MARKOV CHAIN PORTFOLIO LIQUIDITY OPTIMIZATION MODEL
Eder Oliveira Abensur
Universidade Federal do ABC (UFABC), Brazil
Email: eder.abensur@ufabc.edu.br
Submission: 18/10/2013
Accept: 03/11/2013
ABSTRACT
The international financial
crisis of September 2008 and May 2010 showed the importance of liquidity as an
attribute to be considered in portfolio decisions. This study proposes an
optimization model based on available public data, using Markov chain and
Genetic Algorithms concepts as it considers the classic duality of risk versus
return and incorporating liquidity costs. The work intends to propose a
multicriterion nonlinear optimization model using liquidity based on a Markov
chain. The nonlinear model was tested using Genetic Algorithms with twenty
five Brazilian stocks from 2007 to 2009. The results suggest that this is an
innovative development methodology and useful for developing an efficient and
realistic financial portfolio, as it considers many attributes such as risk,
return and liquidity.
Keywords:
Portfolio Optimization, Markov chain, Genetic Algorithms
1.
INTRODUCTION
The
fundamentals of the Modern Finance Theory are represented by articles written
by Markowitz (1952) and Sharpe (1964). Markowitz broke the paradigms of
portfolio selection that considered only the return aspect. His proposed
formulation based on the riskreturn duality, explains why diversification is an
advantage when it comes to portfolio selection and demonstrates that there is
an optimal mix of assets in a portfolio that achieves both maximum return with
a minimum risk.
Markowitz
formulated the variance (or risk) theory of a generic portfolio composed of n assets and showed that it depends on
the variances of individual assets and the covariance’s between pairs of assets
involved, as originally published in the following formula:
Where:
X = asset
participation in the portfolio
σ_{ij}_{ }= covariance between asset i and asset j
n = number of assets
Sharpe
(1964) developed the fundamentals of asset pricing by taking into account the
conclusions of Markowitz portfolio risk. Among its conclusions, he emphasizes
that there is a linear relationship between the rates of return on assets and
their covariance with the market portfolio. This relationship is expressed by
beta (β), a standardized covariance
to the market portfolio variance. Therefore, there is a linear relationship
between the return on assets and β
defined by:
Where:
asset
expected return
R_{F} = riskfree rate
beta of the asset
R_{M} =
market expected return
According
to the Modern Portfolio Theory, the risk of a portfolio can be divided into two
components: (i) a factor that affects a large number
of assets, each with a higher or lower intensity, called systematic and (ii) a
factor that specifically affects a single asset or a small group of assets,
called unsystematic or specific (ROSS; WESTERFIELD; JAFFE, 1999).
Also,
according to the Modern Portfolio Theory, the relevant profitability
differences can only be explained by systematic and unsystematic risks. Any
premium, representing an undesirable feature of the asset would be explained by
a premium of unsystematic risk. Finding a premium that is represented by a
factor not related to unsystematic risk represents an anomaly in the
theoretical model. The literature presents a vast collection of discussions on
possible anomalies to the basic model such as Banz
(1981), Fama and French (1992), Heston
and Sadka (2008), Hogan (2004) and Lewellen (2006) among others.
Over
time, other operational and conceptual problems have been identified in the
original formulation of Markowitz. The most important are:
a) There
are computational difficulties related to solving largescale quadratic
programming problems (KONNO; YAMAZAKI, 1991; YOUNG, 1998; PARRA et al., 2001);
b) Generally,
the portfolios obtained by the original formulation concentrate on few assets,
which is against the idea of diversification (JANA et al., 2009);
c) The
absence of transaction costs and liquidity (or illiquidity) can result in
inefficient portfolios (ARNOTT; WAGNER, 1990; AMIHUD; MENDELSON, 1991);
d) In
large portfolios the model would suggest the purchase of a small fraction of
assets, often lower than the minimum traded in the market (KONNO; YAMAZAKI, 1991);
e) The
resolution of the quadratic programming model is intractable for entire
portfolios with more than 20 assets (KONNO; YAMAZAKI, 1991);
f) The
model assumes there are no difficulties in liquidating the portfolio formed, in
other words, the market would absorb any type and amount of assets allocated by
optimization (POGUE, 1970).
This
study does not intend to discuss liquidity from the perspective of an anomaly,
in accordance with the Modern Portfolio Theory. However, it is a fact that
