SIngle Index method

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Simplifying the Portfolio Optimization

Process via Single Index Model

Yansen AliJune 2008

Industrial Engineering Honors ProgramMcCormick School of Engineering

Northwestern UniversityAdvisor

Professor Sanjay Mehrotra

Industrial Engineering and Management SciencesNorthwestern University


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Abstract

Markowitz (1959) is one of the pioneers of modern portfolio theory. Since he introduced the

basic portfolio optimization model as early as 1956, there had been numerous advances of the

portfolio theory. However, putting the theory into practice remains a challenging area for

practitioners as financial system and settings become more sophisticated. This paper will lookinto the single index model as one potential solution to simplify the calculation of optimal

portfolios and examine its effectiveness under different settings of the model.

Abstract 2

Introduction 3

Markowitz Model 3

Single Index Model 5

Method 1 Unadjusted beta 6

Method 2 Blumes beta 6

Method 3 Vasiceks beta 7

Solution Approach 8

Results & Analysis 9

Conclusions 12

Acknowledgement 12

Appendices 13

References 17


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Introduction

Portfolio optimization has been a highly researched area in operations research. This

paper will attempt to apply one of the foundations in portfolio optimization, the Markowitz

portfolio selection model (Markowitz, 1959) to solve real world problems based on the daily

returns data of 48 different industries. While the optimization model used here is limited in itsscope and complexity, the paper will mostly focus on methods of implementing the Markowitz

model. Thus, the main objective of this paper is to gain important insights on how to develop an

effective computational procedure to determine optimal portfolios. The performance of the

resulting portfolios will then be compared with the industrial standard returns from Dow Jones -

AIG Commodity Index (DJ-AIGCI).

One of the main components in the inputs to portfolio analysis is the correlation structure

of the stocks. When the number of stocks to select from the portfolio is large, the estimation of

the covariance can get very impractical for computation purposes. This project will look into the

single index model which was discussed comprehensively by Elton and Gruber (1987) andanalyze its applicability in solving the Markowitz optimization model using real world data.

The data used in this project are the daily average-value weighted returns of 48 different

industries collected from Kenneth R. French Data Library. For complete breakdown of the 48

industries, please refer to Appendix I. The average value-weighted return is simply the weighted

average of all stock returns, with the weights given by the market value of the stock (price times

shares outstanding) at the end of the previous trading period. The input data points include 3553

daily returns data and the period starts from 01/04/1988 to 12/31/2001 (see Appendix II).

The performance measures are the difference in returns by Markowitz model and theDow Jones returns. The non-parametric pair sign test will provide the required statistical analysis

as the data of difference in returns will later be found to be not normally distributed. Additionally,

some parameters of the Markowitz model will be customized to provide a broader view on the

performance analysis of the single index model as a method of forecasting the correlation

structures for portfolio optimization.

Markowitz Model

The basic solution approach to this problem is to implement the Markowitz model infinding an optimal portfolio selection in each forecast period. There are two objectives behind

Markowitz model; to achieve high returns and to achieve stable returns with low uncertainty. In

my model, the objective function is to maximize total returns, constrained by maximum

allowable risk level. The Markowitz optimization model can be modeled as follows:


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Inputs

ir = return on industry i

k = maximum risk factor

ijcov = covariance between industry i and j

N = portfolio size (number of industries)

Decision Variables

jx = fraction of portfolio to invest in industryj

Objective

Maximize Total Returns: iIi

i xr

Constraints

Budget constraint:

1Ii

ix

Maximum allowable risk:

N

kxx iji

Ii Jj

j

cov

The project used historical data to estimate the inputs to Markowitz model. These inputs

require estimates of the expected return on each stock ir and the covariance between each

possible pair of stocks for the stocks under consideration. The estimation can get very complex

as the portfolio size becomes large. For instance, if the number of stocks in a portfolio is 48, as inour case, we need to estimate N*(N-1)/2 = (48*47)/2 = 1128 correlation coefficients. The large

number of inputs can be computationally impractical due to the large number of estimates that

have to be made. Part of the project was to develop a more efficient estimation model and assess

its performance in terms of total returns created. For this objective, single index model was

applied to simplify the inputs to the Markowitz model.


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Single index model

Single index model assumes that the co-movement between stocks is due the single

common influence by market performance. Hence, the measure of this index can be found by

relating the stock return to the return on a stock market index. The formulation for single index

model can be shown below:

miii rar +=

where

ir= return on stock i

ia = component of stock is return that is independent of the markets performance

m

r = the rate of return on the market index

i= a constant that measures the expected change in irgiven a change in mr

The term ia can be further broken down into i and ie where i is the expected value of ia and

ie is the random element of ia .

