FINANCIAL OPERATIONS RESEARCH: Mean Absolute Deviation And Portfolio Indexing

Transcription

1 [1] FINANCIAL OPERATIONS RESEARCH: Mean Absolute Deviation And Portfolio Indexing David Galica Tony Rauchberger Luca Balestrieri A thesis submitted in partial fulfillment of the requirements for the degree of BACHELOR OF APPLIED SCIENCE Supervisor: R. Kwon Department of Mechanical and Industrial Engineering University of Toronto March 20, 2008

2 ABSTRACT The purpose of this paper is to research and gain insight needed to develop a robust and computationally easy model to select a representative stock market index, while maintaining the same performance characteristics of a larger market. To do this the mean absolute deviation model by Konno and Yamazaki [2] was altered to account for integer selection constraints and the necessity for tracking returns. Tests were run to observe the behavior of the model for the purposes of this application. Using data from the S&P500 index for the year 2005, the model was found to be adequate for tracking, with or without normality of the composite return distributions, and comparable to the mean variance model with some time-saving benefits at larger index sizes. However, the degeneracy of basic variables at index sizes over thirty makes the model problematic for use in practice. 1

3 ACKNOWLEDGEMENTS Exceptional gratitude to our thesis supervisor Professor R. Kwon for his guidance and expertise In recognition of Professor B. Balcioglu and Professor V. Makis for their invaluable assistance. Special thanks to the Rotman School of Management, Business Information Centre Resource Laboratory 2

4 TABLE OF CONTENTS I. LIST OF SYMBOLS... 5 II. LIST OF FIGURES... 6 III. LIST OF TABLES INTRODUCTION CONTEXT PROBLEM PURPOSE BACKGROUND MEAN VARIANCE MEAN ABSOLUTE DEVIATION ISSUES EVALUATION CRITERIA METHODOLOGY EXPLORING THE EFFICIENT FRONTIER Testing Results Evaluation REVISING THE MODEL Models Testing Results Evaluation THE TWO-STAGE MODEL (CONSTRAINING THE MODEL FOR INDEX SIZE) Testing Results Evaluation EXPLORING THE NORMALITY ASSUMPTION Testing Results Evaluation THE EFFECTS OF INDEX SIZE ON THE REVISED MODEL Testing Figure Probability Plot for Periods 41 to Results Evaluation TRACKING FUTURE PERFORMANCE Testing Results Evaluation CONCLUSIONS FURTHER CONSIDERATIONS REFERENCES APPENDIX A: MEAN VARIANCE MODEL APPENDIX B: ORIGINAL MAD MODEL APPENDIX C: OPL MODEL FOR S&P 500 DATA STREAM APPENPDIX D: SENSITIVITY FOR CONSTRAINT

5 APPENDIX E.1: T-TESTS FOR RISK FOR THE ORIGINAL MAD, BIG R AND RT MODELS.. 80 APPENDIX E.2: T-TESTS FOR RETURN FOR THE ORIGINAL MAD, BIG R AND RT MODELS APPENDIX F: INDEX MODEL APPENDIX G.1: TRACKING ERROR APPENDIX G.2: INTERNAL TRACKING ERROR APPENDIX G.3: TRACKING ERROR FOR FIRST 50 FUTURE PERIODS APPENDIX G.4: TRACKING ERROR FOR LAST 50 PERIODS OF APPENDIX H: ADDITIONAL CONSTRAINT APPENDIX I: MEETING MINUTES APPENDIX J: PROJECT SCHEDULE

6 i. LIST OF SYMBOLS Decision Variables: x j = the proportion of total funds invested in stock j z j = binary variable to select investment in stock j (invest = 1) Cost Variables: r jt = the return of asset j in period t r j = the average return of asset j r t = the average return of all j assets in the set at time t R = grand average return of the larger portfolio a jt = r jt - R or a jt = r jt - r j or a jt = r jt r t M = the total amount of money available for investment ρ = the specified return of the index or portfolio S = the chosen size of the portfolio s + = positive return slack from ideal return s - = negative return slack from ideal return σ ij = covariance of rates of return between stock i and stock j w(x) = the absolute deviation of a given stock P i = the return of the index portfolio at time t M i = the return of the market at time t Indices: 5

7 ii. LIST OF FIGURES Figure Normality Graph for Asset ADI Figure Normality Graph for Asset GLW Figure The efficient frontier for S&P 500 data for the year 2005 Figure The straight line in the figure above is the Capital Market Line Figure Relationship between Absolute Deviation and Standard Deviation Figure Portfolio Selection Size vs. Objective Value Figure Portfolio Selection Size vs. Standard Deviation of Error Figure Aggregation of samples at time t Figure a Individual Absolute Deviation Figure b Aggregate Absolute Deviation Figure Graph of Normality vs. Risk-Return Difference from the Mean-Variance Model Figure Graph of Normality vs. the MAD Model Solution Objective and Tracking Error Figure P -Value vs. Period Range Figure Probability Plot for Periods 41 to 140 Figure Normality Graph for Period Figure Time vs. Index Size Figure Objective/Tracking Error vs. Index Size for the Mean Variance Model Figure Objective/Tracking Error vs. Index Size for the Mean Absolute Deviation Model Figure Objective/Tracking Error vs. Index Size for the Big-R Model 6

8 iii. LIST OF TABLES Table Twenty Random Pools Containing Twenty Assets Each (S&P500 Data) Table Portfolio Allocations of Each Model for Random Pool 5 Table Objective Values for Each Model (In Terms of Mean Absolute Deviation) and Proportion Difference from Mean variance Table Tracking Errors for Each Model and Proportion Difference of Objectives From Mean-Variance Table Ticker Symbols of 20 Random Samples Table Summary of Normality Test Data Table Ticker Names of the 200 Random Stocks Chosen from S&P 500, Year 2005 Table The Portfolio Selected By The Unconstrained Big R Model From The 200 Random Stocks 7

9 1. INTRODUCTION The following explains the purpose of this paper and insight into why it is an important issue to explore. 1.1 Context Investing in securities can be both a highly risky and highly lucrative endeavor. A generally well known idea is that of positive correlation between risk and return, in that there is increased risk of potential loss associated with increased potential return and vice versa. When we think of securities, there are three classes which are most easily distinguishable, each associated with a general level of both riskiness and potential return. The first is Cash and Cash Equivalents, which refers to the investments made into things like standard bank accounts and government issued savings bonds. Although their long term returns are not as high as other classes of securities, they are for the most part, essentially risk free. The second class is Fixed Income Securities, which include things like guaranteed investment certificates (GIC s), company issued bonds and preferred shares. In comparison to their Cash and Cash Equivalent counterparts, they are higher in risk, but also have the possibility of greater return. The third and final class is Equity, which includes things like common shares and derivatives. This class does carry with it the most risk and volatility, but carries with it also the possibility of highest return to investors [3]. This is the tradeoff associated with becoming partial owners of publicly traded companies by purchasing their shares. Common shares in particular, have an immense ability to achieve significant returns over long periods of time if healthy companies are chosen to invest in. The problem most often faced however is how to 8

10 choose these healthy companies to obtain these returns and achieve long term financial success. The most basic idea associated with investing in stocks, is that of diversifying the allocation of funds to ensure that the risk of significant financial loss is minimized. There are a vast amount of variables that affect the health of a company, which influence the intrinsic interest of investors of that company, which in turn influences the fluctuations of the share price for that company. When building a portfolio, there are generally two approaches that one can take. The first is active fund management, which seems to be the most intuitive when thinking of investing. The idea is that one wishes to select a portfolio of assets that as a whole, outperforms a given market, or some other larger portfolio with admirable performance. Essentially, a portfolio, usually of larger size, is chosen based on the belief that it will perform better than some benchmark market and hence give returns that are better than that benchmark. The second approach is passive fund management, which in essence is indexing. A portfolio is selected so that it can achieve performance which is on par with a given market or larger portfolio, but it contains only a subset of assets of the larger portfolio. Therefore the subset of assets is an adequate representation of the larger portfolio, but because it is smaller in size, it avoids the excess transaction costs of a portfolio with similar performance, that is larger in size [4]. Transaction costs can be somewhat of a nuisance because in order to trade an individual asset, an account has to be setup in the name of that asset which brings about a transaction cost. Passive funds are quite popular, because although they tend not to be as large as active funds, they are successful in matching the market, while active funds in 9