liquidity or transaction costs are implicitly incorporated by investors in
their investment allocation decisions. In other words, all else being equally
constant, investors prefer more liquid than less liquid assets, particularly in
the shortterm.
Recently
there has been an increased interest in studies of financial models with
parameters modulated by Markov chains in an attempt to reflect the dynamics of
the markets under conditions of financial distress (BAUERLE; RIEDER, 2004;
CAKMAK; OZEKICI, 2006; COSTA; ARAUJO, 2008; REBOREDO, 2002). Liquidity would be
one more attribute of any investments allocation decision, expanding investor
focus beyond the traditional duality of risk vs. return.
The
work is structured as follows: (i) section 2
describes the methodology of the research; (ii) section 3 presents several ways
to measure liquidity, presents empirical evidence on the adopted liquidity
indicator and proposes a measure of liquidity for the portfolio based on Markov
chain concepts; (iii) section 4 shows the Markowitz model; (iv) section 5 shows
the proposed model; (v) in section 6, the tests applied are characterized and the
results presented and (vi) section 7 presents the conclusions of the study.
2.
METHODOLOGY
This
paper is an experimental research evaluation, also involves the use of
standardized techniques of data collection and simulation.
The
job data were collected from daily reports provided by the BMF&Bovespa
from January 2007 to September 2009. The selection, comparison and testing of
hypotheses applied to the chosen liquidity indicator comprised the years 2007
and 2008. To perform the simulations, an application based on an Excel spreadsheet
using the Microsoft Excel® nonlinear programming solver for the Markowitz model
was developed. The proposed model used the search tool Evolver® with Genetic
Algorithms. Portfolios were formed and compared from an arbitrary initial
application of $ 5,000, before Brazilian taxes. The estimated price of these
shares was based on the average behavior of the Brazilian financial market in
the first half of 2009.
Liquidity
was estimated by weighting the frequency F
obtained by dividing the trading days of each action in period (F_{a}) by the number of trading
days in the period (F_{p})
(F = F_{a} / F_{p}).
This frequency was weighted by its respective average IN divided by the maximum number recorded between the studied
securities (F x IN_{avg} / IN_{max})).
The weighted ratio was grouped into quartiles arbitrarily assuming, the average
probability of 1, 2, 3 and 4, respectively, 1.00, 0.75, 0.50 and 0.05. This was
the possible liquidity estimate that could be obtained from the public data
available. The proposed liquid measure was incorporated in a stochastic process
of Markov chain in order to evaluate the probability of trading the shares.
3.
LIQUIDITY MEASURES
The
concept of liquidity can be found in many ways. In accounting, liquidity is
associated with the ease or speed which an asset can be turned into cash. In
economic terms, an asset is considered liquid if its value is both easily
negotiable and experiences little volatility over time.
In
financial terms, liquidity can be defined as the ease which an asset can be
exchanged within a short period of time (trading) without causing significant
changes in its price (transaction cost). It is a systemic phenomenon that
depends on the interaction between economic agents, where one wants to buy the
asset (tangible or intangible) from another.
For
purposes of this study, among the measures selected and analyzed were the known
liquidity indicators highlighted in literature and market practices as follows:
(i) liquiditybased on spread, (ii) liquidity based
on impact on the price, (iii) liquidity based on frequency and (iv) hybrids,
liquidity based on a combination of two or more factors. Table 1 below
summarizes these indicators.
The
indicators in Table 1 were divided according to the type of approach, but they
can also be classified according to their data collection frequency. Intraday
indicators are generate over a short period of time during the trading session
(eg: bid ask spread, effective spread) and others can
be collected on a daily, monthly, quarterly or yearly basis (e.g.: Roll, Holden,
Zeros, Amihud). Moreover, intraday indicators can be
converted into higher frequencies (bid ask spread). Intraday indicators are not
available to the public, so the question is whether the low frequency
indicators can be used. Goyenco, Holden and Trzcinka (2009), concluded that low frequency indicators
provide good liquidity measures.
Table 1 – Analyzed liquidity measures
Measue 
Type 
Description 
Formula 
Bidask 
spread 
Difference between the best selling and buying
offers divided by the average price 

