Hedge Funds puzzles, solved step by step

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

Showing 21–30 of 35 · filtered from 100Clear filters

058A market maker knows that 30% of the orders she receives come from traders who know the true value is exactly Rs 1 above or below the mid. The other 70% are uninformed and buy or sell at random. How wide must her bid-ask spread be just to break even?Market making and trading gamesCoreProp and quant trading firmsVolatility and relative value funds
Try it first
How wide must the full spread be?
Show the worked solution
The spread must be Rs 0.60: a bid Rs 0.30 below the mid and an offer Rs 0.30 above it. Call the half-spread s. Each uninformed order earns her s on average. Each informed order trades on the side where the value really is Rs 1 away, so she loses 1 minus s. Breaking even needs 0.7s = 0.3(1 - s), which gives s = 0.30.
Why do some of the people who trade with you cost you money?
A shop that buys second-hand phones pays the same price to every seller, but the sellers who know their phone is faulty are the keenest to sell. A quote is an offer made to everyone, and the people who know more take it only when it hurts you. The spread is the fee charged to the whole crowd to pay for the few who know more, which is called adverse selectionThe tendency for the counterparties who choose to trade with you to be the ones who know the trade is good for them..
How do you set up the break-even?
Charge a half-spread s on each side. An uninformed order is as likely to be a buy as a sell and carries no information, so on average it earns you s; an informed order buys at your offer only when the value is Rs 1 above the mid, so it costs you 1 minus s. Weight each by how often it arrives and set the total to zero: 0.7s - 0.3(1 - s) = 0, so s = 0.30 and the spread is Rs 0.60.
The relationship
$0.7\,s = 0.3\,(1 - s) \;\Rightarrow\; s = 0.30, \qquad \text{spread} = 2s = 0.60$
s the half-spread, the distance from the mid to the bid or the offer
0.7 the share of orders from uninformed traders
1 - s what an informed trade takes: the Rs 1 value gap less the half-spread crossed
What it says in wordsThe spread earned from uninformed flow must pay for the expected loss to informed flow.
At a half-spread of Rs 0.20 she earns Rs 0.14 an order from uninformed flow but loses Rs 0.24 to informed flow, a loss of Rs 0.10; at Rs 0.30 she earns and loses Rs 0.21 each, the break-even spread of Rs 0.60.
What moves the spread?
Two things: how much of the flow is informed and how much the informed know. In general the half-spread is s = aV, where a is the informed share and V is how far the true value sits from the mid, so doubling either doubles the spread. Here 0.3 x 1 = 0.3. That is why quotes widen ahead of a results announcement, when more of the flow may know something, and in small stocks, where one piece of news moves the value a long way. The model leaves out inventory risk and competing quoters, both of which move a real spread; say so.
Where candidates lose it
The common slip is answering Rs 0.30, which is the half-spread, or quoting a width without saying where the bid and offer sit. Name both sides: bid at the mid minus 0.30, offer at the mid plus 0.30.
The deeper loss is charging each informed trade the full Rs 1 and forgetting that the informed trader also crosses the spread, which gives 0.7s = 0.3 and s of about 0.43. Her loss to an informed order is the value gap less the half-spread.
What the interviewer asks next
Half the flow is now informed. What spread breaks even?
You receive three buy orders in a row. Should you move your mid, and which way?
A rival quotes Rs 0.40 wide. What happens to the mix of flow you receive?
059A stock trades at Rs 1,000 and pays no dividend. A six-month European call struck at Rs 1,000 costs Rs 60 and the matching put costs Rs 45. Six-month simple interest is 3%. Is there an arbitrage, and how do you lock it in?Options and payoffsCoreVolatility and relative value fundsProp and quant trading firms
Try it first
What should the call minus the put be worth?
Show the worked solution
Yes: buy the call, sell the put, short the stock and lend Rs 970.87, which collects about Rs 14.13 today with nothing owed at expiry. Put-call parity says the call less the put should equal the stock less the present value of the strike, 1,000 minus 1,000/1.03, about Rs 29.13. The market prices that gap at Rs 15, so the call is cheap against the put by about Rs 14.
What does a call minus a put actually pay?
Imagine agreeing today to buy a used car in six months at a fixed price. If car prices rise you gain the difference; if they fall you lose it. A long call with a short put at the same strike does exactly that: at expiry it pays the stock price minus the strike, whichever way the stock moved. So it must cost the same as buying the stock today with money borrowed against the strike, 1,000 minus 970.87, which is Rs 29.13. That identity is put-call parityFor European options on the same stock, strike and expiry, call minus put equals the stock price minus the present value of the strike, less the value of any dividends..
