Portfolio Management puzzles, solved step by step

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  1. 039A 60/40 portfolio holds equities with 18% volatility and bonds with 6% volatility, and the two correlate at 0.1. What is the portfolio's volatility, and what share of its risk comes from equities?Portfolio risk mathsHardMulti-assetAsset allocation

    Try it first

    Roughly what share of the portfolio's risk comes from the 60% in equities?

    Show the worked solution

    Volatility is about 11.3%, and equities contribute about 93% of the risk. The variance is 0.36 x 324 plus 0.16 x 36 plus 2 x 0.6 x 0.4 x 0.1 x 18 x 6, which is 116.64 plus 5.76 plus 5.18, or 127.58. The square root is 11.3%. Equities' own term plus half the cross term is 119.2, which is 93% of the total.

    Why does 60% of the money carry more than 90% of the risk?

    Picture a household with two earners, one a salaried clerk and one a commission-only salesperson. The salesperson may bring in 60% of the money but almost all of the month-to-month swings. Risk contribution depends on weight times volatility, and variance squares it, so a sleeve three times as volatile with more capital swamps the other. Equities' variance term is 0.36 times 324, or 116.6; bonds' is 0.16 times 36, or 5.76. Twenty to one before the cross term.

    Capital split 60/40, risk split about 93/760%40%Capital93.5%6.5% bondsRiskEquitiesBondsBonds: 40% of capital,only 6.5% of the riskVariance, % squaredEquities 0.6² x 18² = 116.64Bonds 0.4² x 6² = 5.76Cross term = 5.18Total = 127.58Volatility = 11.30%
    Capital is split 60/40, but equities supply 93.5% of the portfolio's variance and bonds only 6.5%. The portfolio's volatility is 11.30%, so a 60/40 portfolio behaves almost entirely like an equity portfolio with the volume turned down.

    How do you split the risk between the two sleeves?

    Give each asset its own variance term plus half of the cross term. An asset's risk contribution is its weight times its covariance with the whole portfolio, and the contributions add to the total variance. For equities that is 0.6 x (0.6 x 324 + 0.4 x 10.8), which is 0.6 x 198.72, or 119.23. Divided by 127.58, that is 93.5%. Bonds take the remaining 6.5%.

    The relationship
    σp2=we2σe2+wb2σb2+2wewbρ σeσbRCe=we (weσe2+wb ρ σeσb)σp2\sigma_p^2 = w_e^2\sigma_e^2 + w_b^2\sigma_b^2 + 2 w_e w_b \rho\,\sigma_e\sigma_b \qquad RC_e = \frac{w_e\,(w_e\sigma_e^2 + w_b\,\rho\,\sigma_e\sigma_b)}{\sigma_p^2}
    w_e, w_bthe capital weights, 60% and 40%
    \sigma_e, \sigma_bthe volatilities, 18% and 6%
    \rhothe correlation, 0.1
    RC_ethe equity share of portfolio variance
    What it says in wordsPortfolio variance is each asset's own variance plus the cross term, and each asset's share is its weight times its covariance with the portfolio.

    This is the arithmetic behind risk parity, which sizes each sleeve so that the risk contributions are equal rather than the capital. To give bonds half the risk here, the portfolio would hold roughly three times as much bond capital as equity, and often borrow to lift the return. The limitation: correlation is not stable. In some years stocks and bonds fall together, and the 7% can grow quickly.

    Where candidates lose it

    The first slip is saying the portfolio volatility is 0.6 x 18 plus 0.4 x 6, which is 13.2%. That ignores diversification and would be right only at a correlation of one. The second is reporting 60% as the equity risk share because that is the capital share.

    Write the three variance terms, take the square root, then split. The whole point of the question is the gap between 60 and 93, so say it in a sentence.

