Hedge Funds puzzles, solved step by step

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

Showing 41–50 of 50 · filtered from 100Clear filters

078Two traders each arrive at a random time between 9:00 and 10:00, independently, and each waits 15 minutes for the other before leaving. What is the probability that they meet?Continuous probability and distributionsCoreQuant and systematic fundsProp and quant trading firms
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Pick the closest answer before drawing anything.
Show the worked solution
7/16, about 43.8%. Plot one trader's arrival on each axis of a 60 by 60 minute square. They meet when the arrivals are within 15 minutes, a band around the diagonal. They miss in the two corner triangles, each 45 by 45 over two, which together cover 45 squared over 60 squared, or 9/16. So they meet 1 minus 9/16, which is 7/16 of the time.
Why turn two arrival times into a square?
Think of two friends who say they will meet at a cafe sometime in the lunch hour. Every possible pair of arrival times is one point: the first friend's time across, the second's up. Because both times are uniform and independent, every point in the square is equally likely. A probability question about two uniform times becomes an area question, and the areas of triangles need no calculus.
Each pair of arrival times is a point in a 60 by 60 minute square; the traders meet in the green band within 15 minutes of the diagonal and miss in the two grey corners, which cover 2,025 of 3,600 square minutes, so they meet with probability 7/16, or 43.75%.
How do you get the area without integrating?
Find the region where they miss, because it is two clean triangles. They miss when A arrives more than 15 minutes after B, or B more than 15 minutes after A. Each miss region is a right triangle with legs of 45 minutes, so each covers 45 x 45 / 2 = 1,012.5 square minutes out of 3,600. Together that is 2,025 out of 3,600, or 9/16, and the band where they meet is the remaining 7/16.
The relationship
$P(\text{meet}) = 1 - \left(\frac{60 - w}{60}\right)^2 = 1 - \left(\frac{45}{60}\right)^2 = \frac{7}{16}$
w how long each trader waits, here 15 minutes
60 the length of the arrival window in minutes
What it says in wordsThe chance of meeting is one minus the two corner triangles, which fit together into a square of side 45 minutes.
How does the answer move with the waiting time?
The formula shows the shape. Waiting 30 minutes gives 1 minus (30/60) squared, which is 75%, not double 7/16. The chance rises quickly at first and then flattens, because the band is squeezed at the ends of the hour, where someone arriving at 9:55 has only 5 minutes left to be joined. The same geometry answers questions about two orders arriving within a latency window, or two news releases landing in the same trading hour.
Where candidates lose it
The fast wrong answers are 1/4, from 15 over 60, and 1/2, from doubling it because either trader can wait. Both ignore that the window is clipped at 9:00 and 10:00, where a late arrival has less time left to be joined.
The other slip is computing the corner area correctly and then giving 9/16 as the answer. Say out loud which region you computed, miss or meet, before you give the number.
What the interviewer asks next
If one trader waits 15 minutes and the other only 5, what is the probability they meet?
What is the expected gap between the two arrival times?
How would the answer change if arrivals were more likely near 9:30 than at the edges of the hour?
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
Try it first
Before any arithmetic, which do you back?
Show the worked solution
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?
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?
Show the worked solution
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?
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
Try it first
Pick your answer before setting up any equation.
Show the worked solution
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?
Show the worked solution
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?
Show the worked solution
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
091Cards are turned over one at a time from a well-shuffled 52-card deck. On average, at what position does the first ace appear?Counting and combinatoricsCoreQuant and systematic fundsProp and quant trading firms
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What is the expected position of the first ace?
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Position 10.6. The four aces split the 48 non-aces into five gaps: before the first ace, between each pair of aces, and after the last. By symmetry, a given non-ace is equally likely to fall in any of the five gaps, so each gap holds 48/5 = 9.6 cards on average. The first ace comes straight after the first gap, at 9.6 + 1 = 10.6.
Why do the gaps have the same size on average?
Picture four red beads threaded at random among 48 white ones on a string. Pick any one white bead: it is as likely to sit before all four reds as between the first and second, or after the last, because a shuffle favours no order. Each non-ace lands in each of the five gaps with probability 1/5, so the expected size of every gap is 48/5 = 9.6 cards. Any single shuffle has uneven gaps; it is only the average that is equal.
One shuffle leaves uneven gaps of 4, 5, 14, 11, 14 non-aces around its four aces, but averaged over all shuffles every one of the five gaps holds 9.6 cards, so the aces sit at 10.6, 21.2, 31.8, 42.4.
