This course covers fundamental definitions of loss functions, involving risk factors and risk factor changes. These concepts will be illustrated with examples of different value functions. For the quantitative analysis of the losses of a portfolio we introduce risk measures: general overview from variance to expected shortfall. We concentrate in highly important risk measures: value at risk (VaR) and expected shortfall (ES).

Considering a portfolio we analyse the distribution and dependence between different risks. We cover multivariate models and copula models: Sklar's theorem, fundamental copulas, clayton copulas, archimedean copulas, dependence measures. As part of dimension reduction we also study Principal component analysis. Finally, we also look at the tail of the distributions and study extreme value theory.