optimal portfolio construction in stock market

www.cpmr.org.in Opinion: International Journal of Business Management 37

ISSN: 2277-4637 (Online) | ISSN: 2231-5470 (Print) Opinion Vol. 2, No. 2, December 2012

Optimal Portfolio Construction in Stock Market-An Empirical Study on Selected Stocks in

Manufacturing Sectors of India

Dr. Sathya Swaroop Debasish* Jakki Samir Khan**

ABSTRACTThe main focus of this research is to construct anoptimal portfolio in Indian stock market with thehelp of the Sharpe single index model. Portfolioconstruction is an important process of the investorsfor investment in the equity market. A goodcombination of portfolio will give maximum returnfor a particular level of risk. In this research, 14selected stocks from the various manufacturingsectors like Automobiles, Cements, Paints, Textilesand Oil & Refineries sectors have been taken intoconsideration and these stocks are constituent ofthe NSE Nifty index. The daily data for all the stocksfor the period of January 2003 to November 2012have been considered. The proposed methodformulates a unique cut off point (Cut off rate ofreturn) and selects stocks having excess of theirexpected return over risk free rate of returnsurpassing this cut-off point. Percentage ofinvestment in each of selected stocks is then decidedon the basis of respective weights assigned to eachstock depending on respective beta value, stockmovement variance unsystematic risk, return onstock and risk free return vis-a-vis the cut off rate

of return. The study finds that only three companystocks constitute the optimum portfolio and theseare Asian Paints, Tata motors and Hero Motor Corp.with ideal proportion of investment of 1.9 %, 38.88and 58.22% respectively. This research findings andsuggestions would be helpful to investors.

Keywords: Beta, Market variance, Residualvariance, Portfolio construction, Single index model,Optimal portfolio, Risk and return trade off,Diversification, Nifty.

I. INTRODUCTIONThe security analysis and portfolio management hasemerged as the most concerned aspect for rationalinvestment and decision making. Portfolio is acombination of securities such as stocks, bonds andmoney market instruments. The process of blendingtogether the broad assets classes so as to obtainoptimum return with minimum risk is called portfolioconstruction. A portfolio tries to trade off the risk returnpreferences of an investor by not putting all eggs in singlebasket and thus allows for sufficient diversification.Markowitz was the first who laid foundation of “Modernportfolio theory” to quantify risk. He provided analytical

*Reader, Department of Business Administration, Utkal University, Vani Vihar, Bhubaneswar, Odisha, India** Lecturer, Modern Engineering & Management Studies, Balasore, Odisha, India.



tools for analysis and selection of optimal portfolio. Thisportfolio approach won him Nobel Prize in 1990. Thework done by Markowitz was extended by WilliamSharpe. He simplified the amount and type of input datarequired to perform portfolio analysis. He made thenumerous and complex computations easy which wereessential to attain optimal portfolio. This simplificationis achieved through single index model. This modelproposed by Sharpe is the simplest and the most widelyused one.

The study focuses on finding out an optimal portfoliousing Sharpe’s single index model. This paper is builtaround building up an optimal portfolio by balancingthe positive and negative correlation existing betweenthe securities and in turn getting returns closer to theanticipated results. For this study, the stocks from thedifferent Manufacturing sectors are included since thesesectors are of prime importance for real growth of aneconomy.

II. OBJECTIVES OF THE STUDYThe primary objective is to construct an optimal stockportfolio among selected stock from the variousmanufacturing sectors in India.

The specific objectives are:• To present a review of past works relating to

optimum portfolio construction and analysis.• To build an optimum stock portfolio among

selected stocks belong to the manufacturingsectors in India, using Sharpe Single Index model.

• To calculate the proportion of investment to bemade into each of the stock that is included in theoptimal stock portfolio that is constructed usingSharpe index model.

III. REVIEW OF LITERATUREIn an earlier study Campbell, Husiman and Kodedijk(2001) viewed that optimal stock portfolio is one whichallocates financial assets by maximizing expected returnsubject to the constraint that the expected maximumloss should meet the Value at Risk limits set by the risk

