Financial Portfolio Optimization Through a Robust Beta ... · PDF fileFinancial Portfolio Optimization Through a Robust ... The objectives were accomplished by converting Sharpe’s

Financial Portfolio Optimization Through a Robust Beta Analysis

Ajay Shivdasani

A thesis submitted in partial fulfilment of the requirements for the degree of

BACHELOR OF APPLIED SCIENCE

Supervisor: R.H. Kwon

Department of Mechanical and Industrial Engineering University of Toronto

March, 2008

ABSTRACT

All investors face the challenge of minimizing the risk of their portfolio while

maximizing their return. William F. Sharpe developed a model that simplifies this portfolio

selection problem. One of the limitations with Sharpe’s model is that it does not take into

account uncertainty in the market as time progresses, instead the model optimizes for a

single point in time. This is not a practical solution for investors who attempt to invest

today, but who also want to take into consideration future possibilities of their rebalancing

strategy as the market changes.

The focus of the work in this thesis is to (a) develop a model that will take into account

uncertainty in the market, (b) analyze and display the inherent benefits of the new model

over the Sharpe’s original model, and (c) determine any limitations that should be taken into

consideration with the new model.

The objectives were accomplished by converting Sharpe’s Single Index Model in to a

two stage stochastic program. Once the model was formulated, a comparative analysis

between Sharpe’s model and the stochastic model showed that the stochastic model is more

suited for a long term investment. Lastly, an analysis conducted on the minimum return

constraint of the model indicated that this constraint could affect the model’s ability to

maintain its stochastic features.

I

ACKNOWLEDGEMENTS

I would like to thank my supervisor, Professor Roy Kwon, for taking me on as one of

his thesis students. His insights and guidance throughout this process have proven to be

extremely valuable in achieving the objective of this thesis. The knowledge I have gained

through our discussions has allowed me to develop an understanding in portfolio theory and

further enlightened my knowledge and interest in Operations Research. I feel indebted to

him for his support in helping me pursue a research topic that has enhanced my knowledge

in this field.

II

TABLE OF CONTENTS

ACKNOWLEDGEMENTS.......................................................................................................................... I TABLE OF CONTENTS.............................................................................................................................II LIST OF FIGURES ................................................................................................................................... IV LIST OF TABLES .......................................................................................................................................V 1. INTRODUCTION.....................................................................................................................................1

1.1 Motivation.........................................................................................................................................1 1.2 Objective...........................................................................................................................................2 1.3 Research approach ...........................................................................................................................3

2. LITERATURE REVIEW.........................................................................................................................4 2.1 PORTFOLIO THEORY ...............................................................................................................................4

2.1.1 Markowitz’s full covariance model................................................................................................4 2.1.2 Sharpe’s Single Index Model .........................................................................................................8 2.1.3 Market Beta .................................................................................................................................13

2.2 TWO STAGE STOCHASTIC PROGRAMMING ............................................................................15 3. METHODOLOGY..................................................................................................................................18

3.1 OPTIMIZATION MODELING...................................................................................................................18 3.1.1 Programming Sharpe’s Single Index Model into OPL ................................................................18 3.1.2 Formulate a Two Stage Stochastic Model ...................................................................................18 3.1.3 Program Two Stage Stochastic Formulation into OPL...............................................................23

3.2 DATA GENERATION ..............................................................................................................................23 3.2.1 Current Time Period Data...........................................................................................................23 3.2.2 Scenario Data for One Time Later ..............................................................................................24

4. RESULTS & ANALYSIS .......................................................................................................................26 4.1 COMPARATIVE ANALYSIS.....................................................................................................................26

4.1.1. Perfect Information Solution ......................................................................................................27 4.1.2. Deterministic (Expected Value) Solution....................................................................................28 4.1.3 Stochastic Model Solution ...........................................................................................................30 4.1.4 Results..........................................................................................................................................31

4.2 VALUE OF THE STOCHASTIC SOLUTION (VSS) ......................................................................................32 4.3 EFFECT OF THE MINIMUM RETURN CONSTRAINT...................................................................................33

5. CONCLUSION........................................................................................................................................36 6. FURTHER RESEARCH ........................................................................................................................38 7. REFERENCES........................................................................................................................................40 APPENDIX A – OPL CODE FOR SHARPE’S SINGE INDEX MODEL.............................................41 APPENDIX B – OPL CODE FOR THE STOCHASTIC MODEL ........................................................44 APPENDIX C – PERCENT CHANGE TABLES ....................................................................................48 APPENDIX D – INPUT TABLES FOR EACH SCENARIO..................................................................50 APPENDIX E – OPTIMAL INVESTMENT DECISIONS AND OBJECTIVE VALUE CALCULATIONS USING PERFECT INFORMATION .......................................................................52 APPENDIX F – OPTIMAL INVESTMENT DECISIONS AND OBJECTIVE VALUE CALCULATIONS USING DETERMINISTIC MODEL........................................................................54

III

APPENDIX G – OPTIMAL INVESTMENT DECISIONS AND OBJECTIVE VALUE CALCULATIONS USING STOCHASTIC MODEL ..............................................................................56 APPENDIX H – RESULT OF MINIMUM CONSTRAINT ANALYSIS ..............................................58

IV

LIST OF FIGURES

FIGURE 1.1 SECURITY CHARACTERISTIC LINE (TRANDAFIR, CHAPTER 10)......................14 FIGURE 4.1 THE EFFECT OF THE MINIMUM RETURN CONSTRAINT ON THE OBJECTIVE FUNCTION .................................................................................................................................................34

V

LIST OF TABLES

TABLE 3.1 INPUT DATA FOR CURRENT TIME PERIOD................................................................24 TABLE 3.2 PERCENT CHANGE OF BETA FROM CURRENT DATA.............................................25 TABLE 3.3 BETA INPUTS FOR EACH SCENARIO ONE TIME PERIOD LATER........................25 TABLE 4.1 INITIAL INVESTMENT DECISIONS BASED ON PERFECT INFORMATION THAT SCENARIO 2 WILL OCCUR ...................................................................................................................27 TABLE 4.2 OPTIMAL REALLOCATION DECISIONS FOR SCENARIO 2 GIVEN INITIAL INVESTMENT DECISIONS IN TABLE 4.1 ...........................................................................................28 TABLE 4.3 INITIAL INVESTMENT DECISIONS FOR EXPECTED VALUE SOLUTION ...........29 TABLE 4.4 OPTIMAL REALLOCATION DECISIONS FOR SCENARIO 2 GIVEN EXPECTED VALUE SOLUTION...................................................................................................................................29 TABLE 4.5 INITIAL INVESTMENT DECISIONS BASED ON STOCHASTIC MODEL................30 TABLE 4.6 OPTIMAL REALLOCATION DECISIONS FOR SCENARIO 2 GIVEN INITIAL INVESTMENT DECISIONS IN TABLE 4.5 ...........................................................................................31 TABLE 4.7 COMPARISON OF OBJECTIVE VALUES FOR EACH MODEL .................................31 TABLE 4.8 PERCENT INCREASE IN OBJECTIVE VALUE FROM PERFECT INFORMATION SOLUTION..................................................................................................................................................32 TABLE 4.9 EXPECTED OBJECTIVE VALUES AND VSS..................................................................33

1

1. INTRODUCTION

1.1 Motivation

The main purpose for investing in a portfolio of securities is to achieve a desired rate of

return over a specific time period. However, there is a risk involved in investing as there are

no guarantees in achieving a positive rate of return. Thus, for most investors the real goal of

investing is to achieve a desired rate of return while minimizing the risk involved. Most

rational investors use a technique of diversification to achieve this goal. Diversification is

defined as holding a broad portfolio of assets that are unrelated to one another in order to

achieve a desired rate of return.

