Risk bounds for factor models

Recent literature has investigated the risk aggregation of a portfolio X=(Xi)1≤i≤n$X=(X_{i})_{1\leq i\leq n}$ under the sole assumption that the marginal distributions of the risks Xi$X_{i} $ are specified, but not their dependence structure. There exists a range of possible values for any risk measure of S=∑i=1nXi$S=\sum_{i=1}^{n}X_{i}$, and the dependence uncertainty spread, as measured by the difference between the upper and the lower bound on these values, is typically very wide. Obtaining bounds that are more practically useful requires additional information on dependence.Here, we study a partially specified factor model in which each risk Xi$X_{i}$ has a known joint distribution with the common risk factor Z$Z$, but we dispense with the conditional independence assumption that is typically made in fully specified factor models. We derive easy-to-compute bounds on risk measures such as Value-at-Risk (VaR$\mathrm{VaR}$) and law-invariant convex risk measures (e.g. Tail Value-at-Risk (TVaR$\mathrm{TVaR}$)) and demonstrate their asymptotic sharpness. We show that the dependence uncertainty spread is typically reduced substantially and that, contrary to the case in which only marginal information is used, it is not necessarily larger for VaR$\mathrm{VaR}$ than for TVaR$\mathrm{TVaR}$.