O.R. Applications
Genetic algorithms for portfolio selection problems
with minimum transaction lots
Chang-Chun Lin
a,*, Yi-Ting Liu
ba
Department of Information Management, Chang Gung University, 259 Wen-Hwa 1st Road, Kwei-Shan, Tao-Yuan 333, Taiwan, ROC
b
Institute of Information Management, National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu 300, Taiwan, ROC Received 12 March 2006; accepted 11 December 2006
Available online 11 January 2007
Abstract
Conventionally, portfolio selection problems are solved with quadratic or linear programming models. However, the solutions obtained by these methods are in real numbers and difficult to implement because each asset usually has its min-imum transaction lot. Methods considering minmin-imum transaction lots were developed based on some linear portfolio opti-mization models. However, no study has ever investigated the minimum transaction lot problem in portfolio optiopti-mization based on Markowitz’ model, which is probably the most well-known and widely used. Based on Markowitz’ model, this study presents three possible models for portfolio selection problems with minimum transaction lots, and devises corre-sponding genetic algorithms to obtain the solutions. The results of the empirical study show that the portfolios obtained using the proposed algorithms are very close to the efficient frontier, indicating that the proposed method can obtain near optimal and also practically feasible solutions to the portfolio selection problem in an acceptable short time. One model that is based on a fuzzy multi-objective decision-making approach is highly recommended because of its adaptability and simplicity.
Ó 2007 Published by Elsevier B.V.
Keywords: Portfolio selection; Markowitz model; Minimum transaction lots; Genetic algorithm; Fuzzy multi-objective decision making
1. Introduction
Markowitz’ model[1]uses the mean and variance of historical returns to measure the expected return and risk of a portfolio. Conventionally, such portfo-lio selection problems are solved with quadratic or linear programming models under the assumption that the asset weights in the portfolio are real
num-bers, which are difficult to implement. Specifically, each asset has its minimum transaction lot, while the solutions involve only real-number asset weights rather than asset trading units. For example, stocks might be traded at the unit one share, and mutual funds have their individual minimum trading amounts. Thus, the solution obtained by Marko-witz’ model must be integers to be applicable in practice.
Other than Markowitz’ model, Speranza [2], Mansini and Speranza [3,4], and Kellerer et al. [5]
proposed their respective portfolio selection models
0377-2217/$ - see front matter Ó 2007 Published by Elsevier B.V. doi:10.1016/j.ejor.2006.12.024
* Corresponding author. Tel.: +886 32118800; fax: +886
32118020.
E-mail address:chanclin@ms29.hinet.net(C.-C. Lin).
based on Konno and Yamazaki’s mean absolute deviation (MAD) model[6]. Speranza[2]proposed a mixed integer program considering realistic char-acteristics in portfolio selection, such as minimum transaction lots and the maximum number of secu-rities, and suggested a simple two-phase heuristic algorithm to solve the proposed integer program. Mansini and Speranza[4]showed that the portfolio selection problem with minimum transaction lots is an NP-complete problem and proposed three heu-ristic algorithms to solve the problem. Based on the MAD model, Konno and Wijayanayake[7] pro-posed an exact algorithm for portfolio optimization problems under concave transaction costs and min-imum transaction lots. However, minmin-imum transac-tion lots were not the major concern in their study. Later, Mansini and Speranza [8] derived a mean-safety model with side constraints from the MAD model, and proposed an exact algorithm to solve for portfolios under the consideration of transaction costs and minimum transaction lots. However, Markowitz’ model is still the most widespread port-folio selection model. Solving the portport-folio selection problem based on Markowitz’ model and, simulta-neously, considering minimum transaction lots are of practical significance. However, it appears that no methods in the past solving the portfolio selec-tion problem with minimum transacselec-tion lots were based on Markowitz’ model.
Apart from considering minimum transaction lots, Markowitz’ model is intrinsically a multi-objective decision-making (MODM) problem whose decision criteria conflict with each other. The return is required to be maximized and the risk minimized. However, the risk is often high when the return is maximized, and the return is often low when the risk is minimized. Researchers have pro-posed different approaches, such as goal program-ming [10] and multiple objective programming
[11], to solve multi-objective portfolio selection problems. Lee and Lerro [10] pioneered goal pro-gramming in portfolio selection, but their method is not based on Markowitz’ model. Arenas Parra et al. [12] proposed a fuzzy goal programming approach to solve a portfolio selection problem, using a multi-index model to estimate the return and risk of portfolios and treating fuzzy goals as fuzzy numbers. However, their method ignores the existence of minimum transaction lots. Apart from goal programming, fuzzy programming can also be used to solve MODM problems[13], but conven-tional fuzzy programming methods cannot
incorpo-rate objective weights that are usually important for the decision maker (DM) to express his/her prefer-ence regarding return and risk. Lin[14]recently pro-posed a weighted max–min model to incorporate objective weights with fuzzy programming. Promis-ingly, this method can be applied to solving the portfolio selection problem. Using this method, the DM obtains portfolios through setting appro-priate objective weights.
