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045Two stock markets close at different times of day. Their same-day daily returns correlate at 0.30, and market B's return today correlates at 0.25 with market A's return yesterday. All other lagged correlations are zero, neither market's returns are autocorrelated, and both have the same daily variance. Roughly what is the correlation of their monthly returns?AQR Capital ManagementGreenwich · 2022
Try it first
Is the monthly correlation higher, lower or the same as the daily 0.30?
Show the worked solution
About 0.54, close to 0.55. Add up 21 daily returns in each market. Each monthly variance is 21 times the daily variance. The monthly covariance collects 21 same-day terms of 0.30 and 20 lagged terms of 0.25, where A's day falls inside the same month as B's next day. So the correlation is (21 x 0.30 + 20 x 0.25) over 21, which is 0.538, tending to 0.55 for long windows.
Why would daily data understate how much two markets move together?
Two friends in different cities hear the same news; one hears it at lunch, the other the next morning. Compare their moods hour by hour on the same day and they look unrelated; compare their moods over the whole week and they look very similar. When two markets close at different times, news that arrives between the two closes shows up in one market's return today and the other's tomorrow, so same-day daily correlation misses part of the true co-movement. Longer windows add the delayed part back in.
News that arrives after market B has closed moves market A the same day and market B only the next day, which creates the 0.25 lagged correlation. Summing 21 days captures both the same-day 0.30 and 20 of the lagged pairs, so the monthly correlation is about 0.54. How do you add up the covariance terms?
Write each month's return as the sum of its daily returns. The covariance of two sums is the sum of every pairwise covariance. With a unit daily variance, same-day pairs contribute 21 x 0.30, and the pairs of B's day t with A's day t minus 1 contribute 20 x 0.25, since only 20 such pairs sit inside a 21-day month. The variances are 21 each, with no autocorrelation to add. So the monthly correlation is (6.3 + 5.0) over 21, which is 0.538.
The relationshipN trading days in the month, 21 \rho_0 the same-day daily correlation, 0.30 \rho_1 B today against A yesterday, 0.25 \rho_{M} the monthly correlation What it says in wordsOver a month, the same-day and the one-day-lagged co-movement both count, less one lagged pair lost at the month's edge.This matters for portfolio construction. A risk model built on daily same-day correlations between markets in different time zones will think they diversify each other far more than they do over the horizon an investor actually holds. The standard fixes are to use weekly or monthly returns, or to add lagged terms to the daily covariance, as this calculation does. The limitation: with only 12 monthly observations a year, the monthly estimate is itself noisy.
Where candidates lose it
Candidates often say correlation is a property of the two assets and does not change with the horizon, or that monthly data is noisier and so correlations must be lower. Both miss the timing effect, which is the entire question.
Draw the two closing times, say where the news lands, then add up the covariance terms. The interviewer wants to hear that you know lagged cross-correlation, and that you counted 20 lagged pairs rather than 21.
What the interviewer asks next
- How would you estimate the lagged correlation from daily data?
- If market B's returns were also positively autocorrelated, would the monthly correlation go up or down?
- Why do daily correlations between Asian and US markets look lower than weekly ones?
Asked at AQR Capital Management, Quantitative Research, Greenwich, 2022 (Wall Street Oasis):
what would be the difference between the correlation of daily vs monthly returns of a given year
061Forecaster A predicts next year's index return with a bias of plus 1 point and an error standard deviation of 3 points. Forecaster B is unbiased, with an error standard deviation of 3.5 points. Whose mean squared error is lower, and what does an equal blend of the two give if their errors are independent?Bridgewater AssociatesNew York · 2024
Try it first
Which forecaster has the lower mean squared error?
Show the worked solution
A has the lower error, 10 against 12.25, and an equal blend cuts it to about 5.6. Mean squared error is bias squared plus variance: A is 1 plus 9, B is 0 plus 12.25. Averaging the two halves the bias to 0.5 and, because the errors are independent, quarters each variance: 0.25 plus 5.31, or 5.56. The blend beats both forecasters by a wide margin.
How can a biased forecaster beat an unbiased one?