Roll 
spread 
Assesses the effective spread based on price
covariance 

Effective tick 
spread 
Assesses the weighted spread probability divided by
the average 

Holden 
spread 
Weighted average of possible spreads 

LOT 
spread 
Difference between percentage costs of buying and
selling 

Zeros 
spread 
Percentage of days with zero return 

Amihud 
price 
Ratio of the return of a share and its financial
volume 

Amivest 
price 
Ratio of the financial volume of a share and its
return 

Turnover 
frequency 
Ratio of the daily financial traded share and the
number of outstanding shares 

IN^{1} 
hybrid 
Measures the negotiation intensity of a share
combining the number negotiate ratio (n/N)
with its financial volume ratio (v/V)


IN_{p}^{2} 
hybrid 
Combines the IN
of a share with its stock exchange frequency floor 

Source: Lesmond (2005), Goyenco, Holden and Trzcinka
(2009)
^{
1 }Liquidity measure used in the
Brazilian financial market (PAULA LEITE and SANVICENTE, 1995;
BMF&Bovespa,
2012)
^{ 2 }Liquidity measure proposed to evaluate trading
probability of the formed portfolio
The Brazilian stock market releases
and offers the public daily information on the bidask spread for each traded
stock and releases the necessary data to evaluate the negotiability index (IN). The IN has several advantages: (i) available
data; (ii) reliability because the data come from the BMF&Bovespa;
(iii) measuring the intensity of trading action is consistent with the purposes
of this study and (iv) by assessing the quantity and total financial volume
traded in a period of time, IN avoids
price distortion in the analysis of a long time series, for example, cases of
split and bonus. However, IN does not assess the exposure
frequency of the stock during the analyzed period.
Due to the facts presented, a new
liquidity ratio (IN_{P}) was proposed
whose calculation method is shown in Table 1.
3.1.
Empirical
Evidences of the Adopted Liquidity Measure (IN_{P})
It is now of interest to test the
validity of some the predictable behaviors expected from the definitions and
characteristics of the liquidity measure adopted for asset classification (IN_{P}). In particular, it is
expected that there is independence between the IN_{P} of the analyzed assets. If this does not happen, in
other words, if there are any assets that might lead to dependence on trading
then these assets will be identified and other stocks should be negotiated
after them.
Among the 420 listed shares on Bovespa, three groups were formed representing high, medium
and low trading activity (liquidity) in accordance with the IN_{P} and current rules of the
Brazilian stock market. The stratified collection of data was carried out
according to the historical financial information published by Bovespa. The highly traded group of shares (group 1),
called blue chips, were randomly selected into a group of 12 stocks that
represent 40% of Bovespa.
Intermediately traded stocks (group
2) were randomized into a group of 10 that have a 6% share of the total market.
According to the same sources, the selected stocks in the group with low
liquidity (group 3) represent 7 stocks that account for less than 0.05% of
total Brazilian market. The rates for the groups were collected from the Daily
Bulletin of Business of BMF&Bovespa over a period
of 494 consecutive trading days in 2007 and 2008 or 24 months. This period
comprised various typical aspects of business life such as tender offers,
acquisitions, mergers, splits, bonuses, equity contributions, disclosure of
halfyearly results and disclosure of relevant facts. In addition to the above
criteria, the following conditions for the formation of groups were used:
a) Companies
in the process of reorganization or bankruptcy were excluded;
b) Companies
bought or merged during the period of analysis were excluded;
c) Companies
with less than 12 months of participation in BMF&Bovespa
were excluded;
d) The
shares were selected exclusively by liquidity.
Although the samples were
stratified, the active components of each group were randomly selected which
allowed for the formation of a diversified group consisting of common shares (ON)
and preferred shares (PN) in several areas of activity such as: the food and beverage
industry, ceramic industry, garment industry, metallurgical industry, petroleum
industry, financial institutions, energy companies, steel companies,
telecommunication companies, hotels, wholesalers of pharmaceutical products,
insurance companies and construction companies.
The null hypothesis (H0) of there
being no significant correlation between assets was formulated. Evaluations
were conducted on the crosscorrelations between assets of different groups,
and between assets of the same group. The identification of companies and the
results of the correlation analysis on IN_{P}
are shown in Tables 2 to 13.
Table 2 – IN_{p}
correlation_{ } between highly liquid assets in 2007
*denotes significance at the 1% level
Table 3 – IN_{p}
correlation between moderately liquid
assets in 2007
*denotes significance at the 1% level
Table 4 – IN_{p}
correlation_{ } between assets with low liquidity in 2007
*denotes significance at the 1% level
Table 5 – IN_{p}
correlation_{ } between highly and moderately liquid assets in
2007
*denotes significance at the 1% level
