Hedge Funds puzzles, solved step by step

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080A strategy trades 200 stocks, each with an average daily value traded of Rs 50 crore. It may take at most 5% of any day's volume in a stock, and it turns its book over 10 times a year across 250 trading days. Roughly how much capital can it run?Estimation and mental mathsHardMulti-manager platformsLong-short equity funds

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What is the capacity, to the nearest round number?

Show the worked solution

About Rs 12,500 crore. Each stock allows 5% of Rs 50 crore, Rs 2.5 crore a day. Across 200 stocks that is Rs 500 crore a day, and across 250 days Rs 1,25,000 crore a year. A book turned over 10 times a year needs 10 rupees of trading for every rupee of capital, so the book can be Rs 1,25,000 crore divided by 10. Treat it as a ceiling, not a working size.

What does capacity actually mean?

Picture a juice stall that can squeeze 500 glasses a day. How many regular customers can it serve? That depends on how often each one comes. If every regular drinks a glass a day, 500; if each comes once in ten days, 5,000. A strategy's capacity is the capital it can run before its own trading exceeds what the market will absorb, so it is tradable volume divided by how often the book must be traded.

Rs 50 crore of daily value in one stock, at a 5% participation limit, across 200 stocks and 250 days, allows Rs 1,25,000 crore of trading a year, which supports a Rs 12,500 crore book at a turnover of 10; the same volume supports Rs 62,500 crore at 2x and only Rs 2,500 crore at 50x.

How do you build the number out loud in the room?

Go one multiplication at a time and say each one. 5% of Rs 50 crore is Rs 2.5 crore per stock per day. Two hundred stocks make Rs 500 crore a day. Two hundred and fifty days make Rs 1,25,000 crore a year. Dividing by a turnover of 10 gives Rs 12,500 crore, and the step people drop is the last one: capacity is not a day's or a year's volume.

The relationship

K = \frac{N \times \text{ADV} \times p \times D}{\tau} = \frac{200 \times 50 \times 0.05 \times 250}{10} = 12{,}500

N	number of stocks traded, 200
ADV	average daily value traded per stock, Rs 50 crore
p	the most of a day's volume the strategy may take, 5%
D	trading days a year, 250
tau	turnover, how many times a year the book is traded, 10

What it says in wordsCapacity is what the market lets you trade in a year, divided by how many times a year you need to trade your book.

Why is turnover the lever that matters most?

Every input enters in proportion, but the market and the risk team set the others, while the strategy itself sets its turnover, and turnover varies far more across strategies. A strategy that turns over 50 times a year has a capacity of Rs 2,500 crore on this universe; one that turns over twice a year has Rs 62,500 crore. That is why fast strategies close to new money early and slow ones can run large books.

Why is the real capacity lower than this ceiling?

The estimate assumes the strategy trades evenly every day and in every stock. It does not. Trades cluster when signals fire, which is often when others trade too, and the thinner names in the list hit the 5% limit long before the larger ones. The definition of turnover matters as well: if 10 times means buying 10 times the book and also selling it 10 times, the traded value doubles and capacity halves to Rs 6,250 crore. Say which definition you used, and add that trading costs rise before the hard limit, so returns fade well before the ceiling.

Where candidates lose it

The usual loss is stopping at Rs 500 crore a day or Rs 1,25,000 crore a year and calling that capacity. Both measure how much can be traded, not how much capital that trading can support, and the interviewer is waiting for the division by turnover.

The second loss is presenting Rs 12,500 crore as an exact answer. It is a ceiling built on even trading and one definition of turnover; saying so is what makes the estimate believable.

What the interviewer asks next

Half of the 200 names trade only Rs 10 crore a day. What is the capacity now?
How would you define turnover so the estimate is not off by a factor of two?
At what size would you expect returns to start fading, and why before the ceiling?

