← NCO bridge demos paper (PDF) · verification script

The member weights do not depend on the dial

Under the paper's model of grouped assets, turning the dial changes only how much each group holds of its representative. The variance the bridge gives up then has a closed form.

The setting

Assets are grouped, and each group has one representative, the paper's knot. The paper's model of the world, the gateway model, says that every other member of a group is its representative scaled by a beta plus independent noise. So a member's only connection to the rest of the market runs through its own group's representative. Such a market is drawn at random: four groups with two, three, one and two members besides the representative.

Inside a group

The bridge builds a small portfolio for each group, the group's direction, and then decides how much money each group gets. The direction lists a weight for the representative and a weight for each member; only its shape matters, since the second step rescales it. At the dial γ = 0 the direction is the group's own minimum-variance portfolio, as in nested clustered optimization. As γ rises the group is told what the other groups' representatives explain about its own.

The paper's result is that this information changes only one thing: how much of the representative the group holds. The weights on the members are the same at every γ, and the same as they would be if the other groups did not exist. The matrix on the left is the market's correlation with the groups outlined. The bars on the right are every group's direction at three settings of the dial: the member bars are identical across the three, and only the representative bars change.

The direction bars. Bars are grouped by asset, with the representative first in each group. For every member the three bars have the same height; for every representative they differ. The readout gives the largest movement of any member weight across the whole dial, which is rounding error, and confirms that at γ = 1 the portfolio equals the global optimum for the chosen objective. A member bar that differed across the three settings would refute the result. A different seed draws a different market with the same pattern.

How far from optimal is the bridge at each γ?

Because the members never move, the paper can collapse each group to two numbers, its representative and one summary of its members, and write the bridge's excess variance over the optimum in closed form. The chart compares that formula with the excess variance computed directly, on the same random market.

The excess variance. The height is how much variance the bridge gives up relative to the best possible portfolio, with no estimation error. It is largest at NCO on the left, falls as the dial turns, and reaches zero only at the right end. The two curves lie on top of each other, which is the paper's lost-precision identity checked live.

Everything is computed from the sampled covariance in the browser. Betas are drawn around 1 and residual noise is independent, so the gateway model holds exactly.