Portfolio selection is an important issue for researchers and practitioners. Focusing on equity mutual funds, this paper proposes a basic portfolio selection model in which future return rates and future risks of mutual funds are represented by triangular fuzzy numbers. Firstly, a cluster analysis is proposed to categorize the huge amount of equity mutual funds into several groups based on four evaluation indices: rates of return, standard deviation, turnover rate, and Treynor index, in order to aid investors in making the investment decision. The fuzzy optimization model is proposed to determine the optimal investment proportion of each cluster. The portfolio optimization problem is developed in two ways: to maximize the future expected return subject to the given greatest future risk, and to minimize the future risk subject to a required lowest future expected return. The proposed approaches are demonstrated by Taiwan equity mutual funds.
Financial investments are especially important for individual and business financial managers due to current the low interest rate. Among all outlets of investments, mutual funds are one of investors’ favorite tools. Mutual funds have the characteristics of accumulation of risk and return, hence investors with a limited budget can also profit. The objective of managing mutual funds is to disperse investment risk to the smallest degree. From the data given by SITCA (http://www.sitca.org.tw/), the growth of Taiwan’s mutual funds market is increasing very rapidly, and by the end of December 2006, the size of funds reached approximately NT 1967 billions dollars, with a total of 508 funds traded. Among different types of funds, equity mutual funds play an important role in the market share.
Portfolio selection, as originally articulated by Markowitz, 1952 and Markowitz, 1959, has been one of the important fields of research in modern finance. This theory examines the trade-off of risk and return in the “mean–variance” context. According to Markowitz’s mean-variance model, given a specific rate of risk, one can derive the maximum investment return by maximizing the expected return of portfolios; or for a given specific return rate, one can derive the minimum investment risk by minimizing the variance of portfolio. The mean–variance method has been challenged and modified by many studies since it was first proposed decades ago. One other popular work is the market model, proposed by Sharpe, 1966, Sharpe, 1967, Sharpe, 1970 and Litner, 1965, which simplified Markowitz’s model by ignoring the covariance between returns.
Conventional portfolio selection models have an assumption that the future condition of stock market can be accurately predicted by historical data, but no matter how accurate the past data is, this premise will not exist in real financial markets, due to the high volatility of market environments. Therefore, fuzzy set theory, proposed by Zadeh (1978), has become a helpful tool in handling the imprecise conditions and attributes of portfolio selection. This paper considers the uncertainty of future return rates and risk, and thus presents return rates and risk in fuzzy numbers, and solves the optimal asset allocation by fuzzy optimization. The result will provide a more reasonable investment decision that is more suitable for the imprecise financial environment.
Section 2 briefly reviews the background of portfolio selection approaches and performance evaluation techniques. In Section 3 we discuss the portfolio selection model and cluster analysis. We present the numerical results for Taiwanese equity mutual funds in Section 4. Some concluding remarks and comments are provided in final section.
Mutual funds have become the most popular products for diversity of investment, since they are able to disperse investment risks to the smallest degree. In selecting mutual funds, the past performance of funds plays a central role in the expectations of the future performance of funds. Therefore, a number of performance evaluation techniques have been employed.
This paper investigates the performance of Taiwanese listed equity mutual funds, taking 122 funds as our sample data, with the aim of determining the best allocation of money to equity mutual funds in order to gain optimal return rates with tolerable investment risk or to have minimal investment risk with a specified return rate. The criterion of clustering is the performance indices, namely rate of return, standard deviation, turnover rate, and the Treynor index. After clustering, four clusters are determined, namely inferior performance funds, stable funds, aggressive funds, and good performance funds. Among them, aggressive funds and good performance funds dominate the other two, so that only they are included in the portfolio selection model.
The asset allocation for portfolio selection is performed in two ways. Investors who emphasize the gain of return rates may consider the maximization of return rates as the objective, taking risks as the constraint. While risk-averse investors may find that minimizing risk is more important than maximizing return rates, and may consider the minimization of risks as the objective. In addition, to deal with the nature of uncertainty in portfolio selection for future investment, the variables are represented as triangular fuzzy numbers based on fuzzy set theory. According to different confidence levels, the optimal investment proportions can thus be determined. For the future research direction, a multi-objective programming problem could be considered to take into accounts the return rates and risks simultaneously based on the investors’ different preferences.