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?
092A casino offers the St Petersburg game: a fair coin is tossed until the first tail, and if that takes n tosses you are paid Rs 2 to the power n. The casino can pay out at most Rs 1 crore. What is a fair price to play?Expected value and dice gamesHardQuant and systematic fundsProp and quant trading firms
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Roughly what is the capped game worth?
Show the worked solution
About Rs 24.19. Round n pays Rs 2^n with probability 1/2^n, so each round adds exactly Rs 1 of expected value. That holds up to n = 23, since 2^23 is about Rs 84 lakh and 2^24 is above the Rs 1 crore cap. From round 24 on, the payout is stuck at Rs 1 crore, with total probability 1/2^23, which adds about Rs 1.19. So 23 + 1.19, about Rs 24.
Why is the uncapped game worth an infinite amount?
Each round is a doubling bet: the payout doubles while the chance of reaching it halves. Think of a raffle where a ticket twice as valuable is half as likely to win: every prize tier is worth the same to you. Round n pays 2^n with probability 1/2^n, so every round contributes exactly Rs 1 to the expected value, and there are infinitely many rounds. That is the famous paradox: few people would pay even Rs 100, yet the expected value has no bound. The resolution that matters on a desk is not psychology; it is that nobody can pay out an unlimited amount.
Rounds 1 to 23 each add exactly Rs 1 to the expected value, and once the Rs 1 crore cap binds the later rounds add 0.60, 0.30, 0.15 and so on, Rs 1.19 in all, so the capped game is worth about Rs 24.19.
How does the cap change the sum?
Find the round where the cap starts to bind. 2^23 is Rs 83,88,608, under Rs 1 crore; 2^24 is Rs 1,67,77,216, over it. So rounds 1 to 23 each add Rs 1, and every round from 24 on pays the capped Rs 1 crore, which together happen with probability 1/2^23 and add 1,00,00,000 / 83,88,608, about Rs 1.19. The fair price is about Rs 24.19. Nearly all of the textbook infinity lives in outcomes the casino cannot pay.
The relationship
$E = \sum_{n=1}^{23} \frac{2^n}{2^n} + \frac{10^7}{2^{23}} = 23 + 1.19 = 24.19$
2^n the payout if the first tail arrives on toss n
1/2^n the chance the first tail arrives on toss n
10^7 / 2^23 the capped Rs 1 crore times the chance of reaching round 24 or later
What it says in wordsEvery uncapped round is worth one rupee; the capped tail is worth the cap times the chance of getting that far.
What does a bigger casino buy you?
Very little. The value grows only with the logarithm of the cap: each doubling of the casino's bankroll adds about Rs 1. A cap of Rs 1,000 crore, a thousand times larger, lifts the fair price only to about Rs 34. That is the lesson a risk manager takes away: a payoff whose expected value rests on rare, enormous outcomes is worth what the other side can actually pay, and any estimate built on the tail should be checked against who stands behind it.
Where candidates lose it
Answering infinity is the trap for anyone who knows the textbook game. The interviewer added the cap to see whether you can find where it binds and redo the sum, not recite the paradox.
The other loss is dropping the tail beyond the cap and saying Rs 23, or valuing it crudely at the full Rs 1 crore times a guessed chance. The tail is a clean sum: probability 1/2^23 of receiving Rs 1 crore.
What the interviewer asks next
The cap rises to Rs 1,000 crore. What is the fair price now?
With logarithmic utility and wealth of Rs 1 lakh, roughly what would you pay for the uncapped game?
How is a book short deep out-of-the-money options like the casino in this game?
094You buy an at-the-money option at 20% implied volatility and delta-hedge it. Over its life the stock realises 30% volatility. Where does your P&L come from, and does the direction of the moves matter?Options and payoffsHardVolatility and relative value fundsProp and quant trading firms
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What drives the hedged position's P&L?
Show the worked solution
From gamma: the hedged option earns on the size of moves, not their direction, and pays theta for the privilege. Each day the P&L is about half gamma times the squared move, minus theta. Theta is priced for 20% volatility, so a daily move of about 1.26% breaks even. At 30%, typical moves are 1.89%, and the squared gain is 2.25 times the decay. Up or down makes no difference; the path still does.
What is left after the delta hedge?
Think of a street vendor who sells umbrellas and sunglasses from the same cart: she no longer cares whether it rains or shines, only whether the weather changes enough to bring people out. A delta hedge sells enough stock to cancel the option's first-order bet on direction, so what remains is the curvature: the option gains a little more on the way up than the hedge loses, and loses a little less on the way down than the hedge gains. That curvature is gamma, and it pays on moves either way.
The option's value curves above the straight delta hedge line on both sides of Rs 1,000, so a move either way earns the shaded gap; a day at 30% realised volatility earns about Rs 0.71 of it per option against Rs 0.32 of time decay priced at 20%.
How big is the daily P&L in numbers?
Take a stock at Rs 1,000 and a three-month at-the-money call priced at 20% volatility, with zero interest rates. Its gamma is about 0.0040 per rupee and its time decay about Rs 0.32 a day over 252 trading days. A day's move of 1.89%, typical of 30% volatility, is Rs 18.9, and half of 0.0040 times 18.9 squared is a gain of Rs 0.71, against Rs 0.32 of decay: about Rs 0.40 a day per option. The decay is exactly what a move of 1.26%, typical of 20% volatility, would pay back.
The relationship
$\text{P\&L}_{\text{day}} \approx \tfrac{1}{2}\,\Gamma\,(\Delta S)^2 - \Theta = \tfrac{1}{2}\,\Gamma S^2\left(\sigma_r^2 - \sigma_i^2\right)\Delta t$
Gamma how fast the option's delta changes with the stock price
Delta S the day's move in the stock price
Theta the option's daily time decay
sigma_r, sigma_i realised and implied volatility, 30% and 20%
What it says in wordsEach day a hedged long option earns half its gamma times the squared move and pays its time decay; on average that is the gap between realised and implied variance.
Why does direction not matter, but the path does?
The move enters squared, so plus Rs 18.9 and minus Rs 18.9 earn the same. The P&L depends on realised volatility against implied, not on where the stock ends up. But gamma is largest near the strike and shrinks as the stock drifts away, so 30% realised in a trend that carries the stock far from the strike early earns less than 30% realised while the stock chops around the strike. Hedging frequency matters too: hedge rarely and the P&L becomes noisy, even if its average barely changes.
Where candidates lose it
The common loss is saying the option made money because the stock went up. With a delta hedge in place, direction has been sold away; a candidate who talks about direction has not understood what the hedge does.
The second loss is saying a hedged position has no P&L. It has exactly one exposure left, realised against implied volatility, and naming it is the whole answer.
What the interviewer asks next
Realised volatility comes in at 15% instead. What happens to the P&L, and why?
Why is the same realised volatility worth more while the stock stays near the strike?