Think of two watches. One always runs a minute fast but is otherwise steady; the other is right on average but wanders a few minutes either way. If you need to catch a train, the steady fast watch may serve you better. Mean squared error charges for two things, the average miss and the scatter around it, so a small steady bias can cost less than extra scatter. A's bias of 1 adds 1 to its error. B's standard deviation of 3.5 instead of 3 adds 3.25. On this measure, A is the better forecaster.
The relationshipbias the forecaster's average error, in points of return sigma the standard deviation of the error around that average What it says in wordsThe average squared miss is the square of the average miss plus the scatter of the misses.Why does averaging the two help so much?
Because independent errors partly cancel. When A is too high, B is as likely to be too low as too high, so the average of the two misses by less than either. An equal blend halves the bias and, with independent errors, cuts the variance to a quarter of the sum of the two variances. Here that is a quarter of 9 plus 12.25, which is 5.31, plus a bias term of 0.5 squared, 0.25. The total, 5.56, is little more than half of A's 10.
Forecaster A's error of 10 is 1 of bias squared plus 9 of variance, B's 12.25 is all variance, and an equal blend with independent errors cuts the total to 5.56 because averaging halves the bias and quarters each variance. You can do slightly better by leaning towards A. The weight on A that minimises the error is B's variance over the sum of A's mean squared error and B's variance, about 55%, which gives 5.51. The gain over a plain 50/50 is tiny, which is the practical lesson: a simple average of decent, independent forecasts captures almost all of the benefit. The limitation is the word independent. Two economists reading the same data make correlated errors, and then averaging helps far less.
Where candidates lose it
The trap is choosing B on principle because unbiased sounds better. The question asks about mean squared error, and a candidate who does not split it into bias squared plus variance cannot compare the two.
The second loss is averaging the standard deviations for the blend, 3.25, and squaring it. Variances add, not standard deviations, and the blend divides each variance by four. Say independent out loud, because that assumption is doing the work.
What the interviewer asks next
- What if the two forecasters' errors are correlated 0.8? What does the blend give then?
- What weight on A minimises the blend's error, and why is it close to one half?
- Why might a portfolio manager prefer the biased forecaster even though the blend is better?
Asked at Bridgewater Associates, Generalist, New York, 2024 (Wall Street Oasis):
Was asked math questions about forecasting
074An analyst takes ten years of monthly data, builds rolling 12-month fund and market returns sampled every month, regresses one on the other, and reports an alpha with a t-statistic of 3.6 from ordinary least squares. Why is that overstated, and roughly what is an honest t-statistic?AQR Capital ManagementGreenwich · 2022AQR Capital ManagementGreenwich · 2022
Try it first
Roughly how large is the honest t-statistic?
Show the worked solution
Because the observations overlap, the honest t-statistic is close to 1, about 1.04. Ten years of monthly data give 109 rolling 12-month windows, but each shares 11 months with its neighbour. OLS assumes independent errors, so it treats 109 observations as 109 pieces of information. With the overlap, the variance of the estimate is understated about twelvefold, and 3.6 over the square root of 12 is about 1.0.
Which OLS assumption breaks?
Picture asking the same twelve people for their opinion, then swapping one person each month and asking again. You would not claim 109 independent surveys. OLS standard errors assume the regression errors are uncorrelated with each other, and overlapping windows make them strongly correlated, so the standard errors come out far too small. The alpha estimate itself is not biased; what is wrong is the claim about how precisely it is known, which is exactly what the t-statistic reports.
Two 12-month windows one month apart share 11 months, so their correlation is 11 over 12 and falls in equal steps to 1 over 12 at an 11-month gap, which inflates the true variance of the estimate about 12-fold. How do you get to about 1?
Add up the overlap. If monthly returns are independent, two 12-month windows k months apart have a correlation of 12 minus k over 12. The variance of an average of such overlapping sums is inflated by one plus twice the sum of those correlations, which here is 1 plus 2 times 5.5, or 12. The t-statistic shrinks by the square root of the inflation factor, 3.6 divided by the square root of 12. That gives about 1.04: nowhere near significant, and consistent with having only about ten independent years.