The portfolio selection problem with minimum transaction lots is a combinatorial optimization problem whose feasible region is not continuous. Studies have shown that genetic algorithm (GA) is a promising approach to combinatorial optimization problems. Shoaf and Foster [9] applied GA to Markowitz’ portfolio selection problem and found that the time complexity of GA approximates Oðn log nÞ and is better than that of quadratic pro-gramming. However, their approach does not con-sider the constraints of minimum transaction lots. The solution obtained will be much more realistic if the constraints of minimum transaction lots are considered when solving with GAs. Therefore, this study presents and compares three possible decision models with their corresponding GAs to solve the portfolio selection problem with minimum transac-tion lots, an NP-complete MODM problem. The proposed GAs are highly efficient and effective in providing near optimal solutions within a few minutes.
The rest of this paper is organized as follows. Based on Markowitz’ model, Section 2 derives two models for portfolio selection with minimum trans-action lots. Section 3 presents the fuzzy MODM model of Lin[14], and then an integrated model for portfolio selection based on fuzzy MODM and con-sidering minimum transaction lots is proposed. The GAs for solving the proposed models are stated in Section 4. Section 5 presents the experiments with the proposed GAs on the mutual funds in Taiwan and discusses the results. Finally, Section6concludes the paper and suggests possible future research.
2. Portfolio selection and minimum transaction lots Originally an MODM problem, portfolio selec-tion attempts to maximize the rate of return and minimize the portfolio risk simultaneously. In prac-tice, the MODM problem is often degenerated to a single objective one by introducing a preference structure to compromise between the objectives, or simply optimizing one of the two objectives while
bounding the remaining one. The latter approach leads to the well-known Markowitz’ model [1]. Markowitz’ model assumes that rational investors aim to maximize return under a certain risk level or minimize risk above a certain return level. For convenience, the decision maker usually fixes the expected rate of return and then minimizes the port-folio risk under this return constraint. Conse-quently, Markowitz’ model can be expressed as a quadratic programming problem as follows: Model 1 min r2 p¼ Xn i¼1 Xn j¼1 wiwjrij ð1Þ subject to rp¼ Xn i¼1 wiri¼ r; ð2Þ Xn i¼1 wi¼ 1; ð3Þ
where n is the number of assets; rp is the expected
rate of return; r is the required rate of return; riis
the expected rate of return of asset i; rijis the return
covariance between assets i and j, and r2
p is the
re-turn variance of the portfolio. wi represents the
weight of budget invested in asset i. As in the origi-nal Markowitz’ model, wiP0, that is, no short
sell-ing is allowed.
However, in practice, each asset has its minimum transaction lot that should be taken into consider-ation in finding minimum-risk portfolios. Solutions to portfolio selection problems thus must be inte-gers because real-number solutions might be diffi-cult to implement. As a consequence, the portfolio selection models need to be modified to consider the minimum transaction lots. A modified Marko-witz’ model can be formulated as follows to con-sider the minimum transaction lots.
Model 2 min r2 p¼ Xn i¼1 Xn j¼1 wiwjrij; ð1Þ subject to ep¼ Xn i¼1 xiciriP br; ð4Þ Xn i¼1 xici6b; ð5Þ wi¼ cixi Pn i¼1cixi ; i¼ 1; . . . ; n; ð6Þ xi2 Z;
where b represents the budget and ep the expected
portfolio return. ci denotes the unit price of asset
i, and xirepresents the units invested in asset i. Since
the same rate of return can be achieved with differ-ent budgets, Eq.(4)uses portfolio return instead of rate of return to maximally utilize the budget. More return is earned as more capital is invested. Never-theless, constraint(5) requires that the total invest-ment be below the budget. Similar to (3), constraint (6) requires that the sum of weights be unity. Notably, this constraint causes the model to be nonlinear.
Similar to Model 1, Model 2 minimizes the investment risk with respect to a given rate of return r except that the optimal solution must be in inte-gers. However, the modification might make a port-folio with a rate of return equals r unobtainable, namely, the rate of return might not exactly equal r. Owing to constraints (4), the rate of return can only exceed r. In a portfolio selection problem whose solution is in real numbers, the desired rate of return along with its corresponding risk repre-sents a target portfolio to be achieved. The target portfolio can always be achieved in a real-number solution space. However, in an integer solution space, the target portfolio might be unreachable, and the optimal solution conforms to constraint
(4) can be far from the target. Specifically, the risk of the optimal solution to Model 2 might be too high to be acceptable if its rate of return is to exceed the desired r. Conversely, a portfolio with a rate of return slightly less than the desired r might be more appealing if the portfolio risk can accordingly be lowered significantly. This alternative leads to another decision model that minimizes the distance between the obtained and the target portfolios. The model is as follows: Model 3 min d2¼ rp r r r 2 þ ep br br br 2 ; ð7Þ subject to r2 p¼ Xn i¼1 Xn j¼1 wiwjrij; ð1Þ ep ¼ Xn i¼1 xiciri; ð8Þ Xn i¼1 xici6b; ð5Þ wi¼ cixi Pn i¼1cixi ; i¼ 1; . . . ; n; ð6Þ xi2 Z;
where r is the minimum risk corresponding to the desired r. Notably, the risk and return are normal-ized to eliminate the influence of scales. As shown inFig. 1, r*and r*represent the maximal rate of
re-turn and the minimal risk that can be achieved, and rand r are their corresponding risk and rate of return. Therefore, ðr; rÞ and ðr; rÞ represent
the lower and upper ends of the efficient frontier. Obviously, this model is also a nonlinear integer programming problem that is difficult and time-con-suming to solve. Fortunately, genetic algorithms are helpful in solving difficult nonlinear and/or combi-natorial optimization problems. That is why this study uses genetic algorithms to solve the portfolio selection problem with minimum transaction lots. Nevertheless, despite seemingly workable, the mod-els still have some potential problems. The same rate of return might involve different levels of risk in dif-ferent markets or in different time periods. Deter-mining an appropriate value of r to reflect the investor risk preference is a tricky business. Further-more, the desired r for Model 3 must be obtained in advance by solving Markowitz’ model (Model 1). Obtaining the target portfolio might be time-con-suming when the number of assets is large. The next section presents an alternative approach that might be more applicable to solving the portfolio selection problem, which is a fuzzy MODM approach requir-ing no predetermined r and r, and might be promis-ing when applied to different scenarios.