11 many cases fail to even do that [5]. The proof of this statement is a problem all unto itself and is out of the scope of this assignment. When generating portfolios using historical data as an aid and forecasting tool as to future trends in the market, mathematical models are an excellent tool to accomplish this task. One of the most well known models applied to portfolio allocation is the Mean Variance model, created by H.V. Markowitz (Markowitz, Harry. Portfolio Selection The Journal of Finance, Vol. 7, No. 1. March 1952, pp.77-91). The idea behind the model was to create a portfolio based on historical data that minimized risk, which was measured in terms of the mean variance of the generated portfolio, subject to achieving a minimum return. Because mean variance is a measure taking into account how pairs of assets perform relative to one another, it is a very accurate measure of diversification, and hence risk. Because of the complexity of the model, the time required to solve the model for a large scale problem (when the number of assets in the selection pool is greater than 500) is a significant deterrent in its widespread use [2]. As an approximation, a model was created by Konno and Yamazaki, dubbed the MAD Model. The general idea of the model was the same as that of the mean variance (trying to minimize risk, subject to achieving a minimum return), but risk was measured in terms of mean absolute deviation. The models are stated to generate portfolio allocations that are roughly equivalent in terms of risk, based on the assumption that the assets in the market being followed consist of returns that are multivariate normally distributed. 10

12 1.2 Problem Because assuming normality of the returns of each asset in a benchmark market is relatively absurd, as the returns of the market deviate from normality, the MAD will generate portfolio allocations that differ in terms of risk from that of the mean variance. Another issue surrounding the use of the MAD model is that its objective and constraints are tailored more towards the idea of active fund management in that it tries to achieve as high of a rate of return as possible, while minimizing the inherent risk of the generated portfolio, hence when considering the idea of passive fund management and indexing, the standard MAD model is not completely sufficient. A crucial objective of first time investors that do not have a vast amount of initial funds, is how to make significant long term returns, while at the same time avoiding excess risk, by using their limited funds as effectively as possible. Ideally, one would like to diversify their portfolio as much as possible to hedge against risk but minimize the number of assets in that portfolio to avoid transaction costs and hence maximize the amount of funds actively working for them in the market. The literature on the MAD model does not consider the effect the amount of basic variables (i.e. the number of assets chosen to invest in) has on the objective of risk directly. 1.3 Purpose The purpose of this paper is to research and gain insight needed to revise the MAD into a robust and computationally easy model to select a representative index, while maintaining the same performance characteristics of the larger market, using Modern 11

13 Portfolio Theory techniques. A final point will be to analyze the sensitivity of diversification, as measured in terms of risk, to the amount of assets in an index. 12

14 2 BACKGROUND The following will provide an understanding of mean variance and mean absolute deviation in the context of money markets and how this theory relates to indexing. 2.1 Mean Variance The field of operations research has made significant strides in the field of modern portfolio theory, most notably from the work of H.V. Markowitz who applied a mean variance analysis and mathematical programming solution to the problem of portfolio selection. It is important to fully understand the theory which supports Markowitz work on the mean variance model. It requires stock prices for a given period of time that can be found using a historical database. From these prices, a variable symbolizing the rate of return (R j ) per period is easily calculated for each asset j, where j=1,, n. A potential shareholder has varying levels of total funds available or allotted to investing which is denoted by M O. The decision variable x j, is essentially the amount of available money that an investor should devote to asset j [2]. The mathematical model developed utilizes expected return per period of all the stocks that can potentially be chosen for the optimal portfolio. The equation r( x n n 1,..., xn ) = E R j x j = E[ R j ] j= 1 j= 1 computes expected return for each period, which under ideal circumstances, a larger value is favored while simultaneously maintaining a low level of risk [2]. Risk is subjectively interpreted as the level of potential danger of loss present within any type of occurrence. In financial terms, the term is associated with the level of danger or potential x j 13

15 loss that can be incurred caused by a negative rate of return in an individual s investment. It has already been mentioned that Markowitz s model measures the level of risk in terms of mean variance and its objective is to diversify an investor s portfolio allocation. This is accomplished by comparing interaction and relative performance occurring between all pairs of assets through the development of a covariance matrix. This is achieved by Which is equivalent to σ ( x 1,..., xn ) n n = E R j x j E R j x j= 1 j= 1 j 2 n n ( x) = σ σ x x, i= 1 j= 1 Where σ ij is the covariance between the rates of return of assets i and j. The objective measures risk and is formulated into a parametric quadratic program given in Appendix A [2]. The model is minimizing the amount of proportionally weighted covariance between all possible pairs of assets. Intuitively, this is equivalent to minimizing the cross correlation of the generated portfolio. Cross correlation is in essence the degree to which a pair or group of assets fluctuates with respect to one another. The greater the similarity of the performance between the assets, the greater the cross correlation between these assets [6]. In order for the results to be applicable, there are two assumptions in which must be satisfied. The first involves having a multivariate normally distributed periodical rate of return of each of the assets in the pool from which the portfolio is selected and secondly, an investor prefers a smaller valued standard deviation which is safer in terms of risk but will concurrently reduce the potential for larger gains. ij i j 14

16 2.2 Mean Absolute Deviation The Mean Absolute Deviation (MAD) portfolio optimization model, as proposed by Konno and Yamazaki [2] is a linear approximation that generates relatively accurate approximations. The absolute deviation risk function is given by n n w( x) = E R j x j E R j= 1 j= 1 j x j which replaces the standard deviation function [2]. The model uses the proof that σ = 1.25 * MAD, to prove that minimizing the standard deviation (σ), which is viewed as risk, is equivalent to minimizing the mean absolute deviation. Before continuing to describe the model, it is essential to explain several key variables that play an important role for the optimization problem. The random variable R j can be replaced with the actual historical rate of return and is denoted r jt. The model supposes that the estimated rate of returns for R j can be calculated as the average for a specific asset over the total time period which is given by r j = E T [ R j ] = t = 1 r jt / T. [2] The above formula is used in conjunction with the aforementioned formula w(x) to derive the relationship of E n R x E j j j= 1 j= 1 n 1 R j x j = T T n t = 1 l = 1 ( r jt r j ) x j, which is applied to the minimization model as shown in Appendix B [2]. It is evident that the model is minimizing the amount of mean absolute deviation over all possible assets. The proposed model has a number of advantages over Markowitz s mean 15

17 variance model. Firstly, it is no longer necessary to compute a covariance matrix which consequently allows the MAD model to become extremely efficient when updating the data because it does not require computing a new covariance matrix each time new data is available. An important facet of the model is its computational efficiency largely due to the linear nature of the problem because the amount of constraints does not change regardless of the amount of assets included [2]. It is important to mention that by introducing binary constraints limiting the number of assets to include in the generated portfolio, the model can produce an integer solution making it applicable to a realistic stock trading and indexing problem [2]. The MAD is fundamentally a linear approximation of the mean variance model and hence a very useful tool in generating solutions that are considerably accurate but can be solved much more quickly, given the right conditions. 2.3 Issues The preceding two models contain issues that prevent them from properly indexing a market portfolio. Since developing the mean variance model, it was found that significant strides have been taken in dealing with these problems. Most though, are associated with trying to derive a means by which the problem, as proposed by Markowitz, can be solved in a more efficient manner. This is because quadratic and nonlinear programming which was what the Markowitz model used is very time consuming, especially when the problem involves several variables and constraints. Therefore, trying to generate a portfolio of substantial proportions could not be done very quickly using the Markowitz model. Each time the data is run, a new covariance matrix, which can become relatively large as the number of assets in the market portfolio increases, must be 16

18 re-computed and is not practical for investors [2]. The computational burden to solve the model will become too impeding to allow the realization of an optimal solution in acceptable time. It is unlikely to solve a problem containing over 500 assets because the amount of sub-primal nodes that the model must visit and test will be too large for efficient optimization [2]. For the purpose of our research and future applications, the model is required to run on a large market portfolio of considerable size, thus proving the mean variance inadequate for our needs. A second issue involves investors who are not satisfied with the notion of standard deviation representing he amount of risk for a given asset. Subjectively, an investor wishes to make large profits rather than remain symmetric around the mean, signifying that there is a potential to make small or no profits. Lastly, the optimal solution may involve purchasing small amounts of varied assets which incurs increased costs. In reality, an investor wants to maximize his/her returns with the fewest number of stocks possible and thus the need for integer constraints becomes necessary to restrict the selection process to minimal lot size purchases, and try to avoid excess transaction costs associated with each asset [2]. There are two important assumptions that are made by the MAD model which may cause problems for indexing purposes. The most notable assumption made by this model, is that the rates of returns for the various assets under consideration in the large market portfolio are multivariate normally distributed [2]. The supposition is required to ensure that the absolute deviation function proposed in this model is essentially equivalent to the standard deviation function in the mean variance model. It is necessary to maintain this identity because, as mentioned previously, the MAD is a linear approximation of the mean variance model. However, by carefully analyzing the 17

