Distribution Forecasting of Nonlinear Time Series with Associative Memories. A. R. Pasley

Transcription

1 Distribution Forecasting of Nonlinear Time Series with Associative Memories A. R. Pasley This thesis is submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy. University of York York YO10 5DD UK Department of Computer Science September 2003

2 2 Abstract The forecasting of financial variables is a problem which has been investigated ever since economies began to develop. Modern day information technology now provides at the fingertips of forecasters an enormous range of historical data, automated forecasts and news stories. In addition to assisting with traditional financial forecasting problems this has allowed financial time series to be investigated at a much higher frequency. The sampling of financial time series at intra-day frequency introduces further complications to forecasting than traditional problems. However, it makes available much larger sets of historical data. This thesis examines the utility of existing forecasting methodology for high frequency forecasting. It then tries to understand what information is provided by past high frequency data and how data intensive algorithms could use it. This thesis also investigates the area of distribution forecasting where the aim is to produce a full probability density forecast rather than a single value. This is a recent addition to the financial forecasting world, encouraged by modern computing power and the large volumes of historical data now available. High frequency data sets have made possible new approaches to distribution forecasting and it is natural to bring the two together. During this work a new approach to the problem is constructed. An algorithm requiring fast k-nn type search is implemented using AURA, a binary neural network based upon Correlation Matrix Memories. This novel architecture also constructs probability distribution forecasts, the volume of data allowing this to be done in a nonparametric manner. Financial forecasting is part of the larger problem of nonlinear forecasting and the new algorithm is therefore tested not only on financial data but other standard forecasting data sets. In addition to standard statistical error measures the implementation of simulations allows actual measures of profit to be calculated. The creation of distribution forecasts introduces difficulties in forecast evaluation and results are reported from a separate evaluation of distribution accuracy.

3 Contents 1 Introduction and Problem Definition Investment Management Asset Allocation Forecasting Financial Theory Efficient Market Hypothesis Modern Portfolio Theory Is the EMH a reliable model? Risk Management VaR Forecasts for individual assets Distribution Forecasting Aims and Thesis Overview Aims Thesis Overview Field Survey and Review Fundamental Analysis Technical Analysis Averaging Models ARIMA ARCH processes Delay Coordinates for Forecasting Neural Network Approaches Neural Networks Neural Networks for Financial Forecasting

4 CONTENTS Multi Layer Perceptron Other Forms of Neural Network Alternatives to Neural Networks High Frequency Forecasting Generating Distribution Forecasts Parametric Methods Non-Parametric Methods Semi-Parametric Methods (Mixture Models) Static or Dynamic (Evolving) Forecasts Evaluation of Approaches Distribution Evaluation Conclusion Correlation Matrix Memories and AURA Networks Correlation Matrix Memories AURA Networks Problem Solving in AURA Binary Codes in AURA Applications of AURA Construction of a New Forecasting Algorithm Assumptions and Aims Initial Forecasting Algorithm Farmer-Sidorowich Algorithm Requirements to build a distribution Implementation of forecasting algorithm Design Issues Parameter estimation Aims and Objectives Preliminary Experiments and Results Data Sets and Experiment Details Statistical Error Measures for Point Forecasts

5 CONTENTS Results and Analysis Non-Financial Time Series Financial Time Series Distribution Evaluation Visual Examination of the distributions Integral Test Empirical Test Conclusion Success of Current Results Measuring success Results and Simulations Experiments Results: Accuracy of Point Forecasts Larger Data Set Optimal Model Verification Histogram Construction Combining Features The Effect of Input Code Weight Speed Trading Simulations Using a Distribution for Trading Trading Model Implemented and Results Improving the Simulation Returns Evaluation of Probability Distributions Extending Forecasts Further Ahead Extending AURA-FS Forecasts Results Simulations Results Using a Large Bid-Ask Spread Results Using a Smaller Bid-Ask Spread

6 CONTENTS Overall Simulation Results Conclusions and Future Work The Forecasting Architecture Forecasting Accuracy and Potential Applications Financial Data Other Data Sets Risk Management and Distribution Accuracy Future Work Understanding Parameters Multivariate Series A Electricity Load Demand Forecasting 134 A.1 Existing Demand Forecasting Methods A.2 Problem Definition A.2.1 Factors affecting demand A.2.2 Evaluation and Distribution Accuracy A.2.3 Data Sets A.3 Alteration to AURA-FS Algorithm A.4 Results A.4.1 Forecastability A.4.2 NGT Results A.4.3 npower Results A.5 Demand Forecasting Conclusions B Nonparametric Statistical Tests Used 149 B.1 Mann-Whitney B.2 Smirnov

