Integer Program Modeling of Portfolio Optimization with Mental Accounts Using Simulated Tail Distribution

Abstract

Since Markowitz introduced mean–variance portfolio theory, there have been many portfolio selection problems proposed. One of them is the safety-first portfolio optimization considering the downside risk. From behavioral portfolio theory, investors may not consider their portfolios as a whole. Instead, they may consider their portfolios as collections of subportfolios over many mental accounts. In this study, we present a mixed-integer programming model of portfolio optimization considering mental accounts (MAs). In this study, varied MAs are described by different level of risk-aversion. We measure the risk as the probability of a return failing to reach a threshold level, called the downside risk. An investor in each MA specifies the threshold level of return and the probability of failing to reach this return. Usually the portfolio’s returns are assumed as normally distributed, but this move may underestimate the downside risks. Accordingly, we estimate the downside risk by using models utilizing extreme-value theory and copula. We generate scenarios of the tail distribution based on this model, on which the mixed-integer program is applied. In the end, we use historical data to back test our model and the results are consistent with what they expected. These actions result in a better understanding of the relation between investor goals and portfolio production, and portfolio optimization.