3. Fuzzy multi-objective portfolio selection
Conventional MODM approaches involve goal programming, compromise programming and fuzzy programming. Goal programming minimizes the weighted sum of absolute deviations to the objective
goal values subjectively given by the DM. Model 3 can be considered as a goal programming approach. The major problem with goal programming lies in the incommensurability between objectives of differ-ent measures. Differdiffer-ent metrics of objectives lead to incommensurable goal values and deviations from them. Summing up these incommensurable devia-tions is simply unreasonable. Compromise pro-gramming minimizes the sum of weighted distances to the ideal solution, based on some dis-tance metric. Usually, compromise programming uses Minkowsky distance Lpas the distance metric,
and thus, the solution obtained depends on the parameter p. Although the incommensurability problem in compromise programming can be resolved by normalization, the choice of an appro-priate distance parameter p becomes a tricky prob-lem to the DM. Fuzzy programming has been widely and successfully applied to various MODM problems, such as those in the studies of Chang and Chen [15], Abd El-Wahed and Abo-Sinna
[16], and Rasmy et al. [17]. Fuzzy programming helps to resolve the incommensurability problem between objectives, and thus provides an alternative for solving MODM problems.
Since Zadeh [18] proposed fuzzy set theory in 1965, related theories and methodologies have been widely applied to many fields, including operations research, decision science, engineering and artificial intelligence. Fuzzy programming arose when Zimmermann [13]first applied the max–min opera-tor of Bellman and Zadeh[19]to MODM problems. The DM defined a membership function for each objective to represent the achieved level of that objective regarding different objective values. Fuzzy programming thus maximizes the minimal achieved level among the objectives. Narasimhan [20] incor-porated fuzzy sets with goal programming before many related approaches for such problem, fuzzy goal programming, were proposed. Fuzzy program-ming can be regarded as a special case of fuzzy goal programming where the goals to achieve are pre-cisely the optimal values for the objectives.
A fundamental fuzzy goal programming problem with m fuzzy goals can be formulated as follows: Model 4 Find x; ð9Þ to satisfy fjðxÞfPgj; j¼ 1; 2; . . . ; m; ð10Þ subject to Bx 6 b; ð11Þ x P 0; r σ ∗ r − r ∗ σ σ−
where x is the solution in vector form with compo-nents x1; x2; . . . ; xn; fjðxÞ means the jth objective, and
Bx 6 b are the system constraints in vector nota-tion. The notation fP means that the relation P is fuzzy such that the goal gjcan be partially achieved.
The achievement of gjis measured by a membership
function defined for that goal. Any fuzzy goal pro-gramming problem can be expressed with Model 4 without any loss of generality because everyffi con-straint can be replaced with a e6constraint and afP constraint, and every e6constraint can be converted to an equivalent fP constraint.
Since all objectives might not be fully achieved simultaneously, the DM may define a lower toler-ance limit and a corresponding membership func-tion for each objective to determine the achieved level of that objective. A membership function ljðfjÞ for the jth goal can be defined as
ljðfjÞ ¼ 1 if gj6fjðxÞ; fjðxÞlj gjlj if lj 6fjðxÞ < g j; 0 if fjðxÞ < lj; 8 > > < > > : ð12Þ
where gjand ljare the specified goal value and the
lower tolerance limit for the jth objective.
After Zimmermann [13], several alternative methods [21–23]have been proposed to solve fuzzy MODM problems. However, no model seems to consider objective weights. Lin [14] recently pro-posed a weighted max–min model for fuzzy MODM problems. The weighted max–min model can be expressed as follows: Model 5 max k; ð13Þ subject to -jk 6 fjðxÞ lj gj lj ; j¼ 1; 2; . . . ; m; ð14Þ Bx 6 b; ð11Þ x P 0:
where k represents the minimal achieved level among the objectives; -j denotes the weight of the
jth objective.
