Dynamic Asset Allocation and Algorithmic Trading Master Thesis presented by: Victor August Moquist M.Sc. in Applied Economics and Finance (cand.merc) Supervisor: Søren Agergaard Andersen Pages: 80 STU: 154,410 COPENHAGEN BUSINESS SCHOOL November 2014 Executive summary This thesis studies the opportunities arising from algorithmic trading that is gradually becoming available to private investors. For this purpose three portfolio selection models are crossed with three rebalancing strategies. The result is eight individual algorithms that build partly on observed practical application and partly on theoretical portfolio optimisation. The first category of algorithms is non-rebalancing and features a naïve portfolio and a Markowitz portfolio. The second category is fixed period rebalancing algorithms featuring a naïve, a Markowitz and a Gârleanu-Pedersen portfolio model. The third category, are the same three portfolio models with dynamic rebalancing, conditional on increased expected utility. These eight combinations are formulated in matrix algebra for algorithmic application. Subject to transaction costs and predicted returns, the eight algorithms are run in a back-testing simulation. The simulation is run on daily price observation from a portfolio of 58 stocks from the S&P500 index and the U.S. 10 year treasury yield from 1994-2014. In addition, historical price data from 1980-1994 is applied in ARIMA models for long term and short term price forecasting. Four quantitative result measures are drawn from the simulation to be used in a comparative evaluation of the eight algorithms. It is found that automatically rebalanced portfolios can yield a superior utility and that the dynamic algorithms are significantly outperforming other algorithms on transaction costs. It is also shown that the Markowitz portfolio and especially the Gârleanu-Pedersen model are significantly more prone estimation errors from poor return predictability. KEYWORDS: Dynamic asset allocation, algorithmic trading, Portfolio Rebalancing, Mean-variance optimization, Negative exponential utility, ARIMA ACKNOWLEDGEMENT: A special thanks goes to my supervisor Søren Agergaard Andersen for his guidance and patience. Also, I want to thank PhD fellow Niels Joachim Christfort Gormsen for our many interesting academic discussions. Table of Contents CHAPTER 1 - INTRODUCTION ...................................................................................................... 7 1.1 PROBLEM STATEMENT .......................................................................................................... 8 1.2 THESIS STRUCTURE ............................................................................................................... 9 CHAPTER 2 - LITERATURE REVIEW ......................................................................................... 11 2.1 ASSET ALLOCATION ............................................................................................................ 11 2.1.1 Risk Aversion and Utility Functions .............................................................................. 12 2.1.2 Asset Return ................................................................................................................... 15 2.1.3 Single-period Portfolio Models ........................................................................................ 16 2.1.4 Multi-period Portfolio Models ........................................................................................ 24 2.1.5 Summary ......................................................................................................................... 30 2.2 RETURN PREDICTABILITY ................................................................................................... 31 2.2.1 The Random Walk ......................................................................................................... 31 2.2.2 Market Equilibrium Models ............................................................................................ 34 2.2.3 Summary ......................................................................................................................... 36 CHAPTER 3 - METHODOLOGY .................................................................................................... 37 3.1 RESEARCH APPROACH......................................................................................................... 37 3.3.1 Research Design .............................................................................................................. 37 3.1.2 Research Perspective ....................................................................................................... 38 3.1.3 Research Model ............................................................................................................... 39 3.2 ECONOMETRIC TOOLBOX .................................................................................................... 40 3.2.1 Linear Regression ............................................................................................................ 40 3.2.2 Box-Jenkins Methodology ............................................................................................... 42 3.3 DATA ................................................................................................................................... 43 3.3.1 Asset Selection ................................................................................................................ 44 3.4 LIMITATIONS ........................................................................................................................ 46 CHAPTER 4 – MODELLING ALGORITHMS ................................................................................ 49 4.1 GENERAL CONDITIONS ........................................................................................................ 49 4.1.1 Portfolio Returns ............................................................................................................ 49 4.1.2 Transaction Costs ........................................................................................................... 51 4.1.3 Negative Exponential Utility .......................................................................................... 52 4.1.4 Utility Optimisation ........................................................................................................ 54 4.1.5 Summary ......................................................................................................................... 55 4.2 PORTFOLIO ALGORITHMS .................................................................................................... 55 4.2.1 No Rebalance .................................................................................................................. 56 4.2.2 Fixed Period Rebalance .................................................................................................. 57 4.2.3 Automated Rebalance ..................................................................................................... 60 4.2.4 Summary ......................................................................................................................... 62 CHAPTER 5 - EMPIRICAL ANALYSIS ......................................................................................... 63 5.1 RETURN PREDICTION – BOX-JENKINS METHODOLOGY ....................................................... 63 5.1.1 Identification ................................................................................................................... 64 5.1.2 Estimation and Diagnostic Checking .............................................................................. 66 5.1.3 Forecasting ...................................................................................................................... 66 5.1.4 Summary ......................................................................................................................... 69 5.2 ALGORITHM SIMULATION .................................................................................................... 69 5.2.1 No Rebalance .................................................................................................................. 70 5.2.2 Fixed Period Rebalance .................................................................................................. 72 5.2.4 Automated Rebalance ..................................................................................................... 74 5.2.5 Summary ............................................................................................................................ 