LAGRANGIAN RELAXATION APPROACHES TO CARDINALITY CONSTRAINED PORTFOLIO SELECTION. Dexiang Wu

Transcription

1 LAGRANGIAN RELAXATION APPROACHES TO CARDINALITY CONSTRAINED PORTFOLIO SELECTION by Dexiang Wu A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Mechanical & Industrial Engineering University of Toronto Copyright 2016 by Dexiang Wu

2 Abstract LAGRANGIAN RELAXATION APPROACHES TO CARDINALITY CONSTRAINED PORTFOLIO SELECTION Dexiang Wu Doctor of Philosophy Graduate Department of Mechanical & Industrial Engineering University of Toronto 2016 Portfolio selection with cardinality constraint is a process that creates a strict subset of assets from a large selection pool. The advantage of cardinality constraint is that fewer assets can reduce transaction costs and complexity of asset management. Also, this type of constraint can be used to mimic a benchmark portfolio (index) such as S&P 500. In this dissertation we study two different cardinality constrained portfolio selection problems, known as Index Tracking and Financial Planning. Index Tracking is a typical application of the cardinality constrained portfolio selection process and has attracted much attention from portfolio managers. However, replicating unpredictable market indices using limited available resource requires advanced modelling and optimization techniques in practice. This thesis aims to qualitatively investigate and analyze different types of index tracking problems and the associated optimal strategies. Firstly, we construct the tracking portfolio via a constrained clustering approach which considers various practical aspects such as transaction costs, turnover, and sector limits constraints. We show that the portfolio allocation can diversify between different sectors and reduce the portfolio risk fairly well. Next we address a cardinality constrained Financial Planning problem through Stochastic Mixed Integer Programming and extend the network flow structured framework to index tracking problem. Finally, we incorporate the cardinality restriction to a classical mean-variance based tracking model and build the robust counterpart via Robust Optimization. ii

3 All developed models demand problem solvability due to the rapid increase in the number of variables and constraints for tracking real indices such as S&P 500. We design three dual decomposition algorithms, which allow different specific heuristics to be embedded, to quickly obtain high quality solutions for associated models. For example, Tabu Search was applied to solve the scenario sub-problems to speed up the Progressive Hedging algorithm for cardinality constrained financial planning problems. Our designed models are general enough to extend to many other management applications, and our accompanied decomposition algorithms are efficient enough to handle the cardinality constraint in these problems. The generated portfolios illustrate the effectiveness of our selection technologies and designed algorithms in terms of different performance metrics with respect to the market. iii

4 Dedication To Tina and Mandy iv

5 Acknowledgements This dissertation would not have been possible without the support of many remarkable people to whom I would like to express my sincere gratitude. First and foremost, I would like to thank my supervisor, Professor Roy H. Kwon, for his consistent support of my Ph.D study and related research, for his patience, inspiration, and immense knowledge, and for many appropriate advices that improve the quality and contribution of my papers. His guidance helped me in all the time of research and writing of this thesis. I could not have imagined having a better advisor and mentor for my Ph.D study. Besides my supervisor, I want to thank Professor Yuri Lawryshyn and Professor Timothy Chan for their insightful comments and wonderful suggestions to improve my research from various perspectives while serving on my supervising committee. I also want to thank Professor Oleksandr Romanko and Professor Hani Naguib for their time and remarks as members of the examination committee. Also, I would like to thank Professor Seong Moon Kim for taking time out from his busy schedule to serve as my external reviewer. I would like to thank all the members of the University of Toronto Operations Research Group (UTORG), which provides me many excellent opportunities to meet with unique individuals from all over the world. Finally, I appreciate the financial support from CSC that funded parts of my studies. v

6 Contents 1 Introduction and Thesis Outline Background of Portfolio Optimization Research Objective and Contribution Thesis Outline Modern Portfolio Theory and Index Tracking Literature review for MVO and Its Extension Literature review for Index Tracking Lagrangian Relaxation in Literature Metaheuristics in Literature Literature review for LR and Its Extension A Constrained Clustering Approach for Index Tracking Introduction Literature Review for Index Tracking Model Formulations Basic cluster-based index tracking model Model with buy-in threshold and turnover constraints Basic model with sector limits The model with trading and sector diversification constraints Tractability of the cluster-based Models Lagrangian Relaxation Algorithms Computational Results: Tracking the S&P vi

7 4.5.1 Parameter Estimation LR versus SLR Comparison between 4 models Conclusions and Discussion Progressive Hedging for Cardi. Constrained FP Introduction to Financial Planning Problem Model Development Equivalent Cardinality Constrained FP Models Scenario Generation Lagrangian Decomposition Scheme LR method for scenario sub-problem Tabu search for scenario sub-problem Progressive Hedging for FP problem Design a lower bound Progressive Hedging method Numerical experiment Progressive Hedging for Index Tracking problem Conclusions and Discussion Lagrangian Relaxation for CCCP Introduction to CCCP Literature Review Lagrangian Relaxation Scheme Robust Factor model to Index Tracking Nominal Index Tracking Model Robust Multi-Factor Model for Index Tracking Computational Experiments Testing the Three-Factor and Single-Factor models Index Tracking using the S&P100 Index Index Tracking using the S&P500 Index vii

8 6.5.4 Index Tracking using the Russell 1000 Index Index Tracking using the Russell 3000 Index Conclusions and Discussion Conclusion and Future Research Conclusion Future Research Modelling discussion Algorithm discussion Bibliography 144 A Appendix of Chapter A. 1 Numerical example for Heuristic I A. 2 Numerical example for Heuristic II A. 3 Ticker in S&P A. 4 Gap by LR and SLR A. 5 Sector Allocation B Appendix of Chapter B. 1 The pseudocode for LR sub-solver B. 2 The pseudocode for Tabu search sub-solver B. 3 Speed up solving process for sub-problem C Appendix of Chapter C. 1 Parameter generation for the robust tracking model C. 2 LR gap information (S&P500) viii

9 List of Tables 4.1 Model test by Gurobi (q = 10) Time comparison for updating dual in LR method Parameter Setting Sharpe ratio for out-of-samples ( ) Sharpe ratio for out-of-samples ( ) Sharpe ratio for out-of-samples ( ) Sharpe ratio for out-of-samples ( ) Model Comparison - with and without transaction cost term LR method and Gurobi Comparison - instance Computational result (N=50, K=5, S=15) - instance Computational result (N=100, K=10, S=3 - instance 2) Computational result (N=100, K=10, S=10 - instance 3) Computational result in literature Parameter setting for the model and PH algorithm Bound details under different methods for S= Bound details under different methods for S= Bound details under different methods for S= Bound details under different methods for S= Numerical result (N=100, K, S=15) Numerical result (N=100, K, S=30) Numerical result (N=100, K, S=50) Numerical result (N=100, K, S=75) ix

10 5.16 Test different ratios (N=100, K, S) R 2 value for the regression models Ticker symbol across Sectors (SP100) The average TE/TC ratios under different size Tracking ratio comparison Bounds information (SP500) Bounds information (Russell 1000) Bounds information (Russell 3000), TE=4STD Bounds information (Russell 3000), TE=3STD A.1 Ticker symbol across Sectors (SP500) A.2 Gap between LB and UB, B.1 LR method and Gurobi Comparison - instance B.2 LR method and Gurobi Comparison - instance B.3 LR method and Gurobi Comparison - instance B.4 LR under different iteration number B.5 Tabu search under different (L, iter number, M) B.6 LR and Tabu comparison (N=100, K=10, S=15) B.7 LR and Tabu comparison (N=100, K=15, S=15) B.8 LR and Tabu comparison (N=100, K=20, S=15) B.9 LR and Tabu comparison (N=100, K=25, S=15) B.10 LR and Tabu comparison (N=100, K=30, S=15) C.1 Bounds information (SP500) x

11 List of Figures 1.1 Thesis Structure Efficient frontier with and without cardinality constraint Lagrangian Decomposition Scheme for integer programs Gap Comparison between LR and SLR Norm of sector differences between constructed portfolio and S&P Sector diversification Comparison of Performance optimal objective value Comparison of Performance portfolio return Comparison of Performance portfolio variance Comparison of Performance portfolio Sharpe ratio Comparison of Performance Tracking Ratio of out-of-sample period (2007, 2008) Comparison of Performance Tracking Ratio of out-of-sample period (2009, 2011) Network structure with cardinality at stage 0 and Equivalent scenario trees Running time of PH method for different problems Portfolio return vs TE with different q under different σ (SP100) Portfolio variance vs TE with different q under different σ (SP100) Portfolio Sharpe ratio vs TE with different q under different σ (SP100) xi

12 6.4 Robust bound for expected return and variance (SP100) Wealth evolutions for rolling out-of-samples Model comparison - portfolio return Model comparison - portfolio variance Model comparison - portfolio Sharpe ratio Model comparison - Tracking error Tracking Error to Transaction costs ratios (SP100) TE/TC ratios with respect to the trading ratio α Model comparison - Tracking ratio Iteration details (SP500) Bounds and gap comparison by LR method (SP500) Gurobi iteration details for different size q A.1 Portfolio allocation in sectors xii

13 Chapter 1 Introduction and Thesis Outline 1.1 Background of Portfolio Optimization Making a trade-off between the expected rate of return and variance of the rate of return for a portfolio is at the heart of mean-variance optimization (MVO). MVO was initially established by Markowitz in 1952 [107] and provides a foundation for single-period investment theory. The MVO framework offered a rigorous risk management tool for investors and inspired the subsequent Capital Asset Pricing Model (CAPM) in the 1960s [135] and the concept of the Sharpe ratio [134] that can be used to appraise portfolio performance. According to the MVO and the CAPM, risk-averse investors only need to determine their budget allocation to a single fund of risky assets and the risk-free asset to achieve efficient portfolios (see the one-fund theorem in [104]). The single master fund usually refers to specific market indices because theoretically one cannot find a single fund that include all assets in the world, and practically typical indices have relative long-term outperformances than that of the active investments. For example, Zenios reported that the average return of 769 all-equity actively managed funds was 2% to 5% lower than the S&P 500 index during the period [151]. More recently, Standard & Poor s Scorecard has reported that from the 5 years and 10 years before Dec. 31, 2014, more than 88% and 82% of actively managed large-cap funds were outperformed by the S&P 500, respectively [1]. These evidence show that tracking benchmark portfolios as closely as possible is an efficient representative of the one-fund theorem. Therefore, exchange-traded funds (ETFs) that replicate the market indices increased exponentially since the 1990s. The proliferation and 1

14 Chapter 1. Introduction and Thesis Outline 2 demand of market index ETFs such as the SPDR S&P 500 Index ETF is a reflection of the demand in investment in broad markets as opposed to actively managed investments that try to beat the markets. ETFs allow a broader participation in investment in major market indices since it is the ETF company that is responsible for replicating an index, i.e. investing to mimic the risk and return profile of a market index. A key strategic decision of an ETF company is the construction of a portfolio that mimics a given benchmark market index. However, this is not a trivial task and is often referred to as index tracking. An index based ETF attempts to reproduce the performance of a specific index by holding all constituents of the index and trading less frequently, e.g. one or two times a week. To perfectly mimic the target portfolio, all assets in the benchmark are held in the quantities specified by the weightings of the benchmark portfolio. The full replication strategy inherently diversifies the allocation across the entire benchmark index. However, full replication is not practical given the transaction costs this would entail. For example, fully replicating the S&P 500 index would require holding the 500 assets along with weightings for each asset. The weightings are based on market capitalization and change constantly based on the asset prices. Constant re-balancing of the tracking portfolio would result in a prohibitive number of transactions. Also, certain stocks in the index with small market-cap weights have to be held in full replication portfolios, which will result in illiquidity and especially be undesirable for tracking small-cap indices. To overcome these issues, an alternative strategy is to select a strict subset of assets from the benchmark and match the benchmark as closely as possible, and obviously tracking errors between the tracking portfolio and benchmark index will be generated. Practically, cardinality constraints that restrict the portfolio as a subset of the assets constitute the index are crucial for implementing the partial replication strategy. A tracking portfolio with fewer assets can avoid the small fraction holdings and reduce transaction costs compared with the fund who purchases all of the stocks that make up the index. In addition, the tracking portfolio with the cardinality constraint can simplify the complexity of asset management and reduce administrative overhead and administration costs. However, several challenges need to be considered. First, it is not easy to keep a stable and robust tracking portfolio as the movement of the index is unpredictable under uncertain market environment. Secondly, tracking large indices with different practical constraints usually

15 Chapter 1. Introduction and Thesis Outline 3 encounters the bottleneck of solvability. This thesis aims to explore and construct different cardinality constrained index tracking models using optimization fashion and therefore test the one-fund theorem empirically. It is well known that estimator errors for parameters in portfolio selection models can affect the optimal portfolio significantly. Many approaches have been proposed in the literature to prevent under- and over-estimation of parameters and then to enhance the robustness of the solution structure. Recourse-based stochastic programming [24] is a prevalent tool for immunizing against estimator errors. In this approach, a recourse decision is obtained in the second stage to compensate for the effects of the first-stage decision that is fixed ahead for given uncertainty sets. For example, Asset Liability Management (ALM) is an investment strategy that covers the liability over a multi-period horizon [152]. Financial Planning model is another classical topic in financial optimization that uses the network flow structure to match anticipated deposits and liabilities under different future scenarios through multi-stage stochastic programming [115]. Robust optimization is one alternative to immunize against parameter uncertainty and is particularly suitable for portfolio selection models in which risk controls are heavily involved [13]. The strategy of robust optimization refers to the use of a finite worst-case scenario to represent the infiniteness of the uncertainty set while maintaining the same level of complexity as a nominal problem. Moreover, the adaptive features of robust optimization allow us to conveniently merge other techniques such as a factor model, i.e., factorbased MVO selection [66]. Another important optimization stream for portfolio selection is to apply the idea of Value at Risk (VaR) and then Conditional Value at Risk (CVaR) in which greatest concern is on the default risk of an investment. CVaR selection models [126] have received additional attention since they are convex like the MVO model. In this dissertation, we primarily apply the stochastic programming and the robust optimization approaches to study the issue of parameter uncertainty for different index tracking models. Portfolio selection models also have been developed in Operational Research with many different types of practical constraints including buy-in threshold, turnover, tracking error, sector limit, cardinality, and round-lot constraints. Cardinality constraints draw special attention from academics and this thesis not only because they are key to solving index tracking problems but also because they increase the complexity of solving the problem due to the binary

16 Chapter 1. Introduction and Thesis Outline 4 requirement in the model. With the rapid development of computer science and operations research in the last two decades, one can efficiently obtain the optimal portfolio from a large selection pool through MVO-based models within a reasonably short time. Although polynomial iteration-complexity algorithms, e.g. interior-point based methods, are available to large-scale MVO problem since the 1990s, a key practical issue to portfolio managers is that the optimal portfolio allocation may concentrate on a few assets which may result in high portfolio risk, or diversify too broadly and lead to high transaction costs. Thus additional trade-off is between the portfolio size, risk, and managemental cost arises to investors and ETF companies. One way to implement this trade-off is to use cardinality constraints but obtaining the associated solution is non-trivial. Typical solution methods for cardinality constrained portfolio selection in the existing literature can be categorized into two main groups. The first group of methods mainly focuses on cut generation for branch-and-bound algorithms [30, 22] or relies on heuristics designed to satisfy cardinality constraints [10]. The second group either reformulates binary variables as a set of conic constraints or reconstructs the cardinality constraints into a non-convex SDP, and employs the semidefinite relaxation to approximate the non-convex programs [123, 33]. Meanwhile, software packages using branch-and-bound methods are currently available to handle mixed integer conic programming, e.g. SeDuMi [140], MOSEK [113], CPLEX [42], and GUROBI [71]. Their solutions are usually used as benchmarks by researchers who propose new methods. For example, we mainly compare the solutions generated by our proposed Lagrangian methods for different partial tracking models with that from the Gurobi mixed integer solvers in this thesis. Many companies that offer ETFs to the open public are large financial institutions that will invariably use portfolio management systems e.g. computer-based decision support to assist in construction of (tracking) portfolios in modern finance [149]. In particular, optimization-based decision support can be even more relevant for portfolio optimization where in addition to database and statistical modules, an optimization module is present that contains mathematical models and algorithms [17]. But a central challenge for any optimization-based decision support is to have mathematical models that not only can track a given benchmark well, but that can also be solved within a reasonable amount of time [138].

17 Chapter 1. Introduction and Thesis Outline Research Objective and Contribution The main objective of this thesis is to demonstrate that portfolio selection via tracking typical indices are crucial for risk management in investment science. Studying and mimicking the indices is a key step to obtain the efficient portfolio. Meanwhile, with the mathematical and computational developments, more practical restrictions can now be incorporated, and the financial engineering trend to select portfolio is more prevalent and applicable. In this dissertation, two types of well-known financial problems are introduced, modelled realistically, and solved efficiently. We sketched and generalized these financial problems in terms of risk control through advanced mathematical programming. The designed models are accompanied by decomposition algorithms which overcome computational challenges that have prevented previous attempts. The contributions of this thesis can be described from two perspectives. First we developed three financial models: ˆ A cluster-based approach for index tracking. A tracking portfolio model that includes practical constraints controlling the portfolio size, the buy-in thresholds, the transaction costs for re-balancing, and the sector centralization. ˆ A two-stage cardinality constrained financial planning problem with a network flow structure. The designed portfolio model not only contains the constraints that limit the size of the portfolio, the buy-in thresholds, and the transaction cost of cash-flows, but also considers asset return uncertainties via an advanced Stochastic Programming approach. A financial planning framework that extends to index tracking is also examined. ˆ A factor based robust index tracking model which considers a three-dimensional trade-off between the portfolio return, portfolio risk (e.g. variance and tracking error), and portfolio size. The robust factor model takes account of uncertainty in the assets expected return and variance. The designed model can be captured by a general cardinality constrained conic framework. The three above investigations encompass several important characteristics of portfolio design such as portfolio size, sector diversification, re-balancing and transaction costs, and consid-

18 Chapter 1. Introduction and Thesis Outline 6 eration of uncertainties associated with future circumstances of financial markets or investors goals. These developed models combine different risk control tools for portfolio selection. These realistic and sophisticated modelling techniques are highlighted and useful with respect to the market environment through in-sample and out-of-sample analyses. To overcome the large-scale computational difficulties associated with the solution process of these models, we summarize our promising Lagrangian decomposition strategies as follows: ˆ Lagrangian and Semi-Lagrangian relaxation methods to decompose the clustering tracking models across different sectors. A variable neighborhood search heuristic using the LR bound information is embedded into the LR framework to yield a near-optimal solution. ˆ Progressive Hedging which decomposes the cardinality constrained financial planning models across different scenarios. Tabu search and LR methods are designed to quickly solve the hard sub-problems. ˆ A Lagrangian relaxation method to decomposes the factor-based robust index tracking model across different variable space. The proposed solution methods that solve state-of-the-art financial problems, and the effectiveness of the modelling techniques relevant to the developing field of portfolio optimization have been studied and provided in this dissertation. 1.3 Thesis Outline The rest of the thesis is organized as follow: In Chapter 2 we present a literature review of MVO-based portfolio selection models, and then a literature review of index tracking models and its extension. In Chapter 3 we briefly review different types of algorithms for cardinality constrained selection models and draw attention to Lagrangian relaxation methods in the literature for financial problems. In Chapter 4, we consider various characteristics of a not well-known index tracking model and design a Lagrangian based algorithm to approximate high-quality solutions. In Chapter 5, we present a network structure financial planning framework with cardinality constraints that captures various sources of uncertainty through a mixed integer stochastic program with recourse. In Chapter 6 factor-based robust index tracking

19 Chapter 1. Introduction and Thesis Outline 7 is generalized by the proposed cardinality constrained conic program which can be efficiently solved via the proposed Lagrangian algorithm. Chapters 4 to 6 display in-sample and out-ofsample test results that focus on the real financial market, which form the backbone of the thesis. Finally, we conclude our work and discuss future research directions in Chapter 7. We display the thesis structure in the following Figure (1.1): Introduction Ch1 Portfolio optimization and extension Ch2&3 practical constraints uncertainty algorithms Chapter 4 Chapter 5 Chapter 6 Conclusion and discussion Ch7 Figure 1.1: Thesis Structure As shown in the Figure (1.1), the structure of the thesis can be unified from three points of views. First we construct the tracking portfolios via a predominant model and different alternatives. The goal of these investigations in Chapters 4 to 6 is to illustrate and prove the effectiveness of the one-fund theorem in modern finance [104]. Secondly, we implement these index tracking approaches through cardinality constraints and therefore lead to NPhard problems. Therefore, methodologically we unify these projects via a dual decomposition framework that integrates different metaheuristics. We list a more detailed overview for each chapter as follows.

20 Chapter 1. Introduction and Thesis Outline 8 Chapter 2 - Modern Portfolio Theory and Index Tracking We comprehensively review the history of the Mean-Variance Optimization (MVO) model and its extensions in this chapter. Many researchers have proposed modification to the MVO framework after the introduction of Harry Markowitz s Mean-Variance Optimization (MVO) model in We examine these models through a literature review of the current approaches to portfolio selection, and define important characteristics relevant to this thesis. In particular, we survey different index tracking problems such as enhanced indexation and approaches that incorporate the parameter uncertainty in the literature. Chapter 3 - Lagrangian Relaxation in Literature We provide a history of the application of the Lagrangian approach to different management problems, especially relative to the problems in financial optimization. We then explain the mechanism of the dual decomposition through a simple numerical example and review major variations of LR methods in the literature. We point out that LR methods are crucial for solving index tracking problems not only because the metaheuristics can be easily embedded into the dual decomposition scheme but also as the bound information can be used to quickly generate high-quality solutions. Chapter 4 - A Constrained Clustering Approach for Index Tracking We consider the problem of tracking a benchmark target portfolio of financial securities, in particular the S&P 500. Linear integer programming models are developed that seek to track a target portfolio using a strict subset of securities from the benchmark portfolio. The models represent a clustering approach to the selection of securities and also include additional constraints that aim to control risk and transaction costs. Lagrangian and semi-lagrangian methods are developed to compute solutions to the tracking models. The computational results show the effectiveness of the linear tracking models and the computational methods in tracking the S&P 500. Overall, the models and methods presented can serve as the basis of an optimization-based decision support model for creating tracking portfolios. Chapter 5 - Progressive Hedging for Cardinality Constrained FP Problem Cardinality constrained Financial Planning (FP) problems are described using a network flow structure in this chapter. We outline how the special characteristics of this structure can be

21 Chapter 1. Introduction and Thesis Outline 9 used to fully encompass a comprehensive set of real-world portfolio elements and considers market uncertainties. The network structure cardinality constrained Financial Planning problem is formulated as a Stochastic Mixed Integer Program (SMIP). The proposed FP framework can be naturally extended to an index tracking problem. We apply a dual decomposition method, Progressive Hedging (PH), to efficiently accommodate instances with large numbers of scenarios. Solving the scenario sub-problems is crucial for the proposed PH algorithm. Therefore, Lagrangian relaxation and Tabu search methods are designed for handling the scenario subproblem, and numerical results show that our sub-solver reduce the solving time significantly compared with the time information by Gurobi. Moreover, a Lagrangian lower bound was embedded into the PH method and, as a result, better gap information is obtained compared with the gap obtained by Gurobi. Chapter 6 - Lagrangian Relaxation for CCCP We study a class of Cardinality Constrained Conic Programming (CCCP) that is suitable for the robust index tracking problem in this chapter. A robust version of the Fama-French three factor model is developed whereby uncertainty sets for the expected return and factor loading matrix are generated. The resulting model is a mixed integer second-order conic problem. Computational results in tracking the S&P 100 out-of-sample show that the robust model can generate portfolios that have a better tracking error and Sharpe ratio than those generated by the nominal model. We then present a method to approximate the optimal solution by using the bound information generated from its Lagrangian dual. This strategy allows us to decompose the CCCP into two easier subcases and calculate a tight lower bound and feasible upper bound quickly. Meanwhile, sub-gradient cut and fully regular cuts are obtained to exclude sub-optimal points that have been explored in previous iterations. Computational results in tracking the S&P 500 and Russell 1000 show that the proposed method has practical effectiveness for the class of CCCP problem we are addressing. Chapter 7 - Conclusion and Future Research We summarize the conclusion and the findings of the models we investigated in Chapters 4 to 6. The results that we present in this thesis enhance the applicability and adaptation of portfolio optimization in finance. We describe future research directions relevant to the fields of finance,

22 Chapter 1. Introduction and Thesis Outline 10 optimization, and computer science. We also discuss alternative models and methodologies that can be used as points of comparison for with our current work.

23 Chapter 2 Modern Portfolio Theory and Index Tracking From the one-fund theorem [104], we know that any efficient portfolio can be expressed as a combination of a single master fund and a specific risk-free asset. That is, we can obtain all different efficient points via changing the weighting between these two assets, and measure the risk of the market. However, the single master fund is not perfectly available as it requires the fund contains an asset set as large as possible, ideally includes all the assets in the world. In practice investors usually represent the single master fund by using different typical market indices in different countries such as S&P 500 (USA), DAX 100 (German), the Hang Seng (Hong Kong), FTSE 100 (UK), and Nikkei 225 (Japan). These market indices generally consist of excellent companies in associated countries and regions and have good enough performance, and thus are adopted by different investors. For example, risk-averse investors are more prefer to allocate most of their budget on bond indices, while aggressive investors may mainly use stock indices as their benchmark. Also, the performance of an index can affect the decision that whether to invest the foreign market since the index reflects the economic fundamentals of the country. Therefore, although the single master fund seems hard to obtain theoretically, it is possible to approximate the single fund by combining and replicating different indices. Thus efficiently replicating an index is very important to investors and ETF companies. As mentioned in Chapter 1, the strategy of full replication that holds all of the stocks in the same proportions as in the index has a number of disadvantages. For instance, the ineffectiveness to 11

24 Chapter 2. Modern Portfolio Theory and Index Tracking 12 purchase and hold very small fractions of certain stocks, high transaction costs of rebalancing all the positions in the index, and the illiquidity of certain stocks for tracking small-cap indices. Cardinality restriction to the replication process, on the other hand, partially mimic the index but can overcome these issues. Based the MVO and the CAPM, superior risk-adjusted returns are impossible to obtain in an efficient market, and investors only need to follow and replicate the market indices. The goal of this thesis is to support the one-fund theorem and illustrate that partial replication through professional and advanced tracking models are crucial in modern investment science. Specifically, we study three types of index tracking models with cardinality constraints. ˆ First we develop a cluster-based approach for tracking based on a model of Cornuejols and Tutuncu [40]. The cluster-based tracking models avoid using the first moments information, i.e. expected return µ, which are hard to estimate, and keep the problem as a linear mixed integer optimization programs. Numerical result for tracking S&P 500 show the alternative approach is a powerful tool to construct tracking portfolios. ˆ In the second approach, we first incorporate the cardinality constraints to a Financial Planning model by Mulvey and Vladimirou [115], then we extend the network structure framework to index tracking problem. Numerical results show that establish alternative can track S&P 100 successfully under numerous scenarios of the expected returns. ˆ Finally, we consider the cardinality constraint to a traditional MVO-based tracking model, and develop it to a cardinality constrained robust factor-based enhanced-index tracking model via building the robust counterparts for the tracking error and portfolio risk constraints. Numerical results based on S&P 100 show the enhanced ability of the robust portfolios in terms of tracking error and Sharp ratio compared with those generated by the nominal model. Of course, there are different tracking models tailored for indices replication problem which have been extensively developed in the last decade. To clearly see the main development of the modern portfolio theory, we first review the Mean-Variance Optimization (MVO) selection model and its broad extensions, then we focus on the cardinality constrained selection approaches, primarily index tracking models, in the literature. Since the cardinality constraints

25 Chapter 2. Modern Portfolio Theory and Index Tracking 13 increase the complexity of obtaining the tracking portfolio, we will also review the algorithms used in the literature in next chapter. 2.1 Literature review for MVO and Its Extension The goal of investing different tradeable financial instruments in the market is to maximize profit for a given tolerance of loss on his balance sheet. A tradeable financial instrument, e.g. bond, stock, is a legal agreement carrying monetary value and can be circulated between different investors. The process of determining and combining of the weights of the selected securities is called portfolio selection, which leads to a portfolio with lower risk than the assets that compose it when taken individually as these assets are usually affected in opposite directions by unpredicted future events and partial of risk can offset each other. The MVO selection model by Markowitz in 1952 [107] is the first systematic and quantitative treatment that take into account the balance of portfolio return and risk. Suppose that there are n risky asset can be selected. Let r i be the random return of asset i, the expected return of asset i is µ i, and the covariance between assets i and j is σ ij, then for a given weight x the portfolio return r p = n i=1 µ p = n µ i x i, and the portfolio variance is expressed as: i=1 ( [ n ) 2 σp 2 = E (r p µ p ) 2] n = E r i x i µ i x i i=1 i=1 j=1 i=1 r i x i, the expected return of the portfolio ( n ) n n n = E (r i µ i ) x i (r j µ j ) x j = E (r i µ i ) (r j µ j ) x i x j n n = σ ij x i x j i=1 j=1 i=1 j=1 Portfolio variance in (2.1) gives an intuitive and quantitative measure to the loss of an investment. The remained task is to determine the proportion of the wealth to each asset, thus in MVO framework the optimal portfolio weight x is generated by solving following quadratical model: (2.1) min n n σ ij x i x j (2.2) i=1 j=1

26 Chapter 2. Modern Portfolio Theory and Index Tracking 14 s.t. n µ i x i R, (2.3) i=1 n x i = 1, (2.4) i=1 lb i x i ub i, i = 1,, n (2.5) where lb, ub are the lower and upper bounds of the proportion to asset i. lb 0 denotes the short selling is prohibited. A brief story to above model is that one wants to achieve a portfolio with minimum loss i.e. objective (2.2) with designed return i.e. constraint (2.3) under limited budget i.e. constraint (2.4) and (2.5). Finding a solution to the basic MVO model is trivial because of the fact that the covariance matrix always is positive semi-definite (PSD). The efficient frontier which represents a trade-off between portfolio return and risk is produced by generating the corresponding variance under the designed portfolio goal R, see red-circle curve in Figure (2.1). The adaptable properties of the basic MVO allow people to develop the model along various directions. The first influential consequence is what is known as Capital Asset Pricing Model (CAPM), which is a collision between the MVO and factor models, was primarily developed by Sharpe [135], Lintner [101] and Mossin [114] in the 1960s. The factor based MVO model keeps inspiring many researchers to explore suitable factors to interpret the connection between the market and assets. For example, Fama and French [54] extended the CAPM model based on the observation that small-capitalization stocks and value stocks (i.e. stocks with a high book to price ratio) tend to outperform the market as a whole. In the model, three risk factors reflect the sensitivities of each stock to the market excess return (market factor), the excess of value stocks over growth stocks (book-to-market factor), and the excess of small-cap stocks over large-cap stocks (size factor). Black and Litterman [25] used the prior observations of the market equilibrium (market factor) and investor s views (confidence factor), and applied the Bayesian inference to adjust the mean and variance to build a robust coefficient for MVO model. Burmeister, Roll, and Ross [28] presented a macroeconomic factor model that considers five risk terms, which are the investor confidence, interest rate, business cycle, inflation and market index, in interpreting the historical stock returns. It turns out that these models explain the cross-sectional variation in asset returns fairly well. Contemporaneously, Fama et al. [55] pointed out that the market can adjust new information to the asset price rapidly, which offers a strong evidence for the efficient market hypothesis. Many articles then further

27 Chapter 2. Modern Portfolio Theory and Index Tracking 15 demonstrated that the asset price is unpredictable over a short term but may be forecasted by regression analysis in a long run, see [53, 132, 73]. Therefore, the CAPM model suggests that every efficient portfolio should be priced at an equilibrium where a weighted linear combination of the market and the risk-free asset is obtained. This conclusion gives rise to a prominent application i.e. index fund or index tracking in modern finance. Sharpe and Markowitz shared the Nobel Memorial Prize in Economic Sciences in 1990 due to their distinguished work on portfolio allocation and asset pricing, and Fama, Hansen, and Shiller shared the Nobel Memorial Prize in Economic Sciences in 2013 because of their initial finding and contribution to an understanding of long-term market behaviour which is used as theoretical and empirical support for constructing and tracking indices. Some researchers seek to simplify the basic MVO model in terms of the computational complexity or risk measurement. For instance, Konno and Yamazaki [94] found that the MVO model can be converted into a Mean-Absolute Deviation (MAD) model under the condition that the asset returns follow the multivariate normally distribution. Besides MAD framework, VaR and CVaR are important alternative measures for risk management, and associated VaR and CVaR models are also prevalent in the literature. VaR measurement was firstly applied by the Basel Committee on Banking in 1996 and then broadly adopted in the financial industry. Unlike the MVO model, which adopts the symmetric risk measurement for portfolio, VaR and CVaR constraints mainly measure the downside loss of an investment. Since the VaR constraint lacks sub-additivity property and may result in local minima, Rockafellar and Uryasev [126] proposed a CVaR model which captures the average loss to evaluate the credit risk of a portfolio. Both MAD and CVaR models are linear programs which can be efficiently solved for large-scale applications. The basic MVO allows people to incorporate different practical constraints into the selection procedure, which consists of the second extensional stream. Some typical constraints in practice are described as follows: ˆ Buy-in threshold constraint which is used to avoid small fraction investment in the portfolio. This constraint can be implemented by adjusting the values of lb i and ub i for asset i in constraint 2.5.