19 historical rates of returns for several random assets during the specified time period of 252 days, the assumption can be shown to be invalid. Stocks symbols ADI and GLW are noticeably not normally distributed by inspecting their respective normality graphs illustrated in Figures and The charts provide concrete evidence that it is not safe to assume the rates of returns are normal and thus puts into the question the validity of the optimal solutions produced by the model for indexing. As described earlier, passive fund management is achieved through the second assumption that an investor prefers less risk and a lower return, over a higher risk and a higher return. This occurs because the objective function in the model is minimizing the deviation for the mean Histogram of ADI Normal Mean StDev N Frequency ADI Figure Normality Graph for Asset ADI 18

20 60 50 Histogram of GLW Normal Mean StDev N 251 Frequency GLW Figure Normality Graph for Asset GLW 2.4 Evaluation Criteria The necessity for a revised MAD model to improve the quality of portfolio indexing is evident. An improved version the absolute deviation program must demonstrate dominance in a number of facets related to indexing. Most notably, indexing attempts to produce a significantly smaller portfolio that will mimic the risk-return behavior of the larger market by minimizing the difference in performance between the two. The difference in performance between the two is proportional to the difference in the rates of return of the newly created portfolio and of the larger portfolio, which introduces the idea of the incorporating tracking error into the model. This is captured by the following equation 19

21 TE = T t = 1 ( P M ) i T 1 i 2, where P i is the return of the portfolio at time i and M i is the return of the benchmark market at time i. T is the total number of time periods over which the generated portfolio will be tracked. In a real world scenario, asset managers who benchmark a market index require a better measure of portfolio risk than simply monitoring its standard deviation of returns [7]. The notion of tracking error will be accounted for by implicitly minimizing it in the improved model. The term active return is known as the variation between the investor s current portfolio and the portfolio s benchmark return. Therefore the standard deviation of the portfolio s active return is essentially tracking error [7]. There are numerous factors affecting the level of tracking error. For the purposes of the project, there are only several that have a significant impact on the optimal objective value. The first issue that will be examined is the number of stocks included in a portfolio. As the number of stocks within a portfolio increases, the unsystematic risk decreases, thus reducing the amount of tracking error. Secondly, the closer an index fund portfolio is to the benchmark index, the lower the tracking error value will be due to less active returns. A third cause, resulting in the increase of tracking error, is the amount by which a portfolio differs in sector allocation with respect to the benchmark. The objective is to incorporate both the minimization of volatility and tracking error. Volatility is measured in terms of risk where a lower value is preferred over a larger one. This statement follows the notion that an investor is willing to make less money with lower risk, as well as having the possibility to make more money with increased risk. Also, in order for the model to be realistically applied, the optimal solution should be 20

22 established in a speedy manner regardless of the introduction of integer constraints. The most important evaluation criterion is monitoring the actual future performance of the index chosen to validate whether or not the model is yielding equivalent results for investors. 21

23 3 METHODOLOGY The following is a systematic exploration into answering the question of how effective the mean absolute deviation is as an indexing tool. 3.1 Exploring the Efficient Frontier Prior to doing any modeling, the group decided that it was necessary to begin doing some research in the field of portfolio indexing and how to apply financial mathematics and modern portfolio theory to the issue of choosing an index. This research laid the foundation for understanding how and why the assets selected were in fact selected, as well as giving us some insight as to further steps that can be taken, besides simply solving the MAD model. This information will be explained later on in the paper, as results are explained Testing To begin working with the MAD model, it was decided that a larger problem should be undertaken, which similar in scope to the topic of the thesis, but gave us insight as to the field of financial modeling. This way, a significant solution could be found so that a true analysis of the success and level of efficiency of the model could be determined. The idea was that the MAD model could be used to select an optimal allocation of funds given a minimum desired rate of return. This was to be done using a large pool of assets from which the portfolio was selected. It was felt that using an entire market as the initial pool of assets to choose from was too large and would be involve too great of a computational burden, so a viable alternative was to substitute a well known index to use 22

24 as the benchmark market. A unanimous decision was to use Standard and Poor s S&P 500, which is an index of 500 companies in various sectors. The index is based on market capitalization; hence it is no surprise that the 500 companies in the index comprise approximately 75% of the equities market in the United States [8]. Because of its large span over the equities market, the index is used as a tool of stating the overall financial health of the equities market in the Unites States. To model this problem, historical share prices had to be found. The Financial Laboratory at the Rotman School of Management seemed to be the most convenient place where this historical data could be found. Using an MS Excel add-in called DataStream AFQ, historical share prices of a desired index could be queried from the DataStream database, for a desired range of dates. Stock prices were taken from the 252 periods from January 2005 to December 2005 for this index, using the AFQ DataStream. Although the DataStream add-in queried its database quite efficiently, upon reviewing the data retrieved, erroneous data was found. This indicated that a more thorough examination of the data was needed. Because there was approximately 126,000 stock prices in the spreadsheet (500 companies * 252 days), this took a fair amount of time. Upon completion of the check, a yearly average rate of return could be inputted into the model r j as well as a j, the difference between r it (rate of return of asset j at time t) and r j. An OPL model (as shown in Appendix C) was used to solve the problem. The formulation for this problem may be seen in Appendix B. The model here forms a connection to the excel spreadsheet containing the weekly historical data for all 500 companies so that OPL could read the values in directly from the spreadsheet as opposed to manually entering all of the data into the OPL data file. An optimal allocation was determined based on minimizing the summation of the volatility 23

25 of each individual stock (measured as mean absolute deviation), based on a minimal desired rate of return. Because the optimal portfolio generated shows a positive correlation between risk and return, a tradeoff has to be made between risk and return. Hence, because the model generates a value for risk based on return, a return satisfying the minimal constraint had to be chosen, such that the risk value associated with the portfolio that gave this risk was still acceptable. A key note to mention is that at this stage, the model is not being applied to an indexing type problem. The model tries to generate a portfolio that at minimum achieves a particular historical rate of return, as opposed to trying to match a particular return, as is the case in indexing. Also, the model is not constrained to choosing a particular number of assets, since n (2T + 2) is less than zero at n = 500 assets and T = 251 periods [2]; therefore, technically any number from one to the number of assets in the benchmark can be selected Results To find a solution to this problem, an efficient frontier was created. The efficient frontier is a curve that plots return versus risk. Each point on the curve shows the upper bound for possible return, given a value for risk. Therefore, taking into consideration the pool of assets used and the time period studied, a portfolio with a given amount of risk can not yield a return that is higher than that of the point on the curve [7]. Because the model was run several times, an efficient frontier could be created using the data obtained from the model. The curve produced may be seen in Figure below. 24

26 0.04 Efficient Frontier for S&P Rate of Return Deviation Figure The efficient frontier for S&P 500 data for the year 2005 The sharp drop around zero (the grand mean of all returns) is expected, since the mean value allows the objective to choose from the maximum range of stocks with the lowest absolute deviation. Although an efficient frontier was found, the question of which point on the line to choose was still unanswered. Finding this optimal tradeoff value involved finding the greatest possible value of Sharpe s Ratio. This ratio, which is defined as: E S = [ R R f ] E[ R R f ] = σ Var[ R R ] where R is the return of the given portfolio and R f is the return of the risk-free asset. f 25

27 This equation defines a ratio of the average rate of return in excess of the risk free rate, over the standard deviation of the rate of return [9]. This risk free rate, much as the name suggests, is a rate of return which could be obtained by investing in an asset which has essentially no risk associated with it. An example of this could be a government savings bond. Therefore, the point on the curve with the highest value for Sharpe s Ratio is the point that gives us the most appropriate tradeoff for risk versus return. For the purposes of this analysis, zero was chosen as the risk free asset because the average return for the entire S&P 500 index (the source of our data) in 2005 was just below zero. Zero is also a logical choice since not investing at all is also a risk-free option. Because there are several points on the curve, finding the value of Sharpe s Ratio for each would be somewhat time consuming and inefficient. Therefore, to find the point on the curve with the highest ratio value, a Capital Market Line (CML) was drawn. The CML is plotted below in Figure

28 16 x 10-3 Capital Market Line Rate of Return Deviation Figure The straight line in the figure above is the Capital Market Line Here zero is chosen as the risk free rate of return to mirror the poor market performance. Reading from the graph (shown by the rectangle), the best portfolio was selected with a return of and an average deviation of

