Optimal Portfolio Construction : Application of Sharpe ' s Single-Index Model on Dhaka Stock Exchange

This study aims to find whether Sharpe's single-index model of portfolio construction offers better investment alternatives to the investors of the Dhaka Stock Exchange (DSE). For this purpose, month-ended closing price data of 178 companies listed on the DSE, the prime bourse of Bangladesh, and the month-ended index value of DSEX have been used for the period starting from January 2013 to February 2018. The stocks selected for this study belong to 16 industrial sectors, and purposive sampling technique has been used to select these sectors. Sharpe's model formulates a unique cut-off rate and selects the stocks having an excess return-to-beta ratio above that rate. In this study, 54 stocks qualified to be a part of the optimal portfolio. Hence, the proportion of investment to be made on each of the stock is calculated according to the model. The study reveals that three industries occupy a hefty chunk (65.78%) of the proposed investment portfolio. The constructed portfolio offers a monthly return of 2.1489% and carries 1.9516% risk as measured by standard deviation. The beta of the optimal portfolio is only 0.124003. The constructed portfolio outperforms every individual stock as well as the market index in terms of offering the optimal riskreturn combinations. Therefore, this five-and-a-half-decade-old model offers a great opportunity for Bangladeshi investors to optimize return and diversify risk in an efficient manner.


INTRODUCTION 1.1 Research Background
The ground-breaking article published by Markowitz in 1952 marked the beginning of modern portfolio theory.His model is concerned with creating an optimal portfolio of assets by risk-averse investors.According to Markowitz (1952), a risk-averse investor should choose efficient portfolios.An efficient portfolio is one that maximizes return for a given level of risk or minimizes risk for a given level of return.Markowitz's pioneering article, named Portfolio Selection, and his subsequent studies have been a source of inspiration for many researchers and scholars.His model was intended to be pragmatic and implementable.However, it is ironic that the volume of work required to construct an optimal portfolio was staggering, and thus the model was rarely used in practice (Elton, Gruber, & Padberg, 1976).To form a portfolio using Markowitz's model we will need N×(N-1)/2 correlation coefficients, where "N" stands for the number of stocks a firm or an investor follows (Elton, Gruber, Brown, & Goetzmann, 2009).Consequently, the implementation of the model is also quite time consuming and expensive.Recognition of these problems has motivated researchers to develop and simplify the portfolio construction process.In quest of a solution, William F. Sharpe came up with a simplified alternative to the Markowitz's model that significantly reduces the data input and computational requirements.
The model developed by Sharpe, known as Sharpe single-index model, is based on the assumption that co-movement between stocks' return is due to movement in return of a broad market index (Sharpe, 1963).Such a return on a broad market index is taken as a valid substitution for the common macroeconomic factor (Bodie, Kane, & Marcus, 2009).According to Sharpe's model, a single value, known as the cut-off rate, measures the desirability of including security in the optimal portfolio.The portfolio construction process is simplified if the cut-off rate can dictate which securities to include in our optimal portfolio, and which ones to exclude.This simplified model not only allows an investor to pick securities for his/her optimal portfolio but also helps determine how much to invest in each of those securities (Elton, Gruber, & Padberg, 1976).Hence, this five-and-a-half-decade-old model is still considered as one of the simplest and most widely used models for portfolio construction.
Dhaka Stock Exchange (DSE), formerly known as East Pakistan Stock Exchange Association Ltd. (1954)(1955)(1956)(1957)(1958)(1959)(1960)(1961)(1962) and East Pakistan Stock Exchange Ltd.(1962)(1963)(1964), began its journey long before the independence of Bangladesh with a paid-up capital of about BDT 4 billion (Dhaka Stock Exchange, 2017).The introduction of DSE has opened the door to many investors and businesses in fulfilling their investment and financing needs.However, DSE evolved slowly and could not properly fulfill these needs due to various political unrest, corruption, military coups and other massive anomalies (Rahaman, Hasan, & Ahsan, 2013).These evils have always haunted the capital market of Bangladesh that resulted in some turmoils in 1990, 1996, 2011 and 2012 for instance.As a result, investors' trust and confidence in DSE have plummeted.Unless we make the necessary changes to this situation, the capital market can't contribute in a desirable way to the economic growth of Bangladesh.These present a great challenge for the rational investors of Bangladesh in picking a good investment, which necessitates the adoption of a systematic approach in the investment decision.A sound and in-depth knowledge of portfolio analysis can help them to diversify the investment risk without unfavorably affecting the return.
The present study has been carried out with a view to evaluating how well the portfolio constructed using Sharpe's single-index model performs in the Bangladeshi capital market.Besides, the study aims to assist both domestic and foreign investors of DSE in simplifying the construction of an optimal portfolio of equity securities.Although there have been extensive studies on this regard in the developed economies, empirical work on simplifying the portfolio construction and evaluation of its performance is comparatively scarce in the developing economies.Specifically, in Bangladesh, there is a significant lacuna of research in this context.Besides, none of the studies carried out in the developing economies consisted of a wide-ranging collection of stocks representing various industrial sectors.As a result, these studies cannot completely reveal the portfolio optimization potential of Sharpe's model.

