The Impact of Correlation on (Range) Value-at-Risk

The assessment of portfolio risk is often explicitly (e.g., the square root formula under Basel III) or implicitly (e.g., credit risk portfolio models) driven by the marginal distributions of the risky components and the correlations amongst them. We assess the extent by which such practice is prone to model error. In the case of a sum 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 = X_1+X+2+...+X_n when only the marginal distributions of the X_i 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, 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 two or more risks, we derive Range Value-at-Risk (RVaR) bounds under an average correlation constraint (in addition to the knowledge of the marginal distributions). While these bounds are not best-possible in general, we show that they are in the case of a sum of three or more standard uniformly distributed risks. As far as we know, this result is the first that provides an explicit best-possible bound on RVaR for a general sum of three or more risks (uniformly distributed) under a correlation constraint