Joint inference on market and estimation risks in dynamic portfolios

We study the estimation risk induced by univariate and multivariate methods for evaluating the conditional Value-at-Risk (VaR) of a portfolio of assets. The composition of the portfolio can be time-varying and the individual returns are assumed to follow a general multivariate dynamic model. Under sphericity of the innovations distribution, we introduce in the multivariate framework a concept of VaR parameter, and we establish the asymptotic distribution of its estimator. A multivariate Filtered Historical Simulation method, which does not rely on sphericity, is also studied. We derive asymptotic confidence intervals for the conditional VaR, which allow to quantify simultaneously the market and estimation risks. The particular case of minimal variance and minimal VaR portfolios is considered. Potential usefulness, feasibility and drawbacks of the different approaches are illustrated via Monte-Carlo experiments and an empirical study based on stock returns.