Tối ưu hóa danh mục đầu tư phân tích rủi ro tại trung quốc bài học cho việt nam

Tối ưu hóa danh mục đầu tư và
Phân tích rủi ro tại Trung Quốc
Bài học cho Việt Nam

Võ Hồng Đức

2020


TABLE OF CONTENTS

1

INTRODUCTION .............................................................................................. 11

2

RESEARCH OBJECTIVES.............................................................................. 13

3

LITERATURE REVIEW .................................................................................. 13

4

5

6

3.1

MARKOWITZ’S MEAN-VARIANCE OPTIMIZATION .................................. 13

3.2

MEAN-SEMIVARIANCE OPTIMIZATION FRAMEWORK ............................ 14

3.3

RESAMPLING METHODOLOGY ............................................................... 16

3.4

EMPIRICAL STUDIES .............................................................................. 17

DATA AND METHODOLOGY ....................................................................... 25
4.1

ESTIMATES OF PORTFOLIO RISK AND RETURN ....................................... 25

4.2

OPTIMIZATION ...................................................................................... 29

4.3

DATA..................................................................................................... 31

RESULTS ............................................................................................................ 31
5.1

RETURN, RISK, AND RANKING ............................................................... 31

5.2

EFFICIENT FRONTIERS ........................................................................... 35

CONCLUSIONS ................................................................................................. 39

2


LIST OF TABLES

Table 1 Annualized average daily returns, standard deviations, semi-deviations
and their rankings by sectors in the whole period from 2007 to 2016 in China, in percent.
...................................................................................................................................... 33
Table 2 Rankings among sectors using daily, weekly, and monthly data sets in the
period of 2007-2016 ..................................................................................................... 34
Table 3 The average Sortino ratio under four optimization frameworks ............ 37

3


LIST OF FIGURES

Figure 1 The figure shows the Markowitz efficient frontier as well as meansemivariance efficient frontier in the period 2007-2016. The annualized average daily
returns and annualized semideviations of ten selected sectors along with the SSE Index
are presented. ................................................................................................................ 35

Figure 2 The figure shows the resampled efficient frontier as well as resampled

mean-semivariance efficient frontier using 2007-2016 sample. The annualized average
daily returns and annualized semideviations of ten selected sectors along with the SSE
Index are presented. ...................................................................................................... 36

Figure 3 Optimal weights under three framework: (a) traditional mean-variance,
(b) mean-semivariance, and (c) resampled mean- semivariance; daily data of ten sectors
in China, 2007 to 2016. ................................................................................................ 38

4


1

Introduction
The Chinese stock market is enormous and plays a significant role to its economic

growth as well as other global markets. At the beginning of 2016, Shanghai market, one
of the largest equity markets in the world, hits US$3.5 trillion in market capitalization
(Shanghai stock exchange 2017 Factbook). A large number of listed stocks in Shanghai
Stock Exchange (SSE) are from public sector, which is large and represents the whole
economy of China. There are also a number of studies which demonstrate a link between
the stock market development in China and economic growth (Liu & Sinclair, 2008;
Wong & Zhou, 2011). In addition, the fluctuations in the Chinese stock market have
increasingly affected developed stock markets, such as those in the US, the UK,
Germany, and Japan (Yu, Fang, Sun, & Du, 2018). In a recent study, Fang and Bessler
(2018) argued that Chinese market has a powerful impact on most stock markets in Asia
since the nation’s role has improved through its economy’ strong growth as well as
financial openness. This market interdependence is improved partly thanks to the
financial market reform in China after joining the World Trade Organization in 2001
(He, Chen, Yao, & Ou, 2015).

The Chinese stock market becomes more attractive as an investment opportunity
due to its convergence to the global market condition. Currently, foreign investors can
even approach both “A-shares” and “B-shares” on the market, where the first category
was restricted to local investors only before the reforms in December 2002. Yao, Ma,
and He (2014) found that not only the herding behaviour at the beginning of the last
decade mostly disappeared over time, but also the A-share markets seem to become
more and more rational. Carpenter, Lu, and Whitelaw (2015) considered that the level
of market efficiency in China converges to the US. This implies that the stock pricing
and portfolio construction methodology is also very similar between two markets even
though their levels of risk are different.
The convergence to global market encourages application of finance and
investment theories in China, seeking for profitable opportunities. Drew, Naughton, and
Veeraraghavan (2003) built a multi-factor model for Chinese stock market but found
that the market factors could not separately explain the return’s fluctuation. Later, Xu