The relationship
$C - P = S - \frac{K}{1 + r} = 1{,}000 - \frac{1{,}000}{1.03} \approx 29.13$
C, P the call and put prices, Rs 60 and Rs 45
S the stock price today, Rs 1,000
K the strike, Rs 1,000
r six-month simple interest, 3%
What it says in wordsA call less a put is worth the stock less the strike discounted to today.
How do you build the trade?
Buy what is cheap and sell what is dear. The option version of the forward, long call and short put, costs Rs 15; the cash version, long stock bought with borrowed money, is worth Rs 29.13, so buy the option version and sell the cash version. Buy the call for 60, sell the put for 45, short the stock for 1,000 and lend 970.87 at 3%. The cash left over today is Rs 14.13, worth about Rs 14.55 at expiry.
At expiry the long call and short put pay the stock price minus Rs 1,000 while the short stock and the maturing Rs 1,000 loan pay Rs 1,000 minus the stock price, so the package nets to zero at every price and the Rs 14.13 collected at the start is kept.
What could stop the arbitrage working?
Say what you assumed. The lock holds only if the options are European, the stock pays no dividend before expiry, and you can short the stock without a borrowing fee that eats the Rs 14. A dividend lowers the forward and so the fair gap; an American put can be exercised against you early; a hard-to-borrow stock charges a fee. A gap this size in a liquid name usually means one of those is present, so your first question back is which one.
Where candidates lose it
The common loss is comparing the call and the put directly, seeing the call is dearer and deciding it must be the one to sell. Parity says the call should be dearer, by about Rs 29; the trade depends on whether the actual gap is too wide or too narrow.
The second is forgetting to discount the strike and calling zero fair, or using 3% of the stock, Rs 30, which is close but wrong. Say the formula before the numbers, then build the four legs.
What the interviewer asks next
The stock will pay a Rs 20 dividend before expiry. Is there still an arbitrage, and which way?
The call is Rs 80 and the put Rs 45. What trade do you put on now?
Why does parity hold exactly for European options but only as a band for American ones?
064An exporter earns Rs 100 of revenue, 60% of it billed in dollars, and has Rs 85 of costs, all paid in rupees. The rupee weakens from 80 to 88 to the dollar. What happens to its operating margin?Valuation, accounting and macro riddlesCoreLong-short equity fundsGlobal macro funds
Try it first
What is the operating margin after the move?
Show the worked solution
The margin rises from 15% to about 19.8%, and operating profit rises 40%. Each dollar now buys 10% more rupees, which turns Rs 60 of dollar revenue into Rs 66, so revenue reaches Rs 106. Costs are all in rupees and stay at Rs 85, so profit climbs from Rs 15 to Rs 21, and 21 over 106 is 19.8%. A 6% rise in revenue becomes a 40% rise in profit.
Why does a weaker rupee help this company so much?
Think of a tutor in Pune who teaches some students abroad for fees in dollars and pays rent and a salary in rupees. When the rupee weakens, each dollar fee converts into more rupees while the rent does not move. A currency move lands on the revenue billed in the foreign currency and on nothing else, so when every cost is local the whole gain drops into profit. Here the gain is Rs 6, on a profit that started at only Rs 15.
Before the move, Rs 40 of rupee revenue and Rs 60 of dollar revenue less Rs 85 of rupee costs leave Rs 15 of profit, a 15% margin; at 88 to the dollar the dollar revenue is worth Rs 66, costs stay at Rs 85 and profit rises to Rs 21, a 19.8% margin.
How do you size the profit move quickly?
Chain three numbers. The dollar buys 10% more rupees, 60% of revenue is exposed, so revenue rises 6%; a thin 15% margin turns that Rs 6 into a 40% rise in profit. The ratio of the profit change to the revenue change, 40 over 6, almost 7 times, is the company's operating leverageHow much operating profit moves for a given move in revenue, high when most costs are fixed or unaffected by the change. to the currency. An analyst on a long-short book uses that multiplier to judge how much of a stock's move a currency view is really driving.
The relationship
$\text{margin} = \frac{40 + 60 \times \tfrac{88}{80} - 85}{40 + 60 \times \tfrac{88}{80}} = \frac{21}{106} \approx 19.8\%$
40 revenue billed in rupees, unchanged
60 x 88/80 dollar revenue restated at the new rate, Rs 66
85 costs, all in rupees, unchanged
What it says in wordsRestate only the dollar-billed revenue at the new rate, hold rupee costs fixed, and divide profit by the new revenue.
What would you check before trusting the new margin?
