VaR at Its Extremes: Impossibilities and Conditions for One-Sided Random Variables

Battle Count: 98

The paper is primarily relevant to risk management and regulatory capital rather than direct trading strategy development. However, it has indirect relevance: (1) Understanding VaR super-additivity is critical for portfolio risk assessment when holding heavy-tailed positions; (2) The impossibility of VaR sub-additivity for non-negative risks informs the choice of risk measures for position sizing; (3) The framework applies to insurance-linked securities and catastrophe bond portfolios; (4) The results on diversification failure under heavy tails are relevant for tail-risk hedging strategies. The paper does not address trading signals, execution, or alpha generation.

While the theoretical framework is elegant, practical implementation requires: (1) Deep understanding of copula theory, stochastic ordering, and extreme value theory; (2) Verification of NSD requires checking F_S(t) ≤ ∏F_Xi(t) for all t, which may be analytically intractable for complex joint distributions; (3) The SD condition requires analyzing the function Φ(x) = Σ x_i log F_Xi(x_i) for monotonicity, which involves reverse hazard rate computations; (4) For non-identical margins, the framework is more flexible than prior work but verification becomes more involved; (5) The paper provides sufficient conditions (Propositions 3.6, 3.7) that simplify verification, but these are not necessary conditions. No code or algorithms are provided.

The paper is entirely theoretical with complete mathematical proofs provided in the main text and appendix. All definitions, theorems, propositions, and examples are self-contained with explicit formulas. No computational code or data is required. The proofs are constructive and verifiable by any reader with graduate-level probability theory background. The truncated approximation argument (Theorem 2.2) and the NSD/SD framework (Theorem 3.5) are fully specified. Example 3.11 provides explicit verification for seven standard distributions.