بهینه سازی هندسی پرتفوی با نیمه واریانس در مهندسی مالی

In this paper we consider a portfolio optimization problem on maximizing the geometric mean return subject to the lower semivariance as a risk measure in the financial engineering. Its optimal condition and the solving method via the Monte Carlo simulation are given, and a numerical experiment is presented in order to show that the method is efficient.

The geometric mean investment strategy, introduced into the finance and economics literature by Henry Latane [1]in 1959, has recently received some attentions in scholarly circles [2-5]. Ye and Li [6] considered the geometric mean return on portfolio investments with the variance of returns as a risk measure. The variance, however, is a questionable measure of risk for at least two reasons: First, it is an appropriate measure of risk only when the underlying distribution of returns is symmetric. And second, it can be applied straight forwardly as a risk measure only when the underlying distribution of returns is normal. However, both the symmetry and the normality of stock returns are seriously questioned by the empirical evidence on the subject. The lower semivariance of returns, on the hand, is a more plausible measure of risk [7-9] for several reasons: first, investors obviously do not dislike upside volatility; they only disliked own side volatility. Second, the lower semivariance is more useful than the variance when the underlying distribution of returns is asymmetric and just as useful when the underlying distribution is symmetric; in other words, the lower semivariance is at least as useful a measure of risk as the variance. And third,the lower semivariance measures the information provided by two statistics, variance and skewness, thus making it possible to use a one-factor model to estimate required returns. Thus, we will consider the lower semivariance as a risk measure to maximize the geometric mean return on portfolio investments. The paper is organized as follows.Section 2 develops our portfolio optimization model and its optimal condition. In Section 3, a Monte Carlo method is proposed to solve the model and a numerical example is given to show the effectiveness of the model. Conclusion is given in Sections 4.

We have investigated a finance optimization problem which maximizes the geometric mean return subject to the lower semivariance constraint, in fact this is a specific convex stochastic optimization. The optimal condition is given. The frame work for solving this problem is presented via Monte Carlo simulation in order to make the problem be equivalent to an approximating optimization problem. Under some parameters given, we give a numerical experiment by the Matlab software.