Three things that often shrink the gain. Hedges, pricing and imported inputs: a company that sold its dollars forward keeps the old rate until the hedges roll off, buyers abroad may push for lower dollar prices, and any imported materials get dearer too. Each one eats into the Rs 6. The 19.8% is the unhedged, first-round answer; naming those three tells the interviewer you would not stop at it.
Where candidates lose it
The first slip is saying the margin is unchanged because revenue and costs both grow. Only the dollar-billed revenue moves; rupee costs stay where they were, and that asymmetry is the whole point.
The second is applying the 10% to all Rs 100 of revenue, which gives Rs 110 and a 22.7% margin. Only 60% of revenue is billed in dollars, so revenue rises 6%, not 10%.
What the interviewer asks next
Half of the costs are imported and paid in dollars. What is the new margin?
The company hedged half its dollar revenue at 80. What margin does it report this year?
What does the same move do to an importer with the mirror-image structure?
070A portfolio is split equally between two assets, each with 15% annual volatility. Their correlation is 0.2 in normal markets and 0.8 in a crisis. What is the portfolio's volatility in each regime?Portfolio and risk mathsCoreMulti-manager platformsQuant and systematic funds
Try it first
What happens to the portfolio's volatility when correlation rises from 0.2 to 0.8?
Show the worked solution
About 11.6% in normal markets and 14.2% in a crisis. With equal weights and equal volatilities, portfolio variance is 15% squared times (1 + rho)/2. At a correlation of 0.2 that is 225 x 0.6 = 135, a volatility of 11.6%; at 0.8 it is 225 x 0.9 = 202.5, a volatility of 14.2%. About 77% of the diversification benefit disappears just when it is needed.
Where does correlation enter the arithmetic?
Picture an umbrella shop and an ice-cream stall owned by one family. On ordinary days one does well when the other is quiet, and the family income is smoother than either shop's. A city-wide power cut shuts both at once. Portfolio variance is each asset's own variance, weighted, plus a co-movement term that carries the correlation, so when correlation jumps the co-movement term grows and the smoothing shrinks. Here the own terms are 56.25 each and the co-movement term is 2 x 0.5 x 0.5 x rho x 225.
The relationship
$\sigma_p^2 = w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2w_1w_2\rho\,\sigma_1\sigma_2 = 225 \times \frac{1+\rho}{2}$
w1, w2 the weights, 0.5 each
sigma1, sigma2 the volatilities, 15% each
rho the correlation between the two assets
What it says in wordsPortfolio variance is the weighted own variances plus a cross term that grows with correlation.
At a correlation of 0.2 the co-movement term adds 22.5 to a variance of 112.5, giving 11.6% volatility; at 0.8 it adds 90.0, giving 14.2%, so the benefit of holding both assets falls from 3.4 points to 0.8.
What are the two numbers?
Normal markets: 112.5 plus 2 x 0.25 x 0.2 x 225 = 22.5 gives 135, and the square root is 11.6%. Crisis: the co-movement term rises to 90, variance to 202.5, and volatility to 14.2%, a 22.5% jump in risk with no change in either asset's own volatility. The benefit of holding two assets instead of one falls from 3.4 points to 0.8. In a real crisis each asset's volatility usually rises too, so this is the milder case.
What would a risk manager do with this?
Stop sizing positions on correlations measured in calm markets. A risk limit set on a normal-market correlation understates crisis risk by about 22% here, before any rise in the assets' own volatility. Platforms that run many books run stressed-correlation scenarios for this reason, and ask each manager what the book looks like if everything moves together. The honest limitation: nobody knows the crisis correlation in advance; 0.8 is an assumption, and the answer should say so.
Where candidates lose it
The common loss is treating correlation as if it scaled volatility directly, or quoting 15% in both regimes because each asset's volatility has not changed. Correlation enters only through the cross term in the variance.
The second is computing both numbers and missing the point of the question: the benefit you were counting on shrinks exactly in the regime where you need it. Say that sentence.
What the interviewer asks next
At what correlation does the portfolio's volatility reach 13%?
With three equally weighted assets, what is the volatility at a correlation of 0.8?
Why might correlations between assets rise in a sell-off?
072An investment earns 10% a year before fees and charges a 2% annual fee. Over 20 years, what share of the investor's ending wealth does the fee consume?Returns, compounding and feesCoreFund of funds and allocatorsMulti-manager platforms
Try it first
Over 20 years, what share of the ending wealth does the 2% fee take?
Show the worked solution
About 31% of the ending wealth. Rs 1 lakh growing at 10% a year becomes Rs 6.73 lakh in 20 years; at 8% after the fee it becomes Rs 4.66 lakh. The fee takes Rs 2.07 lakh, about 31% of what the investor would otherwise have had, although each year it looks like only a fifth of the return.