7 List of Tables 4.1 Comparison of Distance Metric implemented with standard measures. Distances from other bins would follow a similar pattern Results for Santa Fe Data - Series A Selection of results for Santa Fe Data - Series D Selection of results for Santa Fe Data - Series E Selection of results using ADA for Santa Fe Data - Series C Selection of results using Theil s for Santa Fe Data - Series C Series D: Results of check that the training distributions and test distributions are samples of the same distribution. The Mann-Whitney and Smirnov tests have been carried out for each feature Series C: Results of check that the training distributions and test distributions are samples of the same distribution. The Mann-Whitney and Smirnov tests have been carried out for each feature Comparison with other published results for the same data Error results for AURA-FS and πcmm The parameters for particular Models implemented in AURA-FS The error measures for AURA-FS models for data sets 100A and 100B The time taken by AURA-FS implementing Model C The effect of Γ on the number of transactions produced by trading using model A Results of check that the training distributions and test distributions are samples of the same distribution. The Mann-Whitney and Smirnov tests have been carried out for each feature A comparison of different techniques for forecast extension Simulation returns and transactions generated for Γ = 4 using Model A

8 LIST OF TABLES Simulation returns for Γ = 1 using Model A and data with an artificially fixed Bid-Ask spread A.1 Error measures for various models forecasting NGT Data A.2 NGT Data Set: Results of empirical test into training distributions and test distributions. The Mann-Whitney and Smirnov tests have been carried out for each feature A.3 npower Data Set: Results of empirical test into training distributions and test distributions. The Mann-Whitney and Smirnov tests have been carried out for each feature

9 List of Figures 1.1 Discrete distribution of high frequency returns for Japanese Yen and US Dollar, from over half a million points of data. The returns do not follow a normal distribution, although the logarithmic scale used exaggerates the non-normal behaviour at the tails First-order approximation: The evolutions of the nearest matches to the recent behaviour (current state) are used to construct the forecast First-order approximation: Neighbours to the current state x(t) can be considered members of the domain of F T. Their evolutions make up the range from which the forecast ˆx(t + T ) is constructed Histogram approach to density estimation: The distribution given could be reasonably approximated by the histogram shown A Gating Network Diagram showing how AURA has been used in many practical applications. A back-check operation is performed to remove noise from the CMM recall results Training. The CMM is updated to learn the relationship between the input and the features (clusters). The frequency store holds a historical simulation for each cluster Forecasting. A recall on the CMM produces the nearest neighbours from which the forecast is constructed. Each neighbour has its own distribution in the frequency store Calculating the thresholded output from a CMM column Calculating the thresholded output from a CMM column

10 LIST OF FIGURES Distance measures in two dimensions. The solid line shows the boundary for an arbitrary threshold using Euclidian distance from a state whose values are [x, y ] (the point at the centre of the circle). The dashed line shows the boundary using Manhattan distance where the threshold has been selected to approximate the Euclidian distance with no true negatives. If Euclidian distance is considered the correct metric then false positives will be returned by a k-nn search using Manhattan distance Distance measures in two dimensions. Each solid line shows the boundary for a particular distance from a state whose values are [x, y ]. Beyond a certain point the distance metric implemented is no longer Manhattan Plot of a section of Series A. Time is displayed as the number of measurements taken since the beginning of the displayed section of the series Plot of a section of Series C. Time is displayed as the number of quotes observed since the beginning of the displayed section. The series displayed here covers approximately 27 hours of trading Plot of a section of Series D. Time is displayed as the number of measurements taken since the beginning of the displayed section of the series Plot of a section of Series E. Time is displayed as the number of measurements taken since the beginning of the displayed section of the series Typical forecasts for Series D. The change in time series is change from the most recently observed value. Therefore a no change forecast is aligned at 0 on the horizontal axis. The observed value is indicated by an arrow on the horizontal axis Sample forecasts for Series C. The change in time series is change from the most recently observed value. Therefore a no change forecast is aligned at 0 on the horizontal axis. The observed value is indicated by an arrow on the horizontal axis More sample forecasts for Series C. The change in time series is change from the most recently observed value. Therefore a no change forecast is aligned at 0 on the horizontal axis. The observed value is indicated by an arrow on the horizontal axis Empirical Test - Series D: Comparison of distribution observed during and after training Empirical Test - Series C: Comparison of distribution observed during and after training Theil s results for varying parameters d and η. Φ = 93% and Ψ = 80% ADA results for varying parameters d and η. Φ = 93% and Ψ = 80%

