Modeling and Analysis for Portfolio Optimization in an Emerging Market The Case of Kuwait

Aldeehani Department of Finance and Financial Institutions College of Business Administration Kuwait University Abstract:- In this paper, we study the optimal portfolio selection of stoc

Modeling and Analysis for Portfolio Optimization

in an Emerging Market: The Case of Kuwait

Majid M Aldaihani Department of Industrial and Management Systems Engineering

College of Engineering and Petroleum

Kuwait University P.O Box 5969 Safat 13060 Kuwait Tel: 4811188-7257: Fax: 4816137

KUWAIT Talla M Aldeehani Department of Finance and Financial Institutions College of Business Administration

Kuwait University

Abstract:- In this paper, we study the optimal portfolio selection of stocks in Kuwait Stock Exchange (KSE)

as an emerging market An integer programming mathematical model for portfolio optimization is

developed to balance the tradeoff between the expected return and risk Moving Average (MA) and Random Walk (RW) techniques are used to determine the expected return, while standard deviation and correlation between the selected stocks in the portfolio are used to measure the portfolio risk A quarterly basis strategy and an annual basis strategy are applied to test the model using real data from KSE for the years from 1994

to 2001 The results indicate that there is room for optimization in KSE, if the model uses the annually basis strategy.

Key-word:- Portfolio Selection/Optimization/Mathematical Modeling/Integer Programming

1 Introduction

Investments that provide high returns safely is the

ultimate goal of most of the investors worldwide The

relationship between the return and risk is obvious in

the stock markets, where mostly stocks that grant high

returns are very risky Therefore, investors sometimes

search out portfolios that balance the trade off between

risk and return Markowitz's seminal work on portfolio

selection, 1950s, inspirited researchers to study the

effectiveness of asset portfolio optimization Special

attention was given to stock markets Theoretically

speaking, when the stock market is reasonably

efficient, there is little room for ordinary investors to

make excess returns as information of any kind, public

or private, is of no use in beating the market However,

when the market is inefficient, it is logical to assume

the possibility of beating the market through

manipulation of public information Kuwait Stock

Exchange (KSE) is an emerging market that has been

found by many studies to be inefficient [see for

example Al-Loughani (1995), (2000a) and (2000b),

Al-Loughani and Chapell (2000) and Al-Loughani and

Moosa (1999)] One study of particular interest to this

paper is the work of Loughani, Deehani and Al-Saad (2004) which focuses on portfolio selection

They tested the validity of the Dow-10 investment

portfolio selection strategy in the KSE The results of their work revealed that the risk-adjusted returns of the

Dow-10 portfolio were much higher than the returns of

the market portfolio This is an additional proof of KSE inefficiency

The objective of this research is to develop a mathematical model for portfolio optimization in an identified time horizon for KSE The model, which is based on an integer programming optimization technique, identifies the size of the portfolio (number

of stocks) and the name of stocks in the portfolio The percentage to be invested in each stock is assumed to

be distributed equally among the portfolio stocks The main contribution of this research is to check whether

or not there is room for optimization in Kuwait Stock Exchange The ultimate technical goal of the

optimization model is to find a portfolio which maximizes the expected return subject to a certain limit of risk The proposed model takes into account,

