Portfolio Selection Models: a Review and New Directions

Modern Portfolio Theory (MPT) is based upon the classical Markowitz model which uses variance as a risk measure. A generalisation of this approach leads to mean-risk models, in which a return distribution is characterised by the expected value of return (desired to be large) and a ”risk” value (desired to be kept small). Portfolio choice is made by solving an optimisation problem, in which the portfolio risk is minimised and a desired level of expected return is specified as a constraint. The need to penalise different undesirable aspects of the return distribution led to the proposal of alternative risk measures, notably those penalising only the downside part (adverse) and not the upside (potential). The downside risk considerations constitute the basis of the Post Modern Portfolio Theory (PMPT). Examples of such risk measures are lower partial moments, Value-at-Risk (VaR) and Conditional Value-at-Risk (CVaR). We revisit these risk measures and the resulting mean-risk models. We discuss alternative models for portfolio selection, their choice criteria and the evolution of MPT to PMPT which incorporates: utility maximisation and stochastic dominance.