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060Give an example of two random variables that have zero correlation but are nonetheless completely dependent on each other.Two SigmaNew York · 2025
Try it first
X is -1, 0 or 1 with equal chances and Y is X squared. What is the correlation between X and Y?
Show the worked solution
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
Try it first
What is the ten-day 99% VaR?
Show the worked solution
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
