Hedge Funds puzzles, solved step by step
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020A book holds Rs 60 crore of a stock with 30% volatility and Rs 40 crore of another with 20% volatility, and the two have a correlation of 0.5. What is the book's volatility in rupees, and what share of the risk comes from each position?Man GroupBoston · 2022
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What share of the book's risk comes from the Rs 60 crore position?
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The book's volatility is about Rs 23.1 crore a year, and the Rs 60 crore position carries about 74% of it on 60% of the capital. Stand-alone risks are Rs 18 crore and Rs 8 crore. Book variance is 18 squared plus 8 squared plus 2 x 0.5 x 18 x 8, which is 532, so volatility is Rs 23.07 crore. Each position's contribution is its covariance with the book over the book's volatility: Rs 17.17 crore and Rs 5.90 crore, which add back to the total.
Why is risk not shared out like capital?
Two friends share a taxi. One rides twice as far, straight through the traffic jam; the other gets off after a short hop. Splitting the fare by the number of bags each carries would be absurd. Risk belongs to a position in proportion to how much it moves and how much it moves with everything else, not to how much money sits in it. Here the first stock is larger, more volatile and positively correlated with the second, so it carries far more than its 60% of the capital.
Stand-alone risks of Rs 18 crore and Rs 8 crore add to Rs 26 crore, diversification at a 0.5 correlation removes Rs 2.93 crore, and the book's Rs 23.07 crore of volatility splits 74.4% to the first position and 25.6% to the second, against a 60 to 40 split of capital. The relationshipa, b stand-alone rupee volatilities: 60 x 30% = 18 and 40 x 20% = 8 \rho the correlation, 0.5 RC a position's contribution to book volatility What it says in wordsEach position owns its own variance plus half the shared term, and dividing by the book's volatility turns that into rupees of risk.Position Capital, Rs crore Volatility Stand-alone risk Risk contribution Share of risk A 60 30% 18.0 17.17 74.4% B 40 20% 8.0 5.90 25.6% Book 100 26.0 23.07 100.0% Rs crore of annual volatility. The stand-alone risks add to Rs 26.0 crore, but the book's volatility is Rs 23.07 crore, of which position A contributes 74.4% and position B 25.6%. Why do the contributions add up exactly to the total?
Split the variance. The cross term, 2 x 0.5 x 18 x 8 = 144, is shared equally, 72 to each position. So position A owns 324 + 72 = 396 of the 532 of variance and position B owns 64 + 72 = 136, and dividing each by the book's volatility of 23.07 gives rupee contributions that sum exactly to Rs 23.07 crore. The diversification benefit is the gap between the stand-alone total of Rs 26 crore and the book's Rs 23.07 crore.
What does a risk manager do with the split?
Cut where the risk is, not where the money is. Each rupee in position A carries 28.6 paise of marginal risk against 14.7 paise in position B, so trimming Rs 10 crore from A lowers book volatility by roughly Rs 2.9 crore. Recomputing exactly gives Rs 2.84 crore, close to the estimate. The limitation: the split is a snapshot at one correlation, and when correlations move, both the total and the split move with them.
Where candidates lose it
The quick answer shares risk like capital, 60 and 40, or like stand-alone risk, 18 and 8. The first ignores volatility and the second ignores correlation; neither sums to the book's actual Rs 23 crore of risk.
The second slip is adding the stand-alone risks to get the book's risk, Rs 26 crore. Volatilities do not add unless the correlation is exactly 1; variances do, with the cross term included.
What the interviewer asks next
- If the correlation fell to zero, how would the risk split between the two positions?
- How much of position B would you add to minimise the book's volatility, holding A fixed?
- How do transaction costs change which position you trim first?
Asked at Man Group, Investment Management, Boston, 2022 (Wall Street Oasis):
How do you understand portfolio risk and transaction cost?
060Give an example of two random variables that have zero correlation but are nonetheless completely dependent on each other.Two SigmaNew York · 2025
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X is -1, 0 or 1 with equal chances and Y is X squared. What is the correlation between X and Y?
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Let X be -1, 0 or 1 with equal chances and let Y = X squared. Y is fixed completely once you know X, yet the correlation is exactly zero. The covariance is E[XY] minus E[X]E[Y]; E[X] is 0 and E[XY] is the average of -1, 0 and 1, which is also 0. Correlation measures only straight-line co-movement, and a symmetric U shape has none.
What does correlation actually measure?