28 Chapter 2. Modern Portfolio Theory and Index Tracking 16 ˆ Turnover constraint which is applied to limit the transaction cost for the portfolio construction or re-balance. The most common mathematically implementation is expressed n as a linear turnover form, xi x 0 i α γ, in which x 0 i denotes the initial portfolio i=1 weight, α denotes the unit trading cost and γ denotes the trading budget. This type of constraint can be convexified through convex it into equivalent set of linear constraint (see details in Chapter 4). ˆ Tracking error constraint is useful for index fund manager who is interested in a comparison or small outperformance with a specific benchmark such as S&P 500. This constraint can be formulated as n n σ ij (x i x ib ) (x j x jb ) T E, where x B is the weights of the i=1j=1 benchmark. We will investigate this constraint in Chapter 6. ˆ Cardinality constraint used to control the portfolio size via introducing new binary variable y and modifying the constraint (2.5), is expressed as: lb i y i x i ub i y i, i = 1,, n n y i = q i=1 y i {0, 1}, i = 1,, n (2.6) ˆ Round lot constraint is designed to improve the liquid of the portfolio through dividing the trading shares into small blocks. One can add the following equation into the MVO framework: x i = z i f i = p iz i M C, i = 1,, n, where z i Z is an integer number of rounding lots, f i be fraction of the portfolio wealth, p i denotes the trading price of asset i, M denotes the round lots, and C denotes the total portfolio wealth. ˆ Chance constraint is used to measure the downside risk of an investment. Mathematical expression can be wrote as Pr ( µ T x β ) 1 α, where β is the psychological threshold to a portfolio performance e.g. maximal loss, and α denotes the confidence level. Besides the popular restrictions previously mentioned, we show that sector limit constraint considered in Chapter 4 is also a useful way to diversify the portfolio across sectors. The basic MVO (2.2) - (2.5) with buy-in threshold constraint, turnover constraint and tracking error constraint remains the convex property so it can be efficiently solved by interior point

29 Chapter 2. Modern Portfolio Theory and Index Tracking 17 based algorithms. In contrast, the combination of the basic MVO with cardinality constraint and round lot constraint becomes a quadratic mixed integer programming. Although integer requirement changes the problem to be NP-hard, there are explicable benefits behind these constraints. For example, although the cardinality constraint destroys the smooth of the efficient frontier, such restriction can replicate the efficient portfolio with cheaper cost. One example depicted in Figure (2.1) illustrates this idea. Assume that we select 2 out of 4 assets to build the portfolio, the short selling is allowed. We draw all efficient frontiers for any 2 assets picked which are represented by the dash lines, and take a fractional piece from each EF to sketch the whole efficient frontier under portfolio size that equals 2 i.e. the black-start curve. It is clear to see that the original EF (red-circle curve) only have one capital market line for a given risk-free asset while the EF with the cardinality constraint may draw different tangle lines in different ranges for the same given risk-free asset. One observation is that we can efficiently approximate the market (q = 4) with a smaller size portfolio (q = 2), e.g. R 6%. This example also illustrates the idea of index tracking. Figure 2.1: Efficient frontier with and without cardinality constraint Many articles in the literature offer alternative insights into different practical constraints.

30 Chapter 2. Modern Portfolio Theory and Index Tracking 18 Konno and Kobayashi [93] constructed a reliable stock-bond portfolio via integrating different asset classes into MVO. Adcock and Meade [4] proposed a pure MVO-based portfolio selection model with transaction cost constraints and applied an efficient algorithm that quickly generate the optimal solution. Jobst et al. [84] studied the MVO model with buy-in threshold constraint, round-lot constraint and cardinality constraint in a whole model, and examined the effect of these constraints on the changing efficient frontiers. One key issue of the MVO model is that the optimal portfolio is extremely sensitive to the estimated parameters i.e. expected returns and covariances between assets [36]. That is, a tiny amount of changing in expected return or covariance derive from a short-time price movement will result in significantly different portfolio allocations. For example, Tutuncu and Koenig [144] demonstrated that the efficient frontiers under nominal inputs can be drastically changed within only 5 percentiles for means of monthly log-returns and covariances of these returns. Chopra and Ziemba [36] showed that estimated errors in the expected returns are 9 12 times more important than errors in covariances, which indicates any small increase in covariance matrix may amplify the portfolio Sharpe ratio 10 times. Since the MVO framework involves the estimate of asset return and variance, it is believed that the estimation errors will also affect the optimal portfolio significantly. To address this issue, another important stream of MVO extension has been explored in Operational Research that focuses on finding stable portfolios that are immune to uncertainties over time. This stream is referred to as multi-periods portfolio selection. Hakansson [72] found that the variance of the efficient portfolio over multi-periods is irrelevant to the return under the transformation of a suitable utility function. His findings became the basis of the portfolio choice theory. Therefore, many investment problems only focus on dealing with uncertainty for expected return of asset over multi-period horizon by using stochastic programming with recourse, e.g. Asset Liability Management (ALM) and Financial Planning problems we discussed in Section (1.1) in Chapter 1. In stochastic programming, a recourse decision is obtained in the second stage to compensate for the effects of the first-stage decision that is fixed ahead for a given uncertainty set. One main drawback of applying the stochastic program to the MVO model is that the number of scenario for a small size uncertain set of expected return may be innumerable, and lead to a large-scale problem which may encounter the solvability issue.

31 Chapter 2. Modern Portfolio Theory and Index Tracking 19 Thus, there exist other methods that take account of both first and second central moment information and meanwhile maintaining the tractability for multi-periods MVO selection. Robust programming is one of the alternative methods capable of achieving these goals. Robust optimization has been considered in many applications to mitigate the effects of parameter uncertainty. A comprehensive survey (over 130 references) of robust optimization is given in [19]. The authors listed several important applications in finance, which include multiperiod asset allocation problem as in Ben-Tal et al. [12] where the authors propose a secondorder cone program as a robust counterpart, and Bertsimas and Pachamanova [21] where under specific norms the problem is cast as a linear program. Goldfarb and Iyengar in [66] considered robust mean-variance optimization formulations based on robust factor models and show that the resulting robust problems can be formulated as Second Order Cone Programming (SOCP), which is one category of convex problem. Erdogan, Goldfarb, and Iyengar [51] incorporated transaction costs into the robust MVO problems and the resulting model remains as an SOCP. Cardinality restrictions to robust portfolio selection have also been studied. Sadjadi et al. [131] applied robust optimization to cardinality constrained Mean-Variance problem which resulted in a mixed-integer second-order cone programming and applied genetic algorithms to compute solutions. Nalan et al. [64] also used robust cardinality constrained MVO problems and solved the resulting mixed-integer SOCP instances using a commercial solver. We then review the index tracking problem in the literature in next section. 2.2 Literature review for Index Tracking A market index is a representation of entire market which combines typical top performing constituents together to an aggregate value. Security market indices are useful tools that help investors track the performance of various specific markets, estimate risk, and evaluate the performance of portfolio managers. The value of a market index can be calculated by different methods, such as market capitalization weighted, price-weighted, and equal-weighted. Market capitalization weighted method is a traditional and predominant approach to measuring an index. For example, S&P500 is a market-cap based American stock index which contains 500 large companies traded the US public market. These companies are picked from 10 sectors which

32 Chapter 2. Modern Portfolio Theory and Index Tracking 20 are measured by specific sector indices [1]. Almost all important markets adopt the market-cap weighted method to construct their indices in the world today. These typical examples also include S&P/TSX Composite Index that contains over 220 of the largest Canadian securities, Russell 3000 Index represents over 98% of the investable US equity market in terms of market value, and Nasdaq Composite Index which is heavily weighted towards information technology sector. Price weighted method, on the other hand, puts more weight on the stock with a higher price and reflects the investor s confidence about the economy. A notable example is the Dow Jones Industrial Average, which clearly records most of the disasters in American economic history. Besides above two methods, equal weighted index is another primary index weighted method which assigns index components with equivalent weights. The advantage is that the tracking portfolio can replicate the target index easily but, on the other hand, it may result in a high turnover cost. Because of the impressive average performance over the years, market indices also form a basis of new financial products such as ETF funds. The index-based ETFs are the primary category of the ETF funds. On one hand, perfectly yielding exact same returns to the target s is a major task of the tracking portfolios, and one the other hand, partial replication through cardinality constraints are more efficient for practical purpose. Therefore, the consideration of the trade-off between the tracking error and the portfolio size is necessary to portfolio management. Different tracking error objectives and practical constraints are studied for the index tracking problem in the literature. Beasley et al. [10] considered tracking error that minimizes the return differences between the portfolio and the benchmark, and thus leads to a non-linear tracking model with transaction costs and cardinality constraint to construct the tracking portfolio in testing five major markets in the world. Bertsimas et al. [20] applied mixed integer programming to build a portfolio to track a given benchmark portfolio with the aim of having fewer stocks with limited turnover and transaction costs. Coleman et al. [37] minimized tracking error based on MVO framework with cardinality constraints and showed that the developed model is NP-hard. Cornuejols and Tutuncu [40] presented an index tracking model which maximize the similarity between selected assets and the assets of the target index and represented a clustering-based approach for constructing a tracking portfolio. Karlow and Rossbach [87] applied a VaR constraint to the tracking error term, and added a regularization

33 Chapter 2. Modern Portfolio Theory and Index Tracking 21 term into objective instead of using a cardinality constraint. Recently, the discussion about enhanced indexation arises in the literature. The goal of enhanced tracking portfolio is to generate a small amount of excess return but keep the same or similar risk level. This method combines both active and passive management strategies and thus requires human intelligence to carefully set a parameter trade-off between the tracking error and portfolio risk. Jorion [85] showed that 83% of the stock-based funds have a higher risk than the benchmark via using tracking error constraint in MVO framework. Canakgoz and Beasley [29] considered the enhanced index tracking problem via a mixed integer program where the objective is to allow outperformance of a benchmark, the model includes transaction cost and is tested on eight large market indices. Chavez-Bedoya and Birge [32] studied the enhanced indexation by using a multi-objective non-linear programming approach in which the variance of the tracking error term can be decomposed for optimal portfolio analysis. The issue of parameter uncertainty described in Section 2.1 may also be encountered for index tracking models and has attracted widespread interest from authors. Stoyan and Kwon [139] developed a mixed integer model which includes several discrete choice restrictions such as buy-in thresholds, cardinality constraints, as well as round lots to track the Toronto Stock Exchange (TSX). Kwon and Wu [98] developed a factor-based robust enhanced index tracking model which take account of both tracking error and portfolio risk constraints and examined the model by using Fama and French 3 factor model as the basis of constructing robust counterparts of the nominal tracking model. Lejeune and Samatli-Pac [100] applied a chance-constrained stochastic integer programming approach that partially considers parameter estimation risk for enhanced indexation. Although different tracking models are established, It is still a non-trivial task to obtain the associated optimal solutions. As mentioned before, cardinality constraint and the binary requirement make the problem NP-hard and thus it is necessary to review the methodologies for solving the index tracking problem in next Chapter.

34 Chapter 3 Lagrangian Relaxation in Literature In this chapter, we first briefly review numerous algorithms that can be potentially used for solving our designed index tracking models. Then we illustrate the Lagrangian Relaxation (LR) mechanism via a simple numerical example and summarize the literature review on the LR approaches for different types of OR problems and cardinality constrained portfolio selection models. The LR methods to index tracking problem draw more attention from us. 3.1 Metaheuristics in Literature The optimal or near-optimal solutions for proposed models are important to decision makers. To date, there is no polynomial-complexity algorithm for solving large-scale integer programming, the solution strategies to different types of problems highly depend on the intelligence of designed methods. A heuristic that can generate sufficient good solution to an optimization problem in a short amount of time or under limited computation capacity is called a metaheuristic. Typical metaheuristics for solving mixed integer programming in fields of Operational Research and Computer Science mainly include: ˆ Greedy heuristic. A greedy algorithm is a problem-solving heuristic which attempts to make the best optimal choice at each iteration or stage with the hope of leading to a global optimal solution [39]. The greedy method is powerful tool to solve many hard optimization problems such as activity-selection problem, p-median problem [96] and scheduling [38]. ˆ Lagrangian Relaxation. Lagrangian relaxation is a useful method that can generate a 22

35 Chapter 3. Lagrangian Relaxation in Literature 23 compact bound by relaxing the hard constraints and solving the alternative relative easy. LR methods have been applied different OR problems, e.g. p-median problem, portfolio optimization problems. A detailed description about LR method will be displayed later. ˆ Branch and Bound. Branch-and-bound (B&B) algorithms attempt to search the complete space of candidate solutions via excluding large parts of the search space by using previous generated bounds on the quantity of optimizing the easier sub-problems, e.g. linear programming relaxation, at each iteration. B&B algorithm is an exact method that can guarantee optimal solution or prove that no such solution exists for mixed integer programming. The method was first presented by Land and Doig in 1960 [99] and has become the most commonly used tool for solving NP-hard optimization problems, e.g. travelling salesman problem. However, there are evidence show that pure B&B method usually converges slowly for large-scale discrete problems in practice [146]. ˆ Tabu Search. Tabu Search (TS) is a method that can escape from the local optimum by using a tabu list to prevent the occurrence of the search to previously visited solutions and obtain improved neighbors from the current solution. Originally created by Glover in 1986 [63], TS methods have become an important local search strategy for NP-hard problems due to the good performance for many classes of the optimization problems [44, 127, 31]. ˆ Variable Neighborhood Search. Variable Neighborhood Search (VNS) [75] is another type of metaheuristic method for jumping out from the current local minimum via changing and exploring the generated various neighborhoods. Despite the mechanism of VNS is simple and easy to understand, it proves that VNS algorithms can generate good enough solutions for many NP-hard problems [74, 128]. ˆ Genetic Search. Genetic Algorithm (GA), initially developed by Holland in the 1970s [79], is a search heuristic for optimization problems that generates global or near-global optimal solutions by simulating the selection process of natural evolution system. GA is a fast, useful and reliable technique because that GA can extract the good information hidden in a solution and pass them to its offsprings (new solutions), and hopefully move

36 Chapter 3. Lagrangian Relaxation in Literature 24 towards the global optimality. Typical applications include p-median problem [81], index tracking problem [10, 119] and power generation [120]. ˆ Simulated Annealing. Simulated Annealing (SA) is a probabilistic approach for approximating global optimal solution in a large search space for discrete problems. Inspired from the annealing process in metallurgy, SA algorithms search the optimal solution in a more extensive space at a probability from a given worse solution [88]. SAs have been employed to study the OR problems such as portfolio selection problems [31, 45] and p-median problem [35]. The described metaheuristics above usually borrow the advantages from each other or combine with other techniques such as valid cuts for branch and bound to improve the performance of the methods according to the special structure of the problems [80, 30]. We follow the same fashion in which the metaheuristic are combined together to enhance the solving ability. In next section we primarily focus on Lagrangian relaxation methods because the mathematical advantage allows different techniques be conveniently embedded into the Lagrangian relaxation framework for our developed index tracking models. For instance, Variable Neighborhood Search is used to find the near optimal solution with the help of the Lagrangian dual bound information in Chapter 4 and Tabu Search and LR methods are applied to solve the scenario sub-problems to speed up the whole Progressive Hedging algorithm in Chapter Literature review for LR and Its Extension Lagrangian relaxation (LR) is a technique in optimization well suited for problems where the constraints can be divided into hard and easy constraint sets. In the LR procedure, the hard constraints are pumped into the objective function with assigned weights or penalties, e.g. the Lagrangian multipliers, which makes the relaxes alternative easier to solve than the original problem. Lagrangian relaxation offers a compact bound that can be used to approximate optimal solution for the problem. Since Lagrangian approximation generally can be decomposed into a series of sub-problems, LR is also called Lagrangian Decomposition. We illustrate the idea of Lagrangian relaxation through the following numerical example.

37 Chapter 3. Lagrangian Relaxation in Literature 25 max Z (x) = x 1 + x 2 (3.1) s.t. x 1 2 (3.2) x 2 3 (3.3) 0.3x x (3.4) where constraint (3.4) is relative harder than other two constraints, thus we decompose above problem into two easier subcases by removing the constraint (3.4) into objective with a positive multiplier, i.e. L (x, λ) = x 1 +x 2 λ (0.3x x 2 2.5) = (1 0.3λ) x 1 +(1 0.7λ) x λ. Then each subcase has analytical solution for relaxed primal problem (2, 3), 0 λ 10 7 (x 1, x 2) = 10 (2, ), 7 < λ 10 3 (, ), λ > 10 3 max x 1 2,x 2 3 L ( x, λ ), which is easier than directly solving original problem Z (x). The updating of Lagrangian multiplier λ is bounded according to the following weak dual inequality: min λ 0 L (x, λ) Z (x ) then we go to the next iteration with new λ until the stopping criteria be satisfied. Lagrangian dual L (x, λ) is convex and thus it is useful for solving non-convex problem through iteratively reducing the gaps between the lower and upper bounds. Lagrangian relaxation for integer programming was initially discussed by Geoffrion [61], Geoffrion and McBride [62], Fisher [56] and Cornuejols et al. [41]. LR is used to approximate a difficult problem with a computationally tractable relaxation, of which the solution is a tight bound to the original problem. LR-based algorithms have successfully solved many problems in Operational Research such as multidimensional assignment problems [124], facility location problems [41, 90], and portfolio optimization problems [136]. LR based methods have also been developed along different directions. First, many researchers attempted to reduce the integrality gap by modifying the LR procedure. Cornuejols et al. [41] showed that the maximal integer gap cannot exceed 1/e 36.79% for p-median problem. Narciso et al. [116] presented Lagrangian relaxation with surrogate constraints, numerical results indicated that using surrogates to update multipliers can efficiently improve the convergence process and local bound. Beltran et al. [11] proposed a Semi-Lagrangian Relaxation (SLR) method which can achieve an

38 Chapter 3. Lagrangian Relaxation in Literature 26 improved bound as compared to the LR method, they also produced more accurate solutions compared with the regular LR method via solving the p-median problem. However, surrogate LR and Semi-LR cannot utilize the decomposition advantage for large scale computation. Another direction of development is the augmented Lagrangian methods also known as the method of multipliers [18] in which one penalty term is added to the Lagrangian objective, e.g. L (x, λ) + ρ 2 g (x), to quickly approximate Lagrangian multipliers and therefore speed up the convergence process. The strategy of augmented Lagrangian takes both advantages of Lagrangian relaxation and penalty methods. Progressive Hedging (PH) is one main stream of this type of method to handle the non-anticipativity constraint and to decompose the problem across scenarios by using Lagrangian dual in Stochastic Programming. This approach highly emphasizes the mathematical development and computational effectiveness of different problems in the literature. Rockafellar and Wets [125] proved that the PH method has a linear convergence rate to the linear type of stochastic programs. Helgason and Wallace [77] approximated the scenario solutions to improve the convergence performance by solving the fisheries management problem. They pointed out that exact solutions of subproblems is not required when apply the PH method to solving the non-linear problems. Mulvey and Vladimirou [115] applied the PH algorithm to solve network structured Financial Planning problem, which considers the re-balance cash flows between different stages. Progressive Hedging also has been extensively studied for mixed integer Stochastic Programming. Lokketangen and Woodruff [103] embedded the Tabu search heuristic used for large size scenario subproblems into the PH algorithm, and provided the computational evidence to support the effectiveness of the method by solving the production problem includes uncertain cost structures and demands over multiple-periods. Gade et al. [58] derived a tight bound to evaluate the quality of the solution of the PH algorithm to mixed integer SP. They showed that such bound could be as tight as that obtain from Lagrangian dual, which offers a theoretical support for large application of PH method to mixed integer SP in practice. Crainic et al. [43] proposed a progressive hedging algorithm with metaheuristic to solve a stochastic variant of the fixed-charge capacitated multicommodity network design problem. In their method they built cycle-based neighbourhoods and simultaneously searched the associated γ - residual networks using Tabu heuristic for sub-problems. Watson and Woodruff [147] presented a mathematical

39 Chapter 3. Lagrangian Relaxation in Literature 27 modification of penalty coefficient for the PH algorithm to a class of stochastic resource allocation model. According to the argument of the problem structure, their innovation for the accelerators in regularization function decreased the running time and enlarged the solvability of the PH method to resource allocation problem. Veliz et al. [145] investigated the forest planning problem that incorporates the uncertainty of harvesting and road construction decisions in developing country through a mixed integer SP. They applied the PH procedure to obtain the solution of the realistically sized problem instances. The Lagrangian dual is a key concept for the reviewed LR methods, we draw the dual decomposition scheme in the following Figure (3.1), and we will further discuss variants of Lagrangian relaxation in the literature that relative to different problems in their respective chapters. initial λ 0 Master problem Lagrangian relaxation... sub-problem k updated λ v Dual problem Penalty Adjustment N Aggregation of solutions Satisfy stop criteria? Y Optimal Figure 3.1: Lagrangian Decomposition Scheme for integer programs

40 Chapter 4 A Constrained Clustering Approach for Index Tracking 4.1 Introduction Index tracking is an important passive investing strategy where one seeks a portfolio of securities that emulates a given benchmark portfolio such as the S&P500. Several studies [106, 69, 151] have concluded that the actively managed funds usually cannot outperform broad market indices. For example, Zenios reported that the average return of 769 all-equity actively managed funds was 2% to 5% lower than the S&P 500 index during the period [151]. Full replication of the benchmark portfolio is an obvious strategy for tracking where all assets in the benchmark are held in the quantities as specified by the weightings of the benchmark portfolio, but full replication is not practical given the transaction costs this would entail. For example, fully replicating the S&P500 index would require holding the 500 assets along with weightings for each asset. The weightings are based on market capitalization and so as soon as the prices of assets change the weights change as well. Constant rebalancing of the tracking portfolio would result in a prohibitive amount of transactions. An alternative strategy is to select a strict subset of assets from the benchmark, however, this results in tracking portfolios that do not match the benchmark as closely as in full replication. A well-known measure of this discrepancy is called tracking error and is defined as the difference between returns of the tracking portfolio and benchmark. In general, there will be a trade-off between 28

41 Chapter 4. A Constrained Clustering Approach for Index Tracking 29 tracking error and transactions costs. Models that seek to minimize tracking error have emerged as a popular approach for constructing tracking portfolios [86]. Such models exhibit nonlinearity as it is the variance of tracking error that is often minimized or constrained. A further complication is that in enforcing only a strict subset of assets are selected discrete variables must be introduced. This constraint is called the cardinality constraint and requires binary variables for its implementation. Incorporating this aspect along with tracking error minimization into a model will result in a non-linear integer optimization problem which can present substantial challenges in computing optimal or near-optimal solutions. Furthermore, most tracking models e.g. those minimizing tracking error require estimates of expected return of time series of prices of assets, but it is well known that it is challenging to obtain these estimates and estimation error could result in substantial bias in optimized portfolios that require these estimates. In this chapter, we consider linear mixed integer optimization models for tracking broad market indices such as the S&P 500. The models we consider represent a cluster-based approach for tracking based on a model of Cornuejols and Tutuncu [40]. The cluster-based approach seeks to partition the assets in a benchmark portfolio into disjoint clusters from which a single (representative) asset is selected from each cluster. The set of representatives constitutes the tracking portfolio. The clusters are grouped to maximize similarity among assets in a cluster. The number of clusters to generate is a user controlled parameter and is implemented by a cardinality constraint that explicitly restricts the number of representatives to equal the user specified number of assets to hold. A measure of similarity can be represented by correlations between returns of pairs of assets. One of the advantages of the cluster-based models they only require information about similarity whereas most tracking models e.g. those that use tracking error require information about expected returns in addition to correlation estimations. However, a tracking strategy based only on clustering may producing a tracking portfolio that tracks a benchmark portfolio well in terms of return, but could produce an insufficiently diverse portfolio when tracking a broad market index such as the S&P 500 thereby increasing the risk of the tracking portfolio. A market index such as the S&P 500 consists of approximately 500 large cap stocks from 10 different economic sectors such as energy, information technology, consumer discretionary, consumer staples, materials, financial, utilities, industrials, telecommunication and services, and health care. The sectors represent the broad and diverse

42 Chapter 4. A Constrained Clustering Approach for Index Tracking 30 economy of the United States. A pure clustering solution may result in concentration of assets into just a few sectors. As such, we consider constraints to ensure that a tracking portfolio for the S&P 500 contains reasonable representation from each sector. We also consider some additional important constraints that aim to control transaction costs such as buy-in thresholds and turnover constraints [151]. Buy-in threshold constraints ensure that assets selected will have weights that are not unrealistically small and turnover constraints ensure that the tracking portfolio does not deviate excessively from a current tracking portfolio. Thus, we propose a sector constrained linear clustering approach for tracking the S&P 500 with buy-in thresholds. The models that we propose are linear integer programs and as such can still be challenging to solve for optimal tracking portfolios, but should be substantially easier than solving nonlinear integer models of the tracking problem and with only modest information requirements. We propose Lagrangean and Semi-Lagrangean relaxation methods to solve the models and find that our methods often find optimal or near optimal solutions. Furthermore, the tracking portfolios from our models are shown to track the S&P 500 effectively with sector diversification compared to basic clustering approaches without safeguards for diversification. The rest of the chapter is organized as follows: Section 4.2 briefly surveys the literature on index tracking. In Section 4.3 we formulate the index tracking models with sector limit and other practical constraints. In Section 4.4 we develop the Lagrangian relaxation-based methods for the models. In Section 4.5 computational results are given and we conclude the paper in Section Literature Review for Index Tracking A common approach to the index tracking problem is to formulate it as an integer optimization problem. One of the major challenges is to deal with the cardinality constraint and a diversity of algorithmic methods ranging from evolutionary heuristics to methods based on branch-andbound have been considered to solve models with cardinality restrictions. A general nonlinear tracking model is considered in Beasley et al. [10] with transaction costs and cardinality constraint and is solved using evolutionary heuristics in testing five major markets in the world. Bertsimas et al. [20] considers mixed integer programming to construct a portfolio to track a

43 Chapter 4. A Constrained Clustering Approach for Index Tracking 31 given benchmark portfolio with the aim of having fewer stocks with turnover and transaction costs. Coleman et al. [37] minimize tracking error in the index tracking problem with cardinality constraints and uses a graduated non-convexity algorithm to satisfy the cardinality restriction. Jansen and van Dijk [83] convert the cardinality constraint into a continuous non-convex power function, and apply a diversity method to decide the best stocks and weights of the portfolio. Oh et al. [119] use genetic algorithms to generate the optimal weights for the selected stocks to track a benchmark (where the tracking portfolio has strictly fewer assets) where first stocks are distributed into the sectors with larger market capitalization. Ruiz-Torrubiano and Suarez [129] apply a hybrid approach that uses a genetic algorithm to select the assets that track different market indices with fewer assets and use quadratic programming to determine the weights of the assets selected by the genetic algorithm; other practical constraints such as transaction cost are not included in their model. Stoyan and Kwon [139] develop a two-stage stochastic mixed integer programming with recourse which includes several discrete choice constraints such as buy-in thresholds, cardinality constraints, as well as round lots to track the Toronto Stock Exchange (TSX). Leujene and Samatli-Pac [100] consider a chance constrained stochastic programming formulation for the risk averse indexing problem with cardinality constraints and develop an outer approximation method. Cornuejols and Tutuncu [40] presented an index tracking model which maximize similarity between selected assets and the assets of the target index and represents a clustering-based approach for constructing a tracking portfolio. Chen and Kwon [34] consider a robust version of Cornuejols. Canakgoz and Beasley [29] consider the enhanced index tracking problem via a mixed integer program where the objective is to allow outperformance of a benchmark, the model includes transaction cost and is tested on eight large market indices. Gaivoronski et al. [59] consider different types of risk measurement for index tracking ranging from mean-variance and conditional value at risk (CVaR) models to tracking with fewer numbers of assets. Chavez-Bedoya and Birge [32] consider a multi-objective non-linear programming approach where their model also considers enhanced indexation. The formulation decomposes the variance of the tracking error of the portfolio so that a model with fewer variables is obtained. Most models described above require estimates of expected price or return of assets. In general, it is difficult to estimate expected returns accurately and portfolio optimization models

44 Chapter 4. A Constrained Clustering Approach for Index Tracking 32 can be sensitive to estimation errors of returns [36] and often maximizes the errors found in estimates [108]. In the next section we develop the models for index tracking that are based on [40] which do not require expected return estimates, but only require information about similarity e.g. correlation between the returns of assets. 4.3 Model Formulations Basic cluster-based index tracking model The basic index tracking model we adopt is from Cornuejols and Tutuncu [40]. Suppose the target portfolio has n securities. The model seeks to partition the n securities of the target portfolio into q disjoint groups (clusters) of securities where securities in a group are the most similar to each other. Then, the model will select a representative from each group. The q representatives will constitute the tracking portfolio. The correlation of the returns between pairs of securities is used as the measure of similarity in our experiments, other measures of similarity such as cointegration or covariance can be used as well [5]. Let ρ ij represent the correlation (similarity) between security i and asset j and let q denote the size of tracking portfolio where q < n. For i, j = 1,..., n, let x ij represent whether stock j is a representative of stock i where x ij is 1 if j is the most similar security in the portfolio to i, or 0 otherwise. For j = 1,..., n let y j represent the selection of a security to be part of the tracking portfolio where y j is 1 if security y j is selected or 0 otherwise. Then, the problem of creating a tracking portfolio can be formulated as follows: n n max ρ ij x ij (4.1) s.t. i=1 j=1 n y j = q (4.2) j=1 n x ij = 1, i = 1,, n (4.3) j=1 x ij y j, i = 1,, n, j = 1,, n (4.4) x ij, y j {0, 1} (4.5) The objective (4.1) is to select securities so that total similarity of all groups is maximized.

