Teruyuki Watanabe

"In a real investment, stocks are dealt with based on a block of shares. A block of shares is a minimum unit for trading stocks. However, a conventional portfolio selection problem does not consider about a block of shares. If we deal with stocks according to a block of shares, real allocations of funds to each stock should differ among the cases of different amounts of money. Furthermore, a decision maker should be unable to buy less than one block even if the investing ratio for some stock is much smaller. The objective of this paper is to build a portfolio selection model in consideration of the amount of investing funds and a block of shares. Our model is formulated as an integer quadratic programming problem. In general, an integer nonlinear programming problem is difficult to solve for all but the smallest cases. So we also propose the efficiently approximate model employing a Meta-controlled Boltzmann machine."

A conventional portfolio selection problem, which is based on a mean-variance model, is difficult to solve by using mathematical programming techniques. This difficulty is caused by the fact that the corresponding mathematical programming problems are large-dimensional one, since almost all variance-covariances of return rates are, typically, not zeros. In this paper, we propose an efficient method for solving a portfolio selection problem, a method which uses a Boltzmann machine. In a real-life problem, it is also important to find the optimal combination of a small number of invested securities out of many securities in a market, because of a limited amount of funds to invest into securities. So we also propose a portfolio selection method to obtain the invest ratio of limited number of securities out of huge number of securities using a multi-stage application of the Boltzmann machine.