not like previous methods, variety of stocks’ risks

(correlation, variation, and number of stocks) The

developed optimization model uses only past data

from KSE, including stock name, sector name, date,

and price The model is tested using this real data by

comparing the portfolio performance measures (return

and risk) to the market index Technically, the

proposed model improves the previous methods

Although it has been proved that KSE is an inefficient

market, there is little work in the literature introducing

applications of optimization models for KSE

Al-Loughani and Moosa (2000) tested the efficiency of

Kuwait Stock Market using a moving average rule

They studied the market for the time periods

1986-1990 and 1992-1997 Their results, obtained in the

study, showed some evidence demonstrating that

Kuwait stock market is inefficient Al-Loughani

(1995) studied the application of the Random Walk

rule in thinly traded stock markets He specifically

studied Kuwait stock market and showed that it is

inefficient when sophisticated tests are used Most of

the other research that is conducted on KSE is merely

statistical analysis This is actually a strong motivation

for developing such an optimization technique

Markowitz (1952) was one of the first to formulate

the portfolio selection problem The introduced model

was to minimize the risk, represented by the

covariance, subject to a certain bound of expected

return (see also Markowtiz, 1959) Additionally,

Mansini and Speranza (1999) is a good source for

reviewing the portfolio optimization models It

includes some heuristic algorithms They introduced

methods to find a solution closed to optimal (heuristic

solution) with a reasonable amount of computational

time

2 Kuwait Stock Exchange Properties

Although informal trading of stocks in Kuwait started

in 1952, organized and controlled trading did not begin

until 1983 Compared to all Arab stock markets, KSE

has the highest turnover ratio It is ranked second in

terms of value traded and third in terms of market

capitalization In the year 2002, KSE's market

capitalization was $35.1 billion representing about

45% of all Gulf Cooperation Council (GCC) countries'

stock markets and about 17% of all Arab stock

markets The value trade for the same year was $22.1

billion which represents about 40% of GCC countries

and about 34% of all Arab stock markets At the end

of 2002, there were 96 listed companies, 10 of which

are non-Kuwaiti

Common stock is the only financial security traded in KSE Short selling is not allowed Although not practiced by the vast majority of traders, organized margin trading is available through only one provider Trading is settled through brokers that are prohibited

by law from providing any advice

Ever since the start of its formal operations, KSE can only be described as instable This is due to major financial and political factors These are, the Iraq-Iran War 1980-1988, The AL-Manakh financial crisis that started at the end of the 1970s Its consequences still persist The Gulf War in 1990 added more to the volatility of the market and still persists And lately, the consequences of the war against Iraq in 2003 that still persist These conditions along with other socio-economic factors have made KSE a manipulative market Compared to the regional GCC markets (except Oman), KSE seemed the most volatile (Al-Deehani 2004) Therefore, short-term investment and market manipulation appear to be logical investment strategy for most KSE investors A comprehensive description of KSE main characteristics can be found

in AL-Loughani and Moosa (1999)

3 Problem Formulation

In this section, we describe the problem mathematically Let iN represent the stocks in the market For each stock, there is a standard deviation si and expected return ri The standard deviation of a stock is measured using the previous eight quarters (two years) The expected return of a stock is computed using two forecasting rules The first one is the Moving Average (MA) rule (using 2 quarters) The second one is the Random Walk (RW) rule (using 8 quarters) Both methods are used in the model for comparison purpose For each pair of stocks in the market i and j, there is a correlation corrij which describes the relationship between these two stocks The decision variable in this problem is xi which equals one if the stock i is selected in the portfolio and equals zero otherwise lbsd represents an identified lower bound for the stock standard deviation, marketsd denotes the average standard deviation of the all the stocks in the market, and lbcorr is the lower bound for the portfolio correlation The objective function of the problem is to select a subset from N that maximizes the expected return while at the same time satisfies all the constraints regarding the risk (variation and correlation)

In other words, the proposed mathematical model is used to balance the trade off of average return from a set of stocks that are to be selected in a portfolio with

the risk associated with selecting these stocks The risk

in the model is measured and restricted using three

main constraints These constraints are as follow:

3.1 Correlation

In some instances, the objective of studying the joint

behavior of two stocks is not to use one stock to

predict the other, but to check whether they are related

It is natural to speak of stock A and stock B having a

positive relationship, if large A’s are paired with large

B’s and small A’s are paired with small B’s

Similarly, if large A’s are paired with small B’s and

small A’s are paired with large B’s, then a negative

relationship between the stock is implied This is

actually studied by computing the correlation between

the stocks that are selected in the portfolio It is

required that the selected stocks be limited to a

predetermined limit This definitely helps in avoiding

a sudden collapse of the portfolio

3.2 Variation

One way for evaluating the investment risk in a stock

is to check its variability There is no dispute that for

two stocks with similar expected returns, it is more

safe to invest in the one that has less variation This is

the reason for restricting the average standard

deviation of the portfolio to be less than or equal to the

average standard deviation of the market Furthermore,

an additional constraint is set to limit the standard

deviation of each stock selected in the portfolio

3.3 Portfolio size

It is crucial to identify the required number of stocks

that the portfolio contains The “Dow ten” provides a

pattern for this constraint The number of stocks to be

selected in the portfolio is bound by setting the range

from 5 to 15 stocks (+/- 50% of 10 stocks) Also, the

model is capable of bounding the number of stocks in

each sector separately However, this is found to be

irrelevant in this case study

3.4 Mathematical Model

Before presenting the mathematical formulation in

detail, let us summarize the notations that are used in

the model:

N set of all stocks in KSE

xi binary variable which equals 1 if the

stock “i” is selected in the portfolio and 0 otherwise (Decision Variable)

ri expected return of stock “i” over the

time horizon (2 quarters for the MA technique and 8 for the RW technique) corrij correlation between the stocks i and j

over the time horizon

si standard deviation of stock “i” over

the time horizon lbcorr lower bound for the portfolio

correlation marketsd average standard deviation of all the

stocks in the market for the time horizon

lbsd identified lower bound for the stock

standard deviation over the time horizon

Below is the mathematical model:

) 6 ( }

0

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) 2 (

) 1 ( max

N i x

x

N i lb s

x

market x

s x

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to subject

r x z

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sd i

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Note that only constraint (2) makes the model nonlinear since there are two decision variables multiplied by each other To convert the model from nonlinear programming to an Integer Programming, equation (2) is replaced by the following three constraints:

)9 ,

}

0

)8

) 7 ( , 1

N j y

lb

corr

y

N j y

x

x

ij

i j

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ij

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ij j

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The Solver tool used in solving this mathematical

model uses the Generalized Reduced Gradient (GRG2)

optimization code developed by Leon Lasdon,

University of Texas at Austin, and Allan Waren,

Cleveland State University The optimization code is

designed in such a way that any of the functions may

be nonlinear, any of the bounds may be infinite and

any of the constraints may be absent If there are no

constraints, the problem is solved as an unconstrained

optimization problem Upper and lower bounds on the

variables are optional and, if present, are not treated as

additional constraints, and are handled separately

4 Experimental Tests

Data was collected from Kuwait Stock Exchange

(KSE) for the past five years The data includes date,

stock name, sector name, stock price, and market

index KSE consists of a number of companies that are

categorized, according to their business, under 8 main

sectors The names of the main sectors and the number

of companies in each one are shown in Table 1 below:

Table 1 Sectors in KSE

Sector

Number of Companies

The numbers above vary over time, due to the entering

and leaving of new and out of business companies,

respectively The considered companies in this

research are the ones that are sufficiently represented,

data-wise The model is tested by applying a quarterly

basis strategy and an annual basis strategy In each

one, the portfolio generated by the model is compared

to the market return in the same interval of time

4.1 Quarterly basis strategy

The optimization model is tested using real data from

KSE for the time period from 1994 until 2001 The

model uses past data (8 quarters) for the companies in

the current market in order to estimate the model’s parameters (e.g correlation, standard deviation and expected return) The portfolio is generated on a quarterly basis and compared to the market index As shown in Table 2, the quarters are divided into two types, which are over-market quarters and under-market quarters, according to their performance compared to the market The over-market quarters are the ones where in the generated portfolio provides better return than the market (i.e quarters 2,3,4, and 6) While the under-market quarters are the ones where

in the market provides better return than the generated portfolio (i.e quarters 1, 5 and 7) It is true that the market has been beaten in 4 out of 7 quarters

However, it is important to recognize the percentage over or below the market This can be seen in Figures

1 and 2 On the other hand, the model provides less risk than the market with respect to the average standard deviation This is not an unexpected result since there is a constraint in the model, restricting the generated average standard deviation to be less than or equal to the market average standard deviation Table

3 shows that for all the quarters, Model (MA) and Model (RW) have less Standard Deviation (S.D.) than the Market Standard Deviation (S.D.)