    What the interviewer asks next

    • What equity weight gives equal risk contributions from the two sleeves?
    • How does the equity risk share change if the correlation rises to 0.5?
    • Why might a pension fund still describe itself as 60/40 despite this?
  2. 050Asset A has a correlation of 0.9 with asset B and 0.9 with asset C. What is the lowest possible correlation between B and C?Portfolio risk mathsHardQuantitative asset managementRisk management

    Try it first

    Pick the lowest correlation B and C could have.

    Show the worked solution

    About 0.62. Think of each asset's returns as a vector and correlation as the cosine of the angle between two of them. A correlation of 0.9 is an angle of about 25.8 degrees. B and C are each within 25.8 degrees of A, so they are at most 51.7 degrees apart, and cos 51.7 degrees is 0.62. The formula gives the same: 0.81 minus the square root of 0.19 x 0.19.

    Why can the third correlation not be anything you like?

    If your office is 10 km from your home and the gym is also 10 km from your home, the office and the gym cannot be 50 km apart. Distances have to fit on a map. Correlations work the same way: they are cosines of angles between return vectors, and angles have to fit together in space, so two strong correlations force a third. A correlation matrix that breaks this rule does not describe any real set of assets, and it can make a risk model report a negative variance.

    Correlations are cosines of angles, and two small angles cap the thirdABC25.8°25.8°at most 51.7°cos 25.8° = 0.9 for A-B and A-CB and C at most 51.7° apartmin corr(B, C) = cos 51.7°= 0.81 - 0.19 = 0.62Max is 1: B and C on the same lineRange: 0.62 to 1.00Lower is not a valid correlation matrixSame logic with two links of 0.7:minimum 0.49 - 0.51 = -0.02, so weak linksforce almost nothing on the third.
    B and C each sit 25.8 degrees from A, because the cosine of that angle is 0.9. At most they are 51.7 degrees apart, so their correlation cannot fall below cos 51.7 degrees, which is 0.62.

    How do you get 0.62 without drawing angles?

    Split B and C into a part driven by A and a part independent of A. Each has 0.9 of A in it, and the independent parts carry the remaining variance, 1 minus 0.81, or 0.19. The correlation of B and C is 0.81 from the shared A part plus up to 0.19 either way from their independent parts, depending on whether those parts move together or against each other. So the range is 0.81 minus 0.19 to 0.81 plus 0.19: 0.62 to 1.00. The lowest value comes when the independent parts are perfectly opposed.

    The relationship
    ρBC≥ρABρAC−(1−ρAB2)(1−ρAC2)=0.81−0.19=0.62\rho_{BC} \ge \rho_{AB}\rho_{AC} - \sqrt{(1-\rho_{AB}^2)(1-\rho_{AC}^2)} = 0.81 - 0.19 = 0.62
    \rho_{AB}, \rho_{AC}the given correlations, 0.9 each
    \rho_{BC}the correlation being bounded
    1-\rho^2the share of each asset's variance not explained by A
    What it says in wordsThe shared link through A gives 0.81, and the parts unrelated to A can take away at most 0.19.

    Why this matters on a risk desk: when analysts override individual correlations in a model, for a stress test or a view, they can create a matrix that no real market could produce. The fix is to check the matrix is positive semi-definite, and the lesson generalises. The bound is only strong when the given correlations are high; with two links of 0.7, the minimum for the third is -0.02, which forces almost nothing.

    Where candidates lose it

    The common answer is minus 1, from the idea that correlations are unrelated to each other, or 0.81, from multiplying the two links as if correlation were transitive. The first ignores the geometry; the second gives the answer only for one special case.

    Say the angle picture in one sentence, give 0.62, and show the formula as a check. Then add that the same logic is why a hand-edited correlation matrix must be tested before it goes into a risk model.

    What the interviewer asks next

    • What is the lowest possible correlation between B and C if both links are 0.5?
    • How would you check whether a 10 x 10 correlation matrix is valid?
    • Give a real-world example of three assets where this bound would bind.