You estimate a 500 by 500 sample covariance matrix from 1000 daily returns, and the true covariance is the identity. What ratio of largest to smallest sample eigenvalue should you expect, and what does it do to a mean variance optimiser?

You estimate a 500 by 500 sample covariance matrix from 1000 daily returns, and the true covariance is the identity. What ratio of largest to smallest sample eigenvalue should you expect, and what does it do to a mean variance optimiser?

Approach: Use the Marchenko-Pastur edges at aspect ratio q = N/T, then remember that an optimiser inverts the matrix rather than using it directly.

33.97. With q = N/T = 0.5 the Marchenko-Pastur law puts the sample eigenvalues on [(1 - sqrt(q))^2, (1 + sqrt(q))^2] = [0.0858, 2.914], so the condition number is 2.914/0.0858 = 33.97 while every true eigenvalue equals 1. The eigenvalue spread here is entirely estimation error. A mean variance optimiser inverts the matrix, so it multiplies by 1/0.0858 = 11.7 along the direction the sample says is quietest, and that direction is pure noise, so the optimiser puts its largest positions there. At q = 1 the smallest edge is 0 and the matrix is singular, and for N > T it is singular by construction with N - T zero eigenvalues.

Follow-up: How long a sample would you need for the condition number to fall below 2, and is that history usable?

Key concepts: marchenko-pastur law, eigenvalue spread, condition number, estimation error.