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Weekly Updates[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)
In general, when goals [[??].sub.1], ..., [[??].sub.n] and m constraints [[??].sub.1], ..., [[??].sub.m] are given, the resultant decision can be expressed as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)
The model definition is represented in Figure 1.
[FIGURE 1 OMITTED]
However, optimal fuzzy solution is not easy to interpret. Because of that, the result has to be defuzzified by converting the outcome into corresponding non-fuzzy value. It is appropriate to recommend that the crisp [D.sub.m] is a subset of D with the highest degree of membership function, as the optimal decision.
[[mu].sub.Dx](x) = Max [[mu].sub.[??]](x) [[mu].sub.Dx](x) = Max Min {[[mu].sub.i](x)} (5)
Zimmermann's (2001) FTAA flexible programming model
Before focusing on Zimmermann's FTAA flexible programming model, classical linear programming is introduced. Mathematically speaking, it can be stated as:
Max f(x) = [c.sup.T]x
Such that Ax [less than or equal to] b
x [greater than or equal to] 0
with A [member of] [R.sup.mxn], b [member of] [R.sup.m], c [member of] [R.sup.n] (6)
All coefficient of A, b, and c are crisp numbers and all constraints must be strictly satisfied. [x.sup.*] is called a solution of PL problem if [cx.sup.*] [greater than or equal to] cx for all x [member of] X. It is always considered a special kind of decision model in which its decision spare is satisfied all "goals" and "constricts".
However, as stated before, an investor's desired level of return sometimes cannot be represented by a precise numerical value. To address this issue, the fuzzy set theory is incorporated in a classical linear programming structure (6). Assume that an investor can establish an aspiration level z, that the objective function is achieved and each constrict is modeled as a fuzzy set. It can be expressed as follows:
Find x Such that [c.sup.T] x [??] z Ax [??] b x [??] 0 A [member of] [R.sup.mxn], b [member of] [R.sup.m], x [member of] [R.sup.n] (7)
where: c objective function; A constraint function; z aspiration level; m number of constraints; n number of goals; [R.sup.mxn]: m x x real matrix.
[??] and [??] denote the fuzzified version of [less than or equal to] and [greater than or equal to] and have the linguistic interpretations of "essentially smaller than or equal to" and "essentially greater than equal to".
Since the n-vector x is variable symmetric to both the objective and constraint functions, their coefficients can be substituted by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
The model (7) becomes:
Find x Such that Bx [??] d x [greater than or equal to] 0 B [member of] [R.sup.(m+1)xn], x [member of] [R.sup.n], d [member of] [R.sup.(m+1)] (8)
[[mu].sub.i](x) (MF) for i = 1, ..., m + 1 are assumed to be linear, increasing monotonically from 0 to 1 over the tolerance interval [[d.sub.i], [d.sub.i] + [p.sub.i]] as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)
[p.sub.i] are constants subjectively chosen to represent the admissible violation of constraint and objective. According to Bellman and Zadeh (1970):
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)
The crisp optimal solution is defined along the formulation
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)
Introducing a new variable k in the fuzzy set discussion, it can be illustrated simply as Figure 2.
[FIGURE 2 OMITTED]
Max [lambda]
Such that [lambda][p.sub.i] + [(Bx).sub.i] [less than or equal to] [d.sub.i] + [p.sub.i], i = 1, ..., m + 1 0 [less than or equal to] [lambda] [less than or equal to] 1 x [greater than or equal to] 0 (12)
For all feasible solutions, [[lambda].sup.*] can be interpreted as the degree of achievability of optimal solution of this fuzzy linear programming.
Ramik and Rimanek's (1985) FTAA robust programming model
However, coefficients of functions are sometimes ambiguous in nature. It should be modeled with fuzzy set theory, namely "Robust Programming".
For Ramik and Rimanek's approach, the optimization problem is defined as:
Maximizing (minimizing) the real function of n real variables,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)
where: [[??].sub.ij] & [[??].sub.ij] [member of] [M.sub.Li-Ri], L-R fuzzy numbers (Trapezoid fuzzy numbers); [direct sum] denotes the extended addition.