Lin’s model can readily be applied to the portfo-lio selection problem. Let return and risk be the first and second objectives. Then let br and r*be their
goal values, and br and r their tolerance limits. The membership functions for return and risk can be defined as follows, respectively.
lrðepÞ ¼ 1 if br6ep; epbr brbr if br 6e p< br; 0 if ep< br; 8 > > < > > : ð15Þ lrðrpÞ ¼ 1 if rp6r; rrp rr if r6rp<r; 0 if r <r p: 8 > < > : ð16Þ
Let -rand -rdenote the weights of return and risk,
respectively. The fuzzy programming model for the portfolio selection problem can be formulated as Model 6. Model 6 max k; ð13Þ subject to -rk 6 ep br br br; ð17Þ -rk 6 r r p r r; ð18Þ r2p¼ Xn i¼1 Xn j¼1 wiwjrij; ð1Þ ep¼ Xn i¼1 cirixi; ð8Þ Xn i¼1 xici6b; ð5Þ wi¼ cixi Pn i¼1cixi ; i¼ 1; . . . ; n: ð6Þ xi2 Z:
With Model 6, the DM expresses his/her preferences for the return and risk through assigning the objec-tives weights. DMs with higher risk aversion can give the risk higher weights to raise its achieved le-vel. The portfolio risk can be lowered through sac-rificing the achieved level of return. By contrast, DMs with lower risk aversion give the return higher weights to raise its achieved level. Assume, without losing generality, that -rþ -r¼ 1. -r¼ 1 indicates
that the risk-neutral DM considers only return, and the left-hand side of constraint(18)equals zero, rep-resenting that any risk is acceptable. By contrast, -r¼ 1 indicates that the DM considers only risk
and thus that any return is acceptable. The DM can easily obtain a satisfied efficient portfolio through assigning proper weights to the objectives. Model 6 still represents a time-consuming nonlinear combinatorial optimization problem such that a ge-netic algorithm is necessary for solution.
4. Genetic algorithms
Many optimization problems are very complex and difficult to solve by conventional methods. Therefore, evolutionary algorithms simulating natu-ral processes were developed to solve them [24]. Among them, GA has been successfully in solving many optimization problems. Originally proposed by Holland[25], GA is a stochastic searching tech-nique based on the mechanism of genetics and nat-ural selection [26]. The models for portfolio selection with minimum transaction lots are combi-natorial optimization problems. Many studies have showed that GAs can efficiently find near optimal or even the optimal solutions for many combinato-rial optimization problems. Most GAs, including those in this study, use roulette wheel selection, one-point crossover and one-point mutation [25]. The roulette wheel approach belongs to the fit-ness-proportional selection and can select a new population with respect to the probability distribu-tion based on fitness values. The GAs in this study employs a different evolution mechanism from con-ventional ones. In the original GA[25], all parents are replaced by their offspring to form a new gener-ation, which is called generational replacement. Off-spring may be less fit than their parents because GAs are blind, causing some fitter chromosomes to be lost from the evolutionary process. Therefore, several replacement processes have been examined to resolve this problem. Holland [25] suggested to replace an arbitrarily selected chromosome with a new-born offspring. DeJong[27]proposed a
crowd-ing strategy that selects the parent that is most clo-sely resembles the new-born offspring to die when the offspring is born. Gen and Cheng[28]proposed that both parents and their immediate offspring are all candidates for the new generation to preserve the best chromosome. The GAs in this study follows a similar strategy to prevent the populations from degenerating. The population evolves continually without being replaced by another population. Let PðtÞ be the population at iteration t.Fig. 2describes the basic GA.
Although the GA represents a possible way of solving the models, some problems remain in its implementation. The main problem in applying a GA to constrained optimization problems is how to deal with the constraints. Constraints can be dealt with strategies such as reject, repairing and penalty strategies, and the strategy of modifying genetic operators [28]. The reject strategy excludes infeasible solutions immediately on generation, resulting in an efficient GA. The repairing strategy transforms an infeasible solution into a feasible one through a repairing process. The difficulty in designing a repairing process to comply with the problem weakens the repairing strategy. The pen-alty strategy uses a penpen-alty function to penalize all infeasible solutions, hoping that infeasible solutions might evolve toward feasible. Finally, the strategy of modifying genetic operators aims to devise prob-lem-specific representations and specialized genetic operators to maintain feasibility. Comparatively, the strategies of penalty and modifying genetic oper-ators appear more suitable for this study. Therefore,
begin 0 ← t ; initialize P(t); evaluate )P(t ;
find the best and worst chromosomes of P(t);
while t < a predetermined iteration number do
select two parents from P(t);
generate two offspring by crossover and mutation; evaluate the offspring;
if the offspring is fitter than the worst chromosome of P(t) then
randomly select a parent except the best one and replace it;
find the best and worst chromosomes of P(t);
1 + ← t t ; end end
the strategies of penalty and modifying genetic oper-ators were used in this study to deal with constraints
(4) and (5), respectively.
4.1. Encoding and decoding
The way to encode a solution into a chromosome is a key issue in using GAs. Although the solutions to the proposed models are integers, this study opted to use real numbers, for the sake of opera-tional simplicity, to encode the weight wi instead
of directly encoding the trading units xiwith an
inte-ger. The genes of a chromosome are real numbers between 0 and 1 to represent the weights invested in the assets. However, the summation of these weights might not be 1 in the initialization stage or after genetic operations and thus violate con-straint (6). To overcome this problem, the weights are normalized as follows:
w0i¼Pwni i¼1wi
; ð19Þ
where w0
irepresents the new weight invested in asset
i after normalization. Even though, the normalized weights are still inapplicable, as stated previously. Multiplying wi by the budget b gives the actual
amount that could be invested in asset i. However, there are usually residuals left. The actual units in-vested in asset i is then obtained by
xi¼
bw0i ci
; i¼ 1; . . . ; n: ð20Þ
The residuals make the total investment below the budget b and thus conformant with constraint (5). However, the actual investment weights are prone to change. Eq. (6) is used to compute the actual weights before computing the return variance of the portfolio. The wiof the chromosomes in the
ini-tial population is randomly generated before being normalized using Eq.(19). Two of the three effective constraints in the models, constraints (5) and (6), can be satisfied with the above weight normalizing and budget controlling strategies. However, con-straint (4) in Model 2 might be easier to handle using a penalty strategy, which will be stated in the next subsection.
