The impact of correlation on (Range) Value-at-Risk

The assessment of portfolio risk is often explicitly (e.g. the Basel III square root formula) or implicitly (e.g. credit risk models) driven by the marginal distributions of the risky components and their correlations. We assess the extent by which such practice is prone to model error. In the case of n = 2 risks, we investigate under which conditions the unconstrained Value-at-Risk (VaR) bounds (which are the maximum and minimum VaR for $ S=\sum _{i=1}^{n}X_i $ S=∑i=1nXi when only the marginal distributions of the $ X_i $ Xi are known) coincide with the (constrained) VaR bounds when in addition one has information on some measure of dependence (e.g. Pearson correlation or Spearman's rho). We find that both bounds coincide if the measure of dependence takes value in an interval that we explicitly determine. For probability levels used in risk management practice, we show that using correlation information has typically no effect on the highest possible VaR whereas it can affect the lowest possible VaR. In the case of a general sum of $ n \geqslant 2 $ n⩾2 risks, we derive Range Value-at-Risk (RVaR) bounds under an average correlation constraint and we show they are best-possible in the case of a sum of $ n\geqslant 3 $ n⩾3 standard uniformly distributed risks.