11 LIST OF FIGURES Theil s results for various models. Each graph shows the effect of varying parameters d and η for the stated values of Φ and Ψ ADA results for various models. Each graph shows the effect of varying parameters d and η for the stated values of Φ and Ψ Forecast coverage for varying parameters d and η. Φ = 93% and Ψ = 80% Effect of varying the value of η. Φ = Ψ = 93% and d = Theil s results for varying parameters d and η. Φ = 93% and Ψ = 80%. The data set contained 197, 235 entries ADA results for varying parameters d and η. Φ = 93% and Ψ = 80%. The data set contained 197, 235 entries Effect of varying θ on forecast accuracy Effect of varying Ψ on forecast accuracy Effect of varying ω (the weight given to the vectors constructed to represent each bin) on forecast accuracy Simulation returns (no costs included) for varying parameters d and η. Φ = 93% and Ψ = 80% Simulation returns (costs included) for varying parameters d and η. Φ = 93% and Ψ = 80% Iterative extension using the distribution forecast. All possible values given by the previous forecast are treated as observations to make a possible forecast for one further step ahead. An average of all possible forecasts combines them to a single forecast. The weighted average is calculated by the probability given by the distributions of the series following that path Extending a distribution forecast The effect of the extension k on the accuracy of AURA-FS forecasts Simulation results for varying parameters k and Γ using Model A Number of transactions for varying parameters k and Γ using Model A Simulation results for varying parameters k and Γ using Model A using data with an artificially fixed Bid-Ask spread. The surface colour has been interpolated to aid visual clarity A.1 Forecast and observed values for NGT data A.2 Forecasts compared against observed (target) values A.3 Effect on error of θ A.4 Effect of measuring forecastability on forecast coverage and accuracy

12 LIST OF FIGURES 12 A.5 Empirical Test - npower Data Set. Comparison of distribution observed during and after training

13 Acknowledgements I must first thank Jim Austin who has given me constant support throughout the whole of this research. His advice during regular discussions on the direction of the work has been invaluable and regularly highlighted areas of interest in the literature. His assistance also extended to the more time consuming and mundane task of checking documents and papers before submission in which he continued to offer excellent advice. My thanks must extend to the Advanced Computer Architecture Group as a whole. I have sought assistance from every member of the group at one time or another and all have been keen to help and a source of good advice. I must mention in particular Rob Davis who without financial incentive or physical pressure read through a full draft of the thesis. His comments helped correct many mistakes and omissions but also demonstrated an interest and understanding in the research that will help any future work in this area. Away from the department family, friends and especially my darling Emma have all shown patience and given support throughout which has been equally as important and greatly appreciated. 13

14 Declaration I declare that this thesis has been completed by myself and that except where indicated otherwise the research is entirely my own. Some of the material presented in chapters 4 and 6 has been presented previously at a conference (Pasley & Austin 2002). Some of the material presented in chapters 6 and 7 is to be published in a special issue of the Decision Support Systems journal. 14

15 Chapter 1 Introduction and Problem Definition Throughout history individuals and civilisations have pursued wealth. Whether just to survive or for security, happiness or freedom, in today s global economy, wealth can now be defined simply in terms of money. Money can be used to purchase all sorts of goods and services and although many people argue that money can t buy things like happiness or freedom, money is the driving force behind the lives of many people, even in communist and third world countries. The aim of making (or earning) money has been around for centuries. For most people this has been through making goods or providing services, but people have always been keen to make money through investing. The stock markets of the world have given big business, organisations, governments and individuals the chance to make money without providing goods and services, instead taking the risk that the investment may lose value. People have often thought that only big business and rich individuals have gained from the activities of stock markets but successive governments have introduced measures to increase share ownership. Pension funds also rely heavily on stock market investments and currently more and more people are making their own pension plans. By investing in a company or business venture, the investor is providing funds which may not be available elsewhere for the production of goods or services which can benefit all members of society. In a capitalist economy investing in the stock market not only provides the possibility of personal gain but is part of a process that effects everybody in that economy, helping to provide jobs, research and development, and opportunities for all. This chapter introduces investment management and the practices by which investment professionals work. Future forecasts of financial variables are an essential part of their day to day business and are key to generating excess profit relative to rival investors. Financial forecasting is not an isolated problem. Forecasting is required of many time series which show similar properties and behaviour to financial markets. The area of nonlinear dynamics investigates the underlying process behind apparently random time series from 15