11


and Zhang (2014) successfully applied the Fama-French three-factors model in the
Chinese stock market and argued that the model could explain more than 93 per cent of
the Chinese A-share return’s movements.
However, investing in an emerging market like China faces a huge systematic
downside risk along with attractive returns. A three-week crash of the Chinese stock
market in 2015 blew away about 30 per cent of its market shares, which raised a concern
about a more serious influence on the world economy than the Greek debt crisis in 2011
(Allen, 2015). A strong negative impact from the crash on several Asian markets was
also noted by Fang and Bessler (2018). In addition, Yu et al. (2018) argued that the
large magnitude of risk from the Chinese stock market, especially through downside
periods, has reduced the benefit of diversification.
Among the most original portfolio construction theories, Markowitz meanvariance portfolio optimization (Markowitz, 1952) is commonly used to instruct

investors how they can efficiently allocate their investments. Unfortunately, this famous
theory is also known in literature as producing biases because inputs with massive
estimation error are used. This weakness puts the theory into a serious trouble since
such an error is amplified by the optimization procedure. Consequently, estimates for
the outputs, portfolio’s optimal weights as well as risk and return from this framework
have arguably become less convincing. For example, when an asset’s expected return
is overestimated, it will be allocated much more risk by the classical mean-variance
optimization method than it should be. In addition, resampling method introduced by
Michaud (1989) is an approach that solves the problem. Moreover, since investors do
not shy away from the extremely positive returns, the variance used in the classical
theory leads to another bias and must be replaced by a downside risk measure, such as
semivariance.
In this paper, we will construct the portfolio optimization in China, using classical
Markowitz mean-variance framework, mean-semivariance framework, and applying
Michaud resampling method to the optimization procedure. This paper explores two
research questions. First, are the rankings of risk and returns among sectors significantly

12


changed using different risk- return measures? Second, do resampling method and
mean-semivariance framework actually improve the optimization procedure?
Findings from this study indicate that this new framework improves the
performance of the optimal portfolios, measured by Sortino ratio, and diversification.
2

Research objectives
It is noted that the Chinese stock market becomes more attractive as an investment

opportunity due to its convergence to the global market condition. The convergence to global

market encourages application of finance and investment theories in China, seeking for
profitable opportunities. This observation leads to following research question:

• Are the rankings of risk and returns among sectors significantly changed
using different risk- return measures?
• Do resampling method and mean-semivariance framework actually
improve the optimization procedure?
In order to answer to those questions, this study is conducted to achieve the
following research objectives:
• Ranking risk and return of 10 sectors through mean-variance optimization
framework and mean-semivariance optimization framework.
• A comparision between mean-variance optimization framework and meansemivariance optimization work in terms of optimal portfolio selection.
3
3.1

Literature review
Markowitz’s mean-variance optimization

Mean-variance optimization, which was initially introduced by Harry Markowitz
in 1952, is known as a cornerstone in portfolio selection world. In 1990, Markowitz
shared the Nobel Memorial Prize in Economic Sciences with William Sharpe and
Merton Miller for their contribution in the financial economics theory. The meanvariance optimization framework uses expected return as a measure for reward and
variance as a risk measure, based on historical return, volatility, and covariance matrix.
The outputs of this procedure are optimal portfolios with highest return in each level of

13


risk, along with their proposed weight vectors. This theory is successfully tested by a
number of empirical studies such as Farrar (1962) and Perold (1984). It is also the

background for the famous capital asset pricing model (Sharpe, 1964). Perold (1984)
insisted that the Markowitz framework has been widely accepted as a practical method
for portfolio construction process. However, the classical mean-variance framework has
its own limitations. For example, the assumption of symmetrically and normally
distributed returns.
Ongoing studies on downside risk measures have tried to replace variance by a
more appropriate risk measures. Value-at-risk (VaR) is one of the candidates which is
developed to mean- VaR framework to solve the optimization problem (Campbell,
Huisman, & Koedijk, 2001). Conditional Value-at-risk (CVaR) is later proposed due to
the fact that VaR does not own the subadditivity property, one of criteria of a coherent
risk measure. Recently, Vo et. al. (2018) applied CVaR to seek for the optimal portfolios
in the South East Asian region. In addition, Powell et. al. (2018) constructed new
metrics named EVaR and ECVaR to measure downside volatility of commodity assets
in various economic periods.
3.2

Mean-semivariance optimization framework

Semi-variance has increasingly utilized from studies on downside risk measures
(Harlow, 1991; Sortino & Price, 1994; Sortino & Van Der Meer, 1991). In general, it is
an asymmetric risk measure, which quantifies the deviations below the mean or a
threshold level of return. Estrada (2006) provided an example of using semi-variance
and semideviation as the alternatives. In addition, Estrada (2004) argued semivariance
is superior to variance for the following three reasons. First, investors only dislike
downside movement on the asset returns; they will not feel harmful with upside returns,
which are also included in the measurement of the ‘variance’. As such, the semivariance fits investors’ demand in analyzing risk. Second, the semi-variance is more
statistically helpful than variance when return is asymmetrically distributed, which is
often observed in practice. Finally, semi-variance is a measure that combines variance
and skewness at the same time; hence, we can use single factor models to estimate the
returns.