manager. Similar to the mean-variance approach aperformance index like the Sharpe index is constructed.Furthermore when expected returns are assumed to benormally distributed, it is shown that the model providesalmost identical results to the mean-variance approach.Liow (2001) in his research work examined theinvestment performance of Singapore real estate andproperty stocks over the past 25 years. The analysismade using coefficient of variation (CV) and SharpeIndex (SI) suggest that real estate outperformed propertystocks on a risk-adjusted basis. Results also indicatethat risk adjusted investment performance for residentialproperties remained superior to performance for otherreal estate types and property stocks. Haslem andScheraga (2003) in their study used Data envelopmentanalysis (DEA) to identify the large-cap mutual funds inthe 1999 Morningstar 500 for efficient or inefficient. Anattempt is made to identify the financial variables thatdiffer significantly between efficient and inefficient fundsand to determine the nature of these relationships.According to study findings, there are identified input/output and profile variables that are significantly differentbetween the 1999 morning star 500 large-cap mutualfunds that are DEA performance-efficient and inefficient.The Sharpe index represents the DEA output variable.The findings indicate the variables that are significantlydifferent between performance efficient and inefficientfunds and the nature of their relationships. The variablevalues associated with efficient funds are relativelyconservative in nature, not aggressive. Andrea, Wilfredand Jean (2003) suggest empirical evidence on theefficiency and effectiveness of hedging U.S. basedinternational mutual funds with an ASIA-Pacificinvestment objective. The case for active currency riskmanagement is examined for a passive and a selectivehedge, which is constructed with currency futures in themajor currencies. Both static and dynamic hedgingmodels are used to estimate the risk-minimizing hedgeratio. The results show that currency hedging improvesthe performance in internationally diversified mutualfunds. Such hedging is beneficial even when based onprior optimal hedge ratios and efficiency gains fromhedging, as measured by the percent change in the



Sharpe Index, are greatest under a selective portfoliostrategy that is implemented with an optimal constanthedge ratio.

Moreno, Macro and Olmeda (2005) analyzed, froman investors perspective, the performance of severalrisk forecasting models in obtaining optimal portfolios.Specifically, it studies whether ARCH-type basedmodels obtain portfolios whose risk-adjusted returnsexceed those of the classical Markowitz model. Thesame analysis is performed with models based on theLower Partial Moment (LPM) which take into accountthe asymmetry in the distribution of returns. The resultssuggest that none of the models achieve a clearlysuperior average performance. It is also found thatmodels based on semi variance perform as well as thosebased on the variance, but not better than, even if theevaluation criterion is based on the Reward-to-Semivariance ratio. Ebner and Neumann (2008) explainedthe correlation instabilities in US stock returns and usedthree different estimation approaches to overcome theproblem : (1) moving window least squares, (2) flexibleleast squares and (3) the random walk model. Theresults suggest that a time-varying estimation of returncorrelations fits the data considerably better than timeinvariant estimation and thus, increases the efficiency ofrisk estimation and portfolio selection. Nateson andRajesh (2010) constructed optimal portfolio usingSharpe’s Single Index Model consisting of eight stocksfrom Nifty Nifty stocks six stocks selected from NiftyJunior. The respective portfolio beta’s were calculatedand capital allocation for each stock was alsodetermined. Thus, the analysis of the portfolio providesthe rationale for forming an optimal portfolio of thesecurities instead of buying only a single security.

In the Indian scenario, Varadharajan (2011)constructed an optimal equity portfolio with the help ofSharpe Index model. The study was conducted withthe financial data from April 2006 to March 2011. Thesample size was limited to 19. He took these companiesfrom Banking and Information Technology. The portfoliowas constructed with the top 5 stocks that meet thecriteria to be included in the portfolio according to SharpeIndex Model. The portfolio predominantly consisted of

stocks from the banking sector, and one stock from ITsector. In a recent study Saravanan and Natarajan(2012) attempted to construct an optimal portfolio byusing Sharpe’s Single Index Model. For this purposeNSE Nifty Index has been considered. The daily datafor all the stocks and index for the period of April 2006to December 2011 have been considered. Theyformulated the cut-off point and selected stocks havingexcess of their expected return over risk free rate ofreturn surpassing this cut-off point. Percentage ofinvestment in each of selected stocks is then decidedon the basis of respective weights assigned to each stockdepending on respective beta value, stock movementvariance unsystematic risk, return on stock and risk freereturn vis-à-vis the cut off rate of return. From theempirical analysis, it was concluded that returns on eitherindividual securities or on portfolio comprises ofsecurities of different companies listed in Nifty 50 stocksunder various sectors are asymmetrical andheterogeneous. The optimal portfolio consists of fourstocks selected out of 50 short listed scrips, giving thereturn of 0.116. Further it helps to elicit that return onsecurities of different portfolio is independent of thesystematic risk prevailing in the market.

