Portfolio Optimization Studies on Kuwait Stock Market



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 Al-Loughani, Al-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:

i index of a stock

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:

[pic]

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:

[pic]

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

| |Number of Companies |

|Sector | |

|Banks |8 |

|Investment |14 |

|Insurance |4 |

|Estate |8 |

|Industry |14 |

|Services |11 |

|Food |4 |

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 |Market |Model |Model |

|Period |Return |(MA) |(RW) |

| | |Return |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 |Market |Model |Model |

|Period |S. D. |(MA) |(RW) |

| | |S. D. |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 |Market |Model |Model |

|Period |Return |(MA) |(RW) |

| | |Return |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)

|Quarter 1 |Sector |Company |Expected Return |Standard |Actual Return |

| | | | |Deviation | |

| |Bank |Tamwel |-0.057 |0.095 |-0.069 |

| |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 |

| |Average |-0.0060 |0.1016 |0.0060 |

|Quarter 2 |Sector |Company |Expected Return |Standard |Actual Return |

| | | | |Deviation | |

| |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 |

| |Average |0.0279 |0.1023 |0.0396 |

|Quarter 3 |Sector |Company |Expected Return |Standard |Actual Return |

| | | | |Deviation | |

| |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 |

| |Average |0.0706 |0.1071 |-0.0051 |

|Quarter 4 |Sector |Company |Expected Return |Standard |Actual Return |

| | | | |Deviation | |

| |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 |

| |Average |0.0426 |0.1072 |0.1547 |

|Quarter 5 |Sector |Company |Expected Return |Standard |Actual Return |

| | | | |Deviation | |

| |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 |

| |Average |0.0526 |0.0962 |0.0956 |

|Quarter 6 |Sector |Company |Expected Return |Standard |Actual Return |

| | | | |Deviation | |

| |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 |

| |Average |0.1559 |0.1305 |-0.0384 |

|Quarter 7 |Sector |Company |Expected Return |Standard |Actual Return |

| | | | |Deviation | |

| |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 |

| |Average |0.0823 |0.1256 |0.0447 |

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.

References

[1] Al-Loughani N. E and Moosa I. A. (1999), “Testing the Efficiency of an Emerging Stock Market Using Trading Rules: The Case of Kuwait”, Journal of Gulf and Arabian Peninsula Studies, Volume 95, pp 219-237.

[2] Al-Loughani, N. E. (1995), “Random Walk in Thinly Traded Stock Markets: The Case of Kuwait”, Arab Journal of Administrative Sciences, Volume 3, pp 198-209.

[3] Al-Loughani, N., Al-Deehani, T. and Al-Saad K., "Stock Dividend Yield and Investment Rates of Return in Kuwait Stock Exchange", Journal of King Saud University (Administrative Sciences), Forthcoming 2004.

[4] Al-Loughani, N.E. (2000a), "Recent Trends and Market Inefficiency in the Kuwait Stock Exchange: Evidence from the post-liberation Era". In: Arab Stock Markets: Recent Trends and Performance, Dahel Riad (Ed.), The Arab Planning Institute, Kuwait, The American University in Cairo Press, 2000a, 25-27.

[5] Al-Loughani, N.E. (2000b), "The Analysis of Causal Relationship between Stock Prices and Trading Volume in the Kuwaiti Stock Market", Journal of Economic and Administrative Sciences, 15, 217-237.

[6] Al-Loughani, N.E. and Chappell, D. (2000), "Modelling the Day-of-the-Week Effect in the Kuwait Stock Exchange: A Non-Linear GARCH Representation", Applied Financial Economics (Forthcoming).

[7] Bulter, K.C. and Malaika, S. J. (1992), “Efficiency and Inefficiency in Thinly Traded Stock Markets: Kuwait and Suadi Arabia”, Journal of Banking and Finance, Volume 16, pp 197-210.

[8] Mansini R. and Speranza (1999), “Heuristic Algorithms for the Portfolio Selection Problem with Minimum Transaction Lots”, European Journal of Operations Research, Volume 114, pp 219-233.

[9] Markowitz H. (1952), “Portfolio Selection”, Journal of Finance, Volume 7, pp 77-91.

[10] Markowitz H. (1959), Portfolio Selection: Efficient Diversification of Investments, John Wiley, New York.

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