Portfolio Management puzzles, solved step by step
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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?Multi-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 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 relationshipw_e, w_b the capital weights, 60% and 40% \sigma_e, \sigma_b the volatilities, 18% and 6% \rho the correlation, 0.1 RC_e the 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?
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?Quantitative 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.
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\rho_{AB}, \rho_{AC} the given correlations, 0.9 each \rho_{BC} the correlation being bounded 1-\rho^2 the 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.
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?MSCIMonterrey · 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.
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 relationshipw_A, w_B active weights, +5% and -5% sigma_A, sigma_B volatilities, 30% and 25% rho correlation, 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?
