each month below the average price?
Asked at Man Group, Equity Hedge, London, 2016 (Wall Street Oasis): A car travels a distance of 60 miles at an average speed of 30 mph.
083A binary contract pays Rs 100 if the index closes higher tomorrow and nothing otherwise. The market is 55 bid, 60 offered, and your model says the chance of an up close is 50%. What do you do, and what is your edge?Market making and trading gamesCoreProp and quant trading firmsVolatility and relative value funds
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What is the trade?
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Sell at the 55 bid; the edge is Rs 5 a contract, five points of probability. A contract paying Rs 100 on an up close is worth Rs 100 times the probability, so your model values it at Rs 50. Selling at 55 collects Rs 55 for a liability worth Rs 50 on average. Buying at the 60 offer would give away Rs 10 of expected value. The edge is only as good as the model behind it.
Why is a binary price just a probability?
Think of a bet with a friend: you pay a fixed sum now and get Rs 100 back if it rains tomorrow. If you think rain is a 30% chance, the most you would pay is Rs 30. A contract paying Rs 100 or nothing is worth 100 times the probability of the payout, so a price of 55 is the market saying 55%. Read as probabilities, the quote says between 55% and 60%, and your model says 50%.
With your model value at 50, a bid of 55 and an offer of 60, selling at the bid earns 5 over fair value and buying at the offer loses 10; selling 100 contracts at 55 receives Rs 5,500 against an expected payout of Rs 5,000, an expected profit of Rs 500.
Which side of the quote can you trade?
You buy at the offer and sell at the bid, never the other way round. Buying costs 60 for something your model values at 50, a loss of 10 in expectation, while selling receives 55 for the same thing, a gain of 5. So the trade is to sell at 55. Sell 100 contracts and you receive Rs 5,500; half the time you pay out Rs 10,000 and half the time nothing, so the expected payout is Rs 5,000 and the expected profit Rs 500.
The relationship
$\text{edge} = \text{bid} - 100 \times p = 55 - 100 \times 0.50 = \text{Rs } 5$
bid the price at which you can sell, 55
p your model's probability of an up close, 0.50
100 the payout if the index closes higher
What it says in wordsYour edge on each contract sold is what the market pays you minus what the contract is worth on your numbers.
When would you do nothing?
If your model said 57%, the fair value of 57 would sit inside the quote: selling at 55 loses 2 and buying at 60 loses 3. An edge exists only when your value falls outside the bid and offer, not whenever you disagree with the middle of the market. Then add the honest caveat. A market pricing an up close near 57.5% may know something your model does not, and each contract swings between plus 55 and minus 45, so a 5-point edge needs many independent trades, and a model you trust, before it shows up in the P&L.
Where candidates lose it
The fast wrong answer is to buy because the market is above 50 and so seems to think up is likely. That reads the quote as a forecast to follow rather than a price to trade against, and the candidate who buys at 60 pays 10 more than the contract is worth on their own numbers.
The second loss is trading the wrong side of the quote, selling at 60 or buying at 55. Say which side you would hit before you give the edge.
What the interviewer asks next
Your model says 57%. What do you do?
You are asked to make a two-sided market around your 50%. Where do you quote, and why not 49 to 51?
The contract instead pays Rs 100 for every point the index rises. How does the pricing change?
084You own a stock bought at Rs 500 and sell a call struck at Rs 550 for a premium of Rs 12. At expiry, what is your maximum profit and where is your breakeven?Options and payoffsWarm upVolatility and relative value fundsProp and quant trading firms
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What is the most you can make?
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Maximum profit is Rs 62 and the breakeven is Rs 488. Above the Rs 550 strike the stock is called away, so you keep the Rs 50 rise from 500 to 550 plus the Rs 12 premium. Below 550 the call expires worthless and you keep the premium, which cushions the first Rs 12 of any fall. You lose money only below 500 minus 12, which is Rs 488.
What have you actually sold?
Think of renting out a flat you own with an agreement that the tenant may buy it at a fixed price within the year. You collect rent now, but if flat prices soar, the tenant buys at the agreed price and the extra gain is theirs. A covered call swaps the upside above the strike for cash today: the premium is the rent, and the strike is the agreed sale price.
The covered call earns Rs 12 more than the plain stock at every price up to Rs 550, is capped at a profit of Rs 62 from Rs 550 upwards, breaks even at Rs 488, and falls behind the plain stock above Rs 562.
