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# Portfolio Analysis in US stock market using Markowitz model

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### Portfolio Analysis in US stock market using Markowitz model

1. 1. IJASCSE Vol 1, Issue 3, 2012Oct. 31 Portfolio Analysis in US stock market using Markowitz model Emmanuel, Richard Enduma correlation coefficient (or covariance) Abstract of return for each pair of securities in the set of securities that are The risk management systems now considered for inclusion in the portfolio used in portfolio management are are required as data inputs for doing based on Markowitz mean variance the portfolio analysis. We may optimization. Successful analysis presume that although analysts in depends on the accuracy with which stock broking companies have been risk, market returns and correlation are using this method, but still they don’t predicted. The methods for forecasting describe its application for the public at now normally used for this purpose large. In this paper, we attempt to depend on time-series approaches make the optimal portfolio formation which generally ignore economic using real life data and the objective of content. This paper is trying to suggest the research is to provide an example that explicitly incorporation of of optimal portfolio management using economic variables into the process of real life data. forecasting can improve the reliability of such systems in managing the risk 2. INPUTS REQUIRED by making a provision for a delineation between risks related to changes in For analysing the portfolio using the economic activities and that Markowitz method, we need the attributable to other discontinuities and expected return, standard deviation for shocks. each of the securities for its holding period to be considered for including in 1. INTRODUCTION the portfolio. We also have to know the correlation coefficient or covariance Harry Markowitz (1952), wrote his between each pair of the securities portfolio analysis method in 1952. among all the securities which are to Using his method, an investor can be included in the portfolio. This determine an optimal portfolio with his approach explicitly makes risk specific risk level. Although the method management comprehensively on the given by Markowitz is a method of user by making portfolio construction normalization and detailed steps were in a probabilistic framework. The described by Markowitz (1959) in a results of this analysis are normally book, it is quite difficult to find a presented in the form of the efficient published literature for an example for frontier, which shows expected return its application to real life data based on on portfolio as a strict function of risk . quantitative expectations of analysts or The approach uses three key steps in investors. For each security expected the process return, standard deviation of return and www.ijascse.in Page 1
2. 2. IJASCSE Vol 1, Issue 3, 2012Oct. 31 A portfolio with equal weights has (1) consideration of the specific constant weight on all stocks, where Xi investment alternatives = Xj = 1/n (2) how to perform the optimization; The ‘n’ is the number of stocks. The (3) how to choose the appropriate sum of these weights is equal to one. implementation process. It is a very simple to understand how a The maximum return can be expected particular stock makes contribution to from the resulting portfolio at minimum the expected return or to its covariance risk. of a portfolio. For example, if we Let Xi be the fraction of wealth expect, return of a stock is high, we invested in stock i of the portfolio. can increase the expected return in a Xi: The weight of portfolio on stock i. proportional manner by increasing the Therefore, weight of that stock. ∑ Xip = 1 The part associated with its beta for a i stock’s variance is often called as the rp: The return on the portfolio, given by stock’s: rp = ∑ Xiri  arket risk i  systematic risk E(rp): Expected return of portfolio,  non-diversifiable risk given by And the part associated with the E(rp) =∑ XiE(ri) Cov(rp,W): Covariances in portfolio, the: given by  residual risk Cov(rp,W) = ∑ Xi *Cov(ri,W)  firm specific risk  diversifiable risk The above both are linear in portfolio  non-systematic risk weights but the following is non linear.  idiosyncratic risk Var(rp): Portfolio variance, given by Simply putting, it is wise enough to sell Var(rp) = ∑∑ XiXj ⱷij the stock which has much positive i j higher error and buy the stock which has much negative lower error. In matrix formation: 3. Making of a Portfolio Var(rp ) = Xp’VXp The steps to make initial portfolio, and Where Xp = [ Xp1, Xp2,.........Xpn] and to use technical analysis are as given Cov(rp, rq) = X’pVXq below: Decomposing the formula we obtain: 1) The first step is the collection of the Var(rp)=∑∑XiXjⱷij= Xi2ⱷi2 + ∑∑ XiXjⱷ ij i j≠i historical data. The more the number = (Contribution of own variances) + of data is, the better our calculation is. (contribution of covariance) Let’s compute average and standard deviation on each stock return www.ijascse.in Page 2