16 CHAPTER 1. INTRODUCTION AND PROBLEM DEFINITION 16 many diverse fields of science including electricity demand forecasting, physics laboratory experiments and artificial data. 1.1 Investment Management The opening paragraphs attempted to justify, simply and without political or economic depth, investing on the stock market. Investment can be a complicated process and must be described in detail before financial forecasting can be defined. In dealing with investment, the discussion focuses on investment by big business and investment banks. Investment banks deal in large sums of money and are therefore able to reduce commission and transaction charges and have the finances for top of the range information technology to assist their traders/fund managers. These areas can all cause problems to individual investors and reduce their trading options. The behaviour of private investors is also less well documented or analysed, so research concentrated on investment banks may not be of any value to an ordinary investor. Investment management is the process of money management. It normally consists of several stages, beginning with a definition of objectives through to an evaluation of the performance of the investment. Probably the most important element of this process is the selection of financial instruments (or assets) to invest in. This is the stage that this thesis is concerned with Asset Allocation As just stated, Asset Allocation is the selection of assets to invest in from all potential assets. The number of potential assets is assumed to be very large. In practice the most important part involves measuring the merits of these assets. These measurements are then used in the decision about which assets to invest in. Investors (and investment organisations) make different measurements and calculations but common to all are return and risk. Return The return on a security over a time period is the profit (or loss) gained by holding the security, usually expressed as the percentage change in price. Given the prices at time t and t 1, and assuming no interim payment (e.g. dividend) the return at time t is r t = p t p t 1 p t (1.1) Alternatively, the following formula may also be used r t = ln[p t /p t 1 ] 100 (1.2)

17 CHAPTER 1. INTRODUCTION AND PROBLEM DEFINITION 17 Risk Risk is a measure of how likely future returns are to deviate strongly from their expected value. Traditionally this has been measured as the variance (or standard deviation) of future return(s), known as Volatility. More recently practitioners have used many different measures of risk, based on extensions to Volatility or the use of probabilities. The most widely used of these alternatives is Value at Risk (VaR) (Jorion 1997), a measure which describes the tail of the probability distribution and will be discussed later in this chapter. Different investors and banks have used other forms of risk. There is no general consensus on which should be used, although some make assumptions about the distribution of returns which will be discussed later Forecasting A key part of Asset Allocation is the measurement (or prediction) of return and/or risk. In both cases this is a forecast of a future value of a time-series, although previous values of the time-series may not have been used to construct the forecast. Many forecasting techniques have been researched, tested and used by academics, banks and individuals. Before they can be examined it is useful to have an insight into the economic theory which has led to many of these techniques being developed or discarded. The most common models used are discussed in the next section. Forecasting methods are also determined by the time period over which a forecast is required. Traditionally this varies from daily forecasts to monthly or yearly forecasts and even several years for some financial variables. As the financial markets have developed and incorporated improvements in information technology into their daily business, the need for (and ability to use) intra-day or even tick 1 price forecasts has appeared and grown rapidly. This area of high frequency forecasting requires speed not possible with many traditional forecasting methods, but makes available much larger data sets than previously could be used. This has led to new approaches to forecasting in an already continuously adapting and improving field. 1.2 Financial Theory Efficient Market Hypothesis The Efficient Market Hypothesis (EMH) was the most widely understood and used model for stock market pricing. Although people have debated its validity since its conception, it is 1 A tick refers to a change in a stock s price from one trade to the next. Forecasting tick price data implies producing a forecast of the next price, which is usually within a few minutes of the current price and can be separated by as little as a few seconds.