The expected return, variance and covariance can be estimated as follows when they are

used to represent the joint movement of stocks:

Mean return of stock, miii rr +=

Variance of a stocks return, 2222 eimii +=

Covariance of returns between stocks i and j, 2mjiij =

where 2m = market variance and2

ei =unique risk factor

The single index model will need the estimates of mean return, variance of return and thebeta for each stock, which amounts to 3N + 2 = (3*48 + 2) = 146 estimates, in our case of 48

industries. This is much easier to compute than the previously mentioned estimates of 1128

correlation coefficients. For the purpose of the Markowitz model, I estimated the mean return of

each industry, ir and the market variance,2

m by calculating the average industry returns and

variance of market returns over a specified period, respectively. Finally, we need to estimate beta


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for each stock in order to calculate the covariance needed in the Markowitz model. Beta is

simply a measure of sensitivity of stock to market movement. There are 3 methods of estimating

beta as forecasters of covariance:

1) Forecasts of covariance by estimating betas from prior historical period (unadjustedbeta)

2) Forecasts of covariance by estimating betas from the prior two periods and updatingvia Blumes technique (Blumes beta)

3) Forecasts of covariance by estimating betas from prior historical period and updatingvia Vasiceks technique (Vasiceks beta)

Method 1: Unadjusted beta

The first method simply estimates betas from historical data. The historical beta for each

stock i can be obtained through regression analysis of stock return itr against market

return mtr from a past period, t = 1 to t = T. The calculation of beta for each stock is formally

shown below. The estimation of historical beta is subjected to error and might deviate

significantly from actual beta since actual beta is not perfectly stationary over time. The betas

might change significantly from one period to another and large random error may lead to

substantial forecasting error.

( )( )[ ]( )

=

=

==T

t

mtmt

T

tmtmtitit

m

imi

rr

rrrr

1

2

1

2

Method 2: Blumes beta

Blumes analysis on the behavior of betas over time shows that there is a tendency of

actual betas in the forecast period to move closer to one than the estimated betas from historical

data. Blumes technique attempts to describe this tendency by correcting historical betas to adjust

the betas towards one, assuming that adjustment in one period is a good estimate in the next

period. Consider betas for all stocks i in period 0, 0i and betas for the same stocks i in the

successive period 1, 1i . The betas for period 1 are then regressed against the betas for period 0

to obtain the following equation:


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01 21 ii kk +=

The relationship implies that the beta in period 1 is k1 + k2 times the beta in the period 0.

Therefore, if 1i is A, the estimate of beta in the next period 2i will be (k1+ k2*A) instead of A.

This adjustment sets the average beta to undergo similar trend for subsequent forecastperiods. If there is an increasing trend in average beta for period 1, average beta for period 2 will

consequently increase. This might not reflect the actual beta movement from one period to

another. Hence, Blume further modifies the average beta towards historical mean. This is done

by first calculating the average beta of all stocks for period 1 and 2,1 and 2 . To adjust the

mean of the forecasted beta towards historical mean, the new forecast of beta for each stock i

2i is obtained by subtracting 2 fromthe previously forecast of beta andadding 1 .

Method 3: Vasiceks beta

As mentioned earlier, the average beta tends to move towards one over time. Another

method to capture this tendency is via Vasiceks technique. Vasiceks technique adjusts past

betas towards the average beta by modifying each beta depending on the sampling error about

beta. When the sampling error is large, there is higher chance of larger difference from the

average beta. Therefore, lower weight will be given to betas with larger sampling error. The

following formula demonstrates this idea:

12

1

2

1

2

1

12

1i

2

1

2

1

2

i

i

i

i

++

+=

where

2i = forecast of beta for stock i for period 2 (later period)

1 = average beta across the sample of stocks in period 1 (earlier period)

2

1 = variance of the distribution of historical estimates of beta across the sample

of stocks

1i = estimate of beta for stock i in period 1

2

1i = variance of the estimate of beta for stock i in period 1


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Solution Approach

The goal in this project is to assess the performance of Markowitz portfolio optimization

model under different scenarios summarized below.

1)

Find the optimal portfolio selection for different risk levels (risk factor k).

2) Find the optimal portfolio selection for each rolling period from 01/04/1988 to 12/31/2001.There is a total of 3443 periods with each rolling period consisting of 90 days. I added

transaction cost in linear terms as an extension to the Markowitz model shown below.

