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Quantitative Risk Management / Loss Distributions, VaR, and Coherent Risk Measures

Risk Aggregation and the Sub-additivity Question

When a bank computes capital for its trading book, it is computing the capital for a portfolio — a sum of positions across desks, asset classes, and risk types. A central question is: should diversification reduce the capital requirement, and if so, by how much? This is the sub-additivity question in practice.
We have already seen (in the coherence lesson) that ES is always subadditive, so diversification always reduces ES. VaR need not be — but it is subadditive under a powerful sufficient condition: when the portfolio is linear in elliptically distributed risk factors (Theorem 8.28 / Proposition 8.29). This covers the normal and Student-tt models used in standard market risk models.
The note then asks: what happens when the elliptical assumption fails? The Fréchet problem answers a harder question — how wide is the range of VaR values consistent with given marginal loss distributions, regardless of the dependence structure? The answer, via complete mixability, shows that VaR can be dramatically superadditive in heavy-tailed portfolios, while ES's worst-case is always the comonotonic sum. This is one of the deepest theoretical arguments against VaR.
Learning objectives. By the end of this note you should be able to:
  • state the two industry aggregation formulas (simple summation and correlation-adjusted summation) and identify when each is justified;
  • state Theorem 8.28 on elliptical distributions and VaR subadditivity, and trace why positive homogeneity and law-invariance give the correlation structure;
  • explain what a Fréchet problem is and state the worst-case VaR formula for a two-asset portfolio;
  • describe the concept of complete mixability and why it means VaR‾α(Sd)=ES‾α(Sd)\overline{\text{VaR}}_\alpha(\mathcal{S}_d) = \overline{\text{ES}}_\alpha(\mathcal{S}_d) in the homogeneous case;
  • give at least one practical example where VaR summation underestimates aggregate risk (superadditivity with VaR as the risk measure).

In this lesson

    1. Two industry aggregation rules
    1. When is the correlation formula justified?
    1. VaR summation can underestimate risk
    1. Fréchet problems and dependence uncertainty
    1. Complete mixability and the coincidence of worst-case VaR and ES
    1. The Rearrangement Algorithm

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