In 1963, William F. Sharpe published a portfolio optimization model that simplified

Harry M. Markowitz’s full covariance model. Markowitz’s model quantified the objective of

maximizing return, while minimizing the overall risk of the portfolio. More importantly,

Markowitz’s model was the first to capture the phenomenon of diversification by measuring

the performance of each individual security in relation to all the other securities in the

portfolio. This measurement is known as the covariance between a pair of securities. The

practical limitation with Markowitz’s model is that as the number of securities in the

portfolio increases, the more complex the analysis becomes. This is due to the fact that the

model needs to compute the covariance between each possible pair of assets in the portfolio.

Sharpe’s Single Index Model simplifies Markowitz’s model by proposing that the relationship

between each pair of assets can be indirectly measured by comparing each asset to a

common factor that is shared amongst all the assets, such as the market’s performance. One

2

common statistical index that can capture this relationship between a security and the market

is known as the market beta (β).

Studies and analysis have identified that there are limitations involved with Sharpe’s

model. One of the limitations is that the model determines the optimal portfolio for a single

point in time. Thus, as time passes, the optimal solution may no longer be viable due to

constant changes in the market. This is not very practical for investors who are looking to

minimize the rebalancing of their portfolio as the market changes.

This thesis will specifically look at how investors can use the β of securities to diversify

their portfolio and hence minimize risk for a given level of return. To further develop the

robustness of this study, the thesis will look at how to take into account the dynamic and

uncertain nature of the market beta as time progresses.

1.2 Objective

The objective of this undergraduate thesis study is to generate and determine the

benefits of an alternate formulation to Sharpe’s model. The modifications made to Sharpe’s

original model would address the limitation of producing an optimal portfolio for a single

point in time. The overall aim is to create a tool that is more practical for investors to use by

taking into account uncertainty in the market beta from one time period to another. By

accounting for uncertainty, investors will have the opportunity to determine a portfolio that

is well suited for the long run. This in turn is more practical because it allows investors to

maintain a viable portfolio that they do not have frequently and drastically rebalance as time

progresses. Once the model has been created that incorporates all of the above aspects, a

comparative analysis will be performed to determine the benefits of the newly developed

3

model versus Sharpe’s Single Index Model. The final objective is to identify any limitations

that the newly developed model might face, to gain a better understanding of how it can be

implemented.

1.3 Research approach

The approach taken to achieve this objective begins with developing an understanding of

modern portfolio theory and stochastic programming through a literature review. Once the

foundations behind portfolio theory have been established, a formulation of the new

optimization model will be developed. Developing this new formulation will involve

converting Sharpe’s original model to a two stage stochastic program. Once the model has

been formulated, arbitrary data will be used to test and analyze the model. A comparison

analysis between Sharpe’s model and the newly developed stochastic model will be

conducted to determine the benefits achieved from this study. A further study will be

conducted to determine if any limitations exist within the model.

4

2. LITERATURE REVIEW

2.1 PORTFOLIO THEORY

This section of the topic based literature review discusses the various techniques for

selecting security portfolios, specifically Harry M. Markowitz’s full covariance model and

William F. Sharpe’s Single Index Model. The purpose of this section is to get a general

understanding of the portfolio problem that investors face on a regular basis and the

methods that can assist in determining its solution. Furthermore, the two models described

below establish the context and motivation for the research done in this thesis study.

2.1.1 Markowitz’s full covariance model

In the article entitled A Survey and Comparison of Portfolio Selection Models, Buckner A.

Wallingford provides a very succinct and comprehensive explanation of Markowitz’s classic

work in portfolio theory. Wallingford begins by describing the model that Markowitz’s

published in 1959, which best represented the daily investor’s problem at the time. The

general objective of this problem is to determine a portfolio that provides the maximum

return for a given level of risk or the minimum risk for a given level of return. Since every

investor is unique in terms of their risk attitude, the model is flexible in its ability to find a set

of portfolios that achieves the maximum return for every possible level of risk while

minimizing the risk for every possible level of return. This set of optimal solutions is known

as the efficient set (Wallingford 86).

5

2.1.1.1 Determining the Objective Function of Model

In order to develop the objective function of the model, the return and the risk must be

quantified. The best estimate for the return of a portfolio is the weighted sum of the

expected returns from each of the assets that make up the portfolio. Thus, the expected

return E(R) can be modeled by the equation:

( ) (1)i ii

E R X E=∑

where Xi represents the proportion of the investor’s capital in security i and Ei

represents the expected return for security i

Now that the return has been quantified, the next step is to quantify the risk of the

portfolio. The more likely that the actual return of the portfolio will be closer to the

expected return, the less risky the portfolio is. Since variance is a measure of the amount by

which the actual return is likely to vary from its expected level, it can be used to

quantitatively resemble the risk of the portfolio. In order to compute the overall variance of

a portfolio, it is necessary to know the variance of each individual asset and its covariance

with all the other assets in the portfolio. The covariance between a pair of assets measures

how each asset in the pair moves in relation to one another. A high covariance represents

high risk because the portfolio is composed of assets that tend to move in the same direction

of one another. For example, if one security in the portfolio achieves a negative return,

many assets will follow in its direction and magnify the negative impact. That is why a

diverse set of securities minimizes the overall impact than any one asset has on the overall

portfolio’s performance, thus making it less risky.

6

Thus, the variance of the portfolio can be modeled by

( ) (2)i j iji j

V R X X C=∑∑

Where Xi and Xj are proportions in security i and security j respectively. Cij is the

covariance between securities i and j. When i =j, Cij is the variance of the individual security

i.

By minimizing the variance equation above, Markowitz was the first to model the

phenomenon of diversification. Diversification, as previously mentioned, is the mixing of

different investments within a portfolio to reduce the impact of any one asset on the overall

portfolio

By quantifying the risk and return, the objectives the model could now be represented by

maximizing the following equation

( ) ( ) (3)i i i j iji i j

Z E R V R X E X X C= − = −∑ ∑∑

The objective function is trying to maximize the expected return and the negative of the

variance. By maximizing the negative of the variance, the model is indirectly minimizing the

risk. However, solving the above equation would only result in one possible solution in the

efficient set. In order to incorporate an individual’s utility function, preference of risk vs.

return, a coefficient λ is added in front of E(R). By increasing λ, the more risk an individual

is willing to incur to potentially reap the rewards of increased returns. For every value of λ,

between 0 and infinity, there is a unique solution in the efficient set.

7

Thus, the final objective function can be modeled as:

Maximize (4)i i i j ij

i i j

X E X X Cλ −∑ ∑∑

2.1.1.2 Determining the Constraints

Markowitz’s model assumed that the entire portfolio must be invested and no security

may be held in negative quantities. Thus adding the following two constraints:

1 (5)

0 for all (6)

ii

i

X

X i

=

≥

∑

2.1.1.3 Limitations to the Model

Markowitz’s model does have some theoretical and practical limitations that are

discussed in Wallingford’s article and in Sharpe’s A Simplified Model for Portfolio Analysis.

First of all, the assumption that variance is a good measure of risk is a limitation because

it assumes that deviations above and below the expected return are equally undesirable

(Wallingford 93). For example, it is implied that a deviation of 5% above the expected return

is equivalent to 5% below the expected return. However, in reality a deviation that exceeds

the expected return would be more desirable than a deviation that falls short. This concept is

not incorporated into Markowitz’s model.

Furthermore, Markowitz’s model is a point in time analysis (Wallingford 95). This means

that the model is optimized based on data at a current point in time. It does not take into

account uncertainty in the data as time progresses. Thus at some time period later, the

8

solution may no longer be optimal and running the model could result in a completely

different portfolio.