Inputs

ir = return of industry i

k = maximum risk factor

ijcov = covariance between industry i and j

N = portfolio size (number of industries)

ix0 = fraction of the original portfolio invested in industry i in the previous period

t = transaction cost (in percentage)

Decision Variables

ix = fraction of portfolio to invest in industry i

iy = amount of long transaction in industry i

iz = amount of short transaction in industry i

Objective

Maximize Total Returns: )(*

+

Ii

iii

Ii

i zytxr

Constraints

Budget constraint:


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1Ii

ix

Maximum allowable risk:

N

kxx iji

Ii Jj

j

cov

Buying constraint

ix - ix0 iy Ii

Selling constraint

ix0 - ix iz Ii

3) Apply the three different methods of estimating beta (Unadjusted/Historical beta,Blumes beta and Vasiceks beta) in single index model as inputs to the Markowitz model.

Results and Analysis

I present three sets of results that display the difference between the returns based on

Markowitz model and the returns of Dow Jones - AIG Commodity Index (DJ-AIGCI). The

Markowitz portfolio selections were obtained by solving the portfolio optimization problems for

3443 periods from 01/04/1988 to 08/16/2001. Each rolling period consists of 90 days, which

means that we have to estimate the betas and expected returns using 90-day data for each period.

The portfolio solutions will be used to decide which industry and what amount to invest in each

industry on the 91st day. The forecast period starts from 05/11/1988 to 12/28/2001 and the model

was run with risk factors k = 1, 1.5, 2, 2.5, 3, 3.5 and 4. I repeated the same procedure for the

three methods of beta estimation in single index model (unadjusted beta, blumes beta and

vasiceks beta). The Dow Jones return indexes were selected as the benchmark because they

provide diversification, liquidity, stability and weightings that indicate economic significance.Thus, DJ-AIGCI is a reliable source of comparison with our model even though it does not have

identical commodities as the 48-industries portfolio.

I calculated the paired difference between returns based on the Markowitz model and the

Dow Jones index returns and performed paired sign test to test for a difference between the two

means of returns. The non-parametric paired sign test (function signtest.m on Matlab) was used


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instead of the paired t-test because the distribution of the difference between two means is not

normally distributed and is extremely heavy tailed, as depicted in the Q-Q plots in Appendix VII.

The first set of result in table 1 gives the summary statistics of the difference between

returns of Markowitz model using unadjusted beta and Dow Jones returns (Markowitz returns

Dow Jones returns) with different risk factor k. The paired sign test tests the null hypothesis thatthe median of the differences in the pairs is zero. The results show positive mean difference of

returns for all risk factors k. However, these results are not significant as all the hypothesis tests

based on paired sign test give H = 0, which means that null hypothesis cannot be rejected at

0.05 % significant level. All the tests conclude that there is no significant difference between the

returns by our model for all risk levels and the Dow Jones returns (high p-values). We can see

slight increase in the mean difference of returns and decrease in p-values as the risk levels

increase from 1 to 4.

Table 1: Summary Statistics of Paired Sign Test for Difference between Returns by

Markowitz model with Unadjusted beta and Dow Jones returns

Risk factor , k

Statistic 1 1.5 2 2.5 3 3.5 4

Mean Diff 0.0136 0.0257 0.0320 0.0322 0.0315 0.0335 0.0335

H 0 0 0 0 0 0 0

p-value 0.620 0.303 0.321 0.303 0.222 0.109 0.109

Test Statistic 1360 1346 1347 1346 1341 1331 1331

Z-value 0.50 1.03 0.99 1.03 1.22 1.60 1.60

Table 2 analyzes the difference between returns by Markowitz model with blumes beta

and Dow Jones returns. There is no significant difference between the returns by Markowitz

model with blumes beta and Dow Jones returns for all risk factors k. The mean difference of

returns is increasing with risk level while the p-values decrease with risk level. For k = 4, the p-

value is fairly low (0.093), which indicates that at 0.1 % significant level, we can assume that the

returns of the Markowitz model are greater than the Dow Jones returns.


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Markowitz model with Blumes beta and Dow Jones returns

Risk factor , k

Statistic 1 1.5 2 2.5 3 3.5 4

Mean Diff 0.0145 0.0255 0.0348 0.0352 0.0343 0.0345 0.0360

H 0 0 0 0 0 0 0

p-value 0.593 0.268 0.268 0.237 0.194 0.170 0.093

Test Statistic 1359 1344 1344 1342 1339 1337 1329

Z-value 0.53 1.11 1.11 1.18 1.30 1.37 1.68

The third set of result is displayed in table 3 where a comparison between the returns by

Markowitz model with vasiceks beta and Dow Jones returns is made. Again, there is no

significant difference between the returns by Markowitz model and the Dow Jones returns for all

risk levels. There is also some variation on the mean differences and the p-values for differentrisk levels.