45 Chapter 4. A Constrained Clustering Approach for Index Tracking 33 Constraint (4.2) enforces that the tracking portfolio will have exactly q securities and is called a cardinality constraint. Constraint (4.3) ensures that each security has exactly one representative in the portfolio. Constraint (4.4) prohibits a security to be a representative of any security if it is not selected to be part of the tracking portfolio. The model above only selects securities for the tracking portfolio, but once the model is solved the investment weight for each selected security expressed as proportion of total investment can be calculated. In particular, a weight w j be calculated for each selected asset j using total market value of all securities in the group that security j represents divided by the total market value of all securities in the target portfolio (index), i.e., w j = i V ix ij i V i. For example, if stock 1 represents stock 2 and 3 in the portfolio, we sum the market values of stock 1, 2 and 3, and then divide the sum by the market value of the n securities in the target portfolio. The weight for security 1 in the tracking portfolio would be positive assuming that all securities have positive prices and the weights for securities 2 and 3 would be set to 0 as they would not be in the tracking portfolio. This follows the capitalization-based weighting that is found in the S&P 500 and other major indices. It should be noted that the models presented in this chapter seek to track and not outperform the S&P 500 and so this motivates the use of capitalization-style weightings for the assets selected by the models. The clustering based model utilizes only linear constraints and therefore is a pure 0 1 linear integer program. The quality of the tracking portfolio generated by the model is measured ex-post i.e. tracking error and metrics to measure closeness to the benchmark index portfolio are computed after the tracking portfolio is generated. An alternative would be to explicitly have tracking error minimized as the objective in a tracking model. This has been a popular approach in the practice and literature [86]. However, this would create a non-linearity in the objective as the variance of the difference of the returns of the tracking and benchmark portfolios would need to be minimized and in conjunction with cardinality constraint requirements would result in a quadratic non-linear integer program which is known to be very challenging to solve [121, 22]. Chen and Kwon [34] have shown that the model (4.1) (4.5) can track a benchmark portfolio S&P 100 well where the number of securities in the benchmark portfolio is n = 100. Instances of model (4.1) (4.5) were able to be solved adequately with exact methods. However, there are

46 Chapter 4. A Constrained Clustering Approach for Index Tracking 34 several important practical elements that have not been considered. First, model (4.1) (4.5) above lacks transactions costs. It will be most likely in practice that some tracking portfolio is already extant. It will be important to make sure that a new tracking portfolio is not too different from the currently existing one as substantial differences will result in higher turnover and thus higher transactions costs. Model (4.1) (4.5) will be extended to have turnover constraints that limit transaction costs. Further, tracking portfolios with small positions are also limited by incorporating buy-in thresholds in model (4.1) (4.5). Second, the tracking portfolio generated from model (4.1) (4.5) may track a benchmark well in terms of return, but the portfolio itself may be insufficiently diversified as there is no constraints that limits portfolio risk. This is an important issue when tracking market indices such as the S&P 500 as any tracking portfolio should include securities across the 10 different sectors (Consumer Discretionary, Consumer Staples, Energy, Financials, Health Care, Industrials, Information Technology, Materials, Telecommunications Services, and Utilities) that comprise a market index. Model (4.1) (4.5) can be shown to produce tracking portfolios with securities from only a few e.g. 2 or 3 sectors. This would be problematic for most portfolio managers concerned about risk and diversification. To this end, constraints that ensure sector diversification are incorporated in model (4.1) (4.5) Model with buy-in threshold and turnover constraints We now consider the addition of buy-in threshold and turnover constraints in model (4.1) (4.5). The resulting model is given in the following formulation: max s.t. n n ρ ij x ij (4.6) i=1 j=1 n x ij = 1, i = 1,, n (4.7) j=1 x ij y j, i = 1,, n, j = 1,, n (4.8) n y j = q (4.9) j=1 l j y j n i=1 V ix ij n i=1 V i u j y j, j = 1,, n (4.10)

47 Chapter 4. A Constrained Clustering Approach for Index Tracking 35 w j = n i=1 V ix ij n i=1 V, j = 1,, n (4.11) i n w j 0 w j α γ (4.12) j=1 x ij, y j {0, 1} (4.13) Model (4.6) (4.13) shares the same decision variables and parameters as model (4.1) (4.5) but now has the following additional parameters: α is a proportional transaction cost, γ is the limit on transaction, V i denotes the market capitalization of stock i at current time, wj 0 denotes the proportion of stock j in current portfolio. In addition, model (4.6) (4.13) has the variable w j denoting the proportion of wealth invested in stock j for j = 1,..., n. The buy-in threshold constraints sets the weight of a stock to be i V ix ij i V i which is the standard market capitalization based weight of assets in indices such as the S&P 500 and is set to 0 if asset j is not selected. In the transaction cost constraint, wj 0 w j denotes the turnover of stock j from buying or selling and the cost of turnover of an asset j is proportional to the amount of turnover given by wj 0 w j α. The transaction constraint limits the total proportional turnover (transaction) cost to γ. The absolute value terms in transaction cost constraint can be removed by introducing auxiliary variables z j, after which the model (4.6) (4.13) becomes equivalent to the following model: max s.t. n n ρ ij x ij (4.14) i=1 j=1 n x ij = 1, i = 1,, n (4.15) j=1 x ij y j, i = 1,, n, j = 1,, n (4.16) n y j = q (4.17) j=1 l j y j n i=1 V ix ij n i=1 V i u j y j, j = 1,, n (4.18) w j = n i=1 V ix ij n i=1 V, j = 1,, n (4.19) i n z j γ (4.20) α j=1 z j wj 0 w j, j = 1,, n (4.21) z j ( wj 0 ) w j, j = 1,, n (4.22)

48 Chapter 4. A Constrained Clustering Approach for Index Tracking 36 z j 0, j = 1,, n (4.23) x ij, y j {0, 1} (4.24) However, computational experiments in section show that optimal tracking portfolios from model (4.1) (4.5) and model (4.14) (4.24) are often concentrated in a few sectors which may result in high portfolio variance or lack of diversification. Therefore, constraints that impose diversification in a natural way is considered in next section Basic model with sector limits For simplicity of exposition, we first consider diversification (sector limit) constraints for model (4.1) (4.5) and then consider the addition of these constraints to model (4.14) (4.24). The idea is to classify assets in a tracking model according to what sector an asset belongs to. For example, in the S&P 500 index the constituent assets are classified as belonging to one of 10 sectors collectively representing the broad economy of the United States. A sector represents a segment of the economy such as materials, consumer discretionary, consumer staples, industrials, health care, telecommunication services, financials, utilities, energy, or information technology. In general, we assume that the benchmark index consists of K sectors. Let x ijk is 1 if stock j is the most representative of stock i in sector k, 0 otherwise. y jk is equal to 1 if stock j from sector k is selected to the tracking portfolio, 0 otherwise. K is the number of sectors, and n k denotes the number of assets (stocks) in sector k. The idea of the sector constrained model is to ensure that there is sufficient investment across all sectors by creating sub-portfolios for each sector where each sub-portfolio is sought that maximizes similarity of the sub-portfolio with respect to its sector. Let ρ ijk denote the similarity between assets i and j in sector k. k and k denote the lower and upper bounds on the cardinality of the sub-portfolio from sector k. q k denotes sub-portfolio size of sector k and q denotes total portfolio size. Then, model (4.1) (4.5) modified for sector constraints is as follows: max K n n ρ ijk x ijk (4.25) i=1 j=1k=1

49 Chapter 4. A Constrained Clustering Approach for Index Tracking 37 n s.t. y jk = q k, k = 1,, K (4.26) j=1 k q k k, k = 1,, K (4.27) K q k = q (4.28) k=1 n x ijk = 1, i = 1,, n, k = 1,, K (4.29) j=1 x ijk y jk, i = 1,, n, j = 1,, n, k = 1,, K (4.30) y jk = 0 if j / sector k (4.31) x ijk, y jk {0, 1} (4.32) Model (4.25) (4.32) can be reduced to the following model (4.33) (4.39) since constraint (4.31) forces x ijk = 0 if the asset i does not belong to sector k. max s.t. n k n k K ρ ijk x ijk (4.33) i=1 j=1k=1 n k y jk = q k, k = 1,, K (4.34) j=1 k q k k, k = 1,, K (4.35) K q k = q (4.36) k=1 n k x ijk = 1, i = 1,, n k, k = 1,, K (4.37) j=1 x ijk y jk, i = 1,, n, j = 1,, n k, k = 1,, K (4.38) x ijk, y jk {0, 1} (4.39) The model with trading and sector diversification constraints We now consider a comprehensive version of a cluster-based model for tracking, model (4.40) (4.49), that includes the buy-in thresholds, trading constraints, and the sector diversification constraints as seen in model (4.14) (4.24) and model (4.33) (4.39). max n k n k K ρ ijk x ijk (4.40) i=1 j=1k=1

50 Chapter 4. A Constrained Clustering Approach for Index Tracking 38 s.t. n k y jk = q k, k = 1,, K (4.41) j=1 k q k k, k = 1,, K (4.42) K q k = q (4.43) k=1 n k x ijk = 1, i = 1,, n k, k = 1,, K (4.44) j=1 x ijk y jk, i = 1,, n, j = 1,, n k, k = 1,, K (4.45) nk i=1 l jk y jk V ikx ijk n i=1 V u jk y jk, j = 1,, n k, k = 1,, K (4.46) i w jk = nk i=1 V ikx ijk n i=1 V i, j = 1,, n k, k = 1,, K (4.47) n k K w 0 jk w jk α γ (4.48) j=1k=1 x ijk, y jk {0, 1} (4.49) The parameter w 0 jk denotes the initial proportion of wealth invested in stock j (from sector k) which is needed when considering transaction costs (turnover) in the presence of sector constraints and the decision variable w jk denotes the proportion of wealth invested in stock j (from sector k). The absolute values that appear in the turnover constraints 4.48 can be removed by introducing auxiliary continuous variables z jk that represents the turnover amount for asset j (from sector k) and which represents the aggregate turnover of assets in sector k to get the the following constraints for turnover: w jk = nk i=1 V ikx ijk n i=1 V i, j = 1,, n k, k = 1,, K (4.50) n k z jk = p k, k = 1,, K (4.51) j=1 K p k γ α k=1 (4.52) z jk wjk 0 w jk, j = 1,, n k, k = 1,, K (4.53) z jk ( wjk 0 w jk), j = 1,, nk, k = 1,, K (4.54) z jk 0, j = 1,, n k, k = 1,, K (4.55)

51 Chapter 4. A Constrained Clustering Approach for Index Tracking Tractability of the cluster-based Models The number of variables and constraints in model (4.14) (4.24) and model (4.33) (4.39) is larger than in the base model (4.1) (4.5) and model (4.40) (4.49) contains the largest number of constraints and variables out of all models considered. We solve instances of each of these models including the base model using the commercial solver Gurobi on a 1.58 GHz PC with 2GB of RAM. Random instances of the tracking problems were generated where for each instance q assets will be selected from a benchmark portfolio of n assets where n is chosen as 100, 200, and 500. We randomly generated multivariate normal distribution for different n through mvnrnd function in MATLAB, and calculated the associated correlation matrix ρ ij. Computational results are presented in Table (4.1). Each row in Table (4.1) is for an instance of n assets. Moving across each row from left to right we see that as more constraints are incorporated into model (4.1) (4.5), the objective values decreases. Moving down each column we see that instances with larger n have better objective values for each type of model. Gurobi cannot solve models (4.33) (4.39) and model (4.40) (4.49) when n = 500. This motivates the development of algorithms for model (4.40) (4.49) so that quality solutions for instances of n = 500 are possible. Important and popular market indices such as the S&P 500 have 500 assets and so it will be critical to have methods to deal with indices of this size. Table 4.1: Model test by Gurobi (q = 10) Model (4.1) - (4.5) (4.14) - (4.23) (4.33) - (4.39) (4.40) - (4.49) n Out of memory Out of memory 4.4 Lagrangian Relaxation Algorithms Lagrangian relaxation (LR) for integer programming was initially discussed by Geoffrion [61], Geoffrion and McBride [62], Fisher [56] and Cornuejols et al. [41]. LR is used to approximate a difficult problem with a computationally tractable relaxation, of which the solution is a tight bound to the original problem. Since the Lagrangian approximation usually can be decomposed into a series of sub-problems, LR is also called Lagrangian Decomposition. LR-based methods

52 Chapter 4. A Constrained Clustering Approach for Index Tracking 40 have successfully solved many operations research problems such as multidimensional assignment problems [124], facility location problems [41, 90] and portfolio optimization problems [136]. Many researchers attempt to reduce the integrality gap by modifying the LR procedure. Narciso et al. [116] presented LR with surrogate constraints, numerical results indicated that using surrogates to update multipliers can efficiently improve the convergence process and local bound. Beltran et al. [11] proposed a Semi-Lagrangian Relaxation (SLR) method which can achieve an improved bound as compared to LR; they also produced more accurate solutions for the p-median problem. In this chapter, we applied LR and partial SLR to the developed index tracking model due to the special structure of the coefficient matrix of the constraints, we also observed that partial SLR method can improve the solution process and accelerate the convergence in section We present both Lagrangean relaxation and Semi-Lagrangean relaxation methods for problem (4.40) (4.49). The rationale for a Lagrangian relaxation is that easy and hard constraints in the model can be identified and then the hard constraints put in the objective to get a problem (the Lagrangean dual) whose optimal solution represents the smallest upper bound on the optimal solution of the original problem (4.40) (4.49) but is easier solve for. In particular, 2 constraints in problem (4.40) (4.49),i.e. K k=1 q k = q and K k=1 p k γ α, can be put into the objective function by using the Lagrange multipliers λ and µ, respectively. Then a relaxation (L) of the original problem is the following: L (x, y, z, λ, µ) = max = max = n k n k K (x,y,z) i=1j=1k=1 K (x,y,z) k=1 K k=1 [ max [ nk i=1j=1 n k (x,y,z) i=1j=1 ρ ijk x ijk λ ] n k ρ ijk x ijk λq k µp k ] n k ρ ijk x ijk λq k µp k ( ) ( ) K K q k q µ p k γ α k=1 k=1 + λq + µγ α + λq + µγ α L (x, y, z, λ, µ) can be decomposed across different sectors, and the associated k th sector sub-problem becomes: max (x,y,z) s.t. n k n k ρ ijk x ijk λq k µp k (4.56) i=1 j=1 n k y jk = q k, k = 1,, K (4.57) j=1

53 Chapter 4. A Constrained Clustering Approach for Index Tracking 41 k q k k, k = 1,, K (4.58) n k x ijk = 1, i = 1,, n k, k = 1,, K (4.59) j=1 x ijk y jk, i = 1,, n, j = 1,, n k, k = 1,, K (4.60) nk i=1 l jk y jk V ikx ijk n i=1 V u jk y jk, j = 1,, n k, k = 1,, K (4.61) i w jk = nk i=1 V ikx ijk n i=1 V i, j = 1,, n k, k = 1,, K (4.62) n k z jk = p k, k = 1,, K (4.63) j=1 z jk w 0 jk w jk, j = 1,, n k, k = 1,, K (4.64) z jk ( w 0 jk w jk), j = 1,, nk, k = 1,, K (4.65) x ijk, y jk {0, 1}, z jk 0, i = 1,, n k, j = 1,, n k, k = 1,, K (4.66) Solution to model (4.56) (4.66) is easier than that to model (4.40) (4.49) because we can solve for K times standard model (4.14) - (4.24) but much smaller size under fixed (λ, µ). The dual problem is min L (x, y, z, λ, µ), whose optimal solution will provide the lowest upper (λ,µ 0) bound for problem (4.40) (4.49). The Lagrangean dual will be solved with a Golden Section Search method and sub-gradient method separately with heuristics for feasibility. This forms the basis of the Lagrangian relaxation algorithm for solving problem (4.40) (4.49). We summarize the Lagrangian relaxation algorithm as follows: Lagrangian Relaxation Algorithm Step 0: (Initialization) v 0, λ (v) 1, µ (v) 0 Step 1: (Dual Decomposition) For k K, Solve the corresponding sector sub-problem L ( x, y, z, λ (v), µ (v)) k UBD K k=1 L ( x, y, z, λ (v), µ (v)) k + λ (v) q + µ(v) γ α If ( x (v), y (v), z (v)) is feasible to model (4.40) - (4.49), LBD UBD, STOP. Else find a feasible solution (and associated LBD) by Heuristic I (v = 0) or II (v > 0), gap (v) = UBD LBD LBD

54 Chapter 4. A Constrained Clustering Approach for Index Tracking 42 Step 2: (Lagrangian Multiplier Update) Build Lagrangian dual problem min µ 0 L ( x (v), y (v), z (v), λ, µ ) Update step size t (v) by Golden Section Search (GSS) and Bi-section methods respectively. ( K ) λ (v+1) = λ (v) + t (v) k=1 q k q ( ( K )) µ (v+1) = max 0, µ (v) + t (v) k=1 p k γ α Solve (L) with new multiplier ( λ (v+1), µ (v+1)) Step 3: (Move to next iteration) If gap (v) > ɛ, v < V v = v + 1. GO TO Step 1. Here are some remarks when we implement the LR algorithm: (1) In Step 1, Heuristic I is applied to obtain a initial solution to trigger the iterations. Then a vector q k can be returned by solving (LR) at each iteration, if the solution is infeasible, a more sophisticated heuristic (heuristic II) is applied to satisfy the global constraints, i.e. K i=1 q k = q and K i=1 p k γ α, and the associated lower bound can be updated. Let m (k) denotes the size of sector k. Q = {q k, k = 1,, K } be a vector satisfies the cardinality constraint in model (4.40) (4.49) and Q = {q k, k = 1,, K } be another vector that also satisfies the cardinality constraint in model (4.40) (4.49) but different than Q, Q LR = { qk LR, k = 1,, K } be a vector satisfies the cardinality constraint in model (L). I, I and I LR be the associated index set of Q, Q and Q LR, respectively. We first describe the Heuristic I as follows: Heuristic 1 : Heuristic I for initial lower bound (0) Sort market capitalization of assets in a descending order and put in vector V, Chose the first q assets in V that satisfy sector cardinality bounds and weight bounds; Obtain a Q vector.

55 Chapter 4. A Constrained Clustering Approach for Index Tracking 43 (1) Divide the index of Q into 3 groups: I 1 = {h Q h = 0} sort I 1 in descending order according to {m (h) h I 1 } I 2 = {i Q i 0, i {index set of first l largest Q i }} I 3 = {j Q j 0, j I\I 1 \I 2 }, sort I 3 according to {(Q j, m (j)) j I 3 } Switch portion of indices between I 1, I 2 and I 3 Generate N neighborhood points Q around Q. (2) Solve (L) without constraint K k=1 p k γ α under Q (3) Test transaction cost constraint (TC); Choose solution Q better than Q and satisfy (TC) if it exists, STOP; else GO TO (1). Step (0) in Heuristic I guarantees that a starting solution will satisfy the transaction cost constraint by emphasizing the selection of assets with larger market capitalization. For example, suppose V = (10000, 100, 10) T and associated wj 0 = (0.9891, , )T, if the first asset is not selected to the tracking portfolio, the turnover weight is 98.91% and is much larger than the maximal turnover weights of the second and third assets, so the turnover constraint will be easily violated. We then generate the neighborhood of points around Q in Step (1) by choosing pairs of sectors between sectors as indexed by the subsets I 1, I 2 and I 3, and and swapping pair-wise. The philosophy behind the swap rules is to generate only a small size of neighborhood points such that swaps attempt to distribute the assets to more sectors so that the objective value becomes better. In Step (1), we sort I 3 in increasing order according to {Q j j I 3 }. If elements in {Q j j I 3 } are equal, we then sort I 3 in descending order according to {m (j) j I 3 }. We always select sectors at front position of the index sets I 1, I 2 and I 3, and switch 2 assets between pairs of these three groups in Step (1). If no improvement occurs at the current iteration, the sectors with different positions in the index sets are selected in next iteration. For example, parallel swapping steps include: 1O Pick 2 assets from a th sector in I 2 to b th sector in I 3, obtain a Q ; 2O Or pick 2 assets from a th sector in I 2 to b th sector in I 1, obtain a Q ; 3O Or pick 2 assets from a th sector in I 2, add 1 asset to b th sector in I 3 and 1 asset to c th sector in I 2, obtain a Q ; 4O Or pick 1 asset from a th sector in I 2 and 1 asset from b th sector in I 3, add them to c th sector in I 1, obtain a Q. Here the indices a, b, and c are generally set as small values since

56 Chapter 4. A Constrained Clustering Approach for Index Tracking 44 switching other indices may be inefficient to improve the objective, e.g. a, b, c are set no more than 2 times in our computation. We leave a detailed numerical example that illustrates Heuristic I in the section A. 1 of Appendix A to interested readers. Heuristic 2 : Heuristic II for updated lower bound (1) Adjust a given Q LR vector as follows: Pick { k min { Q LR }} k, if Q LR k m (k), q k = qlr k, else q k = m (k) Repeat above steps unless q K k=1 q k = 0; if q K k=1 q k > 0, add the difference into the sector has maximal number of assets; solve (L) with Q vector; (2) Test transaction cost constraint (TC); If solution satisfies TC, STOP, else, GO TO Step (3); (3) Within each sector k, } do: I 1 = {wjk 0 j {Q k }, sort I 1 in increasing order. } I 2 = {wjk 0 j {m (k)} \ {Q k }, sort I 2 in decreasing order. Switch first assets between I 1 and I 2. Solve (L) with new Q vectors Test TC, if TC satisfied, STOP, else GO TO Step (4) (4) Pick { two sectors that have } large (k 1 ) and small (k 2 ) asset number in Q, do: I 1 = wjk 0 j {m (k 1)}, sort I 1 in increasing order. { } I 2 = wjk 0 j {m (k 2)}, sort I 2 in decreasing order. Swap first assets between I 1 and I 2. Obtain new Q LR vectors, GO TO (1) If the TC cannot be satisfied in Step (3) in Heuristic II, we adjust the portfolio by capital weights in the same sector, and then adjust the portfolio between the sectors in Step (4) if necessary. We selected the sectors with large and small stocks because so as to not lose too much objective value. Like we did in Heuristic I, we always go back to the assets have larger capital weights to adjust the constructed portfolio. It is a trade-off between the Cardinality and Transaction Cost constraints. How to exchange the assets between sectors in Step (4)? One approach is Variable Neighborhood Search (VNS) [75]. We describe the steps here we implemented: (1) Shaking - randomly perturb some assets between max ( Q LR) and min ( Q LR) from current solution; (2) Local search - search the selected neighborhood region, i.e. the new Q vectors. (3) Move or not if an improved solution obtained. Our computational observation is that in most of instances Step (3) and (4) are needed to achieve a feasible solution for transaction cost constraints, which indicates that the cardinality and TC constraints are a computational challenge to satisfy as they run in opposing directions. We also leave a numerical example that illustrates

57 Chapter 4. A Constrained Clustering Approach for Index Tracking 45 Heuristic II in the section A. 2 of Appendix A to readers. (2) In Step 2 in the LR algorithm, step size t (v) was updated by Golden Section Search (GSS) and Bi-section methods respectively. Algorithm 3 : Golden Section Search (GSS) for step size in Step 2 of LR algorithm Set scalars {( A and ) B, A B, } t (v) t (v) = 1 t (v) t (v) 1 = A (B A) 2 t (v) 2 = A (B A) ( K ) λ (v+1) = λ (v) + t (v) k=1 q k q ( ( K )) µ (v+1) = max 0, µ (v) + t (v) k=1 p k γ α Solve (L) with new multiplier ( λ (v+1), µ (v+1)) while (B A) ɛ B = t (v) 2 if L (v) t < L (v) 1 t 2 repeat 1O - 4O or A = t (v) 1 if L (v) t L (v) 1 t 2 1O 2O 3O 4O GSS has been proved that it can perform with a linear convergence rate with τ = to one dimension search problem [9], this feature initially attract us to apply it for dual updating in our algorithm. However, GSS slow down the whole LR algorithm since it tries to obtain the best dual objective at each iteration. On the other hand, bi-section method for one dimension searches in sub-gradient method has been widely used in LR algorithm [61, 56]. The details of the bi-section method are presented below. Algorithm 4 : Bi-section search for step size in Step 2 of LR algorithm Set initial σ, ( K ) λ (v+1) = λ (v) + σt (v) k=1 q k q ( ( K )) µ (v+1) = max 0, µ (v) + σt (v) k=1 p k γ α where t (v) UBD LBD = K i=1 q k q; K i=1 p k γ α Solve (L) with new multiplier ( λ (v+1), µ (v+1)) while L ( x, y, x, λ (v+1), µ (v+1)) UBD and σ ɛ σ =.5σ, repeat 1O - 3O 1O 2O 3O How to determine the step size t v? To illustrate the problem, let s simplify the Lagrangian max ω min x LR (x v, ω v ) = c T x v + (ω v ) T (Bx v b). We know that d v = Bx v b is the

58 Chapter 4. A Constrained Clustering Approach for Index Tracking 46 gradient to Lagrangian function at x v, suppose ω v+1 = ω v + t v d v, then LR ( x v, ω v+1) = c T x v + ( ω v+1) T (Bx v b) = c T x v + (ω v ) T (Bx v b) + t v (d v ) T (Bx v b) = c T x v + (ω v ) T (Bx v b) + t v (Bx v b) T (Bx v b) = LR (x v, ω v ) + t v (Bx v b) T (Bx v b) = LR (x v, ω v ) + t v Bx v b 2 = t v = LR ( x v, ω v+1) LR (x v, ω v ) Bx v b 2 In the bi-section method we initialize t v = BestUB CurrentLB Bx v b 2 = σ(bestub CurrentLB) Bx v b 2 where σ > 1, if the objective LR ( x v, ω v+1) LR (x v, ω v ), the step size is reduced by half in each iteration and the main advantage is that we can quickly update the dual variables. The step size after k iterations is tv 2 k, so it may require exactly log 2 (t v /ɛ) iterations in worst case scenario for dual variable updating. method with GSS and Bi-section in following Table (4.2): We present the numerical comparison of the LR Table 4.2: Time comparison for updating dual in LR method LR method with Bi-section search LR method with Golden section search q Fesi. LB LR UB Gap Time (S) Fesi. LB LR UB Gap Time (S) T gss /T bi % % % % % Aver. / / 7.46% / / 7.57% In our computation we set initial A = 0, B = 8 for GSS and σ = 20 for Bi-section search. All other parameters in LR method are kept same. From Table (4.2), we see that lower and upper bounds for all instances are close to each other. However, the searching time for step size by Bi-section search is much less than that by GSS, the average search time of GSS is 4 times larger than that from Bi-section search. The reason is that the Bi-section search does not require the steps generate best dual objective value, it terminate when a better dual objective is found and start a new outer loop in LR method and speed up the whole algorithm. Therefore, we mainly use Bi-section search for dual variable updating in our computation.

59 Chapter 4. A Constrained Clustering Approach for Index Tracking 47 It is easy to show that solving model (4.40) - (4.49) by LR Algorithm takes the running time of approximate V K T sub O ( n 2), where T sub is the average time of solving a sub-problem, K is the sector number, and V is the iteration number. Since T sub depends on the capacity of the solver, usually T sub is constant on average. If we fixed V, as K increase, the problem can be always solved within a predictable time. Note that in Table (4.2), there still exist large gap between the bounds for q = 150, 200, we hope a tighter LR upper bound so that the gap can be shrank. One possible extension of the LR algorithm is Semi-Lagrangian Relaxation (SLR), a LR approach with more strict feasible region and therefore tighter bound. Due to the decomposition requirement in main algorithm structure, the global constraints cannot be returned into the constraint set. However, other types of constraint can be relaxed and then returned to constraint set and partially satisfy the SLR framework. This procedure is called partial SLR [11] and suitable for our problem. In particular, after relaxed the assignment constraint, we put the assignment constraint back and formulate the partial SLR as follows: L (x, y, z, λ, µ, θ) = max = = n k n k K (x,y,z) i=1j=1k=1 K k=1 K k=1 [ [ max (x,y,z) i=1j=1 max ρ ijk x ijk λ n k n k n k (x,y,z) i=1j=1 ( ) ( ) K K q k q µ p k γ α k=1 k=1 (ρ ijk θ ik ) x ijk λq k µp k ] ] n k P ijk x ijk λq k µp k Then the k th SLR problem can be formulated as follows: ( ) nk θ ik x ijk 1 n k K i=1k=1 + λq + µγ α + n k + λq + µγ α + n k K θ ik i=1k=1 K θ ik i=1k=1 j=1 max (x,y,z) n k n k P ijk x ijk λq k µp k (4.67) i=1 j=1 s.t. (4.57) (4.58), (4.60) (4.66) n k x ijk 1, i = 1,, n k, k = 1,, K (Relaxed assignment) (4.68) j=1 and the dual problem becomes max L (x, y, z, λ, µ, θ). Then, the LR framework can be (λ,µ 0,θ 0) applied to the partial SLR construct. We present the Semi-Lagrangian-based Algorithm as follows:

60 Chapter 4. A Constrained Clustering Approach for Index Tracking 48 Algorithm 5 : Semi-Lagrangian Relaxation Algorithm Step 0: (Initialization) v 0, λ (v) 1, µ (v) 0 θ (v) ik 0, i n k, k K Step 1: (Dual Decomposition) For k K, do P ijk ρ (k) ijk θ(v) ik, i n k, j n k ( Solve the corresponding sector sub-problem L UBD ( ) K k=1 L x, y, z, λ (v), µ (v), θ (v) k ik + λ (v) q + µ(v) γ x, y, z, λ (v), µ (v), θ (v) ik α ) k + K k=1 θ(v) ik If ( x (v), y (v), z (v)) is feasible to model (4.40) - (4.49), LBD UBD, STOP Else find a feasible solution (and associated LBD) by Heuristic I (v = 0) or II (v > 0), gap (v) = UBD LBD LBD Step 2: (Lagrangian Multiplier Update) Build Lagrangian dual problem min µ 0 L ( x (v), y (v), z (v), λ, µ, θ ) Update step size t (v) by Bi-section methods. ( K ) λ (v+1) = λ (v) + t (v) k=1 q k q ( ( K )) µ (v+1) = max 0, µ (v) + t (v) k=1 p k γ α ( ( θ (v+1) ik = max 0, θ (v) ik + nk )) t(v) j=1 x ijk 1 ( ) Solve (Partial SLR) with new multiplier λ (v+1), µ (v+1), θ (v+1) ik Step 3: (Move to next Iteration) If gap (v) > ɛ, v < V v = v + 1. GO TO Step 1.

