Principles of Portfolio Selection
The most appropriate definition for portfolio selection, could be obtained from Zhang, Wang, Chen, and Nie (2007), Tiryaki and Ahlatcioglu (2009) and Huang (2009), which is allocating what proportion of one's wealth to each asset, to achieve investors' goals. It is also noticeable that sometimes these goals affect mathematical definition of objectives. Interestingly results obtained by Liu (2011) conform to claim provided by Markowitz (1952) that the greater the amount of risk that an investor is willing to take on, the greater the potential return.
Various measures are defined to represent risk of portfolio; (Markowitz, 1952) as the pioneer, introduces variance to measure the risk. Konno & Yamazaki (1991) demonstrated that mean absolute deviation risk function eliminates complexity of computing variance. Huang (2009) mentions that people are only averse of negative deviations not any kind of deviation, Therefore he suggested to employ semivariance as measure of risk.
Leung, Daouk, and Chen (2001) and X. Li, Qin, and Kar (2010) order to maximize skewness beside minimizing variance because increasing skewness will decrease the probability of large negative returns, while increasing the probability of large positive returns. As an evidence we can mention an article by Prakash, Chang, and Pactwa (2003) and a recent work by Chen, Luo, Liao, Yu, and Yang (2011), which state that maximizing the skewness of return could efficiently improve performance of the traditional Markowitz mean–variance portfolio model.
Bilbao-Terol, Pérez-Gladish, Arenas-Parra, and Rodríguez-Uría (2006) explains different attitudes toward risk and classifies its measures in three categories, variance-covariance matrix, single-index model and multi-index model. In this paper it is discussed that single-index model outperforms other ones in defining risk of portfolio.
Liquidity of an asset is defined as follows: “Possibility of converting an investment into cash without any significant loss in value.” (Arenas Parra, Bilbao Terol, & Rodrı́guez Urı́a, 2001)
Transaction cost is defined as follows: “In economics and related disciplines, a transaction cost is a cost incurred in making an economic exchange (restated: the cost of participating in a market). For example, most people, when buying or selling a stock, must pay a commission to their broker; that commission is a transaction cost of doing the stock deal.”(Firth, 2010)
Necessity of Using Fuzzy Approach
To clarify the necessity of using fuzzy approach in field of portfolio selection, We mention some portfolio experts' points of view about this matter below.
Markowitz (1952) without any knowledge about fuzzy logic perceived the need for engaging judgment of practical men and mentioned combination of subjective factors with objective ones. Tanaka, Guo, and Türksen (2000) mention that the simulation results show the possibility portfolio is more suitable than the fuzzy probability one in real investment problems. Arenas Parra et al. (2001) suggest a portfolio selection model which takes into account three criteria: return, risk and liquidity. These objectives, in general, are not crisp, so we must deal with them in fuzzy terms. According to what Joseph and Mazouz (2010) mention about large price shocks, it is vital to insert fuzzy approximations because large shocks are not mostly predictable by the means of probabilistic approaches. Liu (2011) states that because of volatility of financial market we cannot assume that future behaves as past did; So we should consider market parameters fuzzy. Huang (2011) expressed that using only random variables is inefficient especially for short-term securities.