Where to sit on the NCO bridge: the analytic conditions, checked live

Four pages: one Monte Carlo experiment, and three that each state a closed-form condition from Nested Clustered Optimization Is One End of a Schur Bridge, and the Interior Is Sometimes Provably Better and then compute it live.

The paper is about a dial. Assets are grouped, and each group has a representative asset. At one end of the dial, γ = 0, each group is allocated from its own covariance alone and the groups are then combined, which is nested clustered optimization. At the other end, γ = 1, the whole covariance is used at once, which is global minimum variance. In between, each group is told a fraction γ of what the other groups' representatives explain about its own.

The question is where on the dial to sit when the covariance is estimated with error. The paper answers with analytic conditions: a closed-form best dial for a noisy estimate, a sign that decides whether small noise moves the best dial inside at all, a formula for how far each group travels along the dial, and a closed form for the variance given up at every setting.

• The bridge by Monte Carlo: a random market, a noisy estimate, and the out-of-sample curve
  The true correlation matrix and a sample estimate of it side by side, the algorithm run on hundreds of such estimates, and the average out-of-sample variance against the dial with its interior minimum. Then the portfolios themselves as bars. There are no formulas, only the algorithm run on data.
• The best dial in closed form, and the sign that decides whether small noise moves it
  When every group is the same, the only decision is how much to hold of the representatives, and the out-of-sample variance is a closed-form function of the dial. With a noisy two-point estimate the best dial is strictly inside, at exactly 2/3 in the simplest case, and with small noise it can go either way, with an explicit threshold.
• How far each group really travels: λ = γr / (1 − γ + γr)
  Setting γ = ½ moves an independent representative halfway to its minimum-variance holding but a nearly redundant one almost not at all. In the limit of two identical representatives the bridge is stuck until the very end of the dial, and an alternative dial keeps the path continuous.
• The member weights do not depend on the dial, so the variance given up has a closed form
  On a random market that satisfies the paper's model, the dial changes how much each group holds of its representative and nothing else; the member weights are the same at every γ. The bridge's distance from the optimum then has a closed form, checked live.

The exact rational checks behind the same claims are in paper/verify_schur_nco_bridge.py.