Cross-sectional analysis — dispersion, relative value, and the common factor

Computed 2026-06-10 by models/graph/cross_section.py → data/cross_section.json. Rendered on docs/charts.html#xsec and the dashboard. The computation is fact (re-runs from the cached data); the diversification reading is a labeled interpretation. Not used in the formal proofs.

Most of the charts elsewhere are time-series (one rate through time). This block is cross-sectional: at each moment it compares the whole cross-section of segments — every credit-rating bucket, every developed sovereign, every muni state — and asks three questions the desk literature asks:

1. How dispersed is the cross-section (and is that compressed or stretched vs its own history)?
2. Which segments are rich or cheap vs their own trailing history (relative value)?
3. How much of the co-movement is one shared factor (the common-factor / PC1 share)?

It runs over four cross-sections, in the order requested.

Method (and why)

For each time-series cross-section we compute, on a common monthly date grid:

• Dispersion over time — the cross-sectional standard deviation / range across segments each month. It compresses in calm regimes and blows out in stress (the credit-dispersion-as-stress gauge). We also z-score the current dispersion vs its own history (compressed / mid-range / elevated).
• Relative-value z-scores — each segment's latest level vs its own trailing history (z-score + percentile). Standard desk relative value: positive z = wide/cheap vs its history (more stress priced), negative = rich/tight.
• Snapshot ranking — the latest month, every segment side by side, ranked.
• Cross-sectional correlation + PC1 share — Pearson correlation of monthly changes (the Collin-Dufresne–Goldstein–Martin basis), plus the first-principal-component variance share via power iteration on the correlation matrix (a correlation matrix's eigenvalues sum to n, so PC1 share = λ₁ ⁄ n). The PC1 share is the empirical signature of a single common factor.

Literature anchors. Collin-Dufresne, Goldstein & Martin (2001), The Determinants of Credit Spread Changes (J. Finance 56(6)) found that monthly credit-spread changes are dominated by one common systematic factor the usual structural variables (leverage, vol, rates, slope) don't explain. Cross-sectional spread dispersion is a standard stress/credit-cycle gauge; sovereign-yield dispersion across a bloc is a fragmentation gauge; z-score/percentile vs own history is desk-standard relative value.

Results (2026-06-10 snapshot)

• US credit's ~91% PC1 share (and ~0.88 average pairwise correlation on monthly OAS changes) is a textbook confirmation of the CDGM common factor: the AAA/BBB/CCC/IG/HY buckets move almost as one. The credit-spread change correlation heatmap (on the charts page) is nearly solid red.
• TRACE breadth (the real trade-tape advance/decline ratio by IG/HY/convertibles) shows the same one-factor risk-on/off (87%).
• Sovereigns load on a global-rates common factor (70%), with the unified snapshot flagging Japan as the most stretched segment vs its own history — consistent with the JGB escaping yield-curve-control.

Bank cross-section (FDIC, 194 institutions). Each bank is z-scored against its peers on three axes — HTM loss/equity, uninsured-deposit %, CRE/Tier-1 — and a composite z ranks system vulnerability with a percentile. The cross-sectional standard deviation of HTM holes ≈ 7.5 pp of equity: the hidden losses are highly unequal across the system (the SVB lesson — the average is benign while the tail is not).

Funding-graph connectors. The cross-layer connectors are re-scored with a bridge_score = z(degree) + z(distinct neighbor-sectors) + 0.5 if the node spans both the financial and structural layers — surfacing the structural keystones that tie the proven financial core to the surrounding webs.

Unified snapshot. Every time-series segment is z-scored vs its own history so credit, sovereign, muni, and breadth read on one scale, ranked from most- to least-stressed.

Stress-test: what the ~91% factor reversing does (added 2026-06-15)

The PC1 share is usually read as a calm-regime fact ("the buckets move together"). The stress-test asks the asymmetric question: what happens on the reversal, when the common factor flips from risk-on to risk-off?

• Diversification is ~illusory, quantified. For n equally-weighted segments with average pairwise correlation ρ and equal vol, portfolio variance ÷ single-name variance = 1/n + (1−1/n)·ρ. For the US-credit ladder (n=5, ρ=0.88): 0.904 — holding all five rating buckets removes only ~9.6% of the variance, vs the ~80% a zero-correlation 5-name basket would remove. The effective number of independent bets ≈ 1/0.904 ≈ 1.1 — the cross-section is ~1.1 bets wearing 5 labels. So on a reversal there is almost no cushion: AAA→BBB→CCC widen together.
• Compression is the pre-stress tell. Dispersion "compresses in calm, blows out in stress." Two of the four cross-sections — municipal and TRACE-breadth — read compressed now, while US credit reads elevated: a mixed regime where the highest-PC1 segments are precisely the ones whose dispersion is most compressed (priced for calm). Compressed dispersion historically precedes the blow-out, not follows it — the cross-section is primed, not relieved.
• The reversal is the forcing event for the self-marked four. The machine-checked self_marked_value U3 proves the four self-marked gaps (bank HTM, AI fair-value marks, private-credit NAV, insurance captives) co-move under a common factor and cannot be netted; a PC1 reversal in credit is exactly that common factor turning — so it prices all four self-marked numbers at once, with no asset-class diversification either. The ~91% credit PC1 and the U3 "no netting" result are two views of one fact.
• Direction + magnitude, honestly. The level of stress on a reversal is not forecastable (the date is unknowable, per spec-unwind-timing); what the math fixes is the shape: a one-factor reversal is undiversified across both the credit cross-section (~1.1 effective bets) and the four self-marked asset classes (U3), so the loss is correlated, simultaneous, and un-hedgeable by holding "many different credits." That is the stress-test's load-bearing claim — graded labeled interpretation on the reversal scenario, fact on the ρ=0.88 / PC1=91% inputs and the diversification arithmetic.

Where the data disagrees with the literature

Mean-variance portfolio theory says holding many different credits diversifies idiosyncratic risk away. When PC1 explains ~91% of the cross-sectional variance, that diversification is illusory at the system level — the segments are one trade. That is the empirical agreement with the project's machine-checked self-marked-value result (models/z3/self_marked_value.py, U1–U4): the gaps across the four self-marked asset classes correlate under a common factor, so there is no netting, and carrying value is forced to converge on a forcing event. The cross-section is not a portfolio; it is a single position wearing many labels.

Reproduce

python3 models/graph/cross_section.py (pure stdlib; recomputes from data/fred_monthly.json, data/yahoo_monthly.json, data/tape_trace.json, data/bank_exposure.json, data/graph.json). Wired into run_all.sh and scripts/new-research.sh. Full charts, heatmap, and tables: docs/charts.html#xsec.

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