For L-R fuzzy numbers,
[[??].sub.ij] = ([m.sub.ij], [n.sub.ij], [[alpha]sub.ij], [[beta]sub.ij]) and [??]sub.ij] = ([p.sub.ij], [q.sub.ij], [[gamma].sub.ij], [[delta].sub.ij])
They assert that [??] [less than or equal to] [??] is valid if,
[[epsilon].sub.L] ([gamma] - [alpha]) [less than or equal to] p - m
[[epsilon].sub.R] ([beta] - [delta]) [less than or equal to] q - n
[[delta].sub.L] ([gamma] - [alpha]) [less than or equal to] p - m
[[delta].sub.R] ([beta] - [delta]) [less than or equal to] q - n (14)
where the above fuzzy numbers are defined as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)
Rewriting the fuzzy linear programming (13) with the above inequalities, the extended operation of the product of the fuzzy numbers and variable x of the constraint function is given as:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (16)
The constraint function (14) can be written as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (17)
Before the implementation of these programming models, it is important to identify the parameters. For Zimmermann's FTAA flexible programming model, the admissible violation of portfolio risk, return and proportion of assets are hypostatized to be 2%, 0.4% and 20% respectively. For Ramik and Rimanek's FTAA robust programming model, [??] and [??] should be non-negative. In order to simplify the computation, [alpha] and [beta] are presumed to be at 0.2%. [gamma] and [delta] are presumed to be at 0.3%. These fuzzy programming models can be solved with solvers in Excel or other optimal tools (i.e. MATLAB). In our study, the solver of Excel is utilized to optimize the asset portfolio.
4. THE DATA SETS FOR ASSET ALLOCATION MODELS
For the localized portfolio in Hong Kong, the Hang Seng Composite Industry Index (HSCII), HSBC Hong Kong Dollar Bond Index and Private Domestic Price Index, from July 2000 to May 2008, are deployed for our study. The corresponding sectors are shown in Tables 1 and 2.
The Hang Seng Composite Industry Index (HSCII) consists of the top 200 stocks from 11 industries in terms of average market capitalization in the past 12 months, including 38 properties & construction companies. It serves as a good indicator for the performance of various sectors of the Hong Kong stock market. In this study, the sector of properties & construction is perceived as indirect property investments.
The Private Domestic Price Index is computed by the Rating and Valuation Department in measuring the price adjustments of private properties with its quality being kept constant. In this study, this figure is used as a proxy for the performance of direct property investments.
Moreover, the HSBC Hong Kong Dollar Bond Index represents Hong Kong Dollar Bonds and measures the performance of Hong Kong Dollar denominated fixed rate debt instruments issued by the Hong Kong SAR Government and other non-government entities.
The return of an investment is the aggregate of the dividend yield and the value appreciation in the period of assessment.
[DELTA][TR.sub.t] = [I.sub.t] - [I.sub.t-1]/[I.sub.t-1] + [D.sub.t] (18)
where: [DELTA][TR.sub.t] is the total return of period t; [I.sub.t] is the index value at the end of period t; [I.sub.t-1] is the index value at the end of period t-1; [D.sub.t] is the dividend yield during the period t.
Note: Dividend yield was missing in some observation periods in some sectors, which was assigned the corresponding average of dividend yield of remaining periods.
5. ANALYSIS
The efficient frontier of portfolio with MPT is established. The monthly expected returns and standard deviations of individual assets involved in portfolio are presented and summarized in Table 3. (1)
The resulting Markowitz MPT optimization frontier is illustrated in Figure 3. It consists of portfolios with the combination of assets that maximize the expected portfolio return for a given level of risk. Portfolios on the frontier are regarded as optimum portfolios. To do this, the Hong Kong dollar bonds are considered a risk-free asset in our study. The selected portfolio consists of 19.43% in Energy, 43.98% in Utilities, 13.50% in Class D, 20.13% in Class E, and 2.97% in Hong Kong dollar bonds. The monthly expected return and standard deviation are 1.6% and 2.89% respectively (Point MP).
[FIGURE 3 OMITTED]
The expected return and portfolio standard deviation from MPT are considered the same as that in the fuzzy linear programming models, which investors wish to achieve. The corresponding model estimations of programming are presented in Tables 4 and 5, and the resultant asset allocations of the three models are illustrated in Table 6.
Some differences can be observed in the allocation of assets under the three programming models (Table 6). On the one hand, the asset allocations in the MPT and FTAA flexible programming model are similar. They have a high allocation of 44% of capital to Utilities stocks, about one-third to direct property investment (Class D and Class E), and close to one-fifth to Energy stocks. In short, both MPT and FTAA Flexible Programming Model emphasize on low-risk and low-return investments. On the other hand, the FTAA robust programming model produces a much different allocation pattern, as compared to the other two. Aside from the allocation of 18.5% of resources to Energy stocks, which is similar to that under MPT and FTAA Flexible Programming Model, the remaining capital is very evenly distributed to every other stock and real estate investment options, ranging from 4.5%-6%. One reason behind such disparities is that the portfolio risk in this model is necessarily not restricted, which allows room for relatively riskier investment. Thus, the resultant asset allocation differs from that of both MPT and FTAA flexible programming model.
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