3. 3. IJASCSE Vol 1, Issue 3, 2012Oct. 31 Calculation of Input Variables: The expected returns are calculated as the 2) Next step is to checking of the least difference between current market square result of market return on LHS price and target of each security, and each stocks return on RHS on the shown as a percentage of current CAPM equation. This can be done market prices. Monthly returns, through the software CAPM Tutor or needed to find the co-variances are E-View to get result calculated for each stock from the 3) Now the covariance table is to be monthly closing prices. The covariance computed. matrix for the 10 stocks is calculated 4) Using CAPM Tutor the frontier line by using excel covariance function and is to be computed the monthly covariance is converted 5) Setting the target for return keeping into annual covariance by multiplying it a certain risk level, initial portfolio is to with 12. Re-balance is taken when be made.. minimum two of all stock optimal 6) The portfolio is to be restructured portfolio weights increased or toward the positive-negative direction decreased by 1 %, compared with 7) Buy stocks iff the return is below previous month. return average We have considered a risk-aversion 8) Sell stocks iff the return is over coefficient A and a skewness- return average preference coefficient B in the cubic 9) Use Markowitz technique of utility function analysis to find the appropriate timing 1 1 U  r   E  r   A Var  r   B  E r  E  r  3 of trading the individual stocks and 2 6   keep restructuring the portfolio . The input data is thus made ready for 4. Application of Markowitz the next step for the analysis. We have portfolio analysis in USA used CAPM tutor to decide the weight, stock market for example Software Computer Auto We have chosen highly liquid Relations Systems Manufactur industries namely Software relations , Under-mean AVT Corp Evans & Ford Motor e computer Systems, Auto manufacture, stock Sutherland Airline, Chemicals, Investment Banks, Computer and Food Suppliers and have chosen Weight 3.7% -0.65% 38.99% stocks in such a way that it is either Over-mean Intel Corp Sun Toyota most under-performed or over- stock Microsystem Motor performed stock based on mean- s Corp variance bell curve. We used monthly last trade data from January 1996 to Weight 0.01% 5.19% 12.14% December 2010, and calculated the price mean and variance (Table-1) We have supposed the cost of trading 0.05% of actual capital movement www.ijascse.in Page 3
4. 4. IJASCSE Vol 1, Issue 3, 2012Oct. 31 5. Portfolio Analysis 7. LIMITATAIONS The software which is used is the excel optimizer by Markowitz and Todd Mean-variance optimization has (2000) explained in the book ‘Mean several limitations which affects its Variance Analysis and Portfolio effectiveness. First, model solutions Choice’. are often sensitive to changes in the inputs. Suppose if there is a small The software requires as input the increase in expected risk then it can above mentioned variables and the sometimes produce an unreasonable lower and upper boundaries for the large shift into stocks. Secondly, the ratio of each security in the portfolio number of stocks that are to be and additional constraints, if any. included in the analysis is normally limited. Last but not the least, The portfolio analysis is being done allocation of optimal assets are as with lower and upper boundaries for good as the predictions of prospective investment in a single stock as zero returns, correlation and risk that go (zero percent) and one (100 percent) into the model. respectively. The additional constraint being specified is that the sum of the 8. CONCLUSION & FUTURE ratios of all securities has to be 1 or SCOPE 100%, for the amount available for investment. We have collected the 30- Markowitz’s portfolio analysis may be day Treasury-Bill rate as the proxy for operational and can be applied to real the risk-free rate and the monthly life portfolio decisions. The optimal return data of the CRSP value- portfolios constructed by this analysis weighted index as a proxy for the represent the optimal policy for the market portfolio investors who want to use this for estimating target price. 