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Weekly Updates1. INTRODUCTION
In an intricate and dynamic market, decision making is a complex human cognitive process with regard to uncertainties such as price and interest volatility. Therefore, institutional investors and practitioners are always immersed in managing their investment portfolios, not only to optimize returns, but more importantly to minimize potential risks. Risk diversification has been a main theme in many previous studies on portfolio management, first seen in the mean-variance modern portfolio theory (MPT) by Markowitz in 1952. However, since the reliability of MPT, as well as other mathematical models in deciding the optimal investment allocation for a real estate-only portfolio (Olaleye, 2008) or a mixed-asset portfolio (Falkenbach, 2009), depends mostly on availability of relevant market data, sometimes they might not be able to accurately reflect real world situations. Thus, these market data do not represent one's decision making as some information are not quantifiable in nature, i.e. the human cognitive process. Expert judgment seems to offer an acceptable alternative to tackle this problem (Su, 2007).
By relaxing the crispness and precision for rigorous modeling and enabling a robust summary of expert knowledge, fuzzy logic systems can assist decision makers in their portfolio selection. The foundation of fuzzy logic for expressing imprecise, vague and uncertain information was first proposed by Zadeh (1964). His model/method has since been developed and widely used in investment, operating design and decision-making, along with information technology, artificial intelligence, management science and urban planning. Ko and Cheng (2003) emphasized that fuzzy logic not only can provide an approximate but also effective descriptions for highly complex, ill-defined, or difficult-to-analyze mathematical systems. The fuzzy approach can capture uncertainty in a realistic state as well.
Yet, the application of fuzzy set theory in real estate investment, especially one's allocation of such in investment portfolios, has been a relatively unexplored area. How does real estate investment, known for its capabilities of hedging against uncertainties such as inflation, influence the dynamics of one's portfolio asset allocation? This paper attempts to fill this gap of knowledge by incorporating fuzzy set theory into the classical asset allocation models, i.e. MPT. In order to relax the crispness and precision in portfolio management, two fuzzy tactical asset allocation (FTAA) models, namely Zimmermann's (2001) FTAA flexible programming model and Ramik and Rimanek's (1985) FTAA robust programming model, are applied in this study. These models incorporate fuzzy set theory and linear programming into the tactical asset allocation process.
Following the introduction, Section 2 reviews the relevant literature on real estate asset allocation. It is followed by a presentation of the methodology, approach and details of the model operation in Section 3. Section 4 provides details of the data set used for the modeling. Then, the results of our models are discussed and compared. Lastly, concluding remarks are provided.
2. LITERATURE REVIEW
This section reviews the previous asset allocation studies and relevant researches. Traditionally, asset allocation is based on the expected mean-variance (EMV) analysis. It relies on the premise that investors would diversify assets so as to optimize expected return while minimizing risk (volatility). An investor will trade-off between expected or anticipated return and risk, subject to various constraints since market imperfections cannot be ignored (Markowitz, 1952). This decision is not merely which securities to own, but how to allocate investors' wealth amongst securities. And asset allocation is a powerful tool for risk reduction (Kaplan, 1998). Yet, as the model relies greatly upon historical data as inputs to proffer recommendations on asset allocation and past performance of elements is an important influence on the decision (French, 2001), those inputs should be adjusted to reflect the understanding of the market (Kaplan, 1998). Without such adjustments, the credibility of the analytical result is compromised.
In order to overcome these problems, expert-knowledge is adopted in modeling. Expert judgments are pervasive and important to analytical systems (Fischhoff, 1989). It could be used in adjusting the expected return and variance in accordance with various factors, which might not be taken into account by mathematical computations, for instance, the MPT. Keeney and von Winterfeldt (1989) noted that human linguistic qualifications are preferable to numerical expressions of knowledge because it can well express or reflect expert's vagueness. It is therefore necessary to integrate such qualification into model calculations. The use of fuzzy logic in expressing imprecise, ambiguous, vague and uncertain information into a scientific approach was introduced by Zadeh (1964) and later enhanced via the introduction of the concept of linguistic (non-numeric) variable in 1970, i.e. "high", "low" (Bellman and Zadeh, 1970).
