Hedge Funds puzzles, solved step by step

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  1. 050A researcher regresses 12-month forward returns on a signal using monthly observations, so consecutive observations overlap by 11 months, and reports a t-statistic of 4.0 from ordinary least squares. Roughly what is the honest t-statistic?Statistics and estimationHardQuant and systematic funds

    Try it first

    The honest t-statistic is closest to

    Show the worked solution

    Roughly 1.2, not 4.0. Consecutive 12-month returns share 11 months, so 240 monthly rows over 20 years hold only about 20 independent observations. OLS standard errors assume independence and come out too small by roughly the square root of the overlap, root 12, about 3.5. Dividing 4.0 by 3.46 gives about 1.15: the result is no longer significant. A Newey-West or Hansen-Hodrick standard error does this properly.

    What does the overlap do to the regression?

    Asking twelve friends for restaurant advice sounds like twelve opinions, but if eleven of them only repeat what the first one said, you have heard about one. Each 12-month return shares 11 months with its neighbour, so the rows are mostly the same data counted again, and the regression thinks it has twelve times more independent evidence than it does. The slope estimate is not biased by the overlap. What breaks is the standard error, because the residuals are strongly correlated from one row to the next, and that breaks one of the {term('OLS assumptions', 'The conditions under which ordinary least squares standard errors are correct, including residuals that are uncorrelated across observations.')}.

    Monthly 12-month windows share 11 of every 12 months123456789101112131415161718MonthObs 1Obs 2Obs 3Obs 4Obs 5Obs 6Each window adds one new month (lime) and repeats 11 months already counted (green)20 years of data240 monthly observationsabout 20 independent ones4.0 / root 12 = 4.0 / 3.46t about 1.2below 2: not significanton this rough correctionPlain OLS standard errors treat all 240 rows as independent, so the t-statistic is too big by about root 12
    Monthly observations of 12-month returns share 11 of every 12 months, so 240 rows over 20 years hold only about 20 independent observations, and the reported t-statistic of 4.0 shrinks to about 1.2 once divided by root 12.

    Why divide by root 12 and not by 12?

    The standard error scales with one over the square root of the number of independent observations. If the effective sample is twelve times smaller, the standard error is root 12, about 3.46, times larger, and the t-statistic is 3.46 times smaller: 4.0 becomes about 1.15. This is a rough correction. The exact factor depends on how persistent the signal is: for a slow-moving signal, such as a valuation ratio, it is close to root 12; for a fast-moving one it can be smaller.

    The relationship
    thonest≈tOLSh=4.012≈1.15t_{\text{honest}} \approx \frac{t_{\text{OLS}}}{\sqrt{h}} = \frac{4.0}{\sqrt{12}} \approx 1.15
    hthe overlap horizon, 12 months
    t_OLSthe t-statistic from plain OLS standard errors, 4.0
    What it says in wordsWith overlapping returns of horizon h, the plain t-statistic is too large by about the square root of h.

    Say how you would fix it properly: use Newey-West standard errors with at least 11 lags, or Hansen-Hodrick errors built for exactly this overlap, or run the regression on non-overlapping annual data and accept the smaller sample. Any of those should give a t-statistic well below 4.0, and a researcher who reports only the OLS number has not yet shown the signal works.

    Where candidates lose it

    The common loss is accepting the 4.0 because the slope looks economically sensible. The overlap does not move the slope; it fakes the precision, and the interviewer wants to see you spot that.

    The second loss is overcorrecting, dividing by 12 instead of root 12. Standard errors shrink with the square root of the sample, so the correction is the square root of the overlap.

    What the interviewer asks next

    • How many Newey-West lags would you use here, and why?
    • Would non-overlapping annual regressions give the same slope but a bigger standard error?
    • Why do long-horizon return predictability studies often report very high R squared values?
  2. 075A stock-selection signal has an information coefficient of 0.05, and you can make 400 independent bets a year with it. What information ratio should you expect, and how many independent bets would you need for an information ratio of 1.5?Statistics and estimationHardQuant and systematic funds

    Try it first

    How many independent bets a year does an IC of 0.05 need for an information ratio of 1.5?

    Show the worked solution

    An information ratio of about 1.0, and about 900 independent bets a year for 1.5. The fundamental law of active management says the information ratio is roughly the information coefficient times the square root of breadth: 0.05 x the square root of 400 = 0.05 x 20 = 1.0. To reach 1.5 the square root must be 30, so breadth must be 900, more than double, because breadth enters under a square root.

    Why do many weak calls add up to a strong result?

    Picture a cricket pundit who calls the winner right 52.5% of the time. On one match that is nearly useless; over hundreds of independent matches, the small edge becomes a steady record. With independent bets, the expected gain grows in proportion to the number of bets while the noise grows only with its square root, so the ratio of the two grows with the square root of the number of bets. An information coefficientThe correlation between a signal's forecasts and the returns that follow; for a simple up or down call it equals twice the hit rate minus one. of 0.05 is roughly that pundit's edge: a hit rate of 52.5%.

    The relationship
    IR≈IC×BR=0.05×400=1.0,BR=(1.50.05)2=900\text{IR} \approx \text{IC} \times \sqrt{\text{BR}} = 0.05 \times \sqrt{400} = 1.0, \qquad \text{BR} = \left(\frac{1.5}{0.05}\right)^2 = 900
    IRthe information ratio: active return per unit of active risk
    ICthe information coefficient, the skill of each forecast
    BRbreadth, the number of independent bets a year
    What it says in wordsExpected information ratio is the skill per bet times the square root of the number of independent bets.
    Skill counts once, breadth counts under a square root0.51.01.5002004006008001,000Independent bets a year (breadth)IC 0.10IC 0.05400 bets: IR 1.0900 bets: IR 1.5225IR = IC xroot of breadthDouble IC =4x the bets
    With an information coefficient of 0.05 the information ratio rises with the square root of breadth, reaching 1.0 at 400 independent bets and 1.5 only at 900, while doubling the coefficient to 0.10 reaches 1.5 with just 225 bets.

    What does the square root mean for building a strategy?

    Skill and breadth are not equal levers. Doubling the information coefficient doubles the information ratio; doubling breadth raises it only by about 41%, so matching a doubling of skill needs four times the bets. Going from 1.0 to 1.5 on breadth alone means 2.25 times as many independent bets, 900 against 400. That is why quant funds chase breadth across many stocks and short horizons, and why a small gain in forecast quality is worth so much.

    What does the law leave out?

    Two things that usually cut the answer. Independence is the hard part: 400 bets on stocks in one sector, or rebalanced so often that they repeat the same view, are far fewer than 400 independent bets. And constraints on position size, shorting and turnover stop a portfolio from fully expressing the signal; a transfer coefficientA number between 0 and 1 measuring how fully a constrained portfolio reflects the signal; it multiplies the fundamental law. of 0.6 would take the expected information ratio from 1.0 to 0.6. State the law, then say which of these you would check first.

    Where candidates lose it

    The common slip is scaling linearly: 1.5 is one and a half times 1.0, so 600 bets. Breadth sits under a square root, so the bets needed rise with the square of the target: 2.25 times, or 900.

    The second loss is treating 400 bets as 400 independent bets without comment. The interviewer wants to hear that correlated positions and portfolio constraints shrink the effective breadth, and that the law is an upper guide rather than a forecast.

    What the interviewer asks next

    • Your 400 bets are 100 stocks rebalanced quarterly with a signal that barely changes. What is the real breadth?
    • What information coefficient would give an information ratio of 1.5 with the original 400 bets?
    • The signal's IC decays by half after one month. How should that change the rebalancing frequency?