Table 6 – INp correlation between assets with high and low liquidity in
2007
*denotes significance at the 1% level
Table 7 – INp correlation between assets with medium and low liquidity
in 2007
*denotes significance at the 1% level
Table 8 – IN_{p}
correlation_{ } between highly liquid assets in 2008
*denotes significance at the 1% level
Table 9 – IN_{p}
correlation_{ } between moderately liquid assets in 2008
*denotes significance at the 1% level
Table 10 – IN_{p}
correlation_{ } between assets with low liquidity in 2008
*denotes significance at the 1% level
Table 11 – IN_{p}
correlation_{ } between assets with high and medium liquidity
in 2008
*denotes significance at the 1% level
Table 12 – IN_{p}
correlation_{ } between assets with high and low liquidity in
2008
*denotes significance at the 1% level
Table 13 – IN_{p}
correlation_{ } between assets with medium and low liquidity
in 2008
*denotes significance at the 1% level
The
data show that in 2007 and 2008, 88% and 84% of cases have no correlation at a
1% level of significance. Therefore, there is evidence that there is no
correlation between the trading of assets in accordance with IN_{P}.
3.2.
Markov
Chain Portfolio Liquidity
All portfolios are formed to be sold
one day. Thus, it is reasonable to assume that portfolios with different
quantities and qualities of assets experience different difficulties when they
come to be sold. Portfolio A,
consisting of a single stock, is likely to face less selling difficulty than
Portfolio B, comprising 1,000
different shares including the stock of Portfolio A. Therefore, the liquidity of a portfolio in terms of selling all
of its assets should incorporate the individual conditions of liquidity of each
asset. In addition, the complete liquidation of a portfolio or a single stock
has a dynamic characteristic, it may occur after several consecutive attempts
(consecutive trading orders) over a period of time. What remains is to define
an appropriate way of measuring the liquidity of a portfolio, it being
understood that liquidity is the ease with which the whole portfolio is traded.
Initially, consider two portfolios, A and B, with two assets in each one. The probabilities of trading these
assets are, respectively, P_{1}
and P_{2}. These
probabilities can be obtained from past observations that associate the quality
of these assets to the amounts traded, or by a subjective estimate originating
from the intuition of experts. The random variable X_{i} represents the possibility of trading of portfolio at
time i with
i = 1,2,3 ,.... n. Assuming that X_{i} are independent events and
the probabilities remain the same throughout the n attempts in the portfolio, then, from the viewpoint of Markov
stochastic processes there are two possible states for the portfolio: (i) S_{1},
the portfolio is full (complete), or (ii) S_{2},
the portfolio is empty (sold or traded). Figure 1 below shows the state diagram
of this situation.
Figure 1 – State diagram of the negotiation portfolio
comprising two assets
A question arises about the scenario
shown: what is the probability of negotiating the portfolio after n attempts?
The stability condition of the
Markov chain requires that the transition probabilities are for n = 1,2 ,..., and all known possible
sequences of states s_{1}, s_{2},
s_{3} ,...., s_{n}_{ +1} with
X_{n}, X_{n}_{
1}, ...., X_{1} are given by:
P (X_{n+1}
= s_{n+1} / X_{1} = s_{1}, X_{2} = s_{2},....,
X_{n}=s_{n})
= P (X_{n+1} = s_{n+1} / X_{n}=s_{n}) (14)
The respective transition matrix of
the examined case of a portfolio with two assets would be:
Similarly, the transition matrix of
a portfolio of n assets would be:
Using matrix calculations and the
notion of Markov stochastic processes, the probability of trading a portfolio
of n assets after two attempts
starting from an empty position is given by:
Based on the concepts of a finite
Markov chain after n trials, there is
the possibility of the convergence of the probability matrix to state of
equilibrium (since
at least one P_{i} <1).
The probabilities of this state are obtained by solving a linear system of
equations. For the case analyzed of a matrix with two states, the probabilities
π_{1} and π_{2} of
liquidating the portfolio at the state of equilibrium would be obtained by
solving the following system of equations (TAHA, 2008)
The
number of iterations to reach each of the states of equilibrium would be 1/π_{1} =1/π_{2}_{ }=20.
Figure 2 shows an example of the convergence of the probability of trading a
portfolio with two stocks with a general trading probability of 0.90. As
expected, according to the theory of Markov chains, the final state is quite
different from the initial condition.
Figure 2  Probability of the
portfolio after n attempts
4.
THE MARKOWITZ MODEL
The
approach developed by Markowitz (1952) assumes that the expected returns of the
examined assets are known and so the allocation of available capital is
possible. He suggests the use of past observations as an alternative to
projecting expected returns.
R_{i} is a random variable representing the rate of return
per period of asset i
with i =
1,2,3 ,,,,, and X_{i} is the
amount of capital to be invested in asset i. The expected return of the
investment for the analyzed period is given by:
The
