Demonstrations: the minimum out-of-sample variance bridge

Every claim, computed live in your browser from the same recursion the paper analyzes.

Four assets is small enough that a browser can run the exact Schur-complementary recursion thousands of times per frame. Each page below demonstrates one result of When the Out-of-Sample-Optimal Schur Portfolio Lies Between HRP and Minimum Variance, with sliders where a noise level or amplitude can be varied. Exact rational certificates for the same claims are in paper/neither_end_certificate.py.

The bridge on market data

Walk-forward bridge on Dow constituents
The out-of-sample variance of the bridge portfolio as a function of the coupling γ, on real returns. Its minimum sits near γ ≈ 0.55 and survives Ledoit–Wolf shrinkage of the inputs.

Lemma 1 — HRP never sees the cross-block
At γ = 0 the recursion reads only the diagonal blocks, so cross-block noise cannot move the weights at all.
Theorems 1 and 2 — the noise cost is γτ²
The expected excess cost factors as γτ²H, and the tempting γ²τ² bound fails for a generic family.
Example 1 — HRP becomes locally optimal at finite noise
The boundary derivative is exactly −1/700 + (8/189)τ² and flips sign at τ* = √(27/800).
Theorem 3 — small noise moves the optimum off the minimum-variance end
1 − γ*(τ) grows like (G′(1)/V₀″(1))τ², with the coefficient 1025/153 for the solved family.

Theorem 4 — unique interior optimum, monotone in the noise
The exact objective in closed form: strictly convex, minimizer interior at every admissible noise level, sliding from 1 to ½.
Example 2 — one-entry noise, interior without symmetry
Noise in a single covariance entry, no common-noise symmetry, and the interior optimum survives.

Theorem 5 — the sign of Ξ decides
Every entry of the covariance noisy. Interiority is decided by an explicit second-moment functional, and D-block-only noise realizes the Ξ < 0 branch where full coupling stays optimal.
Entrywise noise — enumerate every sign state
All 64 or 1024 sign states averaged through the recursion, exchangeability broken, minimizer still interior.
The sample covariance — γ* sweeps the bridge with T
Seeded Monte Carlo in the Antonov–Lipton–López de Prado setting: the median-optimal γ* runs from 0.62 at T = 15 to 0.96 at T = 250, with sample Markowitz as a separate benchmark in the paper.

Theorem 6 — the clipped frontier
The surrogate optimizer is the clipped inverse of t(γ) = −V₀′/G′; one family sweeps the whole bridge, the other clips to HRP at s = 27/800.