Table2: Quarterly Basis Strategy (Return)

Investment Period

Market Return

Model (MA) Return

Model (RW) Return

Q1: 3/00-6/00 2.76% 0.60% -1.00% Q2: 6/00-9/00 1.98% 4.00% 3.00% Q3: 9/00-12/00 -6.65% -0.50% 1.00% Q4: 12/00-3/01 7.86% 15.50% 10.80% Q5: 3/01-6/01 15.89% 9.60% 14.80% Q6: 6/01-9/01 -5.04% -4.00% -4.00% Q7:9/01-12/01 6.81% 4.50% 3.60%

Table 3 Quarterly Basis Strategy (Risk)

Investment Period

Market

S D.

Model (MA)

S D.

Model (RW) S.D.

Q1: 3/00-6/00 14.2% 10.2% 10.9%

Q2: 6/00-9/00 15.5% 10.8% 11.0%

Q3: 9/00-12/00 14.3% 10.0% 10.9%

Q4: 12/00-3/01 13.9% 9.4% 9.9%

Q5: 3/01-6/01 13.9% 9.3% 9.9%

Q6: 6/01-9/01 16.0% 12.5% 12.3%

Q7: 9/01-12/01 16.1% 12.8% 11.5%

Figure 1.Model and Market Performance

(quarterly Basis)

Figure 2 Accumulating portfolios (quarterly basis)

4.2 Annual Basis Strategy

The results presented in the previous strategy can be

improved significantly when we use the annual basis

strategy In the annual basis strategy, the model

annually generates and accumulates four quarterly

basis portfolios in the year and compares the

performance of the model to the market Note that the portfolios are generated as before on a quarterly basis The comparison, however, is done annually In other words, the money invested at date 3/00 cannot be retrieved until 3/01 even though there are four portfolios that are generated in between For example:

If $100 is invested in 3/00, it becomes $100.6 in 06/00,

$104.6 in 09/00, $104.1 in 12/00, and $120.2 in 03/01 Hence the model return is 20.2% from 3/00 to 3/01 as compared to the market compounded return during the same period of time, which is 5.5%

The other comparisons are shown in Table 4 below It

is good to say here that the model always provides a portfolio with less risk than the market with respect to the average standard deviation since there is a

constraint in the model formulated specifically for this purpose Furthermore, note that the model has other constraints for the purpose of limiting the risk of the generated portfolio with regard to the correlation and number of selected stocks

Table 4 Annual Basis Strategy

Investment Period Market Return

Model (MA) Return

Model (RW) Return

3/00-3/01 5.5% 20.2% 14.1% 6/00-6/01 19.0% 31.0% 32.3% 9/00-9/01 10.8% 20.9% 23.3% 12/00-12/01 26.8% 27.0% 26.5%

Table 5 shows the selected stocks in each of the seven generated portfolios For each quarter, the information includes the sector from which the company is selected, the name of the chosen company, the expected return of the company during the quarter, using the moving average technique, the standard deviation during the previous 8 quarters, and the actual return of the company for the same quarter Also, at the bottom of each section of the table, we provide the average of the above mentioned statistical measures

Table 5 Selected Stocks of Portfolios (MA Model)

Expected Return

Standard Deviation

Actual Return

Bank Tamwel -0.057 0.095 -0.069

90.0

100.0

110.0

120.0

130.0

140.0

0 1 2 3 4 5 6 7 8 9

mkt (RW)

(MA)

$

Qtrs

-10.00%

-5.00%

0.00%

5.00%

10.00%

15.00%

20.00%

Quarter

MA RW

Investment Sahel -0.005 0.058 0.051 Insurance Kuwait 0.034 0.078 -0.026 Estate Salheia 0.000 0.137 -0.062 Estate AlMsaleh -0.024 0.120 -0.079 Industry Tabreed -0.006 0.136 0.176 Industry Caibellat 0.011 0.107 -0.032 Industry Bobyan -0.017 0.133 0.063 Food Agtheia 0.010 0.050 0.031