61 Chapter 4. A Constrained Clustering Approach for Index Tracking 49 The feasible lower bound is generated by the same Heuristics as LR algorithm. In Step 2, sub-gradient method with Bi-section search [61], [56] was applied to calculate the dual variable ( ) λ (v+1), µ (v+1), θ (v+1) ik for SLR algorithm. The computation is terminated if optimal solution obtained in Step 1 or gap tolerance or iteration number reached in Steps 3. As mentioned in [11], partial SLR cannot guarantee a tighter bound. However, it returns a better bound than LR in some instances in our computation, we will compare the result from LR and SLR in next section. 4.5 Computational Results: Tracking the S&P500 In this section we give the computational results from using the LR and SLR methods to solve model (4.40) (4.49). The S&P 500 index is used as the target benchmark Parameter Estimation To generate the correlation matrix ρ ij for S&P500, we collected the historical price information of all components of S&P500, and calculated the daily returns by r it = P it P i,t 1 P i,t 1, where P it, P i,t 1 are the adjusted closing prices at time t and t 1. Then daily returns were used to calculate the mean returns of assets and covariance matrix between different assets: µ i = 1 T T r it, cov ij = 1 T t=1 T (r it µ i ) (r jt µ j ) t=1 Here we use one year s daily return (T=252) to generate correlation matrix, i.e. ρ ij = cov ij covii cov jj, for all models, and we calculate the correlation matrices by using data from 4 time intervals which were [ ], [ ], [ ] and [ ] respectively. Some stocks in the S&P500 index may be replaced by some other stocks outside of the index since they do not satisfy the selection criteria of S&P500 in the designed time period, we retrieved the stocks that were moved out into the designed intervals and the associated price information. For example, ABK was replaced by LO in June 10, 2008, and then we used the price information of ABK rather than the data of LO to calculate that before Usually this replacement is rarely and the components of S&P500 are stable. According to Global Industry Classification Standard (GICS) Sector criterion [3], the com-

62 Chapter 4. A Constrained Clustering Approach for Index Tracking 50 ponents of S&P500 index are selected from 10 main sectors in US market and we indicate sector 1 10 represent Consumer Discretionary, Consumer Staples, Energy, Financials, Health Care, Industrials, Information Technology, Materials, Telecommunications Services, and Utilities in this research. Sector size vector m(k) = [ ] T at the time of this research. We adjusted the number of stocks in each sector for designed intervals if necessary and computed the associated correlation matrix for the models include the sector limit constraint. Ticker across Sectors in S&P500 are displayed in the Table (A.1) of Appendix A. We normalized the marker value of each component to calculate the component weight, and used these weights as previous proportion, i.e. wj 0, for transaction cost constraint in model (4.14) - (4.24) and model (4.40) - (4.49). All necessary data were obtained from the Financial Research and Trading Lab at University of Toronto. All models were computed by Gurobi with a MATLAB interface Gurobi Mex [150]. We set the initial ( λ 0, µ 0) = (1, 0) for LR and ( λ 0, µ 0, θ 0) = (1, 0, 0) for SLR, and Table (4.3) gave the parameter setting when we implement the algorithm. Table 4.3: Parameter Setting α.001 γ.05 k 0 k Maximum stock number of sector k l jk.001 u jk 1 wj 0 Normalize the market capitalization of component of SP LR versus SLR We computed solutions for model (4.40) - (4.49) over portfolio sizes ranging from 10 to 350 in increments of 10 assets, the upper bound (UB) decreased and the lower bound (LB) increased iteratively in the LR algorithm and ideally a global optimal solution was achieved when the UB equals LB. Although the LR method cannot guarantee the optimal solution for every instance, it returned a minimal UB when the computation was terminated, and a bound associated with the heuristic can be used to approximate the optimal solution. Figure (4.1) depicted the computational comparison by LR and partial SLR, where the

63 Chapter 4. A Constrained Clustering Approach for Index Tracking 51 maximal gaps between the lower and upper bound were 2.37% and 4.59% respectively. Most of the gaps were under 0.5%, especially in the practical interval, [50 200] (see Table (A.2) in Appendix A). In some cases, SLR returned a better bound and a smaller gap than LR (see q = 20, 80), and in other cases SLR was worse than LR (see q = 250). However, the running time by SLR (average 1.83 hrs) was generally smaller than LR (average 2.87 hrs). We main used a partial SLR algorithm to approximate the optimal solution in the next section, since it returned a better solution relative to LR in the practice region q [10, 200]. Figure 4.1: Gap Comparison between LR and SLR

64 Chapter 4. A Constrained Clustering Approach for Index Tracking Comparison between 4 models Differences of portfolio efficiency and allocation We denote model (4.1) - (4.5) as the model (1), model (4.14) - (4.23) as the model (2), model (4.33) - (4.39) as the model (3), and model (4.40) - (4.49) as the model (4) in the rest of chapter. Four different portfolios were constructed by model (1) - (4). We illustrated the models with portfolio size q equal 10, 30, 100, which represent the low, medium and high density separately. Figure A.1 in Appendix A shows the details about the sector difference between the 4 portfolios. S&P500 index is collected from 10 sectors where sector 1-10 represent Consumer Discretionary, Consumer Staples, Energy, Financials, Health Care, Industrials, Information Technology, Materials, Telecommunications Services, and Utilities respectively. We varied the portfolio size q and compared the associated portfolios by different models. Interesting results include: (1) The tendency of sector diversification. The computational results for all period intervals demonstrated that without sector limit constraint, the portfolio allocation are concentrated in fewer sectors (see q = 10, 30). This sector diversification can explain the reason why the portfolio with sector limit has a lower variance in next section. (2) The model with sector limit has a constant sector weight with respect to the changes of size q. Although Bertsimas and Shioda [22] pointed out that the investment in different sector must be limited, they have not explored how to decide the best sector investment fraction in their model. In this chapter, our numerical results shown that the optimal sector weights were consistent to the sector weights of the target index. Figure (4.2) have shown the norm value of the difference in the sector weights between the tracking portfolios and target S&P500. TC in all figures represents transaction cost and turnover constraints in model (2), and sector in all figures refers to the sector limit constraints in model (3). It is clear to see that when the sector limited constraint is considered, i.e. model (3) and model (4), the sector weight of the constructed portfolios was more close to the S&P500 than the model without the sector limit constraint, i.e. model (1) and model (2). Figure (4.2) was drawn based on all computational results under different q from 10 to 100. Because of the space limitation, we listed the numerical result (q = 10, 30, 100) about sector weight on Figure A.1 in Appendix A.

65 Chapter 4. A Constrained Clustering Approach for Index Tracking 53 Figure 4.2: Norm of sector differences between constructed portfolio and S&P500 Figure (4.3) illustrated the sector diversification process. For a small portfolio size (q=10), the stocks only distributed in 5 sectors when the sector limit constraint was not incorporated (model 1 and 2) while the stocks will distributed in 10 sectors if we considered sector limit (model (3) and (4)). This same situation existed when portfolio size increased to q=30, 100. One major advantage of sector limit constraint is that the diversification in sectors can reduce the portfolio risk. The sector limit constraint can make the investment allocation even across 10 sectors, i.e. the maximal sector fraction without sector limit is constantly larger than the fractions with sector limit. For instance, people will invest 59.27% by model (1) and 47.23% by model (2), the largest weight of their budget, to the sector of financials if only transaction costs and buy-in threshold constraints were incorporated into model (1). In contrast, the maximum sector weight is only 17.47% for the sector of Information Technology by model (3) and 18.39% by model (4). More comparison result that relative to diversification process will be discussed in next Section.

66 Chapter 4. A Constrained Clustering Approach for Index Tracking 54 Figure 4.3: Sector diversification

67 Chapter 4. A Constrained Clustering Approach for Index Tracking 55 Comparison of Performance Metrics In this section we compare the performance of the portfolios constructed by Model (1)-(4). The performance metrics include optimal objective values, portfolio return, portfolio variance, portfolio Sharpe ratio and tracking ratio. Intuitively, the objective value is the first consideration of the comparison between different models since it denotes the similarity of the constructed portfolio with the original index that is tracked. The portfolio return is an important aspect of the performance of the generated tracking portfolios, and the portfolio variance is a prevalent risk measurement of the constructed portfolios. The Sharpe ratio [135], or the information ratio, which measure the risk/return efficiency of excess return was the third comparison because it can describe the trade-off between the excess return to the market and the associated portfolio risk. Finally the tracking ratio was used to compare the tracking quality of the portfolio during different out-of-sample period under different restriction. Figures 5-9 show the numerical result with respect to different portfolio size q from 10 to 100 per 10 units for different time periods. As shown in Figure (4.4), the optimal objective value increased with respect to portfolio size q. Model (1) gives the greatest objective value while model (4) presents the smallest value, which is reasonable since model (4) includes all types of constraint. The objective value of model with sector limit is less than that without sector limit. For example, for any given specific q, the value of model (1) is larger than that of the model (3) and the value of model (2) is larger than that of model (4). This is obvious as more strict constraints are added into the underlying model. Compared with model (2) and (3), we can see that the sector limit constraint affected the objective value more significant than the transaction costs and buy-in threshold constraint, i.e. the value of model (3) decreased faster than value of model (2). One explanation is that the sector limit constraint is a global restriction which dominates the local constraints such as transaction costs. When the local constraints were incorporated, the objective value changed progressively (see the lines about model (1) and (2)). In contrast, the objective value can changed dramatically with the impact of the global constraint (see the lines about model (1) and model (3)).

68 Chapter 4. A Constrained Clustering Approach for Index Tracking 56 Figure 4.4: Comparison of Performance optimal objective value Figure (4.5) presented the tendency of the portfolio return by different models in response to the changing of portfolio size. The straight line in each plot in Figure (4.5) indicates the yearly return of market index, S&P500. The main goal of the tracking portfolio is to match the return of the market index, as can be seen from Figure 6, the portfolio returns under different models moved close to the return of the target when the portfolio size became larger. For example, the returns with q = 10 deviated further from the straight line than the returns with q = 100 in The reason is that when more stocks were allowed to hold, more chance of the full replication could be achieved. The portfolio returns resulted from sector limit constraint (see the lines about model (3) and model (4)) were close to each other. Likewise, the portfolio returns without sector limit constraint (see the lines about model (1) and model (2)) approached each other. An interesting observation is that the path of the model (2) matches the path of

69 Chapter 4. A Constrained Clustering Approach for Index Tracking 57 model (1) in every sub-figure, while the lines of model (3) did not follow that of model (1). As we mentioned, the local constraints such as transaction costs may slowly affect the solution structure with the change of the size, so the path of model (2) was close to path of the underlying model (1). On the other hand, the global constraint such as sector limit may lead to a totally different solution, which created different portfolio returns compare with the returns of model (1). Overall portfolio return changes with respect to the different restrictions and they are close to the return line of the target index. Figure 4.5: Comparison of Performance portfolio return The tendency of the portfolio variance under different models with respect to the portfolio size is plotted in Figure (4.6). The straight line indicates the yearly variance of market index, S&P500. The smaller the value of the variance is, the better the portfolio performs. Model (1) and (4) produced the upper bound and lower bound of the variance. It can been seen from

70 Chapter 4. A Constrained Clustering Approach for Index Tracking 58 the data in the Figure (4.6), the variance value by model (1) was 3 to 7 times higher than the variance value by the model (3). It is apparent from the figure that the portfolio variance with sector limit constraint (see the lines about model (3) and model (4)) was less than the variance without sector limit constraint (see the lines about model (1) and model (2)). The reason is that the sector diversification process distributed the limited number of the stocks into different independent sectors and hedged against the potential risk. Figure 4.6: Comparison of Performance portfolio variance The portfolios by model (2) tended to perform worse than the portfolios by model (3) in terms of the portfolio risk. This indicated that compared with the limitation of the total transaction cost, the sector diversification is a more efficient strategy to control the risk of tracking portfolio. Therefore, the portfolio variance can decrease when the sector limit constraint was incorporated into mode (2), i.e. the line of model (2) moved down to the line of mode (4) in

71 Chapter 4. A Constrained Clustering Approach for Index Tracking and Interestingly, the portfolio variance increased if the transaction costs and buy-in threshold constraints were added into the model (3). The reason is that the solution structure of each sector sub-problem became worse as the local constraints were incorporated, which result in the higher portfolio variance, i.e. the line of model (3) moved up to the line of mode (4) for every sub-figure. The Sharpe ratio was calculated as the difference in returns between a tracking portfolio and the market divided by the standard deviation of the difference in variance between the portfolio and the market. The higher Sharpe ratio value, the better performance of the portfolio occurred. The straight line indicates the yearly Sharpe ratio of market index, S&P500. From the Figure (4.7), we can see that the difference of the Sharp Ratios between model (3) and model (1) was larger than the difference of the Sharp Ratios between model (2) and model (1), which indicated that the sector limit constraint improved the Sharp Ratios more better than the transaction costs and buy-in threshold constraints for the same underlying model. For example, for q = 20 in 2007, the Sharp Ratio difference between model (3) and model (1) was 0.8 but the Sharp Ratio difference between model (2) and model (1) was -0.5, which means the sector limit constraint increased the Sharp Ratio of model (1) but the transaction costs and buy-in threshold constraints decreased the Sharp Ratio of model (1). All the Sharp Ratio values were negative in 2009 when the financial market dropped sharply. The Sharp Ratio value of model (1) was close to the Sharp Ratio value of the target, and the model (3) returned most negative values. However, as more local constraints were incorporated, the Sharp Ratio values were increased, i.e. the line of model (3) moved up to the line of model (4) in the sub-figure of Overall, from the Sharp Ratio perspective the portfolios with the sector limit constraint had over-performance to the portfolios without the sector limit constraint. The reason is that the sector limit constraint improved the denominator part of the Sharp Ratio given that the excess returns were close to each other.

72 Chapter 4. A Constrained Clustering Approach for Index Tracking 60 Figure 4.7: Comparison of Performance portfolio Sharpe ratio Next we calculate the Sharpe Ratio for the period of out-of-samples and compare the difference between Sharpe ratios of in-samples and out-of-samples. The periods of in-samples are the intervals of [ ], [ ], [ ] and [ ] and the associated outof-samples are the daily returns in intervals of , , and respectively. The numerical results shown in the following Table (4.4) 7 with respect to the portfolio size. The diff columns are the difference of Sharpe ratio by out-of sample (even columns) subtracts the value that from in-samples (Figure 4.7), a positive number indicates a portfolio still keep good performance during the out-of-samples, and a negative value means the portfolio constructed by the Model has a relative underperformance in the associated period without any re-balance.

73 Chapter 4. A Constrained Clustering Approach for Index Tracking 61 Table 4.4: Sharpe ratio for out-of-samples ( ) q Model (1) diff Model (2) diff Model (3) diff Model (4) diff Aver From Table (4.4), we see that the Sharpe ratio values for out-of-sample by Model (3) and (4) are generally larger than that from Model (1) and (2), which indicate the model with sector limit has better performance for out-of-samples. In terms of robustness of the Sharpe ratio testing, Model (3) decreased the most value ( averagely) while Model (4) increased 14.64% averagely. This results shown that the transaction cost constraints in Model (4) can improve the solution quality. Overall, portfolio by Model (4) has a best performance for both in-samples and out-of-samples testing. Table 4.5: Sharpe ratio for out-of-samples ( ) q Model (1) diff Model (2) diff Model (3) diff Model (4) diff Aver Table (4.5) lists the Sharpe ratios for out-of-samples during , a main period during the financial crisis. We can see that all instances have negative value, which indicate the portfolio has underperformance during the out-of-samples. Model (2) has the most negative value of Sharpe ratio for out-of-sample testing, while Model (4) has the smallest negative value. This shown that the portfolio by Model (4) has relative better performance. The difference

74 Chapter 4. A Constrained Clustering Approach for Index Tracking 62 value by Model (1) looks better than other models because it generates lower Sharpe ratios for in-samples (See subfigure on Figure 4.7) and associated difference may be low after subtraction. Table 4.6: Sharpe ratio for out-of-samples ( ) q Model (1) diff Model (2) diff Model (3) diff Model (4) diff Aver Table (4.6) shows the Sharpe ratio of out-of-samples at period of Again Model (4) generated largest average ratio values and Model (3) had a largest difference value. These instances have shown the benefit of the sector limit constraint for out-of-sample testing, i.e. the sector limit constraint can improve the portfolio s Sharpe ratio values. Table 4.7: Sharpe ratio for out-of-samples ( ) q Model (1) diff Model (2) diff Model (3) diff Model (4) diff Aver Finally we test the Sharpe ratio for out-of-samples at period of in Table (4.7). We see that the average Sharpe ratio of Model (4) and Model (2) are close to each other, but better than the average value from Model (3). Meanwhile the difference value by Model (1) and (3) are negative, and Model (3) has the largest average difference. Here we point out that for some data structure, transaction cost and turnover constraints can determine a portfolio with good average performance (see Model (2) in Table (4.7)) while for some other

75 Chapter 4. A Constrained Clustering Approach for Index Tracking 63 data structure, sector limit constraint generate a better portfolio, e.g. the Model (3) in Table (4.4). Model (4) generally have good performance for out-of-sample testing as seen Table (4.4) to Table (4.7) since it incorporate both transaction cost and sector limit constraint sets. Similar to the definition in Cornuejols and Tutuncu [40], we calculated the tracking ratio by following formula: R 0t = n i=1 V it/ n i=1 V i0 q j=1 w jv jt / q j=1 w jv j0 where n i=1 V it n i=1 V i0 indicates the target index s movement after investment, q j=1 w jv jt q j=1 w jv j0 denotes the portfolio s performance during the out-of-sample period. The ideal tracking ratio, R 0t, is 1, a higher value over than 1 means underperformance with respect to the target index, and a lower value less than 1 indicates excessive return. The straight line indicates the portfolio perfectly tracked the market index, S&P500. The out-of-sample periods were tested where the durations are 6 months and 12 months respectively, there was no re-balance during the tracking period after investment. Figure 4.8: Comparison of Performance Tracking Ratio of out-of-sample period (2007, 2008)

76 Chapter 4. A Constrained Clustering Approach for Index Tracking 64 Figure 4.9: Comparison of Performance Tracking Ratio of out-of-sample period (2009, 2011) Figure (4.8) and Figure 4.9 displayed the out-of-sample tracking ratios for four periods. As shown in Figure (4.8), the tracking portfolios might have a better tracking performance in the near future (6 months) than the longer future (12 months). For example, all portfolios were superior to the market index during , i.e. all R 0,6 < 1, while some portfolios had underperformance than the market during , i.e. all R 0,12 > 1. Another observation was that the models with the sector limit constraint (lines of model (3) and model (4)) had a stable performance than the models without sector limit constraint (lines of model (1) and model (2)). Taken together, the analysis from Figure provided important insights into the portfolio performance under different restriction. The sector limit constraint, as a global restriction, can change the solution structure so that the objective value changed significantly. As a result, the performance of portfolio with sector limit was better than the corresponding portfolio without sector limit in terms of the comparison of the portfolio variances and Sharp ratios.

77 Chapter 4. A Constrained Clustering Approach for Index Tracking Conclusions and Discussion In this chapter we have investigated portfolio tracking models that are linear mixed integer optimization problems that represent a constrained clustering approach for tracking a benchmark index, in particular the S&P 500. Motivated by real investment cases transaction costs and sector limits constraints were added to a base clustering model. We then developed both a Lagrangian Relaxation (LR) algorithm and the partial Semi-Lagrangian Relaxation (SLR) algorithm to solve the tracking problem with constrains. Numerical results have shown that both of the methods can achieve high quality solutions. Through the computational results we observe: (1) the sector limit constraint can diversify stocks into different sectors and then reduce the portfolio variance efficiently; (2) the optimal sector weights are consistent to the sector weights of the target index if the sector limit constraint is incorporated. In general, the constrained clustering approach tracked the S&P 500 effectively and the models and methods in this chapter can be used to effectively track any market index.

78 Chapter 5 Progressive Hedging for Cardinality Constrained Financial Planning Problem In Chapter 4, we explored Lagrangian relaxation algorithms for index tracking problem. Index tracking is a prevalent passive investing strategy to emulate the movement of the market indices. It provides an optimization tool for choosing a limited number of assets to represent a target index. We concluded that cardinality constraint is important for portfolio selection because practically, it can improve the model to suit more requirements, and theoretically, it may change the property of the problem. In this chapter we will study the Financial Planning problem with cardinality constraint. 5.1 Introduction to Financial Planning Problem Financial Planning (FP) problem is a portfolio selection process that achieves specific goals with limited resource. For example, the pension funds involve revision of portfolio investment to maximize profit and meet liabilities over time periods. For instance, the arbitrage trade in currency market can be formulated through a network in the form of a loop, and the arbitrage opportunity can be detected if the product of the multipliers for arcs on the loop is greater than unity. Other examples include asset allocation for portfolio selection and international 66

79 Chapter 5. Progressive Hedging for Cardi. Constrained FP 67 cash management in [115]. Cash (decision variable) flows on the arcs between different nodes, and transaction cost accumulates if better nodes are invested. The benefit of the FP problem with a network structure is that the decision process is straightforward and visible. Taking uncertainty into consideration is critical in FP problem. Stochastic programming (SP) is a popular tool to prevent the uncertainty of the parameters such as expected return of the asset in FP model. Uncertainty is fixed in the first decision stage, and recourse action is allowed at some cost to restore the feasibility after a realization of uncertainty is observed in next decision stage. Compared with other strategies such as robust optimization for the quantification of the uncertainty for model parameters, SP can return a solution trade-off between different scenarios. However, the problem size increases exponentially with respect to the time period, asset set and scenario number. The FP problem is formulated as a Linear Programming (LP) and the authors solve the model by progressive hedging algorithm which has a linear convergence rate. When we tested the FP problem in [115], we found that in some instances optimal portfolio allocations often are concentrated in a few assets which may result in high portfolio variance, while in some other instances the optimal portfolio distributes across a large range of assets which will result in high transaction costs. The potential disadvantages motivated us to incorporate the cardinality constraint to improve the model. The main contributions of this chapter include: ˆ We developed the FP problem (LP) into a Stochastic Mixed Integer programming (SMIP), and decomposed the associated SMIP across different scenario. ˆ Lagrangian relaxation and Tabu search methods were used to solve the scenario subproblem, and the numerical result showed that our sub-solver reduce the solving time efficiently compared with the time information by Gurobi. ˆ Progressive Hedging Algorithm was applied for SMIP and instances with large scenario number (S = 75) were tested. Moreover, a Lagrangian lower bound was embedded into the PH method and better gap information was obtained compared with the gap by Gurobi. The rest of the chapter is organized as follows: In Section 5.2, we formulate a series of equivalent Financial Planning problem. In Section 5.3, we decompose the FP problem correspond to

80 Chapter 5. Progressive Hedging for Cardi. Constrained FP 68 scenarios and design Lagrangian relaxation and Tabu search methods for scenario sub-problem. Section 5.4 describes details of the Progressive Hedging Algorithm. We also generate, in this section, a lower bound for the problem. In Section 5.5, we extend the FP framework into index tracking problem and present additional numerical result. The final section summarizes the current work and proposes possible areas in need of further research. 5.2 Model Development Equivalent Cardinality Constrained FP Models We develop the network structure Financial Planning problem in [115] by adding cardinality constraint. Suppose that K assets are selected from the asset set N where Cardi(N ) = N. Figure (5.1) shows the network structure of financial planning as a 0 1 stochastic programming. At first stage we can pump initial budget b i to each node and choose the initial cardinality number for the asset set. At the second stage, we rebalance the portfolio under different scenarios. We apply progressive hedging to force all arcs in different scenarios at stage 0 into a unique arc. Figure 5.1: Network structure with cardinality at stage 0 and 1 We first describe the parameters and decision variables relative to above figure and our model as follows: ˆ Parameters:

81 Chapter 5. Progressive Hedging for Cardi. Constrained FP 69 b i > 0 denotes the initial investment to the node i N. c ij > 0 denotes transaction cost ratio on arc (i, j) where i j N at stage 0. R s i denotes the total return Rs i to the asset i under scenario s. c s ij > 0 denotes transaction cost ratio on arc (i, j) where i j N at stage 1 under scenario s. p s denotes the probability that the scenario s may occur at at stage 1. ˆ Decision variables: x ij = amount of cash flow on the arc (i, j) at stage 0. If x ii > 0 means assets i is selected and be directed to the portfolio. If x ii = 0 => y s ij = 0, i N, (i, j) A s, s S, Note that x ij is the initial wealth come out from node i, it will be scaled by c ij when it goes into node j. g i = 1 denotes asset i is chose at stage 0, 0 otherwise. If g i = 1 means the value of node i is bounded by a lower and upper bound. If g i = 0 => x ii = 0, then there is no value to node i which can be switched into next stage, and there is no any arc come from node i at stage 1 (see Figure 5.1). y s ij = the amount of investment flow on the arc (i, j) under scenario s. Then we formulate the whole problem as follows: min c ij x ij p s ( ) 1 c s ij y s ij (j,i) A 0 s S (i,j) A 1s (5.1) s.t. b i + x ji x ij, i N (j,i) A 0,j i (i,j) A 0 (5.2) li x g i x i u x i g i, i N (5.3) g i = K (5.4) i N g i {0, 1}, i N (5.5) Ri s ( ) x i + 1 c s ji y s ji yij, s i N, s S (j,i) A 1s,j i (i,j) A 1s (5.6) l ys ij g i y s ij u ys ij g i, i N, (i, j) A 1s, s S (5.7)

82 Chapter 5. Progressive Hedging for Cardi. Constrained FP 70 where the arc set A 0 and A 1s denote the network at stage 0 and network at stage 1 under scenario s respectively. Both arc set A 0 and A 1s include all arcs between the node i and j, they also include an arc between i 0 and i 1 which connect different stage. l x i, ux i are the lower and upper bound flow to x i on node i N. l ys ij, uys ij are the lower and upper bound flow to ys ij on arc (i, j) A 1s for any scenario s. The objective of model (5.1) - (5.7) means we minimize the total transaction cost at stage 0 and maximize the total expected net wealth of the network at stage 1. The constraint (5.2) means for any node i, the total cash flow in no less than the total cash flow out at stage 0, constraint (5.3) - (5.5) denote the cardinality number of the portfolio, if some g i = 0, then the node value is forced to 0 by constraint (5.3). Constraint (5.6) means the total cash flow out from the fixed network cannot exceed the total cash flow in since if any node is unselected at stage 0, there is no transaction arc to any other nodes from the unselected node at stage 1, which is bounded by constraint (5.7). There exist a remarkable difference between the objective functions of the proposed model (5.1) - (5.7) and the FP model in [115], that is, we move the transaction ratio c ij and c s ij into the objective function and change the equality sign into inequality sign in constraint 5.2. The main advantage of this operation for the transaction ratio is that c ij can be used to adjust the penalty coefficient during iterations in the Progressive Hedging framework. However, we need to test if this change significantly affect the objective value. We displayed the comparison results that show how close between the two models in Table 5.1. We run 26 instances and all instances were obtained the optimal solution by using Gurobi mixed integer solver. 22 pairs of instances had the same objective value with optimal positions (see the last column). The average difference is 0.82%, which indicates the modified model does not change significantly if we add the transaction cost into the objective function. We decompose the model (5.1) - (5.7) across different scenarios by studying the nonanticipativity constraints. The left side of Figure 5.2 shows a simple scenario tree with 3 stages and 2 time period, therefore, the total scenario number S = 3 2 = 9. Now if we split the scenario tree to the right side of Figure 5.2, and force the variables in brackets are same, which means the variables in brackets followed the same historical path, then these two scenario trees are exactly same. This type of constraint is called Non-anticipativity constraint in [130].

83 Chapter 5. Progressive Hedging for Cardi. Constrained FP 71 Table 5.1: Model Comparison - with and without transaction cost term (N, S) K Model (5.1) - (5.7) Model (5.1) - (5.7) obj diff relative norm no trans. term col.6 - col.4 obj diff obj diff Best LB Fesi UB Best LB Fesi UB % % 0 (10, 3) % % % % % 0 (10, 15) % % % % 2 (50, 3) % % % % (50, 15) % % % % 0 (100, 3) % % % % 0 (100, 15) % % % 0 Average % Figure 5.2: Equivalent scenario trees

84 Chapter 5. Progressive Hedging for Cardi. Constrained FP 72 For our two stage FP problem, the first stage decision variables (x ij, g i ) T can be split into (x s ij, gs i )T across scenario s, then the model (5.1) - (5.7) can be reformulated as the following equivalent problem: min p s s S s.t. b i + c ij x s ( ) ij 1 c s ij y s ij (i,j) A os (i,j) A 1s (5.8) x s ji x s ij, i N, s S (i,j) A 0s (5.9) (j,i) A 0s,j i li x gi s x s i u x i gi s, i N, s S (5.10) gi s = K, s S (5.11) i N g s i {0, 1}, i N, s S (5.12) x s ij = x ij, (i, j) A 0s, s S (5.13) gi s = g i, i N, s S (5.14) Ri s x s ( ) i + 1 c s ji y s ji yij, s i N, s S (j,i) A 1s,j i (i,j) A 1s (5.15) l ys ij gs i y s ij u ys ij gs i, i N, (i, j) A 1s, s S (5.16) where x ij = s S ps x s ij, (i, j) A0s, s S and g i = s S ps gi s, i N, s S. Two networks A 0s and A 1s are running separately for any scenario now, and the non-anticipativity constraints (5.13) and (5.14) force every A 0s into A 0, which makes the model (5.8) - (5.16) is equivalent the model (5.1) - (5.7). The non-anticipativity constraints (5.13) and (5.14) is crucial for solving the problem (5.8) - (5.16) since it connect the split variable at stage Scenario Generation We generate two types of scenario for different parameters in the developed models, i.e. transaction cost ratio c s ij on the arc (i, j) and node expected return Rs i. We reasonably assume the transaction cost ratio is deterministic and can be predicted in the near future, then c ij can be assigned to for every postulated economic scenario. For example, assume that current transaction cost ratio is 5%, if the market goes up too quick, then the market regulator will increase the transaction cost ratio to 8% to cool the market; if the market plunge rapidly, then the government will decrease the transaction cost ratio to 2% to stimulate the trading activ-

85 Chapter 5. Progressive Hedging for Cardi. Constrained FP 73 ities; otherwise, the transaction cost ratio will keep as 5%. In our model, we assign different transaction cost ratios for different market stages in Table 5.7. Obtaining the discrete outcomes R s i for node i in future is more difficult and various techniques have evolved for generating scenarios for stochastic programs [82, 70, 122, 76]. Hoyland and Wallace [82] presented a moment matching method to obtain discrete outcomes whose s- tatistical properties are as close as possible to the specified distribution. Define K to be the set of all specified statistical properties and S V ALk to be the value of the specified property k K. For example, statistical property can be expressed moments information such as mean, variance/covariance, skewness (third central moment) and kurtosis (fourth central moment) from observations. Let f k (x, p) denote the mathematical expression about statistical property k in terms of x and p. Therefore, the model is given by min x,p s.t. w k (f k (x, p) S V ALk ) 2 (5.17) k K p l L t l = 1, t = 1,, T (5.18) p l 0, l L t, t = 1,, T (5.19) where w k is the weight of statistical property k. That is, an optimization problem is formulated to minimize the norm distance between the statistical properties of the constructed tree and those specified by the decision maker. The main advantage of this method is that it can capture any moment of the new series of data which consist of historical price and the aggregation of all possible movements of the node. The moment matching method has been developed by extensive research [82, 95, 105, 70]. In many cases, the moment matching method requires the distribution or description of the functions of marginal, which is not easy. In this section, we present a revised moment matching method which does not require the property of marginal as one whole optimization program. Our model captures mean, variance and covariance between the assets since these moments are the most important statistical specifications. Assume that the history return vector h i,[0,s] which represents the return of security i in the past s periods is observed at the current time, and scenario tree for T time periods need to be built in future. At any time point t [1, T ] for

86 Chapter 5. Progressive Hedging for Cardi. Constrained FP 74 any asset i, the first two central moments can be calculated as follows: E ( u t ) L t i = p lx t l=1 il, i, t (5.20) u i,[0,s+t ] = su i,[0,s] + T t=1 E ( u t ) i, i (5.21) s + T 1 (s + T 1) var i,[0,s+t ] = s (s + T 1) covar ij,[0,s+t ] = s m=1 ( hi,m u i,[0,s+t ] ) 2 + T t=1 m=1 + T t=1 ( hi,m u i,[0,s+t ] ) ( hj,m u j,[0,s+t ] ) ( E ( u t i ) ui,[0,s+t ] ) 2, i (5.22) (5.23) ( E ( u t i ) ui,[0,s+t ] ) ( E ( u t j ) uj,[0,s+t ] ), {(i, j) i j} lb t il xt il ubt il, i, l Lt, t = 1,, T (5.24) where constraint (5.20) denotes the expected return of asset i at time period t, constraint (5.21) refers to the first central moment of asset i with addition of new T periods, and constraints (5.22) and (5.23) calculate the second central moments, i.e. covariance matrix, for asset i in new time series. constraint (5.24) denotes the boundary conditions of variable x t il for asset i at scenario l in time period t. One primary issue for scenario generation is that the existence of the arbitrage opportunity which may lead to an unrealistic decision. Klaassen proposed an approach to detect and exclude the arbitrage opportunities through the dual argument, and numerical examples are shown in his work [89]. Since arbitrage opportunities may also exist in the scenario set generated by (5.20) - (5.24), follow the same argument process in [89], we preclude the arbitrage scenarios by adding following dual constraints: π0 t L t l=1 πt l xt il = Lt l=1 xt il, i, t (5.25) π t l 0, l Lt, t = 1,, T (5.26) L t l=1 θt l ( 1 + x t il ) = 1, i, t (5.27) θ t l 0, l Lt, t = 1,, T (5.28) where π, θ are the dual variable vectors for 2 types of individual arbitrage opportunities described in [89], constraints (5.25) - (5.26) deal with the case where the possible non-negative payoff with zero investment, while constraints (5.27) - (5.28) handle the case where you can obtain some reward immediately without any risk in future.