a)
Does Sharpe's model of portfolio construction offer better risk-return combination to the investors of DSE? b) Does the portfolio construction model propounded by Sharpe simplify the investment decision for the investors of DSE?

Research Purpose
The primary objective of this study is to find whether Sharpe's single-index model of portfolio construction offers better investment options to the investors of DSE.

2.
THEORETICAL FRAMEWORK Before Markowitz's monumental work on portfolio selection, investors concentrated on evaluating the risk and return characteristics of individual stocks in constructing their portfolios.The rule of thumb was to pick a stock that offered the best opportunity of earning reward with the least amount of risk (Paudel & Koirala, 2007).Markowitz revolutionized the investment criteria by

Sharpe's Single-Index Model
William Sharpe (1963) simplified variant of the Markowitz Model, universally known as the Sharpe's single-index model, assumes that the co-movement between stocks' return is due to movement in return of a broad market index, which is DSEX in our case.Hence, the primary equation underlying the single-index model is: Where,   = Expected return on security i,   =Intercept of the straight line or alpha coefficient (Constant),   = Slope of a straight line or beta coefficient,   = The rate of return on market index, and   = Error term.
Alpha co-efficient (  ) and error term (  ) are two components of a random variable denoted by   .Since the error term (  ) has an expected value of zero, therefore the mean return on security can be expressed as: To perform the portfolio optimization, the measurement of spread and co-movement of return statistics is also required.The mathematical equations required regarding this respect are: Where,   2 = Total variance of a security's return,   2   2 = Market-related variance,   2 = Variance of a stock's movement that is not associated with the movement of the market index, i.e. the stock's unsystematic risk.
Where,   =The covariance of returns between securities i and j,   = Beta of security i,   = Beta of security j, and   2 = Variance of market return.The monthly return of the selected securities is calculated using the following equation.
Where, R it = Monthly return on stock i at time t, P it = Monthly closing price of the stock i at time t, and P it−1 = Monthly closing price of the stock i at time t-1.
In the same manner, the return characteristics of the benchmark market index (DSEX) is calculated using the following equation.
Where, R mt = Monthly return on the market index at time t, I it = Monthly closing market index value at time t, and I it−1 = Monthly closing market index value at time t-1.
The excess return yielded by risky security is calculated by subtracting the risk-free rate of return, which is 0.4784 percent per month (or 5.74 percent per annum) in our case, from the expected return of that risky security.The sensitivity of security's return to the movement in the market's return, popularly known as systematic risk or beta coefficient, is required for analyzing the risk of a security.The beta coefficient can be calculated as follows: Where,   = Covariance of the stock i return with the market return, and   2 = Variance of the market return.
Based on the equations depicted above, it is just a matter of a few steps to reach our optimal portfolio.Sharpe's single-index model greatly simplifies the portfolio construction process by taking a single number to quantify the desirability of a stock's inclusion in the optimal portfolio.If one is ready to accept the standard form the single-index model as describing the co-movement between securities, such a number exists.In this instance, the desirability of any stock is directly related to the excess return to the Beta ratio (Elton, Gruber, Brown, & Goetzmann, 2009): Where,   = Expected return on stock i,   = Return on a riskless asset, and   = Expected change in the rate of return on stock i associated with a 1% change in the market return.The excess return to Beta ratio measures the additional return on security for each unit of systematic risk or non-diversifiable risk.In a colloquial manner, this ratio expresses the relationship between the potential risk and reward.
Once we have the excess return to the Beta ratio for each of the stocks under consideration, the following step is to rank those stocks in descending order of their respective ratios.Since we are assuming that short selling is prohibited, any stock with a negative excess return to Beta ratio has been excluded from further analyses.
The stocks to be included in the optimal portfolio depend on an unprecedented cutoff rate such that all stocks with higher ratios of (  −   )/ are included and all stocks with lower ratios are excluded.This cut-off point is indicated by C*, and is computed from the characteristics of all the stocks that belong to the optimal portfolio.To determine C*, it is required to estimate its value as if there were different numbers of stocks in the optimal portfolio.If we nominate   as a candidate for C*.The value of   is calculated when i securities are assumed to belong to the optimal portfolio.For a portfolio of i stocks,   is given by: Where,   2 = The variance in the market index, and   2 = The variance of a stock's movement that is not associated with the movement of the market index.This is commonly referred to as a stock's unsystematic risk.After estimating the   of all the stocks, the highest   value is selected as the cut-off point (C*), which is then compared with the excess return to the Beta ratio of each stock.All stocks used in the calculation of   have an excess return to Beta above   and all stocks not used to calculate   have an excess return to Beta below   .There will always be one and only one   with this property and it is the cut-off point, C*.After determining which stocks to be included in the optimal portfolio, the investors must find out the optimum percentage of capital to be invested in each of them.The percentage invested in each stock is: Where The former expression indicates the weights on each stock, and they sum up to one.The latter expression determines the relative investment in each stock.The residual variance on each security   2 plays a vital role in determining the amount to be invested in each security.
After determining the weights on each security, beta and alpha on a portfolio are estimated to find out the portfolio risk and return.Beta on a portfolio,   , is a weighted average of the individual   on each stock in the portfolio and is denoted by: Similarly, the Alpha on the portfolio,   , is calculated as: The return on the investor's portfolio can be expressed as: Finally the risk of the investor's portfolio,   , as: All the equations (eq.1-15) cited above are sourced from the book of Elton, Gruber, Brown, & Goetzmann (2009).

