12.9.2023 Update
Came across a related paper
Kim, Jang Ho; Kim, Woo Chang; Lee, Yongjae; Choi, Bong-Geun; Fabozzi, Frank J. (2023) “Robustness in Portfolio Optimization,” Journal of Portfolio Management, online published.
Stock Portfolio Analysis Using Markowitz Model
Indah Nur Safitri , Sudradjata, Eman Lesmanaa
Department of Mathematics, Faculty of Mathematics and Natural Sciences, Padjadjaran University,
INDONESIA.
International Journal of Quantitative Research and Modeling
Vol. 1, No. 1, pp. 47-58, 2020
Available online at http://ijqrm.rescollacomm.com/index.php/ijqrm/index
---------------------
16.2.2012
This paper provides the procedure optimal portfolio development using real life data on equity shares and Markowitz Portfolio Analysis.
__________________________________________________
INTRODUCTION
Harry Markowitz (1952) published the portfolio analysis method in 1952. Using this method, an optimal portfolio can be determined for an investor who can specify his risk level. Expected return and standard deviation of return for each security and correlation coefficient (or covariance) of return for each pair of securities in the set of securities that are considered for inclusion in the portfolio are required as data inputs for doing the portfolio analysis. Even though the method proposed by Markowitz is a normative method and detailed implementation steps were described by Markowitz (1959) in a book, the implications of the method were better captured in the equilibrium condition for the risky asset market (Harrington, 1983) and its application in portfolio formation and revision was relatively neglected. It is difficult to find in the published literature an example for the application of the Markowitz portfolio analysis to real life data based on quantitative expectations of investors or analysts. We may presume that analysts in stock broking companies and mutual funds and other professional investment organizations may be using the analytical method, but still descriptions of its application are not made available for the public at large. In this paper, the optimal portfolio formation using real life data subject to two different constraint sets is attempted. The objective of the research is to provide an example of optimal portfolio development using real life data.
INPUTS REQUIRED FOR PORTFOLIO ANALYSIS
For performing the portfolio analysis using the Markowitz method, we need the expected return for the period of holding for each of the securities to be considered for inclusion in the portfolio. We also require the standard deviation of the return for each security. In addition we have to know the covariance (or correlation coefficient) between each pair of securities among all securities from which we have to form the portfolio.
The model proposed by Markowitz points out to the need for estimating expected returns in quantitative terms. But this line of enquiry (estimating expected returns over a period of time) was not pursued further adequately in the literature. That may be one of the reasons, why papers outlining the application of the model to real life data were in short supply. Analysts were giving their anticipation regarding the performance of various securities in twelve months or one year ahead even in 1920s. But Benjamin Graham (1940), known as Dean of Wall Street, was not in favor of such analysis. This analysis slowly developed into prediction of target prices 12 months ahead for many securities. These target price predictions can be used to determine the expected returns for one year holding period. Using the target price predictions to determine 12-month expected returns and then using these expected returns to form the optimal portfolios is a feasible and rational line of approach. This approach to quantitative investing is proposed and initiated in this paper.
To estimate standard deviations and covariances, past data can be used (Grinold and Kahn, 2004). The historical risk measures of securities are more stable in comparison to historical expected return measures.
RESEARCH ON TARGET PRICES
Research on target prices is of recent origin. Bradshaw (2002) has examined the frequency with which analysts have used target prices to justify their stock recommendations. He reported that in two thirds of the sample reports that were examined by him, analysts used target prices. The target prices were determined using price multiple heuristics, with PEG (price earnings growth ratio) as one of the important rule for specifying the price-earning (P/E) multiple.
Asquith et al. (2004) have examined the performance of target prices set by analysts of All-American Analyst award winners for the period 1997-99. They examined whether the price of the security crossed its target price within 12 months after the recommendation. When this definition of accuracy was used, the authors have found that 54% of the price targets were achieved or exceeded. Even in the case of remaining 46% of the securities or recommendations, on average 84% of the price target was found to be achieved. This performance is very creditable. But we have to notice that these price targets were targets of award winners, where the award itself was based on their performance. So, to generalize the findings, we require studies of more representative samples.
Bradshaw and Brown (2005) have examined the accuracy of 12-months-ahead target price forecasts over the period 1997-2002. They reported that on an average 24 to 45 percent of forecasts were met. Analysts have shown more skill in forecasting company earnings compared to forecasting target prices. This study generated interest in study of success rate of target price forecasts.
Gleason et al. (2006) have examined the performance of target prices over the period 1997-2003. According to this study, the buy recommendations have an average target return of 28 percent. They analyzed results over quintiles. In the most accurate quintile, 57% of the targets were achieved or exceeded within the 12 month period. In the least accurate quintile, the success rate was found to be 49%. The interesting finding of the study is that the return that would have been earned by selling each of the securities with buy recommendations at their maximum prices within the 12 months is 42.49% even in the case of lowest quintile. One needs to compare this 42.49% with average target return of 28%. These studies do provide evidence that target price estimates have utility to investors for their decision making. They also provide the evidence that investors, traders and fund managers are encouraging analysts to provide target prices and many analysts are providing them.
