The dependence band reported with every capital aggregate: what it covers on planted dependence structures

Every capital figure below the simple sum asserts a diversification benefit that rests on a dependence the bank cannot observe in the joint tail. We study a protocol that never shows the aggregate alone: each aggregate carries a band of eight rows over the same marginals, from a lower bound to the sharp worst case; only the simple sum is labelled capital, and an Euler allocation of the expected shortfall adds up to the aggregate. Nine tests accompany the band, and three cannot run without the rows beyond the Gaussian row of the base matrix. We prove what holds and run the band on the six committed aggregates of a synthetic bank and on ten planted dependence structures of known capital. The band runs from CLP 118.5 to 823.3 billion against a headline of CLP 658.0 billion, 205 Monte Carlo standard errors wide; the Gaussian row is CLP 492.0 billion with a standard error of 2.5. The sharp range contains all ten planted capitals, at a median width of 1.26 times the capital. The interior rows of the base matrix contain 2 of the ten, against 75 percent predicted, and 8 with the rows of the market-credit stress. The Gaussian copula alone is within 5 percent for 2. Two of nine predictions were refuted: the coverage by the interior rows and the share of tests that need the other rows. The band exposes the dependence uncertainty; it does not reduce it, and it is conditional on the marginals. Repository and reproducible pipeline (MIT): https://github.com/fsantibanezleal/CAOS_Contraste .