6. RESULTS AND FINDING Mean variance findings are so important in portfolio theory and in 1200000 technical analysis that they bring the 1000000 common mathematical trunk of a 800000 portfolio tree. 600000 From the view point of theory, because market is random, the skewed 400000 distribution becomes simply noise of 200000 market. The technical analysis, on the other hand, particularly in momentum analysis, keeps the distortion as an Graph 1 : Performance of a few investment opportunity. So, it might not stocks in Time series be possible to be complicated with www.ijascse.in Page 4
5. 5. IJASCSE Vol 1, Issue 3, 2012Oct. 31 each other. However, in the world of real trading, performance is itself the most important matter in any case, so it is better to utilize the each specific character. Finally, the investigation tells that the adroit utilization of technical analysis would contribute high-performance and stabilization in real trading. I used the example of mean variance investigation, but technical tool application and comprehension are surely key factor of an individual performance. The software for portfolio analysis, the Todd’s program can be operated with 256 companies. In any particular case, brokers normally do not give more than 256 buy recommendations at any point in time. Hence, the software program is not a limitation. But certainly there is scope to improve the software, as more investors may use the methodology, and thereby need easy to use and efficient software combined with more facilities to come out with various measurements. Table 1 An example of selected 10 stocks in USA stock market Symbo Company Name LAST Mean Varian Stdev Bell lGM General Motors Corporation 80.1 64.48 77.27 ce 8.790 Positio 1.77 HMC Honda Motor Co., Ltd. 70.56 974 25 75.73 75.13 004 338 n -9 8.66 ESCC Evans & Sutherland Computer 11.6 484 25 17.29 30.56 075 5.528 78 - 0.59 DELL Corporation DELL Computer 57.68 817 25 36.08 87.91 043 9.376 149 2.30 1.02 7 WCO MCI Worldcom 43.18 401 75 45.70 121.9 632 11.04 37 -4 6 ACNA Air Canada M 10.6 229 75 5.490 4.332 327 2.081 231 2.46 0.22 AMR AMR Corporation F 2530 169 27.79 12.01 692 3.465 512 80.63 7 SUNW Sun Microsystems, Inc. 96.1 872 33.68 607.1 035 24.64 595 2.53 5 BAC Bank of America Corporation 2550 026 64.11 109.2 716 10.45 085 -4 BK Bank of New York Company, Inc. 38.68 34.65 428 15.05 709 3.880 327 1.04 1.35 75 055 585 186 00 www.ijascse.in Page 5
6. 6. IJASCSE Vol 1, Issue 3, 2012Oct. 31 Ibbotson, Roger, and Paul Kaplan. 9. References “Does Asset Allocation Explain 40%, 90%, or 100% of Alexander, Carol. “Volatility and Performance?” unpublished paper, Correlation Forecasting,” pp. 233-260 Ibbotson Associates, April 1999. in The Handbook of Risk Management and Analysis, Lamm, R.M. Hedge Funds as an Asset Edited by Carol Alexander, New York: Class: Still Attractive Despite Last John Wiley & Sons, 1996. Year’s Turbulence. Bankers Trust research Beckers, Stan. “A Survey of Risk report, January 22, 1999. Measurement Theory and Practice,” pp. 171-192, in Lummer, Scott L., Mark W. Riepe, and The Handbook of Risk Management Laurence B. Siegle, “Taming Your and Analysis, edited by Carol Optimizer: A Alexander, New York: John Wiley & Users Guide to the Pitfalls of Mean- Sons, 1996 Variance Optimization,” in Global Asset Allocation: Brinson, Gary P., L. Randolph Hood, Techniques for Optimizing Portfolio and Gilbert L. Beehower, Management, ed. by Jess Lederman “Determinants of and Robert Klein. Portfolio Performance,” Financial New York: John Wiley and Sons, Analysts Journal, July/August 1986. 1994. Brinson, Gary P., L. Brian D. Singer, Markowitz, H. Portfolio Selection: and Gilbert L. Beehower Brinson, Efficient Diversification of Investments. “Determinants of New York: Portfolio Performance II: An Update,” John Wiley and Sons, 1959. Financial Analysts Journal, May/June 1991. Connor, Gregory. “The Three Types of Factor Models: A Comparison of Their Explanatory Power,” Financial Analysts Journal, May-June, 1996, pp. 42-46. Chopra, Vijay and William Ziembra. “The Effect of Errors in Means, Variances, and Covariances on Portfolio Choices,” Journal of Portfolio Management, Fall 1993, pp. 51-58. www.ijascse.in Page 6