87 Chapter 5. Progressive Hedging for Cardi. Constrained FP 75 We assign a weight vector (w i1, w i2 ) for the first and second central moments respectively, and minimize the statistical properties distance between the constructed distribution and specification. Then we formulate the overall optimization problem: ( ) 2 ( ) ] 2 [w i1 ui,[0,s+t ] u i,[0,s] + wi2 vari,[0,s+t ] var i,[0,s] min x,p,π,θ N i=1 + N i=1 N s.t j=1,i j wi2 w j2 ( covarij,[0,s+t ] covar ij,[0,s] ) 2 (5.29) We do not need to describe the prosperities of marginal since they are dependent variables in the system (5.29). The parameter, (w i1, w i2 ), can be expressed as decision maker s attitude about future. For example, setting the ratio w i2 /w i1 = 1/10 denotes the first moment is 10 times important than the second moment for asset i, and vice versa. We apply the proposed model (5.29) to generate scenarios for R s i via employing historical market data in Section Lagrangian Decomposition Scheme Solving model (5.8) - (5.16) is difficult because (I) the problem size increase quickly with respect to the size of network and scenario number; (II) the model includes binary variables which destroy the convex property of the problem. However, high-quality solutions for large scale instances can be obtained through Progressive Hedging, a specific Lagrangian technique, that handle with the non-anticipativity constraint. We relax non-anticipativity constraints (5.13) and (5.14) by assigning Lagrangian multiplier λ s ij and πs i respectively, and add the proximal term by penalty ρ, and then we formulate the augmented Lagrangian: ALAG (x, g, y, λ, π) = p s c ij x s ( ) ij 1 c s ij y s ij s S (i,j) A os (i,j) A 1s + p s λ s ( ) ij x s ρ ( ) ij x ij + x s 2 2 ij x ij s S (i,j) A os (i,j) A os + ( ) p s πi s (gi s g i ) + ρ (gi s g 2 i ) 2 s S i N i N = p s ( cij + λ s ) ij ρx ij x s ij + ρ ( ) x s 2 2 ij s S (i,j) A os

88 Chapter 5. Progressive Hedging for Cardi. Constrained FP 76 where = + ( ) ( p s 0 + πi s ρg i + ρ ) gi s 2 s S i N p s ( ) 1 c s ij y s ij + s S ρ 2 (x ij) 2 λ s ij x ij + (i,j) A os i N (i,j) A 1s ( ρ 2 ) πs i gi. We simplify the second quadratic term (g s i g i) 2 = (g s i )2 2g s i g i + (g i ) 2 = g s i 2gs i g i + g i, for binary variable g s i {0, 1}. ALAG (x, g, y, λ, π) is non-separable across scenario because of the cross product x ij x s ij and g i g s i in the quadratic terms, however, if we fixed the x ij and g i by previous iterative solution, i.e. x v 1 ij and g v 1 i, the problem become fully decomposed within iteration v: ( ALAG P H x, g, y, λ, π, x v 1, g v 1) = p s ( ) c ij + λ s ij ρx v 1 ij x s ij + ρ 2 s S (i,j) A os + ( ) ( p s 0 + πi s ρg v 1 i + ρ ) gi s 2 s S i N p s ( ) 1 c s ij y s ij + s S (i,j) A 1s For each scenario s at iteration v, setting C s ij = c ij +λ s ij ρxv 1 ij and F s i ( x s ij ) 2 = 0+π s i ρgv 1 i + ρ 2, then the problem can be decomposed into following scenario sub-problems which maintain the FP structure: min Cijx s s ij + ρ ( ) x s 2 2 ij + Fi s gi s ( ) 1 c s ij y s ij (i,j) A os i N (i,j) A 1s (5.30) s.t. b i + x s ji x s ij, i N, s S (j,i) A 0s,j i (i,j) A 0s (5.31) li x gi s x s i u x i gi s, i N, s S (5.32) gi s = K, s S (5.33) i N gi s {0, 1}, i N, s S (5.34) Ri s x s ( ) i + 1 c s ji y s ji yij, s i N, s S (j,i) A 1s,j i (i,j) A 1s (5.35) l ys ij gs i y s ij u ys ij gs i, i N, (i, j) A 1s, s S (5.36) The time for solving model (5.30) - (5.36) is non-trivial for proposed Progressive Hedging method in Section 5.4. However, one observation is that the current advanced solver consumed

89 Chapter 5. Progressive Hedging for Cardi. Constrained FP 77 long time to deal with scenario sub-problem with respect to large asset number N (see Table 5.2). We design two methods, Lagrangian Relaxation and Tabu Search, to speed up the process of solving scenario sub-problem in Section and Section LR method for scenario sub-problem Observe that binary variable g i connect both variable x ij and y ij in constraint (5.32) and (5.36), so one strategy is to relax constraint (5.32) and (5.36). The relaxed problem will become much easier than the model (5.30) - (5.36) since the relaxed problem can be separated into continuous part (only x ij and y ij ) and integer part (only g i ), also we reduce the number of constraints dramatically, which can save much of time for coefficient matrix construction. We assign ωi s 0 and ωi s+ 0 to constraint li xgs i xs i 0 and ux i gs i + xs i 0 respectively and assign θij s 0 and θij s+ 0 to constraint l ys ij gs i ys ij 0 and uys ij gs i + ys ij 0 respectively. Then we construct following Lagrangian objective for subproblem: = = sub LR (x, g, y, ω, θ) Cijx s s ij + ρ ( ) x s 2 2 ij + Fi s gi s ( ) 1 c s ij y s ij (i,j) A os i N (i,j) A 1s + ( ω s i (li x gi s x s i ) + ωi s+ ( u x i gi s + x s i ) ) i N + ( ( ) ( )) θij s l ys ij gs i yij s + θij s+ u ys ij gs i + yij s (i,j) A 1s (i,j) A os + i N + (i,j) A 1s ( C s ij + diag ( ω s+ i Fi s + li x ωi s u x i ωi s+ + j N ( ) c s ij + θij s+ θij s 1 yij s ωi s )) x s ij + ρ ( ) x s 2 2 ij ( ) l ys ij θs ij u ys ij θs+ ij gi s As can be seen, the above sub LR (x, g, y, ω, θ) can be decomposed into two separated parts. The first part is a smaller size linear integer programming, sub LR IP (g, ω, θ): min s.t. i N Fi s + li x ωi s u x i ωi s+ + j N ( ) l ys ij θs ij u ys ij θs+ ij gi s (5.37) gi s = K, s S (5.38) i N

90 Chapter 5. Progressive Hedging for Cardi. Constrained FP 78 gi s {0, 1}, i N, s S (5.39) Solve the model (5.37) - (5.39) is not hard, we just sort the coefficient of objective and choose the first K nodes with the minimal coefficient. The second part can be seen as a quadratic programming, sub LR QP (x, y, ω, θ): min (i,j) A os s.t. b i + R s i x s i + ( C s ij + diag ( ω s+ i (j,i) A 0s,j i x s ji (j,i) A 1s,j i ωi s )) x s ij + ρ ( ) x s 2 ( ) 2 ij + c s ij + θij s+ θij s 1 yij s (i,j) A 1s (5.40) x s ij, i N, s S (5.41) (i,j) A 0s ( ) 1 c s ji y s ji yij, s i N, s S (5.42) (i,j) A 1s The solution from model (5.40) - (5.42) under g s i from model (5.37) - (5.39) is a feasible solution to model (5.30) - (5.36), then a high quality upper bound is constructed at each iteration with the solution information from (5.37) - (5.42). The dual problem of sub LR (x, g, y, ω, θ) is a convex problem, max ω 0,θ 0 min x,g,y sub LR (x, g, y, ω, θ) which can be efficiently solved. The dual problem returns either an optimal solution or maximal lower bound to model (5.30) - (5.36). The pseudocode of LR algorithm is displayed in Appendix B. 1 for further interesting reading. We present the numerical comparison between the LR method and Gurobi in Table 5.2. From Table 5.2 we see that the solution of LR method is close to the solution from Gurobi, the worst gap is 2.68% (s = 9) and the average difference is 1.38%. However, the average solving time of LR method (92.97 seconds) is only half of the solving time by Gurobi ( seconds). This is a tradeoff between the solving time and accuracy of the solution, we save the solving time for sub-problems so that we can speed up the Progressive Hedging strategy later. More numerical instances listed in Table B.1 - B.3 in Appendix B. 1 have shown that our LR method can significantly reduce the solving time while keeping the high quality of the solutions.

91 Chapter 5. Progressive Hedging for Cardi. Constrained FP 79 Scenario subcase N=50 K=5 S=15 Table 5.2: LR method and Gurobi Comparison - instance 1 Gurobi Best LB Feasi. UB Gap Time (S) LR Method Best LB Feasi. UB Gap Time (S) Gap to Gurobi col (7-3)./col 3 s= % % % s= % % % s= % % % s= % % % s= % % % s= % % % s= % % % s= % % % s= % % % s= % % % s= % % % s= % % % s= % % % s= % % % s= % % % Aver % % % Tabu search for scenario sub-problem The objective value of model (5.30) - (5.36) can be divided into 3 parts: ˆ Z (x) = (i,j) A os Cs ij xs ij + ρ 2 ( x s ij) 2 where C s ij is the transaction cost ratio on arc (i, j) at stage 0 under scenario s. The formula tells us that we should open the arc with negative C s ij and reduce the flow on the arc with positive Cs ij as much as possible. For a specific arc (i, j) with negative Cij s, we know that: C Z(x s ij s ij) = b i + ρ 2 (b i) 2, if Cij s < ρb i (Cs ij) 2 2ρ, if ρb i Cij s < 0 Z(x s ij ) will be used to the swap process because open an arc (i, j) with a negative Cs ij indicates the node j will be selected as a searched candidate. ˆ Z (g) = i N F i sgs i where F i s is the transaction cost on arc ( i 0, i 1) from stage 0 to stage 1 under scenario s, we will compare the ratio F s i R s i ˆ Z (y) = (i,j) A 1s (1 c s ij stage 1 under scenario s. ) y s ij where cs ij to determine a better node. is the transaction cost ratio on arc (i, j) at

92 Chapter 5. Progressive Hedging for Cardi. Constrained FP 80 For a given g satisfy (5.33) and (5.34), an optimal flow x (g) is determined by minimizing Z (x) subject to (5.31) and (5.32) and then a corresponding optimal flow y (x, g) can be determined by maximizing Z (y) subject to (5.35) and (5.36). Thus a current feasible point (x, g, y ) is fixed. Intuitively we can distribute the first K assets which have the highest returns to get the initial g. However this may not global optimal because the transaction costs (i,j) A os Cs ij xs ij or F s i may be too high so that the current selection is inefficient. Therefore we need to swap some assets and move to a better neighbor point (x, g, y ). Assume that we swap one asset each time and hope to get a better objective value. The details of the Tabu heuristic framework is described in Appendix B. 2. Rather than randomly swap between K and S, we use 3 cases in Step 1 to include the node which can improve the network structure. In case 1 we close the arcs with high transaction ratio and open the arcs with low transaction ratio so that the structure in stage 1 can be improved, and then hopefully it can reduce the objective globally. In case 2, we move to the node with a higher return because this movement may increase the profit in stage 2 dramatically and can offset the increased cost in stage 1. Case 3 is the opposite situation of case 2, we move to the node which may decrease the cost in stage 1 and offset the decreased profit in stage 2. Compared with the sub-problems in [43], the main difference between the Tabu methods is that they built cycle-based neighbourhoods and searched the associated γ - residual networks by Tabu heuristic, while we construct the path-based neighbourhoods and search them. This is determined by the nature of the problems. For their problem, adding or deleting one arc of the network will not affect the iterative decisions too much since there are many other alternative arcs can be chosen, so they build a cycle which is consist of many arcs and evaluate the flow perturbation of the cycle. For our problem, any arc between different stages will have significant effects on the objective value, so we must evaluate the path-based neighbourhood rather than the cycle-based neighbourhood. We first solve the small size sub-problems (N=50, K=5, S=15) and list the result in the following Table 5.3. And then increase the problem size N to 100, also we list 2 different randomly instances with different scenario number in Table 5.4 and 5.5.

93 Chapter 5. Progressive Hedging for Cardi. Constrained FP 81 Table 5.3: Computational result (N=50, K=5, S=15) - instance 1 N=50, Gurobi LR method(iter# = 200) Tabu method(l=5, iter#=10) K=5,S Best LB Feasi. UB Gap Time Best LB Feasi. UB Gap Time Gap to Feasi. UB Time Gap to (%) (s) (%) (s) Gurobi (s) Gurobi % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % Aver % % From Table 5.3 we see that all methods obtained the optimal solution for each scenario sub-problem, the running time by Tabu method is generally larger than the LR method. Table 5.4: Computational result (N=100, K=10, S=3 - instance 2) N=100, Gurobi LR method(iter# = 200) Tabu method(l=5, iter#=10) K=10,S Best LB Feasi. UB Gap Time Gap Time Gap to Best LB Feasi. UB Feasi. UB Time Gap to (%) (s) (%) (s) Gurobi (s) Gurobi % % % % % % Aver % % Table 5.5: Computational result (N=100, K=10, S=10 - instance 3) N=100, Gurobi LR method(iter# = 200) Tabu method(l=5, iter#=10) K=10,S Best LB Feasi. UB Gap Time Gap Time Gap to Best LB Feasi. UB Feasi. UB Time Gap to (%) (s) (%) (s) Gurobi (s) Gurobi % % % % % % % % % % % % % % % % % % % % Aver % % From Table 5.4 and 5.5, we see that the Tabu method has better solutions than LR method, however, the running time is longer than LR method. Different parameters about both methods are tested to speed up the solving process in Appendix B. 3. We embedded LR and Tabu methods into Progressive Hedging algorithm for whole problem in next Section.

94 Chapter 5. Progressive Hedging for Cardi. Constrained FP Progressive Hedging for FP problem Design a lower bound Lagrangian relaxation technique is used to achieve a quality lower bound for model (5.8) - (5.16) in this section. First we covert the equality non-anticipativity constraints (5.13) and (5.14) into the following equivalent inequality constraints (5.43) - (5.46): x s ij + p s x s ij 0, (i, j) A 0s, s S (5.43) s S x s ij p s x s ij 0, (i, j) A 0s, s S (5.44) s S g s i + s S p s g s i 0, i N, s S (5.45) g s i s S p s g s i 0, i N, s S (5.46) Then we assign µ s ij 0 and µ s+ ij 0 to constraint (5.43) and (5.44) respectively and φ s i 0 and φ s+ i 0 to constraint (5.45) and (5.46) respectively. Then the Lagrangian objective for the whole programs is: LR LB (x, g, y, µ, φ) = p s c ij x s ( ) ij 1 c s ij y s ij s S (i,j) A os (i,j) A 1s + p s ( µ s ij x s ij + ) ( ij) p s x s ij + µ s+ ij x s ij p s x s s S (i,j) A os s S s S ( ) ( )) φ s i + ( p s s S i N = p s ( s S (i,j) A ( os g s i + s S p s g s i c ij + µ s+ ij µ s ij + φ s+ i p s s S ( φ s+ i g s i s S p s g s i ( µ s+ ij φ s ) ) i gi s ) ) µ s ij x s ij + ( p s φ s+ i φ s i p s s S i N s S Setting Uij s = c ij + µ s+ ij µs ij ( ) ps µ s+ ij µ s ij and Φ s i = φs+ i s S the primal problem becomes to minimize LR LB (x, g, y, µ, φ), i.e.: min x,g,y s S p s Uijx s s ij + Φ s i gi s (i,j) A os i N ) (i,j) A 1s (i,j) A 1s ( 1 c s ij ) y s ij φ s i p s ( φ s+ i s S ( 1 c s ij ) y s ij φ s ) i, (5.47)

95 Chapter 5. Progressive Hedging for Cardi. Constrained FP 83 s.t. (5.31) (5.36) (5.48) (5.47) can be decomposed across different scenario, and the associated dual problem is max µ 0,φ 0 min x,g,y LR LB (x, g, y, µ, φ). The dual variables can be updated by sub-gradient method that we did for the scenario sub-problem in Section For each iteration, a Lagrangian lower bound and a feasible upper bound are generated at the same time in the proposed Progressive Hedging algorithm in next section Progressive Hedging method Note that maximize the final wealth implies minimize the transaction cost in the constraint, we propose Progressive Hedging algorithm to adjust the cost ratios on the arcs so that the first stage decision variables can converge as much as possible. We adjust the linear coefficient of the first decision variable iteratively, the process can be seen as follows: at first stage we choose portfolio components arbitrary, and the corresponding transaction cost can occur, i.e. coefficient C s ij = c ij, to arc (i, j), as the scenarios have been revealed, we adjust C s ij to arc (i, j) and F s i to arc (i 0, i 1 ) at the same time in object function. If g s i in some scenarios is 0 and most of other scenarios are 1, we award the arcs come into the node i and penalize the arcs goes out from node i in that scenarios so that more values can remain in the node i, meanwhile F s i decreased so that the flow can pass on arc (i 0, i 1 ). Conversely if g s i in some scenarios is 1 and most of other scenarios are 0, we penalize the arcs come into the node i and award the arcs goes out from node i in that scenarios so that more values can leave node i, meanwhile F s i increased so that the flow can leave the arc (i 0, i 1 ). We implement this strategy in following algorithm. Progressive hedging Algorithm Step 0: (Initialization) λ s,v ij π s,v i v 0 0, (i, j) A 0s, s S constraint (5.13) and (5.14) 0, i N, s S µ s,v ij, µ s+,v ij 0, (i, j) A 0s, s S φ s,v i, φ s+,v i ρ v ρ 0 constraint (5.43) - (5.46) 0, i N, s S For all s S, do

96 Chapter 5. Progressive Hedging for Cardi. Constrained FP 84 C s,v ij c ij, (i, j) A 0s F s,v i 0, i N Solve the corresponding FP sub-problem (5.30) - (5.36) by LR method and Tabu search method x v ij s S ps x s,v ij, (i, j) A0s, s S g v i s S ps g s,v i, i N, s S Calculate and evaluate g M,v (First K nodes with largest probability in g v i ). Calculate and evaluate x M,v (aggregation value of x s,v ij under g M,v ). feasible upper bound ( x M,v, g M,v) Step 1: (Coefficient adjustment) C s,v ij F s,v i v v + 1. For all s S, do c ij + λ s,v 1 ij 0 + π s,v 1 i ρ v 1 x v 1 ij, (i, j) A 0s ρ v 1 g v 1 i + ρv 1 2, i N Solve the corresponding FP sub-problem (5.30) - (5.36) by LR method and Tabu search method x v ij s S ps x s,v ij, (i, j) A0s, s S g v i s S ps g s,v i, i N, s S Calculate and evaluate g M,v (First K nodes with largest probability in g v i ). Calculate and evaluate x M,v (aggregation value of x s,v ij under g M,v ). U s,v ij Update minimal upper bound if ( x M,v, g M,v) gives current best. c ij + µ s+,v ij Φ s,v i 0 + φ s+,v i µ s,v ij For all s S, do p s ( s S φ s,v i p s s S µ s+,v ij ( φ s+,v i ) µ s,v ij φ s,v i, (i, j) A 0s ), i N Generate lower bound by solving the corresponding FP sub-problem (5.47). Calculate gap v = (UB LB) UB. Step 2: (Lagrangian multiplier and penalty update) λ s,v ij λ s,v 1 ij + ρ v 1 ( x s,v ij π s,v i π s,v 1 i + ρ v 1 (g s,v i ) x v 1 ij, (i, j) A 0s, s S g v i ), i N, s S

97 Chapter 5. Progressive Hedging for Cardi. Constrained FP 85 ρ v αρ v 1 ( ) µ v ij max 0, µ v 1 ij + t v µd v µ (gradient method) ( ) φ v i max 0, φ v 1 i + t v φ dv φ (gradient method) Step 3: (Move to next iteration) Calculate δ v = s,v v s S ps x ij x ij, gap v = (UB LB) g i g UB. i If δ v ɛ or gap v η, GO TO Step 1. The aggregation operator, i.e. g v i, define the opening or closing probability of the arc (i 0, i 1 ), we open the first K node who has the largest probabilities and close others in g v i, then the aggregation of x s,v ij under g M,v is a feasible solution because for any s S, x s,v ij point that satisfy constraint (5.9) and x M,v = s S ps x s,v ij (5.9). Therefore the objective under ( x M,v, g M,v) is a feasible upper bound. is a feasible is also a feasible point to constraint Numerical experiment We test large size instances in this section. First we list the computational result for different types of problem in literature in Table 5.6. Gade et al. (2013) [58] Crainic et al. (2011) [43] Watson and Woodruff (2011) [147] Veliz et al. (2011) [145] Lokketangen and Woodruff (1996) [103] Takriti et al. (1996) [142] Table 5.6: Computational result in literature # of variable # of # of Maximal Time # of Total integer variable constraint scenario iteration (min) 1,200 (binary) 16,194 24, / 10,800 (binary) 874, , (binary) 50,544 (binary) N (binary) 2,400 (binary) 405 1, (Aver.) (Aver.) 77, , / N + N 2 2N 10 / ,800 /

98 Chapter 5. Progressive Hedging for Cardi. Constrained FP 86 From Table 5.6, we see that the largest scenario number equal 324 in [145], however, the total variable number is only 77,760. Crainic et al. [43] solved their problem with 90 scenarios and 874,800 variables, which is impressive. For our FP problem, we have ( 2N 2 + N ) S variables and ( N 2 + 3N + 1 ) S constraints. We set N = 100, and test the performance of PH method under S = 15, 30, 50 and 75. Then our PH problem includes 1,507,500 variables and 772,575 constraints for largest size instance. Table 5.7 list the parameter setting of our computation for the models and PH algorithm, the parameter R s i are generated by moments matching method in Section All instances were implemented on a 2.66 GHz computer with 3 GB memory available. Table 5.7: Parameter setting for the model and PH algorithm For Model For PH algorithm b i 100 Outer loop iteration # 7 li x 10 Iteration # for sub-gradient method 60 u x i sum(b i ) Iteration # for Tabu search method 10 c 0 ij.05 ρ log (arc#) (1 + D0) c s ij rand( ) D0 The inconsistency level l ys ij 0 u ys ij sum(b i ) The initial ρ 0 is determined by the inconsistency level D0, i.e. the number of arcs and nodes for which there is non-consensus amongst the scenario solutions [43]. Table 5.8: Bound details under different methods for S=15 (N,K,S) Solve by Gurobi PH with LR sub-solver PH with Tabu sub-solver 1 1 Best LB Feasi. UB Gap (%) Time (s) Best LB Feasi. UB Gap (%) Time (s) Feasi. UB Time (s) UB LR UB T abu T LR T T abu (100,5,15) % % (100,10,15) % % (100,15,15) % 5.78% (100,20,15) % 34.92% (100,25,15) % 43.64% (100,30,15) % 74.72% (100,35,15) % 94.34% (100,40,15) % 15.33% Aver % 25.33% The running time for Gurobi is set as 10 hrs in Table 5.8, and the scenario sub-problems are solved by LR method and Tabu search method separately. It is clear to see that the gaps decreased as K increased, but the gap via PH shrank more quickly than Gurobi. PH methods returned a better solution than Gurobi when K = 20. From K = 25 to 40, Gurobi cannot return a quality solution and the solution from PH methods still be considerable from the practice

99 Chapter 5. Progressive Hedging for Cardi. Constrained FP 87 point of view. Compared with the two PH methods with different sub-solver, the Tabu search can return a better solution in a shorter time when K = 5. When K is greater than 5, the solution is close each other but the running time of Tabu search is larger than LR method. Table 5.9: Bound details under different methods for S=30 (N,K,S) Solve by Gurobi PH with LR sub-solver PH with Tabu sub-solver 1 1 Best LB Feasi. UB Gap (%) Time (s) Best LB Feasi. UB Gap (%) Time (s) Feasi. UB Time (s) UB LR UB T abu T LR T T abu (100,10,30) NaN NaN % 44.06% (100,15,30) NaN NaN % 49.43% (100,20,30) NaN NaN % 42.67% Aver. NaN NaN % 45.21% From Table 5.9, we can see that Gurobi cannot return the lower bound for S = 30, and the quality of the feasible solution is poor. On the other hand, both PH methods can return reasonable bounds within a limited time. The solutions between LR and Tabu search are close, but the Tabu search consumed averagely 45% more time than LR method. Table 5.10: Bound details under different methods for S=50 (N,K,S) Solve by Gurobi PH with LR sub-solver PH with Tabu sub-solver 1 1 Best LB Feasi. UB Gap (%) Time (s) Best LB Feasi. UB Gap (%) Time (s) Feasi. UB Time (s) UB LR UB T abu T LR T T abu (100,20,50) / / / 16 hrs % 31.33% (100,25,50) % 45.19% (100,30,50) % 72.97% Aver % 49.60% For instances S = 50, Gurobi fail to solve the problem in 16 hrs in Table Meanwhile, the LR methods return the considerable bounds consistently. The average gap between lower and upper bounds is 4.96% and the average running time is around 50 hrs. From the last two columns, we can see that the Tabu search method is more time expensive than the LR method. The same tendency of the solution occurred for larger scenario S = 75 in Table Table 5.11: Bound details under different methods for S=75 (N,K,S) Solve by Gurobi PH with LR sub-solver PH with Tabu sub-solver 1 1 Best LB Feasi. UB Gap (%) Time (s) Best LB Feasi. UB Gap (%) Time (s) Feasi. UB Time (s) UB LR UB T abu T LR T T abu (100,20,75) / / / / % 43.65% (100,25,75) % 40.33% (100,30,75) % 39.43% Aver % 41.08% In a nutshell, our numerical results showed that the proposed PH method with different sub-solvers has consistent performance. More numerical testing will list for the index tracking problem with network structure.

100 Chapter 5. Progressive Hedging for Cardi. Constrained FP Progressive Hedging for Index Tracking problem We extend the FP framework to index tracking problem in this section. The objective function of FP problem, (5.8), is modified as follows: min i I i Nx 0 + p s yi s I1 s + c ij x ij ( p s s S i N (j,i) A 0 s S (i,j) A 1s ) ( ) 1 c s ij yij s where I 0 is the market value of the target index at stage 0 and Is 1 is the market value of the target index at stage 1 under scenarios s. The objective can be seen as a trade-off between the goals that minimize the tracking errors for both stages and maximize the final wealth at last stage. Different types of decision maker may emphasis different aspects, and we can assign a weights vector (α, β) to the goals. For example, setting the ratio β/α = 1/10 denotes the goal of minimizing the tracking error is 10 times important than the goal of maximizing the final wealth, and vice versa. We first set the ratio β/α = 1/1 and test more ratios in this section late. The objective can be linearized by introducing new variables X, X and Y s, Y s : x i I 0 = X + X yi s i N I1 s = Y s + Y s i N x i I 0 = X X Stage 0, yi s I1 s = Y s Y s Stage 1. i N X, X 0 i N Y s, Y s 0 Then a network structure index tracking model can be formulated as follows: min p s X s + X s + Y s + Y s + c ij x s ( ) ij 1 c s ij y s ij (5.49) s S (i,j) A os (i,j) A 1s s.t. (5.9), (5.10), (5.11), (5.12), (5.15), (5.16) x s ij = x ij, (i, j) A 0s, s S (5.50) gi s = g i, i N, s S (5.51) X s = E ( X s ), s S (5.52) X s = E (X s ), s S (5.53) x s i I 0 = X s X s, s S (5.54) i N yi s Is 1 = Y s Y s, s S (5.55) i N

101 Chapter 5. Progressive Hedging for Cardi. Constrained FP 89 X s, X s 0, Y s, Y s 0 s S (5.56) Constraints (5.50) - (5.53) denote the non-anticipativity constraints for different variables. After taken off the absolute sign, the whole program becomes an SMIP, and the same Progressive Hedging procedure can be applied to above index tracking model. Similar to FP problem, we test the index tracking model correspond to different scenarios. The initial investment b i to node i is scaled by the market value weights. For example, suppose that the total market value of the index SP100 at stage 0 is 10,000, i.e. 100 N = 10, 000, and the weight of the first asset is according to the real data, then the initial cash on the first node is Numerical result list in Table N=100 S=15 K Best LB Table 5.12: Numerical result (N=100, K, S=15) Solve by Gurobi Feasi. UB Gap (%) Time (s) Best LB PH with LR sub-solver Feasi. UB Gap (%) Time (s) Aver (18 hrs) (7.3 hrs) The running time set as 18 hrs for Gurobi, we see that the average running time by PH method is 7.3 hrs from Table Meanwhile, the objective values by PH method are close to Gurobi when K = 10, 15 and PH method return better solutions for instance K = 20, 25. PH method also returns better lower bounds for all instances, which makes the average gap (8.24% averagely) is much better than the gaps by Gurobi (671.45% averagely).