Why does a fifth of each year's return become almost a third of the wealth?
Think of a mango tree whose fruit you replant. If a neighbour takes one mango in every five each year, you lose those mangoes and every tree they would have grown. A fee taken every year removes money that would itself have compounded for the remaining years, so its cost grows faster than the fee rate suggests. The share lost is 1 minus (1.08/1.10) to the 20th: each year the investor keeps 98.2% of what the gross path would have held, and 0.982 to the 20th is about 0.69.
The relationship
$\text{share lost} = 1 - \left(\frac{1.08}{1.10}\right)^{20} = 1 - \frac{4.66}{6.73} \approx 30.7\%$
1.10 the gross growth factor each year
1.08 the growth factor after the 2% fee
20 the number of years
What it says in wordsThe fraction of wealth the fee takes is one minus the ratio of the two growth paths after 20 years.
Rs 1 lakh grows to Rs 6.73 lakh in 20 years at 10% but only Rs 4.66 lakh at 8% after the fee, a gap of Rs 2.07 lakh that widens every year and ends at about 31% of the gross wealth.
How do you estimate it without a calculator?
Use the rule of 72. At 10% money doubles about every 7.2 years, so 20 years is about 2.8 doublings, near 7 times; at 8% it doubles every 9 years, about 2.2 doublings, near 4.6 times. The exact numbers are 6.73 and 4.66, so the estimate lands within a few per cent, and 4.6 over 6.9 already tells you roughly a third is gone. Say the estimate first, then the exact figure.
What does an allocator take from this?
That fees should be judged against the return they leave, over the holding period, not as a percentage in a single year. A manager charging 2% needs to beat a cheaper alternative by about 2 points a year, every year, just to leave the investor level; over 20 years the difference is 31% of the pot. Hedge fund fees often add a share of profits on top, which widens the gap further. The illustration assumes a steady 10%; real returns vary, but the compounding of the fee does not depend on that.
Where candidates lose it
The two fast answers are 2% and 20%: the fee rate, or the fee as a share of one year's return. Both treat the fee as a one-year cost and ignore that every rupee taken would have compounded for the years that follow.
The second loss is working out 4.66 and 6.73 and then reporting the gap as a share of the smaller number, 44%. The question asks what share of the investor's potential wealth the fee consumed, so divide by the gross figure.
What the interviewer asks next
The fee is 1% instead of 2%. What share of ending wealth does it take over 20 years?
Add a 20% share of profits on top of the 2% fee. Roughly what does the investor keep?
Why do fees matter more for a 30-year retirement saver than for a 3-year investor?
073A thousand fund managers have no skill at all: each has a 50% chance of beating the market in any year, independently. How many will beat it five years running, and what does that say about track records?Statistics and estimationCoreQuant and systematic funds
Try it first
How many of the 1,000 unskilled managers beat the market five years in a row?
Show the worked solution
About 31 managers, 1,000 halved five times. Each year roughly half the unbeaten managers beat the market by luck, so 500 survive year one, 250 year two, then 125, 62.5 and 31.25. A perfect five-year record is something luck hands to about 3 managers in every 100, so in a large crowd it cannot on its own separate skill from chance.
Why does a crowd produce streaks even without skill?
Fill a stadium with a thousand people and ask each to toss a coin five times. Someone will throw five heads, and about 31 will. A result that is rare for one person is almost certain somewhere in a large group, so the question is never whether a flawless record exists but how many you would expect by chance. Each manager's chance is 1 in 32; across 1,000 managers that is 31.25 expected perfect records.
Starting from 1,000 unskilled managers, half fall away each year, leaving 500, 250, 125, 62.5 and finally about 31 with a flawless five-year record produced by chance alone.
The relationship
$E[\text{perfect records}] = 1{,}000 \times \left(\tfrac12\right)^5 = 31.25$
1,000 the number of managers
1/2 each manager's chance of beating the market in a year
5 the number of years
What it says in wordsMultiply the crowd by the chance that one member gets the streak.
If some managers really are skilled, how much does a perfect record tell you?
Suppose 5% of the thousand are skilled and beat the market 60% of the time. They produce about 3.9 perfect records, while the 950 unskilled produce about 29.7, so a manager with five perfect years is skilled only about 12% of the time. A 60% manager has only a 8% chance of five perfect years, so most skilled managers do not have flawless records either. The record is weak evidence in both directions.
What should you look at instead?
