1. 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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2. IJASCSE Vol 1, Issue 3, 2012
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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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3. IJASCSE Vol 1, Issue 3, 2012
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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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4. IJASCSE Vol 1, Issue 3, 2012
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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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5. IJASCSE Vol 1, Issue 3, 2012
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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
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6. IJASCSE Vol 1, Issue 3, 2012
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Ibbotson, Roger, and Paul Kaplan.
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