102 Chapter 5. Progressive Hedging for Cardi. Constrained FP 90 Table 5.13: Numerical result (N=100, K, S=30) N=100 S=30 Solve by Gurobi PH with LR sub-solver K Best Feasi. Gap Time Best Feasi. Gap Time LB UB (%) (s) LB UB (%) (s) 10 NaN NaN NaN NaN NaN NaN Aver. NaN NaN (18 hrs) (57 hrs) From Table 5.13, we obtain the same tendency that our PH method returns higher quality solutions than that by Gurobi. As scenario increased, Gurobi can only return heuristic solutions within the setting time, and such solutions are not practical for FP problem. N=100 S=50 K Best LB Table 5.14: Numerical result (N=100, K, S=50) Solve by Gurobi Feasi. UB Gap (%) Time (s) Best LB PH with LR sub-solver Feasi. UB Gap (%) Time (s) 10 NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN Aver. NaN NaN NaN NaN (78 hrs) N=100 S=75 K Best LB Table 5.15: Numerical result (N=100, K, S=75) Solve by Gurobi Feasi. UB Gap (%) Time (s) Best LB PH with LR sub-solver Feasi. UB Gap (%) Time (s) 10 NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN NaN Aver. NaN NaN NaN NaN (174 hrs) Note: NaN denote out of memory. From Table 5.14 and 5.15, Gurobi cannot start the solving process because of memory issue for loading the large size coefficient matrix, while the PH method returns practical solutions consistently, and the gap between the lower and upper bound is relatively small (around 5%).

103 Chapter 5. Progressive Hedging for Cardi. Constrained FP 91 We compare the PH running time for FP and index tracking problems in Figure 5.3, we can see that the running times is nearly linear increase correspond to the scenario number. The running time for index tracking are larger than the time of FP problem, this is reasonable since more variables and constraints are included in the index tracking problem, and different goals need to be satisfied. Figure 5.3: Running time of PH method for different problems Next we test more instances for different β/α ratios. We set β/α equals 1/10, 1/1, and 10/1 respectively to represent different weights on different goals. Table 5.16 list the gap between bounds and solving time for different K and S. Table 5.16: Test different ratios (N=100, K, S) β/α = 1/10 β/α = 1/1 β/α = 10/1 N=100 (K,S) Best LB Feasi. UB Gap (%) Time (hrs) Best LB Feasi. UB Gap (%) Time (hrs) Best LB Feasi. UB Gap (%) Time (hrs) (10,15) (15,15) (20,15) (25,15) (10,30) (15,30) (20,30) (10,50) (15,50) (20,50) Aver. / / / / / /

104 Chapter 5. Progressive Hedging for Cardi. Constrained FP 92 From Table 5.16 we see that the average gaps are 0.44%, 0.83% and 1.18% for ascent β/α ratios. Meanwhile, the average solving time are hrs, hrs and hrs. This can be explained that for the index tracking the PH algorithm mainly adjust the node value on the network while for the FP problem needs to adjust both the value on the nodes and arcs during the iterations, which makes the gap of β/α = 10/1 larger than gap of ratio β/α = 1/10. Again The running time linearly increased with respect to the scenario number S. Overall the PH method can return the high-quality solution for index tracking problem. 5.6 Conclusions and Discussion We incorporated cardinality restriction into Financial Planning problem (LP) and developed it into an SMIP. Inspired by real application, we decomposed the SMIP corresponding to scenarios and effectively solved large size instances for FP and index tracking problems by proposed Progressive Hedging Algorithm. Subgradient and Tabu search methods are applied to Progressive Hedging framework to speed up the solving process. Numerical experiments showed that our method can efficiently solve the SMIP with large scenario number.

105 Chapter 6 Lagrangian Relaxation for Cardinality Constrained Conic Programming In Chapter 5, we applied the stochastic mixed integer programming to protect against the uncertainty of asset returns in Financial Planning problem. In this section we will study the robust optimization that can also immune to the parameter uncertainties of both return and variance for index tracking problem which can be captured by presented Cardinality Constrained Conic Programming. 6.1 Introduction to CCCP Given a variable vector (x, t, y) R n+l+n, cardinality constrained conic programming (CCCP) can be written as: min c T x + d T y (6.1) s.t. A i x + b i t i, i = 1,, L (6.2) Ex + Gt f (6.3) e T y = q (6.4) l j y j x j u j y j, j = 1,, n (6.5) 93

106 Chapter 6. Lagrangian Relaxation for CCCP 94 y {0, 1} (6.6) where c, d R n, E R m n, G R m L, f R m, e R n where all components equal 1, and l j, u j denote the lower and upper bounds for variable x j. denotes the standard Euclidean norm, i.e. z = z T z. Constraint (6.2) indicates that variable (x, t i ) lies in ith Lorentz cone with dimension (p i + 1) and parameters A i R pi n, b i R p i, t i R. Without cardinality restriction for variable x, i.e. constraints (6.4) - (6.6), the CCCP reduces back to a Second-Order Cone Programming (SOCP), which has been well-studied in literature [102, 6]. Model (6.1) - (6.6) is a primary class of Mixed-Integer Second-Order Cone Programming (MISOCP) and has significant influence on theory and application [15], and it is one generalization of mixed 0-1 linear or quadratic programs. This problem is particularly interesting to us from both methodology and application points of view. First the proposed CCCP is nonconvex and therefore an NP-hard problem because of the binary requirement (6.6), and finding an optimal or near-optimal solution of large-scale CCCP within a reasonable time has proven to be a puzzle for researchers in optimization for years. Numerical approaches, which can be generally categorized by exact or inexact methods, emerged to globally shrink the bounds gap and achieve a good solution. Exact methods typically explore only part of variable space by using pruning rules and ordering heuristics to avoid visiting all variable space, and meanwhile maintaining the feasibility. Inexact strategies mainly utilize the local search techniques to e- valuate a small neighborhood of current solutions, and quickly move to a better solution by following a promising direction. There are substantial similarities and fundamental distinctions between the exact and inexact methods. Both method groups try to convert the NP-hard problem to tractable sub-cases so that associated relaxed bounds and feasible solution can be iteratively improved. The difference between them is that the exact method can theoretically guarantee the optimal solution but requires exponential running time, while the inexact method can quickly produce a reasonable solution but cannot guarantee the optimal solution. Secondly the proposed CCCP is an important mathematical tool to handle various problems in real-life. For example, for portfolio selection in finance, the cardinality constraints (6.4) - (6.6) control the portfolio size while the conic constraint (6.2) is usually used to restrict or minimize the portfolio variance. Investors are struggling to decide a trade-off between the size and risk of a portfolio. On one hand the risk constraint (6.2) may be easily violated if the

107 Chapter 6. Lagrangian Relaxation for CCCP 95 portfolio concentrates on a few assets i.e. q is small. On the other hand fully replicating the market (large q) is inefficient due to transaction costs, management fees, and other concerns which are captured by constraint (6.3). Moreover, Model (6.1) - (6.6) can be seen as a naturally extension of robust optimization since the parameter uncertainty sets can be formulated as the conic form, and the cardinality constraint restricts the number of non-zero components in x. Motivated by the real application, we design a Lagrangian based inexact approach that can quickly approximate the optimal solution for the CCCP, and then use the CCCP framework to deal with a type of index tracking problem under uncertain environment. More specifically, we decompose the variable space into continuous and integer parts by relaxing connected constraint (6.5), and as a result, the associated Lagrangian subproblems, i.e. one SOCP and one 0-1 knapsack problem, can be solved efficiently. Moreover a sub-gradient cut and fully regular cuts are generated at each iteration to shrink the feasible set of {0, 1} n structure. Computational observation shows that the sub-gradient cut can significantly speed up the solving process. The proposed Lagrangian relaxation scheme enriches the solving methodology for CCCP, and we show the effectiveness of the LR method through a comparison with Gurobi s mixed integer SOCP solver. To the best of our knowledge, our work is the first paper to focus on the relaxation of the boundary constraint (6.5) for the CCCP in current literature. We organize the rest of chapter as follows: We display a literature review for CCCP and associated applications and methodologies in Section 6.2, and then propose a Lagrangian decomposition scheme in Section 6.3. In Section 6.5, we compare our computational results with those from Gurobi s mixed integer SOCP solver to illustrate the effectiveness of the LR method. Finally, Section 6.6 concludes our work. 6.2 Literature Review Second-Order Cone Programming (SOCP), i.e. Model (6.1) - (6.3), includes linear programming (LP), convex quadratic programming as special cases, and it also is one special case of semidefinite programming (SDP). Therefore, SOCP strengthens the ties between the linear programs and non-linear convex programs, and it attracts many researchers to solve SOCP efficiently in last two decades. The extension of existing primal-dual based methods, i.e. inte-

108 Chapter 6. Lagrangian Relaxation for CCCP 96 rior point method and active set method, from LP to SOCP is a natural transition, and such transitions have been proven successfully for solving large size optimization problems. Lobo et al. [102] showed that many engineering problems can be generalized as SOCP and presented a primal-dual based interior point method which generally requires 5-50 iterations in their work. Alizadeh and Goldfarb [6] studied the algebraic properties of Jordan frames for the second-order cone, and adopted a norm-2 centrality measure to obtain a polynomial time interior point method. They pointed out that numerical stability of the method is available on testing of both real application and randomly generated problems. Also by using the concepts of Jordan algebra, Tsuchiya [143] analyzed the complexity of variants of primal-dual path following methods for SOCP via extension of Nesterov and Todd (NT) direction [117, 118] and HRVW/KSH/M direction [91, 110, 78] from that of SDP. His work proved that both types of algorithm reserve polynomial iteration-complexity which is relevant to the number of the second-order cones. Kuo and Mittelmann [97] extended and developed the interior point method in [7] to SOCP, and displayed the robustness of their method through the comparison with different solvers on testing extreme instances for many Operations Research (OR) problems. As a matter of fact, there is a large amount of research showing that interior-point based algorithms have polynomial time complexity for SOCP, LP and SDP [111, 112, 133]. Meanwhile, software packages based interior-point methods are currently available to efficiently handle SOCPs or convex programs, e.g. SeDuMi [140], MOSEK [113], CPLEX [42], CVX [68], and GUROBI [71]. Active-set based extension for SOCP also draws a lot of attention from authors. Erdougan and Iyengar [52] studied a single-cone SOCP by dualizing its nonnegativity constraint to obtain Lagrangian subproblems where the nullspace of coefficient matrices are projected onto associated orthonormal basis. They compared their active set method with SeDuMi for the randomly generated network flow problems. Goldberg and Leyffer [65] recently designed a two-phase active set method which firstly identified the active cones and then applied Newton-like method to quickly obtain the solutions of sub-socps. Numerical comparison with interior-point based solvers were displayed in their work. Although polynomial analysis is not available, active set methods exhibit nice property that interior point methods lack, i.e. it can obtain vertices for MISOCP relaxation in the nodes of the branch-and-bound search tree. Aside from the inte-

109 Chapter 6. Lagrangian Relaxation for CCCP 97 rior point method and active set method, simplex based approaches for SOCPs were studied in [148, 67], and the method using polyhedral approximations of the second-order cone was investigated in [14]. While both theories and methodologies for SOCPs have been well-established, MISOCP is relative new but more attractive since it has more broad applications. A comprehensive survey about MISOCPs was compiled by Benson and Saglam [15]. They formulated numerous examples in fields of operations management, engineering, and machine learning as MISOCPs, and reviewed different approaches that solve MISOCP in the literature. Another interesting application was introduced by Miyashiro and Takano in [109]. The authors improved the fitting ability of a multiple linear regression model via MISOCP formulations to select a limited number of factors. The benefits by extending the strong dual theory from LP to convex programs does not exist for integer programming, and thus solution methods for MISOCP will be more challenging and will mainly rely on the heuristic methods such as branch-and-cut algorithms that solve the SOCP or SDP relaxations to reduce the number of nodes visited in the search tree. Different advanced cuts for MISOCP were explored in the present literature. Cezik and Iyengar [30] generated the linear cuts based on Chvatal-Gomory (C-G) procedure and convex quadratic cuts (e.g. lift-and-project cut) through tighter relaxations to approach the convex hull of the solution set. However, updating the dual vector in self-dual cone for C-G linear cut generation is not clear in their work. Atamturk and Narayanan [8] showed that their conic mixed-integer rounding cuts can efficiently reduce the root nodes of the branch-and-bound tree for solving MISOCP problems. Drewes and Pokutta [49] derived a strong binary symmetric cut for a special class of MISOCP where binary variables only occur within the conic constraints by extending the Sherali Adams framework [137]. Besides the cut generation based algorithms, there are also other inexact methods such as outer approximation approach using subgradient linearizations [50], non-linear reformulation to original MISOCP by smoothing and regularization [16] for solving MISOCP. The proposed CCCP can be seen as one special case of general MISOCP where binary vector y affects the continuous variable x only in the linear polyhedrons, so the reviewed above methods for general MISOCP could be also applied to our CCCP. Moreover, the special structure of the constraint set allows us to adopt the decomposition advantages which widely used in LP

110 Chapter 6. Lagrangian Relaxation for CCCP 98 and mixed-integer LP and benefit of SDP relaxation for binary variable. Cardinality restriction (6.4) and binary requirement (6.6) are crucial for the presented CCCP. Numerous studies seek to deal with these two hard constraints and the methodologies can be classified into two categories. The first group either reformulates binary variable y j {0, 1} as constraints yj 2 y j = 0, y j [0, 1] or reconstruct the cardinality constraint into a non-convex SDP form, and utilizes the semidefinite relaxation to the developed non-convex programs. Poljak et al. [123] explored the equivalence of quadratic and semidefinite relaxations for 0-1 quadratic programming, and applied their technique to different practical problems. Galli and Letchford [60] derived an equivalent SDP relaxation that may generate tighter Lagrangian bound for the 0-1 QP in their work. d Aspremont et al. [47] employed a l 2 -norm to approximate the cardinality and explain the robustness and sparsity of the solution. Chen et al. [33] suggested that l p [0,1] -norm regularization can achieve better performance for the sparse portfolio management. However, l-norm regularization cannot control the size exactly and the associated solution cannot guaranteed the dualized rank one constraint. In contrast with the SDP form reformulation, another main category of methodologies focuses on the cut generation in branch-and-bound tree or heuristics design for satisfying of cardinality constraint (6.4). Bienstock [23] replaced constraint (6.4) with tighter constraint j (x j/u j ) q and generated a valid cut for branch-and-bound algorithm. They visualized the advantage of his method via simple numerical examples. Bertsimas and Shioda [22] continuously investigated this alternative to quadratic programs by applying Lemke s method to sub-problems in branch-and-bound tree, and compared its performance with that of CPLEX solver. In their work, the = sign can be relaxed to a sign because the optimal solution is always obtained in the surface of convex hull of feasible set. However, such sign relaxation is prohibited for our methodology in Section 6.3. Chang et al. [31] presented three types of heuristic algorithm to handle the cardinality constraint set (6.4) - (6.6), but no comparison was made with the optimal solution. Cui et al. [46] applied the factor model to simplify the parameters of MIQCQP, and obtained a better SOCP relaxation bound for Lagrangian subproblems. However, the subproblems may be still hard to solve since they are keeping a MIQCQP. Also the accuracy of parameter generation via factor model for MIQCQP needs to be examined through historical backtesting.

111 Chapter 6. Lagrangian Relaxation for CCCP 99 We adopt ideas of decomposition and cut generation to develop our strategy for solving the proposed CCCP. To the best of our knowledge, this strategy for the CCCP have not been studied in formal literature. The method is described as follows. The CCCP is firstly divided into to two independent but easier parts via dualizing the connected constraint (6.5), and then be unified by adjusting the dual variables for relaxed constraint in the dual space. Both subproblems can be efficiently solved, e.g. the first part remains SOCP while the second parts is linear 0-1 knapsack problem. Meanwhile, a sub-gradient cut and fully regular cuts are used to exclude the sub-optimal points that have been explored during previous iterations. Tadonki and Vial [141] pointed out that the boundary constraint (6.5) makes problem hard and handled the MIQP through relaxing the hard constraint to generate C-G cuts for 0/1 sub-problems. We also focus on the boundary constraint but our strategy is fundamentally different than they used. The authors fixed variable x and then obtained associated variable y, i.e. (x, y) while we use an reversed solving direction, i.e. (x, y), also the authors did not show the dual updating process but we do. Another main difference is that the authors used the traditional C-G cuts to speed up the procedure of solving sub-problem, while our cut based on the weak dual theory is totally new for the sub-problem. We show the specific details for proposed method in next Section. 6.3 Lagrangian Relaxation Scheme Observe that constraint (6.5) connect the continuous variable x and binary variable y, so we relax constraint (6.5) and decompose the model (6.1) - (6.6) into continuous part (SOCP associated with variable (x, t)) and integer part (only y). Both parts are easily to solve since the SOCP is a convex problem and integer part is a Knapsack problem with an easy constraint. We assign π j 0 and π + j 0 to constraint lb j y j x j 0 and ub j y j + x j 0 respectively. Then we construct following Lagarangian term: L ( x, y, π, π +) = n c jx j + n d jy j + n j=1 j=1 j=1 π j (lb jy j x j ) + n j=1 π+ j ( ub jy j + x j ) = n ( ) c j + π j + π j x j + n ( ) d j + πj lb j π j + ub j y j j=1 j=1 = ( c + π + π ) T x + ( d + π lb π + ub ) T y = C T x x + C T y y

112 Chapter 6. Lagrangian Relaxation for CCCP 100 where C x and C y are the adjusted coefficient vector associated variable x and y. Then above L (x, y, π, π + ) can be decomposed into two separated Lagarangian sub-problems. The first part can be seen as a Second-Order Conic programming, i.e. LR SOCP (x, π, π + ), as follows: min (c + π + π ) T x (6.7) s.t. (6.2), (6.3) Model (6.7) can be efficiently solved by convex analysis and the associated methodology since it is a SOCP. For example, interior point method is used to solve LR SOCP (x, π, π + ) in our LR algorithm. The second part can be seen as a linear integer programming, i.e. LR IP (y, π, π + ): min (d + π lb π + ub) T y s.t. (6.4), (6.6) (6.8) Model (6.8) is a 0-1 knapsack problem with relative smaller size and thus can also be efficiently solved by commercial solver because of its linear form structure. Moreover, different types of inequalities are generated and added into sub-problem (6.8) to exclude infeasible and inefficient feasible points of original problem. The first one is sub-gradient cut that dreives from the weak dual theory where: L ( x, y, π, π +) = C T x x + C T y y C T x x + C T y y = L ( x, y, π, π +) in which L (x, y, π, π + ) is the Lagrangian objective in last iteration ν 1. We partition { } the index set I of y from iteration ν 1 into I (v 1) 0 = j I y (v 1) j = 0 and I (v 1) 1 = { } j I y (v 1) j = 1. Note that the right hand side L (x, y, π, π + ) may not enforceable in practice, we replace it by L (x, y, π, π + ) +, where 0 is used to strengthen the lower bound. Then after solving the LR SOCP (x, π, π + ) part, a sub-gradient cut can be generated at current iteration: C T y y L ( x, y, π, π +) + C T x x (6.9) can be set as a small constant positive or varied iteratively. We observed that constant positive cannot or slightly improve the convergence time in our computation. In practice, we set as follows: = max ( [ { 0, ɛ min C (x) j + C (y) j } { j I (v 1) 0 max C (x) j + C (y) j }]) j I (v 1) where ɛ is a scale to adjust the lower bound enhancement. The empirical value of ɛ decreased as iteration number increased since the strengthening became harder and harder as the bounds converged to each other. For instance, ɛ = 1 when v V/3, and ɛ = 10 4 when v V/3 where 1

113 Chapter 6. Lagrangian Relaxation for CCCP 101 V is designed iteration tolerance. While inequality (6.9) is used to strengthen the Lagrangian lower bound, the following inequalities are applied to swap the nodes and test the infeasibility of original model. First at current iteration, the following inequality need to be satisfied if better solution exist: y j 1 (6.10) j I (v 1) 0 Inequality (6.10) indicates that there at least one node in I (v 1) 0 and one node in I (v 1) 1 exchange each other. If the = sign in (6.4) is relaxed to sign, we cannot guarantee that always pairs of node are switched between the sets in I (v 1) 0 and I (v 1) 1. Therefore, we prohibit the sign relaxation in our method. Inequality (6.9) and (6.10) can exclude the sub-optimal points that have been explored during the whole iteration process. Second, given a fixed y by model (6.8) at current iteration ν we slove the rest part of original model (6.1) - (6.6) to produce a feasible solution as follows. min c T x (6.11) s.t. A i x + b i t i, i = 1,, L (6.12) Ex + Gt f (6.13) x j [ l j y j, u j y j ], j = 1,, n (6.14) If the resulted SOCP (6.11) - (6.14) is infeasible, we add following inequality to sub-problem (6.8) and resolve above SOCP to obtain a feasible solution to original problem: y j q 1 (6.15) j I (v) 1 Observation shown that these types of cut can speed up the whole LR method significantly. A lower bound of original problem (6.1) - (6.6) can be obtained by solving the associated dual problem of L (x, y, π, π + ) at current iteration as follows: max min L ( x, y, π, π +) π,π + 0 x,y The dual problem returns either an optimal solution or maximal lower bound to model (6.1) - (6.6). We then design following algorithm for solving original CCCP: Lagrangian Relaxation algorithm Step 0: (Initialization) ν 0, LBD, UBD,

114 Chapter 6. Lagrangian Relaxation for CCCP 102 π,v i Step 1: (Dual Decomposition) 0, π +,v i 1, i N C (v) x c + π + π Solve LR SOCP (x, π, π + ), i.e. model (6.7), for given C (v) x, C (v) y d + π lb π + ub Add inequalities (6.9) and (6.10) to model (6.8) Add inequality (6.15) to model (6.8) if necessary. Solve LR IP (y, π, π + ) with added inequalities for given C (v) y, Update LBD max ( LBD, L ( x (v), y (v), π,v, π +,v)) If ( x (v), y (v)) is feasible to constraint (6.5), Update UBD min (UBD, L (x v, y v, π,v, π +,v )). STOP. Else find a fesible solution x (v) adj in model (6.7) under y(v) adj from model (6.8), and calculate UBD (v) adj to model (6.1) - (6.6). ( ) Update UBD min UBD, UBD (v) adj. GO TO Step 2. Step 2: (Lagrangean multiplier update) Build Lagrangian dual problem max π,π + 0 L ( x (v), y (v), π, π +) ( ( )) 1O π,v+1 i max 0, π,v i + αt (v) lb (v) i y (v) i x (v) i, i N ( ( )) 2O π +,v+1 i max 0, π +,v i + αt (v) ub (v) i y (v) i + x (v) i, i N 2 where t (v) = (UBD LBD) / lb(v) y (v) x (v) ub (v) y (v) + x (v) 3O Solve LR IP ( y, π,v+1, π +,v+1) and LR SOCP ( x, π,v+1, π +,v+1) Step 3: (Stop criteria) While L ( x v, y v, π,v+1, π +,v+1) < L (x v, y v, π,v, π +,v ) α =.5α, repeat 1O - 3O Calculate Gap v = (UBD LBD) / UBD If Gap v > ɛ or v < V, v = v + 1. GO TO Step 1.

115 Chapter 6. Lagrangian Relaxation for CCCP In practice we set t v = (UBD LBD) / lb(v) y (v) x (v) ub (v) y (v) + x (v), if LR ( x v, ω v+1) LR (x v, ω v ), then α =.5α and recalculate the lower bound until LR ( x v, ω v+1) > LR (x v, ω v ). Numerical experiments for models and designed method are shown in next section. 6.4 Robust Factor model to Index Tracking Nominal Index Tracking Model In this section we develop the nominal enhanced index tracking model. Let µ denote the vector of expected returns of assets and Σ the covariance matrix, x is vector of portfolio weights, x BM is the vector of weights of a benchmark index. Then the difference in expected returns (or excess returns) between the tracking portfolio and the benchmark is µ T (x x BM ), and the standard deviation of excess returns (tracking error) is (x x BM ) T Σ (x x BM ). Here we adopt the index tracking model from [40]. In this formulation, a portfolio x is sought that maximizes expected return subject to a limit on portfolio risk and tracking error. The model is given as: max µ T x (6.16) s.t. Σ 1 2 x σ (6.17) Σ 1 2 (x xbm ) T E (6.18) e T x = 1 (6.19) x 0 (6.20) where e R n is a vector all of whose components equal 1, denotes the norm value of a vector. σ denotes the tracking portfolio risk limit and T E is the tracking error limit. For example, T E = 5% means that a tracking portfolio may not have standard deviation of excess returns of more than 5%. A cardinality constraint can be added into Model (6.16) - (6.20) to control the portfolio size exactly: max µ T x (6.21) s.t. Σ 1 2 x σ (6.22)

116 Chapter 6. Lagrangian Relaxation for CCCP 104 Σ 1 2 (x xbm ) T E (6.23) e T x = 1 (6.24) e T y = q (6.25) lb i y i x i ub i y i, i (6.26) x 0, y {0, 1} (6.27) where lb, ub are the lower and upper bounds on the tracking portfolio weights, q is the portfolio size. Model (6.21) - (6.27) can eaily be seen to be a Mixed-Integer Second-Order Cone Programming (MISOCP) as the risk and tracking error constraints are quadratic with all other constraints linear and a linear objective function and binary integer restrictions. Next, we illustrate the nominal model by solving several instances. All instances were solved to optimality by using the mixed integer solver in Gurobi which is based on branch-and-bound to obtain zero gap between lower and upper bounds [71]. In particular, the effect of the tracking error constraint (6.23) and the risk control constraint (6.22) are investigated by repeatedly solving the model with increasing values for the parameter T E under different σ value. We used daily returns from June 30, 2005 to December 31, 2007 to generate the parameters (µ, Σ) for the model. We fixed σ with a large value, e.g. σ = 80 max (diag (Σ)), then increase T E with a given portfolio size, then we change σ to a smaller value, e.g. σ = max (diag (Σ)), and repeat the same computational process by changing T E value. The parameters (µ, Σ) were estimated through linear regression, specifically a three-factor model was applied for our estimation (see details in Section 6.5.1). We computed instances over different q sizes that represent low, medium and high density portfolios. For example, we chose the portfolio size from q [5, 65] as we found that tracking portfolios will be very close to the index when q over 75, but this will generate higher transaction costs due to holding more assets. We compared the portfolio return, variance and Sharpe ratio with different portfolio tracking sizes q in Figure (6.1) - (6.3).

117 Chapter 6. Lagrangian Relaxation for CCCP 105 Figure 6.1: Portfolio return vs TE with different q under different σ (SP100) Figure (6.1) shows the portfolio return over T E and q under different σ. Moderate tracking portfolio sizes (q = 15, 35) have higher return than larger or smaller sizes (q = 65 or 5). When q = 65 the effects of diversification become stronger reducing return. From the figure we see that the portfolio return sublinearly increases which means the marginal return decreases with respect to T E value. However, the portfolio returns are generally better than the return of benchmark used i.e. the S&P100 (0.35 ). As the parameter σ decreased to max (diag (Σ)), the risk control constraint (6.22) dominate the tracking error constraint (6.23), and therefore the portfolio return cannot be improved via changing the T E value after 0.8*10e-4. Figure 6.2: Portfolio variance vs TE with different q under different σ (SP100) The portfolio variance increases approximately linearly with respect to T E, see left side on

118 Chapter 6. Lagrangian Relaxation for CCCP 106 Figure (6.2), for moderate portfolio sizes. The tracking portfolio with size q = 65 has lower variance due to the diversification effects of having more assets. The results suggest that if larger T E values are allowed, the portfolio return can be improved, however, the portfolio variance may increase quicker than the improvement of return. The variance of S&P100 is lowest out all portfolios most likely due to the diversification effect from having more assets. In particular, the S&P100 variance is 0.06 and associated standard deviation is Again we note that parameter σ can also significantly affect the portfolio variance, see right side on Figure (6.2). The portfolio variance will be invariant to T E after 0.6*10e-4 if σ set too small. Figure 6.3: Portfolio Sharpe ratio vs TE with different q under different σ (SP100) We combine the portfolio return on Figure (6.1) and variance on Figure (6.2) to obtain the portfolio Sharpe ratio on Figure (6.3). From Figure (6.3), the portfolio Sharpe ratio increases with respect to increasing T E, however, the marginal portfolio Sharpe ratio decreases since the marginal variance dominates the marginal return. Thus, by controlling the T E one can improve portfolio performance, but increasing T E as shown above can lead to increased portfolio volatility in Figure (6.2). Thus, the one must be careful about setting T E and σ too high if one cares about risk. Also, the results suggest that one way to help attain enhanced indexing is to not set q too high. We see that the Sharpe ratio of the benchmark index S&P 100 ( in left side and in right side) is worse than the Sharpe ratio of the tracking portfolios. From both sub-figures, we see that the portfolio Sharpe ratios can be significantly affected by both model parameters T E and σ. In the next section we develop the robust counterpart to model (6.21) - (6.27) to errors in

119 Chapter 6. Lagrangian Relaxation for CCCP 107 parameter estimation Robust Multi-Factor Model for Index Tracking We follow as in Goldfarb and Iyengar [66] by employing a robust factor modeling approach. Suppose the return vector r is given by the model: r = µ + V T f + ɛ where µ R n is the vector of mean returns, f N (0, F ) R m, is the vector of returns of the factors that drive the market, V R m n is the matrix of factor loadings of the n assets, and ɛ N (0, D) is the vector of residual returns where D = diag(d), d = [d i ], i = 1,, n. In practice we may need F 0. The assumptions for factor model include: ˆ residual returns ɛ i and ɛ j are independent, i.e. cov ([ɛ i ɛ j ]) = 0 for i j; ˆ residual return ɛ i and factor return f j are independent, i.e. cov ([ɛ i f j ]) = 0. Then E (r) = µ, σ ij = Vi T F V j, i j, σ ii = σi 2 = Vi T F V i + d i, σ i = F V i + d i, or written in matrix form Σ = V T F V + D. Given a weight vector x, the risk of a portfolio, i.e. x T Σx, can be split as a combination of a systematic risk, i.e. x T V T F V x, and a individual risk, i.e. x T Dx, within a portfolio [92]. Then building uncertainty sets around Σ is equivalent to build the uncertainty sets for terms V T F V and D separately. We assume that the market is stable, i.e. F is constant, then generators of uncertainty for parameters (µ, Σ) comes from the generators of uncertainty for the parameters (µ, V, D). We follow as in [66] and design the uncertainty sets for parameters (µ, V, D) separately as follows: V T i ˆ The uncertainty sets S m and S d for parameters D and µ are defined as intervals: S d = { D : D = diag(d), d i [ ] } d i, d i, i = 1,, n { ] } S m = µ : µ = µ 0 + ξ, ξ i [γ i, γ i, i = 1,, n (6.28) (6.29) ˆ The uncertainty set for parameter V belongs to an ellipsoid: { } S v = V : V = V 0 + W, W i g ρ i, i = 1,, n (6.30) where W i is the ith column of strength matrix W around V 0 and W i g = Wi T GW i is an elliptic norm, G 0 denotes the coordinate system that may not be perpendicular. We can always generate a matrix G 0 to maintain the strict convexity of the problem. Then the robust counterpart for objective (6.21): max min µ T x = max min (µ 0 + ξ) T ( x = max µ0 + γ ) T x x µ S m x ξ γ x

120 Chapter 6. Lagrangian Relaxation for CCCP 108 For the constraint (6.22) that measure the portfolio risk: Σ x 2 σ2 x T Σx σ 2 x T ( V T F V + D ) x σ 2 x T V T F V x + x T Dx σ 2 Then the robust counterpart for above constraint: max x T V T F V x + x T Dx σ 2 maxx T V T F V x + maxx T Dx σ 2 V S v,d S d V S v D S d maxx T V T F V x v V S v maxx T Dx δ D S d v + δ σ 2 maxx T V T F V x v V S v 2D1/2 x 1 δ 1 + δ v + δ σ 2 We use the sum of v and δ to represent the total risk since the terms x T V T F V x and x T Dx are independent. For the robust term max V S v x T V T F V x v, Goldfarb and Iyengar [66] show that it can be converted into a collection of linear and second-order conic constraints through Lemma 1 below. Lemma 1. Let r, v > 0, y 0, y R m and F, G R m m be positive definite matrices. Then the constraint max {y: y g r} y 0 + y 2 f v (6.31) is equivalent to either of the following: (i) there exist τ, σ 0, and t R m + that satisfy v τ + e T t σ r 2 στ 1 λ max (H) w 2 i (1 σλ i ) t i, i = 1,, m where QΛQ T is the spectral decomposition of H = G 1/2 F G 1/2, Λ = diag (λ i ), and w = Q T H 1/2 G 1/2 y 0 ; (ii) there exist τ 0, and s R m + that satisfy r 2 τ ( v e T s ) u 2 i (1 τθ i ) s i, i = 1,, m τ 1 λ max (K)

