It almost goes without saying that one purpose of wealth allocation is the diversification of risks. That being so, the utility of a risk-averse investor is increased when the wealth is allocated to an asset mix instead of holding a single risky asset. Concepts and methods for the assessment of market risks have been presented in Part II of this book. The risk measures derived from the statistical models presented have been applied on an ex post basis – that is, for a given portfolio allocation and its implied loss function, the riskiness of the investment can be assessed for a given confidence level. In this chapter and the next, approaches to portfolio optimization are presented that directly address the issue of asset allocation such that the purpose of risk diversification is directly taken into account.
So far, the term ‘diversification’ has been used rather loosely and the approaches presented have not questioned the view that the variance–covariance matrix of the returns is an appropriate statistic for measuring diversification, except in the chapter on modelling dependencies (i.e., copulae). Diversification entails at least two dimensions. The first dimension addresses the question of the underlying common characteristic with respect to which the assets are diverse. The second addresses the question how to measure the degree of diversification with respect to this characteristic. For instance, a risk-averse long-only investor ...