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Weekly UpdatesThe respective performance indicators of portfolios generated by the two fuzzy models are presented in Table 7 below.
As shown in Table 7, both the portfolio return and standard deviation of the FTAA flexible programming model are pretty much identical as that in the MPT portfolio. Meanwhile, the result obtained from the FTAA robust programming model suggests a portfolio with slightly lower returns and slightly higher risks than the other two portfolios. It implies that fuzzy programming portfolios, under certain rules, are able to perform just as well as the traditional method in terms of minimizing risks. However, the portfolio derived from Ramik and Rimanek's FTAA robust programming model presents a unique situation in which lower returns are obtained from riskier investments. This anomaly can be explained by the performance of stock options other than the ones chosen by the MPT/FTAA flexible programming model (Table 3). Albeit with higher expected monthly returns for some investment options over Utilities and Energy stocks, the disparities are not enough to compensate the risks involved. The risks for some stock options are considered too high that they are even less preferable to direct real estate investments from the perspective of an average risk-averse investor. This leads to a scenario in which risky investment does not necessarily bring higher rewards and diversification of assets does not necessarily minimize risks.
In the previous analysis, the objective and constraints (aspiration level, portfolio risk and expected return) are envisaged to be fuzzy in programming models. It is observed that fuzzy programming models can perform well in minimizing portfolio risk given the aspired level of return. It can provide an intuitive way to capture the ambiguous and vague information in an intricate and dynamic market. The efficiency of asset allocation can be improved with the inclusion of expert-knowledge. In the real world, the expert/investors would prefer a possible range of information rather than a precise function value. Compared to MPT, a main advantage of the FTAA models is that institutional investors and practitioners can describe their aspired level in terms of fuzzy instead of a precise formulation and handle tolerance violation easily while the same expected return can be achieved. In a fuzzy environment, these models will select a set of feasible alternatives, which satisfy the objective(s) and constraints.
6. CONCLUDING REMARKS
This paper incorporates the fuzzy concept in linear programming to obtain the best possible outcome in portfolios, when direct real estate investment is included. Despite not as liquid as other investment options on the market, real estate helps hedge uncertainties, such as inflation and interest rate volatilities, which change the complexion of one's investment behaviour. The findings suggest that the fuzzy tactical asset allocation (FTAA flexible programming model), with the inclusion of expert judgments which contain information usually not found in historical data, is able to produce a portfolio just as efficient as traditional asset allocation models while minimizing the possible issues induced by imprecision and vagueness of information. Meanwhile, the FTAA robust programming model proffers a more evenly-distributed portfolio, yet surprisingly with higher risks and lower returns. Aside from the lack of emphasis on portfolio risks minimization, one reason attributed to such anomaly is the low level of returns among high-risk stocks not selected by MPT and FTAA flexible programming models. It results in a unique situation where portfolio diversification does not necessarily guarantee an efficient investment decision. In addition, investors should pay attention to the potential drawbacks for implementing FTAA models when too many constraints are incorporated. It could complicate the optimization process and thus renders these models difficult to use for laymen. Hence, further studies should focus on the development of the determination approach of membership functions.
ACKNOWLEDGEMENT
This study was funded by the Hong Kong Polytechnic University's Internal Grant. The authors would also like to thank Mr. Ka-hung YU for his assistance.
Received 18 February 2009; accepted 22 April 2009
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Eddie Chi Man HUI [1] [mail], Otto Muk Fai LAU [2] and Kak Keung LO [3]
[1] Department of Building and Real Estate, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong E-mail: bscmhui@inet.polyu.edu.hk
[2] Department of Building and Real Estate, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong E-mail: nokia_rokia@hotmail.com
[3] Department of Building and Real Estate, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong E-mail: bskklo@inet.polyu.edu.hk
(1) For the correlation coefficients and covariances, as required in the computations of the programming models, refer to Appendices 1 and 2.
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