Utility-Based Investment Principles

Abstract

In the previous chapter we measured the quality of an investment in terms of the expected value E[V ₁] and the variance Var(V ₁) of the future portfolio value V ₁ and determined portfolio weights (subject to constraints) that maximize a suitable trade-off $\mathrm{E}[{V }_{1}] - c\mathrm{Var}({V }_{1})/(2{V }_{0})$ between a large expected value and a small variance. Attractive features of this approach are that the probability distribution of V ₁ does not have to be specified in detail and that explicit expressions for the optimal portfolio weights are found that have intuitive interpretations. We saw that this approach makes perfect sense if we consider portfolio values V ₁ that can be expressed as linear combinations of asset returns whose joint distribution is a multivariate normal distribution. However, unless there are good reasons to assume a multivariate normal distribution (or, more generally, as will be made clear in Chap. 9, an elliptical distribution), solutions provided by the quadratic investment principles can be rather misleading. Here we want to allow for a probability distribution of any kind, and this calls for more general investment principles that are not only based on the variance and expected value of V ₁.