4.2. Fitness functions
Three GAs were implemented to verify the effec-tiveness of Models 2 and 3 and 6. The primary dif-ferences between these models lie in their objective
functions, leading to different fitness functions in their respective GAs. With the encoding and decod-ing scheme, the constraints in Models 2 and 3 are resolved, leaving constraint (4) in Model 2 to be dealt with a penalty strategy. The penalty function that impels the solutions to satisfy constraint(4) is formulated as follows to make the actual return exceed the expected return.
pðxÞ ¼ br Pn i¼1 xiciri if br >P n i¼1 xiciri; 0 otherwise: 8 < : ð21Þ
Incorporating the objective function (1) and the penalty function(21), the fitness function for Model 2 can be defined as,
fitness¼ expðkðr2
pþ MpðxÞÞÞ; ð22Þ
where k is a positive constant and M a large positive number. The negative exponent transforms the min-imization problem into an equivalent maxmin-imization problem for the GA to solve. The exponential func-tion with constant k confines the range of the fitness and thus alleviates the selection pressure of chromo-somes with higher fitness, to prevent the GA from premature convergence. The large positive number M forces the solution to meet constraint (4)before minimizing the portfolio risk.
Model 3 is also a minimization problem such that the fitness function for this problem is defined sim-ilarly as,
fitness¼ expðkd2Þ: ð23Þ The fitness function for Model 6, a maximization problem, is simply its objective function k, and thus fitness¼ min ep br -rðbr brÞ ; r r p -rðr rÞ : ð24Þ 5. Empirical study
This study used Taiwanese mutual fund data from the year 1997 to 2000 to test the proposed models and GAs. The monthly rates of return were used to determine the mean rates of return of the mutual funds and the return covariances between the assets in each year. The monthly rates of return were used instead of weekly ones because monthly horizon is more likely to be the real investment hori-zon than weekly horihori-zon when investing in mutual funds. Studies[29,30]have shown that portfolio per-formance indexes such as Sharpe’s measure do
depend on sampling horizon and thus its estimated risk measure. However, while a simple experiment has been conducted to find out that the observed risks of weekly data tend to be significantly larger than those of monthly data, whether sampling hori-zon affects the obtained portfolios remains uninves-tigated. Therefore, this study has chosen to adopt monthly data.
Also, the r*, r, r*and r for each year’s data
were obtained with Model 1 and LINGO.Table 1
shows the characteristic values of the terminal port-folios on each year’s efficient frontier. To measure and compare the effectiveness of the proposed mod-els, a number of target portfolios spread on the effi-cient frontiers are selected for the GAs to approach. For Models 2 and 3, let the desired rate of return r be arþ ð1 aÞr. Eleven values of a raging from 0
to 1 and differenced by 0.1 were used. Model 2 was then used to obtain the optimal r with respect to these target rates of return. Table 2 shows the 11 target portfolios for each year. However, the desired rate of return disappears in Model 6. The solution to this model depends upon the objective weights ð-r;-rÞ. Since each target portfolio corresponds
to a specific set ofð-r;-rÞ, the corresponding
objec-tive weights are derived inversely from the 11 target portfolios to be used in Model 6 and its correspond-ing GA. From Eqs.(17) and (18), for a target port-folio on the efficient frontier there would be -r
-r
¼lr
lr
: ð25Þ
Let -r¼ 1 -r. Eq.(25)leads to,
-r ð1 -rÞ ¼ brp br br br r r p r r : ð26Þ
Thus, the -rfor a target portfolio withðr; rÞ can be
obtained by -r¼ r r r r rp r r rþ r r r r : ð27Þ
Table 2 also shows the corresponding objective weights of the target portfolios.