The most significant limitation, which is discussed in Sharpe’s article, is the increased

complexity that this model faces as the number of securities in the portfolio grows. To

determine the variance of the portfolio, the covariance between each possible pair of assets

must be computed, which is represented in a covariance matrix. Thus, increasing the number

of assets results in a larger covariance matrix, which in turn results in a more complex

computation. “In the standard case, if N securities are analyzed this matrix will have ½

(N2+N) elements” (Sharpe 281).

Although Markowitz’s was able to develop a comprehensive technique that was the first

to incorporate diversification, it had many limitations that needed to be resolved.

2.1.2 Sharpe’s Single Index Model

Due to Markowitz’s model and its practical limitations, William F. Sharpe published an

article in 1963, A Simplified Model for Portfolio Analysis, which proposes a simplified model to

the portfolio analysis problem. This model conversely known as either the Diagonal or

Single Index Model, essentially simplifies the problem by proposing that the relationship

between each possible pair of assets in the portfolio can be indirectly measured through each

security’s relationship to a common factor. This common factor is some index that

represents the market. This index could be the level of the stock market as a whole, a price

index or any other factor that is most influential on the returns from securities (Sharpe 281).

Instead of solving a dense covariance matrix, which is required in Markowitz’s model,

the assumption above results in only a diagonal covariance matrix, hence the term the

9

Sharpe’s Diagonal Model. The diagonalization of the covariance matrix reduces the

complexity of the problem drastically.

This model has two important qualities: it is a model that maintains its simplicity, while

not removing the existence of interrelationships among assets; and there is considerable

evidence that it can capture a large part of the interrelationships (Sharpe 281).

Based on Sharpe’s assumption, the return of any security can be determined through a

linear regression of asset returns against market index returns (Berardi, Corradin, and

Sommacampagna 3). This can be modeled by:

(1)i i i iR A B I C= + +

where:

Ri is the return of Security i

I represents the market index

Ai is an additive constant

Ci is the error component (a random variable with a mean of zero and variance Qi)

Bi represents the sensitivity of Ri to the market index, I

2.1.2.1 Incorporating Market Beta into Sharpe’s Model

A common measure that can be used represent the sensitivity of a securities return to the

market index, is a statistical value known as the market beta. The β of a security measures

the co-movement in returns between the market and the security over a given period of

time.

10

In Chapter 13 – Portfolio Optimization, of the Optimization Modeling with Lingo textbook,

an adaptation of Sharpe’s Single Index Model is described. This version of the model

incorporates β into the formulation. The formulation is as follows:

2

2

0

0

Decision Variables:= Proportion of portfolio invested in stock

Parameters: = The market factor = E(M)= Var (M)

= Random movement to specific stock = Var ( )

= the market beta for sto

i

i

i i

i

X i

MmSe iS eb ck

= A desired rate of return to be achieved = the initial alpha value for stock

= number of assets in portfolioi

iR

inα

2 22 20

1

1

1

+ 0

( ) (2.1)

. .

( ) (2.2)

1 (2.3)

( ) ( )

n

i

i

n

i

n

i

i i

Minimize

V R Z S Xi S

S T

Xibi Z

Xi

E R Xi b m Rα

+

=

=

=

=

=

=

= ≥

∑

∑

∑

1

(2.4)n

i=∑

The objective of this model is to minimize the overall variance of the portfolio, which is

represented by equation (2.1). Equation (2.2) defines the parameter Z, which is used in the

objective function. Z represents the expected beta of the portfolio. Similar to Markowitz’s

model, equation (2.3) represents the constraint that the entire portfolio must be invested.

11

The third constraint of this model (2.4) is that the expected return of the portfolio must

achieve a minimum value of R.

The expected return of portfolio, E(R), resembles equation (1) in the section 1.1.1.1. In

this model Ei is substituted by:

+ 0( ) (3)i ib mα .

Equation (3) is derived from Sharpe’s assumption that the return of any security can

modeled by equation (1). Ai is represented by the alpha value of security i, Bi is resembled

by the market beta of security i, and I is equivalent to the expected market return. There is

no parameter representing the error component, Ci, because the expected value of Ci is

equivalent to 0.

Sharpe’s model encompasses the major concepts that Markowitz developed in his model.

Essentially the model is trying to minimize the variance of portfolio for a given level of

return. Like Markowitz’s model, this model is still flexible to an individual’s risk attitude by

having the ability to alter the value of R, the minimum level of expected return.

2.1.2.2 Advantages of Sharpe’s Model

The apparent advantage to Sharpe’s model is that it considerably simplifies the portfolio

problem in comparison to Markowitz’s full covariance model. Not only does it diagonalize

the full covariance matrix, but it also simplifies the computational techniques required to

solving the problem. As a result, “the computation time is reduced to about one percent of

that required by Markowitz Model” (Wallingford 97)

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2.1.2.3 Limitations of Sharpe’s Model

A tradeoff to simplifying Markowitz’s Model is that it ignores much of the information

that is contained in the covariance matrix, and hence accuracy of the optimization is

sacrificed. Although this is an apparent tradeoff, Sharpe explains that his analysis would

indicate that accuracy is not sacrificed to a large extent. His analysis, which he points out is

not conclusive, would indicate that the Diagonal Model is still able to represent the

relationships among securities well (Sharpe 292).

In Insignificant Betas and the Efficacy of the Sharpe Diagonal Model for Portfolio Selection,

Frankfurter and Lamoureux point out that the selection algorithm of Sharpe’s model has the

tendency to select securities with the lowest βs. When a security’s β is statistically

indistinguishable from zero, it essentially means that the performance of the security has a

very low to no relationship with the performance of the market. The issue is that the

relationship between the market and each individual security is supposed to indirectly

measure each security’s covariance with all the other securities in the portfolio. Thus, the

question arises whether the model is still viable if it is selecting securities that have very low

βs and in turn have no relation to the market (Frankfurter and Lamoureux 853). To improve

the performance of Sharpe’s model, the two of them propose a simple heuristic of excluding

any stock with an insignificant β from the set of stocks that would be analyzed by the model.

Similar to Markowitz’s model, Sharpe’s Diagonal Model is a single point in time analysis.

This deterministic model does not take into account uncertainty in the βs over a period of

time. As a result, it makes the optimization valid for one point in time. This is not very

practical for investors who are interested in long term investments.

13

Sharpe was able to develop a simplified and more practical model in response to

Markowitz’s full covariance model. However there are still many limitations to the model

that could be resolved through modifications and heuristics as shown in Frankfurter and

Lamoureux’s article.

2.1.3 Market Beta

The β of an asset is “a measure of comovement between the rate of return on the stock

with the rate of return on the market” (Welch 160). In other words, it describes whether a

stock will move in the same or opposite direction of the market.

A security with a β greater than 1 would indicate that when the stock market’s return

does well, that security will have even a higher rate of return. For example, if a security has a

β of +2 and the market’s return over a given time period is -20%, it would be expected that

the return of the security would double in magnitude and move in the same direction, hence

-40%. If the market beta is negative in value it indicates that the security will move in the

opposite direction of the market. A market beta of 0 indicates that security does not move

at all in relation to the stock market.

2.1.3.1 How to Determine the Market Beta of a Security

The beta of an individual security in comparison to its market can be determined

graphically. By plotting the historical returns of the individual security (Ri) vs. the historical

returns of the market (Rm), a best-fit line between the two series can be constructed. This

line is commonly referred to as the security characteristic line (SCL) (Figure 1.1). The slope

of the SCL line is the market beta. The intercept of the SCL is known as the alpha of the

security and it represents the return on the individual asset in excess of the risk free return.