18 CHAPTER 1. INTRODUCTION AND PROBLEM DEFINITION 18 only in the last decade that it has lost general acceptance. Combined with Modern Portfolio Theory (discussed later), it is the basis for many pricing theories including the Capital Asset Pricing Model (CAPM) (Sharpe 1964) and Arbitrage Pricing Theory (APT) (Ross 1976). An efficient market is one where: All information is freely available All investors know how to interpret available information If a market is efficient, from these assumptions several conclusions can be made Securities are priced to take into account all public information. They only move when new information is received and therefore cannot be gamed Yesterday s news doesn t affect today s prices. Returns are independent and the system has no memory Because returns are independent they follow a random walk and therefore have a normal distribution This is the random walk version of the EMH. The assumption that all investors know how to interpret available information is crucial and introduced the idea of a rational investor. One of the first definitions of such an investor (Osborne 1964) stated that investors valued stocks based on their expected value (or return) only and that they behaved in a rational and unbiased manner. The version of the EMH described above is the strong form which simply stated that markets could not be gamed, excess profit could not be gained by analysis of public information. A semistrong version of the EMH was developed in which prices reflect all public information. Security analysts produce a large number of independent estimates, which produce a fair aggregate value on the market. This suggested the market is efficient because of security analysis, and price changes are random because of changes in both micro- and macro-economics. By the mid-1970s, the semistrong version of EMH had been accepted by both the academic and investment communities Modern Portfolio Theory Modern Portfolio Theory (MPT) was developed around the same time as the EMH by Markowitz and most of its theories, conclusions and simplifications rely on the EMH holding true. MPT first introduced the concept of using variance to measure risk and it is this measure of risk that MPT refers to. For this measure of risk to be used, returns must be normally distributed so that the variance is finite.

19 CHAPTER 1. INTRODUCTION AND PROBLEM DEFINITION 19 The EMH defined the idea of a rational investor. MPT defines this more precisely using risk. It states that investors desire the portfolio with the highest expected return for a given level of risk, i.e. they are risk averse. This approach is known as mean/variance efficiency. This definition for investor behaviour brought about the Sharpe Ratio (SR) (Sharpe 1994) which is effectively another risk measure. Weigend 1997) SR = This definition of SR is taken from (Choey & average excess return standard deviation of excess return (1.3) CAPM The Capital Asset Pricing Model builds upon this definition of a rational investor and assumes an efficient market to build a model of investor behaviour. It defines the optimal portfolio for all investors and explains how they will behave if they are rational. It also explains why diversification reduces risk, the idea that a portfolio of two risky securities can have a lower risk than either of the two securities. This in theory allowed investors to build a portfolio of high-risk high-return assets where the overall risk for the portfolio would be low. CAPM has been well researched and there is no need for it to be investigated or explained here. However, it makes interesting assumptions and simplifications which should be noted. The market is efficient, therefore the conclusions of the EMH are valid There are no transaction costs, commissions or taxes Everyone can borrow and lend at a risk-free rate of interest, interpreted in the US as the 90-day T-Bill rate All investors desire mean/variance efficiency The validity of these statements, especially the EMH will now be discussed Is the EMH a reliable model? For a long time EMH was universally accepted despite there being no proof other than that traders were finding it difficult to beat the market. One condition suggested by EMH which could be tested is that of a normal distribution of returns. One of the first recognised studies was by Osborne (Osborne 1964), a believer in EMH. He could only describe returns as approximately normal, noticing extra observations in the tails of the distribution. Many people accepted this but felt that it didn t affect the validity of the EMH. Mandelbrot (Mandelbrot 1964) concluded they had a stable paretian distribution. Fama (Fama 1965) then finished a more complete study and suggested that returns are characterised by narrower peaks and being negatively skewed, a condition known as leptokurtosis.

