Portfolio Analysis in US stock market using Markowitz model

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					                                    IJASCSE Vol 1, Issue 3, 2012
Oct. 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


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                                         IJASCSE Vol 1, Issue 3, 2012
Oct. 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

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                                      IJASCSE Vol 1, Issue 3, 2012
Oct. 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

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                                  IJASCSE Vol 1, Issue 3, 2012
Oct. 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

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                                    IJASCSE Vol 1, Issue 3, 2012
Oct. 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                 1.77
                                                                             8.790 Positio
      HMC Honda Motor Co., Ltd.                  75.73
                                           70.56 974
                                            25                      75.13
                                                                    004        8.66
                                                                             338 n -9
      ESCC Evans & Sutherland Computer           17.29
                                            11.6 484
                                           25                       30.56
                                                                    075      5.528
                                                                               78    -
                                                                                     0.59
            DELL Computer
      DELL Corporation                           36.08
                                           57.68 817
                                            25                      87.91
                                                                    043       9.376
                                                                             149     7
                                                                                     1.02
                                                                                      2.30
      WCO MCI Worldcom                           45.70
                                           43.18 401
                                           75                       121.9
                                                                    632      11.04
                                                                              37     -4
                                                                                     6
      ACNA Air Canada
      M                                          5.490
                                            10.6 229
                                           75                       4.332
                                                                    327      2.081
                                                                             231      2.46
                                                                                     0.22
      AMR AMR Corporation
      F                                          27.79
                                            2530 169                12.01
                                                                    692      3.465
                                                                             512     80.63
                                                                                      7
      SUNW Sun Microsystems, Inc.                33.68
                                            96.1 872                607.1
                                                                    035      24.64
                                                                             595      2.53
                                                                                      5
      BAC Bank of America Corporation            64.11
                                            2550 026                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
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                                 IJASCSE Vol 1, Issue 3, 2012
Oct. 31


                                                  Ibbotson, Roger, and Paul Kaplan.
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