Think of a household's electricity bill against the outside temperature. It is high in the coldest months, when the heater runs, high in the hottest, when the air conditioner runs, and low in between. Correlation asks only whether one variable tends to rise along a straight line as the other rises, so a U-shaped link, however tight, can score zero. Over a year balanced around a mild middle, temperature explains the bill almost completely and a straight-line measure misses all of it.
The three equally likely points (-1, 1), (0, 0) and (1, 1) lie exactly on Y = X squared, yet the best straight line through them is flat at 2/3, so the covariance and the correlation are exactly zero. How do you prove the covariance is zero?
Write out the three cases. X averages 0, Y averages 2/3, and XY takes the values -1, 0 and 1, which also average 0, so the covariance E[XY] - E[X]E[Y] is exactly 0. Symmetry does the work: every point to the right of the axis has a mirror image on the left with the same Y, so the upward slope on one side cancels the downward slope on the other. A continuous version works the same way: X normal with mean zero and Y equal to X squared.
The relationshipE[XY] the average of X times Y over the three cases E[X] the average of X, zero by symmetry E[Y] the average of Y, 2/3 What it says in wordsThe covariance is the average product less the product of the averages, and both pieces are zero here.Where does this bite on a desk?
Anywhere a payoff depends on the size of a move rather than its direction. A long straddleA call and a put bought at the same strike and expiry, which gains from a large move in either direction. gains from a big move either way, so over moves balanced around zero its return shows little correlation with the stock's return while being driven entirely by it. A risk report built only on correlations would call that position unrelated to the stock. Zero correlation means no straight-line link; only independence means no link at all.
Where candidates lose it
Some candidates reach for two variables that simply look unrelated, which misses the point: the question asks for complete dependence alongside zero correlation. Others say that zero correlation means independence, which is the exact confusion the question exists to catch.
Give the three-point example, compute the covariance out loud, and then name one place on a desk where the difference matters.
What the interviewer asks next
- If X is uniform on 0 to 1 instead, are X and X squared still uncorrelated?
- Are independent variables always uncorrelated? Prove it in one line.
- For which joint distribution does zero correlation imply independence?
Asked at Two Sigma, Generalist, New York, 2025 (Wall Street Oasis):
Come up with two uncorrelated but dependent variables.
085A book's one-day 99% VaR is Rs 10 crore. What is the ten-day 99% VaR under the usual scaling rule, and what has to be true for that rule to hold?AQR Capital ManagementGreenwich · 2022
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What is the ten-day 99% VaR?
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About Rs 31.6 crore: Rs 10 crore times the square root of 10. If daily P&L is independent from day to day, with the same volatility and a mean near zero, variances add, so ten-day volatility is root 10 times daily volatility, and a normal quantile scales the same way. The rule also needs the positions held unchanged for ten days and a distribution that keeps its shape over the horizon.
Why not ten times the one-day number?
Picture ten friends each tossing a coin for Rs 100. The worst case is the group losing Rs 1,000, but the typical spread of the group's total is nowhere near ten times one person's, because some win while others lose. Independent daily moves partly cancel, so their variances add while their volatilities do not, and volatility grows with the square root of the number of days. Ten times would need every bad day to line up in the same direction, which is exactly what independence rules out. Value at riskThe loss a book should not exceed over a set horizon at a set confidence level, for example one day at 99%. inherits that square root when the distribution is normal.
Starting from Rs 10 crore for one day, a straight line reaches Rs 100 crore at ten days only if every bad day lines up, while the square-root curve for independent days reaches Rs 31.6 crore. The relationshipVaR_1 the one-day 99% VaR, Rs 10 crore sqrt(10) the growth in volatility over ten independent days What it says in wordsOver ten independent days the spread of P&L grows by the square root of ten, and so does a normal VaR.What has to be true for the rule to hold?
List the assumptions, because that is the real question. Returns must be independent across days, volatility constant, the mean close to zero, the positions unchanged, and the distribution one that keeps its shape when summed, as the normal does. Break any one and the rule drifts. Positive autocorrelation, where bad days follow bad days, makes the true ten-day number larger. A book that is cut after losses makes it smaller. Fat tails make the one-day 99% quantile a poor guide to the ten-day one.
Which way does the error usually run?
In a calm market the rule is a fair approximation. In stress it tends to understate, because volatility rises and losses cluster just when the ten-day horizon matters. The square root of time is a scaling convenience, not a law, so a risk team checks it against ten-day P&L measured directly. The same assumption sits behind the desk habit of multiplying daily volatility by 16 to get an annual figure, 16 being roughly the square root of the trading days in a year; stretch it to 250 days here and you get Rs 158 crore, a number few would trust.