14

Figure 1.1 Security Characteristic Line (Trandafir, Chapter 10)

15

2.2 TWO STAGE STOCHASTIC PROGRAMMING

Stochastic programming is an approach to modeling optimization problems in which

inputs of the model are uncertain or random. In contrast, deterministic problems

incorporate models that have known parameters. Stochastic programming is more realistic

in a sense that most real world problems include parameters that are unknown at the time of

optimization (Shapiro and Philpott).

One of the most widely applied and studied stochastic programming models are two-

stage (linear) programs (Shapiro and Philpott). In a two stage stochastic program, a set of

decisions have to initially be made without full information on some random events. Once

the full information is received, the second stage of the stochastic program is to take

corrective, recourse actions based on this information.

In Birge’s Introduction to Stochastic Programming he introduces the foundations of two stage

stochastic programming through a well-known example known as the Farmer’s problem. A

farmer needs to determine the proportion of his land that should be devoted to three

specific crops to maximize his profits. Each crop has a specific cost/acre to be planted and a

specific selling price/ton to be sold at. The farmer faces the issue of uncertainty of the yield

(tons produced per acre planted) for each crop due to the weather. Thus, the farmer has to

divide the proportion of acres he has available to each crop without knowing the yield that

will be produced. This decision is known as the first stage of the stochastic program. The

second stage of the program is to determine the amount of each crop that is yielded,

purchased, and sold based on the uncertain scenarios that could occur. Finding the optimal

solution that is ideal for all possible scenarios is impossible. Thus, the farmer must initially

16

determine the proportion of land that he wants to allocate to each crop which will balance or

hedge against all the different scenarios that could occur.

The general formulation of two-stage stochastic model can be written as follows:

Minimize[ ( , )] (1.1)

Subject To (1.2)

0 (1.3)

Tc x E Q x

Ax bx

ξ ξ+

=≥

The first-stage decisions are represented by the vector x. These values are determined

before any random event occurs.

The expected value for the second stage function or recourse function can be

represented by:

[ ( , )] (2)E Q xξ ξ

Where Q(x,ξ) represents the optimal second stage value for each possible scenario

represented by the random vector ξ. The reason Q(x,ξ) is a function of the decision vector x

is because the optimal solution of the second stage is based on full information, which is

received after the first-stage decisions are made. Therefore, the recourse actions are

dependent on the set of first stage decisions.

17

Q(x,ξ) is the optimal solution of the following nonlinear program:

Minimize (3.1)

Subject To( ) ( ) ( ) (3.2)

0 (3.3)

Tq y

T x w y hyξ ξ ξ+ =≥

The second stage problem seeks an optimal decision vector y for given values of the first

stage decision vector x.

To determine the expected value of the second stage optimization, which is represented

by equation (2), each individual scenario’s optimal value, Q(x,ξi), must weighted based on

the expected probability that the scenario ξi is going to occur. The sum of the weighted

scenarios represents the expected value of the second stage objective. The expected value of

the second stage objective can be modeled as:

1

[ ( , )] ( , ) (4)N

i ii

E Q x Q xξ π ξ=

= ∑ξ

Where iπ represents the expected probability that scenario i will occur and N is the

number of scenarios.

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3. METHODOLOGY

3.1 OPTIMIZATION MODELING

3.1.1 Programming Sharpe’s Single Index Model into OPL

The first step of this study was to program Sharpe’s Model into an optimization software

package, which was ILog OPL. The goal of the first step was twofold: to learn the syntax

and programming language used in OPL and to develop a further understanding of Sharpe’s

model. Moreover, this step was necessary because further on in the study optimizations

would be processed using Sharpe’s original model to analyze and compare its differences to

the modified stochastic model.

The OPL code developed for this model is located in Appendix A of the report. To test

that the model was working correctly, the sample problem provided on p. 385 of the

Optimization Modeling With Lingo textbook was inputted and optimized using the OPL

program. The output from OPL was compared to and matched the results presented in the

textbook.

3.1.2 Formulate a Two Stage Stochastic Model

The second step in this study was to develop a formulation that altered Sharpe’s model

to incorporate uncertainty in the market beta over a given period of time. One common way

to incorporate uncertainty into a deterministic model is to convert it into a two stage

stochastic program.

The stochastic model developed during this study is based on the formulation of

Sharpe’s model provided in the Optimization Modeling with Lingo textbook. The two stage

19

stochastic component was implemented using the aid of the farmer’s problem provided in

Birge’s textbook. The final formulation is as follows:

20

Decision Variables:= Proportion of portfolio initially invested in stock = The porportion of portfolio that is invested in each stock one time period later for each scenario

Parameters: =

iX iYij i j

m

2

2

0

0

number of scenarios = number of assets in portfolio

= The Market Factor = E(M) = E(M) in Scenario = Var (M)

= Random movement to stock = Var ( )

= the initial market beta for stock

j

i

i i

i

nMmm jSe iS eb i

= the market beta for stock under scenario after one time period has elapsed = the initial alpha value for stock = the alpha value for stock under scenario after one time period has

ij

i

ij

b i ji

i jαα

1

elapsed

= probability that scenario occurs. 1

= A desired rate of return to be achieved

m

j j

j

j

R

π π=

=∑

2 2 22 20

1 1 1

1

1

( ( ) ) (1.1)

. .

1 (1.2)

1 for all = 1..

n m n

i j i iji j i

n

in

i

Minimize

Z S Xi S U V

S T

Xi

Wij j m

π+ +

= = =

=

=

−

=

=

∑ ∑ ∑

∑

∑

+ 0

1

+ 0

+

(1.3)

( ) (1.4)

( ) for all = 1.. (1.5)(

n

i i

i

i i

ij ij

Xi b m R

Xi b m Ui i nYij b m

α

αα

=

≥

=

∑

1

) for all =1.. for =1.. (1.6)

( ) (1.7)

j

n

i

Vij i n j m

Xibi Z=

=

=∑

21

Since this model represents a two stage stochastic program, there are two sets of

decision variables. The first stage decision variables, Xi, determine the proportion of the

total investment that should be initially allocated to each asset in the portfolio. The second

stage decision variables, Yij, determine how the total investment should be reallocated from

the initial decisions for each of the scenarios that could occur one time period later.

The objective function (1.1) of the model described above has two components to it,

which represents the first stage and the second stage of the stochastic program. The first

component is:

2 2 220

1

(2)n

i

i

Z S Xi S+

=∑

The above component is the exact same as the objective function in Sharpe’s Model

(equation (2.1) in section 2.1.2). It represents the initial variance of the overall portfolio

based on the first stage decisions. The objective of the second stage is to try to keep the

composition of the portfolio as similar as possible to the initial set of decisions. Changing

the composition of the portfolio regularly can be difficult and costly for an investor. The

second stage indirectly models this goal by minimizing the difference in returns between the

initial investment and each possible scenario that could occur. This is determined by

calculating:

(3)i ijU V−

Ui represents the initial expected return for security i and is defined through constraint

(1.5). Vij represents the expected return for security i under scenario j and is defined

through constraint (1.6). Since the objective function is trying to minimize the difference in

22

returns, the optimization would favour scenarios in which Vij is greater than Ui. This is not

accurate as any difference above or below Ui should be considered equal in value if its

magnitude is the same. Thus, to ensure that only the magnitude in difference is considered,

the difference between Ui and Vij is squared. This evolves equation (3) to:

2 ( ) (i ijU V− 4)

The above equation now represents the value of the objective function for each

individual scenario in the second stage recourse function. To determine the expected value

of the overall second stage function, each individual scenario’s value must be weighted based

on the expected probability that the scenario is going to occur. The sum of the weighted

scenarios represents the expected value of the second stage objective. The expected value of

the second stage objective can be modeled as:

2

1 1

( ( ) ) (5)m n

j i ij

j i

U Vπ= =

−∑ ∑

Comparable to Sharpe’s model, the first constraint (1.2) implies that the entire portfolio

must be invested in the first stage decision. The second constraint (1.3) implies that the

entire portfolio must be invested for each scenario in the second stage decision. Again,

similar to Sharpe’s model the first stage decision should achieve a minimum initial expected

return of R, which is modeled through equation (1.4). As mentioned above equations (1.5)

and (1.6) define Ui and Vij, respectively. Similar to equation (2.2) in Sharpe’s model, equation

(1.7) in this formulation defines Z, the expected beta of the initial portfolio.