The application of fuzzy set theory to decision-making and evaluation in different real estate issues has been well-documented, for instance in the geographical information system (GIS) for spatial analysis (Sui, 1992; Zeng and Zhou, 2001), as well as in housing sales performance predictions utilizing a computer-based decision support tool for investors and contractors (Perng et al., 2005). Furthermore, it has also been applied to property appraisal and the estimation of appropriate market value (Bagnoli and Smith, 1998; Pagourtzi et al., 2003). However, little research has been done on deploying fuzzy set theory in devising an optimum investment strategy, particularly when direct real estate investment is being considered as well as other financial products available on the market. In addition to stocks and bonds, real estate is a crucial element in an investment portfolio because, while lacking liquidity, it processes hedging effect against market uncertainties such as inflation and interest volatility, which could change the complexion of one's consideration in the asset allocation of investment portfolios. This study aims to explore the possibility in incorporating expert knowledge through fuzzy set theory to optimize investment portfolios involving direct real estate investment. Two fuzzy mathematical programming, namely "Flexible" and "Robust" models, are utilized in this study. These models concern the aspired level of objective function, the degree of constraints, and the vagueness of coefficients in linear programming.
3. METHODOLOGY
This study focuses on the portfolio selection problem and the incorporation of fuzzy set theory with expert-knowledge in traditional approaches for optimizing the diversification benefit under risk tolerance. In general, most investors who are risk-aversers prefer risk to be as low as possible in their investment strategy. One of the most appropriate and popular approaches, Modern portfolio theory (MPT), is utilized to constitute an optimized portfolio using the concept of asset allocation. It can find the portfolio which will minimize risk and maximize expected return.
Modern Portfolio Theory (MPT)
A portfolio under MPT can be modeled as a quadratic programming function, similar to linear programming. Suppose there are n assets involved in the asset portfolio, the optimization for asset allocation in which the portfolio risk is minimized for a given level of expected return, can be expressed in an objective function and a few constraints as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)
where: [x.sub.i] is the proportion of portfolio allocated to asset i; [[mu].sub.p] is the expected portfolio return; [[mu].sub.i] is the expected return on asset i; [[mu].sub.o] is the given level of expected return; [[sigma].sub.ij] is the covariance between asset i returns and asset j returns. In Markowitz's MPT optimization, investors are predominantly risk-averse and obtain an optimized asset portfolio with the highest possible return at a given level of risk ([[sigma].sup.2.sub.p]) subject to some constraints.
Fuzzy Set Theory
The desired level of return for an investor sometimes cannot be depicted by a precise numerical value. It is regarded as the linguistic vagueness. In order to address it, this study incorporates the fuzzy set theory into linear programming, usually perceived as a powerful tool in decision-making.
Fuzzy set theory can be viewed as the extension of a crisp set theory, which deals with the lexical uncertainty and provide assistance in making decisions. It can be expressed as follows.
A = (x, [[mu].sub.A](x)|x [member of] U) or A = [summation over (i=1)] [[mu].sub.A]([x.sub.i])/[x.sub.i] (2)
where: [[mu].sub.A] (x) is a membership function (MF) of x in set A in the universe of discourse and its value maps to the space [0,1], [mu](x) : X [right arrow] [0,1].
Note: The symbol [SIGMA] implies union, but not addition.
The membership function represents the degree of belonging for x in set A in the interval [0,1]. The value of zero implies no membership and the value of one implies a complete membership. The MF can be formed as triangular, trapezoidal and bell-shape types.
Definition of the fuzzy decision
Linear programming in a fuzzy environment was first formulated by Bellman and Zadeh in 1970. The decision can be viewed as an intersection of objective functions and constricts, as well as the optimal solution for the objective.
Suppose that a fuzzy goal [??] and a fuzzy constraint [??] are given in alternative space. A fuzzy set decision [??] is formed by the intersection of [??] and [??] , which can be expressed as follows:
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