121 Chapter 6. Lagrangian Relaxation for CCCP 109 where P ΘP T is the spectral decomposition of K = F 1/2 G 1 F 1/2, Θ = diag (θ i ), and u = P T F 1/2 y 0. Lemma 1 is proved using the S - procedure which has broad application in engineering science [26]. For details of the proof of the Lemma 1 see [66]. Therefore by using Lemma 1 constraint max V S v x T V T F V x v can be transformed into the following convex constraint set by part (ii) of Lemma 1: maxx T V T F V x v V S v max V x 2 f v V S v u = P T F 1/2 V 0 x 2ρ T x τ v + e T s τ + v et s 2u i v τθ i s i v τθ i + s i, i = 1,, m (6.32) v τλ max (K) 0 τ 0 where K = P ΘP T is the spectral decomposition of K = F 1/2 G 1 F 1/2, Θ = diag (θ). Function x f = x T F x denotes a norm on R m. Note that radius r = ρ T x = ρ T x in the first norm constraint in (6.32) because short selling is prohibited, i.e. x 0. For the constraint (6.23) that measure the tracking error: Σ 1 2 (x xbm ) T E (x x BM ) T Σ (x x BM ) T E 2 z T Σz T E 2 (6.33) z = x x BM Analogously the robust counterpart of z T Σz T E 2 in (6.33) can be obtained by using Lemma 1. The associated convex constraints are constructed as follows: max V S v,d S d z T V T F V z + z T Dz T E 2 max V S v z T V T F V z + max D S d z T Dz T E 2

122 Chapter 6. Lagrangian Relaxation for CCCP 110 maxz T V T F V z l V S v maxz T Dz ζ D S d l + ζ T E 2 max V z 2 f l V S v 2D1/2 z 1 ζ 1 + ζ l + ζ T E 2 w = P T F 1/2 V 0 z 2ρ T z τ l + e T s τ + l et s 2w i l τθ i s i l τθ i + s i, i = 1,, m l τλ max (K) 0 (6.34) τ 0 2D1/2 z 1 ζ 1 + ζ l + ζ T E 2 The absolute value sign in the radius r = ρ T z = n i=1 ρ i z i in the first norm constraint in (6.34) should be removed since variable z could be negative. We replace z i as follows: z i = z + i + z i z i = z + i z i = x i x BM i z + i 0, z i 0 Finally the robust counterpart using the factor model for problem (6.21) - (6.27) can be formulated as follows: max ( µ0 + γ ) T x (6.35) s.t. u = P T F 1/2 V 0 x (6.36) w = P T F 1/2 ( V 0 z + z ) (6.37) z + z = x x BM (6.38) [ ] 2ρ T x τ v + e T s τ + v et s (6.39) [ ] 2u i v τθ i s i v τθ i + s i, i = 1,, m (6.40)

123 Chapter 6. Lagrangian Relaxation for CCCP 111 [ ] 2ρ T (z + + z ) τ l + e T s τ + l et s (6.41) [ ] 2w i l τθ i s i l τθ i + s i, i = 1,, m (6.42) [ ] 1/2 2D x 1 δ 1 + δ (6.43) v + δ σ 2 (6.44) [ ] 1/2 2D (z + z ) 1 ζ 1 + ζ (6.45) l + ζ T E 2 (6.46) v τλ max (K) 0 (6.47) l τλ max (K) 0 (6.48) e T x = 1 (6.49) e T y = q (6.50) lb i y i x i ub i y i, i = 1,, n (6.51) x 0, τ 0, z + 0, z 0 (6.52) y {0, 1} (6.53) The dimension of different type of variables are x, z +, z R n, v, δ, l, ζ, τ R, u, w, s R m, y B n. Therefore, model (6.35) - (6.53) keeps the same CCCP structure as norminal tracking model but includes more variables and cone constraints. In practice we apply Fama-French 3 factor model which is an advanced extension of CAPM model to calculate the numerical values of the parameters, details see [66, 54]. We then test both nominal model (6.21) - (6.27) and the robust counterpart (6.35) - (6.53) by commercial solver Gurobi on an AMD Dual-Core laptop with 2GB of RAM. One interesting observation is that Gurobi take much longer running time to model (6.35) - (6.52) than that to model (6.21) - (6.27) in many instances. For example, for the case that N = 500, q = 70, there still exist 10% gap between lower and upper bounds after 1000 seconds running time for model (6.35) - (6.53) while Gurobi return the optimal solution, i.e. gap equals 0, within 10 seconds for model (6.21) - (6.27), and such instances are common in our testing. The factored robust procedure for index tracking problem simplify the parameter estimation but it may increase the solving time since more conic constraints are included into the model (2m + 2 from 2), which

124 Chapter 6. Lagrangian Relaxation for CCCP 112 make the problem harder and harder. This disadvantage motivated us to apply Lagrangian Relaxation method we designed to speed up the solving process and keep the quality of the solution in next section. 6.5 Computational Experiments Testing the Three-Factor and Single-Factor models We use as the basis of the robust factor model the Fama and French 3 factor model [54] which can be seen as the extension of Sharpe s one factor CAPM model. The Fama-French 3 factor model is based on the observation that small capitalization stocks and value stocks (i.e. stocks with high book to price ratio) tend to outperform the market as a whole. In the model, three risk factors reflect the sensitivities of each stock to the market excess return (market factor), the excess of value stocks over growth stocks (book-to-market factor), and the excess of small cap stocks over large cap stocks (size factor). The one and three-factor models are presented as follows: r it r ft = α i + β im (r Mt r ft ) + ε it (6.54) r it r ft = α i + β im (r Mt r ft ) + β is SMB t + β ih HML t + ε it (6.55) where r it, r ft, r Mt denote the return of asset i, risk-free return, and market return at time t respectively; r it r ft, r Mt r ft denote excess return of asset i and the excess return of market return M over the risk-free rate the market at time t, respectively; SMB t denotes the excess returns of small capitalization stocks over large capitalization stocks at time t; HML t denotes the excess return of value stocks over growth stocks at time t; ε it denotes the residual term of asset i at time t. The regression coefficients are: α i = consistent excess return; β im = the sensitivity of stock i to movements of the market; β is = the sensitivity of stock i to movements in small stocks; β ih = the sensitivity of stock i to movements in value stocks; To fit the observations, r i r f, as best as possible, one approach is to minimize ε i for stock i. For the linear regression model min Ax b, we have analytic solution x = ( A T A ) 1 A T b. Where for single factor model, A = [1, r M r f ] and b = [r i r f ] and for three factor

125 Chapter 6. Lagrangian Relaxation for CCCP 113 model, A = [1, r M r f, SMB, HML] and b = [r i r f ]. After solving for regression coefficients of (6.54) and (6.55), we calculate R 2 i = 1 SS residual,i SS total,i = 1 ε i 2 2, which represent the (T 1)var(r i r f) percentage of the variance in the excess return of stock i, to compare how good the estimated parameters (regression coefficients) fit the observations. We collected data on 5810 stocks which are trading in the US NYSE and NASDAQ exchanges to see which factor model is more suitable. The daily prices of the stock were downloaded from a Bloomberg work station, and the risk factors were downloaded from the data library of Kenneth French s web page [2]. The S&P500 index used as the market and 1-month TBill rate used as the risk free asset in Fama-French three factor model. The stocks without adequate price information were deleted and then linear regressions were implemented for above models. Different time periods of historical data were used to test and the R 2 value are listed in following Table 6.1: Table 6.1: R 2 value for the regression models R 2 stock # stock # stock # stock # stock # stock # (single) (3 factor) (single) (3 factor) (single) (3 factor) 90% % % % % % % % % % From Table 6.1, we can see that the three factor model can explain more of the variability of excess returns than the single factor model as expected, and our numerical results for the R 2 values obtained for the three factor model is higher than that of single factor model for specific stocks i for different time periods. In general the R 2 value associated with a regression with the three factor model is 5% better on average than the values from the single factor model, and in best case the three factor model is 25% better than the single factor model. Therefore we applied the three factor model (6.55) as the basis to form the uncertainty sets of expected return and covariance for the factor loadings in (6.35) - (6.53) as in [66]. The details of this

126 Chapter 6. Lagrangian Relaxation for CCCP 114 construction is in Appendix (C. 1). We also note that any other notable multi-factor models can better interpret risks can also be applied to our proposed robust factor index tracking model. For example, D ecclesia and Zenios [48] showed that 98% of the variability can be explained via identifying multi risk factors of returns of the Italian bond market. Burmeister et al. [28] presented a macroeconomic factor model which includes five risk terms in interpreting the historical stock returns. We did not test for stationarity of returns, and assumed that the market was stable i.e. F covariance of factors, in the event of non-stationarity of variance this could affect the estimation of betas (factor loadings). However, our approach was to let this be handled by the robust optimization over different factor loading matrices captured by the uncertainty set S v Index Tracking using the S&P100 Index In this section, we illustrate the factor-based robust enhanced index tracking model by tracking the S&P100 index. Comparisons of the robust model versus the nominal model illustrate the benefits of robustness. First, in-sample data about the components S&P100 are collected to construct the nominal covariance matrix. We collected the historical price information of all components of S&P100, and calculated the daily return r it = P i,t P i,t 1 P i,t 1, where P i,t, P i,t 1 are the adjusted closing prices at time t and t 1. Then, daily returns were used to calculate the mean returns of assets and covariance matrix of returns of the assets: µ i = 1 T T r it, cov ij = 1 T t=1 T (r it µ i ) (r jt µ j ) t=1 Daily prices between June 30, 2005 and December 31, 2007 (630 samples) were collected and used as in-sample data, and daily prices for each end of month between January 1, 2008 and December 31, 2008 were used to build out-of-samples for the nominal and robust models. Some stocks in the S&P100 index can be replaced by some other stocks outside of the index since they may not satisfy the selection criteria of S&P100 in the designed time period, we retrieved the stocks that were moved out in the time periods used above and obtained the associated price information. Usually this replacement was rare and the components of S&P100 were stable, we check the changing history of the composition of the S&P100 and there is no replacement between June 30, 2005 and December 31, 2008, the period we collected data for. For some

127 Chapter 6. Lagrangian Relaxation for CCCP 115 stocks if there is no adequate data from the Bloomberg work station, we deleted that assets from the index, for example, 7 assets (5% of total market value) were deleted in period of year 2006 and 2007 and 5 assets (2% of total market value) are deleted in year of 2007 and 2008 due to lack of data, this reduction did not significantly impact the total market value of S&P100. Table (6.2) lists the tickers we used for our research grouped them across different sectors: Table 6.2: Ticker symbol across Sectors (SP100) Sector (total number) Ticker Symbol 1: Consumer Discretionary (12) AMZN, CMCSA, DIS, FOXA, GM, HD, LOW, MCD, NKE, SBUX, TGT, TWX 2: Consumer Staples (8) COST, CVS, FB, KO, MDLZ, PEP, WAG, WMT 3: Energy (10) APA, APC, COP, CVX, DVN, HAL, NOV, OXY, SLB, XOM 4: Financials (14) AIG, ALL, AXP, BAC, BK, BRK/B, COF, GS, JPM, MET, PM, SPG, USB, WFC 5: Health Care (13) ABBV, ABT, AMGN, BAX, BIIB, BMY, GILD, JNJ, LLY, MDT, MRK, PFE, UNH 6: Industrials (13) CAT, EMR, FDX, GD, GE, HON, LMT, MMM, NSC, RTN, UNP, UPS, UTX 7: Information Technology (15) AAPL, ACN, BA, CSCO, EBAY, EMC, GOOG, HPQ, IBM, INTC, MA, MSFT, ORCL, QCOM, TXN 8: Materials (6) CL, DD, DOW, FCX, MO, MON 9: Telecommunications Services (2) V, VZ 10: Utilities (7) C, EXC, F, MS, PG, SO, T The Fama-French 3 factor model is used to generate the parameters µ 0, V 0, G, ρ i, γ i, d i and the associated uncertainty sets for µ and V 0 see Appendix (C. 1) for details on the construction and we set ω = 0.95 which represents the joint confidence level. Figure (6.4) shows the worst bound for the expected return µ under the given uncertainty set (6.29) and the worst bound for covariance σ i under the given uncertainty set (6.28) and (6.30). We can see that almost all robust expected returns are below the nominal expected return from the historical data, and all robust covariances are above the nominal covariance computed from the historical data.

128 Chapter 6. Lagrangian Relaxation for CCCP 116 Figure 6.4: Robust bound for expected return and variance (SP100) Robust v.s. nominal portfolio performance We then use the computed tracking portfolios to test rolling out-of-samples and compare the performance of the portfolios. The 4 rolling periods are 2008, 2009, 2010, and 2011 respectively. The rolling process is described as follows. We select two year s daily data, e.g. year 2006 and 2007, as in samples to construct the portfolio and then test the next one year s performance, e.g. year 2008, without re-balance. After that we replace the in-samples as daily data from year 2007 and 2008, and test portfolio performance in year 2009, and so on. Both nominal model (6.21) - (6.27) and robust counterparts (6.35) - (6.53) are solved by Gurobi. For the initial test, we set lb i = 1 n and ub i = 0.7. σ equals 8 times of the maximal standard deviation in the assets in SP100, and T E equals 5 times of standard deviation of SP100. The portfolio size was set at q = 25.

129 Chapter 6. Lagrangian Relaxation for CCCP 117 Figure 6.5: Wealth evolutions for rolling out-of-samples Figure (6.5) shows the portfolio return evolution for the out-of-sample period, there is no rebalancing during the out-of-sample test. The returns from portfolios generated by the robust factor model is reasonably close to the S&P100 index see (6.5) and are relatively stable without large drops. The portfolio returns generated by the nominal model may be sensitive to perturbations of the coefficient and exhibits wider divergence in returns. For example, when the market starts to decrease during time periods 2 to 4, the portfolio generated by the nominal model drops more rapidly than the index but the robust portfolio exhibits good performance and actually dominates the performance of the S&P Index during most of this period of market decline. During time periods 7 to 9, the portfolios by both models avoided the market plunge and the performance by robust factor model were generally better than that of nominal model. These examples shown that the robust factor model protected against the uncertainty of market movement successfully. During periods 8 to 11 a market recovery is seen and the returns from

130 Chapter 6. Lagrangian Relaxation for CCCP 118 the robust portfolios actually lag the returns from the S&P 100 index and nominal portfolio, but then these latter two portfolios drop more steeply in the period from 11 to 12 of decline. This indicates that robustness protects well against large drops but may not accelerate as fast in periods of steady market increases. Similar robust mechanism that protect against downside risk can be seen in other sub-figures in Figure (6.5). For example, when market rapidly increased in year 2009, 2010 and 2011 which represent different parameter structures to the models, the portfolios by factor robust model still displayed the relative stable return performance compared with that from nominal model. It is clear to see that the path of robust model in period 3 to 5 in third sub-figure moved down slower than that of nominal model and target index. Next we vary the portfolio size q from 10 to 75 in increments of 5 and solve both nominal and robust models under different portfolio sizes. The mixed integer solver in Gurobi for MISOCP is mainly based on the branch-and-bound algorithm which tries to shrink the gap between the SOCP relaxed lower bound and its feasible upper bound. For the instances of tracking S&P100, we set the running time for Gurobi as 100 seconds, relative optimality gap equals 10e- 08. In our computation, the hardest instance consumed 50 seconds to satisfy the gap tolerance, which indicates all instances can obtain the optimal portfolio within 50 seconds due to the suitable problem size that Gurobi can quickly handled. The performance metrics include: daily portfolio return, daily portfolio variance, and daily portfolio Sharpe ratio. We compare these performance metrics by using in-sample and out-of sample data. There is no re-balancing of portfolios during a testing period. For example, 630 in-sample daily returns from June 30, 2005 to December 31, 2007 were used to generate data and then traking portfolios were tested out-of-sample from December 31, 2007 to December 30, 2008 which is a period in which a large market decline was experience. The size of uncertainty set is controlled by the joint confidence level ω in equations (C.5) and (C.6) in Appendix (C. 1). To our experience, for a very high joint confidence level, e.g. ω = 0.99, we have a high confidence that the solution of robust model protect against the uncertainty of parameter, but the feasible region of robust model may be restricted and more instances will be infeasible when portfolio size is small, e.g. q = 15. On the other hand, for a low joint confidence level, e.g. ω = 0.55, more small size instances have solution but the confidence

131 Chapter 6. Lagrangian Relaxation for CCCP 119 that the parameters lie in the designed ellipsoid is low. Therefore, we set a reasonable joint confidence level ω = 0.95 in our computation. The parameters (µ, Σ) in nominal model (6.21) - (6.27) are approximated by the three factor model (6.55) where µ = µ 0, Σ = V T 0 F V 0 + D 0, and then are used in the robust model as well. We also used the linear regression to approximate the out-of-samples and then calculate the associated out-of-sample performance. Figure (6.6) - (6.12) shows these comparison between two models. Figure 6.6: Model comparison - portfolio return It is clear to see the trend that the portfolio returns by the nominal model decreased as size increased from Figure (6.6). All instances obtained the optimal solution by applying Gurobi mixed integer solver. The portfolio returns decreased as more the portfolio sizes are allowed because of the diversification process, that is, the more assets are allocated, the less risk is taken to the portfolio, and thus the smaller portfolio returns. Meanwhile the portfolio return by robust model for both in-sample and out-of-sample seem unchanged too much with respect to portfolio size, however they are generally better than the returns generated by the nominal models for out-of-sample. Figure (6.6) shown that the robust model can protect against the downside risk in estimation of expected return vector µ 0 due to market uncertainty. We can also

132 Chapter 6. Lagrangian Relaxation for CCCP 120 see that portfolio return by robust counterpart in the out-of-sample period (averagely 0.63 ) is better than the index return in the same out-of-sample period ( 0.10%). Figure 6.7: Model comparison - portfolio variance From Figure (6.7), we can easily see the diversification process of portfolios generated by nominal model as q gets larger, i.e. as portfolio sizes get larger, the portfolio variance decreased. Portfolio variance for portfolios generated by the robust model for in-sample and out-of-sample are lower than that from the nominal model for corresponding in-sample and out-of-sample periods, which indicates the cardinality constraint had an impact on the conic constraints that represent the portfolio risk in that variance was reduced. The variance of the S&P100 index has the lowest value in the in-sample period, and the value in the out-of-sample period is still lower than robust models due to the diversification effect of of having more assets. The average portfolio variance by robust model in the out-of-sample period is 0.54 averagely, meanwhile the SP100 variance equals 0.22 in the same out-of-sample period.

133 Chapter 6. Lagrangian Relaxation for CCCP 121 Figure 6.8: Model comparison - portfolio Sharpe ratio The portfolio Sharpe ratio is defined as E(rport) E(r f) var(rport) where r f is the return of 10 year U.S Treasury bonds. From Figure (6.8), the Sharpe ratio generated by nominal models decreased as the portfolio size increased, which means the portfolio return decreased more quickly than the reduction of portfolio variance across the size. The Sharpe ratio by nominal model for in-sample are better than that by robust factor model for in-sample, this is reasonable since robust counterpart consider the worst scenario for parameters. On the other hand, the Sharpe ratio generated by robust factor models for out-of-sample are better than those generated by nominal models out-of-sample, this is crucial since we want to reduce the negative effect of market uncertainty. Therefore, Figure (6.8) indicates that the portfolios generated by robust models are more stable than those from the nominal models across different portfolio sizes q, this illustrates the benefit of cardinality constraint in the robust factor model. The average portfolio Sharpe ratio of portfolios generated by robust models is in the out-of-sample period and the Sharpe ratio of S&P100 in the same out-of-sample period is

134 Chapter 6. Lagrangian Relaxation for CCCP 122 Figure 6.9: Model comparison - Tracking error After solving both the nominal index tracking model and its factored robust counterparts, the tracking errors are calculated by (x x BM ) T Σ (x x BM ), which represent the variance difference between the portfolio and the target index. As can been seen in Figure (6.9), the tracking errors by portfolios from the robust model are generally smaller than those from portfolios generated by the nominal model with respect to size for in-sample and for outof-sample. This trend can be guaranteed since we generate the worst scenario bound for the parameters and the corresponding tracking error by robust model is also the lower bound for the tracking error by nominal model. We then test the tracking error to transaction costs efficient frontier that generated by both nominal and robust model. Suppose the initial portfolio wealth is b 0, e.g. one dollar, and trading ratio per dollar is α = 0.5%. From the initial portfolio that starts at January 1, 2008, we update the tracking error and associated tracking cost due to the rebalancing of the portfolio per month, and calculate the tracking error to transaction costs ratio (TE/TC ratio) as follows: (x x BM ) T Σ (x x BM ) i α b 1 i b0 i / i b0 i where b 1 is the new portfolio wealth before charging the transaction costs, which can be calcu-

135 Chapter 6. Lagrangian Relaxation for CCCP 123 lated by b 0 (1 + µ) x. We update the in-samples via keeping the same length size when rolling up along the time horizon. The tracking error to transaction costs ratios for nominal and robust models displayed in the following Figure (6.10): Figure 6.10: Tracking Error to Transaction costs ratios (SP100) The left sub-figure in Figure (6.10) shows the changing of nominal TE/TC ratio with respect to the size and time periods. From the left sub-figure we see that in some periods (period 1, 4, 6) the TE/TC ratios apparently decreased with respect to the increasing of the portfolio size, but in some other periods (period 9 and 12) this trend is not obviously. This can be explained from two points of view. First with the smaller size, the portfolio tracking error may quite large and dominate the occurrence of the transaction costs. While more assets are allowed to invest, the tracking error decreased but the transaction costs may become larger, which lead to an unsmooth decreasing curve for periods 1, 4 and 6. Secondly the portfolio allocation may be dramatically changed as the market significantly dropped in September 2008, therefore, the transaction costs may have happened more frequently and pulled the TE/TC curve down. For example, we see that for any size the TE/TC ratios in period 9 are far lower than that from period 6. Our numerical result showed that the tracking errors in these two periods keep in the same order of magnitude but the transaction cost of period 9 is 15 times higher than that in period 6 on average. Therefore, the nominal TE/TC ratios may be affected by both portfolio size and the uncertainty from the market. The right sub-figure, on the other hand, shows the robust TE/TC ratio according to the size under different rolling periods. In contrast with nominal TE/TC ratio, the robust TE/TC

136 Chapter 6. Lagrangian Relaxation for CCCP 124 ratios are nearly non-decreased (see period 1, 4, 9, 12), which indicates that the transaction costs plays the same important role as the tracking error if we apply the rolling up strategy. However, our numerical result showed that the tracking errors without rolling up testing keep the same order of magnitude as that by the rolling up way. Therefore, it is unnecessary to re-balance the robust portfolio frequently in terms of TE/TC ratio consideration. Next we investigate the changing of TE/TC ratio with respect to the trading ratio α under different size. The efficient frontier is exhibited in the figure 6.11, the left and right sides denote the trend of the efficient frontier under different sizes, i.e. q = 25, 75 represent the different strength of partial replication respectively, and the upper and lower sides indicate the trend of efficient frontier by nominal model and its robust counterpart. Figure 6.11: TE/TC ratios with respect to the trading ratio α It s not surprising that all sub-figures followed a similar decreasing pattern corresponding to the increasing of α because the tracking errors are bounded in both models but the rebalance

137 Chapter 6. Lagrangian Relaxation for CCCP 125 cost will keeping rising no matter a swapping occurred or not if the trading ratio α goes up according to the TE/TC ratio equation we used. A more detailed insight can be seen as follows. From the upper to the bottom, we see that the TE/TC ratio by the nominal model is generally higher than that by the robust model for any fixed trading ratio α, the main reason is that the rebalance cost of the nominal portfolio is much higher than that generated by robust counterpart (see columns 1 and 3 in Table 6.3). From the left to the right, the nominal TE/TC ratio under smaller size (q = 25) is larger than that with a size equals 75, while the robust TE/TC ratio at same smaller size is lower than the corresponding ratio at the same larger size level. To clearly see the reason, we list the average values of the indicators over the 12 rolling periods under α = 0.1% in the Table (6.3): Table 6.3: The average TE/TC ratios under different size Nominal Robust TE e e e e-04 TC e e e e-05 TE/TC From the Table (6.3), the nominal model generated overall higher transaction costs than that from the robust counterpart. As more assets are diversified, the average nominal tracking error reduced quicker than the average reduction of the trading cost, and therefore we see the sharp jump of the nominal TE/TC ratio with respect to the size. The robust tracking error, on the other hand, reduced slowly while the transaction costs keep the similar order of magnitude on average, which may lead to the similar but much smaller TE/TC ratio compared with the nominal model. Another way to measure the tracking performance is by the tracking ratio. Similar to the definition of tracking ratio in Cornuejols and Tutuncu [40], we calculate the tracking ratio through the following formula: n i=1 V it R 0t = M I M P = n i=1 V it/ n i=1 V i0 q j=1 x jv jt / q j=1 x jv j0 where M I = n i=1 V indicates the target index s movement after investment, M P = i0 q j=1 x jv jt q j=1 x jv j0 denotes the movement of portfolio s market value during the out-of-sample period. The ideal tracking ratio, R 0t, is 1, a value over 1 means underperformance with respect to the target

138 Chapter 6. Lagrangian Relaxation for CCCP 126 index, and a value less than 1 indicates excessive return. Figure (6.12) display the comparison of tracking ratios of portfolios generated from the nominal and robust models. Figure 6.12: Model comparison - Tracking ratio The straight line indicates that a portfolio perfectly tracks the market index, S&P100. There was no rebalance during the tracking period after investment. From Figure (6.12), the tracking ratios by robust model are more closer to 1 than that from the nominal model with respect to size for out-of-sample testing, which indicate the factored robust tracking model has better tracking performance during period from December 31, 2007 to December 30, 2008, a main period in financial crisis.

139 Chapter 6. Lagrangian Relaxation for CCCP 127 Table 6.4: Tracking ratio comparison N = 93 move move port M I M I move port M q index M I (nomi. M P 1 ) M P 1 M P 1 1 I M I (rob. M P 2 ) M P 2 M P Aver After obtaining the portfolios by proposed models, we test the movement of index and portfolios in out-of-samples period in terms of market value. Table (6.4) shows more details about the market value movements of index and the portfolios with respect to size. It is clear to see that the movement of the target index is constant to size while the movement of portfolios by different models are varying with respect to size. For example, under q = 25, n i=1 V it = indicates that the market value of the index at time t is 65.75% of the market n i=1 V i0 q j=1 xno min al j V jt q j=1 xno min al value of the index at time 0, or the index value decreased 34.25% at the end of the out-ofsample period. Meanwhile, = denotes the market value of the nominal j V j0 portfolio dropped 41.81% in the same out-of-sample period, and the associated tracking ratio Rno min al 0t = = denotes the speed of the value shrinkage of the nominal portfolio q j=1 xrobust j V jt q j=1 xrobust is faster than that of the index. On the other hand, = denotes the j V j0 market value of the robust portfolio dropped 33.68% at the end of the out-of-sample period, which indicates the downward descent in terms of market value is 8.13% (41.81% 33.68%) less than the descent of the nominal portfolio at the same period, and the associated tracking ratio R robust 0t = = denotes the decreasing speed of the market value of the robust portfolio is also less than the downside speed of the market value of the index market. The columns with M I M P 1 values indicate how close is a constructed portfolio to the index, and the ideal value is 0. As shown in the Table (6.4), the portfolios generated by robust model are relative closer to the S&P100 compared with those by the nominal model.