The parameter setting of the proposed GAs is as follows. For each year’s data, the number of popu-lation and the chromosome length both equal the fund number. The crossover and mutation rates were set to 1 and 0.05, respectively. The number of iterations was 5000. The values of k and the M were set to 0.005 and 100, respectively. The budget
Table 1
Characteristics of efficient frontiers
Year Number of funds r*(%) r(%) r*(%) r(%)
1997 129 6.663 0.480 0.007 15.296
1998 160 3.674 0.480 0.014 7.362
1999 197 9.158 0.388 0.014 8.925
2000 204 0.935 0.422 0.007 1.557
Table 2
Characteristics of target portfolios
Year -r r (%) r(%) 1997 0.000 0.480 0.007 0.097 1.098 1.076 0.189 1.716 2.210 0.277 2.335 3.344 0.364 2.953 4.600 0.450 3.571 5.955 0.536 4.189 7.342 0.620 4.808 8.745 0.704 5.426 10.159 0.787 6.044 11.582 1.000 6.663 15.296 1998 0.000 0.480 0.014 0.097 0.799 0.538 0.192 1.119 1.171 0.284 1.438 1.806 0.374 1.758 2.442 0.462 2.077 3.077 0.547 2.396 3.713 0.631 2.716 4.348 0.712 3.035 4.984 0.795 3.355 5.657 1.000 3.674 7.362 1999 0.000 0.388 0.014 0.096 1.265 0.581 0.188 2.142 1.232 0.275 3.019 1.883 0.358 3.896 2.534 0.437 4.773 3.185 0.512 5.650 3.836 0.584 6.527 4.487 0.656 7.404 5.181 0.788 8.281 6.767 1.000 9.158 8.925 2000 0.000 0.422 0.007 0.092 0.474 0.024 0.169 0.525 0.036 0.251 0.576 0.168 0.337 0.627 0.337 0.430 0.679 0.530 0.530 0.730 0.732 0.636 0.781 0.936 0.749 0.832 1.143 0.871 0.884 1.350 1.000 0.935 1.557
b was set to 200, and the unit price ciis an integer
that was randomly assigned a value 1–5, because a
budget that is too large or unit prices that are too small ease the problem by increasing the number
Table 3
Comparisons between models
Year Model 4 Model 5 Model 9
rp(%) rp(%) Deviation (%) rp(%) rp(%) Deviation (%) rp(%) rp(%) Deviation (%)
1997 0.494 0.012 0.015 0.494 0.013 0.015 0.486 0.012 0.008 1.155 1.250 0.182 1.117 1.177 0.102 1.159 1.249 0.183 1.789 2.453 0.253 1.729 2.330 0.121 1.769 2.400 0.197 2.404 3.561 0.228 2.357 3.457 0.115 2.378 3.523 0.184 3.073 5.026 0.442 2.979 4.766 0.168 2.981 4.908 0.310 3.703 6.476 0.537 3.555 6.149 0.195 3.611 6.235 0.282 4.299 7.821 0.491 4.191 7.543 0.201 4.168 7.599 0.258 4.873 9.167 0.427 4.804 8.946 0.200 4.755 8.879 0.144 5.514 10.509 0.361 5.420 10.318 0.159 5.403 10.239 0.083 6.091 11.750 0.175 6.049 11.680 0.098 6.007 11.613 0.049 6.663 15.296 0.000 6.650 15.280 0.021 6.663 15.296 0.000 Average 0.283 0.127 0.154 1998 0.508 0.048 0.044 0.499 0.046 0.037 0.503 0.043 0.037 0.853 0.701 0.172 0.806 0.623 0.086 0.854 0.710 0.181 1.196 1.401 0.242 1.138 1.272 0.102 1.175 1.336 0.174 1.538 2.071 0.283 1.461 1.922 0.118 1.512 2.027 0.233 1.865 2.707 0.287 1.767 2.552 0.111 1.811 2.643 0.208 2.190 3.418 0.359 2.117 3.259 0.186 2.106 3.271 0.196 2.504 4.013 0.319 2.429 3.853 0.144 2.444 3.893 0.187 2.845 4.683 0.359 2.752 4.521 0.176 2.696 4.451 0.105 3.154 5.339 0.374 3.064 5.086 0.106 2.995 5.075 0.099 3.457 6.274 0.626 3.357 5.767 0.110 3.240 5.524 0.176 3.674 7.362 0.000 3.674 7.362 0.000 3.674 7.362 0.000 Average 0.279 0.107 0.145 1999 0.415 0.016 0.027 0.388 0.016 0.002 0.415 0.016 0.027 1.454 0.866 0.342 1.253 0.708 0.127 1.406 0.840 0.294 2.394 1.620 0.463 2.183 1.407 0.179 2.328 1.542 0.361 3.357 2.375 0.597 2.987 2.071 0.190 3.200 2.259 0.417 4.320 3.234 0.818 3.832 2.734 0.209 3.784 2.978 0.458 5.138 4.053 0.941 4.761 3.554 0.369 4.760 3.626 0.441 5.979 4.606 0.838 5.479 4.229 0.428 5.458 4.267 0.471 7.031 5.440 1.078 6.372 4.900 0.441 6.281 4.791 0.391 7.855 6.038 0.968 7.339 5.647 0.470 7.015 5.393 0.443 8.980 8.659 2.017 7.955 6.737 0.327 7.929 6.056 0.793 9.158 8.925 0.000 9.158 8.925 0.000 9.158 8.925 0.000 Average 0.735 0.249 0.372 2000 0.434 0.019 0.016 0.436 0.020 0.019 0.400 0.013 0.023 0.486 0.051 0.029 0.485 0.050 0.028 0.487 0.051 0.031 0.538 0.168 0.133 0.487 0.051 0.040 0.538 0.168 0.133 0.610 0.569 0.403 0.538 0.154 0.041 0.571 0.427 0.259 0.682 0.789 0.455 0.601 0.481 0.146 0.598 0.507 0.173 0.765 1.040 0.517 0.654 0.640 0.113 0.667 0.722 0.192 0.807 1.182 0.457 0.706 0.873 0.144 0.688 0.780 0.064 0.908 1.476 0.554 0.763 0.962 0.032 0.698 0.813 0.149 0.932 1.547 0.417 0.830 1.277 0.135 0.715 0.971 0.208 0.935 1.557 0.214 0.830 1.277 0.090 0.715 0.971 0.414 0.935 1.557 0.000 0.935 1.557 0.000 0.935 1.557 0.000 Average 0.290 0.072 0.150
of feasible combinations and thus might be unable to verify the effectiveness of the proposed methods. The experiments were conducted on a PC with a Pentium IV 1.8 GHz CPU, and the average execu-tion time was about 3–5 minutes. For the data of each year, 10 trials with each GA were conducted for each target portfolio.Table 3shows the average results of each of the 10 trials. The deviation between the obtained and target portfolios are mea-sured in terms of Euclidean distance.