How do you find the two numbers quickly?
Take the two regions separately. At or above Rs 550 the position is worth 550 plus the 12 kept, against 500 paid, so Rs 62 whatever the stock does. Below 550 the call is worthless and the position is just the stock plus 12, so it loses money only once the stock falls more than 12 below the purchase price, at Rs 488. At a stock price of 520 you make 32: 20 on the stock and 12 of premium. At 450 you lose 38 instead of 50.
The relationship
$\text{profit} = \min(S_T, K) - S_0 + c \qquad \max = 550 - 500 + 12 = 62 \qquad \text{breakeven} = 500 - 12 = 488$
S_T the stock price at expiry
K the strike of the call sold, Rs 550
S_0 the price paid for the stock, Rs 500
c the premium received, Rs 12
What it says in wordsYou keep the stock's value up to the strike, plus the premium, minus what you paid for the stock.
What is the trade-off in plain terms?
The premium improves every outcome below Rs 562 and worsens every outcome above it. Above Rs 562 the plain stock position beats the covered call, because the gain you gave away exceeds the premium you took in. On the downside the cushion is thin: the stock can fall all the way from 500 and the premium covers only 12 of it. That is the limitation to state: a covered call is income with a cap, not protection.
Where candidates lose it
The common slip is quoting the premium, Rs 12, as the maximum profit, which forgets that you still own the stock and keep its rise up to the strike. The opposite slip is saying unlimited, which forgets the call you sold.
For the breakeven, candidates often add the premium to the purchase price and say Rs 512. The premium is money received, so it lowers the breakeven, to Rs 488.
What the interviewer asks next
At what stock price are the covered call and the plain stock worth the same at expiry?
Why might the same call fetch more than Rs 12 just before a results announcement?
Which position gives the same payoff at expiry as a covered call without owning the stock?
085A book's one-day 99% VaR is Rs 10 crore. What is the ten-day 99% VaR under the usual scaling rule, and what has to be true for that rule to hold?Portfolio and risk mathsWarm upAQR Capital ManagementGreenwich · 2022
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What is the ten-day 99% VaR?
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About Rs 31.6 crore: Rs 10 crore times the square root of 10. If daily P&L is independent from day to day, with the same volatility and a mean near zero, variances add, so ten-day volatility is root 10 times daily volatility, and a normal quantile scales the same way. The rule also needs the positions held unchanged for ten days and a distribution that keeps its shape over the horizon.
Why not ten times the one-day number?
Picture ten friends each tossing a coin for Rs 100. The worst case is the group losing Rs 1,000, but the typical spread of the group's total is nowhere near ten times one person's, because some win while others lose. Independent daily moves partly cancel, so their variances add while their volatilities do not, and volatility grows with the square root of the number of days. Ten times would need every bad day to line up in the same direction, which is exactly what independence rules out. Value at riskThe loss a book should not exceed over a set horizon at a set confidence level, for example one day at 99%. inherits that square root when the distribution is normal.
Starting from Rs 10 crore for one day, a straight line reaches Rs 100 crore at ten days only if every bad day lines up, while the square-root curve for independent days reaches Rs 31.6 crore.
The relationship
$\text{VaR}_{10} = \text{VaR}_{1} \times \sqrt{10} = 10 \times 3.162 = 31.6$
VaR_1 the one-day 99% VaR, Rs 10 crore
sqrt(10) the growth in volatility over ten independent days
What it says in wordsOver ten independent days the spread of P&L grows by the square root of ten, and so does a normal VaR.
What has to be true for the rule to hold?
List the assumptions, because that is the real question. Returns must be independent across days, volatility constant, the mean close to zero, the positions unchanged, and the distribution one that keeps its shape when summed, as the normal does. Break any one and the rule drifts. Positive autocorrelation, where bad days follow bad days, makes the true ten-day number larger. A book that is cut after losses makes it smaller. Fat tails make the one-day 99% quantile a poor guide to the ten-day one.
Which way does the error usually run?
In a calm market the rule is a fair approximation. In stress it tends to understate, because volatility rises and losses cluster just when the ten-day horizon matters. The square root of time is a scaling convenience, not a law, so a risk team checks it against ten-day P&L measured directly. The same assumption sits behind the desk habit of multiplying daily volatility by 16 to get an annual figure, 16 being roughly the square root of the trading days in a year; stretch it to 250 days here and you get Rs 158 crore, a number few would trust.