14


Mean-semivariance framework is supported by both strong background theory
and empirical studies. Markowitz, the father of mean-variance optimization framework,
argued that semi- variance appears to generate better optimal portfolios than those based
on variance framework and considered that semi-variance is “more plausible than
variance as a measure of risk” (Markowitz, 1959, 1991). The mean-semivariance
framework attracts academic and empirical studies. Estrada (2002, 2004, 2006, 2007)
constructed a series of papers discuss a number of theoretical frameworks on downside
risk basis, including mean-semivariance optimization framework. The author also
considered that the usual beta can be substituted by ‘downside beta’ and suggested using
the D-CAPM as an alternative of the CAPM. The author also stated that meansemivariance framework is particularly appropriate for emerging markets (Estrada,
2004). Boasson, Boasson, and Zhou (2011) used monthly data from seven exchangetraded index funds from 2002 to 2007 to construct mean-semivariance efficient frontier
and recommended its application in insurance and banking sectors. In addition, PlaSantamaria and Bravo (2013) utilized daily data of Dow Jones stocks over the period
2005-2009 to prove that the mean-semivariance is empirically more suitable to reflect
the downside risks than a classical mean-variance optimization.
With respect to technique issues, although the mean-semivariance framework
gains more and more trusts from academic community, it could not be easily developed
due to mathematical problems. In 1993, Markowitz solved the mean-semivariance
optimization by transforming into quadratic problem using simulated securities
(Markowitz, Todd, Xu, & Yamane, 1993). Foo and Eng (2000) calculated these figures
based on lower partial moments (LPM) constructed by Harlow and Rao (1989). Estrada
(2008) suggested a heuristic approach that creates a symmetric and exogenous semicovariance matrix to solve the optimization problem.
This paper adopts the method introduced in Ballestero (2005), which uses
Sharpe’s beta regression equation (Sharpe, 1964) connecting every asset return to the
whole market. A semi- variance matrix and a quadric objective function are constructed,
however, heuristics are not required. This technique is also used in some empirical
studies on mean-semivariance framework (Boasson et al., 2011).

15


3.3

Resampling methodology

Resampling methodology, which is generally considered as an enhanced meanvariance optimization from Markowitz (1952), was developed by Michaud (1998) on
the basis of a simulation framework. A key objective of this method is to limit the effect
of input estimation errors on the optimal portfolio weights and, as such, to achieve more
robust portfolios through a balanced and diversified asset allocation. The key distinction
separating Michaud resampling method from the original Markowitz optimization is
that the resampling utilizes the data from a stochastic process rather than from a
predetermined data set. This requires various repeats of random sample selection based
on Monte Carlo simulation methodology developed by Metropolis and Ulam (1949).
On a theoretical consideration, Michaud’s resampling method shows its superior
in improving performance of optimal portfolios compared to the classical meanvariance optimization. Markowitz and Usmen (2006) created a simulated battle where
a Bayesian player, representing classical mean-variance optimization, was in
competition with Resampling player, who follows the method developed by Michaud
(1998). The authors found that the Resampling player won ten out of ten times. Harvey,
Liechty, and Liechty (2008) added that the Resampling player will show the advantages
when the return distribution is not the same as the historical distribution.
On a practical consideration, empirical studies also demonstrated that Michaud’s
resampling method will improve performance. Using US risk-free asset and 10 global
stock index returns, Fletcher and Hillier (2001) suggested that resampling method
provides a higher Sharpe performance of optimal portfolios than the traditional meanvariance framework. Cardoso (2015) found a similar result from a number of selected
individual stocks in S&P500 where non-normally distributed resampling is captured.
The resampling method owns two valuable features for the long-term investors:
diversification and stability (Fernandes & Ornelas, 2009; Kohli, 2005). Using various
asset classes from US equity to Euro government bond, Delcourt and Petitjean (2011)

found that the resampled optimization will result in a more stable and diversified
optimal portfolios. Mansor, Baharum, and Kamil (2006) run the model for Malaysian
stock market and found that the method eliminates estimation error when using daily
16


and weekly data. In relation to stability, Asumeng-Denteh (2004) argued that
resampling reduces portfolio rebalancing and, as a consequence, transaction costs
incurred in trading while keeping the portfolio optimal. Galloppo (2010) proposed other
benefits of resampling method beyond the optimization process. When testing these socalled post-modern models including Tracking Error Minimization Model, Mean
Absolute Minimization Model, and Shortfall Probability Model, the author found an
improvement in the performance thanks to resampling method.
3.4

