  In the world of modern financial theory, portfolio construction has
traditionally operated under at least one of two central assumptions: the
constraints are derived from a utility function and/or the multivariate
probability distribution of the underlying asset returns is fully known. In
practice, both the performance criteria and the informational structure are
markedly different: risk-taking agents are mandated to build portfolios by
primarily constraining the tails of the portfolio return to satisfy VaR, stress
testing, or expected shortfall (CVaR) conditions, and are largely ignorant
about the remaining properties of the probability distributions. As an
alternative, we derive the shape of portfolio distributions which have maximum
entropy subject to real-world left-tail constraints and other expectations. Two
consequences are (i) the left-tail constraints are sufficiently powerful to
overide other considerations in the conventional theory, rendering individual
portfolio components of limited relevance; and (ii) the "barbell" payoff
(maximal certainty/low risk on one side, maximum uncertainty on the other)
emerges naturally from this construction.