IV. RESEARCH METHODOLOGYThis is a descriptive study on the construction ofportfolio of stocks. The data taken for the study issecondary in nature. The data has been collected fromthe official website of National Stock Exchange (NSE),namely www.nse-india.com. The study is conducted withthe financial data for the past ten years from January2003 to November 2012. The sample size of the studyis limited to daily stock price series of 14 selected stocksthat belong to the five manufacturing sectors namelyAutomobiles, Cements, Paints, Textiles and Oil &Refineries sectors, and these stocks are also part of the50 stocks that constitute NSE Nifty. The samplingtechnique adopted is purposive sampling.

4.1 ReturnThe daily return on each of the selected stocks iscalculated with the following formula.



1

1itit

it

PRP −

= −

where Pt, Pit–1 are the share price at time t and t-1for security i.

4.2 Standard DeviationThe second phase in the context of testing of Sharpe’smodel for selection of appropriate securities in portfoliois used, the average returns of individual returns orportfolio are adjusted to that of risk free return (here6.5 percent is considered as risk free rate based on theportfolio on 91-day Government of India treasury bills).Therefore to estimates the coefficients with risk freeadjusted average return on individual / portfolio and onmarket risk, the following model is used. The selectionof any stock is directly related to its excess return –beta ratio:

( )i f

i

R Rβ−

where Ri = the expected return on stock i; Rf = thereturn on a riskless asset and βi = the expected changein the rate of return on stock i associated with one unitchange in the market return.

The excess return is the difference between theexpected return on the stock and the riskless rate ofinterest such as the rate offered on the governmentsecurity or Treasury bill. The excess return to beta ratiomeasures the additional return on a security (excess ofthe riskless assets return) per unit of systematic risk ornon-diversifiable risk. This ratio provides a relationshipbetween potential risk and reward.

Ranking of the stocks is done on the basis of theirexcess return to beta. Portfolio managers would like toinclude stocks with higher ratios. The selection of thestocks depends on a unique cut –off rate such that allstocks with higher ratios of ( Ri - Rf)/ βi are includedand the stocks with lower ratios are left out. The cut-off point is denoted by C*.

21 2

22

1 21

i fNm i i

eii

N im i

ei

R Rx

Cσ β

σβσσ

=

=

− Σ

=

+ Σ

The highest Ci value is taken as the cut – off pointC*. The stocks ranked above C* have high excessreturn to beta than the cut – off Ci and all the stockbelow C* has low excess returns to beta. If the numberof stock is large, there is no need to calculate the Civalues for all the stocks after the ranking has been done.It can be calculated until the C* value is found and aftercalculating for one or two stocks below it the calculationscan be terminated.

The Ci can be stated with mathematically equivalentway:

( )ip i fi

i

R RC

β

β

−=

where βip = The expected changes in the rate ofreturn on stock i associated with 1 percent in the returnon the optimal portfolio; Rp = The expected return onthe optimal portfolio and βip and Rp cannot bedetermined until the optimal portfolio is found. To findthe optimal portfolio, the formula in above should beused. Securities are added to the portfolio as long as:

i fi

i

R RC

β−

>

Now,

( )i f ip p fR R R Rβ− > −

The right hand side is the expected excess returnon a particular stock based on the expectedperformance of the optimum portfolio. The term on theleft hand side is the expected excess returns on theindividual stock. Thus, the portfolio manager believesthat a particular stock will perform better than theexpected return base on its relationship to optimalportfolio.



4.3 Construction of the Optimal PortfolioAfter determining the securities to be selected, theinvestors should find out how much should be investedin each security. The percentage of funds to be investedin each security can be estimated as follows:

( )1

ii N

i i

ZXZ=

=Σ

2 *i fii

ei i

R RZ Cβ

σ β−

= −

The first expression indicates the weights on eachsecurity and they sum up to one. The second shows therelative investment in each security. The residual varianceor the unsystematic risk has a role in determining theamount to be invested in each security.

V. FINDINGS & ANALYSISThe results of the Sharpe Single index model for eachof the 14 selected stocks are presented in table-1. Itcan be seen from the table that Asian paints yieldedthe maximum return (0.2441) among the companiesselected and Ashok Leyland yielded lowest return of-0.1557. The results of the Sharpe Single index modelfor each of the 14 selected stocks are presented intable-1.