You think implied volatility is too low but have no view on direction. What position expresses that?
096You roll a fair die repeatedly and keep a running total. What is the probability that the total is ever exactly 10? What does the answer approach for large targets?Random walks and Markov chainsHardQuant and systematic fundsProp and quant trading firms
Try it first
Roughly what is the chance the running total ever hits exactly 10?
Show the worked solution
About 0.289, and it settles at 2/7, about 0.286, for large targets. Let p(n) be the chance the total ever equals n. To hit n, the total must first land on one of n - 1 to n - 6 and then roll exactly the gap, each with chance 1/6, so p(n) is the average of the six values before it, with p(0) = 1. Working up gives p(10) = 0.2893. Totals advance 3.5 a roll on average, so they land on 1 number in 3.5.
How do you set up the recursion?
Think about the last roll before the total reaches n. The total can land exactly on n only by first landing on one of n - 1 down to n - 6 and then rolling exactly the gap, each with chance 1/6, so p(n) = (1/6)[p(n - 1) + ... + p(n - 6)]. Start with p(0) = 1, because you begin at zero, and p of any negative number = 0. Then p(1) = 1/6, p(2) = 7/36, p(3) = 0.227, and so on up to p(10) = 0.2893.
The chance of ever landing on a total climbs from 1/6 at 1 to a peak of 0.360 at 6, then wobbles and settles onto 2/7, about 0.286; the target of 10 is hit with probability 0.289.
Why does the answer settle at 2/7?
Picture stepping stones across a river, where each stride covers 1 to 6 stones with equal chance. Over a long walk you touch about one stone in every 3.5, because that is your average stride. The running total advances 3.5 per roll on average, so in the long run it lands on a fraction 1/3.5 = 2/7 of all numbers, and each far-off target is hit with probability close to 2/7. The early values wobble: p(6) is the highest, 0.360, because 6 is the last total a single roll from zero can reach directly, and the wobbles die out by about 20.
The relationship
$p(n) = \frac{1}{6}\sum_{k=1}^{6} p(n-k), \quad p(0) = 1 \qquad \lim_{n\to\infty} p(n) = \frac{1}{E[\text{roll}]} = \frac{1}{3.5} = \frac{2}{7}$
p(n) the chance the running total ever equals n
p(n - k) the chance of standing k below the target, one roll away
E[roll] the average roll of a fair die, 3.5
What it says in wordsEach total's chance is the average of the six before it, and in the long run the totals land on one number in every 3.5.
Why would a quant interviewer ask for a table or code here?
Because the recursion is dynamic programmingSolving a problem by building up answers to smaller versions of it and reusing them, instead of recomputing from scratch.: each value reuses the six before it, so a table of ten numbers is faster and safer than listing every sequence of rolls that sums to 10. Say the recursion, compute a few terms out loud, and give the limit with its reason; that is the complete answer. Do not try to enumerate paths: there are 492 ordered ways to reach 10 with rolls of 1 to 6, each with its own probability.
Where candidates lose it
The quick answers are 1/6, reasoning that some roll must land on 10 with one chance in six, and 2/7 stated as exact. The first ignores that most runs skip straight over 10; the second is close but is the long-run limit, and 10 is not yet far enough out for it to be exact.
The second loss is trying to count paths. Set up the recursion in one line instead and let it do the counting.
What the interviewer asks next
What is the probability that the running total ever equals exactly 6?
With a coin that adds 1 or 2 instead of a die, what does the hit probability approach?
How would you write this as a dynamic programme in a few lines of code?
097A fund makes 50% in year one on Rs 100 crore, then takes in Rs 400 crore of new money and loses 20% in year two. What are its time-weighted and money-weighted returns, and did its investors make money?Returns, compounding and feesHardFund of funds and allocatorsMulti-manager platforms
Try it first
The fund reports its two-year return. Which statement is right?
Show the worked solution
The time-weighted return is +20%; the money-weighted return is about -10.2% a year, and investors as a group lost Rs 60 crore. Rs 100 crore grows to Rs 150 crore, then Rs 400 crore arrives and the Rs 550 crore falls 20% to Rs 440 crore. Chaining 1.5 x 0.8 gives +20%, the manager's record. But Rs 500 crore went in and Rs 440 crore remains, because most of the money arrived just before the loss.
What does each measure answer?
Think of a restaurant that gets a glowing review after a good year, triples its tables, and then has a poor year. Its record over the two years looks fine, but most of the diners ate in the poor year. The time-weighted return measures the manager, chaining each period's return so the timing of money coming in or out does not count; the money-weighted return measures the investors, weighting each period by the money actually exposed to it.
The fund earns 50% on Rs 100 crore, takes in Rs 400 crore, then loses 20% of Rs 550 crore, so the manager's time-weighted record is +20% while investors put in Rs 500 crore, hold Rs 440 crore, and earned about -10.2% a year.
How do you compute the money-weighted return?
Treat the investors' money as a set of cash flows and find the single annual rate that links them: Rs 100 crore in at the start, Rs 400 crore in after a year, Rs 440 crore out at the end. Solve 100(1 + r) squared + 400(1 + r) = 440; the root is 1 + r = 0.8983, so the money-weighted return is about -10.2% a year. It is negative because most of the money sat through the losing year: Rs 550 crore lost 20%, Rs 110 crore, while the good year earned only Rs 50 crore on Rs 100 crore.
The relationship
$100(1+r)^2 + 400(1+r) = 440 \;\Rightarrow\; 1 + r = \frac{-4 + \sqrt{33.6}}{2} = 0.898$
100 Rs crore invested at the start
400 Rs crore added after one year
440 Rs crore the investors hold at the end
r the money-weighted return, the internal rate of return on those flows
What it says in wordsThe money-weighted return is the one rate that grows every rupee invested, from the day it arrived, into what the investors hold at the end.
Which number should an allocator look at?
Both, for different questions. To judge the manager's skill, use the time-weighted +20%, because the manager did not choose when investors arrived. To judge whether the fund was good for the people in it, or whether money chased performance, use the money-weighted figure. A wide gap between the two, as here, is a warning about hot money: investors piling in after a strong year and bearing the next year's loss on a much bigger base. Had all Rs 500 crore arrived at the start, both measures would agree and the investors would hold Rs 600 crore.
Where candidates lose it
The common loss is reporting +20% and saying the investors made money. The chained return deliberately ignores that most rupees arrived just before the loss, and the interviewer built the numbers so the two measures point in opposite directions.
The second loss is averaging the two yearly returns, +50% and -20%, to get +15% a year. That is neither measure: the time-weighted annual figure is the square root of 1.2 minus 1, about 9.5% a year.
What the interviewer asks next
What is the time-weighted return expressed as an annual rate?
The Rs 400 crore had arrived at the start of year one instead. What are both returns now?
Why do fund fact sheets report time-weighted returns?