29 3.1.3 Evaluation The tangent line spans from the intercept point on the efficient frontier here zero, to the point associated with the return for the risk free asset [7]. The point of intercept on the efficient frontier corresponds to the portfolio giving the appropriate tradeoff value for risk versus return, which is determined by the highest value for Sharpe s Ratio of any point on the efficient frontier curve. Here, zero is chosen as the risk free rate of return to mirror the poor market performance. Reading from the graph (shown by the rectangle), the best portfolio was selected with a return of and an average deviation of This is the optimal value for Sharpe s Ratio. Taking the return at that point, the model was run once more to obtain the portfolio listing the assets included and the respective proportion of funds to be allocated to them, which gives this optimal value for Sharpe s Ratio. The final set of assets selected and the proportion of funds allocated to each is listed in Appendix D in the column labeled Original. 28

30 3.2 Revising the Model In order for a mathematical model to generate a solution that is truly optimal, it must first be ensured that the model encompasses all of the required information and constraints, and that it uses as its objective function, a measure that truly defines the goal of the overall system. With this being said, it is no wonder that models need to be tailored to specific situations such that they truly reflect the exact environment in which the model is being run. Models being used for financial modeling are somewhat standardized. The reason being is that the majority of models simply state that a portfolio allocation is to be generated based on historical market performance, such that the risk of this portfolio is minimized, but a minimal desired historical rate of return is achieved. As was stated in the beginning of this paper, this is the idea behind active fund management. Because however, the theme of this thesis assignment deals with indexing, or passive fund management, there is somewhat of a discrepancy. The issue does not lie in the idea of minimizing risk of course. It lies, on the other hand, with the performance (measured in terms of historical rate of return) with which the portfolio allocation is generated. The idea with indexing is to achieve a return on investment that matches the return of the benchmark market which is being followed, while minimizing the number of assets required to do so in the generated portfolio. The return of the generated portfolio at a given time period can be higher or in fact lower than that of the benchmark market at that time period, but the objective here is to minimize the deviation of the returns at any time period. As was mentioned above tracking error, which is essentially the standard deviation between the returns of a portfolio and its benchmark market over a set number 29

31 of time periods for which the performance of the two are being monitored. Ideally, there would be no deviation between the portfolio and benchmark market at any time period, in which case the tracking error would be zero. Intuitively, this seems to become more prevalent as the number of assets in the portfolio approaches the number in the benchmark index Models In light of the idea of indexing, the standard MAD and mean-variance models have been modified to take into account the altered constraints on returns. Both models were altered in a similar way. Because the generated portfolios can technically have returns that are higher or lower than that of the benchmark market, constraints for the return of the market over the entire modeling period have been set to equal that of the overall rate of return of the benchmark market over the entire modeling period. Because, as stated before, the return of the portfolio may not realistically match that of the market perfectly, two slack variables were included in the constraint to absorb the value of the portfolio return above (S+) or below (S-) that of the market. To ensure that these slack values for return are minimized, the positive slack values are included within the objective function to ensure that they are to be minimized along with the risk of the portfolio. The idea that the original MAD model could be further modified to improve its capability of indexing was further explored. The possibility for improvement lied in the objective function measuring absolute deviation, which picks individual stocks such that their level of historical volatility is minimized. The desire was to modify the objective slightly to better the model s capability of selecting index portfolios, but at the same time, not change the objective too much and diverge significantly from the mean absolute 30

32 deviation idea. Two models arose from this idea. The first was dubbed the Big R Model. The model was similar to that of the original MAD. The Big R model however, chooses individual assets such that the summation of the absolute deviation of their return at time t from that of the grand average of the market, as opposed to the average return of each individual stock, was minimized. This is shown in formula below. 1 T n ( r R) T t = 1 j= 1 jt x j The reasoning behind the model selection was that choosing assets that achieve returns close to that of the overall market over some historical time period is beneficial from a risk point of view because the volatility of an asset over time will hopefully be minimized, and the return of an asset will match the overall return that is trying to be matched. The second model was dubbed the Rt Model. Again, the model was similar to that of the original MAD. The objective of the Rt however was to choose individual assets such that the summation of the absolute deviation of their return at some time period from the return of the market at that time period, over all historical time periods being modeled. The objective of this model may be seen below. 1 T n ( rjt rt ) T t = 1 j= 1 The reasoning behind the second model was that choosing assets whose volatility matches the volatility of the overall market is very beneficial in terms of minimizing tracking error, and hopefully will generate a portfolio of assets that continuously each mimic the overall market in the future. x j 31

33 3.2.2 Testing Because the two tweaked models were generated based mainly on reasoning and intuition, to properly ensure their validity of use, the success of their generated portfolios were tested, alongside the original MAD model. The two standard measures with which the models could be tested are the risk and return of the portfolios generated by the individual models. Because the MAD model is known to be an approximation to Markowitz s mean variance model, the level of success of each of three models being tested was measured in terms of how the portfolios generated deviated in comparison to both the risk and return of the portfolio generated by the mean variance model. To solidify any judgments that would be made regarding the validity of the models, it was essential that the experiments be performed several times so that the judgments could be confirmed statistically. An issue in devising an appropriate test was that, because the success of a model in terms of its comparability to the mean variance was based on the assumption of normality of the returns of each asset in the pool from which the portfolio was generated, a fewer number of samples could seriously skew the results due to non-normality of the market. Another issue had to do with correlation. Because an asset s daily price is often related to the price at which it was listed in previous days, an asset s return can demonstrate autocorrelation within itself over the amount of historical time over which the prices are recorded. Because pairs or group of assets may also fluctuate similarly, for example for those within the same market sector, there may be cross correlation between two or more asset s returns. An asset whose return values can be at least partially attributed to correlation, can no longer be assumed to be perfectly independent from all other assets within the market. This can lead to 32

34 complications in terms of choosing assets on an individual basis. To help combat this issue, twenty samples, each containing twenty assets from the S&P 500 were taken, based on their daily stock prices from the period of January 2005 to December Each pool was chosen at random. The twenty pools each containing twenty assets are shown in Table Table Twenty Random Pools Containing Twenty Assets Each (S&P500 Data) Results Using the twenty randomly chosen samples, the original MAD, the two revised models and the mean variance model was run on each sample. As a standard, each portfolio allocation was to create an allocation for five assets. Hence, each model was to include five assets in its portfolio for each of the twenty samples that it was to be run on. As an example, the portfolio allocations made by each of the four models worked with for sample five are shown in Table

35 The first set of tests run was on the risk of the generated portfolios. Because the objective of each model is not measured in the same terms as any of the others, each of the objective values returned had to be converted into a common, comparable measure. The group consensus was to use mean absolute deviation as the common objective measure. To convert into comparable terms, the portfolio allocations provided by both the Big R and Rt models were substituted into the MAD model as constraints, in order to calculate their mean absolute deviation objective. For the mean variance however, the mathematical relationship of statistical equivalence absolute deviation to standard deviation was used, where σ is equal to the mean variance [2]. n n 2 w( x) = σ ( x), where σ ( x) = σij xi x j π i= 1 j= 1 Figure Relationship between Absolute Deviation and Standard Deviation After each of the four objective values for each sample was converted into mean absolute deviation, comparison of each model s objective value to that of the mean variance could be done. A comparison gave the proportional difference that each of the: original MAD, Big R, and Rt, had in comparison to the mean variance. Because the success in terms of risk of each of the three models at each sample was measured in terms of proportional difference, the values for a particular model could be compared across all samples. Table shows the objective values in terms of mean absolute deviation for each of the four models, as well as the percent difference from the mean variance for the remaining three models. A key note is that the values assigned to the slack variables S+ and S- for each model were set to zero for each and every trial run. The second set of tests was on the return characteristics of the generated portfolios. Here, the tracking error was calculated for each of the four models, 34

36 comparing them to the entire S&P 500 for the year of Then, the percent difference in terms of tracking error was calculated between the mean variance model and each of: the original MAD, Big R, and Rt models. Again, once the success in terms of returns for each of the three models was measured in terms of percent difference from the mean variance, these percent difference values could be compared across all time for each of the three models individually. Table shows the tracking errors for each of the four models, as well as the percent difference of tracking error from the mean variance for each of the remaining three models. Table Portfolio Allocations of Each Model for Random Pool 5 35

37 Table Objective Values for Each Model (In Terms of Mean Absolute Deviation) and Proportion Difference from Mean variance 36

38 Table Tracking Errors for Each Model and Proportion Difference of Objectives From Mean-Variance Evaluation Each of the three models being tested had their success levels for both the risk and return tests measured in terms of the percent difference of their respective value from that of the mean variance. Because of this, the success values for both the risk and return tests for any of the three tested models could be compared across samples. Using this knowledge, 37