Table 1

Calculated values of Return, Beta & Excess return toBeta ratio for the selected stock in the Indian

Manufacturing sectors

Company scrip Return (Ri) Beta (βββββ) Excess Returnto Beta ratio

(Ri-Rf/βββββi)

Bajaj-Auto 0.0313 0.5624 -0.3057

Hero Moto corporation 0.1924 0.6020 0.5398

M & M 0.1839 0.9207 0.0492

TATA Motors 0.0304 1.0704 2.9587

Ashok Leyland -0.1557 0.9011 -0.0531

ACC 0.1969 0.7740 0.0569

Ambuja Cement 0.0053 0.5264 -15.9993

Asian Paints 0.2441 0.2885 0.4292

Aditya Birla Textiles 0.2045 0.7706 0.2049

Grasim 0.2246 0.7192 0.1591

ONGC -0.0327 0.8626 -0.1377

BPCL 0.0415 0.5934 -0.1188

Reliance Refineries 0.0986 0.9991 -0.2539

Hindustan Petroleum -0.0067 0.5689 -0.5201

It can be seen from the table that Asian paints yieldedthe maximum return (0.2441) among the companiesselected and Ashok Leyland yielded lowest return of -0.1557. The returns on stock investment are negativefor three companies and positive for the remaining eleven.Further, beta is a measure of the systematic riskassociated with stock returns and higher beta valuesignify that the volatility in stock return is high and thusnot always desirable. It can be seen from table-1 thatwith the exception of Tata Motors ( with beta of 1.0704),the other beta values are less than 1.0. The lowest betais observed for Asian Paints with value of 0.2885.

According to the Sharpe model the excess returnof any stock is directly related to its excess return tobeta ratio. It measures the additional return on a security(excess of the risk less asset return) per unit of systematicrisk. The ratio provides a relationship between potentialrisk and reward. For the calculation of this ratio, therisk free return (Rf) is taken as the rate of return on the91- days Treasury bill which is found to be 6.5% forthe period under study. Ranking of the stocks are doneon the basis of their excess return to beta. Based on theexcess return to beta ratio the scrip’s are ranked from 1to 14, with TATA Motors being in the first rank andAmbuja Cement being in the last.

5.1 Cut-off pointThe selection of the stocks depends on a unique cut-offrate such that all stocks with higher ratios of excessreturn to beta are included and stocks with lower ratioare left out. The cumulated values of Ci start decliningafter a particular Ci and that point is taken as the cut-offpoint and that stock ratio is the Cut-off ratio C. Thehighest value of Ci is taken as the cut-off point that is



C*. From table-2 it is seen that Asian Paints has thehighest the cut-off rate of C*= 0.379. All the stockshaving Ci greater than C* can be included in the portfolio.

With this criterion , only three stocks namely Hero MotorCorp. , Tata Motors and Asian paints qualify to beincluded in the optimal portfolio.

VI. CONSTRUCTION OFOPTIMAL PORTFOLIO

After determining the securities to be included in theoptimal portfolio, we have to determine the proportionof investment in each of these stocks. Only those stockswith Excess return to beta ratio ( column -2 to table-2)more than C* (0.379) are to be selected in the optimalportfolio. It can be observed from table-2 that only threestocks qualify to be included in the optimal portfolio onthis criterion. These are Tata Motors, Hero MotorCorporation and Asian paints with cut-off point (Ci) of0.342, 0.376 and 0.379, respectively as displayed intable-3.

Table 3

Values of cut-off point and Investment proportionfor the stocks included in the Optimal portfolio

Company Cut-off Proportion ofpoint Investment

Hero motor corporation 0.376 59.22%

Tata motors 0.342 38.88%

Asian paints 0.379 1.90%

By using Sharpe index model, we are able to findout the proportion of investments to be made for eachof the three stocks included in the optimal portfolio.The maximum investment should be made in Hero MotoCorporation (previously Hero-Honda) with a proportion

Table 2Calculated Values of Cut-off point for the selected companies

Company Ri-Rf/βββββi βββββ2/(δδδδδei)2 {Ri-Rf/ ΣΣΣΣΣ{Ri-Rf/ ΣβΣβΣβΣβΣβ2/(δδδδδei)2 Ci(δδδδδei)2}/βββββi (δδδδδei)2}/βββββi

