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What is the most negative pairwise correlation that five assets can all share?

Answer with a number. Fractions, powers and expressions like 23/6 or C(52,5) are read correctly in practice.

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Answer

−0.25-0.25

Use the same eigenvalue: equicorrelation at level ρ\rho across nn assets has smallest eigenvalue 1+(n−1)ρ1 + (n-1)\rho, so validity needs 1+4ρ≥01 + 4\rho \ge 0 and hence ρ≥−0.25\rho \ge -0.25. The bound tightens as the universe grows, which is the intuition worth carrying: many assets cannot all hedge each other, since a large set of mutually offsetting positions would collectively have negative variance. In the limit the floor goes to zero, so a large universe is nearly forced to be non-negatively correlated on average.

Worked solution

  1. Formula
    λmin⁡=1+(n−1)ρ≥0\lambda_{\min} = 1 + (n-1)\rho \ge 0
  2. Substitute
    1+4ρ≥01 + 4\rho \ge 0
  3. Solve
    ρ≥−0.25\rho \ge -0.25
  4. Answer
    −0.25-0.25

Sanity check. The floor rises toward zero as nn grows: many assets simply cannot all pull against one another.

Takeaway: The equicorrelation floor is −1/(n−1)-1/(n-1).

Answer it in practice – your answer is marked and recorded.

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