23

3.1.3 Program Two Stage Stochastic Formulation into OPL

Once the formulation was developed, the next step was to transfer it into OPL to test

and validate it. The OPL code developed for this model is located in Appendix B of the

report. A quality control tool was developed in Microsoft Excel to ensure that the

formulation was accurately transferred into OPL. The outputs for the decision variables Xi

and Yij generated by OPL are inputted into the quality control tool, along with all the input

parameters that OPL requires to run the optimization. The quality control tool then uses

these inputs to generate the expected value of the objective function. This ensures that the

formulated objective function was accurately conveyed and optimized in OPL. Furthermore

the quality control tool determines whether that optimal solution satisfies constraint (1.2) by

summing the Xis and verifying that it adds to 1. Similarly, the quality control tool determines

if the optimal solution for each scenario’s decision variables sums to 1. The quality control

tool also uses the output from OPL to calculate the initial expected return, which can then

be used to check whether equation (1.4) is satisfied. Once all the quality control checks

passed, it indicated the formulation had been successfully transferred into OPL.

3.2 DATA GENERATION

In order to conduct any analyses with the stochastic model, a set of arbitrary data needed

to be generated. For the purpose of this study, data for five arbitrary stocks was generated.

3.2.1 Current Time Period Data

The first set of data that was required as inputs into the model are the current market

conditions. The data used to represent the current state of the five stocks and market is as

follows:

24

Market Data:Mo So

119.15% 0.1623019

Asset Data:

Asset Si Alpha (αi ) Beta (βi)

ATT 0.08 0.364 0.49GMC 0.13 0.764 -0.21USX 0.17 -0.581 1.52

CSCO 0.03 0.154 0.67ABX 0.06 0.987 -0.16

Table 3.1 Input Data for Current Time Period

3.2.2 Scenario Data for One Time Later

Data for a series of six scenarios was then generated to describe possible changes that

could occur in the market one time period later. It is assumed that each of these scenarios is

equally probable to occur.

3.2.2.1 Generating Percent Changes

The data for each of the scenario dependent inputs, α, β, and Mo, was determined by

randomly generating percentages. Each percentage generated would represent the percent

change by which one of the inputs in the current data would change one time period later.

25

An example of a percent change table for the market beta is described below:

Beta

ScenariosAsset S1 S2 S3 S4 S5 S6

ATT 0% -12% 17% -11% 56% -3%GMC 0% -14% 49% -6% -26% -25%USX 0% -37% -16% -58% 5% -88%CSCO 0% -25% -10% -10% 18% -24%ABX 0% 6% 4% 25% -63% 24%

Table 3.2 Percent Change of Beta from Current Data

Please note that Scenario 1 has 0% change for all its values because Scenario 1 was not

randomly generated. It was fixed to 0% so that it would represent a scenario in which there

is no change from the current time period. Please refer to Appendix C for all of the percent

change tables.

3.2.2.2 Converting Percent Changes to Scenario Inputs

Each scenario’s inputs were calculated by multiplying the current data by (1+percent

change). An example of the input data for one time period later is as follows:

Beta

ScenariosAsset S1 S2 S3 S4 S5 S6

ATT 49% 43% 57% 44% 76% 48%GMC -21% -18% -31% -20% -15% -15%USX 152% 96% 128% 64% 160% 19%CSCO 67% 50% 60% 60% 79% 51%ABX -16% -17% -16% -20% -6% -20%

Table 3.3 Beta Inputs for Each Scenario One Time Period Later

Please refer to Appendix D for all of the input tables for each scenario.

26

4. RESULTS & ANALYSIS

Once the final formulation of the model was completed and data had been generated,

the next step was to conduct a series of analyses to determine the benefits of the stochastic

model. Using the arbitrary data, a comparative analysis is conducted between the

deterministic, stochastic, and perfect information model. The analysis attempts to compare

the achieved objective value for each scenario between the deterministic and stochastic

model. The perfect information model is used as a benchmark for comparing the other two

models against.

In addition, the value of the stochastic solution (VSS) is calculated to determine

whether the expected objective value of the stochastic solution is an improvement over the

expected objective value of the deterministic solution. For the comparative and VSS analysis,

the minimum return constraint, R, was fixed at 89%.

The purpose of the first two analyses was to establish the benefits of the stochastic

model. The goal of the final analysis was to understand one of the limitations of the

stochastic model through the minimum return constraint.

4.1 COMPARATIVE ANALYSIS

In this comparative analysis the objective function’s value is calculated for each scenario

using each of the three models. Before the results of this analysis are shown, an explanation

is provided to describe how the objective value for each model is calculated.

27

4.1.1. Perfect Information Solution

A perfect information solution determines the first stage decision, while knowing with

100% certainty that a specific scenario is going to occur. This is the absolute optimal

solution that can be determined for any of the six scenarios. Its optimality is based on the

fact that the investor initially knows what the state of the market is going to be at the time of

rebalancing. With this knowledge, the investor can then find the best initial investment and

rebalancing strategy that will minimize the objective function.

A sample calculation for the objective value using perfect information for Scenario 2 is

shown. The probability value of 100% is inputted into the stochastic model for Scenario 2.

As a result, the initial investment decisions are:

Assets Xi

ATT 0.412GMC 0.000USX 0.050CSCO 0.052ABX 0.487

Table 4.1 Initial Investment Decisions Based on Perfect Information that

Scenario 2 Will Occur

28

Based on this initial investment, the optimal reallocation strategy once Scenario 2 has

been realized is:

Assets Yi2


Table 4.2 Optimal Reallocation Decisions for Scenario 2 Given Initial Investment

Decisions in Table 4.1

Based on the two investment decisions described above, the optimal objective value is

calculated using the objective function. This value is determined to be 0.013428. Please refer

to Appendix E for the optimal decisions and objective value calculations using perfect

information for each scenario.

The perfect information solution can be used as a benchmark to see how close the

deterministic and stochastic solutions are to the ideal optimal value.

4.1.2. Deterministic (Expected Value) Solution

The deterministic solution generates the initial investment decision by optimizing strictly

on the current time period’s data. The assumption in this methodology is that the investor

believes the current time period’s data will continue to be the expected data one time period

later. This is essentially optimizing without taking into account uncertainty in the data and

assuming that the current data will continue to be the expected values one time period later.

Hence, it is given the term deterministic or expected value solution. However, this is an

29

unrealistic assumption, since it is likely that the market is going to change. In this case, the

investor must rebalance the portfolio.

Suppose Scenario 2 occurs one time period later, as described in the example in section

4.1.1. In this methodology, the deterministic solution is solved using the current time

period’s data. The initial investment portfolio is as follows:

Assets XiATT 0.234GMC 0.000USX 0.060CSCO 0.195ABX 0.511

Table 4.3 Initial Investment Decisions for Expected Value Solution

If Scenario 2 is realized one time period later, the optimal reallocation policy, given the

initial investment decisions described in table 4.3, is as follows:

Assets Yi2ATT 0.273GMC 0.000USX 0.000CSCO 0.246ABX 0.481

Table 4.4 Optimal Reallocation Decisions for Scenario 2 given Expected Value

Solution

30

Based on the above two decisions in Table 4.3 and Table 4.4, the objective function of

the model can now be calculated to have a value of 0.01686. It is apparent that this value is

lower than the value for the model with perfect information. This result occurs because with

perfect information the model optimizes according to the scenario that will actually occur,

whereas the deterministic model optimizes according to the expected value scenario which

does not actually occur. Please refer to Appendix F for the optimal decisions and objective

value calculations based on the deterministic solution.