Regardless of the year, the average deviations lead to an obvious and consistent conclusion that Model 5 obtains portfolios closest to the target. Compared with Model 3 and Model 6 obtains port-folios farther from the target than those by Model 3 yet much better than those by Model 2. As stated above, constraint(4) in Model 2 confines the solu-tions to the area above the horizontal line rp¼ r
and below the efficient frontier. Compared with other models, the smaller feasible region for Model 2 leads to poorer solutions. Unless the DM insists that the rp of the obtained portfolio must exceed
the desired r, Model 2 does not seem to be a good approach to the portfolio selection problem. Also,
Figs. 3–6show that the GA obtains portfolios very close to the efficient frontiers, indicating that the proposed approach is efficient in finding efficient portfolios under the constraint of minimum transac-tion lots.Figs. 3 and 4 show that the GAs obtain integer solutions whoseðr; rÞ are very close to the target values. However, Figs. 5 and 6 reveal that the differences between the integer and target solu-tions grow large as the fund number increases. Fund number may be the main cause of the solution degeneration because the fund number increases the population and chromosome sizes and thus the solution space such that the GA failed to find efficient integer solutions within 5000 iterations.
Another cause might be that compared with those of 1999 and 2000, more mutual funds of 1997 and 1998 approach their respective efficient frontiers so that more combinations of assets near the efficient frontiers can be found, increasing the possibility of finding better solutions. Consequently, increasing the program running time or reducing the fund number might help when the number of assets is large.
Fig. 7shows the experimental results obtained by Model 6 for the year 2000 after the fund number is
0.0 2.0 4.0 6.0 8.0 0.0 5.0 10.0 15.0 20.0 Risk (%) Rate of return (%) Target Model 2 Model 3 Model 6
Fig. 3. Model performance with funds in 1997.
0.0 1.0 2.0 3.0 4.0 0.0 2.0 4.0 6.0 8.0 10.0 Risk (%) Rate of return (%) Target Model 2 Model 4 Model 6
Fig. 4. Model performance with funds in 1998.
0.0 2.0 4.0 6.0 8.0 10.0 0.0 2.0 4.0 6.0 8.0 10.0 Risk (%) Rate of return (%) Target Model 2 Model 4 Model 6
Fig. 5. Model performance with funds in 1999.
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 Risk (%) Rate of return (%) Target Model 2 Model 4 Model 6
reduced from 204 to 47 by simply eliminating funds with negative expected rate of return. Though the eliminating method is naı¨ve, the efficient frontier does not seem to change, and the performance was remarkably improved after the asset number was reduced. Nevertheless, the naı¨ve and rough method is not recommended because the efficient frontier might change in other cases. A more sophis-ticated method remains to be investigated to retain the efficient frontier while reducing the asset number.
According to the experimental results, all the proposed models obtained the solutions within a few minutes, with an average budget utilization rate above 96%. Models 3 and 6 appear more effective than Model 2. However, as Models 2 and 3 depends on the specification of a desired rate of return and additionally requires solving for the minimum risk corresponding to the specified rate of return using Markowitz’ model. Besides, the clients need to assure themselves that their desired rates of return are sufficient to reflect their preference for risk. On the other hand, Model 6 (the fuzzy MODM approach) requires only the weights regarding return and risk, which can be applied in various situations and time periods if only the client’s pref-erence for risk aversion remains unchanged. There-fore, Model 6 is recommended for practical use. Notably, the number of assets significantly influ-ences the performance of the proposed GAs even if the program execution time is prolonged. An effective way of reducing the asset number without changing the efficient frontier is crucial in finding quality solutions.
6. Conclusion
This study proposes decision models for portfolio selection problems with minimum transaction lots
and uses genetic algorithms to solve the models. The proposed models include one directly modified from the well-known Markowitz’ model, one mini-mizing the distance between the target and obtained portfolios, and one derived from a fuzzy multi-objec-tive decision making approach. Genetic algorithms were devised to solve the portfolio selection problem formulated by the models. The results of empirical studies show that the genetic algorithms for these models can obtain near-optimal within a reasonably short time. The obtained solutions not only are appli-cable in practice, but also exhibit high mean-variance efficiency. The model incorporates a fuzzy multi-objective decision-making approach is recom-mended because of its adaptability and simplicity. With this approach, the DM is able to express his/ her preferences for the return and risk through assigning the weights of return and risk. Notably, sifting assets ahead can not only save the computa-tion time, but also improve the solucomputa-tion quality. Con-sequently, how to sift the assets without changing the efficient frontier and the solution quality is not only one of the possible future works, but is critical to solving the portfolio selection problem. Another possible future research direction is how to adjust existing portfolios to look after both transaction costs and the workability and efficiency of portfolios. Acknowledgements
The authors thank the National Science Council of the Republic of China, Taiwan for financially supporting the research under Contract No. NSC 91-2416-H-168-005.
