Comparing Density Forecasts in a Risk Management Context

We compare multivariate and univariate approaches to assessing the accuracy of competing density forecasts of a portfolio return in the downside part of the support. We argue that the common practice to perform multivariate forecast comparisons can be problematic in the context of assessing portfolio risk, since better multivariate forecasts do not necessarily correspond to better aggregate portfolio return forecasts. This is illustrated by examples involving (skew) elliptical distributions and an application to daily returns of a number of US stock prices. Additionally, time-varying test statistics and Value-at-Risk forecasts provide empirical evidence for regime changes over the last decades.