Empirical studies

Portfolio selection problem has been one of the most importance topics of research
in modern finance. Put it differently, portfolio optimization problem has received a lot
of attention from both researchers and practitioners over the last six decades. Tiryaki
and Ahlatcioglu (2009) for example proposed to construct a portfolio using analytical
hierarchy process methodology whereas Dia (2009) and Ismail et al. (2012) performed
a portfolio selection using data envelopment analysis. Konno (1991) proposed the mean
absolute deviation (MAD) from the mean as the risk measure. The model is however
equivalent to the Markowitz model when they possess a multivariate normal distribution
of the returns. On the other hand, Young (1998) introduced a model which maximizes
the minimum return (Maximin) or minimizes the maximum loss (minimax). According
to Young (1998), the minimax formulation might be a more appropriate method
compared to the mean-variance formulation when data are log-normally distributed.
Papahristodoulou and Dotzauer (2004) formulated two different linear programming
models based on minimization of MAD and Maximin formulations. These models were

then compared to the classical quadratic programming formulation to test to what extent
all these formulations provide similar portfolios. The results from this study showed
that the Maximin formulation yields the highest return and risk while the quadratic
formulation provides the lowest risk. In addition, all the three formulations were found
to outperform the top equity fund portfolios in Sweden and performed much better than
the market portfolio. Another linear programming model for the portfolio selection
problem is presented by Kamil and Ibrahim (2007). In this study, the problem was
modeled as a mean-risk bicriteria portfolio optimization problem with the mean

17


absolute negative deviation of annual return from the average annual return is used as
the downside risk. In order to evaluate the performance of the proposed model, the
authors compared the results from the proposed model with the results from meanvariance model and MAD model. According to their results, the proposed model
provides better returns than the mean- variance and MAD models. Chiodi et al. (2003)
presented a model for the problem of selecting a portfolio of mutual funds when entering
and management commissions are taken into account. This problem was formulated as
a mixed integer linear programming (MILP) model using mean semi-absolute
deviation. The authors have also designed some heuristic approaches to solve the
portfolio problem. The results of the computational experiments proved that the
problem can be solved using heuristics effectively and efficiently. The study, however,
could be extended to consider leaving commissions which might become a relevant
feature of the problem.
Ibrahim et al. (2008) proposed single-stage and two-stage stochastic programming
models with the objective of the models are to minimize the maximum downside
deviation from the expected return. The purpose of this study is to compare the optimal
portfolio of the two models. The results showed that the two-stage model outperforms
the single stage model in both out-of-sample and in-sample analysis. However, the
authors noticed that the models had lost the trend information due to the use of original

historical data treated as future return scenarios.
Based on the goal programming (GP) approach, Pendaraki et al. (2005) proposed
a methodology for the construction of equity mutual fund portfolio. The proposed
methodology was conducted in two stages. In the first stage, the UTADIS classification
method was used to evaluate and select a limited set of the mutual funds. In the second
stage, a goal programming model was employed to determine the proportion of the
selected mutual funds in the final portfolios. The proposed methodology has been
applied on data of Greek equity mutual funds with promising results. The method
however, could also be extended to consider other types of mutual funds.
In order to construct an optimal mutual fund portfolio for an investor, Sharma and
Sharma (2006) employed lexicographic goal programming approach with specific

18


parameters such as standard deviation, portfolio beta, expected annual return and
expense ratio were taken into account. The objective of the GP model is to minimize a
weighted sum of deviations from the target goals. In this study, the distances of all
possible solutions from the ideal solution were measured by using Euclidean distance
method. Although the model is flexible enough to accommodate other constraints, the
performance of the model depends on the appropriate weights in a priority structure.
Based on Sharpe’s single index model, Bilbao-Terol et al. (2006) have formulated
a new model for portfolio selection. In this work, imprecise future beta of each asset
was represented through a fuzzy trapezoidal number constructed on the basis of
statistical data and the relevant knowledge of experts. The authors have modeled the
problem using fuzzy compromise programming and introduced the fuzzy ideal solution
concept. The main feature of this model, as pointed out by the authors is its sensitivity
to the analyst’s opinion as well as to the investors’ preferences.
Based on the combination of chance constrained programming and compromise
programming, Abdelaziz et al. (2007) proposed a chance constrained compromise

programming to convert the multi-objective stochastic programming portfolio model
into a deterministic one. This study assumes that the parameters associated with the
objectives are random and normally distributed. A numerical example was carried out
to illustrate that the proposed model could be effectively and efficiently used in practice.
Masmoudi and Ben Abdelaziz (2012) addressed a problem of portfolio selection
where the cost of not achieving an acceptable expected rate of return was minimized.
The problem was modeled as a bi-objective stochastic programming where the first
objective function was to maximize the return and the second objective function was to
optimize the risk. The certainty equivalent program was obtained through a combination
of a GP and recourse approach. The model results were illustrated through a case study
using data from S&P100 securities.
By taking into account stochastic and fuzzy uncertainties, Messaoudi and Rebai
(2013) developed a novel fuzzy goal programming model for solving a stochastic multiobjective portfolio selection problem. In this model, fuzzy chance- constrained goals
were described along with the imprecise importance relations among them. The

19


proposed model was then utilized to build a new portfolio selection model that
considered the tradeoff between expected return, Value-at- Risk the price earnings ratio
and the flexibility of investor’s preferences. However, the applicability of the proposed
model on real world data had not been tested in this study.
Xidonas et al. (2010) developed a multi-objective MILP model for equity portfolio
construction and selection. In order to generate Pareto optimal portfolios, the authors
utilized the novel version of the ε-constraint method. Additionally, an interactive
filtering process was also proposed to guide the decision maker in selecting among a
number of Pareto optimal portfolios his/her most preferred. The proposed methodology
could be a useful tool in helping the investors to construct and design their portfolios.
Stoyan and Kwon (2011) presented a complex Stochastic- Goal Mixed-Integer
Programming (SGMIP) approach for an integrated stock and bond portfolio problem.

