AppliedMultiple choiceFree solution

Can three assets be pairwise correlated at −0.5-0.5?

  1. AYes, but only just – it is exactly the boundary
  2. BNo, correlations cannot be negative between three assets at once
  3. CYes, and any value down to −1-1 would work equally well
  4. DIt depends on the individual volatilities of the three assets
Show the answer and worked solution

Answer: A – Yes, but only just – it is exactly the boundary

For the equicorrelation matrix the eigenvalues are available in closed form: 1+(n−1)ρ1 + (n-1)\rho once and 1−ρ1 - \rho with multiplicity n−1n-1. With n=3n = 3 and ρ=−0.5\rho = -0.5 the first is 1+2(−0.5)=01 + 2(-0.5) = 0, so the matrix is positive semi-definite but singular: it sits exactly on the boundary. Any more negative and the smallest eigenvalue turns negative, which would mean some portfolio has negative variance, an impossibility. The general floor is ρ≥−1/(n−1)\rho \ge -1/(n-1), and the volatilities never enter because a correlation matrix is scale-free.

Worked solution

  1. Formula
    λ=1+(n−1)ρ once,1−ρ with multiplicity n−1\lambda = 1 + (n-1)\rho \ \text{once}, \qquad 1 - \rho \ \text{with multiplicity } n-1
  2. Substitute
    n=3, ρ=−0.5n = 3, \ \rho = -0.5
  3. Solve
    1+2(−0.5)=01 + 2(-0.5) = 0
  4. 1−(−0.5)=1.5 (twice)1 - (-0.5) = 1.5 \ \text{(twice)}
  5. Answer
    PSD but singular: exactly the boundary\text{PSD but singular: exactly the boundary}

Sanity check. Any more negative and some portfolio would have negative variance, which is the general floor hoge−1/(n−1) ho ge -1/(n-1).

  • A. Correct: 1+2(−0.5)=01 + 2(-0.5) = 0, so the matrix is positive semi-definite but singular. Any more negative and a portfolio would have negative variance.
  • B. They can; the question is how negative.
  • C. Below −0.5-0.5 the smallest eigenvalue turns negative.
  • D. A correlation matrix is scale-free; volatilities do not enter.

Takeaway: nn assets can be pairwise equicorrelated only down to −1/(n−1)-1/(n-1).

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Can three assets be pairwise correlated at −0.5? · QuantMax