20 CHAPTER 1. INTRODUCTION AND PROBLEM DEFINITION 20 More recently, a complete study (Turner & Wiegel 1990) of prices from 1928 to 1990 reported similar results as Fama. Many recent papers since have moved away from a normal distribution, including (Sterge 1989) making the observation that very large (three or more standard deviations from the norm) price changes can be expected to occur two to three times as often as predicted by normality. The Efficient Market Hypothesis also suggests that returns are independent, there is no memory effect. Research by (Peters 1991) contradicts this, finding that leading world indexes such as the S&P500 have a long term memory which is nonperiodic but averages roughly 4 years. The EMH was developed to explain and allow the use of simple linear methods and standard statistical analysis. Even 10 years ago it was being taught as fact in MBA courses, despite many people objecting to it at this time and the build up of evidence against it. It is certainly bad practice to develop a model to fit existing methods especially when the model becomes so difficult to prove. It is best summed up by (Refenes, Burgess & Bentz 1997) who state It is remarkable that the EMH should have gained such empirical support based upon a testing methodology that started by assuming it is true, and then adopted tests which would rarely have the power to refute it! Experimentation on supplied data It is easy to accept the findings reported by other researchers on the distribution of financial returns. Historically, examining the distribution of financial data would be a problematic task but today the availability of large data sets (and the power to process them) makes building up a distribution from historical data a simple task. It is not difficult to check that the results reported in the literature are repeated in the data used in this thesis. To do this an experiment was carried out on a high frequency set of exchange rates between the Japanese Yen and US Dollar. Further details on this data set appear in chapter 6, here concentration is only on the distribution of the returns. It is a large data set, containing over half a million points. The results are presented as a discrete distribution (histogram) in 1.1, with a logarithmic scale used to display all areas of interest in the same graph. The results of this test are in agreement with most previous work in investigating returns. Too many observations fall at the tails of the distribution. The knowledge that the distribution does not follow a normal curve is useful in choosing methods to construct distributions. This is discussed in further detail in Chapter Risk Management Closely linked to investment management is the increasingly important area of risk management, which attempts to measure the likelihood and size of the potential losses an investment could return. The aim is to provide an easy to understand method of risk measurement that

21 CHAPTER 1. INTRODUCTION AND PROBLEM DEFINITION 21 Discrete distribution of return Frequency Return (%) Figure 1.1: Discrete distribution of high frequency returns for Japanese Yen and US Dollar, from over half a million points of data. The returns do not follow a normal distribution, although the logarithmic scale used exaggerates the non-normal behaviour at the tails. can be used by traders and management. Effective risk management can allow investment banks to set limits on the risks taken by all areas of the business and to take into account risk when rewarding traders. Such information could also be made generally available to investors. A recent addition to modern risk management is the Value at Risk measure, which many banks now use internally and/or externally to communicate the level of risk a portfolio or fund is taking VaR Forecasts for individual assets Philippe Jorion has written an excellent introduction to Value at Risk (VaR) (Jorion 1997). VaR is measured for a particular confidence level and a particular length of time and is intended to report the largest loss deemed likely. A portfolio could be described as having a VaR of 20m at a 95% confidence level. This would imply that there was only a 5% chance of the portfolio making a loss greater than 20m over the given time frame. VaR is calculated from the tail of the probability distribution which can be considered discrete or continuous. Two main techniques to calculate the distribution exist, using the actual empirical distribution (nonparametric) or approximating the distribution with a parametric model. Further comments on parametric and nonparametric methods will be made later in this work. In practice the simplest model to use is that of a normal distribution. It is believed that the normal distribution is widely used despite the evidence that financial returns do not follow such behaviour. The VaR measure is dependent upon the tail of the distribution, which is exactly where returns are thought to deviate strongly from a normal distribution. In particular, the results of the experiment reported in figure 1.1 suggest that

22 CHAPTER 1. INTRODUCTION AND PROBLEM DEFINITION 22 using the normal distribution to calculate VaR on that data set would be flawed. Jorion also argues the importance of good risk management (and therefore measurement) in general and its help in regulating traders. He suggests that incidents such as the collapse of Barings Bank and the bankruptcy of Orange County, USA 2 only occurred due to an absence of enforced risk management policies and that Regular publication of a VaR number would have effectively communicated market risks to nontechnical investors. Improved risk measures could give individual traders/investors an advantage over others to gain excess return or reduce risks taken over other portfolios. Accurate VaR values however, may give a more industry wide benefit in avoiding financial disasters. To this end J. P. Morgan made its RiskMetrics system available free on the internet. RiskMetrics collects large volumes of data suitable for VaR calculations. Even without such help, the cost of making VaR forecasts is well within the reach of most investment banks and other trading institutions. The total risk on a security (or portfolio of securities) is often considered to consist of many individual types of risk. For example, interest rate risk is the risk associated with adverse movements in interest rates and credit risk the potential risk associated with debtors defaulting on payments. There are many risk measures which just measure one particular type of risk. For example, the duration measure used in finance is a calculation of interest rate risk. The overall risk is calculated by combining the individual risk components. This requires several risk models and it is difficult to ensure all possible risk elements have been accounted for. VaR is more general than a single risk measure, incorporating all risks on an asset and is suitable for all types of asset. This is partly why VaR has become the most commonly used risk measure, supported in the USA by rating agencies such as Moody s and Standard and Poor s, the Financial Accounting Standards Board, and the Securities and Exchange Commission. 1.3 Distribution Forecasting Most literature concerns the generation of point forecasts, although probability forecasts (Clemen, Murphy & Winkler 1995) and interval forecasts (Ait-Sahalia & Lo 1998) are widely discussed. A probability forecast aims to measure the likelihood that a future observation will fall above (or below) a certain value. An interval forecast aims to forecast an upper and lower bound which a future observation will fall inbetween with a specified confidence, typically 90% or 95%. An alternative method of forecasting return and/or risk is to forecast the full probability distribution of the return or profit and loss (P&L). From this distribution all measures of risk can be calculated, as well as a forecast for the return, if required. It is worth noting that a VaR forecast effectively requires a forecast of part (the tail) of the distribution. If a distribution forecast is accurate it is easy to take from it an accurate VaR value. 2 The Orange County investment pool, a portfolio of $7.5 billion belonging to municipal investors (including the county, cities and schools), lost $1.64 billion in 1994.