References
[1] H. Markowitz, Portfolio selection, Journal of Finance 7 (1952) 77–91.
[2] M.G. Speranza, A heuristic algorithm for a portfolio optimization model applied to the Milan Stock Market, Computers and Operations Research 23 (1996) 433–441. [3] R. Mansini, M.G. Speranza, On selection a portfolio with
fixed costs and minimum transaction lots, Report no. 134, Dip. Metodi Quantitativi, University of Brescia, Italy, 1997. [4] R. Mansini, M.G. Speranza, Heuristic algorithms for the portfolio selection problem with minimum transaction lots, European Journal of Operational Research 114 (1999) 219– 233.
[5] H. Kellerer, R. Mansini, M.G. Speranza, Selecting portfolios with fixed costs and minimum transaction lots, Annals of Operations Research 99 (2000) 287–304.
[6] H. Konno, H. Yamazaki, Mean-absolute deviation portfolio optimization model and its application to Tokyo Stock Market, Management Science 37 (1991) 519–531.
0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.5 1.0 1.5 2.0 Risk (%) Rate of return (%) Target Before reduction After reduction
[7] H. Konno, A. Wijayanayake, Portfolio optimization problem under concave transaction costs and minimal transaction unit constraints, Mathematical Programming 89 (2001) 233–250. [8] R. Mansini, M.G. Speranza, An exact approach for portfolio
selection with transaction costs and rounds, IIE Transac-tions 37 (2005) 919–929.
[9] J. Shoaf, J.A. Foster, The efficient set GA for stock portfolios, In: Proceedings of the Decision Science Institute, Orlando, 1996, pp. 571–573.
[10] S.M. Lee, A.J. Lerro, Optimizing the portfolio selection for mutual funds, Journal of Finance 28 (1973) 1087–2001. [11] K. Konno, H. Shirakawa, H. Yamazaki, A mean-absolute
deviation-skewness portfolio optimization model, Annals of Operations Research 45 (1993) 205–220.
[12] M. Arenas Parra, A. Bilbao Terol, M.V. Rodrı´guez Urı´a, A fuzzy goal programming approach to portfolio selection, European Journal of Operational Research 133 (2001) 287– 297.
[13] H.-J. Zimmermann, Fuzzy programming and linear pro-gramming with multiple objective functions, Fuzzy Sets and Systems 1 (1978) 45–55.
[14] C.C. Lin, A weighted max–min model for fuzzy goal programming, Fuzzy Sets and Systems 142 (2004) 407–420. [15] P. Chang, Y. Chen, A fuzzy multicriteria decision making
method for technology transfer strategy selection in biotech-nology, Fuzzy Sets and Systems 63 (1994) 131–139. [16] W.F. Abd El-Wahed, M.A. Abo-Sinna, A hybrid fuzzy-goal
programming approach to multiple objective decision mak-ing problems, Fuzzy Sets and Systems 119 (2001) 71–85. [17] M.H. Rasmy, S.M. Lee, W.F. Abd El-Wahed, A.M. Ragab,
M.M. El-Sherbiny, An expert system for multiobjective decision making application of fuzzy linguistic preferences and goal programming, Fuzzy Sets and Systems 127 (2002) 209–220.
[18] L.A. Zadeh, Fuzzy sets, Information and Control 8 (1965) 338–353.
[19] R. Bellman, L.A. Zadeh, Decision-making in a fuzzy environment, Management Sciences 17B (1970) 141– 164.
[20] R. Narasimhan, Goal programming in a fuzzy environment, Decision Sciences 11 (1980) 325–338.
[21] R. Narasimhan, On fuzzy goal programming-some com-ments, Decision Sciences 12 (1981) 532–538.
[22] E.L. Hannan, Linear programming with multiple fuzzy goals, Fuzzy Sets and Systems 6 (1981) 235–248.
[23] R.N. Tiwari, S. Dharmar, J.R. Rao, Priority structure in fuzzy goal programming, Fuzzy Sets and Systems 19 (1986) 251–259.
[24] D.T. Pham, D. Karaboga, Intelligent Optimization Tech-niques: Genetic Algorithms, Tabu Search, Simulated Annealing and Neural Networks, Springer-Verlag, London, 2000.
[25] J.H. Holland, Adaptation in Natural and Artificial Systems, University of Michigan Press, Ann Arbor, 1975.
[26] D.E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning, Addison-Wesley, Reading, MA, 1989.
[27] K. DeJong, An Analysis of the Behavior of a Class of Genetic Adaptive Systems, Ph.D. Thesis, University of Michigan, USA, 1975.
[28] M. Gen, R. Cheng, Genetic Algorithms and Engineering Design, Wiley, New York, 1997.
[29] H. Levy, Portfolio performance and the investment horizon, Management Science 18 (1972) B645–B653.
[30] S.N. Chen, C.F. Lee, The sampling relationship between Sharpe’s performance measure and its risk proxy: sample size, investment horizon and market conditions, Manage-ment Science 27 (1981) 607–618.
