Problem Formulation of Portfolio Optimization with Loans and Its Solution by Artificial Bee Colony Algorithm and Convex-Hull Mapping

Portfolio optimization that allows the borrowed money from a loan to be invested in risk assets is formulated as a constrained optimization problem. Then it is proved that the optimal solution of the portfolio optimization problem borrows money up to the limit if it uses the loan. Based on the knowledge, a revised artificial bee colony algorithm using a mapping-based constraint-handling technique is proposed to solve the portfolio optimization problem. The effectiveness of investments with loans is demonstrated by asset data for the Japanese stock market.