The portfolio model integrates uncertainty in security prices and involves several realworld trading constraints as well as other important portfolio elements such as liquidity,
management costs, portfolio size and diversity. An algorithm to solve the model that
consists of a decomposition, warm-start, and iterative procedure has also been proposed.
This study contributes a significant finding as the proposed algorithm is able to solve
the problem of practical size in an efficient manner.
A very recent study by Tamiz et al. (2013) investigated the problem of portfolio
selection for international mutual funds. The authors employed three variants of GP,
namely, Weighted, Lexicographic and MinMax approach to model the portfolio
problem. Seven factors were considered to be treated as objectives in the GP models in
which three were specific to mutual funds, three were taken from macroeconomics and
one factor represented regional and country preferences. The results of this study,
although very promising were not globally conclusive as they were based on certain
factors such as target values, priority levels and other sets of penalized unwanted
deviational variables.
Considering the increasing importance of investment in financial portfolios, Amiri
et al. (2011) developed a new model called Nadir Compromising Programming (NCP)
model by using an extended of Compromise Programming (CP). This model which can

20


be used to optimize multi-objective problems was formulated on the basis of the nadir
values of each objective. In order to compare the performance of the CP method and
the proposed method, the authors conducted a case study by selecting a portfolio with
35 stocks from the Iran stock exchange. The results obtained confirmed that in spite of
being feasible and optimal, the NCP model was more consistent with decision maker
purposes.
Kırış and Ustun (2012) built a multi objective portfolio optimization model which
combined Markowitz’s model with the objective of the expected performance value of
portfolio and cardinality constraints. This model is classified as a multi-objective mixed

integer nonlinear programming. The proposed model was solved by utilizing
reservation level driven Tchebycheff procedure.
Fernández and Gómez (2007) reported the efficiency of applying Hopfield neural
networks in solving constrained Markowitz problem. In their problem, there are
cardinality and boundary constraints that are based on the Markowitz model. They
compared Hopfield neural network to metaheuristic algorithm such as genetic
algorithms (GA), simulated annealing (SA) and Tabu search (TS). Deng et al. (2012)
studied the portfolio optimization problem based on Markowitz’s model under
cardinality constraints applying improved particle swarm optimization method. This
problem was investigated over five Stock Exchanges including Dax 100, Hang Seng
31, Mikkei 220, S&P 100, and ETSE 100 from Mars 1992 to September 1997. In most
cases, the performance of particle swarm optimization (POS) method is better than
genetic, simulated annealing, and Tabu search algorithms. It was shown that, compared
to the different methods, POS has an effective and strong performance, particularly in
low-risk investments.
Castellano and Cerqueti (2014) investigated the portfolio optimization problem in
presence of risky assets which are traded less frequently and contain lower liquidity.
For a dynamic modeling of non-cash assets, a pure jump process was applied, which
finally led to development of optimum portfolio. The theoretical models in this study
were analyzed through Monte Carlo simulation which provides a useful financial
prospect.

21


Huang (2008) introduced mean-semi variance model to choose an optimum
portfolio. His solution was based on genetic algorithm (GA). Yan et al. (2007) applied
a combination of particle swarm optimization and genetic algorithm in multi-period
portfolio using semi-deviation as risk measure. They showed that a combination of
particle swarm optimization method and GA is more efficient than the mere use of each

one.
Konno (1991) proposed a new model that uses mean absolute deviation risk for
portfolio optimization. Vercher et al. (2012) proposed a probabilistic model for the
portfolio selection problem which is based on a multi-objective optimization model that
are related to probabilistic mean-downside risk-skewness model.
Speranza (1996) considers the portfolio risk as linear combination of meanabsolute deviation (MAD) and proceeded to resolve the problem by applying a
combination of integer linear programming and other heuristic methods. The results
were tested over 20 securities.
Yu et al. (2008) introduced a new radial basis function (RBF) neural network in
order to choose an optimum portfolio based on mean-variance-skewness model.
According to this RBF neural network, an investor can choose a portfolio
simultaneously in line with its risk preferences and mean-variance-skewness objectives.
Yang (2006) applied genetic algorithm (GA) along with a dynamic portfolio
optimization system in order to improve the portfolio returns. In addition to GA, they
have applied mean-variance elliptic methods. The information applied in this research
includes the indices of total returns from six different Stock Exchanges (Canada,
France, USA, Germany, Japan and UK). The information related to 60-month period
were applied as historical data in order to determine stock weights. Finally, GA is
proved to have higher return and lower risk compared to the other two methods. So was
multistage GA compared to single stage GA.
Xia et al (2000) proposed a model for portfolio selection in which expected returns
were ranked, while Crama and Schyns (2003) tested the use of a mixed integer quadratic
model to solve a complex portfolio selection problem. Deng et al (2005) subsequently
applied the New Mini Max approach to model those portfolio selection problems that
22