RESEARCH METHOD 3.1 Research Design
In the pursuit of constructing an optimal portfolio of equity securities traded on DSE, monthly closing price and benchmark market index (DSEX) data have been used for the period starting from January 2013 to February 2018.The secondary data used in this study are collected from DSE Library.

Population and Sample
The study was initially aimed at all, 228 to be specific, the "A" category companies offering equity securities that belong to sixteen industrial sectors.The industries covered in this study are listed in Table 1.Purposive sampling technique has been applied to select these industrial sectors.The companies which are regular at holding annual general meetings and have declared a dividend at the rate of ten percent or above in the last calendar year are classified as "A" category companies (Dhaka Stock Exchange, 2018).Due to the unavailability of data during the aboved study period, 50 companies have been excluded from the portfolio construction, which leads to a final sample size of 178 companies.The average monthly cutoff yield of 91-day Treasury Bill has been used as a proxy for the risk-free rate of return (Bangladesh Bank, n.d.).
The 178 sample companies selected following the above-mentioned criteria constitutes 61.59 percent of all the equity securities (289) from the sixteen industrial sectors (refer to Table 1).A number of statistical techniques have been applied to analyze the risk-return characteristics and to construct the optimal portfolio of equity securities using Microsoft Excel 2016.In addition to substantial data coverage, the study covers a period of five years and two months, which reduces the possibility of sampling error and the influence of temporary fluctuations to distort the study outcome.Hence, the sample size can be assumed as adequate to represent the population and make smart investment decisions.

