Mutual Fund Mastery puzzles, solved step by step

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All topicsCompounding and time value9Statistics, correlation and diversification8Bond maths and duration10Performance measurement and returns8Costs and fee drag8Valuation riddles11Logic and numeracy brainteasers6Estimation and market sizing7Probability and expected value8NAV, units and fund mechanics7Risk, volatility and drawdown8Behavioural traps6Withdrawals and after-tax arithmetic4
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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?

  2. 067A fund has a market beta of 1.1 and a size-factor loading of 0.3. Over the year cash paid 6%, the market beat cash by 4%, the size factor (small minus large) returned 4%, and the fund returned 15%. What is its alpha after the factors?Statistics, correlation and diversificationCoreState StreetCambridge · 2019

    Try it first

    What is the fund's alpha after both factors?

    Show the worked solution

    Alpha is 3.4%. The fund's exposures alone would have earned cash of 6%, plus 1.1 x 4% = 4.4% for its market beta, plus 0.3 x 4% = 1.2% for its tilt to small companies: 11.6% in all. It returned 15%, so 3.4% is what the factors do not explain. It beat the market's 10% by 5 points, but 1.6 of those were paid-for risk.

    Why is beating the market by 5 points not 5 points of skill?

    A delivery rider who earns more than others in the monsoon may simply be taking the rainy-day shifts that pay extra. A fund that takes more market risk, or leans towards small companies, is paid for those exposures in years when they do well, and that pay is not skill. A factor modelA way of explaining a fund return as cash plus a set of exposures, each times the return of a common driver such as the market or small companies, with what is left over called alpha. prices each exposure. This fund has a beta of 1.1, so it gets 1.1 times the market's 4% premium over cash: 4.4%. It has a size loadingHow strongly a fund return moves with the gap between small company and large company returns. A positive loading means a tilt towards small companies. of 0.3, so it gets 0.3 times the 4% that small companies beat large ones by: 1.2%.

    Peel off what the fund was paid for, and what is left is alpha0%5%10%15%6.0Cash+4.4Market 1.1 x 4+1.2Size 0.3 x 4+3.4Alpha15.0%Fund returnexpected from risk: 11.6%Three readingsBeat the market+5.015% vs 10%CAPM alpha+4.6after beta onlyFactor alpha+3.4after beta and sizeEach step strips outa reward for risk
    Cash of 6.0%, a market contribution of 4.4% and a size contribution of 1.2% explain 11.6% of the fund's 15% return, which leaves 3.4% of alpha, well below the 5 points by which it beat the market.
    The relationship
    α=R−[rf+βm(Rm−rf)+s⋅SMB]=15−[6+1.1(4)+0.3(4)]=3.4%\alpha = R - \left[r_f + \beta_m (R_m - r_f) + s \cdot SMB\right] = 15 - [6 + 1.1(4) + 0.3(4)] = 3.4\%
    Rthe fund's return, 15%
    r_fthe cash rate, 6%
    beta_mmarket beta, 1.1
    R_m - r_fthe market's return over cash, 4%
    sthe size loading, 0.3
    SMBsmall minus big: small companies' return over large ones, 4%
    What it says in wordsAlpha is what remains after cash and every priced exposure have been paid.

    What changes as you add each factor?

    Each step strips out a reward that anyone could have bought cheaply. Against the market alone, the fund is 5.0 points ahead. After beta, the CAPMThe capital asset pricing model, which explains returns with one factor, the market, scaled by beta. alpha is 4.6%. After the size tilt as well, alpha is 3.4%, so about a third of the apparent outperformance was a small-company bet that a cheap index fund could have delivered. Add a value or momentum factor and the alpha could shrink further, or grow if the fund leaned against a factor that did well.

    The limits are worth stating plainly. One year of data says almost nothing; loadings and alpha are estimated from many periods of returns, and a 3.4% alpha with typical noise needs years before it is distinguishable from luck. The answer also depends on which factors you include and how they are built, so two research teams can report different alphas for the same fund.

    Where candidates lose it

    The first trap is quoting 5%, the gap to the market. That treats a fund with more risk and a small-company tilt as if it were the index, and gives the manager credit for exposures.

    The second is stopping at 4.6% after beta. The question names a size loading because the interviewer wants to see you price every exposure the fund carries before you call anything skill.

    What the interviewer asks next

    • The size factor returns -4% next year. What would the same fund need to return to show the same alpha?
    • Why might a fund with negative alpha still be a reasonable holding?
    • How many years of monthly data would you want before trusting a 3.4% alpha estimate?

    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