4.1.3 Stochastic Model Solution

The stochastic solution determines the initial investment by taking into account that any

of the six scenarios are equally likely to occur. The stochastic model simultaneously

determines the initial investment strategy and the reallocation policies for all of the possible

scenarios given the initial investment. It essentially tries to find an initial investment strategy

so that if any of the scenarios occur, the reallocation strategy does not differ drastically from

the initial investment.

For example, suppose Scenario 2 occurs one time period later. Before Scenario 2 occurs,

the initial investment strategy using the stochastic model is determined by taking into

account all possible scenarios that could occur. The initial investment strategy is:

Assets Xi


Table 4.5 Initial Investment Decisions Based on Stochastic Model

31

The stochastic model has also determined the optimal reallocation strategy if Scenario 2 was

to occur based on the initial decision. This strategy is as follows:

Assets Yi2


Table 4.6 Optimal Reallocation Decisions for Scenario 2 Given Initial Investment

Decisions in Table 4.5

Based on the above two decisions generated in the stochastic model, the objective

function of the model can now be calculated to have a value of 0.014243. This solution

outperforms the deterministic model. This is because Scenario 2 was taken into

consideration in the stochastic model, whereas the deterministic model did not. Please refer

to Appendix G for the optimal investment decisions and objective value calculations using

the stochastic model.

4.1.4 Results

The following table summarizes the objective value for each scenario using each model:

ScenarioS1 S2 S3 S4 S5 S6

Expected Value Solution 0.0031462 0.0168516 0.0036580 0.0048395 0.0031464 0.0087812Stochastic Solution 0.0034707 0.00347110.0142436 0.0034721 0.0038868 0.0050054Perfect Information Solution 0.0031462 0.0134283 0.0032563 0.0033193 0.0031464 0.0035027

Table 4.7 Comparison of Objective Values for Each Model

32

From this table it is apparent that four out of the six scenarios, which are highlighted in

green, perform better using the stochastic solution in comparison to the expected value

solution. The stochastic model has a higher objective value in Scenario 1 because that

scenario represents that the current data is actually maintained one time period later. Hence,

the deterministic model is acting similar to the perfect information model in this situation.

Further analysis shows that the inputs to Scenario 5 were very similar to Scenario 1, which

explains the better performance by the deterministic model.

To further understand the benefit of the stochastic solution the percent increase from

the ideal objective value was calculated using Table 4.7.

ScenarioS1 S2 S3 S4 S5 S6 Avg

Expected Value Solution 0.00% 25.49% 12.34% 45.80% 0.00% 150.70% 39.05%Stochastic Solution 10.32% 10.32%6.07% 6.63% 17.10% 42.90% 15.55%Perfect Information Solution 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00%

Table 4.8 Percent Increase in Objective Value from Perfect Information Solution

This analysis determined that on average the stochastic solution was 15.6% greater than

the perfect information solution, whereas the expected solution was 39.1%.

4.2 VALUE OF THE STOCHASTIC SOLUTION (VSS)

The VSS is a theoretical analysis that shows the difference in the expected value of the

stochastic solution over the expected value of the deterministic solution.

To compute the expected value of the deterministic model, the deterministic model is

solved to get an initial investment strategy. The stochastic model is then solved with this

33

fixed initial investment decision. This can be done by adding the following constraint to the

stochastic model described in section 4.1:

= for all 1.. (1)Xi Di i n=

Where Di represents the initial investment strategy for Asset i in the deterministic

solution. The objective value generated from this model is the expected value of the

deterministic solution.

The expected value of the stochastic solution is the objective value generated from

running the stochastic model without any fixed initial investment decision. The expected

values for the two models are:

Expected Value Solution 0.00674Stochastic Solution 0.00559Difference (VSS) 0.00115

Table 4.9 Expected Objective Values and VSS

The VSS describes the expected gain in value that will be achieved by using the

stochastic model. On average the objective value of the stochastic solution is expected to be

0.00115 units less than the deterministic solution.

4.3 EFFECT OF THE MINIMUM RETURN CONSTRAINT

The final analysis done in the study was to determine the effect of the minimum return

constraint, R, has on the performance of the stochastic model. To do this analysis all of the

inputs into the stochastic model were fixed based on the data generated in section 3.2. The

only input that varied is the constant R. The objective function and its components were

34

computed as R was increased in increments of 10%, starting at 50%. The last return analyzed

was 123% because any return greater than this value would result in an infeasible solution.

The following figure displays the result generated:

The Effect of the Minimum Return Constraint on the Objective Function

0.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0.800

0.900

0% 20% 40% 60% 80% 100% 120% 140%Minimum Return (R)

Ob

ject

ive

Val

ue

Objective Value Initial Variance E[Difference in Returns]2

Figure 4.1 The Effect of the Minimum Return Constraint on the Objective Function

35

From this figure it is apparent that R does in fact have a significant impact on the

stochastic model (Please refer to Appendix H for the data table for Figure 4.1). When R

ranges from 50% to 89%, R has the same effect on the model. This is because the model is

able to generate the exact same optimal solution, which achieves an expected return of 89%.

As R increases past the 89% range, the value of the objective function increases rapidly. This

can be partly attributed to the variance component increasing as the concept of

diversification diminishes. The model is forced to invest more into the one or two stocks

that could potentially achieve the minimum return. As a result the portfolio is becoming less

diverse and riskier.

As displayed on the graph, the difference in returns component plays a large role in the

objective function increasing at rapid rate. This result displayed by the difference in returns

signifies the negative effect that an increased R has on the stochastic model. An increased R

diminishes the inherent value added by using a stochastic model. As R increases, the model

applies less importance to the different scenarios and the reallocation strategy when

determining its initial investment decision. Its initial investment decision becomes more

forced towards a result that will meet this expected return constraint instead of considering

the scenarios, becoming more like the deterministic model.

36

5. CONCLUSION

The purpose of thesis study was to determine a way in which to improve on Sharpe’s

Single Index Model by taking into account uncertainty in the changing market conditions.

Sharpe’s model tries to minimize the current risk of portfolio for a given level of return

based on the current time period’s data. This is not very realistic for investors who need to

determine their investment strategy today but also need to plan ahead for the future.

Investors want to ensure that as time progresses the rebalancing of their initial investment

portfolio is minimal when adapting to the changes in the market.

This improvement was achieved in this thesis by converting Sharpe’s model into a two

stage stochastic program. This new model determines an initial investment strategy that takes

into consideration a variety of scenarios that could happen one time period later when it is

time to rebalance the portfolio. The model determines how to invest today to best balance

the minimization of both the current risk and reallocation of the portfolio one time period

later.

A series of analyses was conducted to determine if this benefit of planning for the future

was captured by the stochastic model. By comparing the objective values generated by each

model, it is apparent that in majority of the scenarios the stochastic model will make a better

initial investment strategy. In this study specifically, two thirds of the scenarios proved to

produce better results. Not only does the stochastic program perform better in more

scenarios, but the difference in magnitude by which it outperforms the deterministic model

is important. This can be seen by its significant difference in deviation from the ideal perfect

information solution. On average the stochastic solution was 15.6% worse than the perfect

37

information solution, whereas the expected solution was 39.1%. The VSS computation

further advances the conclusion that the stochastic model outperforms the deterministic

model. It indicates that the expected objective value of the stochastic solution is expected to

be a 17% improvement over the deterministic solution.