RESEARCH RESULT AND ANALYSIS 4.1 Descriptive Analysis
Risk and return have always been the decisive factors in any investment model, and Sharpe's single-index model is of no exception.To apply this model, expected return and risk of each instrument are required.Thus, the month-ended closing price data of the 178 sample stocks as well as the month-ended index value of DSEX, for the period ranging from January 2013 to February 2018, have been used to estimate various risk and return characteristics.The mean monthly return and risk in terms of standard deviation, variance, covariance with market return, and the beta coefficient for each of the selected stocks have been calculated and depicted in Appendix-A.
The risk-return combinations offered by each of the 178 selected stocks have been portrayed in Picture-1.Analysis of the return data on the vertical axis reveals that most of the stocks yielded a monthly return from -0.27 percent to 1.3 percent.However, the average risk carried by the stocks is about 11.5 percent (Picture 1).Although Picture 1 paints an overall picture of the risk and returns Dhaka Stock Exchange Sharpe's Single-Index Model Optimal Portfolio Investment composition offered by the stocks, we cannot pinpoint any precise information from it.An in-depth examination of the data in Appendix-A can be quite useful in this regard.
As revealed by Appendix A, the stock of Northern Jute Manufacturing Co. Ltd. yields the highest monthly return (8.227 percent) followed by Desh Garments Ltd. (5.439 percent) and Renwick Jajneswar & Co (Bd) Ltd. (4.892 percent).In terms of risk, the stock of Legacy Footwear Ltd. has the highest standard deviation (29.845 percent) followed by Grameenphone Ltd. (27.499 percent) and Northern Jute Manufacturing Co. Ltd. (26.969 percent).It is also found that 114 stocks offered a return below the market rate, which is 0.681 percent in our study, and surprisingly none of the stocks in this study had a beta value above the market.
Picture 2 Risk-Return Combination of Each Stock (Secondary Data Processed, 2018) According to Sharpe's single-index model, the expected return ( ̅  ) has two components: a unique or firm-specific part (  ) and a market-related part (    ).Likewise, a security's variance (  2 ) has the same two components: a unique risk (  2 ) and a market-related risk (  2   2 ).The stocks having larger beta value also have a greater part of return that is dependent on the market's performance.Breakdowns of risk & return will guide us to glean some idea about how much this model is going to benefit the investors in their quest for optimizing risk and return.Besides, these variables will be used in further calculations related to portfolio construction and performance evaluation.Hence, the decomposition of each firm's return and risk into unique and market-related parts are presented in Appendix-B.
By analyzing the decomposition of return, we can infer that the contribution of a stocks independent component is the dominant factor than its market-related component.Similarly, the breakdown of risk leads us to the identical conclusion: the contribution of a security's unsystematic variance in its total variance is the dominant factor than its systematic variance.The result suggests that the stocks selected for this study offer a significant opportunity for reduction of unsystematic risk, which we will be able to verify when we construct the optimal portfolio and measure its performance.

Return
Standard Deviation ratio for each stock has been calculated and the stocks are ranked in descending order according to their respective ratios (Appendix-C).The mean monthly cutoff yield of 91-day Treasury Bill, which is 0.4784 percent, has been used as a proxy for the risk-free rate of return.Of the 178 stocks studied, only 70 stocks have a positive excess return-to-beta ratio.Since short selling is assumed to be prohibited, the rest of the stocks have been excluded from further analysis.Among the stocks studied, Renwick Jajneswar & Co (Bd) Ltd. secured the top spot with an excess return-to-beta ratio of 6.18333.Legacy Footwear Ltd. and Grameenphone Ltd. were in the second and third spot with the respective ratio of 4.21675 and 1.12589 (Appendix-C).Among the top scorers, only Renwick Jajneswar & Co (Bd) Ltd. secured the third position in terms of highest return generation.These outcomes suggest that the significantly lower beta value of Legacy Footwear Ltd. and Grameenphone Ltd. is the reason why these companies made their way to the top of the chart in terms of the excess return-to-beta ratio.

Validity and Reliability
This study only used secondary data from DSEX to defined the optimal investment portfolio proposed model, therefore, the validity and reliability test is not needed in this study.

Research Analysis
According to the single-index model, the selection of stock in the optimal portfolio depends on a unique cut-off rate ( * ) and its comparison with the excess return-to-beta ratio of each stock studied.All stocks used for the estimation of  * have an excess return-to-beta ratio above the cut-off rate and the stocks excluded from the estimation have an excess return-to-beta ratio below that rate.The detailed calculation of the cut-off rate is portrayed in Table 2. From Table 2, it is apparent that the highest   value is 0.01351 for stock number 78, i.e.MJL Bangladesh Limited; hence, the cutoff rate ( * ) is 0.01351.Sixteen stocks have an excess return-tobeta ratio below the  * , therefore are excluded from the optimal portfolio.So according to Sharpe's single-index model, the remaining fifty-four stocks, having an excess return-to-beta ratio above the  * , qualify to be a part of the optimal portfolio.