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

2^n	the payout if the first tail arrives on toss n
1/2^n	the chance the first tail arrives on toss n
10^7 / 2^23	the capped Rs 1 crore times the chance of reaching round 24 or later

Gamma	how fast the option's delta changes with the stock price
Delta S	the day's move in the stock price
Theta	the option's daily time decay
sigma_r, sigma_i	realised and implied volatility, 30% and 20%

p(n)	the chance the running total ever equals n
p(n - k)	the chance of standing k below the target, one roll away
E[roll]	the average roll of a fair die, 3.5

Try it first

What does capacity actually mean?

How do you build the number out loud in the room?

Why is turnover the lever that matters most?

Why is the real capacity lower than this ceiling?

Where candidates lose it

What the interviewer asks next

Try it first

Why switch from probabilities to odds?

How do the three calls combine?

What assumption carries all of this?

Where candidates lose it

What the interviewer asks next

Try it first

Why is the uncapped game worth an infinite amount?

How does the cap change the sum?

What does a bigger casino buy you?

Where candidates lose it

What the interviewer asks next

Try it first

What is left after the delta hedge?

How big is the daily P&L in numbers?

Why does direction not matter, but the path does?

Where candidates lose it

What the interviewer asks next

Try it first

How do you set up the recursion?

Why does the answer settle at 2/7?

Why would a quant interviewer ask for a table or code here?

Where candidates lose it

What the interviewer asks next

Try it first

What does each measure answer?

How do you compute the money-weighted return?

Which number should an allocator look at?

Where candidates lose it

What the interviewer asks next