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  1. 001A fund's returns have an R squared of 0.81 against its benchmark index. The fund's volatility is 20% a year and the index's is 18%. What are the correlation, the beta and the fund's residual volatility?Statistics and forecastingWarm upPerformance analysisAsset management

    Try it first

    Before you work it: how much of the fund's 20% volatility does the index fail to explain?

    Show the worked solution

    Correlation 0.9, beta 1.0 and residual volatility of about 8.7%. Correlation is the square root of R squared, so 0.9. Beta is correlation times the ratio of volatilities, 0.9 x 20 / 18, which is exactly 1.0. The unexplained 19% of the fund's variance of 400 is 76, and the square root of 76 is 8.7%: the fund's own risk, on top of what the index explains.

    Why does 81% explained still leave so much unexplained?

    Think of a household's monthly spending. If rent explains most of how the bill moves, the groceries, travel and surprises that make up the rest can still swing it by a lot. The fund is the same. R squared splits variance, and variance is volatility squared, so a small share of variance becomes a much larger share once you take the square root back. The fund's variance is 20 squared, 400. The index explains 81% of it, which is 324. The other 76 belongs to the fund alone, and the square root of 76 is 8.72%.

    81% of the variance is explained, yet 8.7 points of volatility are the fund's ownIndex returnFund returnslope (beta) = 1.0correlation 0.9, R squared 0.81Fund variance = 20 x 20 = 400index: 324 (81%)76own: 76 (19%)Take square roots to get back to volatilityTotal20.0Index part, 1.0 x 1818.0Fund's own8.718 + 8.7 is not 20. Volatilities add in squares:18 x 18 + 8.7 x 8.7 = 324 + 76 = 400
    The index explains 324 of the fund's variance of 400 and leaves 76 unexplained; in volatility terms that is 18 points from the index and 8.7 points of the fund's own, which combine to 20 only because volatilities add in squares.

    How do you get the correlation and the beta from R squared?

    In a regression on a single index, R squared is simply the correlation squared, so the correlation is the square root of 0.81, which is 0.9. Beta is the correlation scaled by how volatile the fund is relative to the index: 0.9 times 20 over 18 is exactly 1.0. So the fund moves one for one with the index on average, and carries about 8.7 points of volatility the index does not explain. Mention the sign: the root could be minus 0.9, but a long-only equity fund with a positive slope takes the positive root.

    The relationship
    ρ=R2=0.9β=ρ σfσi=0.9×2018=1.0σε=σf1−R2=200.19≈8.7%\rho=\sqrt{R^2}=0.9 \qquad \beta=\rho\,\frac{\sigma_f}{\sigma_i}=0.9\times\frac{20}{18}=1.0 \qquad \sigma_\varepsilon=\sigma_f\sqrt{1-R^2}=20\sqrt{0.19}\approx 8.7\%
    R^2the share of the fund's variance the index explains, 0.81
    \rhothe correlation between fund and index returns
    \sigma_f, \sigma_ithe volatilities of the fund, 20%, and the index, 18%
    \sigma_\varepsilonthe residual volatility, the part of the fund's risk the index does not explain
    What it says in wordsCorrelation is the root of R squared, beta rescales it by the volatility ratio, and the residual volatility is the fund's volatility times the root of the unexplained share.

    What does the residual number tell a portfolio manager?

    With a beta of 1.0, the residual volatility is the fund's tracking errorThe volatility of the difference between a fund's return and its benchmark's return. against the index. An R squared of 0.81 sounds index-like, but 8.7 points of tracking error is a genuinely active book: almost half as volatile as the market itself. Say the limitation too. The split assumes the relationship is linear and stable over the sample; a fund whose beta drifted during the period shows a lower R squared for reasons that have nothing to do with stock picking.

    Where candidates lose it

    The common slip is treating R squared as a share of volatility and answering 19% of 20%, which is 3.8%. The interviewer is checking whether you know that variances add and volatilities do not, which is the same fact that sits under every portfolio risk calculation.

    The second slip is computing beta as 0.9 and stopping, forgetting that beta needs the volatility ratio. Say the three formulas in order and the numbers follow.

    What the interviewer asks next

    • If the fund's beta were 1.2 with the same volatilities, what R squared would that imply?
    • How would you tell whether the 8.7 points are skill or just unintended sector bets?
    • Why might R squared against a style index be much higher than against the broad market?