duality returnrisk is characterized by the expectation the investor has of
obtaining maximum return for minimum risk. The risk measure used was the
standard deviation of returns in a given period:
One
interpretation of the Markowitz model as a quadratic programming problem is
given by Konno and Yamazaki (1991):
Where
M_{0} is the total available
capital for investment, ρ is the minimum rate of return desired by the
investor, μ_{i} is the maximum amount of money that can be invested
in asset i,
R_{i} = E[R_{i}]
and σ_{ij}=E[(R_{i}
 r_{i})(R_{j}
 r_{j})].
5.
THE PROPOSED MODEL
The proposed approaches adopt the
same assumptions as Markowitz model, plus the liquidity condition, based on
Markov chains. The nonlinear models proposed aim to form a portfolio that
simultaneously, minimize risk and maximize liquidity, after k sequential attempts of trading,
exceeding a minimum rate of return and deducting the operating costs of trading
(α). The risk of the proposed optimization model (P_{1}) is based on the
covariance matrix of Markowitz.
The
objective function was developed using the concept of goal programming (Hillier
and Lieberman, 2005). The risk goal (R_{g}) used was a small value, but close to zero (eg: 0.1). A natural candidate for the liquidity goal is the
probability at the state of equilibrium explained in equation 16 (π_{e}
= 0.5).
The
model incorporates real practices of the financial market such as fees, taxes
and dividend payments there by making them more realistic. Besides these
features the following assumptions are made:
a) The planning horizon of the investor is the short
term;
b) The planning horizon consists of a single continuous
period;
c) The investor is risk averse, so, the higher the risk
the higher the expected return;
d) Variable and fixed operating costs were considered.
Model P1
Metaheuristics are powerful search
engines inspired by models of human life or nature. They can achieve good
solutions in a short computational time for problems that have no exact
mathematical solution. Metaheuristics are more complex simulations that have
the ability to incorporate patterns of human behavior during the simulation
process, such as adaptation and learning, allowing for the selection of
superior solutions. For this reason, some metaheuristics are considered to be
artificial intelligence (e.g.: genetic algorithms). Examples of metaheuristics:
(i) Genetic Algorithms (GA), (ii) Ant System, (iii) Tabu Search, (iv) Simulated Annealing (SA) and (v) Hybrids.
Financial decisions in the short term, such as the portfolio, are inserted in
the context of optimization.
Genetic
Algorithms were chosen as search engine to select the best combination of
stocks for the portfolio by the proposed model. In GA, the term chromosome
typically refers to a candidate solution. Functionally, the genetic algorithm uses
the following operators (Holland, 1975):
a) Reproduction
The
initial solution is formed by a sequence of bits that represent the
characteristics of the product. The selection operator selects a subset of m chromosomes of size M of the population that can reproduce,
on average, better adapted chromosomes produce more offspring than the less
well adapted. Generally, the size of the chromosome is maintained in successive
generations.
b) Crossover
The
operator of the crossover exchange parts of chromosomes positions specifically
chosen for the formation of new offspring.
c) Mutation
The mutation operator changes the
values of some attributes at random.
6.
RESULTS
Liquidity
was estimated by weighting the frequency F
obtained by dividing the trading days of each action in period (F_{a}) by the number of trading
days in the period (F_{p})
(F = F_{a} / F_{p}).
This frequency was weighted by its respective average IN divided by the maximum number recorded between the studied
securities (F x IN_{avg}
/ IN_{max})). The weighted ratio was
grouped into quartiles arbitrarily assuming, the average probability of 1, 2, 3
and 4, respectively, 1.00, 0.75, 0.50 and 0.05. This was the possible liquidity
estimate that could be obtained from the public data available. The list of
shares participating in the simulations with their respective quartiles and
trading probabilities are presented in Table 14 below.
The
Brazilian financial market defines the validity of a buy or sell order by the
number of days and not by the number of attempts. Twenty attempts were adopted
as an intermediate value between the minimum and maximum used by the market.
As
suggested by Markowitz (1952), for demonstration purposes, the average
performance of the 1^{st} half of 2009 was used to estimate the
profitability of each stock.
Table 14 – List of stocks and probabilities
Ranking 
Stock 
Probability 
Quartil 
Ranking 
Stock 
Probability 
Quartil 
1 
Ambev 
0.75 
3 
14 
MMX 
0.50 
2 
2 
Bradesco 
0.75 
3 
15 
Celesc 
0.50 
2 
3 
Banco
do Brasil 
1.00 
4 
16 
OHL 
0.50 
2 
4 
CESP 
0.75 
3 
17 
P. Seguro 
0.50 
2 
5 
Gerdau 
1.00 
4 
18 
Random 
0.50 
2 
6 
Net 
1.00 
4 
19 
Copasa 
0.75 
3 
7 
Petrobras 
1.00 
4 
20 
Marco Polo 
0.75 
3 
8 
CSN 
1.00 
4 
21 
Klabin 
1.00 
4 
9 
Telemar 
0.50 
2 
22 
Caf Brasil 
0.05 
1 
10 
Usiminas 
1.00 
4 
23 
Sergen 
0.05 
1 
11 
Vale 
1.00 
4 
24 
Hercules 
0.05 
1 
12 
Comgás 
0.50 
2 
25 
Marisol 
0.05 
1 
13 
TAM 
0.75 
3 