39 a t-test was used to measure the success of models relative to each other. Two, twosample t-tests were performed for all combinations of models; one for proportion differences from the mean-variance objective and one for tracking error. Assuming results came from the same population, t-tests calculated a confidence interval on the difference of means for each of the model combinations. Results show that both the Big R model and the original MAD model performed better than the Rt model on approximating mean-variance risk; however, all three models were approximately equivalent in approximating mean-variance tracking. Appendix E.1 and E.2 shows the t- test results as calculated in Minitab statistical software. The extent of this testing essentially proved that the Big R model that was derived is statistically comparable in success terms to the original MAD for indexing purposes; on the other hand, at a significance level of α = 0.05, there is no improvement on the original MAD model, contrary to expectations. 38

40 3.3 The Two-Stage Model (Constraining the Model for Index Size) At this stage, once the subtleties of the model have been understood, another stage of the portfolio model can be introduced. This model is similar to the model as described in Appendix B but it is constrained in terms of the number of assets that are to have positive allocations within the portfolio. This ties back to the objective of this thesis project, which is to create a portfolio constrained in size, which matches the performance of portfolio or market much larger in size Testing The formulation for the model that was used to generate a solution for this problem may be seen in Appendix F. As may be noticed, this model is quite similar to that of the previous model, as described in Appendix B. However, there are a few differences which are quite noteworthy. Firstly, because the second portfolio is a subset of the first, which is unconstrained in terms of size, there is a constraint which limits the amount of assets that can be invested in. This entailed the creation of a binary variable for each asset, which is true (set to equal 1) if the asset corresponding to that variable is included in the portfolio. Because integer decision variables were incorporated into the model, it now uses the branch and bound technique to generate an optimal solution. As is known from experience, the computational burden associated with solving an integer program is much more significant than that of a linear program without integer decision variables. Hence, because the unconstrained portfolio significantly reduces the asset pool that the constrained may choose from, solving a portfolio from that subset that has performance similar to that of the market is much more efficient from a computational point of view 39

41 because the constrained portfolio has as a selection pool, the group of assets chosen by the unconstrained model. Another important difference is that because this constrained portfolio is trying to mimic the performance of the unconstrained portfolio, not only the volatility in each stock, but the difference in performance between the constrained portfolio and that of the unconstrained is to be minimized. This hints at the idea of the incorporation of tracking error into the model, as is described in the background of this paper Results The objective equation tries to incorporate both minimization of volatility (as was the sole objective of the first portfolio) and tracking error. This is done by computing the deviation between the rate of return of the constrained portfolio at a give time, and the overall average of the rate of return of the unconstrained portfolio. The OPL model file that was used to solve this problem may be seen in Appendix C. Because it is quite obvious that the rate of return at a given time for the constrained portfolio is dependant on the number of assets in that portfolio, the objective value and hence the assets chosen with associated proportions are affected by this input variable. Therefore, the model was run several times with varying numbers of assets to be included in the model to observe the sensitivity of asset selection and allocation relative to the number of assets to be included in the portfolio. The results from these trials may be seen in the Figures and

42 6.4 x 10-3 Objective Sensitivity to Portfolio Size Objective Value Portfolio Size Figure Portfolio Selection Size vs. Objective Value 0.08 Standard Deviation of Error from Original Portfolio Proportions 0.07 Standard Deviation of Error Portfolio Size Figure Portfolio Selection Size vs. Standard Deviation of Error 41

43 3.3.3 Evaluation Results of these trials show that the objective is highly sensitive to changes in z j, the binary selection variable for asset j, at lower levels and not sensitive at all when levels are closer to the total portfolio size. Figure above demonstrates the exponential relationship between portfolio size and minimum absolute deviation from R, that as the number of stocks selected in the portfolio increases the objective can minimize deviation with much less trouble. As expected, results have also shown that proportions are more easily maintained with sizes of z j closer to the total portfolio size. Figure above demonstrates the roughly exponential relationship between z j and the standard deviation of proportion error. Here, instead of a linear loss in proportion size as expected, there seems to be a balancing that occurs for risk and return at lower integer constraint sizes, making it difficult to hold the same stocks in their original proportions. 42

44 3.4 Exploring the Normality Assumption One of the main disadvantages of using the mean-absolute deviation model for generating a high quality index is the assumption of normality necessary for its equivalence to the mean-variance model. For the model to be equivalent, returns of the market data must be multivariate normally distributed; in other words, each individual asset distribution R must be normally distributed in M = (R 1, R 2,..., R J ) (p.523 Konno and Yamazaki). In reality however, the distribution returns tend to have autocorrelation, forward skewness, kurtosis, and changes in the underlying distribution affecting mean and variance. These perturbations in return distributions are positively correlated with sales, operating leverage, and other market factors [10]. In order to evaluate the effectiveness of the MAD model in providing an equivalent index to the mean-variance model in risk and return measures, it was necessary to devise and execute a test which could measure the normality of distributions in a given test sample and the effect it has on the risk-return characteristics of the MAD solution index with respect to those of the mean-variance solution. The hypothesis made was the following: the less normal the sample return data was, the more the objective and tracking of the solution would deviate from the mean-variance model; moreover, it was expected that the objective and tracking error would be significantly worse. To discover if this was the case, the following test methodology was devised and executed. 43

45 3.4.1 Testing The main difficulty in testing multivariate normality of the sample data is that returns over a period T are auto-correlated, meaning that the level of performance of a given stock at period t will affect the performance of the stock at period t+1. The reasoning behind this statement is that it seems that the performance of the economy seems to follow trend like patterns over discrete periods of time. The data obtained from the Standard & Poor s 500 index for the year 2005 was highly auto-correlated, likely because we used daily stock prices at each period t, as opposed to a weekly or monthly aggregate. Once the data is auto-correlated it is impossible to measure the normality of each stock distribution; therefore, it was determined that this problem could be solved either by using data for a given stock from a single period in history for approximately 20 years or simply by taking aggregate samples of all stocks in the sample at each period t. Using the former is problematic, given that a solution with data over a period of years is not realistic in practice; therefore, the latter method was chosen. In agreement with Central Limit Theorem, taking aggregate samples of data from the same population of any given distribution will yield a distribution of sample means that approaches a normal distribution. Data was aggregated in the following manner where r t is the sample aggregate data point at time t and r jt is the return for stock j at period t: 1 r = n t r jt n j= 1 Figure Aggregation of samples at time t An autocorrelation test of each sample was executed in MINITAB 15 to determine if the sample means for each time t were auto-correlated. At the α = 0.05 level of significance 44

46 all samples passed; as a result, an Anderson-Darling test was then executed to test the normality of the aggregate sample. In preliminary tests of sample distributions for returns it was observed that shifts in the underlying distributions were affecting the mean and variance of samples, creating multi-modal effects and unusual bulges in the distribution curve. These distribution shifts are due to broader market forces that typically effect the entire population of market returns. These shifts can be dealt with in practice by moving average or time series weighting [11]. These methods are outside the scope of this paper; however, for the purposes of this experiment, the shifting mean and standard deviation problem are addressed in a much more straightforward manner. By taking an Anderson-Darling normality test of the aggregate samples for 100 time periods out of the 251 available for the year 2005, a period of stability was obtained simply by shifting the time frame along to find the period of maximum normality. This test is explained in detail in section Once a period of stability was maintained, an Anderson-Darling test was executed in MINITAB 15 for the 100 aggregate sample data points as shown in Figure for all sets of random stock samples. Twenty samples, containing 20 stocks each, were chosen randomly from the S&P data by generating 20 sets of random numbers from 1 to 500, using the MINITAB software. The ticker symbols for the 20 random sets selected are shown in Table below. Once the stocks were chosen, the model was executed for each of the 20 random sets using ILOG OPL optimization software. After determining the most appropriate model for indexing as previously determined in section 3.2, it was assumed that the model would be run using the original MAD model with integer constraints as seen in 45

47 Appendix B; however, this model requires that each stock j be multivariate normally distributed, which is not the case. As mentioned above, the Anderson-Darling test was performed on the aggregate distribution, for all r t over the 100 periods. In order to discover a meaningful relationship between normality and the quality of the MAD solution it was necessary to change the model so that it minimized the absolute deviation of the aggregate returns r t rather than the individual multivariate returns r jt. In order to do this, the assumption was made that each sample mean r t for the constrained solution would follow approximately the same distribution of r t for the original sample containing 20 stocks. The objective of minimizing deviation was altered from the definition of absolute deviation in Figure a to the aggregate absolute deviation in Figure b, somewhat similar to the Big-R Model mentioned in section 3.2, where R is the grand mean of all returns for all stocks over all periods. n ( rjt rj ) j= 1 x j n j= 1 r jt x j R Figure a - Individual Absolute Deviation Figure b - Aggregate Absolute Deviation This adjusted MAD model, as well as the mean-variance model, was run for all sample sets with a static constraint value of five for the index size in all tests. The objective values, times, and returns were recorded for each run and tracking error was then calculated by the following formula, where R mt is the average return of the market at period t and R it is the return of the solution index at period t: 46