How do you turn gap sizes into a position?
The first ace sits immediately after the first gap. Expected position = expected size of the first gap + 1 = 9.6 + 1 = 10.6. The same logic puts the second ace at 21.2, the third at 31.8 and the last at 42.4, evenly spaced 10.6 apart. A quick check: the last ace at 42.4 plus a final gap of 9.6 reaches exactly 52.
The relationship
$E[\text{first ace}] = \frac{n - k}{k + 1} + 1 = \frac{n + 1}{k + 1} = \frac{53}{5} = 10.6$
n cards in the deck, 52
k special cards, here the 4 aces
(n - k)/(k + 1) the expected number of ordinary cards in each of the k + 1 gaps
What it says in wordsWith n cards and k special ones, the first special card is expected at position (n + 1) divided by (k + 1).
Why is 13 the wrong instinct?
Thirteen is the average wait when every draw has a fresh 1-in-13 chance, as if each card were put back and the deck reshuffled. Drawing without replacement makes the aces easier to reach, because every non-ace you turn over raises the share of aces in what is left. The general rule, (n + 1)/(k + 1), is worth keeping: the first spade is expected at 53/14, about card 3.79, and the same shape answers how many trades you expect to check before finding the first of a handful of booking errors in a batch.
Where candidates lose it
The fast wrong answer is 13, borrowed from a wait with replacement. The interviewer wants to see you notice that the deck is finite, so the first ace must appear by card 49 at the latest and is pulled earlier than a with-replacement wait would suggest.
The second loss is starting the direct sum, position times the chance the first ace is there, which is correct but slow and easy to get wrong at a whiteboard. The gap argument gets there in one line.
What the interviewer asks next
What is the expected position of the last ace?
What is the expected number of cards between the first and second ace?
How many cards do you expect to turn before the first spade?
093Four people must cross a narrow bridge at night with one torch. They take 1, 2, 5 and 10 minutes to cross, at most two can cross at a time, a pair moves at the slower person's pace, and the torch must be carried on every crossing. What is the fastest time for all four to get across?Logic and brainteasersCoreProp and quant trading firmsLong-short equity funds
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What is the fastest crossing?
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17 minutes. 1 and 2 cross (2 minutes), 1 returns (1), 5 and 10 cross together (10), 2 returns (2), and 1 and 2 cross again (2). The total is 17. The obvious plan, with the fastest person escorting everyone, takes 19, because the 5 and the 10 each cost a separate crossing. Pairing the two slowest hides the 5 inside the 10.
Why is the obvious plan not the fastest?
The natural plan uses the quickest person as a shuttle: 1 walks each person over and comes back. That costs 2 + 1 + 5 + 1 + 10 = 19. Every slow person who crosses on a separate trip pays their own time in full, so the 5 and the 10 together cost 15 minutes of crossing. Think of sending two slow parcels in one courier van instead of two: the second rides along for free.
The shuttle plan spends 5 and 10 minutes on separate crossings and takes 19 minutes, while sending 5 and 10 together costs 10 minutes for both and brings the total to 17, even after the 2-minute person makes one extra return trip.
How does the 17-minute plan pay for pairing the slow two?
Pairing 5 and 10 costs 10 instead of 15, a saving of 5. But someone fast must already be waiting on the far side to bring the torch back afterwards. That setup costs one extra return by the 2-minute person and an extra crossing of the pair 1 and 2, so the net saving is 2 minutes: 19 down to 17. The general rule: pair the two slowest when twice the second-fastest time is less than the fastest plus the second-slowest, here 4 against 6.
The relationship
$\underbrace{2 + 1 + 10 + 2 + 2}_{\text{pair the slow two}} = 17 \qquad \underbrace{2 + 1 + 5 + 1 + 10}_{\text{shuttle}} = 19$
10 the one crossing that carries both the 5-minute and the 10-minute person
2 + 2 the extra cost of having the 2-minute person bring the torch back and cross again
What it says in wordsPairing the two slowest saves 5 minutes of crossing at a cost of 3 extra minutes of torch returns and re-crossing.
How do you show 17 is the minimum?
Five crossings are unavoidable: three over and two back. If the 5 and the 10 cross separately, the forward trips cost at least 10 + 5 + 2 and the two returns at least 1 each, which is 19; if they cross together, the person who returns next must already be across, which forces the returns to be the 1 and the 2, and the best total is 17. Talking through that bound is what separates a remembered answer from a reasoned one, and it is what lets you handle the follow-up with different times.
Where candidates lose it
19 minutes is the answer most people reach, and it comes so quickly that they stop there. The interviewer is waiting to see whether you ask what the 5-minute person costs and whether that cost can be hidden.
The second loss is finding 17 by trial and error without the reason. Name the saving, 5 minutes from pairing, and its price, the extra return and re-crossing; that is what makes the answer hold when the interviewer changes the times.
What the interviewer asks next
The times are 1, 5, 6 and 10. Does pairing the two slowest still help?
Add a fifth person who takes 20 minutes. What is the fastest crossing now?
