schur

← NCO bridge demos paper (PDF) · verification script

How far each group really travels: λ = γr / (1 − γ + γr)

A common dial does not damp every group equally. The paper gives the fraction of the journey each group has made in closed form, and it explains why the bridge stalls when two representatives nearly coincide.

The setting

Assets are grouped, and each group has a representative, the paper's knot. The bridge has one dial γ. At γ = 0 each group is allocated from its own covariance alone, as in nested clustered optimization. At γ = 1 each group is also told everything the other groups' representatives explain about its own representative, and the result is the global minimum-variance portfolio. In between, each group is told a fraction γ of that.

What a group does with the information is to change how much it holds of its representative. The paper shows that as γ runs from 0 to 1, the holding of the representative travels from its NCO value to its minimum-variance value, and that the fraction of the journey completed at a given γ is not γ. It is λ = γr / (1 − γ + γr), where r is the share of the representative's variance that the other representatives cannot explain. If the representative is independent of the others, r = 1 and λ = γ. If it is nearly a copy of a combination of the others, r is small, and almost nothing happens until γ is very close to 1.

The fraction travelled. Each curve is one group. The height is how far, as a fraction, that group's representative has travelled from its NCO holding toward its minimum-variance holding. Setting the dial to ½ moves an independent representative halfway, but a representative with r = 0.01 only one percent of the way.
Every curve crosses the dashed half-way line at γ = 1/(1 + r); a crossing anywhere else would refute the formula. For r = 0.01 that is γ ≈ 0.99, so nearly the whole journey happens in the last hundredth of the dial.

The extreme case: two groups whose representatives are the same asset

Take two groups. Each has a representative and one independent asset, all with variance 1. Now make the two representatives almost perfect copies of each other, with correlation 1/(1 + ε). Effectively there are three assets: one representative held twice, and two independent assets. The best portfolio puts a third in each independent asset and a sixth in each copy of the representative, for a variance of 1/3 as ε goes to 0.

NCO cannot see that the two representatives are nearly the same asset. Inside each group the representative looks like an ordinary asset, so NCO holds close to a quarter in everything, with variance near 3/8. The bridge repairs this only as the dial approaches 1, and the smaller ε is, the later the repair starts. The paper proposes an alternative dial λ that moves every group's representative by the same fraction of its journey, and that path stays smooth no matter how small ε is. Both paths are built from, and scored on, the same covariance with the chosen ε, so they share their two ends exactly; what changes with ε is the shape between them.

-2.00
The weights. The four assets are K1, its independent partner, K2, its independent partner. NCO holds a quarter of each. The optimum holds a sixth of each representative and a third of each independent asset. When ε is small the bridge leaves the NCO weights late in the dial.
The variance along the two dials. Both curves start at NCO's variance for this ε and end at the optimum for this ε; the two grey lines mark those values, and they tend to 3/8 and 1/3 as ε shrinks. The solid curve is the bridge against its dial γ; as ε falls toward 10−6 it stays flat across almost the whole dial and drops only at the right end. The dashed curve is the alternative dial λ, and it keeps its smooth shape at every ε. The paper's phrase for the solid curve's behaviour is that the two limits, ε → 0 and γ → 1, do not commute.

Both curves are the paper's two-tier formulas with a two-by-two representative covariance, computed live. The verification script checks the same limits.