48 TE = T t= 1 ( R R ) mt T 1 it 2 Before the data could be compared with the mean-variance model, the objective would have to be in similar units. The main problem is that the effects of making adjustments to the MAD model objective are unknown. Since the model is solving a different objective, the mathematical equivalence requires further examination, beyond the scope of section 3.2. The assumption was made that since all tests were run on the same model, the differences from the mean-variance model would be constant to account for the alteration, and would not influence the relative differences between samples with varying p-values. However, to be consistent with the literature, the original objective was recalculated for each sample using solutions from the altered model. The mean-variance solution objective σ(x) was converted to absolute deviation units w(x) using the relationship in Figure Table Ticker Symbols of 20 Random Samples 47

49 3.4.2 Results Observations were recorded for p-value, objective value, tracking error, as well as the proportional differences between the mean-variance and the MAD model for all 20 random sample sets. Proportional differences between the mean-variance and MAD objective values are labeled as Risk and proportional differences between tracking errors are labeled as Return. Negative proportional differences indicate that MAD solution values are higher (or worse) than the mean-variance solution values. The data recorded, summarized in Table below, is sorted by p-value. Higher p-values indicate random sets that are more normal. The zero p-value listed in the table for random data set number 13 indicates that the p-value was lower than and therefore negligible. In order to observe a trend or relationship between normality and deviations from the mean-variance model in risk-return characteristics, results were graphed using MATLAB with p-value on the x-axis and deviations on the y-axis. The graph is shown below in Figure The graph above does not show any relationship between increasing normality and increasing equivalence to the mean-variance model, as predicted by Konno and Yamazaki [2]. Even at high p-values, where normality is highest, proportion differences from mean-variance tracking error are still off by 20 percent. The variability in risk and return deviations seems unpredictable for all p-values; in fact, MAD objective values appear to be equally distributed on either side of the mean-variance objective values at all levels of normality. 48

50 While there appears to be no relationship of the mean-variance model s equivalence to the MAD under increasing normality conditions, the relationship normality has to the MAD solution objective and tracking may prove to be important, if it exists. The hypothesis was made that if the data is more normal then tracking error would improve, since the model would be more effective at tracking a symmetrical mean; however, the objective value may be slightly worse for higher normality, since asymmetrical distributions provides the opportunity to choose smaller deviations on the skewed portion of the population. In order to observe the relationship between normality and the MAD solution, objective and tracking error were graphed in the y-axis against p- value in the x-axis. This graph is displayed in Figure below. As before, and contrary to predictions, there appears to be no clear relationship between the normality of the data and the solution. Although not obvious, as normality increases there appears to be a slight upward trend in the tracking error and a slight downward trend in the objective, as predicted; however, tracking at lower p-values is a mixed bag. Perhaps the normality of the data helps to control the potential tracking range; nevertheless, there is clearly no statistical significance in any of these relationships. 49

51 Table Summary of Normality Test Data 50

52 0.2 NORMALITY vs. RISK/RETURN DIFFERENCE FROM MEAN-VARIANCE Proportion Deviation From Mean-Variance Risk/Return Risk Return P-Value of Sample Figure Graph of Normality vs. Risk-Return Difference from the Mean-Variance Model 51

53 NORMALITY vs. OBJECTIVE / TRACKING ERROR Objective Tracking Objective / Tracking Error P-Value of Sample Figure Graph of Normality vs. the MAD Model Solution Objective and Tracking Error Evaluation The lack of any measurable relationship between the normality of a sample set and coherence with the mean-variance model is very suspicious. There are a number of factors which may have played a considerable role in contaminating the test results. For example, constraining the solution at five provides a significant variation in the quality of the solution. This variation could possibly have been enough to offset the effects of nonnormality. Another factor affecting the significance of the results is the composition of the solution index returns at each period t. The solution set of j = 5 data points r jt within the 52

54 aggregate sample r t may have had a significantly different P-value than the larger set of 20 data points comprising r t in the original sample. This is a likely source of contamination given the wide range of p-values among the 20 sample sets. The wide range of p-values for the chosen time range of 100 periods indicates that the stocks contained in r t, the aggregate sample data points, have a large effect on the sample normality. It was assumed from the outset that aggregate data for a given time range would have roughly the same normality no matter which subset of stocks were contained in the aggregation; this assumption proved to be questionable given the results. Another opportunity for test corruption has to do with the altered MAD model used for the experiment. The adjustments made to the model may have had effects on its ability to keep equivalence to the mean-variance under normality conditions. The MAD model has an objective of minimizing absolute deviations for individual stock distributions, similar to the mean-variance model in that it has an objective of minimizing co-variances between individual stock distributions; whereas the adjusted MAD has the objective of minimizing absolute deviations for an aggregate whole. The effects of this alteration are unclear and should be further investigated. Regardless of the mean-absolute deviation shortcomings as an approximation to mean-variance, there is no observable relationship with normality here. However, the problem of distribution inequality provokes an interesting question: if return data is not similarly distributed in practice, containing distinctly unique sub-distributions, then is minimizing covariance between dissimilar sub-distributions a sufficient means of minimizing risk? This question, however, is beyond the scope of this paper. 53

55 3.5 The Effects of Index Size on the Revised Model The final step after revising the model is to determine the effects of integer constraints on the risk-return performance characteristics of the chosen index. An analysis on the revised model was therefore conducted in order to provide quantitative evidence on an optimal portfolio size Testing The main objective tested was to establish a fixed number of stocks that an investor should hold in order to duplicate market returns and minimize risk effectively, given the historical data that is currently being used. The tracking error and objective value (risk measure) are the two main variables that will be correlated to the index size. The mean variance, MAD and Big R will be executed and evaluated. In order to ensure robust results that represent realistic situations, historical data for 200 stocks will be entered into the model. This will allow for a better representation of the benchmark market while simultaneously maintaining a moderate time for an optimal solution. The stocks were randomly chosen from the S&P 500 using a random number generator in Minitab 15. In order to guarantee consistency between the mean-variance and mean absolute deviation models, the normality assumption must be taken into serious consideration, as was stated several times throughout this paper. The problem was accounted for by first calculating the average rate of return for each period (r t ) and decomposing the data into nine approximately equal periodical sections (each of 100 operating days). A normality graph and p value indicating the fit of data to the normal distribution was found for each section of data using the program BESTFIT. A larger p value parameter demonstrates 54

56 a higher level of normality which exists within a given data set. The results generated by this preliminary test can be seen in Figure and reveals that section 41 to 140 has the highest p-value. A probability plot and normality graph are further utilized to visually support the level of normality as seen in Figures and respectively. Given this information, it is evident that periods 41 through 140 are the most normally distributed time and will be used for the testing. Integer constraints were incorporated into the model to control the amount of stocks in which the model must choose for the optimal portfolio with respect to either risk or return, respectively. The tests for risk and return were both initially left unconstrained to determine what the models would naturally choose as the optimal number of stocks to hold. The number of assets held within the portfolio was then restricted to 10 and subsequently increased by 10 until it reached 190. It is important to retrieve relevant information derived by running the formulations. As mentioned previously, tracking error and objective values will be recorded alongside the amount of time required for the model to generate an allocation. The results from running the tests were tabulated and documented. 55

57 P-Value vs. Period Range P-Value P-Value Period Range Figure P -Value vs. Period Range 56

58 Probability Plot of Normal Percent Mean StDev N 100 AD P-Value Figure Probability Plot for Periods 41 to Histogram of Normal Mean StDev N 100 Frequency Figure Normality Graph for Period

59 3.5.2 Results The results were recorded and organized according to each model for varying index sizes. It is important to notice the relationship between the time it took for each model to find an optimal solution and the number of assets allocated to the portfolio. This relationship is shown in Figure Mean Variance The original MAD model has an objective in standard deviation units and was converted to absolute deviation units according to the formula in Figure The original unconstrained mean variance model provided an adjusted objective value of and a tracking error of This model selected 32 stocks in seconds to represent the optimal solution. The entire set of results can be examined using Figure which outlines the effect that integer constraints and index size has on the revised model. It is important to mention that for index size constraints above 32, the mean variance model makes positive allocations that are extremely close to zero and thus are considered irrelevant. Mean Absolute Deviation The original unconstrained mean absolute deviation model provided an objective value of and a tracking error of This model selected 28 stocks in seconds to represent the optimal solution. The entire set of results can be examined in Figure which outlines the effect that integer constraints and index size has on the revised model. Big R Revised Model 58