State the general rule for when to pair the two slowest walkers.
099You borrow in yen at 1% and invest in rupee deposits at 7% for a year. By how much can the rupee's value in yen fall over the year before the trade loses money?Valuation, accounting and macro riddlesCoreLong-short equity fundsGlobal macro funds
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How big a fall in the rupee can the trade absorb?
Show the worked solution
About 5.6%. Borrow 1 of yen value, convert it, and the rupee deposit grows to 1.07 in yen terms at today's rate. You owe 1.01. If the rupee's yen value falls by d, the deposit is worth 1.07 x (1 - d), which equals 1.01 when d = 1 - 1.01/1.07, about 5.6%. The cushion is slightly below the 6-point gap because the fall hits the interest as well as the principal.
Where does the carry trade's income come from?
Think of borrowing from a relative at 1% to put the money in a deposit paying 7%, except that the relative wants to be repaid in a different currency. The 6 points look like easy income until the currency moves. A carry trade earns the interest gap only if the currency you hold does not fall by more than about that gap over the holding period. The rate gap is the cushion; the exchange rate is the risk.
Per Rs 100 of yen borrowed, the trade makes Rs 6 if the rupee holds its value, breaks even at a 5.6% fall, already loses Rs 0.42 at a 6% fall, and loses Rs 4.70 at a 10% fall.
Why is the cushion 5.6% and not 6%?
Work in yen per unit borrowed. At the end you hold 1.07 of rupee deposit, measured at today's exchange rate, and you owe 1.01 of yen; the rupee can lose value until 1.07 x (1 - d) = 1.01, so d = 5.6%. The fall applies to the whole 1.07, including the 0.07 of interest, which is why the cushion is a little smaller than 6. Borrow Rs 100 crore worth of yen and a 5.6% fall leaves you exactly even; a 10% fall loses about Rs 4.7 crore.
The relationship
$1.07\,(1 - d) = 1.01 \;\Rightarrow\; d = 1 - \frac{1.01}{1.07} = 5.6\%$
1.07 what each unit becomes in the rupee deposit at 7%
1.01 what you owe on each unit of yen borrowed at 1%
d the fall in the rupee's value in yen over the year
What it says in wordsThe trade breaks even when the grown rupee deposit, after the currency fall, buys back exactly the yen you owe.
What does the forward market say about this?
Covered interest parityThe rule that the forward exchange rate must offset the interest rate gap between two currencies, or else borrowing in one, lending in the other and locking in the forward would be an arbitrage. sets the one-year forward so that the rupee buys about 5.6% fewer yen forward than today, exactly the fall that wipes out the carry. So an unhedged carry trade is a bet that the rupee will fall by less than the forward already implies. The limitation to state: carry trades tend to earn small amounts steadily and then lose sharply when many holders unwind at once, so the spread of outcomes has a fat left tail that the breakeven alone does not show.
Where candidates lose it
The quick answer is 6%, the gap between the two rates. It misses that the currency loss applies to the interest earned as well as to the principal, so the true cushion is slightly smaller, and at exactly a 6% fall the trade is already losing.
The second loss is treating the 6-point gap as a return and forgetting the currency altogether. Name the exchange rate as the risk before doing any arithmetic.
What the interviewer asks next
The rupee weakens 10% against the yen over the year. What is the loss on Rs 100 crore of yen borrowed?
What one-year forward rate does covered interest parity imply, and what does it say about the trade?
Why do carry trades tend to lose money suddenly rather than gradually?

w	how long each trader waits, here 15 minutes
60	the length of the arrival window in minutes

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

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

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

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Why turn two arrival times into a square?

How do you get the area without integrating?

How does the answer move with the waiting time?

Where candidates lose it

What the interviewer asks next

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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?

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What the interviewer asks next

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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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Why is the answer not 5 minutes?

How do you get exactly 25?

What does the shape tell a trader?

Where candidates lose it

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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?

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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 do the gaps have the same size on average?

How do you turn gap sizes into a position?

Why is 13 the wrong instinct?

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Why is the obvious plan not the fastest?

How does the 17-minute plan pay for pairing the slow two?

How do you show 17 is the minimum?

Where candidates lose it

What the interviewer asks next

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Where does the carry trade's income come from?

Why is the cushion 5.6% and not 6%?

What does the forward market say about this?

Where candidates lose it

What the interviewer asks next