76 5.3 QUANTITATIVE COMPARISON .............................................................................................. 77 5.3.1 Utility .............................................................................................................................. 77 5.3.2 Mean and Variance ......................................................................................................... 78 5.3.3 Portfolio Weights and Delta ........................................................................................... 79 5.3.4 Transactions .................................................................................................................... 79 5.3.5 Summary ......................................................................................................................... 80 CHAPTER 6 - DISCUSSION ............................................................................................................ 83 6.1 ASSUMPTIONS AND LIMITATIONS ......................................................................................... 83 6.1.1 Negative Exponential Utility .......................................................................................... 83 6.1.2 Transaction Costs as a Function of Covariance ............................................................. 84 6.1.3 Predicted Returns and Estimation Errors ...................................................................... 85 6.1.4 Technological limitations ................................................................................................ 85 6.2 APPLICABILITY .................................................................................................................... 86 6.3 RESEARCH EXTENSIONS ...................................................................................................... 86 CHAPTER 7 - CONCLUSION .......................................................................................................... 89 REFERENCES ................................................................................................................................... 91 APPENDIX A .................................................................................................................................... 95 Uncorrelated random variable .................................................................................................... 95 APPENDIX B .................................................................................................................................... 97 List of stocks in risky asset portfolio .......................................................................................... 97 Descriptive Statistics................................................................................................................... 98 Covariance Matrix ...................................................................................................................... 99 APPENDIX C .................................................................................................................................. 111 Augmented Dickey-Fuller ......................................................................................................... 111 Outliers ..................................................................................................................................... 112 Correlogram: ACF and PACF.................................................................................................. 114 Squared residuals of forecasted stocks ...................................................................................... 116 APPENDIX D .................................................................................................................................. 117 Optimal Delta plot .................................................................................................................... 117 DYNAMIC ASSET ALLOCATION AND ALGORITHMIC TRADING 7 Chapter 1 - Introduction In the Financial Times James Angel of Georgetown University commented “There’s an old bumper sticker that says, ‘to err is human, to really foul up requires a computer” (FT, 2012, p.3). This was his response to yet another high frequency trading algorithm, which had sent the American stock markets into a complete turmoil. Despite repeated incidents of flash crashes in the financial markets caused by algorithmic trading, this relatively new trading method is becoming more common. Since its origin, algorithmic trading has been exclusively for large investment institutions, who fight for low latency and smart order routing. However, brokers are now building platforms for private as well as small investors to take part in the universe of algorithmic trading (Forbes, 2013). Today, companies such as Quantopian and Straticator are designing coding languages for the less advanced programmer to build private trading algorithms – this process is illustrated in figure 1.a below. In addition, these companies provide a platform for back-testing the algorithms in order to see how the algorithm would have performed historically. This thesis seeks to explore automation of trading for the private and small investor. Figure 1.a – Algorithmic trading service by Straticator.com (www.straticator.com/algo-trading, 2014) The study will take off where the Nobel Prize winning article by Harry Markowitz (1952) ended: “In this paper we have considered the second stage in the process of selecting a portfolio. This stage starts with the relevant beliefs about the securities involved and ends with the selection of a portfolio.” (Markowitz, 1952, p 91). The third stage in this study deals with portfolio maintenance when asset allocation models are brought into a multi staged universe. In other words, after going through the first stage of picking assets and secondly selecting a 8 INTRODUCTION portfolio strategy, the investor will be concerned with keeping the portfolio in line with the strategy. 1.1 Problem Statement Applying algorithmic trading in portfolio management is challenged by empirical findings condemning active trading. As an example, Barber and Oden (2000) find that U.S. households who turn over 75% of their portfolio annually underperform by 3.7% relative to the market index. Thus, before private investors embark on the new era in financial trading it is relevant to pose the question: How can current asset allocation practices benefit from automated rebalancing through algorithmic trading? The answer to this research question lays in the characteristics of the individual investor and the application and can therefore be somewhat ambiguous. Thus, the answer shall be sought by predefining stock pickings, asset allocation model, risk aversion, time horizon etc. to isolate the effect from various rebalancing strategies. Optimization problems can always be discussed verbally and logical reasoning can be used to reach some solution for the optimization problem. The challenge however, is to formulate the optimization problem in a way that allows a computer scientist to write a program that can solve the problem given some formal descriptions and conditions. This study is far from the first to study rebalancing strategies. Most recently Sujit Das, Dimitri Kaznachey and Mukul Goyal (2014) have proposed a framework for the optimal fixed rebalancing frequency. David Smith and William Desormeau (2006) make a similar study and Carsten Sørensen and Anders Trolle (2010) have proposed dynamic portfolio management, allowing continually optimizing portfolio weights. However, this study will be distant to these papers through the focus on algorithmic application. DYNAMIC ASSET ALLOCATION AND ALGORITHMIC TRADING 9 1.2 Thesis Structure Chapter two will bring a thorough introduction to literature on asset allocation and return prediction models. Chapter three will explain the general methodology applied throughout the study. Building on the described theory in chapter two, the fourth chapter adjusts three portfolio selection models to matrix algebra for algorithmic application. Thus, from chapter four, eight discrete time models with rebalancing applications are proposed in algorithmic form. These algorithms are then back-tested on the collected vast stock price observations from 1980 to 2014. Chapter five will both illustrate and describe the findings of each algorithm and make a summary table of quantitative performance measures. Following chapter six provides a discussion about the impacts of limitations and assumption and also the applicability of the findings and model extensions. Chapter seven will conclude on the study and list the most relevant findings and challenges.