Markowitz model with Vasiceks beta and Dow Jones returns

Risk factor , k

Statistic 1 1.5 2 2.5 3 3.5 4

Mean Diff 0.0127 0.0192 0.0285 0.0292 0.0283 0.0297 0.0313

H 0 0 0 0 0 0 0

p-value 0.647 0.492 0.285 0.237 0.285 0.285 0.147

Test Statistic 1361 1355 1345 1342 1345 1345 1335

Z-value 0.46 0.69 1.07 1.18 1.07 1.07 1.45

Overall, there is no significant difference between returns based on the Markowitz model

and the returns by Dow Jones. Further modification to the Markowitz model needs to be done to

improve the performance and the relevance of portfolio optimization model to handle the real

world data. Nevertheless, the study is able to provide some insights on the effectiveness of the

single-index model to estimate the inputs to the model. The results indicate that the Markowitz

model with inputs based on single-index model is able to produce comparable returns with DJ-

AIGCI. Moreover, out of the three methods to estimate beta for single index model, blumestechnique appears to give the best result with highest mean difference in returns and lowest p-

values. The increasing trend in returns as risk factor increases is also more evident in Markowitz

returns with blumes beta. Hence, if we use blumes beta to calculate the inputs to the Markowitz

optimization model, we are more likely to get significantly higher returns than the Dow Jones

returns.


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Conclusions

After completing the study on the computational method in portfolio analysis, I was able

to conclude that the single index model worked well in estimating the inputs to the basic

Markowitz optimization model. This is shown by the comparable returns produced by the

Markowitz model and Dow Jones AIG-Commodity Index. Although the comparison is notcompletely accurate as their portfolio stocks are not wholly identical, the Dow Jones returns data

are still useful to reflect economic influence. Also, it is shown that blumes technique worked

better than the other two methods to estimate beta in single index model.

There are certainly areas of improvement in this project, especially in the development of

more advanced extension to the Markowitz optimization model. The current simplistic model

can be the reason why the portfolio selection is small for all cases; the optimal portfolio only

selects a maximum of 2 industries in all periods (see Appendix V). This paper has also not

looked in depth for potential biases in the single index model, which might affect the end results.

From this project, I found that computational procedure can get as complicated as

formulating a solution method, especially when large scale data is involved. It is imperative that

one plans a robust and systematic method in solving a theoretical model. When the right method

is applied, it can lead to a better utility of the model and more efficient implementations.

Acknowledgement

I would like to thank Prof Sanjay Mehrotra for all his help and guidance throughout the

project and Michael Chen, Northwestern University for finding me the required data and goingthrough the basic of portfolio analysis with me.


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Appendices

Appendix I: Breakdown of the 48 Industries

Refer to Excel Spreadsheet

Source: Kenneth R. French Data Library, 2007

Appendix II: Daily Average Value Weighted Returns for 48 Industries

Refer to Excel Spreadsheet

Source: Kenneth R. French Data Library, 2007

Appendix III: Markowitz Model Formulation in Ampl

set Industry;

param k > 0, default 1; # maximum risk factor

param varM; # market variance

param beta {Industry}; # beta for each industry

param x0{Industry} >= 0, default 0; #investment in the previous period

param t > 0, default 0.005 ; # transaction cost

param cov {i in Industry,j in Industry} = beta[i]*beta[j]*varM; # Calculate covariance

param m {Industry}; # mean return of random variable r

var x{Industry} >= 0, = 0;

var z{Industry} >= 0;

maximize returns: sum{i in Industry}( m[i] *x[i] - t*(z[i]+y[i]) );


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subject to risk:sum{i in Industry, j in Industry} x[j]*cov[j,i]*x[i]


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Appendix VII: Q-Q plots of difference between returns based on the Markowitz model and

the Dow Jones index returns

The plots show similar pattern where the distribution is not normally distributed and heavy tailed.

These plots also include all returns with different risk factor k.

Fig 1: Returns under Unadjusted beta Dow Jones Returns

Fig 2: Returns under Blumes beta Dow Jones Returns


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Fig 3: Returns under Vasiceks beta Dow Jones Returns


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References

"Dow Jones AIG-Commodity Index."Dow Jones Indexes. 2008. Dow Jones.

.

Elton, Edwin J., and Martin J. Gruber.Modern Portfolio Theory and Investment Analysis. 3rd ed.

John Wiley & Sons, 1987. 95-128.

Fourer, Robert, David M. Gay, and Brian W. Kernighan.AMPL: a Modeling Language for

Mathematical Programming. 2nd ed. Thomson - Brooks/Cole, 2003.

French, Kenneth R. "Data Library." Kenneth R. French. 2007..

Markowitz, Harry M. Portfolio Selection: Efficient Diversification of Investments. John Wiley &

Sons, New Jersey, 1959.

"Two-Sample Paired Sign Test." PROPHET StatGuide. Northwestern University Medical

School..

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