Risk budgets are frequently used to estimate and allocate the risk of a portfolio by decomposing the total portfolio risk into the risk contribution of each component position. Many approaches to portfolio allocation use ex post methods for constructing risk budgets and take the variance as a...

Financial risk managers routinely use non-linear time series models to predict the downside risk of the capital under management. They also need to evaluate the adequacy of their model using so-called backtesting procedures. The latter involve hypothesis testing and evaluation of loss functions....