portfolio analysis in us stock market using markowitz model

IJASCSE Vol 1, Issue 3, 2012

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Oct. 31

Portfolio Analysis in US stock market using Markowitz model

Emmanuel, Richard Enduma

Abstract

The risk management systems now

used in portfolio management are

based on Markowitz mean variance

optimization. Successful analysis

depends on the accuracy with which

risk, market returns and correlation are

predicted. The methods for forecasting

now normally used for this purpose

depend on time-series approaches

which generally ignore economic

content. This paper is trying to suggest

that explicitly incorporation of

economic variables into the process of

forecasting can improve the reliability

of such systems in managing the risk

by making a provision for a delineation

between risks related to changes in

economic activities and that

attributable to other discontinuities and

shocks.

1. INTRODUCTION

Harry Markowitz (1952), wrote his

portfolio analysis method in 1952.

Using his method, an investor can

determine an optimal portfolio with his

specific risk level. Although the method

given by Markowitz is a method of

normalization and detailed steps were

described by Markowitz (1959) in a

book, it is quite difficult to find a

published literature for an example for

its application to real life data based on

quantitative expectations of analysts or

investors. For each security expected

return, standard deviation of return 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. We may

presume that although analysts in

stock broking companies have been

using this method, but still they don’t

describe its application for the public at

large. In this paper, we attempt to

make the optimal portfolio formation

using real life data and the objective of

the research is to provide an example

of optimal portfolio management using

real life data.

2. INPUTS REQUIRED

For analysing the portfolio using the

Markowitz method, we need the

expected return, standard deviation for

each of the securities for its holding

period to be considered for including in

the portfolio. We also have to know the

correlation coefficient or covariance

between each pair of the securities

among all the securities which are to

be included in the portfolio. This

approach explicitly makes risk

management comprehensively on the

user by making portfolio construction

in a probabilistic framework. The

results of this analysis are normally

presented in the form of the efficient

frontier, which shows expected return

on portfolio as a strict function of risk .

The approach uses three key steps in

the process


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(1) consideration of the specific

investment alternatives

(2) how to perform the optimization;

(3) how to choose the appropriate

implementation process.

The maximum return can be expected

from the resulting portfolio at minimum

risk.

Let Xi be the fraction of wealth

invested in stock i of the portfolio.

Xi: The weight of portfolio on stock i.

Therefore,

∑ Xip = 1

i

rp: The return on the portfolio, given by

rp = ∑ Xiri

i

E(rp): Expected return of portfolio,

given by

E(rp) =∑ XiE(ri)

Cov(rp,W): Covariances in portfolio,

given by

Cov(rp,W) = ∑ Xi *Cov(ri,W)

The above both are linear in portfolio

weights but the following is non linear.

Var(rp): Portfolio variance, given by

Var(rp) = ∑∑ XiXj ⱷij

i j

In matrix formation:

Var(rp ) = Xp’VXp

Where Xp = [ Xp1, Xp2,.........Xpn] and

Cov(rp, rq) = X’pVXq

Decomposing the formula we obtain: Var(rp)=∑∑XiXjⱷij= Xi2ⱷi2 + ∑∑ XiXjⱷ ij

i j≠i

= (Contribution of own variances) +

(contribution of covariance)

A portfolio with equal weights has

constant weight on all stocks, where Xi

= Xj = 1/n

The ‘n’ is the number of stocks. The

sum of these weights is equal to one.

It is a very simple to understand how a

particular stock makes contribution to

the expected return or to its covariance

of a portfolio. For example, if we

expect, return of a stock is high, we

can increase the expected return in a

proportional manner by increasing the

weight of that stock.

The part associated with its beta for a

stock’s variance is often called as the

stock’s:

arket risk

systematic risk

non-diversifiable risk

And the part associated with the

the:

residual risk

firm specific risk

diversifiable risk

non-systematic risk

idiosyncratic risk

Simply putting, it is wise enough to sell

the stock which has much positive

higher error and buy the stock which

has much negative lower error.

3. Making of a Portfolio

The steps to make initial portfolio, and

to use technical analysis are as given

below:

1) The first step is the collection of the

historical data. The more the number

of data is, the better our calculation is.

Let’s compute average and standard

deviation on each stock return


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2) Next step is to checking of the least

square result of market return on LHS

and each stocks return on RHS on the

CAPM equation. This can be done

through the software CAPM Tutor or

E-View to get result

3) Now the covariance table is to be

computed.

4) Using CAPM Tutor the frontier line

is to be computed

5) Setting the target for return keeping

a certain risk level, initial portfolio is to

be made..

6) The portfolio is to be restructured

toward the positive-negative direction

7) Buy stocks iff the return is below

return average

8) Sell stocks iff the return is over

return average

9) Use Markowitz technique of

analysis to find the appropriate timing

of trading the individual stocks and

keep restructuring the portfolio

4. Application of Markowitz

portfolio analysis in USA

stock market

We have chosen highly liquid

industries namely Software relations ,

computer Systems, Auto manufacture,

Airline, Chemicals, Investment Banks,

and Food Suppliers and have chosen

stocks in such a way that it is either

most under-performed or over-

performed stock based on mean-

variance bell curve. We used monthly

last trade data from January 1996 to

December 2010, and calculated the

price mean and variance (Table-1)

Calculation of Input Variables: The

expected returns are calculated as the

difference between current market

price and target of each security,

shown as a percentage of current

market prices. Monthly returns,

needed to find the co-variances are

calculated for each stock from the

monthly closing prices. The covariance

matrix for the 10 stocks is calculated

by using excel covariance function and

the monthly covariance is converted

into annual covariance by multiplying it

with 12. Re-balance is taken when

minimum two of all stock optimal

portfolio weights increased or

decreased by 1 %, compared with

previous month.