Quarter 2

Expected Return

Standard Deviation

Actual Return

Bank Khalej 0.032 0.072 0.081 Bank Tejare 0.102 0.128 0.118 Investment Sahel 0.020 0.063 -0.038 Estate Salheia -0.038 0.135 -0.131 Estate AlMsaleh -0.026 0.114 0.114 Industry Tabreed 0.060 0.138 0.175 Industry Sakb 0.087 0.110 0.043 Food Agtheia -0.014 0.058 -0.045

Quarter 3

Expected Return

Standard Deviation

Actual Return

Bank Tejare 0.144 0.133 0.057 Bank Awsat 0.120 0.097 -0.061 Bank Tamwel 0.025 0.096 0.000 Insurance Ahleia 0.017 0.131 -0.030 Industry Sena'at 0.033 0.103 0.047 Industry Sakb 0.125 0.104 0.000 Industry Bobyan 0.108 0.143 -0.133 Food Agtheia -0.007 0.048 0.079

Quarter 4

Expected Return

Standard Deviation

Actual Return

Bank Tejare 0.088 0.134 0.269 Bank Tamwel 0.060 0.095 0.093 Investement Tashelat 0.136 0.114 0.259 Insurance Khalej -0.009 0.100 0.055 Insurance Ahleia -0.036 0.128 0.154 Estate Ajial 0.064 0.134 -0.087 Industry Sakb 0.021 0.102 0.274 Food Agtheia 0.017 0.052 0.221

Quarter 5

Expected Return

Standard Deviation

Actual Return

Bank Tamwel 0.047 0.095 0.049 Insurance Khalej 0.010 0.100 0.034 Insurance Ahleia 0.062 0.128 -0.053 Industry Sena'at 0.001 0.107 0.219 Industry Caibellat 0.052 0.095 0.021 Industry Sakb 0.137 0.102 0.269

Services Ta'alemeia -0.038 0.092 0.118 Food Agtheia 0.150 0.052 0.108

Quarter 6

Expected Return

Standard Deviation

Actual Return

Bank Ahle 0.291 0.159 0.046 Investment Kuwaiteia 0.191 0.151 -0.238 Insurance Warba 0.097 0.121 -0.097 Estate Salheia 0.001 0.081 0.075 Industry Esment 0.259 0.155 -0.080 Industry Bahreia 0.157 0.144 -0.070 Industry Bobyan 0.086 0.150 -0.030 Food Agtheia 0.165 0.083 0.087

Quarter 7

Expected Return

Standard Deviation

Actual Return

Investment Sahel 0.06 0.079 0.053 Investment Markaz 0.089 0.141 0.127 Insurance Ahleia 0.009 0.115 -0.039 Industry Sofon 0.169 0.145 0.092 Industry Bahreia 0.122 0.145 0.045 Industry Sakb 0.058 0.155 0.080 Industry Bobyan 0.053 0.145 -0.021 Food Agtheia 0.098 0.080 0.020

5 Conclusion and Future Work

It has been shown, as a major contribution of this

paper, that an integer programming optimization

model can identify a stock portfolio that is able to

outperform the KSE market index in terms of risk and

return Another concluding remark in this research is

that although the quarterly basis strategy provided an

optimized portfolio that did not outperform the market

index in 3 out of 7 quarters, the annual basis strategy

outperformed it for all four tested years

As shown in Table 3, the introduced model has

significantly outperformed the market in all four years

tested when the annually basis strategy and moving

average rule are used Also, it is noticed that the

market is beaten by the model with respect to return

and risk The model provides higher return with lower

risk when compared to the market Consequently, it

suggests that there is room for implanting optimization

techniques in Kuwait Stock Market This conclusion

supports the work done by Al-Loughani (2000) and

Al-Loughani et al (2004), which provide evidence of

the weak efficiency of KSE

In light of the data used, and before we can generalize

on the results of this paper, further research on portfolio selection in emerging markets is encouraged

to include a larger data sample, different constraints and different markets The main implication of this research for practitioners is the possibility of using this model to select a portfolio that can produce higher returns without increasing risk

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