Where candidates lose it
Rs 100 crore is the reflex answer, adding ten daily VaRs as if every day were the worst day. The interviewer is testing whether you know that independent risks add in variance.
The second loss is giving Rs 31.6 crore and stopping. The question asks what must be true; independence, constant volatility, unchanged positions and a stable distribution are the answer the interviewer is listening for.
What the interviewer asks next
Daily returns have positive autocorrelation. Is the true ten-day VaR above or below Rs 31.6 crore?
Scale the one-day figure to 250 trading days. What do you get, and would you trust it?
Why does square-root scaling work poorly for a book that is long deep out-of-the-money options?
Asked at AQR Capital Management, Quantitative Research, Greenwich, 2022 (Wall Street Oasis): Specific statistics questions on financial concepts. daily vs monthly return, VAR
086A stock ticks up or down by Rs 1 each minute with equal probability, starting at Rs 50. On average, how many minutes pass before it first touches Rs 45 or Rs 55?Random walks and Markov chainsCoreQuant and systematic fundsProp and quant trading firms
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Pick your answer before setting up any equation.
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25 minutes. Let E(k) be the expected minutes to exit from price k. Each minute costs one and moves the price up or down with equal chance, so E(k) = 1 + half E(k + 1) + half E(k - 1), with E(45) = E(55) = 0. The solution is E(k) = (k - 45)(55 - k), the product of the distances to the two barriers. From Rs 50 that is 5 x 5 = 25.
Why is the answer not 5 minutes?
Think of someone pacing a corridor, taking one step forward or back on each coin toss. After 25 steps they are not 25 steps away; typically they are about 5 away, because the steps keep undoing each other. A fair random walk covers distance like the square root of time, so reaching a barrier 5 away takes on the order of 5 squared, 25 steps, not 5.
Three sample paths from Rs 50 leave the Rs 45 to Rs 55 channel after 11, 23 and 45 minutes; averaged over all paths the exit takes (50 - 45) x (55 - 50) = 25 minutes, and from Rs 48 or Rs 46 it takes 21 or 9.
How do you get exactly 25?
Set up the one-step equation. From any price k strictly between the barriers, you spend one minute and then stand at k + 1 or k - 1 with equal chance. E(k) = 1 + half E(k + 1) + half E(k - 1) says the second difference of E is always minus 2, so E is a downward parabola that is zero at both barriers. The only such parabola is (k - 45)(55 - k). Check a point: from Rs 46 it gives 1 x 9 = 9 minutes, and it passes the one-step test, since 1 plus half of E(47), which is 16, plus half of E(45), which is 0, is 9.
The relationship
$E(k) = (k - a)(b - k) \qquad E(50) = (50 - 45)(55 - 50) = 25$
a, b the lower and upper barriers, Rs 45 and Rs 55
k the starting price
E(k) the expected number of one-minute steps before either barrier is touched
What it says in wordsFor a fair walk, the expected time to leave a channel is the distance to the floor times the distance to the ceiling.
What does the shape tell a trader?
Starting in the middle is the slowest place to be, and an off-centre start is much faster: from Rs 48 the answer is 3 x 7 = 21 minutes, and from Rs 46 only 9. Doubling both distances quadruples the expected time: barriers at Rs 40 and Rs 60 give 100 minutes. That is the arithmetic behind why widening a stop and a profit target together makes a trade live much longer. It holds only for a fair walk: if the stock ticks up 60% of the time, the exit comes sooner, about 19.2 minutes, and mostly at the top.
Where candidates lose it
The quick wrong answer is 5 minutes, which treats the walk as if it moved steadily towards one barrier. A fair walk wanders, and the interviewer is checking whether you know that distance grows with the square root of time.
The second loss is reaching 25 from the square-root intuition without being able to show it. Write the one-step equation and the parabola; that is what turns a good guess into an answer.
What the interviewer asks next
What is the probability the stock touches Rs 55 before Rs 45?
From Rs 50, the barriers move to Rs 40 and Rs 55. What is the expected time now?
The stock ticks up with probability 0.6. Why does the expected time fall?
087A fund charges 2 and 20 and earns a gross return of 12% on Rs 1,000 crore. Investors push the management fee down to 1%. What performance fee keeps the manager's total fee income unchanged at that return?Returns, compounding and feesCoreTwo SigmaNew York · 2026
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What performance fee keeps the manager whole?
