VaR at Its Extremes: Impossibilities and Conditions for One-Sided Random Variables
Abstract
We investigate the extremal aggregation behavior of Value-at-Risk (VaR) -- that is, its additivity properties across all probability levels -- for sums of one-sided random variables. For risks supported on \([0,\infty)\), we show that VaR sub-additivity is impossible except in the degenerate case of exact additivity, which holds only under co-monotonicity. To characterize when VaR is instead fully super-additive, we introduce two structural conditions: negative simplex dependence (NSD) for the joint distribution and simplex dominance (SD) for a margin-dependent functional. Together, these conditions provide a unified and easily verifiable framework that accommodates non-identical margins, heavy-tailed laws, and a wide spectrum of negative dependence structures. All results extend to random variables with arbitrary finite lower or upper endpoints, yielding sharp constraints on when strict sub- or super-additivity can occur.
Summary
This paper investigates the conditions under which Value-at-Risk (VaR) exhibits sub-additivity or super-additivity, particularly focusing on one-sided random variables. The main research question is to determine the necessary and sufficient conditions for VaR to be sub-additive or super-additive across all probability levels, especially when dealing with risks supported on non-negative intervals. The authors demonstrate that VaR sub-additivity is impossible for risks supported on [0, ∞) except in the trivial case of exact additivity, which only occurs under co-monotonicity. To characterize VaR super-additivity, the paper introduces two key structural conditions: negative simplex dependence (NSD) and simplex dominance (SD). NSD ensures that the probability of the sum of the random variables being less than a threshold is less than or equal to the product of the individual probabilities. SD, on the other hand, requires a margin-dependent functional to be minimized when all its arguments are equal to the sum of the inputs. The authors prove that these conditions provide a unified framework for determining VaR super-additivity, even with non-identical margins, heavy-tailed distributions, and various negative dependence structures. The results are extended to random variables with arbitrary finite lower or upper endpoints, yielding constraints on when strict sub- or super-additivity can occur. This research matters because it provides a comprehensive understanding of VaR's behavior under extreme conditions, offering practical criteria for risk management in scenarios involving heavy tails and negatively dependent risks.
Key Insights
- •VaR sub-additivity is impossible for risks supported on [0, ∞) unless the random variables are co-monotonic and VaR is exactly additive.
- •The paper introduces negative simplex dependence (NSD) and simplex dominance (SD) as sufficient conditions for VaR super-additivity. These conditions offer a unified framework that accommodates non-identical margins and heavy-tailed laws.
- •Theorem 3.4 establishes that if a random vector is NSD with continuous marginals and a specific margin-dependent functional (Φ) is SD, then the vector is VaR super-additive.
- •Proposition 3.5 shows that negative lower orthant dependence (NLOD) implies NSD, providing an easier-to-verify condition for NSD.
- •Example 3.1 demonstrates that neither a particular dependence structure nor the mere non-integrability of margins is sufficient on its own to guarantee VaR super-additivity. Rather, it requires analyzing how the joint distribution interacts with the full set of marginal distributions.
- •Example 3.8 provides an example of a VaR super-additive random vector that satisfies NSD but not NLOD, demonstrating the strength of NSD.
- •Proposition 4.1 shows that for random variables with finite lower end-points, VaR sub-additivity is only possible under additivity and co-monotonicity. Similarly, for random variables with finite upper end-points, VaR super-additivity is only possible under additivity and co-monotonicity.
Practical Implications
- •The findings can be used by risk managers in financial institutions and insurance companies to better understand and manage the risks associated with portfolios of assets or liabilities, especially when dealing with heavy-tailed or negatively dependent risks.
- •Practitioners can use the NSD and SD conditions to assess whether VaR is likely to be super-additive in a given situation, which can inform decisions about diversification and capital allocation.
- •The results can help regulators evaluate the appropriateness of VaR as a risk measure in different contexts, especially when dealing with extreme risks.
- •Future research directions include exploring alternative dependence structures that satisfy NSD and SD, as well as developing more efficient algorithms for verifying these conditions in high-dimensional settings.
- •The extension of the results to random variables with arbitrary finite endpoints opens up new avenues for research on VaR's behavior in different risk management scenarios.