Once
the portfolio is classified in terms of attributes and levels, an initial
population of size M is randomly
generated. For purposes of this research, a convergence was found with the
following configuration parameters of genetic algorithm: (i)
M = 52; (ii) a uniform rate of crossover equal to
50%; (iii) a mutation rate of 10%, (iv) stopping criterion after 75,000
iterations and (v) the Evolver® internal method recipe. The best results and comparisons between the
models are shown in Table 15 below. Figure 3 below
shows the evolution of the variation in the profitability of the portfolios (Markowitz and P1) during the
period June to December 2009.
Table 15 – Features of portfolios (Ibovespa
= ρ = 4,20%^{3} a
month between Jan and June/2009)




# stocks 
6 
6 
65 
Stocks 
6  8  18 
2  3  4  16 19 20 
^{4} 
Estimated Profitability 
4.26% 
4.63% 
4.20% 
Observerd Profitability^{5} 
4.75% 
5.69% 
4.90% 
Risk (β) 
0.2601 
0.6937 
1.0000^{6} 
Liquidity (Markov) 
0.0184 
0.4998 
1.0000^{6} 
^{3} the average performance of the 1^{st} half of 2009
^{4} 65 stocks on average according to BMF&Bovespa
^{5} from June to Dec/2009
^{6} according to market portfolio definition
Figure 3 – Portfolio profitability evolution
As
expected, the Markowitz model created portfolio with lower volatility (0.2601) relative
to the Ibovespa market portfolio according to β
Sharpe. The proposed model, in turn, have created high volatile portfolio
(0.6937) , however, the difference in quality of the selected assets is
significant. The Markowitz model allocated values to shares 22 and 23 that have
low liquidity while the proposed model P1, as expected, avoided selecting these
assets.
The
share of less liquid stocks (22 and 23) in portfolio of Markowitz reduced the
potential returns of the portfolio because of lower trading frequency of
trading and the updating of their prices at auctions. The portfolio formed by
the P1 model, incorporated the most liquid stocks and is more realistic in
terms of potential trading shares and include well know Brazilian companies
such as Bradesco and Marco Polo.
7.
FINAL CONSIDERATIONS
The
multicriteria optimization model generated in this study incorporated an
innovative measure of liquidity based on the probability of trading the shares
included in a stochastic process of Markov chains. This includes two important
aspects: (i) an approach to the dynamism of a market
that trades shares in several attempts, and (ii) the introduction of liquidity
and transaction costs in the decisions. The work reinforced the conclusion
obtained in other studies that the absence of transaction costs can generate
inefficient or unrealistic portfolios.
This study considered
two possible states of negotiation (negotiated or not negotiated). However,
other states of the partial liquidation of the portfolio could be simulated,
but this would require a large computational effort for implementing. This
opens the way to new lines of research on the subject.
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