USE OF TARGET PRICES IN PORTFOLIO FORMATION
If target prices have information content that is useful to earn return over 12-month horizon, portfolios can be formed using the target prices as the basis. The expected return can be determined as the difference between the target price and the current market price on the date of portfolio analysis and this can be expressed as percentage of current market price on the date of portfolio formation. If the investor/trader has this information with him, an optimal portfolio can be specified for him using Markowitz portfolio analysis.
APPLICATION OF MARKOWITZ PORTFOLIO ANALYSIS IN PRACTICE
Markowitz portfolio analysis gives as output an efficient frontier on which each portfolio is the highest return earning portfolio for a specified level of risk. It basically calculates the standard deviation and return for each of the feasible portfolios and identifies the efficient frontier, the boundary of the feasible portfolios of increasing returns. The financial planners help the investors/traders to arrive at the risk level that they can assume. If the investor/trader specifies his risk level in terms of standard deviation of the portfolio return, the appropriate portfolio for him can be identified using the efficient frontier. Hence the final portfolio selection for an investor/trader requires the combination of portfolio analysis and financial planning.
APPLICATION OF MARKOWITZ PORTFOLIO ANALYSIS IN INDIAN STOCK MARKET
Sources of Data: Valueline is a monthly bulletin published by Sharekhan (2005) a broking firm in India. The bulletin contains the target price information and the market price on the date of publication for various stocks researched and recommended by the firm. The data from the bulletin of July 2005, which was made available on the website of the firm for public access, is selected for getting the data of expected returns. Target price data was available for 43 companies. Covariance is to be calculated using 25 months closing price data. The monthly closing price data was taken from Prowess, an electronic data base of balance sheet and share price data of Indian companies published by Centre for Monitoring Indian Economy (CMIE, Mumbai). Out of the total 43 companies, for two companies, data was not available for the full 25 months. These two companies were dropped from the set of securities considered for forming the portfolio. Hence, the final list of stocks considered for portfolio analysis contains 41 companies.
Calculation of Input Variables: The expected returns were calculated as the difference between target price and current market price of each security, expressed as a percentage of current market price. Monthly returns, required to determine the covariances, were calculated for each company from the monthly closing prices. The covariance matrix for the 41 stocks was calculated using excel covariance function. The monthly covariance between each pair of securities was converted into annual covariance by multiplying it with 12. The input data of expected returns and covariance matrix were thus made ready for the next step in the analysis.
Portfolio Analysis: The software used is the excel optimizer by Markowitz and Todd (2000) described in the book ‘Mean Variance Analysis and Portfolio Choice’. The software was supplied by Todd on request by the author. The software can handle up to 256 securities.
The software requires as input the expected returns of each security, covariance matrix for the set of securities from which the portfolio is to be formed, lower and upper bounds for the proportion of each security in the portfolio and additional constraints if any.
In the first alternative, the portfolio analysis was done with lower and upper boundary for investment in a single security as zero (zero percent) and one (100 percent) respectively. The additional constraint specified is that the sum of the proportions of all securities has to be one or 100%, the amount available for investment. In the second alternative, the analysis was done with the constraint for individual security holding for mutual funds in India, which is a maximum of 10% of the portfolio in a single security. In this case, the lower and upper bounds are 0 and 0.1. The constraint that the sum of all proportions add to 1 or 100% remains. The results are reported in Tables 1 to 4.
RESULTS AND FINDINGS
The 12 month target prices and current market price on 30th June 2005 for the companies included in the set considered for analysis are shown in Table 1. The expected returns for the following 12 months determined from them are shown in column 5 of the Table 1. The covariance matrix for the set of securities is shown in Table 4.
The output of the portfolio analysis for alternative 1, lower bound zero and upper bound 1 for each security, is shown in Table 2. Corner portfolios describe the efficient frontier. Between any two adjacent corner portfolios, the efficient frontier is a straight line, a weighted average of the two corner portfolios. The analysis returned 23 corner portfolios. The minimum return portfolio has an expected return of 13.54% and standard deviation of 14.35%. The maximum return portfolio has an expected return of 95.96% and standard deviation of 36.12%.
Investor has to decide the risk level (standard deviation) he wants to bear to select the optimal portfolio from this efficient frontier. This action involves consultation with financial planners. For illustration, if the investor chooses a risk level of 20.27%, the corner portfolio number ‘9’ becomes the optimal portfolio. The expected return of this portfolio is 55.98%. The portfolio is a combination of 9 shares. The proportion or percentage recommended for investment in various securities being:
1. X(2) = 3%
2. X(3) = 13%
3. X(9) = 30%
4 X(14) = 3%
5. X(16) = 35%
6. X(17) = 4%
7. X(34) = 9%
8. X(38) = 2%
9. X(40) = 1%
The total adds up to 100%. The names of companies represented by identifiers X(2), X(3) etc. can be read from Table 1.