We have considered a risk-aversion

coefficient A and a skewness-

preference coefficient B in the cubic

utility function

31 1

2 6U r E r A Var r B E r E r

.

The input data is thus made ready for

the next step for the analysis. We have

used CAPM tutor to decide the weight,

for example

We have supposed the cost of trading

0.05% of actual capital movement

Software

Relations

Computer

Systems

Auto

Manufactur

e Under-mean AVT Corp Evans & Ford Motor

stock Sutherland

Computer

Weight 3.7% -0.65% 38.99%

Over-mean

Intel Corp Sun Toyota

stock Microsystem

s

Motor

Corp

Weight 0.01% 5.19% 12.14%


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5. Portfolio Analysis

The software which is used is the

excel optimizer by Markowitz and Todd

(2000) explained in the book ‘Mean

Variance Analysis and Portfolio

Choice’.

The software requires as input the

above mentioned variables and the

lower and upper boundaries for the

ratio of each security in the portfolio

and additional constraints, if any.

The portfolio analysis is being done

with lower and upper boundaries for

investment in a single stock as zero

(zero percent) and one (100 percent)

respectively. The additional constraint

being specified is that the sum of the

ratios of all securities has to be 1 or

100%, for the amount available for

investment. We have collected the 30-

day Treasury-Bill rate as the proxy for

the risk-free rate and the monthly

return data of the CRSP value-

weighted index as a proxy for the

market portfolio

6. RESULTS AND FINDING

1200000 1000000 800000 600000 400000 200000

Graph 1 : Performance of a few

stocks in Time series

7. LIMITATAIONS

Mean-variance optimization has

several limitations which affects its

effectiveness. First, model solutions

are often sensitive to changes in the

inputs. Suppose if there is a small

increase in expected risk then it can

sometimes produce an unreasonable

large shift into stocks. Secondly, the

number of stocks that are to be

included in the analysis is normally

limited. Last but not the least,

allocation of optimal assets are as

good as the predictions of prospective

returns, correlation and risk that go

into the model.

8. CONCLUSION & FUTURE

SCOPE

Markowitz’s portfolio analysis may be

operational and can be applied to real

life portfolio decisions. The optimal

portfolios constructed by this analysis

represent the optimal policy for the

investors who want to use this for

estimating target price.

Mean variance findings are so

important in portfolio theory and in

technical analysis that they bring the

common mathematical trunk of a

portfolio tree.

From the view point of theory, because

market is random, the skewed

distribution becomes simply noise of

market. The technical analysis, on the

other hand, particularly in momentum

analysis, keeps the distortion as an

investment opportunity. So, it might not

be possible to be complicated with


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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

l

Company Name LAST Mean Varian

ce

Stdev Bell

Positio

n

GM General Motors Corporation 80.1

25

64.48

974

77.27

004

8.790

338

1.77

9 HMC Honda Motor Co., Ltd. 70.56

25

75.73

484

75.13

075

8.66

78

-

0.59

7

ESCC Evans & Sutherland Computer

Corporation

11.6

25

17.29

817

30.56

043

5.528

149

-

1.02

6

DELL DELL Computer 57.68

75

36.08

401

87.91

632

9.376

37

2.30

4 WCO

M

MCI Worldcom 43.18

75

45.70

229

121.9

327

11.04

231

-

0.22

8

ACNA

F

Air Canada 10.6

25

5.490

169

4.332

692

2.081

512

2.46

7 AMR AMR Corporation 30 27.79

872

12.01

035

3.465

595

0.63

5 SUNW Sun Microsystems, Inc. 96.1

25

33.68

026

607.1

716

24.64

085

2.53

4 BAC Bank of America Corporation 50 64.11

428

109.2

709

10.45

327

-

1.35

0

BK Bank of New York Company, Inc. 38.68

75

34.65

055

15.05

585

3.880

186

1.04

0


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9. References

Alexander, Carol. “Volatility and

Correlation Forecasting,” pp. 233-260

in The Handbook

of Risk Management and Analysis,

Edited by Carol Alexander, New York:

John Wiley & Sons, 1996.

Beckers, Stan. “A Survey of Risk

Measurement Theory and Practice,”

pp. 171-192, in

The Handbook of Risk Management

and Analysis, edited by Carol

Alexander, New York: John Wiley &

Sons, 1996

Brinson, Gary P., L. Randolph Hood,

and Gilbert L. Beehower,

“Determinants of

Portfolio Performance,” Financial

Analysts Journal, July/August 1986.

Brinson, Gary P., L. Brian D. Singer,

and Gilbert L. Beehower Brinson,

“Determinants of

Portfolio Performance II: An Update,”

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.

Ibbotson, Roger, and Paul Kaplan.

“Does Asset Allocation Explain 40%,

90%, or 100% of

Performance?” unpublished paper,

Ibbotson Associates, April 1999.

Lamm, R.M. Hedge Funds as an Asset

Class: Still Attractive Despite Last

Year’s

Turbulence. Bankers Trust research

report, January 22, 1999.

Lummer, Scott L., Mark W. Riepe, and

Laurence B. Siegle, “Taming Your

Optimizer: A

Users Guide to the Pitfalls of Mean-

Variance Optimization,” in Global

Asset Allocation:

Techniques for Optimizing Portfolio

Management, ed. by Jess Lederman

and Robert Klein.

New York: John Wiley and Sons,

1994.

Markowitz, H. Portfolio Selection:

Efficient Diversification of Investments.

New York:

John Wiley and Sons, 1959.

portfolio analysis in us stock market using markowitz model

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