You cluster 2,000 assets by return correlation and have no labels. Give three quantitative ways to decide whether a clustering is any good, state what each measures, and say which one detects a clustering that is fitted to noise.

You cluster 2,000 assets by return correlation and have no labels. Give three quantitative ways to decide whether a clustering is any good, state what each measures, and say which one detects a clustering that is fitted to noise.

Approach: Separate internal geometry measures from resampling measures, then ask which of them a random relabelling of noise could also pass.

Use a silhouette score for internal geometry, a resampling stability measure for reproducibility, and an external test that the clusters predict something not used to build them, and only the stability and external measures detect a clustering fitted to noise. The silhouette score compares each point's mean distance to its own cluster with its mean distance to the nearest other cluster, so it is bounded in [-1, 1] and rewards compact well-separated groups. It is computed on the same data that produced the clustering, so a method with enough freedom scores well on pure noise, which is the failure it cannot catch. Stability resamples the assets or the time window, reclusters, and measures agreement between the two partitions with the adjusted Rand index or the variation of information, with the adjustment being what corrects for the agreement chance alone produces. A partition driven by sampling noise reproduces poorly and the index collapses toward zero. The external test is the one that matters on a desk: hold out a later period and check that within-cluster correlation exceeds between-cluster correlation there, or that a cluster-based hedge reduces variance out of sample. Running the same pipeline on a phase-randomised surrogate that preserves the marginal distributions but destroys the cross-sectional structure gives the null distribution each score should be judged against.

Follow-up: How would you build a surrogate return panel that preserves each asset's own autocorrelation and volatility while destroying the cross-asset structure?

Key concepts: silhouette score, cluster stability, resampling, external validation.