We present an integrated methodological approach for selecting portfolios. The proposed methodology is focused on incorporation of investor’s preferences in the Mean-Risk framework. We propose a risk measure calculated with the downside part of the portfolio return distribution which, we argue, captures better the practical behavior of the loss-averse investor. We establish its properties, study the link with stochastic dominance criteria, point out the relations with Conditional Value at Risk and Lower Partial Moment of first order, and give the explicit formula for the case of scenario-based portfolio optimization. The proposed methodology involves two stages: firstly, the investment opportunity set (efficient frontier) is determined, and secondly, one single preferred efficient portfolio is selected, namely the one having the highest Expected Utility value. Three classes of utility functions with loss aversion corresponding to three types of investors are considered. The empirical study is targeted on assessing the differences between the efficient frontier of the proposed model and the classical Mean-Variance, Mean-CVaR and Mean-LPM1 frontiers. We firstly analyze the loss of welfare incurred by using another model instead of the proposed one and measure the corresponding gain/loss of utility. Secondly, we assess how much the portfolios really differ in terms of their compositions using a dissimilarity index based on the 1-norm. We describe and interpret the optimal solutions obtained and emphasize the role and influence of loss aversion parameters values and of constraints. Three types of constraints are studied: no short selling allowed, a certain degree of diversification imposed, and short selling allowed. 
Portfolio optimization under loss aversion