Trading Risk: Measuring It, and Surviving It

Value at Risk compresses a loss distribution into one number, and there is an arithmetic case where it reports that diversification made things worse. This path builds that failure and the coherence axioms explaining it, then attacks the deeper problem: the distribution itself is estimated and wrong in known directions that all understate danger and all fail together. Leverage turns those errors into forced exits. Sizing is the only decision that changes an outcome, and its arithmetic is unforgiving.