In this chapter portfolio optimization methods are presented in which some sort of risk measure and its level are directly factored into the weightings of the financial instruments. These approaches can be distinguished from Markowitz portfolios in the sense that now a certain VaR or ES level is not the result of an efficient portfolio allocation, but an objective. Hence the user can ask questions such as: ‘How should the portfolio weights be chosen such that the resulting 99% VaR is 3.5%?’. A thorough and concise comparison between mean–VaR and mean–variance portfolios is provided in Alexander and Baptista (1999, 2002, 2004, 2008) as well as in De Giorgi (2002). This approach has recently been investigated by Durand et al. (2011). In the light of the Basel Accord, these kinds of portfolio optimization attracted interest from practitioners, because the capital adequacy requirements can be derived from these kinds of risk measures (see Alexander and Baptista 2002, for instance). Portfolio optimizations involving the expected shortfall have been analysed by Uryasev and Rockafellar (1999) and Rockafellar and Uryasev (2000, 2001).
In a similar vein, portfolios can be optimized with respect to their draw-down. This type of optimization is applicable when a portfolio is benchmarked against a broader market index.
In the next two sections, the mean–VaR and mean-ES portfolio approaches are presented and contrasted with the classical mean–variance ...