In Table 3 are shown the results of portfolio analysis when restrictions on investment imposed on mutual fund portfolios in India are specified in the constraints. The restriction is that upper bound, the proportion invested in any single company’s equity shares, is to be less than 10% of the NAV of the scheme. Accordingly lower bound is specified as zero and upper bound is specified as 0.10. 52 corner portfolios form the efficient frontier in this alternative. The minimum return portfolio has an expected return of 14.02% and standard deviation of 15.59%. The maximum return portfolio has an expected return of 50.64% and standard deviation of 29.35%. It is interesting to compare risk-return characteristics of the maximum return portfolio of alternative 2 with the portfolio selected as an illustration in alternative 1 (55.98% and 20.27%). The expected return is more and standard deviation is lower in the latter case. Thus the constraints imposed through regulation on mutual fund investment are generating an inferior or suboptimal portfolio in this case.
The performance of these two portfolios is compared over one year period from July 05 to June 2006. The mutual fund portfolio (Exp. Ret: 50.64% and Risk: 29.35%) shows a return of 58.4% with 23.13% standard deviation. The other portfolio (Exp. Ret: 55.98% and Risk 20.27%) shows a return of 21.25% with a standard deviation of 21%. As the returns are expected to be more unstable and risk measures are expected to be relatively more stable, the observed performance can be rationalized in such a simple comparison of performance of the two portfolios over one period. Empirical studies to evaluate the superiority of one-year horizon optimal portfolios formed using quantitative methods have to use number of one year periods in the sample.
CONCLUSION AND FUTURE SCOPE FOR RESEARCH
Markowitz’s portfolio analysis can be operationalized and applied to real life portfolio decisions. The 12-month ahead target prices being published for various securities by security analysts can be used as the input for determining expected returns over the next 12 months. The optimal portfolios generated by the portfolio analysis represent the optimal policy for the investor who wants to use the target price estimates rationally.
Acceptance of the methodology for developing and revising portfolios based on target prices provides scope for further research into improving the estimates of the inputs used for portfolio analysis. Also research is to be done to evaluate the performance of the optimal portfolios, in comparison to portfolios formed without using quantitative portfolio analysis models, over a long period of time.
Review of literature reveals that research into the utility of target prices is initiated. Research needs to be extended to find out which target price finding methods are working better. Regarding covariance estimates, Grinold and Kahn (2004) have mentioned that there is possibility of estimation errors in case historical data over a lower number of monthly periods in comparison to number of securities considered for portfolio analysis are used. They suggest structural models. Researchers have to come out with useful models which investors can use on the basis of published data.
Regarding the software for portfolio analysis, the Todd’s program can handle 256 companies. In any particular country, brokers do not normally come out with more than 256 buy recommendations at any point in time. Hence, the software program may not be a limitation. But certainly there will be scope to improve the software, as more and more investors use the methodology, and thereby need efficient and easy to use software with more facilities to come out with various measurements.
REFERENCES
Bradshaw, Mark T. “The Use of Target Prices to Justify Sell-Side Analysts' Stock Recommendations.” Accounting Horizons, March 2002, Vol. 16, no. 1, pp. 27-41.
Bradshaw, Mark T. and Brown, Lawrence D. “Do Sell-side Analysts Exhibit Differential Target Price Forecasting Ability?” Working Paper, 2005, Available at SSRN_ID926400_code175449.pdf.
Graham, Benjamin, and Dodd, David. Security Analysis, 2nd Edition, New York: McGraw-Hill Book Co., 1940.
Grinold, Richard C., and Kahn, Ronald N. Active Portfolio Management, 2nd Edition, New Delhi: Tata McGraw-Hill Pub. Co., 2004.
Harrington, Diana R. Modern Portfolio Theory & Capital Asset Pricing Model, Englewood Cliffs: Prentice-Hall Inc.,1983.
Markowitz, Harry. “Portfolio Selection.” Journal of Finance, March 1952, Vol. 7, no.1, pp. 77-91.
Markowitz, Harry. Portfolio Selection: Efficient Diversification of Investments, New York: John Wiley & Sons, 1959.
Markowitz, Harry, and Todd, Peter G. Mean Variance Analysis in Portfolio Choice and Capital Markets, Revised Issue, New Hope: Frank J. Fabozzi Associates, 2000.
Sharekhan. “Stock Ideas Standing (as on June 30, 2005).” Valueline. July 2005, p.3, Available at http://www.sharekhan.com/articles/ValueLine_july2005.pdf.
_______________________________________________________________________________
Paper presented in research conference - New York Economic Association
Article originally published by me on Knol.
http://knol.google.com/k/ portfolio-analysis-application-and-evaluation-in-indian-stock-market
(Knol not available for public access from 1 May 2012)
Ud. 12.9.2023
Pub 16.2.2012