140 Chapter 6. Lagrangian Relaxation for CCCP Index Tracking using the S&P500 Index We test the proposed LR method with the estimated parameters from real data in this section. We applied the same data processing shown in Section to generate the parameters for the models. Table (A.1) listed the tickers we used for our research grouped them across different sectors in Appendix A. We deleted the ticker without enough public data, and retrieved the ticker if any replacement occurred during selected period. We then solved the model (6.35) - (6.53) by Gurobi directly and compared the numerical results with that obtained from the LR method. We changed the portfolio size q from 20 to 300 per 5 interval and solved the instances one by one. We first listed the gap information for the instances that q 100, which represents the practical region, in Table (6.5), then we showed all numerical details in Table (C.1) in Appendix (C. 2). Table 6.5: Bounds information (SP500) q Gurobi Obj Gap to Gap by Time LB by LR Fesi. UB [1000 s] Gurobi LR by LR % 2.87% % 2.78% % 2.15% % 1.42% % 0.95% % 0.12% % 0.06% % 0.48% % 0.06% % 0.04% % 0.08% % 0.11% % 0.20% % 0.50% * % 0.53% % 1.78% % 1.00% Average / / / 0.15% 0.89% The running time by Gurobi was set as 1000 seconds. From Table (6.5), we see that the solution by LR method is close to the solution form Gurobi, the average gap is 0.15%. Meanwhile the running time of LR method is slightly longer than the time by Gurobi (averagely 1715 vs 1000). It should not be surprised to see that the LR method can quickly converge to near

141 Chapter 6. Lagrangian Relaxation for CCCP 129 optimal solution within a short running time. For instance, for q = 45 and 50, the LR method consumed no more than half of the time that from Gurobi. The possible reason is that the generated inequalities (6.9) and (6.10) improved the iteration procedure. We will show this speed up process soon. The average gap by LR method is 0.89% which indicates the solution is close to the global optimal. In some instances e.g. q = 90, the objective value by LR method is slightly better than Gurobi objective. Next we detailed the comparison of LR method with and without inequalities (6.9) and (6.10) in the first three sub-figures in Figure (6.13), and showed a more precisely iteration process by setting different initial dual variable π + for instance q = 50 in the last sub-figure. Figure 6.13: Iteration details (SP500) As shown in Figure (6.13), the LR gaps usually can shrink to the Gurobi solution after iterations, and LR method with designed cuts (6.9) and (6.10) converged quicker than LR without such cuts. For example, it only required 20 iterations to reach a small gap by

142 Chapter 6. Lagrangian Relaxation for CCCP 130 LR with the cuts while 80 iterations consumed by LR without the cuts to obtain similar gap scalar. In general, LR with designed cuts can save 60% iterations than that by LR without the cuts in our computation. Therefore we apply the designed cuts and set the iteration limit V = 100 for the LR method. Some other parameters for both models and LR methods are set as follows. x BM is the normalization of the the market capitalization of component in S&P500, σ = 8 max (diag ( )), T E = 7 ST D S&P 500, lb set as 1/n and ub = 1. The gap stop criterion ɛ = 1/10 4, the initial dual variable π = 0 and π + = 1, other initial dual value π + that can speed up the LR process can be applied. For example, we observe that if some elements in π + set as 0 and others equal 1, higher precision of solution can be obtained. We then summary the bounds and gap information by Figure (6.14) for the Table (C.1) in Appendix (C. 2). Figure 6.14: Bounds and gap comparison by LR method (SP500) The left side on Figure (6.14) list the lower and upper bounds by LR method with respect to size. We see that in most instances, solutions by LR method are close to Gurobi. LR method can generally shrink the gap between the lower and upper bounds under 5% in the range that q [20, 200] [255, 300]. Although the gap trend increased in the range q [205, 250], the LR solution still close to Gurobi solution, which indicate high quality solution can be obtained. Some instances with better objective value have been marked in Table (C.1) in Appendix (C. 2), i.e. q = 90, 125, 215, 240, 275. From the right side on Figure (6.14) we see that the average gap to Gurobi is 0.21%. Meanwhile the average gap by LR method is 4.16% and the average solving time by LR methos is 1500 seconds. 39 out of 57 instances with relative small gap that less than 5%, and 6 out

143 Chapter 6. Lagrangian Relaxation for CCCP 131 of 57 instances have large gap that over 10% (worst gap equals 13.08%). These hard instances mainly lie in the unpractical range q [220, 250] Index Tracking using the Russell 1000 Index The Russell 1000 Index is another important market-cap based index which represents near 90% of the total market capitalization in US equity market. It has been used to build different index ETFs, e.g. the ishares Russell 1000 Index and the Vanguard Russell 1000 Index ETF. Because the Russell 1000 Index includes more companies than that in S&P 500, it can broadly diversify across the whole market but may also be computationally expensive using the partial replication such as the tracking models we developed in Section 6.4. Therefore we next apply the LR method described in Section 6.3 for tracking the Russell 1000 Index. Similar parameter generation process described in Section was applied for Russell 1000 Index. Table (6.6) listed the comparison between the solution from the LR method and Gurobi. The running time for both methods were set as 3600 seconds. As can be seen, the gaps by LR methods are better than the Gurobi gaps for small size q, e.g. q = 35, 50, meanwhile the gaps of LR method are close to Gurobi gaps for large size q, e.g. q 95. This is reasonable since as q increased, the feasible region of robust model are loosed and the both gaps are improved. Moreover, for the instance that q = 35, 50, 95, The gaps and feasible objective by LR method are superior to that from Gurobi, which indicates the LR method can converged quicker than the mixed integer solver based on branch and bound method in Gurobi within the setting time. Table 6.6: Bounds information (Russell 1000) q Gurobi LB Gurobi Obj Gurobi Gap LB by LR Fesi. UB LR Gap 35* % % 50* % % % % % % 95* % % % % % % % % Average / / % / / %

144 Chapter 6. Lagrangian Relaxation for CCCP Index Tracking using the Russell 3000 Index In this section, we apply our LR method to test Russell 3000 Index, which represents approximately 98% of the investable US equity market. Similarly categorized the S&P 500 in Table (A.1), the assets of Russell 3000 are selected from 10 sectors but with different ticker symbols and sector weights. After deleting the assets without adequate data for the factor based robust index tracking model (6.35) - (6.53), the total number of assets remains as 2359 which accounts 95% of the index value. We now set σ = max (diag ( )), T E = 4 ST D R3000, and other parameters keep the same as we did for S&P500. We first showed the Gurobi iteration details for solving the model under different q in Figure (6.15). Figure 6.15: Gurobi iteration details for different size q We set the running time for Gurobi as 6 hours, 2 instances (q = 30 and 50) used up the running time and other 4 instances (q = 70, 110, 150 and 190) terminated due to out of memory. It is clear to see that the gap cannot be significantly improved in our computation after 5000 seconds. For example, we found that for instances q = 30, the boundary gap remained unchanged as 18% after 3600 seconds. In some other instances (q = 50), the boundary gap

145 Chapter 6. Lagrangian Relaxation for CCCP 133 after 1 hour and 6 hours running was 13.1% and 12.1% respectively, which indicated 0.2% improvement per hour. One more unexpected question is that most of the instances encountered memory capacity problem and some instances still leave large gaps before the solver crashed, e.g. gap equals 23% for q = 70 in the figure. However, our decomposition-based LR method does not have this issue. Based on the experience on Figure (6.15), we set the running time for the solver as 7200 seconds. After solving the model by both approaches, we calculated the relative gaps equal the difference between the upper and lower bounds divided by the upper bound, and the gaps to Gurobi solution by using the difference between the LR and Gurobi feasible objectives to divide the Gurobi feasible objective value. We listed the computational results in Table (6.7) as follows: Table 6.7: Bounds information (Russell 3000), TE=4STD q Gurobi (7200 s) LR method Gap to Time by LB (1e-03) UB (1e-03) Gap LB (1e-03) UB (1e-03) Gap Gurobi LR % % 2.86% % % 4.41% % % 4.73% % % 2.89% % % 4.02% % % % % % 1.80% % % 0.77% % % -0.57% % % -2.31% % % 0.00% % % -0.20% % % -0.35% % % -0.85% % % -0.07% % % -0.58% % % -0.99% % % -3.59% % % -1.01% Aver % % -0.21% The average running time by LR method is around 2500 seconds which represents 65 percent of running time saving. The average gaps are 7.56% by Gurobi solver and 4.74% by our LR method, and the feasible solutions are close each other (-0.21% on average). Specifically, in the range q 100, our LR method can generally obtain the smaller gaps and better feasible

146 Chapter 6. Lagrangian Relaxation for CCCP 134 solutions compared with that from Gurobi. Although there exists a larger boundary gap for the instance q = 40, our LR method generated better lower and upper bounds, and the LR feasible solution is 4.73% better than that by Gurobi. For instance q = 70 on the other hand, we obtained smaller gap but worse feasible solution which probably because the prosolve process of the solver generated a high quality initial solution. Regarding to the range 110 q 200, both methods returned similar gaps and feasible solution, the possible reason is that when larger portfolio size is allowed, both methods approached the optimal or near-optimal solution within the setting time or iterations, and the convergence became slowly and slowly. Now if we shrank the tracking error T E = 3 ST D R3000 and other parameters remain same, we found that both methods are infeasible at q 140, and we showed the computational results for range 140 q 200 in Table (6.8): Table 6.8: Bounds information (Russell 3000), TE=3STD q Gurobi (7200 s) LR method Gap to Time by LB (1e-03) UB (1e-03) Gap LB (1e-03) UB (1e-03) Gap Gurobi LR % % 1.17% % % 0.90% % % 0.29% % % 0.95% % % -0.01% % % 0.12% % % -0.33% Aver % % 0.44% As shown in Table (6.8), the LR method have constant better performance than Gurobi s in terms of consuming time and boundary gaps. Our LR method saved 47% of running time on average to obtain similar gaps that Gurobi achieved (4.68% vs 6.05%). More importantly, 11 out of 26 instances in Table (6.7) and (6.8) returned better objective values (at least 1% better) by LR approach, which indicates the developed LR method is efficient for solving CCCP and our method can be seen as complementary to branch and cut based algorithm. In a nutshell, our LR method is much quicker than Gurobi to generate a better solution and accociated acceptable boundary gap for practical smaller size, and our LR method can also return a near-optimal solution within a reasonable time for larger portfolio size, e.g. within a reasonable time q = 200 for tracking Russell 3000.

147 Chapter 6. Lagrangian Relaxation for CCCP Conclusions and Discussion We designed a Lagrangian decomposition approach for the proposed CCCP in this section. We also generated two types of valid cuts that can speed up the LR algorithm. Index tracking problem can be seen as one application of CCCP framework. A factor-based robust enhanced index tracking model was developed and a robust three factor model of risk of Fama and French was used as the basis of constructing robust counterparts of the nominal tracking model. We highlight our contributions as follows. First, computational results using the S&P100 index as a benchmark have shown that the robust counterpart has better tracking performance and Sharpe ratios than portfolios generated by nominal models out-of-sample. Second, computational results from tracking the S&P 500, Russell 1000 and Russell 3000 demonstrated the effectiveness for the class of CCCP problem we considered. That is, (1) the feasible solution by the LR method is at least close to the solution from Gurobi; (2) the average gap by the LR method is lower than that by Gurobi (see tracking Russell 1000 and Russell 3000), better solution can be obtained in some instances (see tracking Russell 3000). Extending the proposed LR method to different types of problem, e.g. robust p-median problem, will be the subject of future research.

148 Chapter 7 Conclusion and Future Research 7.1 Conclusion In this thesis index tracking and cardinality constrained financial planning problems under uncertain environment were studied through different modelling approaches. Different models involved different investment goals and restrictions but each of the models incorporated the same type of cardinality constraints. As described in Chapter 1, portfolio selection models with cardinality constraints as a part of their decision support system are considered reasonably in practice but NP-hard. To best understand and provide the insights to the developed models, the LR-based algorithms with specific heuristic were applied to deal with computational treats and generate the optimal portfolios in associated chapters. Therefore, the main contribution in this document is that we investigate cardinality constrained portfolio selection models and provide a detailed analysis of three applications for which mathematical programming and financial modelling have been closely combined together to produce effective solving methodologies and managemental strategies. All these work can be used to support the one-fund theorem in practice. We summarize our main outcomes and results of this thesis involves the design and implementation as follows: ˆ We studied different portfolio selection models which contain a comprehensive set of practical managing constraints. Among these managing characteristics, limiting the portfolio size proved to be the most difficult and drew the largest attention in the design. For example, in Chapter 4 we incorporated the cardinality, buy-in threshold, turnover and 136

149 Chapter 7. Conclusion and Future Research 137 sector limit constraints into one index tracking model, in Chapter 5 we learned the cardinality, cash flow re-balance, and transaction costs constraints together in a stochastic programming framework, and in Chapter 6 we considered the cardinality, portfolio risk control and tracking error constraints into a robust index tracking model. Our detailed investigation of practical constraints offered a much clearer insight into the behaviour of portfolio management. ˆ We investigated two different approaches to capture numerous financial uncertainties involved with security return, risk, and other investment goals. in Chapter 5 we used the stochastic mixed integer programming technology to facilitate future uncertainties related to asset returns and index values, while in Chapter 6 we applied the robust optimization modelling structure to protect against the model parameter uncertainties included asset returns and variances. Our numerical results based on real data showed that both technologies can deal with the parameter uncertainty issue derive from the market volatility fairly well. ˆ We efficiently solved the portfolio selection models constructed in 4 to 6 by using a unified dual decomposition framework which embedded specific heuristic. In Chapter 4 we applied the Variable Neighborhood Search heuristic to obtain high-quality solutions for the index tracking problem by utilizing the bound information from the Semi-Lagrangian relaxation. In Chapter 5 we used the Progressive Hedging algorithm that allows designed Tabu Search and LR sub-solvers be embedded to generate the solution for cardinality constrained financial planning problems. In Chapter 6 we also applied the LR algorithm to decompose the factor based robust index tracking problem and generated the high-quality solutions. Overall, our competitive results with respect to various benchmarks showed that the effectiveness of the LR methods and can be used as an alternative of handling the solution for the large-scale applications. Further investigations will provide more insight towards these approaches and the results may improve or broaden the scope of this document. To fully exploit different advantageous characteristics of the proposed models, there are several other features of the developed models can be highlighted in next section.

150 Chapter 7. Conclusion and Future Research Future Research In this section we discuss different future directions that may extend or continue to develop based on this document and the studies in the field of financial engineering and optimization Modelling discussion One further modelling development is to deal with the uncertain parameters for the model in Chapter 4. Robust optimization is an applicable approach that may be integrated or used independently to model the problems considered. The robust counterpart for objective function (4.1) can be formulated as follows: max α (7.1) n s.t. max min ρ ijx ij α (7.2) x ρ i=1 j=1 The tractability of the robust counterpart (7.1) - (7.2) depends on the structure of uncertainty set. For example, the robust version will maintain linear form integer programming if we assume the uncertain ρ ij lies in a box type of perturbation set, i.e. ρ ij { ρ ij + ς ij ρ ij ς ij 1 }. However, such formulation may be too conservative to obtain enough manageable flexibility. Thus elliptical uncertainty set, ς 2 1, is more reasonable but it is hard to get the statistical property of ρ ij directly. To overcome this drawback, one strategy is to calculate the statistical n property of ρ ij in the transformation space by Fisher z-transformation [57]. Let z ij = 1 ( ) ln ρij 1 ρ ij Suppose that r T = (r 1, r 2,, r n ) N (µ, Σ), and observation (r 1t, r 2t,, r nt ) are independent for t = 1,, T, then random variable z N 1 ( ( ) ) 2 ln 1+ρ 1 1 ρ, T 3 where ρ is the true correlation coefficient and T is sample size. Building the robustness for z is relative easier than that for ρ, and we can retrieve ρ by setting (7.3) Then substituting (7.4) into (4.1): ρ ij = e2z ij 1 e 2z ij + 1 (7.4)

151 Chapter 7. Conclusion and Future Research 139 n n e 2z ij 1 max e 2z ij + 1 x ij i=1 j=1 n n ( ) 2 max 1 e 2z x ij ij + 1 i=1 j=1 n n n n ( 2 max x ij + max e 2z ij + 1 i=1 j=1 i=1 j=1 n n ( ) 2 n + max e 2z x ij ij + 1 i=1 j=1 n n ( ) 2 max e 2z x ij ij + 1 i=1 j=1 n n ( ) 2 min e 2z x ij ij + 1 i=1 j=1 ) x ij Note that n is equivalent to: i=1j=1 n x ij = n by summing the constraint (4.3) n times. Now model (4.1) - (4.5) min s.t. n i=1 j=1 n Z ij x ij (7.5) n y j = q (7.6) j=1 n x ij = 1, i = 1,, n (7.7) j=1 x ij y j, i = 1,, n, j = 1,, n (7.8) x ij, y j {0, 1} (7.9) 2 where Z ij =. Numerical results on SP100 have shown that the solution of model (7.5) e 2zij +1 - (7.9) are exactly same with the solution of the basic index tracking model (4.1) - (4.5). The relationship of robustness of parameter between the models shown in following theorem: Theorem 1. Building robustness for parameter Z ij of model (7.5) - (7.9) in Fisher z-transformation space is equivalent to build the robustness for parameter ρ ij of model (4.1) - (4.5) in original space. ( Proof. z ij = 1 2 ln 1+ρij ( ) 1+ρij 2z ij = ln 1 ρ ij 1 ρ ij )

152 Chapter 7. Conclusion and Future Research 140 e 2z ij + 1 = 1+ρ ij 1 ρ ij + 1 = 2 1 ρ ij 2 e 2z ij +1 = Z ij = 2 1 ρ ij 2 = 1 ρ ij Z ij and ρ ij are one-to-one corresponding relation, so robust solution for model (7.5) - (7.9) is equal the robust solution for model (4.1) - (4.5). Now the remaining task is to study the robust counterpart for model (7.5) - (7.9). Suppose that random vector r T = (r 1, r 2,, r n ) N (µ, Σ) is a multivariate normal distribution, then ( ( ) ) z N 2 ln 1, 1 T 3 ( ( ) ) z N ln, 1 T 3 ( ( ) ) 1 + e 2z 4 log N ln, 1 T 3 ( ( ) ) ( ( ) ) 1 + e 2z log N ln + 1, = log N ln e, 1 T 3 1 T 3 ( ( ) ) ( ( ) ) e 2z log N ln e, = log N ln, T 3 e (1 + ) T 3 ( ( ) ) ( ( ) ) (1 ) 4 e 2z log N ln + ln 2, = log N ln, + 1 e (1 + ) T 3 e (1 + ) T 3 ( ) 2 Therefore Z ij = has a log-normal distribution with mean ln 2(1 ij ) e 2zij +1 e(1+ i ) and variance 4 T 3. However, the objective (7.5) n n i=1 j=1 ρ ijx ij, which represent the linear combination of log-normal distribution, in general is not a log-normal distribution. Thus the probability constraint is unlikely to apply to objective function (7.5). Here we build the robustness for objective (7.5) by following the deriving steps in [26]: Given the current observation Z ij under current ρ ij, the real output may perturb Z ij around Z ij with a probability. We can then describe for any Z ij lie in following ellipsoid ε ij with a center Z ij and P R n n, ς R n n : Z ij + Z ij ε ij = { Z ij + P ij ς ij ς 2 1 } where P ij is the standard deviation and ς ij is the length for component Z ij in ε ij. Any weight matrix X = [x ij ] in ellipsoid ε ij can be mapped by the relationship: ς ij = { P T X } ij P T X 2

153 Chapter 7. Conclusion and Future Research 141 in which P X is the dot product of matrix P and X. For the objective (7.5): minmax x = min x = min x = min x = min x = min x Z n i=1 j=1 sup n Z ij x ij n n Z ij + Z ij ε ij i=1 j=1 (Z ij + Z ij ) x ij n n sup Z ij x ij + ςp T P T X ς P 2 1 ij i=1 j=1 n n Z ij x ij + XT P P T X P T X i=1 j=1 2 n n P T X Z ij x ij + P T X i=1 j=1 2 n n Z ij x ij + P T X 2 i=1 j=1 Then the robust counterpart of model (7.5) - (7.9) can be formulated as follows: min φ (7.10) s.t. n n Z ij x ij + P T X 2 φ (7.11) ( 2(1 ) e(1+ ) i=1 j=1 n y j = q (7.12) j=1 n x ij = 1, i = 1,, n (7.13) j=1 x ij y j, i = 1,, n, j = 1,, n (7.14) x ij, y j {0, 1}, φ R (7.15) where Z and P are the mean value vector and standard deviation matrix of random variable ( ) ) ) 4 Z. Since Z log N ln, T 3, we can get Z = ln and P = 4 T 3I in the ( 2(1 ) e(1+ ) robust index tracking model (7.10) - (7.15). Solving the proposed robust index tracking model is non-trivial but the LR methods can be still applied to obtain the bound information. Although the selection models presented in Chapter 5 and Chapter 6 were developed to capture the important characteristics of the portfolio selection under uncertain environment,

154 Chapter 7. Conclusion and Future Research 142 it would be interesting to incorporate some more detailed aspects which we investigated in Chapter 4 into the models. For instance, sector limit constraint can be considered into the Financial Planning problem to simplify the network structure. We test the TE/TC ratio to show the advantage of the factor based robust model for out-of-sample testing in Section (6.5.2), but we can also incorporate the transaction costs constraint into the developed index tracking models and formulate the problem as one whole optimization program. Another aspect that can further be studied is that applying different scenario generation techniques such as Monte Carlo Simulation to mimic the parameter uncertainty of the stochastic financial planning model. As mentioned in the abstract, the models we developed can also extend to other management applications, for example, the factor based robust model could be used to study the facility location problem where decision maker needs to determine the location of the potential facilities so that the uncertain demand can be satisfied Algorithm discussion We have developed three LR-based decomposition algorithms which embedded different specific heuristics to solve a set of NP-hard problems. Although we have compared our methods with the most well-known MIP solver which based branch and bound algorithm with sophisticated cuts, it is possible to improve the results and computational time via combining different information technologies. The first development direction is to embed a Message Passage Interface (MPI) code to parallelize the sub-problems so that the computational time can be significantly reduced. This step is particularly useful to Progressive Hedging algorithm in which there exist numerous scenarios for a parameter in financial models. The second direction involves decomposition strategy from different angles, we applied the dual decomposition through this document but the primal decomposition in [139] might offer additional insights for the understanding of the models. Finally, we can combine the LR framework and different cutting-planes to speed up the convergence process in Chapter 6. Cardinality constraint studied in this thesis is one important approach to limit the portfolio size. Another alternative to obtain sparse portfolio is norm regularization. For example, Burmeister et al. [27] applied the trading budget constraint which can be represented by l 1 -norm to approximate the cardinality constraint, i.e., x 0 K x 1 ε where x 0 = i x i 0.

155 Chapter 7. Conclusion and Future Research 143 The authors tested different alternative trading costs and found that the size in the low to median range level generally has a smaller replicating error. To achieve a designed portfolio size, we can adjust the penalty parameters in an algorithm via solving a sequence of continuous approximation and obtain a suitable budget ε. For example, with l 1 -norm regularization the developed index tracking model in Chapter 6 can be reduced to the SOCPs, which can be efficiently handled by interior point method, in each iteration and finally generates a sparse portfolio that closes to or equals the required size. Therefore, this method can also handle the large-scale computation and can be used as a potential comparison benchmark for our LR method in this thesis.

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170 List of Publications ˆ Part of Chapter 6 of this thesis is published in the Journal of Optimization and Engineering, with the reference of Roy H. Kwon and Dexiang Wu. Factor-based robust index tracking. Journal of Optimization and Engineering, April Online available. ˆ Chapter 4 of this thesis is submitted to The European Journal of Operational Research. 158

171 Appendix A Appendix of Chapter 4 A. 1 Numerical example for Heuristic I To quickly generate an initial feasible solution, a numerical example based on S&P500 is used to illustrate the Heuristic I as follows: Set q = 10, α = 0.001, γ ( = 0.5. ) Sector size vector m = (0) After sorting the marker value and choosing the first q assets, we obtain: ( ) T q k = and associated K i=1 p k γ α = , and L (q k) = { } { } (1) I 1 = 4 2 5, q I1 = { } { } { I 2 = 7 6 3, q I2 = 1 1 1, q I2 equal each other, sort I 2 by m I2 = { } { } I 3 = , m I2 = P =, N = 100, by 1O - 4O: 1O Pick 2 assets from sector 4 (A = 1) in I 1, add to sector 7 (B = 1) in I 2, then new pt1: ( ) ( ) q fesi k = T, L q fesi k = O Pick 2 assets from sector 4 (A = 1) in I 1, add to sector 1 (B = 1) in I 3, then new pt2: ( ) ( ) q fesi k = T, L q fesi k = O Pick 2 assets from sector 4 (A = 1) in I 1, add 1 asset to sector 7 (B = 1) in I 2 and 1 asset to sector 1 (C = 1) in I 3, then new pt3: ( ) ( q fesi k = T, L q fesi k T. ) = O Pick 1 asset from sector 4 (A = 1) in I 1 and 1 asset from sector 7 (B = 1) in I 2, add them to sector 1 (C = 1) in I 3, then new pt4: ( ) ( q fesi k = T, L (2) Solve (L) without constraint K k=1 p k γ α q fesi k under qfesi k vectors in P ; ) = }. 159

172 Appendix A. Appendix of Chapter (3) Test transaction cost constraint (TC); Obj = ( ) L q fesi k by 2O and 3O are better than L (q k ), and both solutions satisfy the TC. STOP. Then we can generate the initial q fesi k = ( ) T, i.e. we add 1 assets to sector 1 from sector 4 and add 1 asset to sector 1 from sector 7. Then the associated K i=1 p k γ α = The objective value under q fesi k is , which is better than the value from Step (0) and the constraint K i=1 p k γ α still be satisfied. The initial feasible objective is lower than that value ( ) by LR method at q=10 on the first sub-figure of Figure (4.4). A. 2 Numerical example for Heuristic II ( ) Sector size vector m = T. Set q = 90, α = 0.001, ( ) γ = 0.5. From Heuristic I we get an initial feasible q fesi T that k = across sectors, ( and its feasible objective ) Then a Lagragian vector is obtained by solving (L) under qk LR = T, but associated K k=1 qlr k = 45 < q = 90, so we go to Heuristic II to adjust qk LR and get a feasible solution. (1) Since K k=1 qlr k < q, we adjust qk LR as follows: Pick k th sector in qk LR that has minimal value, i.e. k = 10, q10 LR = 2 Check if qk LR m (k), i.e. q10 LR = 2 < m (10) = 35, then q fesi k = qk LR, i.e. qfesi 10 = 2 Else if qk LR > m (k), q fesi k = m (k) Repeat above steps, after check all sectors K k=1 qfesi k = 45 < q = 90; then put the difference q K i=1 qfesi k = 45 into the sector have the maximal asset number, i.e. sector 1. We obtain a q fesi k vector as follows by Step (1): m qk LR = , go to Step (2) q fesi k (2) Solve (L) without K k=1 p k γ α, K k=1 p k γ α = 44 > 0, GO TO (3) (3) Set = 2, For k = 1, do: { }} I 1 = {w j1 0 j q fesi 1, sort I 1, i.e. market weights of 49 assets ascently { }} I 2 = {w j1 0 j {m (1)} \, sort I 2, i.e. market weights of 33 assets, descently; q fesi 1 Pick first one number of assets in I 1 and Pick first two number of assets in I 1 and I 2, switch and obtain new neighbor point I 2, switch and obtain new neighbor point For k = 2 : 10, do the same above swap steps. Then we totally generate 20 new q fesi k in Step (3), and we test all new pts for TC, there is no pts satisfy K k=1 p k γ α, go to Step (4)

173 Appendix A. Appendix of Chapter (4) Pick sector 1 (k 1 ) and sector 10 (k 2 ) I 1 = { w 0 j1 j {m (1)}}, sort I 1, i.e. market weights of 82 assets, ascently I 2 = { w 0 j,10 j {m (10)}}, sort I 2, i.e. market weights of 35 assets, descently; Pick first 2 number of assets in I 1 and I 2, Set {y 1:2,1 = y 1:2,10 j, I 1 } and {y 1:2,10 = y 1:2,1 j, I 2 } Obtain a new q k vector Pick sector 1 (k 1 ) and sector 3 (k 2 ) I 1 = { w 0 j1 j {m (1)}}, sort I 1, i.e. market weights of 82 assets, ascently I 2 = { w 0 j3 j {m (3)}}, sort I 2, i.e. market weights of 37 assets, descently; Pick first 2 number of assets in I 1 and I 2, Set {y 1:2,1 = y 1:2,3 j, I 1 } and {y 1:2,3 = y 1:2,1 j, I 2 } Obtain a new q k vector Repeat above steps and we can get 120 new q k vectors. Then some vectors cannot maintain q K k=1 qfesi k = 0, therefore go to Step (1) to adjust the q k vectors. ( ) We test all pts and obtain a better solution with q fesi k = and satisfy TC K k=1 p k γ α = Then we get a feasible objective , which is higher than the initial objective value by Heuristic I. T,

174 Appendix A. Appendix of Chapter A. 3 Ticker in S&P500 Sector (total number) 1: Consumer Discretionary (82) 2: Consumer Staples (41) 3: Energy (41) 4: Financials (81) 5: Health Care (51) 6: Industrials (62) Table A.1: Ticker symbol across Sectors (SP500) Ticker Symbol ANF, AMZN, APOL, AN, AZO, BEAM, BBBY, BBY, BIG, HRB, BWA, CVC, KMX, CCL, CBS, COH, CMCSA, DHI, DRI, DV, DTV, DISCA, DLTR, EXPE, FDO, F, GME, GCI, GPS, GPC, GT, HOG, HAR, HAS, HD, IGT, IPG, JCI, KSS, LEG, LEN, LTD, LOW, M, MAR, MAT, MCD, MHP, NWL, NWSA, NKE, JWN, CMG, ORLY, OMC, JCP, RL, PHM, ROST, SNI, SHLD, SHW, SNA, SWK, SPLS, SBUX, HOT, TGT, TIF, TWX, TWC, TJX, TRIP, URBN, VFC, VIAB, DIS, WPO, WHR, WYN, WYNN, YUM MO, ADM, AVP, BFB, CPB, CLX, KO, CCE, CL, CAG, STZ, COST, CVS, DF, DPS, EL, GIS, HNZ, HRL, K, KMB, KFT, KR, LO, MKC, MJN, TAP, PEP, PM, PG, RAI, SWY, SLE, SJM, SVU, SYY, HSY, TSN, WMT, WAG, WFM APC, APA, BHI, COG, CAM, CHK, CVX, COP, CNX, DNR, DVN, DO, EP, EOG, XOM, FTI, HAL, HP, HES, MRO, MPC, ANR, MUR, NBR, NOV, NFX, NE, NBL, OXY, BTU, PXD, RRC, RDC, SLB, SWN, SE, SUN, TSO, VLO, WMB, WPX ACE, AFL, ALL, AXP, AIG, AMP, AON, AIV, AIZ, AVB, BAC, BK, BBT, BRK.B, BLK, BXP, COF, CBG, SCHW, CB, CINF, C, CME, CMA, DFS, ETFC, EFX, EQR, FII, FITB, FHN, BEN, GNW, GS, HIG, HCP, HCN, HST, HCBK, HBAN, ICE, IVZ, JPM, KEY, KIM, LM, LUK, LNC, L, MTB, MMC, MET, MCO, MS, NDAQ, NTRS, NYX, PBCT, PCL, PNC, PFG, PGR, PLD, PRU, PSA, RF, SPG, SLM, STT, STI, TROW, TRV, TMK, USB, UNM, VTR, VNO, WFC, WY, XL, ZION ABT, AET, AGN, ABC, AMGN, BCR, BAX, BDX, BIIB, BSX, BMY, CAH, CFN, CELG, CERN, CI, CVH, COV, DVA, XRAY, EW, ESRX, FRX, GILD, HSP, HUM, ISRG, JNJ, LH, LIFE, LLY, MCK, MHS, MDT, MRK, MYL, PDCO, PKI, PRGO, PFE, DGX, STJ, SYK, THC, TMO, UNH, VAR, WAT, WPI, WLP, ZMH MMM, APH, AVY, BA, CHRW, CAT, CTAS, GLW, CSX, CMI, DHR, DE, RRD, DOV, DNB, ETN, EMR, EXPD, FAST, FDX, FSLR, FLS, FLR, GD, GE, GR, GWW, HON, ITW, IRM, XYL, JEC, CBE, JOY, LLL, LMT, MAS, NSC, NOC, PCAR, IR, PLL, PH, PBI, PCP, PCLN, PWR, RTN, RSG, RHI, ROK, COL, ROP, R, LUV, SRCL, TXT, TYC, UNP, UPS, UTX, WM Continued on next page

175 Appendix A. Appendix of Chapter Sector (total number) 7: Information Technology (70) 8: Materials (29) 9: Telecommunications Services (8) 10: Utilities (35) Table A.1 continued from previous page Ticker Symbol ACN, ADBE, AMD, A, AKAM, ALTR, ADI, AAPL, AMAT, ADSK, ADP, BMC, BRCM, CA, CSCO, CTXS, CTSH, CSC, DELL, EBAY, EA, EMC, FFIV, FIS, FISV, FLIR, GOOG, HRS, HPQ, INTC, IBM, INTU, JBL, JDSU, JNPR, KLAC, LXK, LLTC, LSI, MA, MCHP, MU, MSFT, MOLX, MMI, MSI, NTAP, NFLX, NVLS, NVDA, ORCL, PAYX, QCOM, RHT, SAI, CRM, SNDK, SYMC, TEL, TDC, TER, TXN, TSS, VRSN, V, WDC, WU, XRX, XLNX, YHOO APD, ARG, AA, ATI, BLL, BMS, CF, CLF, DOW, DD, EMN, ECL, FMC, FCX, IFF, IP, MWV, MON, MOS, NEM, NUE, OI, PPG, PX, SEE, SIAL, TIE, X, VMC AMT, T, CTL, FTR, PCS, S, VZ, WIN AES, GAS, AEE, AEP, CNP, CMS, ED, CEG, D, DTE, DUK, EIX, ETR, EQT, EXC, FE, TEG, NEE, NI, NU, NRG, OKE, POM, PCG, PNW, PPL, PGN, PEG, QEP, SCG, SRE, SO, TE, WEC, XEL

176 Appendix A. Appendix of Chapter A. 4 Gap by LR and SLR Table A.2: Gap between LB and UB, q Best UB Gap Time Best UB Gap Time feasi. LB by LR (%) (hour) feasi. LB by SLR (%) (hour) Aver A. 5 Sector Allocation

177 Appendix A. Appendix of Chapter Figure A.1: Portfolio allocation in sectors