Methodology — Co-crash Map & Downside Asymmetry

Transparency is a feature. This page explains what the co-crash map and the correlation matrix's day-set views show and why — in plain terms. The precise formulas and numeric conventions are documented in our internal methodology; the summary here is deliberate, not an omission.

Average correlation can hide the thing a concentrated investor actually fears: names that look loosely related on ordinary days but crash together. These tools answer that with exact counts and split views instead of a single blended number.

The day-set views on the correlation matrix

The matrix's toggle re-measures every pairwise correlation on three subsets of days, defined once for the whole product by the benchmark you supplied: up days (benchmark rose), down days (benchmark fell), and stress days (the benchmark's worst decile — the same day list the diversification ladder's stress rung uses, by construction, so the two never disagree). These views use plain sample correlations on those days — deliberately not the covariance estimator selected above, so a shrunk cell and its stress view differ by regime, never by estimator.

Honesty devices: each view shows how many days every cell rests on, greys cells inside that view's own noise floor, and withholds a view entirely when the day set is too small to mean anything (with the reason). Two things to know:

Correlations measured on selected days shift mechanically. Even if nothing about your names changed, conditioning on the benchmark's behavior moves the measured number (a well-documented statistical pitfall — Boyer, Gibson & Loretan, 1997). That is why differences between views come with confidence intervals below, not bare numbers.
Non-US listings are excluded from the up/down split. Their home-market close is hours away from the benchmark's, so a same-day split would dress that timing offset up as asymmetry. The rule is deterministic and shown on the view; there is deliberately no "±1-day matching" option — it inflates these statistics roughly threefold against unadjusted baselines.

The downside-asymmetry list

For each pair we report Δ = (down-day correlation) − (up-day correlation), ranked. A positive Δ means the pair co-moves harder when the market falls — the diversification you see on calm days overstates what you'll have in a sell-off. Every Δ carries a confidence interval from a shared block-bootstrap (deterministic: the same book and window always reproduce the same intervals), and stars mark pairs that survive a false-discovery-rate control across all pairs at once.

"Zero starred" never means "no asymmetry." With this much data, only asymmetries above a certain size are detectable at all — the panel shows that minimum detectable Δ explicitly, so an empty flag list reads as "not detected at this sample size."

The co-crash map

For each pair, we count the days when both names sat in their own worst decile — exact counts with the actual dates, so every number can be audited against memory. Each count is compared against two model baselines:

Independence — the joint days you'd expect if the two names' worst days were unrelated coin flips.
Gaussian with their correlation — the joint days a normal-distribution world with this pair's own measured correlation would produce, computed exactly (via Owen's 1956 bivariate-normal result, not simulation).

A count above the second baseline's band means the pair shares more crash days than its correlation alone explains — tail linkage that an average correlation number cannot show. Both baselines are model outputs and labeled as such; none of this is a return forecast.

Each pair is measured on its own overlapping history (a recently listed name shortens only its own pairs, and every row shows its window), the deeper 5%-tail view is offered only for US-listed pairs with roughly a decade of shared history, and all of it runs on at most the trailing ten years of served prices — counts here cannot see 2008 or 2011, and we say so rather than imply otherwise.