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  1. 096A manager holds the index but overweights stock A by 5 percentage points and underweights stock B by 5 points. A has 30% volatility, B has 25%, and they correlate at 0.6. What tracking error does this pair of bets create?Portfolio risk mathsHardMSCIMonterrey · 2013

    Try it first

    Before working it: is the tracking error above or below the 1.5% that the A bet alone would create?

    Show the worked solution

    A tracking error of about 1.25% a year. Tracking error is the volatility of the active weights. A's bet contributes (5% x 30%) squared, 2.25; B's contributes (5% x 25%) squared, 1.5625; and because the bets are opposite on correlated stocks, the covariance term is minus 2.25. The sum is 1.5625, whose square root is 1.25%.

    Why does adding a second bet reduce the risk?

    Buying an umbrella and selling a raincoat leaves you with little net exposure to rain, because both move with the weather. Overweighting A and underweighting B, when the two stocks tend to move together, is partly a hedge: when both rise, the gain on A is partly offset by the shortfall on B. Tracking error measures the risk left after that offset.

    Two opposite bets on correlated stocks partly cancelActive weightsStock A, vol 30%+5 pointsStock B, vol 25%-5 pointscorrelation 0.6one bet up, one down:the covariance termturns negativeVariance terms, squared per cent+2.25A's own+1.56B's own-2.25A with B1.56TotalTE = 1.25%Tracking errorrho 0.6, opposite bets1.25%rho 0, opposite bets1.95%rho 0.6, same direction2.46%
    The variance of the active bets is A's own term 2.25 plus B's 1.5625 minus a covariance term of 2.25, which leaves 1.5625 and a tracking error of 1.25%, against 1.95% if the stocks were unrelated and 2.46% if both bets pointed the same way.
    The relationship
    TE2=wA2σA2+wB2σB2+2 wAwB ρ σAσB=2.25+1.5625−2.25TE^2 = w_A^2\sigma_A^2 + w_B^2\sigma_B^2 + 2\,w_A w_B\,\rho\,\sigma_A\sigma_B = 2.25 + 1.5625 - 2.25
    w_A, w_Bactive weights, +5% and -5%
    sigma_A, sigma_Bvolatilities, 30% and 25%
    rhocorrelation, 0.6
    What it says in wordsTracking error is the portfolio volatility formula applied to the active weights instead of the holdings.

    What is the neat coincidence, and what does it hide?

    Here the covariance term exactly cancels A's own variance, so the answer equals B's bet alone, 5% x 25% = 1.25%. That is a coincidence of these numbers, not a rule: the covariance term is 2 x 0.6 x 30 x 25, which happens to equal 30 squared. Change the correlation to 0.5 and the answer moves. The general lesson holds, though: tracking error depends on the size of the bets and on how much they cancel.

    Say the limits. Correlations are estimated and unstable, so a pair that looks like a hedge in calm markets can decouple when a stock-specific event hits. Real tracking error also includes every other small active weight, and many small bets can add up to more than one large one.

    Where candidates lose it

    The common slip is adding the two bets' risks, 1.5% plus 1.25%, as if they were independent and in the same direction. That ignores both the correlation and the opposite signs of the weights.

    The quieter slip is getting the sign of the covariance term wrong. The weights have opposite signs, so the term is negative; say that out loud before plugging in numbers.

    What the interviewer asks next

    • At what correlation would the tracking error be zero?
    • What tracking error would a 2% overweight in a stock with 40% volatility add on its own?
    • How would you decompose a portfolio's tracking error into contributions from each bet?

    Asked at MSCI, Financial Tools, Monterrey, 2013 (Wall Street Oasis): What's the tracking error formula?