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About 27.3%. At 2 and 20 the manager earns Rs 20 crore of management fee and 20% of the remaining Rs 100 crore gain, Rs 20 crore: Rs 40 crore in all. At 1%, the management fee is Rs 10 crore and the gain after it is Rs 110 crore. To keep Rs 40 crore the performance fee must bring in Rs 30 crore, which is 30/110, or 27.3%. The two deals match only at a 12% gross return.
What does the manager earn today?
Think of a tailor who charges a fixed stitching fee plus a share of whatever the finished suit sells for above cost. Cut the fixed fee and the share must rise to keep the same income, but the share now applies to a slightly larger base. At 2 and 20 on Rs 1,000 crore earning 12%, the manager takes Rs 20 crore of management fee plus 20% of the Rs 100 crore gain left after it, Rs 40 crore in total. Investors keep Rs 80 crore, a net return of 8%.
Under 2 and 20 the manager's Rs 40 crore is Rs 20 crore fixed plus 20% of Rs 100 crore; under a 1% fixed fee it is Rs 10 crore plus 27.3% of Rs 110 crore, because the smaller fixed fee leaves a larger gain for the performance fee.
How do you find the new performance fee?
Keep the total at Rs 40 crore. The management fee falls to Rs 10 crore, so the performance fee must bring in Rs 30 crore, and it is charged on a gain of Rs 110 crore, not Rs 100 crore. 30 divided by 110 is 27.3%. The base grows because less has been taken off the top before the performance fee is worked out. State the assumptions as you go: no hurdle rate, and no earlier losses to recover below a high-water markThe highest value an investor has paid a performance fee on; no new performance fee is charged until the fund climbs back above it..
The relationship
$f = \frac{40 - 10}{120 - 10} = \frac{30}{110} = 27.3\%$
40 the manager's total fee income under 2 and 20, Rs crore
10 the new 1% management fee, Rs crore
120 the gross gain, 12% of Rs 1,000 crore
What it says in wordsThe new performance fee is the income still needed, divided by the gain left after the new management fee.
Is the new deal really the same for investors?
Only at a 12% gross return. The new deal pays the manager less in poor years and more in good ones, so it moves risk from the investors to the manager. At a 4% gross return the old deal pays Rs 24 crore and the new one Rs 18.2 crore; at 20% the old pays Rs 56 crore and the new one Rs 61.8 crore. That is why allocators push for a lower fixed fee even at the price of a higher share: they would rather pay for performance than for size.
Gross return 2 and 20, Rs crore 1 and 27.3, Rs crore Who gains from the switch
4% 24.0 18.2 Investors
12% 40.0 40.0 Neither
20% 56.0 61.8 Manager
Manager's fee income on Rs 1,000 crore under each deal: the two match at a 12% gross return, the new deal pays Rs 5.8 crore less at 4% and Rs 5.8 crore more at 20%.
Where candidates lose it
The common slip is 30%: dividing the Rs 30 crore needed by the old Rs 100 crore base, forgetting that a smaller management fee leaves a larger gain for the performance fee to work on. The other is 25%, dividing by the gross Rs 120 crore.
The second loss is stopping at 27.3% as though the two deals were identical. They match only at a 12% return, and the interviewer wants to hear who comes out ahead in good years and in bad ones.
What the interviewer asks next
At what gross return does the manager prefer the new deal?
Add a 5% hurdle to the new deal. What performance fee keeps the manager whole now?
How does a high-water mark change what the performance fee is worth to the manager?
Asked at Two Sigma, Equity Capital Markets, New York, 2026 (Wall Street Oasis): the 2/20 rule, and if one part of this equation changed, how would the other variable make up for it
088A stock's true model is: stock return = 0.5 x market return + 1.0 x sector return + noise. Regressing the sector's return on the market gives a slope of 0.4. If you regress the stock on the market alone, what slope do you get?Statistics and estimationCoreQuant and systematic funds
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What does the market-only regression report?
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About 0.9. The market reaches the stock by two paths: directly, with a coefficient of 0.5, and through the sector, which moves 0.4 for each unit of market and passes all of it on with a coefficient of 1.0. A regression on the market alone cannot separate the two and reports the total, 0.5 + 1.0 x 0.4 = 0.9. The extra 0.4 is omitted variable bias.
Why does leaving the sector out change the market slope?
Suppose you measure how much ice cream sales rise on hot days, but hot days also tend to be holidays, and holidays sell ice cream too. Leave holidays out and the heat gets the credit for both. A regression gives a left-out variable's effect to whichever included variable moves with it, in proportion to how strongly the two move together. Here the sector moves with the market, so the market's slope absorbs part of the sector's effect.