The relationshipVIF variance inflation from the overlap (12-k)/12 the correlation of two windows k months apart t* the corrected t-statistic What it says in wordsOverlap inflates the true variance about twelvefold, so divide the naive t-statistic by the square root of twelve.How would you fix it? Three answers, in order of simplicity. Run the regression on non-overlapping monthly returns, which uses all the data with independent errors. Or keep the overlap and use standard errors that allow for autocorrelation, such as Newey-West or Hansen-Hodrick with 11 lags. Or sample the 12-month returns once a year, which leaves only ten points but honest ones. The interviewer wants to hear that the fix is about the standard errors, not the coefficient.
Where candidates lose it
The trap is defending 3.6 because the sample has over a hundred observations. The count of rows is not the count of independent pieces of information, and that is the whole point of the question.
The second loss is saying the alpha estimate itself is biased. It is not; the problem is its precision. Name the broken assumption, uncorrelated errors, and the fix, corrected standard errors or non-overlapping data.
What the interviewer asks next
- What other OLS assumptions matter most for return regressions, and how would you check them?
- Why do overlapping returns make long-horizon predictability look stronger than it is?
- How many lags would you use in Newey-West standard errors here, and why?
Asked at AQR Capital Management, Investments, Greenwich, 2022 (Wall Street Oasis):
they asked about how to fix ols assumptions
Asked at AQR Capital Management, Investments, Greenwich, 2022 (Wall Street Oasis):How would you fix violations of the OLS assumptions?
095A fund returned 17% in a year when the risk-free rate was 6%, the market's excess return 8%, the size factor SMB 2% and the value factor HML 3%. Its loadings are 1.1 on the market, 0.4 on size and minus 0.3 on value. How much of the return do its factor exposures explain, and what is its alpha?State StreetCambridge · 2019
Try it first
What is the fund's alpha?
Show the worked solution
The factors explain 14.7%, leaving alpha of 2.3%. Start from the 6% risk-free rate and add each loading times its factor return: 1.1 x 8 = 8.8 for the market, 0.4 x 2 = 0.8 for size, and minus 0.3 x 3 = minus 0.9 for value. The fund's 11 points over cash are mostly priced exposure; 2.3 points is left for skill.
Why not measure the fund against cash or the market alone?
A tutor whose students all score well may simply have been given the strongest students. To judge the teaching, you first adjust for who walked in. A fund with a beta above 1 and a small-cap tilt should earn more than the market in a year when the market and small caps did well, so its exposures must be priced before anything is called skill. The Fama and French three-factor model does exactly that with market, size and value.
From a 6% risk-free rate, market exposure adds 8.8, the size tilt 0.8 and the value tilt takes away 0.9, so the factors explain 14.7% and only 2.3 points of the fund's 17% remain as alpha. The relationshipbeta_M, beta_S, beta_V the fund's loadings on market, size and value: 1.1, 0.4, minus 0.3 MKT, SMB, HML the factor returns that year: 8, 2 and 3 points alpha the part no factor explains What it says in wordsEach exposure earns its factor's return; the leftover is alpha.What does the split tell you about the manager?
Against the plain market's 14%, the fund beat by 3 points. Of those 3 points, 0.7 came from priced tilts in net terms, more beta and small caps less the growth tilt, and 2.3 is left over. The negative value loading matters too: a growth-leaning fund lost 0.9 points in a year value did well, so part of the manager's alpha was earned while swimming against a factor.
Then state the limits. One year of alpha is mostly noise; you would want a regression over many years with a standard error before calling 2.3 points skill. And the answer depends on the model: add momentum or quality factors and some of the 2.3 may turn out to be another priced exposure.
Where candidates lose it
The common slip is calling 11% or 3% the alpha, measuring against cash or the market without adjusting for beta and tilts. The interviewer is testing whether you price every exposure before crediting skill.
The second slip is sign handling on the value loading. A negative loading in a year the factor rose is a cost, minus 0.9, not a gain.
What the interviewer asks next
- If HML had been minus 3% that year, what would the alpha be?
- How would you tell whether 2.3% of alpha is statistically meaningful?
- Why might adding a momentum factor change the answer?
Asked at State Street, Investment Banking, Cambridge, 2019 (Wall Street Oasis):
some basic market knowledge, such as factor model (Fama French), portfolio optimization, risk analysis