23 CHAPTER 1. INTRODUCTION AND PROBLEM DEFINITION 23 Distribution forecasts have only recently been considered in the literature and are still only a small (but growing) area of research. The main reason for this is that distributions weren t demanded by practice, but this has changed recently with the growth in risk management, the new ideas of underlying market dynamics, and new trading and pricing models that depend on good density estimates. The growth of VaR as a risk measure in particular has required investigation into distributions. Other reasons including uncertainty in how to generate a distribution and how to evaluate a distribution have restricted research in the area. Many methods also require computational power that previously was unavailable or expensive. The power of modern computers has allowed work with larger data sets and methods which require complex calculations. Work by Breckling, Eberlein and Kokic (Eberlein, Breckling & Kokic 2000) highlights the advantages of distributions and their ability to allow calculation of all measures of risk. Weigend and Shi (Weigend & Shi 2000) also generate distribution forecasts. However the field of distribution forecasts is growing all the time as risk management advances. Diebold (Diebold, Gunther & Tay 1998) report that The booming area of financial risk management is effectively dedicated to providing density forecasts of portfolio values. How useful a distribution forecast is and how it should be used is related to an individuals loss function 3. This suggests that a density forecast must be tailored to particular requirements depending on how it is to be used, but fortunately this is not the case. (Diebold et al. 1998) provides a proof that Regardless of loss function, we know that the correct density is weakly superior to all forecasts. The fact that distributions are only now being researched is probably due to complexity in calculating the distribution and the lack of standard methods to evaluate it. These are areas that will be looked at during this research. It is important to note that a major implication of rejecting the EMH is that the distribution may change over time and if the system is chaotic this change is almost certain. The distribution must be re-forecast over time in the same way as most algorithms continuously update point forecasts at each time step. 1.4 Aims and Thesis Overview It is now accepted by most academics and professionals that the EMH doesn t hold, because the assumptions it makes are not correct. There is no basis on which to assume returns are independent and have a normal distribution or that investors all behave in the same manner acting upon the same information. Most forecasting techniques are based upon some or all of these conditions. As I will demonstrate in the next chapter it is only the more recent research which has moved away from the EMH. Why the EMH should have failed to explain the behaviour of stock markets is not part of 3 The loss function contains a value for all possible combinations of forecasts and observed values. Each value is the cost (or benefit) to the investor of acting upon a particular forecast when a particular observation is subsequently made.

24 CHAPTER 1. INTRODUCTION AND PROBLEM DEFINITION 24 this report, but to add weight to decisions I make later, I will add my thoughts on the EMH. It is unrealistic to expect all investors to behave in the same manner. In comparison with private investors, fund managers and other professionals have a much greater array of information available to them, from analysts, other contacts in the industry and better access to top of the range computer technology. The aims, especially in terms of time-scale vary greatly between different corporations and individuals. The EMH relies on all investors being rational, which is hardly human nature. These are all contradictions of assumptions required for the EMH and most pricing models. Many other researchers have questioned the assumptions and simplifications of the EMH. These doubters often only offer personal opinion rather than hard fact, but are often convincing. Peters (Peters 1991) provides conclusive evidence against the EMH, having analysed various capital markets. He finds fractal structure and nonperiodic cycles, suggesting prices are chaotic in nature. This is the only proposed alternative to the EMH and has become a large area of research. If stock prices are generated by a deterministic nonlinear system, this requires a completely different approach to forecasting. The CAPM does however provide a good framework for examining the risk of a portfolio based upon the risk of the individual assets it contains, based upon measurements of the covariance between different securities. This area of the CAPM requires fewer assumptions than that of a rational investor, and is particularly useful. The knowledge that methods exist to construct an optimal portfolio based upon several asset forecasts and their expected correlations allows a concentration on forecasting individual time series at the beginning of this research Aims The difference between high frequency forecasting and traditional financial forecasting is to be investigated and its impact upon potential algorithms and models determined. From this, the initial aim of this work is to design and implement a forecasting algorithm and architecture that: Is based upon modern understanding of financial markets Incorporates existing techniques where sound Produces probability distribution forecasts rather than simple point forecasts Can be applied to high frequency data Can be applied to other nonlinear problems Produces forecasts of sufficient accuracy that they would be of interest to an investment organisation or could be profitably traded upon under real market conditions After an algorithm meeting the first four of these requirements has been developed it will be tested on suitable financial data and other nonlinear problems so that success or failure