The market reaches the stock directly with a coefficient of 0.5 and through the sector with 0.4 x 1.0 = 0.4, so a regression of the stock on the market alone reports 0.9, of which 0.4 is the sector's effect credited to the market.
How do you compute the bias?
Write the sector as 0.4 x market plus a part unrelated to the market, then substitute. Stock = 0.5 x market + 1.0 x (0.4 x market + other) + noise = 0.9 x market + (1.0 x other + noise). The bracket is unrelated to the market, so a regression on the market alone recovers 0.9. The bias is the omitted coefficient times the slope of the omitted variable on the included one, 1.0 x 0.4. A simulation of 20,000 days with these coefficients gives a slope of 0.897, matching the algebra.
The relationship
$\hat\beta_{\text{short}} = \beta_M + \beta_S\,\delta = 0.5 + 1.0 \times 0.4 = 0.9$
beta_M the stock's true direct loading on the market, 0.5
beta_S the stock's loading on the sector that was left out, 1.0
delta the slope of the sector's return on the market's, 0.4
What it says in wordsThe short regression's slope is the true slope plus the left-out variable's effect times how much that variable moves with the one you kept.
Is 0.9 wrong, or answering a different question?
It depends on what you use it for. If you want to hedge the stock with the market alone, 0.9 is the right hedge ratio, because it captures everything the market drags along with it. If you want the stock's exposure holding the sector fixed, say to build a sector-neutral book, 0.9 overstates it and 0.5 is the number you need. The bias can also run the other way: if the sector moved against the market, or the stock loaded negatively on the sector, the short slope would sit below 0.5. Naming both uses is what the interviewer is listening for.
Where candidates lose it
The fast wrong answer is 0.5: candidates assume a regression recovers the true coefficient whatever else is left out. It does so only when the omitted variable is unrelated to the included one.
The second loss is getting 0.9 and calling it simply wrong. It is the correct total effect of the market and the right number for a market-only hedge; it is wrong only as an estimate of the direct effect.
What the interviewer asks next
What slope do you get if the sector's slope on the market is minus 0.4?
You add the sector to the regression. What happens to the standard error of the market coefficient if the two are highly correlated?
How does this bias show up when you estimate a stock's factor exposures with too few factors?
089The equity risk premium is 4.5%, and a market's fair multiple is 1 divided by (real yield + premium - real growth). Real yields rise from 1.5% to 2.5% while expected real growth rises from 2.0% to 2.5%. What happens to the fair multiple?Valuation, accounting and macro riddlesCoreCitadelNew York · 2026
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Where does the fair multiple go?
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The fair multiple falls from 25x to about 22.2x, a compression of about 11%. The denominator is the real yield plus the premium minus growth: 1.5 + 4.5 - 2.0 = 4.0% before, and 2.5 + 4.5 - 2.5 = 4.5% after. Real yields rose by a full point and growth by only half a point, so the net rate rose by half a point, and the multiple, its inverse, fell.
Why is the multiple one over a spread?
Think of a shop that pays you rent forever, rising a little each year. What you would pay for it depends on the return you demand minus how fast the rent grows. For earnings paid out and growing forever, price over earnings is one divided by the required return minus growth, so the multiple depends only on the gap between the two. In real terms the required return is the real yield plus the equity risk premiumThe extra return investors demand for holding shares rather than government bonds.. The question treats all earnings as paid out, which is the assumption to name.
The net rate in the denominator rises from 4.0% to 4.5% because real yields climbed a full point while growth climbed half a point, so the fair multiple falls from 25x to 22.2x, about 11% lower.
How do you work it out quickly?
Compute the denominator before and after. Before: 1.5 + 4.5 - 2.0 = 4.0%, a multiple of 25x. After: 2.5 + 4.5 - 2.5 = 4.5%, a multiple of 22.2x. Rates went up by 1.0 point and growth by 0.5, so the spread widened by 0.5 point. A 0.5-point rise on a 4.0% base is a 12.5% rise in the denominator, and the multiple falls by 1 minus 1/1.125, about 11.1%.
The relationship
$\frac{P}{E} = \frac{1}{r + \text{ERP} - g} \qquad \frac{1}{4.0\%} = 25\times \qquad \frac{1}{4.5\%} = 22.2\times$
r the real yield on government bonds
ERP the equity risk premium, 4.5%
g expected real growth of earnings
What it says in wordsThe fair multiple is one over the net rate: what investors demand minus how fast the earnings grow.
What does this teach beyond the arithmetic?