60 The unconstrained revised mean absolute deviation model provided an objective value of and a tracking error of This model selected 28 stocks in seconds to represent the optimal solution. The entire set of results using integer constraints and varying index sizes can be examined in Figure Time vs. Index Size Time (Seconds) Mean Variance Mean Absolute Deviation Big R Index Size Figure Time vs. Index Size 59

61 Objective/Tracking Error vs. Index Size for the Mean Variance Model Objective Tracking Error Index Size Figure Objective/Tracking Error vs. Index Size for the Mean Variance Model 0 Objective/Tracking Error Objective/Tracking Error vs. Index Size for the Mean Absolute Deviation Model Objective Tracking Error Objective/Tracking Error Index Size Figure Objective/Tracking Error vs. Index Size for the Mean Absolute Deviation Model 60

62 Objective/Tracking Error vs. Index Size for the 'Big R' Model Objective Tracking Error Objective/Tracking Error Index Size Figure Objective/Tracking Error vs. Index Size for the Big-R Model Evaluation After organizing and examining the results obtained, the next step is to evaluate the models. It is important to realize that when the original and Big R model were left unconstrained, they produced identical solutions. This reinforces the idea that under realistic circumstances the Big R model performance is equivalent to the original model, as was determined in the t-test from section 3.2. However, for this data sample, both models exhibit a performance improvement over the mean-variance model with regard to the adjusted objective; however, with regard to tracking, the Big R and original MAD models are measurably inferior. This comparison however, does not hold generally, as was illustrated in section 3.2. Furthermore, it should be noticed that the time it took Big 61

63 R to produce an optimal solution was shorter than that of the original MAD. The difference in time is extremely small when the models were run on a market portfolio of 200 assets. This small differential may increase with the introduction of more assets in the market portfolio and further testing would be required. Both MAD models displayed a considerable improvement of almost half the computational time it took the mean variance; however, at lower index size constraint levels this was no longer true (Figure ). The relationship between index size and time reveals several characteristics. The times remain fairly stable and small up to 110 stocks. After this point, the times for all models rise and exhibit a higher variation for differing index sizes. The continuous behavior of low values for the mean variance model is suspicious because the computational burden for quadratic programs becomes enormous for larger problems [2]. A possible explanation for these results is that once integer constraints were introduced, the solution of the relaxation, or the initial nodes being explored, were close to optimal, thus allowing the model to solve quickly for all the index sizes. This hypothesis seems most probable when the model is constrained to choose a smaller set of stocks (for example 20 or 10) from the pool of 200. Quite notably, the most interesting results occur at 10 stocks where the times for the MAD and Big R models jump up immensely in an exponential manner. The constraint limiting the portfolio size to only 10 stocks puts a large strain on the model. This occurs because it is extremely difficult for the model to select such a small number of assets that will precisely mimic the large market. It is important to mention that Big R took considerably more time than the original MAD. It is evident that all three curves in Figures , and stabilize at the same index size of 30. This value is extremely close with the number of stocks that the 62

64 unconstrained models selected as noted earlier. These findings are consistent with corporate finance literature which states that there is only a minor decrease of variation in a portfolio greater than 20 or 30 stocks [12]. The value of 20 to 30 stocks is an optimal value because it is relatively small number of assets and ensures diversification while simultaneously reducing variation [12]. Using modern portfolio theory methodology, such as mean variance or mean absolute deviation, is problematic for index sizes over 30 because of the inherent degeneracy of the model. In order to force quantity into a basic degenerate variable, a lower limit was integrated into the model to force all degenerate variables to select a nominal allocation amount. This was done by replacing the integer constraints in the model in Appendix F with the following constraints x x j j z > z j j M min where min is a minimum specified amount for any basic variable. The result of this alteration once it was entered into the model was that all basic solution variables became equal to min and the other allocations were adjusted proportionately. Due to the algorithm s nature of finding the smallest allocation set to meet the objective, it would not be very useful in practice for an index size over 30 unless minimum allocations were equally forced upon all other constituents. 63

65 3.6 Tracking Future Performance The progress made to date has been quite significant in terms of learning how successful the original MAD and altered models are in terms of performance when compared to the mean variance model. Although this has been extremely useful, the next obvious step is to actually choose one of the models that has been worked with thus far to create an index portfolio that will try and achieve future returns which match that of a benchmark market or portfolio Testing All of the tests to this point have been run using random subsets of the S&P 500 representing the benchmark as opposed to an entire market. This was completed for the simple reason that the computational time required to solve a model using an entire market as the benchmark is too great given the amount of tests and number of times the models were run on the data. As done so far, 100 periods from 2005 were used as the first segment of historical data from which daily stock prices were known. This 100 day period was day 41 to day 140 of To commence the test, a random subset of two hundred stocks was selected from the S&P 500. This random set was the same set that was used in the testing done in section 3.5. The ticker names for these stocks can be seen in Table As stated throughout the paper, the equivalence of the models using absolute deviation depends on the normality of the returns for the pool of stocks from which the index is chosen. Section 3.5 describes how the period of one hundred days portraying returns that are normally distributed was found using the set of two hundred random stocks chosen. The 64

66 one hundred day period found from the testing was also used for the future tracking problem, as described in this section. Having tested the models (section 3.2), the unanimous decision amongst the group was to use the Big R model to create the index. Given the analysis done in the previous section, which showed the sensitivity of the approximated models solutions to the index size, it was felt that running an unconstrained model in terms of the number of assets included in the portfolio, was justified. The reasoning behind this was that as observed in the sensitivity test when the index size was constrained to include values in excess of approximately thirty stocks, the model still chose approximately thirty assets. The remainder of the assets that had positive allocations were degenerate; that is, they had proportions that were so minute that they were technically greater than zero, but because the value was so close to zero on the positive side (ex: 1 x 10-8 ), they essentially had no effect on the objective or tracking values. Thus, these assets chosen by the model in excess of the thirty when unconstrained were so insignificant that their proportions satisfied the constraints of the model, but did not impact the solution. 65

67 Table Ticker Names of The 200 Random Stocks Chosen From S&P 500, Year Results Running the model using 2005 as the historical data from which the index was to be created, the stock ticker names and respective proportions of the generated portfolio were found as seen in Table The future returns of the generated portfolio were plotted over the remainder of 2005 (days to ) and over the year of 2006 (days to ), along with the performance of the benchmark of two hundred random stocks from which the portfolio was chosen. To simplify matters, each stock in the benchmark was given an equal proportion with respect to its weighting on the 66

68 benchmark s return (each stock had a proportion of 1/200 = 0.005). This graph, along with the portfolio s associated tracking error may be seen in Appendix G.1. Table The Portfolio Selected By The Unconstrained Big R Model From The 200 Random Stocks Evaluation Looking at the performance of the generated portfolio in comparison to that of the benchmark, it seems that the portfolio does relatively well. There are of course some periods in which the index performs slightly better or worse than the benchmark. The reasoning behind this is relatively simple. The mathematical models used to generate these portfolios are very successful aids, but they base their allocations on historical performance only. As stated earlier, from observation, markets do seem to follow trends over discrete periods of time. However, there are simply too many variables that affect how a market, or a subset of that market perform and these variables can cause very significant fluctuations in terms of how a particular company or group of companies perform. To illustrate this, Appendix G.2 contains the internal tracking error of the stationary model, which covers the tracking of the generated portfolio with the market over the historical period used by the model. Intuitively, looking at the graph shows that 67

69 the internal tracking over the 100 day period in 2005 is not very good. As seen in section 3.4, this behaviour may be due to the non-normality of the asset returns in the selection pool, which can have significant impacts on the successful approximation of a model using absolute deviation, to that of mean variance. The behaviour may also be attributed to the plethora of unpredictable factors that affect market performance as stated above. Looking at the first 50 future periods for tracking, as in Appendix G.3 may help justify this statement, as the tracking seems to be much better than that of the internal tracking. The same may be said when looking at the future tracking over the last 50 day period of 2006, as in Appendix G.4. An ideal solution would be to use both qualitative and quantitative information to generate an index portfolio. Ideally, a model could be used as a tool to generate an index based on historical data. Once the allocation has been formed, one can go back and subjectively examine the relative proportions of each allocation. Then, based on thorough knowledge of current economic trends and data, one can make alterations as need be that will hopefully improve generated portfolio s tracking. One important improvement notable of mentioning would be to have strict guidelines as to the number and overall proportion of funds to invest in publicly listed companies of particular market sectors. This is extremely important when talking about diversification. 68

