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  1. 017Two funds' daily returns are negatively correlated within any given month, yet their yearly returns are positively correlated. How can both be true?Statistics, correlation and diversificationHardSquarepoint CapitalMontreal · 2024

    Try it first

    Which explanation fits?

    Show the worked solution

    A shared driver that moves slowly lifts or sinks both funds together across years, while short-term noise pushes them in opposite directions day to day. Within a month the slow driver barely changes, so daily correlation reflects only the opposing noise. Across years it dominates. With daily noise of 0.8% at -0.5 correlation and a shared yearly drift of 20% standard deviation, yearly correlation comes out at about +0.57.

    What kind of situation produces this?

    Think of two ice cream stalls on the same beach. On any given day, a customer who buys from one does not buy from the other, so their daily sales move against each other. Across years, both do well in hot summers and badly in wet ones. Correlation is not one fixed number between two things; it depends on which driver dominates at the horizon you measure, and different drivers dominate at different horizons. For funds, the slow driver might be the economy's earnings cycle, which both portfolios share; the fast one might be money rotating between their two styles day to day.

    Opposite day to day, together year to yearDaily returns in one month: fund A and fund BFund AFund BOpposite signs on 16 of 21 daysSample daily correlation -0.49; the model sets -0.5+1%-1%Yearly covariance of A and B+0.040Shared drift-0.008Daily noise+0.032NetCorrelation = 0.032 / 0.0560 = +0.57
    In a simulated month the two funds move in opposite directions on 16 of 21 days, yet across years the shared drift adds 0.040 of covariance against 0.008 removed by the opposing noise, so yearly returns are positively correlated at about 0.57.

    Can you show it with numbers?

    Build each fund's yearly return from two parts. A shared drift, the same for both within a year but different from year to year, with a standard deviation of 20%. Plus daily noise of 0.8% a day for each fund, correlated at -0.5 between them, over 250 trading days. Within a month the drift is a constant, so it drops out of any correlation measured around the month's average, and the daily figure is the noise's -0.5. Across years, the drift contributes 0.2 squared, 0.040, to covariance, and the noise contributes -0.5 times 250 times 0.008 squared, -0.008, leaving +0.032.

    The relationship
    ρyear=σF2+ρd n σd2σF2+n σd2=0.040−0.0080.040+0.016=0.0320.056≈0.57\rho_{year} = \frac{\sigma_F^2 + \rho_d\, n\, \sigma_d^2}{\sigma_F^2 + n\,\sigma_d^2} = \frac{0.040 - 0.008}{0.040 + 0.016} = \frac{0.032}{0.056} \approx 0.57
    \sigma_Fthe standard deviation of the shared yearly drift, 20%
    \rho_dthe correlation of daily noise, -0.5
    ntrading days in a year, 250
    \sigma_deach fund's daily noise, 0.8%
    What it says in wordsYearly correlation is the shared drift's variance less the summed opposing noise, divided by each fund's total yearly variance.

    Give the condition and a second mechanism. The sign flips only if the shared drift's variance is larger than the summed noise covariance; with a drift of 10% instead of 20%, covariance would be 0.010 minus 0.008 and correlation barely positive. A second route is timing: if one fund's holdings are priced with a lag, its daily moves look unrelated or even opposite to the other's, while over a year the lags wash out. Either way, the lesson for a fund analyst is that a diversification benefit measured on daily data may not exist at the horizon a client actually holds.

    Where candidates lose it

    The common failure is saying it is impossible, on the belief that correlation is a fixed property of two assets. The interviewer is checking whether you know correlation is horizon dependent and can name what drives each horizon.

    The second failure is waving at small samples. Twelve monthly points are noisy, but noise is not an explanation; the strong answer builds a two-component model and states when the sign flips.

    What the interviewer asks next

    • Using the same model, at what size of shared drift would yearly correlation be exactly zero?
    • Why might two funds that look like good diversifiers on daily data fail to diversify in a bear market?
    • How would stale prices in one fund distort its measured volatility as well as its correlation?

    Asked at Squarepoint Capital, Hedge Fund, Montreal, 2024 (Wall Street Oasis): correlation can be negative intra-month but positive across a year, how?