Higher yields do not hurt equities one for one if growth rises with them. What matters is whether real yields rise faster or slower than expected growth: faster compresses multiples, slower expands them. Had growth also risen a full point, to 3.0%, the net rate would be 4.0% again and the multiple 25x. The limitation to state is sensitivity: near a 4% net rate, a half-point move shifts the multiple by about 2.8 turns one way and 3.6 the other, so small errors in the premium or the growth guess swamp the answer.
Where candidates lose it
The quick wrong answer is that nothing happens because both rates went up. The question is built so that growth rises by only half as much as yields, and it is the spread, not the level, that sets the multiple.
The second loss is dropping the growth change and answering 20x. Write the denominator out in full, before and after; it takes ten seconds and removes both errors.
What the interviewer asks next
Growth rises by a full point, to 3.0%. What is the multiple now?
The equity risk premium also falls to 4.0%. What is the net effect?
Why do shares whose value sits far in the future fall more than the market when real yields rise?
Asked at Citadel, Software, New York, 2026 (Wall Street Oasis): real yields rising faster than growth expectations predicts equity multiple compression
090Each of your analysts calls the direction of a stock correctly 70% of the time, independently of the others, and your prior is 50/50. Two analysts disagree. What is your probability now that the stock goes up? What if a third analyst then sides with the one who said up?Conditional probability and BayesHardQuant and systematic fundsProp and quant trading firms
Try it first
Where are you after the disagreement, and after the third call?
Show the worked solution
50% after the disagreement, and 70% once the third analyst sides with up. Work in odds. Each analyst's call multiplies the odds by 0.7/0.3 = 7/3 in the direction called. One up and one down multiply by 7/3 and 3/7, which cancel, leaving the prior of 1:1. The third call multiplies by 7/3 again: odds of 7:3, a probability of 70%. A two-to-one split is worth one analyst, not two thirds.
Why switch from probabilities to odds?
Think of two friends who read the weather equally well and disagree about rain: you are back where you started, however good they are. With independent signals, each one multiplies your odds by its likelihood ratioHow much more likely a piece of evidence is if the claim is true than if it is false., so in odds form Bayes' rule is just multiplication. An analyst who is right 70% of the time says up with chance 0.7 if the stock will rise and 0.3 if it will fall, so an up call multiplies the odds by 7/3 and a down call by 3/7.
Starting from 50%, analyst A's up call moves the chance of up to 70%, analyst B's down call returns it to 50%, and analyst C's up call moves it to 70% again, not to the 67% a vote count suggests; three up calls and none down would give 92.7%.
How do the three calls combine?
Start at 1:1. The first analyst says up: 7:3, or 70%. The second says down: 7:3 times 3:7 is 1:1, back to 50%. The third says up: 7:3 again, 70%. Only the net count of calls matters, not the total. Three for up and none against would give 343:27, about 92.7%, which shows how much the one dissenter costs.
The relationship
$\text{odds}_{\text{post}} = 1 \times \tfrac{7}{3} \times \tfrac{3}{7} \times \tfrac{7}{3} = \tfrac{7}{3} \quad\Rightarrow\quad p = \frac{7}{7+3} = 70\%$
1 the prior odds of up against down, 50/50
7/3 the likelihood ratio of an up call from a 70% accurate analyst
3/7 the likelihood ratio of a down call
What it says in wordsMultiply the prior odds by one likelihood ratio per independent call, then turn the odds back into a probability.
What assumption carries all of this?
Independence. If the analysts read the same research and speak to the same management teams, their errors are correlated, and a second agreeing call adds much less than a factor of 7/3. In the extreme where the second analyst simply copies the first, it adds nothing at all. The rule also assumes each analyst is right 70% of the time whichever way the stock moves; an analyst who calls up too often tells you more when calling down. State independence as the assumption, then say how you would test it: by checking how often the analysts' past calls agreed with each other.
Where candidates lose it
The vote-counting answer, two out of three so 67%, is the usual loss. It treats a majority as a probability, when a two-to-one split carries exactly one net call of evidence, 70%.
The second loss comes after the disagreement: candidates reach for something like 58%, feeling that two good analysts must add something. With equal accuracy and opposite calls, the evidence cancels exactly.
What the interviewer asks next
One analyst is right 80% of the time and the other 60%. The better one says up, the other down. Where are you?
Five analysts split three to two. What is your probability?
How would you estimate how correlated your analysts' calls are, and how would you adjust for it?