Constructing the Optimal Portfolio
Sharpe's single-index model not only identifies the stocks to be included in the optimal portfolio but also recommends the proportion of fund to be invested in each of them.Thus, the investors can reduce the unsystematic risk and construct a highly diversified portfolio.The following chart (Table 3) depicts the proportion of investment to be made on the fifty-four stocks qualified for the optimal portfolio.As shown in Table 3, the largest chunk of investment (8.1526%)should be made in British American Tobacco Bangladesh Company Ltd., followed by Pharma Aids Ltd. and GlaxoSmithKline (GSK) Bangladesh Limited with an investment proportion of 4.9961% and 3.9825%, respectively.A sector-wise study on the fifty-four stocks in our optimal portfolio, shown in Picture 2, reveals that most of them belong to three industries: Pharmaceuticals & Chemicals, Engineering, and Food & Allied.These three industrial sectors account for 65.78 percent of the total investment to be made in the optimal portfolio, while the rest of the twelve sectors represent only 34.22 percent of the total investment.Picture 3 Sector-Wise Proportion of Investment (Secondary Data Processed, 2018) From the above pie chart (Picture 2), it is evident that Pharmaceuticals & Chemicals sector occupies the highest proportion of fund to be invested in the optimal portfolio, followed by Engineering and Food & Allied industry.Among the sixteen industrial sectors selected for this study, all but Insurance sector have at least one representative stock in the final portfolio.Hence, the final portfolio has stocks from fifteen industrial sectors that react differently to the changes in economic forces.Travel & Leisure, 0,03%

Performance of Optimal Portfolio
Once we have decided about the stocks along with their respective weights in the optimal portfolio, the final task is to evaluate its overall performance.Therefore, the performance of the constructed portfolio in terms of risk and return is calculated, and the key results are summarized in Table 4.For detailed calculations, refer to Appendix-D.As apparent from the above table (Table 4), the constructed portfolio offers a generous monthly return of 2.1489 percent.However, the portfolio risk, measured by the standard deviation, is only 1.9516 percent.The portfolio has a CV of 90.8184 percent, a rate that is considerably lower than any individual stock in the optimal portfolio.The portfolio beta, expressing the systematic risk, also leads us to the identical conclusion.A comparison of the portfolio beta with that of the market indicates a substantial absence of systematic risk in the constructed one.The results of this study conform to the earlier researches conducted on portfolio optimization.Since the portfolio formed using Sharpe's single-index model offers the best risk-return combinations possible in the given context, the portfolio can be termed as the optimal portfolio and the stocks consisting this portfolio as the efficient stocks.

RESEARCH CONCLUSION AND LIMITATION 5.1 Conclusion
In addition to simplifying the portfolio construction process, Sharpe's single-index model reduces the number of inputs required to form the optimal portfolio.If we were to form a portfolio using the same number of stocks, Markowitz's model would require 16,109 inputs.Whereas, Sharpe's model required only 536 inputs, which is about 2,905 percent less than that of its predecessor.The findings of the study reveal a huge possibility of risk reduction through diversification while achieving a substantial return for Bangladeshi investors.The constructed portfolio outperforms every individual stock as well as the market index in terms of offering the optimal risk-return combinations.Therefore, this five-and-a-half-decade-old model offers a great opportunity for Bangladeshi investors to optimize return and diversify risk in an efficient manner.Nevertheless, investors should not slack off at this point.They need to evaluate the performance of each stock and make necessary amendments in the optimal portfolio at regular intervals.Otherwise, the risk may go up and return may go down, evaporating the benefits of portfolio optimization.

Table 1
Sector-wise Representation of Sample Stocks

Table 2
Calculation of The Cut-Off Rate

Table 3
Weight of Stocks in the Optimal Portfolio

Table 4
Evaluation of Portfolio Performance