Financial Portfolio Optimization Through a Robust Beta Analysis

Transcription

1 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

2 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.

3 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.

4 II TABLE OF CONTENTS ACKNOWLEDGEMENTS... I TABLE OF CONTENTS...II LIST OF FIGURES... IV LIST OF TABLES...V 1. INTRODUCTION Motivation Objective Research approach LITERATURE REVIEW PORTFOLIO THEORY Markowitz s full covariance model Sharpe s Single Index Model Market Beta TWO STAGE STOCHASTIC PROGRAMMING METHODOLOGY OPTIMIZATION MODELING Programming Sharpe s Single Index Model into OPL Formulate a Two Stage Stochastic Model Program Two Stage Stochastic Formulation into OPL DATA GENERATION Current Time Period Data Scenario Data for One Time Later RESULTS & ANALYSIS COMPARATIVE ANALYSIS Perfect Information Solution Deterministic (Expected Value) Solution Stochastic Model Solution Results VALUE OF THE STOCHASTIC SOLUTION (VSS) EFFECT OF THE MINIMUM RETURN CONSTRAINT CONCLUSION FURTHER RESEARCH 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

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

6 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

7 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 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 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

8 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

9 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

10 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.

11 4 2.1 PORTFOLIO THEORY 2. LITERATURE REVIEW 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 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).

12 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: ER ( ) = XE i i (1) i 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.

13 6 Thus, the variance of the portfolio can be modeled by V( R) XXC i j ij (2) = i j 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 Z = E( R) V( R) = XE i i XXC i j ij (3) i i j 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.

14 7 Thus, the final objective function can be modeled as: Maximize λ XE i i i j ij i i j XXC (4) 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: i Xi = 1 (5) Xi 0 for all i (6) 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

15 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 (N 2 +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 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

16 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: Ri = Ai+ BI i + Ci (1) 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 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.

17 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: Decision Variables: Xi = Proportion of portfolio invested in stock i Parameters: M = The market factor m0 = E(M) S i = Var (M) ei= Random movement to specific stock i S = Var ( ei) bi = the market beta for stock i R = A desired rate of return to be achieved α i = the initial alpha value for stock i n = number of assets in portfolio Minimize 2 n ( ) = 0 + i (2.1) V R Z S Xi S ST.. n i= 1 n i= 1 i= 1 ( Xibi) = Z (2.2) Xi = 1 (2.3) n ER ( ) Xi( α i + bm i 0) R = i= 1 (2.4) 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.

18 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 In this model Ei is substituted by: ( i + bm i 0) (3) α. 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 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)

19 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.

20 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 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 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.

21 14 Figure 1.1 Security Characteristic Line (Trandafir, Chapter 10)

22 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 twostage (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

23 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 T c x+ Eξ[ Q( x, ξ )] (1.1) Subject To Ax = b (1.2) x 0 (1.3) 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: Eξ[ Q( x, ξ )] (2) 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.

24 17 Q(x,ξ) is the optimal solution of the following nonlinear program: Minimize T q y Subject To (3.1) T( ξ) x+ w( ξ) y = h( ξ) (3.2) y 0 (3.3) 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: N Eξ[ Q( x, ξ )] = πiq( x, ξi) (4) i= 1 Where number of scenarios. π i represents the expected probability that scenario i will occur and N is the

25 OPTIMIZATION MODELING 3. METHODOLOGY 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 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

26 19 stochastic component was implemented using the aid of the farmer s problem provided in Birge s textbook. The final formulation is as follows:

27 Decision Variables: Xi 20 = Proportion of portfolio initially invested in stock i Yij = The porportion of portfolio that is invested in each stock i one time period later for each scenario j Parameters: m = number of scenarios n = number of assets in portfolio M = The Market Factor m0 = E(M) mj = E(M) in Scenario j S i = Var (M) ei= Random movement to stock i S = Var ( ei) bi = the initial market beta for stock i bij = the market beta for stock i under scenario j after one time period has elapsed α i = the initial alpha value for stock i α ij = the alpha value for stock i under scenario j after one time period has elapsed π j = probability that scenario j occurs. π j = 1 j= 1 R = A desired rate of return to be achieved m Minimize i= 1 i= 1 n m n i + j i ij ZS XiS π ( ( U V ) ) (1.1) ST.. n n n i= 1 j= 1 i= 1 Xi = 1 (1.2) Wij = 1 for all j = 1.. m (1.3) ( i + i 0) (1.4) i= 1 ( i + i 0) for all = 1.. (1.5) Yij( α Xi α b m R Xi α b m = Ui i n n i= 1 ij + bijmj ) = Vij for all i=1.. n for j=1.. m (1.6) ( Xibi) = Z (1.7)

28 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: n i ZS i= 1 XiS (2) 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: Ui V ij (3) 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

29 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 ( Ui Vij) ( 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: m π n 2 j( ( Ui Vij) ) (5) j= 1 i= 1 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.

30 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 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:

31 24 Market Data: Mo So % Asset Data: Asset Si Alpha (α i ) Beta (β i) ATT GMC USX CSCO ABX Table 3.1 Input Data for Current Time Period 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 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.

32 25 An example of a percent change table for the market beta is described below: Beta Scenarios Asset 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 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 Scenarios Asset 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.

33 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.

34 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 GMC USX CSCO ABX Table 4.1 Initial Investment Decisions Based on Perfect Information that Scenario 2 Will Occur

35 28 Based on this initial investment, the optimal reallocation strategy once Scenario 2 has been realized is: Assets Yi2 ATT GMC USX CSCO ABX 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 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 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

36 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 In this methodology, the deterministic solution is solved using the current time period s data. The initial investment portfolio is as follows: Assets Xi ATT GMC USX CSCO ABX 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 Yi2 ATT GMC USX CSCO ABX Table 4.4 Optimal Reallocation Decisions for Scenario 2 given Expected Value Solution

37 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 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 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 ATT GMC USX CSCO ABX Table 4.5 Initial Investment Decisions Based on Stochastic Model

38 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 ATT GMC USX CSCO ABX 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 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 Results The following table summarizes the objective value for each scenario using each model: Scenario S1 S2 S3 S4 S5 S6 Expected Value Solution Stochastic Solution Perfect Information Solution Table 4.7 Comparison of Objective Values for Each Model

39 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. Scenario S1 S2 S3 S4 S5 S6 Avg Expected Value Solution 0.00% 25.49% 12.34% 45.80% 0.00% % 39.05% Stochastic Solution 10.32% 6.07% 6.63% 17.10% 10.32% 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

40 33 fixed initial investment decision. This can be done by adding the following constraint to the stochastic model described in section 4.1: Xi = Di for all i = 1.. n (1) 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 Stochastic Solution Difference (VSS) 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 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

41 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 Objective Value % 20% 40% 60% 80% 100% 120% 140% Minimum Return (R) Objective Value Initial Variance E[Difference in Returns]2 Figure 4.1 The Effect of the Minimum Return Constraint on the Objective Function

42 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.

43 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