Where candidates lose it
Rs 100 crore is the reflex answer, adding ten daily VaRs as if every day were the worst day. The interviewer is testing whether you know that independent risks add in variance.
The second loss is giving Rs 31.6 crore and stopping. The question asks what must be true; independence, constant volatility, unchanged positions and a stable distribution are the answer the interviewer is listening for.
What the interviewer asks next
- Daily returns have positive autocorrelation. Is the true ten-day VaR above or below Rs 31.6 crore?
- Scale the one-day figure to 250 trading days. What do you get, and would you trust it?
- Why does square-root scaling work poorly for a book that is long deep out-of-the-money options?
Asked at AQR Capital Management, Quantitative Research, Greenwich, 2022 (Wall Street Oasis):
Specific statistics questions on financial concepts. daily vs monthly return, VAR
095Daily returns have an autocorrelation of 0.1 at a one-day lag, decaying geometrically: 0.01 at two days, 0.001 at three, and so on. By how much is the true volatility over a 21-day month higher than daily volatility times the square root of 21?AQR Capital ManagementGreenwich · 2022
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How much higher is the true monthly volatility?
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About 10% higher. The variance of a sum of 21 daily returns is 21 daily variances plus twice every covariance between pairs of days. With autocorrelation 0.1 at lag one, 0.01 at lag two and so on, the covariance terms add about 4.42 daily variances, lifting the total from 21 to about 25.4, a 21% rise. Volatility, the square root, rises by about 10%.
Why does a small autocorrelation matter over a month?
Think of a queue where each person who joins makes it slightly more likely that the next person joins too. Each nudge is small, but over a month of days they add up to longer queues than pure chance would give. The variance of a sum counts every pair of days, and there are 20 neighbouring pairs in a 21-day month, each counted twice, so a lag-one correlation of 0.1 alone adds 4 daily variances to the 21.
The square-root rule counts 21 daily variances for a month, but one-day lags add 4.0, two-day lags 0.38 and longer lags 0.04, for 25.42, so monthly volatility is 5.04 daily volatilities rather than 4.58, about 10% higher. How do you do the sum?
Write the monthly variance as the daily variance times 21, plus twice the sum over every lag. Lag one appears in 20 pairs with correlation 0.1, lag two in 19 pairs with 0.01, lag three in 18 with 0.001: twice (2.0 + 0.19 + 0.018 and so on) is about 4.42. The total is 25.42 daily variances instead of 21, a ratio of 1.210, and the volatility ratio is its square root, 1.100. Over a long horizon the ratio tends to (1 + 0.1)/(1 - 0.1) = 1.222, so a month is already close to the limit.
The relationshipsigma^2 the variance of one day's return n days in the month, 21 rho_k the autocorrelation at a lag of k days, 0.1 to the power k n - k how many pairs of days in the month are k days apart What it says in wordsA month's variance is the sum of the daily variances plus twice every covariance between pairs of days, weighted by how many such pairs the month holds.What does this mean for risk and performance numbers?
Scaling daily volatility by the square root of time silently assumes zero autocorrelation. Positive autocorrelation, which shows up in trend-following returns and in portfolios of thinly traded assets priced from stale quotes, makes the square-root rule understate longer-horizon risk. Negative autocorrelation, typical of mean-reverting strategies, does the reverse: at minus 0.1 the monthly variance is only 0.83 times the rule's. The same effect is why correlations between two assets measured on daily returns can differ from those measured on monthly returns: prices that react to the same news on different days look less related day by day than month by month.
Where candidates lose it
Candidates either dismiss 0.1 as too small to matter or report 21% as the answer. The first ignores that the correlation enters twenty times over; the second forgets that volatility is the square root of variance.
The other loss is refusing to estimate without doing every lag. The lag-one term alone gives 21 + 4.0, close to the full 25.42; say that the higher lags add about 0.42 and that you checked.
What the interviewer asks next
- What is the ratio if the lag-one autocorrelation is minus 0.1 instead?
- A fund holds thinly traded assets priced from stale quotes. Which way does its reported volatility err?
- Why can the correlation between two assets look higher on monthly returns than on daily returns?
Asked at AQR Capital Management, Quantitative Research, Greenwich, 2022 (Wall Street Oasis):
describe what covariance means what would be the difference between the correlation of daily vs monthly returns of a given year
