Portfolio Management puzzles, solved step by step

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  1. 024A fund's measured alpha is 4% a year, with a standard error of 3%. Across all funds, true alphas average zero with a spread (standard deviation) of 1.5%. What is your best estimate of this fund's true alpha?Statistics and forecastingHardFund selectionQuantitative asset management

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    Best estimate of the true alpha?

    Show the worked solution

    About 0.8% a year. Combine the fund's noisy measurement with what you know about funds in general, weighting each by its precision. The measurement's variance is 3 squared, 9; the spread of true alphas has variance 1.5 squared, 2.25. The measurement gets 2.25 / 11.25, a weight of 0.2, so the estimate is 0.2 x 4% + 0.8 x 0%, which is 0.8%. A noisy 4% deserves heavy shrinkage toward zero.

    Why not take the 4% at face value?

    A new restaurant with two five-star reviews is probably good, but you would not bet it beats every restaurant in town with thousands of reviews. When a measurement is noisy compared with how much things really differ, most of an extreme reading is luck, so the best estimate sits much closer to the average. Here true alphas across funds rarely stray far from zero, a spread of 1.5, while one fund's measured alpha can miss the truth by 3. A reading of 4% is far more likely to be a modest fund that got lucky than a fund with a true 4%.

    A noisy 4% alpha, pulled toward the zero average of all funds-6%-4%-2%0%2%4%6%8%10%12%True alpha, per cent a yearbest estimate: 0.8%all funds: 0%, spread 1.5measured: 4%, spread 34% pulled 80% of the way back toward 0%
    The prior for all funds is centred on zero with a spread of 1.5 and the fund's measurement on 4% with a spread of 3, so the combined estimate lands at 0.8%, with a spread of 1.34, pulled 80% of the way back to zero.
    The relationship
    α^=τ2τ2+s2 αobs=2.252.25+9×4%=0.8%\hat\alpha=\frac{\tau^2}{\tau^2+s^2}\,\alpha_{obs}=\frac{2.25}{2.25+9}\times 4\%=0.8\%
    \tauthe spread of true alphas across funds, 1.5%
    sthe standard error of this fund's measured alpha, 3%
    \alpha_{obs}the measured alpha, 4%
    What it says in wordsThe estimate is the measured alpha scaled by how much of the total variance comes from real differences between funds.

    How does a fund selector use this?

    By refusing to rank funds on raw past alpha. Shrinkage keeps the ranking but compresses it, so a fund with a long, steady record keeps more of its measured alpha than one with a short, volatile record showing the same number. A fund whose standard error was 1% instead of 3% would keep 2.25 / 3.25, about 69% of its 4%. Say the limitations: the result depends on the assumed spread of true alphas, which itself is estimated, and on the average being zero; if the fund belongs to a peer group with a known positive or negative average, shrink toward that instead.

    Where candidates lose it

    Most candidates either take the 4% as measured, ignoring the noise, or answer zero, ignoring the evidence. The interviewer wants the weighted middle and the reason for the weights.

    The arithmetic slip is weighting by standard deviations, 1.5 and 3, instead of variances, which gives one third and 1.3%. Precision is one over the variance, so square before you weight.

    What the interviewer asks next

    • What if the fund had 20 years of data and a standard error of 1%?
    • How would you estimate the spread of true alphas across funds?
    • Why does this argument make top-quartile rankings unstable from one year to the next?