have random uncertainty in the input data. Hao et al (2009) applied the GA to study
new examples of the MV model based on Markowitz’s theory for portfolio selection
problems with random investment returns. It was found that the asymmetric returns

changed the variance into an inefficient criterion for risk assessment.
Jia and Dyer (1996) pointed out the fact that these conditions rarely exist in realworld problems. Therefore, the MV objective function considering other risk criteria
cannot be sufficiently appropriate. However, other risk criteria that consider the
preferences of the investors in different conditions could be more suitable. Moreover,
in real- world problems, investors increase some constraints of their optimization
model, such as the size of the portfolio, the minimum and maximum amount of
investment in an asset. Such constraints may create a nonlinear integer programming
model, which may be more difficult to solve than the original problem.
Gharakhani and Sadjadi (2011) applied multi-period robust optimization and
stochastic programming to determine the rate of return in future planning, and used data
simulation in order to evaluate the performance of their proposed model. They
subsequently introduced a new approach with uncertain data in portfolio modeling, and
analyzed this using a different robust method. They solved this formulation with the
help of GA, and compared it with various benchmarks (Sadjadi et al. 2012). A
comprehensive review of the 60-year history of portfolio optimization was presented
and analyzed by Kolm et al (2014).
Numerous researchers have worked on properties and calculations of semivariance and hajne has improved the mean semi-variance models. Fine et al (2001)
examined such models in various states, presented three models based on minimizing
the risk using the criterion of semi variance and discussed the evaluation of the
performance of the mean-semi variance model under uncertainty by entering the fuzzy
return. Ceira (2002) showed that consideration of risks as a factor that does not impose
a fee on portfolio selection led to an ineffective and in some cases illogical solution.
They emphasized that in models aimed at maximizing returns and reducing risk, the risk
aversion coefficient must have considerable value. The weighted upper limits on the
constraints of portfolio models were set by Saxena et al (2002) who did not consider

23


the risk of the portfolio, as calculated from previous data, and took advantage of a

scenario-based approach. They pointed out that use of weighted constraints in the MV
model revealed the systematic weight of investment risk in portfolio selection, which is
not possible in the normal MV model.
Over the last few decades the pioneer model proposed by Markowitz,
mathematical programming techniques have become necessary tools to support
financial decision-making process and applied a lot in real life situation. There are
several mathematical tools used in general to find the best solution in portfolio
optimization. Such as Forecasting, Simulation, Statistical Model, Mathematical
programming models. Among these models mathematical programming is the best
option to the decision maker to find the optimal solution.
Also several exact method based techniques had been applied to solve the portfolio
optimization models, such as integer programming method (Bonami, 2009), goal
programming method (Pendaraki, 2005), lexicographic goal programming approach
(Sharma, 2006). Some meta-heuristics based approaches are also used such as simulated
annealing (Crama, 2003), genetic algorithm (Zhu, 2011), particle swarm optimization
(Cura, 2009), ant colony optimization (Deng et al. 2010). Biswas et al (2013) had
discussed about some advanced optimization techniques.
Azmi and Tamiz (2010) reviewed lexicographic, weighted, minmax and fuzzy
goal programming models and discussed the issues concerning multi-period returns,
extended factors and measurement of risk. Metaxiotis and Liagkouras (2012) and
Ponsich, et al. (2013) analyzed the current state of research in portfolio optimiza- tion
with a focus on Multi Objective Evolutionary Algorithms in which the lack of many
real- life constraints as well as ineffectiveness of Pareto ranking schemes in the presence
of many objectives are indicated. Mansini et al. (2014) focused on linear programming
solvable mod- els in the portfolio optimization classifying the models according to
decision variables used in the integration of real features. Kolm et al. (2014) discussed
practical advances in MVPO and pointed out new research directions such as
diversification methods and multi-period optimi- zation. Aouni, et al. (2014) reviewed
the lexicographic, weighted, polynomial, stochastic and fuzzy goal programming

24


models and pointed out the lack in developing computerized deci- sion support systems
to accomplish a helpful tool to facilitate the decision-making process in portfolio
optimization. Doering et al. (2016) focused on recent contributions of metaheuris- tics
in the sense of an introduction to this topic supported with a numerical example.
Masmoudi and Abdelaziz (2018) focused on deterministic and stochastic multiobjective pro- gramming models comparing the different assumptions and proposed
solutions in portfolio optimization. Zhang et al. (2018) reviewed various extensions of
Markowitz‘s MV model such as dynamic, robust, fuzzy portfolio optimization with
practical factors and pointed out that combined forecasting theory with portfolio
selection would be a promising future re- search direction to deal with uncertainty.
Aouni et al. (2018) reviewed multiple criteria decision aid methods for portfolio
selection with a focus on exact solution methods on the con- struction and optimization
of portfolios as well as on the analysis and the evaluation of specific securities.