Although the stochastic model developed in this study proves to offer inherent benefits,

it is also important to consider the limitations of the model. For one, the limitations of

Sharpe’s model, except the single point in time analysis, still exists within this model. An

additional limitation that was identified for the stochastic model, was the effect that the

minimum return constraint, R, had on the model. Analysis showed that as R increased, the

model becomes more similar to a deterministic model. This is because the tight constraints

force the model to make an initial investment strategy with minimal consideration of what

will occur one time period later.

The objective of producing a new robust model was achieved. The analysis also showed

how this model would be beneficial and more realistic for investors to use. Although the

model does not provide the optimal solution for today, it provides a better long-term

solution, which is what most investors are looking to achieve. This model produces an initial

investment strategy that reduces the chances of drastic rebalancing strategies that could

occur due to changes in the market. Rebalancing a portfolio can be costly and also difficult

to do for some investors. Through this study the practical improvements of incorporating

uncertainty into Sharpe’s model has been proven. It is felt and hoped that further research

and testing will be done to produce further evidence of the benefits and limitations of the

stochastic model.

38

6. FURTHER RESEARCH

The analyses conducted in this study have just touched the surface in understanding the

benefits and limitations of the stochastic model developed. To further validate the benefits

of the stochastic solution more scenarios of data could be generated and optimized. The

result of the optimizations could be used to test the consistency of the model. Furthermore

rigorous statistical studies could be used to prove that the stochastic model is better than the

deterministic model.

Another interesting analysis that could be conducted would be to test the models using

actual historical data. Using historical data, an optimal portfolio can be generated for both

the deterministic and stochastic model. The portfolios can be then be tracked to see how

they actually perform. It can also be determined which portfolio would have generated a

better initial investment strategy when it is time to rebalance the portfolio. To conduct this

analysis, inputs derived from the historical data are critical to the study. An in depth analysis

would need to be conducted to determine if the inputs into the model appropriately

represent the historical market conditions.

In addition, further research could be conducted to determine the limitations of the

stochastic model. For example, the efficiency and accuracy of the model can be tested as the

number of assets in the portfolio increases. It is expected that as the number of asset

increases, the efficiency of model will decrease. However, the model may produce better

results as it has a larger selection of assets to choose from to determine its initial investment

strategy. Studies would have to be conducted to prove these hypotheses. Through this

39

further research a better understanding of the model’s benefits, limitations, and sensitivities

can be determined.

40

7. REFERENCES

Berardi, Andrea, Stefano Corradin, and Cristina Sommacampagna. "Estimating Value at Risk with the Kalman Filter." International Centre for Economic Research (2002). 12 Nov. 2007. < http://www.icer.it/workshop/Berardi_Corradin_Sommacampagna.pdf>

Birge, John R., and Francois Louveaux. Introduction to Stochastic Programming. Springer, 1997.

Frankfurter, George M., and Christopher G. Lamoreux. "Insignificant Betas and the Efficacy of the Sharpe Diagonal Model for Portfolio Selection." Blackwell Synergy (1989). 13 Nov. 2007. <http://www.blackwell-synergy.com/doi/pdf/10.1111/j.1540-5915.1990.tb01254.x?cookieSet=1>.

Philpott, Andy, and Alexander Shapiro. "A Tutorial on Stochastic Programming" Stochastic Programming Community. 09 Apr 2007. Community on Stochastic Programming. 27 Feb 2008. <http://stoprog.org/index.html?SPTutorial/SPTutorial.html>

Scharge, Linus. Optimization Modeling with Lingo. Chicago: Lindo Systems Inc, 2003.

Sharpe, William F. "A Simplified Model fo Portfolio Analysis." Management Science 9.2 (1963): 277-293. JSTOR. University of Toronto. 10 Nov. 2007. < http://www.jstor.org>

Trandafir, Mircea. “Chapter 10 Index Models” Department of Economics. 09 Sep 2007. University of Maryland. 26 Feb 2008. < http://www.econ.umd.edu/~trandafi/econ435>.

Wallingford, Buckner A. "A Survey and Comparison of Portfolio Selection Models." The Journal of Financial and Quantitative Analysis 2.2 (1967): 85-106. JSTOR. University of Toronto. 16 Jan. 2008. < http://www.jstor.org>

Welch, Ivo. Introduction to Corporate Finance Textbook. Providence: Brown University, 2006.

41

APPENDIX A – OPL CODE FOR SHARPE’S SINGE INDEX MODEL

42

/********************************************* * OPL 5.5 Model * Author: Ajay Shivdasani * Creation Date: 18/02/2008 at 11:34 AM *********************************************/

{string} Assets = ...; float Beta0[Assets] = ...; float Alpha0[Assets] = ...; float STDEV [Assets] = ...; float m0 = ...; float s0 = ...; float Return= ...; dvar float+ Xi[Assets]; dvar float+ Z; //Minimize Variance minimize (Z)^2*s0^2 + sum(a in Assets) (Xi[a]^2*STDEV[a]^2); subject to { //Z Constraint sum(a in Assets)Xi[a]*Beta0[a]==Z; //Budget Constraint ctInitialInvestment: sum ( a in Assets)Xi[a]==1; // Return Requierement ctInitialReturn: sum(a in Assets)Xi[a]*(Alpha0[a]+Beta0[a]*m0) >= Return; //initial investment

must achieve an expected return }

43

/********************************************* * OPL 5.5 Data * Author: Ajay Shivdasani * Creation Date: 19/02/2008 at 11:36 AM *********************************************/ SheetConnection sheet("InputData v0.07.xls"); Assets from SheetRead(sheet,"Assets!Assets"); Beta0 from SheetRead(sheet,"Assets!Beta0"); Alpha0 from SheetRead(sheet,"Assets!Alpha0"); STDEV from SheetRead(sheet,"Assets!STDEV"); m0 from SheetRead(sheet,"Assets!mo"); s0 from SheetRead(sheet,"Assets!so"); Return from SheetRead(sheet,"Assets!Return");

44

APPENDIX B – OPL CODE FOR THE STOCHASTIC MODEL

45

/********************************************* * OPL 5.5 Model * Author: Ajay Shivdasani * Creation Date: 05/02/2008 at 9:18 PM *********************************************/ {string} Assets = ...; {string} Betas = ...; {string} Alphas = ...; {string} Markets = ...;

float Beta0[Assets] = ...; float Alpha0[Assets] = ...; float STDEV [Assets] = ...; float m0 = ...; float s0 = ...; float Return= ...; float BetaData[Assets][Betas] = ...; float AlphaData[Assets][Alphas]=...; float MarketData[Markets]=...; float Prob[Betas]=...; dvar float+ Xi[Assets]; dvar float+ Yi[Assets][Betas]; dvar float+ Ui[Assets]; dvar float+ Vi[Assets][Betas]; dvar float+ Z; minimize //First Stage - Minimize Initial Variance (Z)^2*s0^2 + sum(a in Assets) (Xi[a]^2*STDEV[a]^2) // Second Stage minimize difference in return + sum(b in Betas) Prob[b] * sum(a in Assets)(Ui[a]-Vi[a][b])^2; subject to { ctPortfolioBeta: //Define Value of Z, which represent The Expected Portfolio Beta sum(a in Assets)Xi[a]*Beta0[a]==Z; //in first stage decision the complete portfolio must be invested ctInitialInvestment: sum ( a in Assets)Xi[a]==1; //for each scenario decision complete portfolio must be invested ctScenarioInvestment: forall (b in Betas) sum (a in Assets)Yi[a][b]==1;