70 4 CONCLUSIONS The applications of mean absolute deviation optimization to indexing have been explored systematically with regard to its adherence to the mean variance model and its effectiveness in accurately tracking a given market with the desired size quantity. It was determined that the time saving benefits of modeling a large amount of market data into two stages had a large time-saving benefit; however, the second stage solution set is restricted to the pool provided by the first stage, limiting the resulting index to the smaller sizes permitted by this method. Revising the model to fit the objectives of indexing and tracking provided insight into the original MAD model s inherent tracking ability, as well as its ease of finding an integer solution with index size constraints given a moderate number of stocks in the pool. However, the ability of the MAD model to provide an index with objectives aligned with those of Modern Portfolio Theory was not found to be a function of the normality of the contributing market data. Other factors affecting the quality of the model are unclear. Lastly, the effect of index constraints on the model proved to be a problem with the MAD model only at very small index sizes, much smaller than would be used in practice. Solution time and index size played a moderate role in the time or solution quality of the resulting index; however, this role was much less of a nuisance than originally anticipated. However, the big problem with both the mean variance and MAD models with regard to constraining index size is the degeneracy of basic variables that occurs at index size specifications over approximately 30 stocks. If the optimal index for tracking the market includes equally distributed nominal allocations for degenerate values, then the model is a successful means of indexing. However, since these added amounts would 69

71 logically increase the objective value, and therefore increase the risk level of the index, there is likely a more effective way to have positive allocations and still maintain good tracking. One way to eliminate degeneracy may be through diversification; however, much more analysis must be done to discover a way to accommodate this issue. 4.1 Further Considerations The inclusion of several additional realistic constraints such as minimum transaction lot sizes (Appendix G), transaction costs and initial funds available for investing is essential to find an optimal solution that will better mimic the real world. The first constraint addresses the issue of minimum transaction lot sizes where investors generally practice purchasing bundles of shares. A possible approach to minimum transaction lots which has been previously studied is setting a minimum amount of money that must be invested in any stock given it is the optimal choice [13]. A diverse method that will be attempted to represent minimum lot sizes is by limiting the actual number of shares to purchase for a given stock to round lots (or multiples of 100). Therefore the constraint will restrict the model to purchase a minimum of 100 shares for a particular stock given it will produce an optimal outcome. However, the model is not constrained to choosing only 100 shares of a particular stock. As mentioned earlier, it may select any multiple of 100 such as 200, 300 or even 800. It must be mentioned that applying a minimum lot size causes a reduction in portfolio diversification [13]. This issue can be counterbalanced with the introduction of additional subjective constraints for an investor to impose minimum purchases of stocks pertaining to specific sectors within the economy. The next constraint that will be discussed is modeling fixed transaction costs. When buying or selling securities in the real market, a fixed brokerage fee is added for each security to the 70

72 total transaction cost [13]. This fee is varies depending on the type of brokerage service an investor has an account with. There are several drawbacks to incurring a fixed transaction cost into the model. It reduces the number of securities held thus increasing the overall risk of the portfolio and also the level of diversification is reduced [13]. The constraint that can be modeled must ensure that the summation of cost to purchase the necessary number of stocks and transaction fees does not surpass the initial monetary funds available to invest. Overall the constraints will help to simulate real world practices and thus provide a closer representation of the real world. Additionally, an important factor to consider is the market size from which the model selects the optimal index. Due to computational time constraints, a market size of 200 was used for testing. It would be beneficial for future work to conduct the same tests on a large sized market of say 500 to 1000 stocks, or by indexing in two stages as in section 3.3. By doing this, there is a larger variety of possibilities for the model to select for optimal indexing which could ultimately produce an improved answer. Potential drawbacks to conducting the tests on a larger market would be the time incurred to find an optimal solution once integer constraints are introduced. 71

73 5 REFERENCES [1] Department of Psychiatry. University of Toronto Logo. Photo. University of Western Ontario. 3 Oct < [2] Konno, Hiroshi, and Yamazaki, Hiroaki. Mean-Absolute Deviation Portfolio Optimization Model and Its Applications to Tokyo Stock Market. Management Science 37.5 (May, 1991): [3] Asset Classes. Internet: et_classes.html, 2007 [Nov. 7, 2007]. [4] Fabozzi, Frank J.. Financial Modeling of the Equity Market. Hoboken, New Jersey: John Wiley & Sons Inc., 2006, pp [5] Liang-Chuan Wu, Seng-Cho Chou, Chau-Chen Yang, and Chorng-Shyong Ong. Enhanced Index Investing Based on Goal Programming. The Journal of Portfolio Management, vol. 33, no. 3, pp , Spring [6] Ka Yui Leung, Charles, Equilibrium Correlation of Asset Price and Return, Discussion Papers, p. 3, [7] Frank J. Fabozzi, Frank J. Jones, and Raman Vardharaj. Determinants of Tracking Error for Equity Portfolios. The Journal of Investing, vol.13, no.2, pp , [8] Equity Indices: S&P 500. Internet: sp/en/us/page.topic/indices_500/2,3,2,2,0,0,0,0,0,0,0,0,0,0,0,0.html, 2007 [Nov. 11, 2007]. [9] B. Scherer and D. Martin. Introduction to Modern Portfolio Optimization. New York, NY: Springer, 2005, pp [10] George W. Blazenko, Corporate Leverage and the Distribution of Equity Returns. Journal of Business, Finance, and Accounting, Vol. 23, no.8, pp october [11] Mehmet Horasanlõ & Neslihan Fidan, Portfolio Selection by Using Time Varying Covariance Matrices. Journal of Economic and Social Research, vol. 9, no.2, pp.1-22 [12] A. Brealey, Richard, Principles of Corporate Finance. Toronto: McGraw-Hill Ryerson, 1992, pp [13] Grazia M. Speranza, Hans Keller, and Renata Mansini. Selecting Portfolios with Fixed Costs and Minimum Transaction Lots. Annals of Operations Research, vol. 99, no. 1-4, pp , December

74 [14] D.M. Blitzer. S&P Canadian Indices Index Methodology. Standard and Poor s. [Online document], 2007 Mar, [cited 2007 Nov 2], Available HTTP: 73

75 APPENDIX A: Mean Variance Model Minimize n n i= 1 j= 1 σ x ij i x j Subject to n r x ρ M j j o n x j = j=1 M o 0 x j u j, j=1,., n Where: r j σ ij = E[ R j ] = E[( R i r )( R i j r j )] ρ represents minimum rate of return u j is the maximum amount of money that can be invested into S j 74

76 APPENDIX B: ORIGINAL MAD MODEL Decision Variables: x j = the proportion of total funds invested in stock j Cost Variables: r jt = the return of asset j in period t r j = the average return of asset j a jt = r jt - r j M = the total amount of money available for investment ρ = the specified return of the index or portfolio Indices: Objective: Minimize average absolute deviation Constraints: 1. Equate y t to the absolute value of the proportionate sum of deviations at period t 2. The sum of returns for each investment j is greater than the total specified return 75

77 3. Sum of money invested in each asset is equal to the total amount available to invest 76

78 APPENDIX C: OPL MODEL FOR S&P 500 DATA STREAM I. OPL Model File: int NumPeriods =...; int NumStocks =...; range Periods = 1..NumPeriods; range Stocks = 1..NumStocks; float R =...; int M =...; float a[stocks][periods] =...; float r[stocks] =...; dvar float+ x[stocks]; dvar float+ y[periods]; constraint ct1; constraint ct2; constraint ct3; minimize (sum (t in Periods)(y[t]))/NumPeriods; subject to{ ct1 = { forall (t in Periods) (y[t] + (sum (j in Stocks) a[j][t] * x[j]) >= 0); forall (t in Periods) (y[t] - (sum (j in Stocks) a[j][t] * x[j]) >= 0); } ct2 = { (sum (j in Stocks) r[j] * x[j]) >= R * M; } ct3 = { (sum (j in Stocks) x[j]) == M; } } 77

79 execute{ for(k in Stocks) writeln("r[",k,"] = ",r[k]); for(l in Stocks) writeln("x[",l,"] = ",x[l]); } II. OPL Data File: SheetConnection sheet("2005.xls"); NumPeriods = 251; NumStocks = 477; R = ; M = 1; a from SheetRead(sheet,"Ajt!A1:IQ477"); r from SheetRead(sheet,"Rjt!IS1:IS477"); 78

80 APPENPDIX D: SENSITIVITY FOR CONSTRAINT The figure below shows the results of various solutions of the IP model for n = 20 stocks, using different constraints labeled above each result. Note here that the lowest level of proportion errors occurred by minimizing the slack around the second constraint in Appendix B. 79