4

Data and methodology
Indices for ten sectors in China are used in this study. These sectors include basic

material, consumer goods, consumer services, financials, health-care, industry, oil and
gas, technology, telecommunications, and utilities. The data is obtained from
Datastream for the period from 2007 to 2016.
4.1 Estimates of portfolio risk and return
Portfolio return
For each period, individual sector returns are computed from their index levels
using log forms. The portfolio expected return is simply the weighted average of these
individual sector returns. Write rp as the expected return of a portfolio with N assets, ri
as the expected return of the asset ith in that portfolio, and ωi is the weight of the asset

ith in that portfolio; the calculation of portfolio expected return is given as followed:
𝑁

𝑟𝑝 = ∑ 𝜔𝑖 𝑟𝑖
𝑖=1

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Variance and portfolio variance
Variance of an individual asset is the average of squared differences between
observed returns and their mean. It estimates the distance of a set of returns away from
the average value.
𝑛

1
𝜎 = ∑(𝑟𝑖 − 𝑟̅ )2
𝑛
2

𝑖=1

where 𝜎2 is the variance of return, 𝑟̅ is the mean return, r𝑖 is the ith return in n
observations in the sample set.
A covariance of a pair of assets quantifies the relationship between them. When
the return of the first asset tends to increase matching an increase in the second asset’s
return, the covariance
is positive; otherwise, it is negative. When the returns of the two assets move
independently, the covariance is positive; otherwise, it is negative. When the returns of
the two assets move independently, the covariance is approximately zero. Let us denote

𝜎𝑥𝑦 as the covariance between the asset x and asset y, both assets have n observations,
𝑥̅ and 𝑦̅ as mean return of each asset, xi and yi as the ith returns of asset x and asset y,
the covariance is given as following:
𝑛

𝜎𝑥𝑦

1
=
∑(𝑥𝑖 − 𝑥̅ )(𝑦𝑖 − 𝑦̅)
𝑛−1
𝑖=1

Furthermore, the portfolio variance measures the dispersion of portfolio returns
and its mean value considering the joint effects between constituent assets. It is defined
by the following formula:
𝑁

𝑛

𝜎𝑝2 = ∑ ∑ 𝜔𝑖 𝜔𝑗 𝜎𝑖𝑗
𝑖=1 𝑗=1

where 𝜎𝑝2 is variance of the portfolio with N assets, 𝜎𝑖𝑗 is the covariance between
the asset ith and asset jth, 𝜔𝑖 and 𝜔𝑗 are the weightes of the assets ith and asset jth in the
portfolio.

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Semivariance and portfolio semivariance
Semi-variance is determined by weighted average of square deviations from a
threshold level used only the observation which is below that level. Through a meansemivariance efficient frontier construction, Ballestero (2005) proposed a semivariance matrix computation based on Sharpe’s beta (Sharpe, 1964) regression basis.
From Ballestero (2005), we apply the following computation in our study
2
𝜎𝑠𝑒𝑚𝑖
(<) is the below-the mean semi-variance;
2
𝜎𝑠𝑒𝑚𝑖
(>) is the above-the mean semi-variance;

𝜔𝑖 is the weight of the jth asset in the portfolio;
𝑟̅ j is the mean value (or expected value) of the jth asset;
𝑟̃ jt is the return of the jth asset, tth observation;
T is the number of observations in the sample;
N is the number of assets in the portfolio;
p(t) is the probability of occurance of the event t. When all the observations from
1 to T have equal probability, then 𝑝(𝑡) =

1
𝑇

The below-the-mean semi-variance is presented as:

where the following inequality is satisfied:

Assuming the beta regression equation holds, the return for jth asset is given as:
𝑟̃𝑗 = 𝛼𝑗 + 𝛽𝑗 𝑟̃𝑚 + 𝜀̃𝑗
Where 𝛽𝑗 is the beta of asset jth and the slope of the regression line:

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𝛽𝑗 =

𝜎𝑖𝑗
2
𝜎𝑚

We denote; rm as the market return; and 𝜎2 is the variance of market portfolio.
Then:

Adding up the individual assets in the portfolio:

where

Replacing (1) by (2):

When the number of assets becomes infinity:

2
2
As definition 𝜎𝑠𝑒𝑚𝑖(>)
+ 𝜎𝑠𝑒𝑚𝑖(<)
= 𝜎 2 , so we can do the limit for downside

semi-variance:

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Sortino ratio
Sortino ratio is a crucial indicator in estimating asset and portfolio performance,
first introduced by Frank Sortino in 1980 (Sortino & Hopelain, 1980). Unlike Sharpe
ratio, the Sortino ratio uses a downside risk measure as its denominator as followed:
𝑆𝑜𝑟𝑡𝑖𝑛𝑜 𝑟𝑎𝑡𝑖𝑜 =

𝑟𝑖 − 𝑟𝑓
𝐷𝑜𝑤𝑛𝑠𝑖𝑑𝑒 𝑟𝑖𝑠𝑘

where ri is the return of an observed asset, rf is the risk-free rate. In this paper, we
calculate Sortino measures for our optimal portfolio using several different frameworks.
We use semi-deviation as downside risk. For simplicity, we set the risk-free rate at 4
per cent, captured from a survey for 51 countries in 2013 (Fernandez, Aguirreamalloa,
& Linares, 2013).
4.2 Optimization
Markowitz mean-variance optimization
Since most nations in the world place bans on short selling in financial market, we
set a constraint of non-negative weights within an optimal portfolio. In addition, many
mutual funds also set their weights higher than a certain level for diversification
purpose. For example, each weight must not be lower than five percent. In this paper,
we only apply the non-negative-weight constraint since the frontier will be less efficient
with additional constraints. Further details on Markowitz mean- variance optimization
are discussed in details in Vo et. al (2018).
Mean-semivariance optimization
Optimization in the mean-semivariance framework applies the same principle of
original Markowitz mean-variance optimization. That is, investors attempt to minimize
the risk for each level of expected return. In this case, the semideviation accounts for
risk instead of the classical standard deviation.
Although Markowitz denied applying mean-semivariance framework into

optimization process due to computational infeasibility in the past, Markowitz admitted
that this measure is potential to enhance the quality of an optimal portfolio. The most

29


difficult part in calculation process is the pairwise covariances among the assets in
“downside” perspective, which has been solved in the previous section of this paper.
The mathematical function and constraints are given as follows:
The objective:

Constraints:

Where ri is the return of the ith asset; r0 is a predetermined portfolio return; 𝜔𝑖 and
𝜔𝑗 are the weightes of the assets ith and asset jth in the portfolio; 𝜎𝑖𝑗 is the covariance
2
between the asset ith and asset jth; 𝛽𝑖 and 𝛽𝑗 are the betas of the ith and jth assets; 𝜎𝑚,𝑠𝑒𝑚𝑖

is the above-the-mean semivariance of market portfolio.
Resampling methodology
Since the standard mean-variance optimization utilizes historical parameters to
estimate the expected returns, variances, and covariances between assets in the future,
it also creates estimation errors, which could be extremely significant. Michaud (1989)
argued that this method accidentally does “errors maximization” instead of maximizing
return and minimizing risk. The idea of resampling method fights against the errormaximization bias of the mean-variance procedure.
Following Michaud (1989), this paper uses the parametric resampling in order to
build the out-of-sample model. We use the Monte Carlo simulation to set 500 resamples
of daily returns in the given period. We also assume that the return follows a geometric

30

Brownian motion, then the resamples will follow a multivariate normal distribution with
given mean and standard deviation.
4.3 Data
Indices for ten sectors in China are used in this study. These sectors include basic
material, consumer goods, consumer services, financials, health-care, industry, oil and
gas, technology, telecommunications, and utilities. The data is obtained from
Datastream for the period from 2007 to 2016.
5

Results

5.1 Return, risk, and ranking
Table 1 shows the annualized return, standard deviation, and semideviation of
each sector in the whole period 2007-2016 in China. Risks and returns for 10 sectors in
China are then ranked based on these estimated figures. Two subperiods 2007-2009 and
2010-2016 are also considered and findings are available at the Appendix.
For the period 2007-2009, Healthcare sector is the best industry, which has the
highest return and the lowest standard deviation and the lowest semi-deviation. The
Telecom sector seems to be the worst sector according to risk while Financials has the
worst performance. For the period from 2010 to 2016, Healthcare sector is still the best
performance, but the sector lost its minimum risk to Oil&Gas and Consumer Goods
sectors.1 In total, for the extended period from 2007 to 2016, Healthcare sector is the
best sector Interestingly, the same results are found in Vietnam and Singapore (Vo et
al., 2018). Oil&Gas sector is the worst in term of performance. Technology and

1

It is noted that China conducted its extensive economic reforms in the 1970s which led to a dramatic

reduction in public expenditures and undermined the public health and health care systems of the country.
However, in 2009, the government started recognizing and reversed course again and established several social
health insurance schemes. It is reported that China has now expanded social health insurance to the vast majority
of its 1.4 billion citizens, but public spending remains low in comparison with the total demand from its people
for the services. As a result, the reliance on private financing generates inequalities in access to health care which
is getting more popular in China in these days.

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