46

//initial investment must achieve an expected return ctInitialReturn: sum(a in Assets)Xi[a]*(Alpha0[a]+Beta0[a]*m0) >= Return; //return for each asset based on initial investment ctZi: forall (a in Assets) Xi[a]*(Alpha0[a]+Beta0[a]*m0) == Ui[a]; //return for each asset based on scneario investment ctYi: forall (a in Assets) forall (b in Betas) Yi[a][b]*(AlphaData[a][b]+BetaData[a][b]*MarketData[b])==Vi[a][b]; }

47

/********************************************* * OPL 5.5 Data * Author: Ajay Shivdasani * Creation Date: 05/02/2008 at 9:18 PM *********************************************/ SheetConnection sheet("InputData v0.07.xls"); Assets from SheetRead(sheet,"Assets!Assets"); Betas from SheetRead(sheet,"Beta!Betas"); Alphas from SheetRead(sheet,"Alpha!Alphas"); Markets from SheetRead(sheet,"Market!Markets");

Beta0 from SheetRead(sheet,"Assets!Beta0"); Alpha0 from SheetRead(sheet,"Assets!Alpha0"); STDEV from SheetRead(sheet,"Assets!STDEV"); m0 from SheetRead(sheet,"Assets!mo"); s0 from SheetRead(sheet,"Assets!so"); Return from SheetRead(sheet,"Assets!Return"); BetaData from SheetRead(sheet,"Beta!BetaData"); AlphaData from SheetRead(sheet,"Alpha!AlphaData"); MarketData from SheetRead(sheet,"Market!MarketData"); Prob from SheetRead(sheet,"Probabilities!Prob");

48

APPENDIX C – PERCENT CHANGE TABLES

49

Percent Change of β from Current DataScenarios

Asset S1 S2 S3 S4 S5 S6ATT 0.00% -11.67% 16.85% -11.00% 55.63% -2.88%GMC 0.00% -14.40% 48.56% -5.51% -26.29% -25.28%USX 0.00% -37.25% -16.09% -57.86% 5.03% -87.62%CSCO 0.00% -25.05% -10.12% -10.28% 18.20% -23.84%ABX 0.00% 6.47% 4.40% 24.54% -62.95% 24.07%

Percent Change of α from Current DataScenarios

Asset S1 S2 S3 S4 S5 S6ATT 0.00% -12.80% 8.02% -13.37% 37.77% -16.67%GMC 0.00% 90.20% -29.18% -66.58% -73.60% 31.04%USX 0.00% 2.79% 40.86% -13.15% 30.43% 48.37%CSCO 0.00% -14.28% -100.70% -17.50% -25.19% -64.69%ABX 0.00% -10.23% -7.18% 41.22% -2.92% 0.79%

Percent Change of Mo from Current DataAsset S1 S2 S3 S4 S5 S6

Mo 0.00% -39.25% 33.60% -6.57% 9.68% 54.21%

50

APPENDIX D – INPUT TABLES FOR EACH SCENARIO

51

Inputs for β for each scenarioScenarios

Asset S1 S2 S3 S4 S5 S6ATT 0.49 0.43 0.57 0.44 0.76 0.48GMC -0.21 -0.18 -0.31 -0.20 -0.15 -0.15USX 1.52 0.96 1.28 0.64 1.60 0.19CSCO 0.67 0.50 0.60 0.60 0.79 0.51ABX -0.16 -0.17 -0.16 -0.20 -0.06 -0.20

Inputs for α for each scenarioScenarios

Asset S1 S2 S3 S4 S5 S6ATT 0.36 0.32 0.39 0.31 0.50 0.30GMC 0.76 1.45 0.54 0.26 0.20 1.00USX -0.58 -0.60 -0.82 -0.50 -0.76 -0.86CSCO 0.15 0.13 0.00 0.13 0.12 0.05ABX 0.99 0.89 0.92 1.39 0.96 0.99

Inputs for Mo for each scenarioAsset S1 S2 S3 S4 S5 S6

Mo 1.19 0.72 1.59 1.11 1.31 1.84

52

APPENDIX E – OPTIMAL INVESTMENT DECISIONS AND OBJECTIVE VALUE CALCULATIONS USING PERFECT

INFORMATION

Perfect Inf

Perfect Inf

Perfect Inf

ormation That Scenario 1 Will Occur: Perfect Information That Scenario 2 Will Occur:

Assets Xi Yi1 Portfolio Variance: 0.00314615 Assets Xi Yi2 Portfolio Variance: 0.00336962ATT 0.234 0.234 Difference in Returns^2 0.000 ATT 0.412 0.543 Difference in Returns^2 0.01006 GMC 0.000 0.000 Objective Value 0.0031462 GMC 0.000 0.000 Objective Value 0.0134283USX 0.060 0.060 USX 0.050 0.000CSCO 0.195 0.195 CSCO 0.052 0.000ABX 0.511 0.511 ABX 0.487 0.457





53

54

APPENDIX F – OPTIMAL INVESTMENT DECISIONS AND OBJECTIVE VALUE CALCULATIONS USING DETERMINISTIC

MODEL

55

Initial Investment Decisions & Reallocation Decisions for Each ScenarioAssets Xi Yi1 Yi2 Yi3 Yi4 Yi5 Yi6ATT 0.234 0.234 0.273 0.164 0.265 0.148 0.185GMC 0.000 0.000 0.000 0.000 0.000 0.172 0.000USX 0.060 0.060 0.000 0.054 0.174 0.056 0.000CSCO 0.195 0.195 0.246 0.182 0.220 0.161 0.182ABX 0.511 0.511 0.481 0.600 0.342 0.463 0.632

Initial Portfolio Variance

Yi1 Yi2 Yi3 Yi4 Yi5 Yi60.00315 0.00315 0.00315 0.00315 0.00315 0.00315

Difference in ReturnsYi1 Yi2 Yi3 Yi4 Yi5 Yi60.00000 0.01371 0.00051 0.00169 0.00000 0.00564

Objective ValueYi1 Yi2 Yi3 Yi4 Yi5 Yi60.00315 0.01685 0.00366 0.00484 0.00315 0.00878

56

APPENDIX G – OPTIMAL INVESTMENT DECISIONS AND OBJECTIVE VALUE CALCULATIONS USING STOCHASTIC

MODEL

57

Initial Investment Decisions & Reallocation Decisions for Each ScenarioAssets Xi Yi1 Yi2 Yi3 Yi4 Yi5 Yi6ATT 0.448 0.448 0.586 0.325 0.524 0.283 0.361GMC 0.000 0.000 0.000 0.022 0.000 0.221 0.002USX 0.032 0.032 0.000 0.032 0.096 0.029 0.000CSCO 0.068 0.068 0.000 0.068 0.075 0.056 0.066ABX 0.453 0.453 0.414 0.553 0.305 0.410 0.571

Initial Portfolio VarianceYi1 Yi2 Yi3 Yi4 Yi5 Yi6

0.00347 0.00347 0.00347 0.00347 0.00347 0.00347

Difference in ReturnsYi1 Yi2 Yi3 Yi4 Yi5 Yi6

0.000 0.011 0.000 0.000 0.000 0.002

Objective ValueYi1 Yi2 Yi3 Yi4 Yi5 Yi6

0.0034707 0.0142436 0.0034721 0.0038868 0.0034711 0.0050054

58

APPENDIX H – RESULT OF MINIMUM CONSTRAINT ANALYSIS

59

Effect of R on the Objective Function

Return Objective Value Initial VarianceE[Difference in

Returns]2

0.5 0.003 0.002 0.001000.6 0.003 0.002 0.001000.7 0.003 0.002 0.001000.8 0.003 0.002 0.001000.9 0.007 0.004 0.002641 0.045 0.017 0.02802

1.1 0.228 0.039 0.188721.2 0.586 0.076 0.510071.23 0.764 0.091 0.67254

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