5/6	the chance one roll of a die is not a six
35/36	the chance one roll of two dice is not a double six
4, 24	the number of tries in each bet

v	the speed on the return leg, in mph
2	hours already spent on the outward leg
60/v	hours the return leg takes

bid	the price at which you can sell, 55
p	your model's probability of an up close, 0.50
100	the payout if the index closes higher

S_T	the stock price at expiry
K	the strike of the call sold, Rs 550
S_0	the price paid for the stock, Rs 500
c	the premium received, Rs 12

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Why does the proportional argument give the same answer for both?

How does the complement settle it?

Why is the gap so small, and what is it worth as a bet?

Where candidates lose it

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Why is 90 mph the wrong instinct?

How do you prove it cannot be done?

Where does the same mistake show up on a desk?

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Why is a binary price just a probability?

Which side of the quote can you trade?

When would you do nothing?

Where candidates lose it

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What have you actually sold?

How do you find the two numbers quickly?

What is the trade-off in plain terms?

Where candidates lose it

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Why not ten times the one-day number?

What has to be true for the rule to hold?

Which way does the error usually run?

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Why is the answer not 5 minutes?

How do you get exactly 25?

What does the shape tell a trader?

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What does the manager earn today?

How do you find the new performance fee?

Is the new deal really the same for investors?

Where candidates lose it

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Why does leaving the sector out change the market slope?

How do you compute the bias?

Is 0.9 wrong, or answering a different question?

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Why is the multiple one over a spread?

How do you work it out quickly?

What does this teach beyond the arithmetic?

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Why switch from probabilities to odds?

How do the three calls combine?

What assumption carries all of this?

Where candidates lose it

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