Tata motors 2.958 101.996 301.779 301.779 101.996 0.342

Hero honda 0.539 181.519 97.986 399.766 283.515 0.376

Asian paints 0.429 65.866 28.276 428.043 349.382 0.379

Aditya birla 0.204 -2831.004 -580.153 -152.110 -2481.622 0.089

Grasim 0.159 -895.575 -142.521 -294.632 -3377.198 0.113

Acc 0.056 322.250 18.364 -276.268 -3054.947 0.121

M & m 0.049 297.918 14.678 -261.589 -2757.028 0.132

Ashok leyland -0.053 -309.04983 16.43428 -245.15525 -3066.078 0.107

Bpcl -0.118 -1402.156 166.621 -78.534 -4468.234 0.021

Ongc -0.137 -221.085 30.462 -48.071 -4689.320 0.0123

Reliance refinieries -0.253 -2520.389 640.143 592.072 -7209.710 -0.092

Bajaj auto -0.305 -512.542 156.708 748.780 -7722.253 -0.108

Hindustan petroleum -0.520 127.691 -66.416 682.364 -7594.562 -0.1

Ambuja cement -15.99 522.379 -8357.741 -7675.377 -7072.183 1.22



of 59.22%, followed by TATA Motors and Asian Paintswith investment proportion of 38.88 % and 1.90%,respectively. Among three securities selected for theinvestment two companies belongs to automobile sectorand one company is from paints sector. Evidently, thecompanies chosen for the investments are growing at asteady rate in the recent years.

Figure 1

VII. REFERENCES1. Andrea LD, Wifred LD and Jean LH. (2003).

Benefits from Asia pacific mutual fund investmentswith currency hedging, Review of quantitativefinance and accounting 21(1):49-59.

2. Bilbao,A., Arenas,N., Jiménez,M. andGladish,B.P. and Rodriguez, MV. (2005). Anextension of Sharpe’s single-index model:portfolio selection with expert betas. Journal ofthe Operational Research Society 57: 1442–1451

3. Campbell R., Husiman R. and Kodedijk K.(2001) . Optimal portfolio selection in a Value-at-Risk framework. Journal of Banking &Finance, 25: 1789-1804.

4. Davidsson,M. (2010). Expected Return andPortfolio Rebalancing. International Journal ofEconomics and Finance 3 (3):123-136

5. Ebner,M. and Neumann,T. (2008). Time-varyingfactor models for equity portfolio construction,The European Journal of Finance 14(5): 381-395.

6. Haslem, J.A. and Scheraga, C.A. (2003). DataEnvelopment Analysis of Morningstar’s Large-Cap Mutual Funds. Available at SSRN: http://ssrn.com/abstract=2080478 or http://dx.doi.org/10.2139/ssrn.2080478s

7. Heck,J.L., Dellva,W. and DeMaskey,A. (2003).Benefits from Asia-Pacific mutual fund investmentswith currency hedging. Review of QuantitativeFinance and Accounting, 21(1): 49-64.

8. Kwok WY, Xiao QY, Heung W (2007). Assetallocation by using the Sharpe rule, Journal ofAsset Management. 8(2):133-152.

9. Liow,K.H. (2001) “The long-term investmentperformance of Singapore real estate andproperty stocks”, Journal of Property Investment& Finance. 19 ( 2): 156 – 174

CONCLUDING REMARKSRisk and return play an important role in making anyinvestment decisions. This study aims at analyzing theopportunity that are available for investors as per asreturns are concerned and the investment of riskthereof. Out of 14 companies taken for the study, 3companies are showing negative return and the other11 companies are showing positive returns. Withregard to beta values, out of 14 companies selected, only one company stock showed beta above 1,indicating that the investments in this stock isoutperforming than the broader market. Finally out ofthe 14 manufacturing sector stocks that are includedin NSE Nifty, only three stocks namely Hero MotorCorp., Tata Motors and Asian Paints are included inthe Optimal Portfolio constructed in this study withmaximum suggested investment in Hero MotorCorporation. Our study is based on the Sharpe Singleindex model and thus limited to the lacunas of thismodel.



10. Moreno,D., Marco,P. and Olmeda,I. (2005),Risk forecasting models and optimal portfolioselection. Applied Economics 37(11): 1267-1281.

11. Pui L.L. and Wong,W.K. (2008).On testing theequality of multiple Sharpe ratios with applicationon the evaluation of iShares, The Journal ofRisk. 10, 3; : 15-21

12. Puri H. and Saxena S. (2012). Construction andevaluation of optimal portfolio using Sharpe’s

Single index model. Journal of Accounting andFinance, 26 (1): 33-49.

13. Varadharajan P (2011), Portfolio constructionusing the Sharpe index model with reference tobanking and information technology sectors,Prime Journal of Business Administration andManagement 1(12) : 392-398.

14. Ward, D.J. and Griepentrog, G L.(1993). Riskand Return in Defaulted Bonds. FinancialAnalysts Journal 49(3): 44-61

optimal portfolio construction in stock market

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