Longer records, more decisions per year and a reason. Skill shows up more reliably in many independent decisions than in a handful of annual outcomes, and in a process that explains where the edge comes from. Allocators also check how many managers were in the starting pool, because the funds still reporting are the ones that survived; the ones that were closed after bad years have dropped out of the data. That is survivorship biasThe distortion that comes from studying only the survivors of a process, whose results look better than those of the whole starting group., and it makes every surviving record look stronger than it is.
Where candidates lose it
The first loss is saying none, or very few, because five in a row sounds impressive. The interviewer wants the crowd arithmetic: rare for one, expected for many.
The second is stopping at 31 without the conclusion. The number is only half the answer; say what it means for reading a track record, and name survivorship bias.
What the interviewer asks next
How many of the 1,000 beat the market in at least four of the five years?
How many years of beating the market would one unskilled manager in 1,000 be expected to reach?
How would you design a test that separates a 60% manager from a 50% one?
074A company has a market value of Rs 1,000 crore, earnings of Rs 100 crore, and Rs 300 crore of net cash that earns 6% after tax. What is the P/E of the operating business on its own?Valuation, accounting and macro riddlesCoreLong-short equity fundsGlobal macro funds
Try it first
What is the P/E of the operating business alone?
Show the worked solution
About 8.5x, not 10x. Rs 300 crore of the Rs 1,000 crore market value is cash, so the operating business is valued at Rs 700 crore. The cash earns 6% after tax, Rs 18 crore, so the operating business earns Rs 82 crore. Rs 700 crore over Rs 82 crore is about 8.5 times earnings. The headline 10x blends a cheaper business with cash priced at about 16.7 times its income.
Why is the headline P/E misleading here?
Imagine buying a shop for Rs 10 lakh that has Rs 3 lakh sitting in its till. You are really paying Rs 7 lakh for the shop itself. A company's price and earnings both include its cash, and cash is valued very differently from a business: here it earns 6% after tax, so it is worth about 16.7 times its income. Blending the two gives the headline 10x, which is neither the price of the cash nor the price of the business.
Removing Rs 300 crore of cash from the Rs 1,000 crore market value leaves Rs 700 crore for the operating business, and removing the Rs 18 crore it earns leaves Rs 82 crore of operating earnings, a multiple of about 8.5x against the headline 10x.
How do you work it?
Strip the cash out of both the numerator and the denominator. The price of the operating business is 1,000 minus 300, Rs 700 crore; its earnings are 100 minus 6% of 300, Rs 82 crore; and 700 over 82 is 8.54x. Taking the cash out of only the price gives 7.0x, which is too low because the Rs 100 crore still contains the cash's interest. The check: 700 plus 300 is 1,000 and 82 plus 18 is 100.
The relationship
$\text{P/E}_{\text{ops}} = \frac{1{,}000 - 300}{100 - 0.06 \times 300} = \frac{700}{82} \approx 8.5\times$
1,000 market value, Rs crore
300 net cash, Rs crore
0.06 x 300 the after-tax income the cash earns, Rs 18 crore
What it says in wordsTake the cash out of the price and its income out of the earnings, then divide.
When would you not strip out all the cash?
When the cash is not really free. Cash needed to run the business, cash trapped abroad that would be taxed on the way home, or cash a management team is likely to spend badly may deserve less than full value. A long-short analyst uses the ex-cash multiple to compare this business with peers that hold no cash, and then asks how much of the Rs 300 crore shareholders will actually see. That judgement moves the answer between about 8.5x and the headline 10x.
Where candidates lose it
The first slip is quoting 10x and moving on, which treats the cash as if it were part of the operating business. The interviewer put Rs 300 crore of cash in precisely so that the headline multiple would mislead.
The second is subtracting the cash from the price but leaving its interest in the earnings, which gives 7.0x. Cash must come out of both sides.
What the interviewer asks next
The cash earns only 3% after tax. What is the operating P/E now?
The company holds Rs 300 crore of net debt instead, at 6% after tax. What is the operating P/E?
Why might the market value the company's cash at less than Rs 300 crore?
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
Try it first
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?

s	the half-spread, the distance from the mid to the bid or the offer
0.7	the share of orders from uninformed traders
1 - s	what an informed trade takes: the Rs 1 value gap less the half-spread crossed

40	revenue billed in rupees, unchanged
60 x 88/80	dollar revenue restated at the new rate, Rs 66
85	costs, all in rupees, unchanged

w1, w2	the weights, 0.5 each
sigma1, sigma2	the volatilities, 15% each
rho	the correlation between the two assets

1.10	the gross growth factor each year
1.08	the growth factor after the 2% fee
20	the number of years

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