25 CHAPTER 1. INTRODUCTION AND PROBLEM DEFINITION 25 relative to the final aims can be reported. appropriately measure the accuracy of any forecasts produced. Therefore it is a further aim of this work to Thesis Overview In Chapter 2 the different approaches to financial forecasting are introduced and the theory behind their design explained. In particular weaknesses of existing methods are identified. In Chapter 3 AURA networks (a form of associative memory) are introduced in detail. In Chapter 4 the design of a new algorithm (AURA-FS) is described. The role of AURA networks within AURA-FS is clearly defined. Design issues and practical considerations for using AURA-FS are discussed. Chapters 5 and 6 detail the extensive range of experiments run and contain all the results collected. These chapters also contain evaluation of the results and many individual conclusions. Chapter 7 introduces a modification to AURA-FS and contains details of the experiments run to test the new algorithm, results found and conclusions drawn. Chapter 8 gives overall conclusions and briefly suggests future directions for the work. Another thread of this work investigated electricity demand forecasting. This is a separate piece of work which does not fit neatly within the incremental advancements reported in Chapters 5, 6 and 7 but is included as Appendix A. Appendix B supplies definitions for some of the statistical tests used throughout the thesis.

26 Chapter 2 Field Survey and Review I have discussed the aims of financial forecasting and the theory which has influenced the design and implementation of forecasting techniques. Now individual techniques are explained and evaluated. Notation Standard notation for time series is to use an indexed variable y. If N observations have been made, they are represented y 1, y 2,..., y N. ŷ t is a forecast for the series y at time t, and unless otherwise stated it is assumed that values y 1, y 2,..., y t 1 have been observed and used to make the forecast. A forecast ŷ t made after observing y t 1 is a next-step forecast. This thesis follows this convention for describing time series unless otherwise stated. 2.1 Fundamental Analysis Fundamental analysis is the use of economic theory to build a model to explain how the data (usually in this case, prices, return or risk) is generated. The models are the result of analysis of a wide range of interactions between different elements of the market and macroeconomic trends. Econometric modelling is the only common and widely understood technique for financial forecasting which falls into the category of fundamental analysis. Econometric analysis is the use of economic theory to build models which rely on measurements from individual companies and/or global economies. These models quickly become very complex, and to produce more accurate forecasts have to become more complex to model well the dynamics of the system being modelled. They usually assume linear processes and a normal distribution of returns, concepts already questioned in this report. To remove these assumptions or extend models will only further complicate the final models. This makes econometric modelling difficult for high frequency forecasting. However, econometric models have been known to outperform technical analysis and other 26

27 CHAPTER 2. FIELD SURVEY AND REVIEW 27 methods for long term forecasts (Howrey, Klein & McCarthy 1974, Armstrong 1978). The emphasis of this research is on high frequency forecasting, and it was felt that little could be learned by an investigation into fundamental analysis. 2.2 Technical Analysis Strictly technical analysis involves only using historical information of the market under consideration and not assuming any relationships between individual elements of the markets. This definition does get stretched, especially with the introduction of neural networks, but there is still a big divide between Fundamental and Technical Analysis. Many different technical analysis techniques for financial forecasting have been investigated and only a selection will be covered in this section. It is common for an investor (or analyst) to do some preliminary investigation into the data set(s) to decide which method would be most appropriate. That analysts tend to use several techniques suggest that no method is an ideal solution that can be applied to all types of financial asset Averaging Models Averaging models produce a forecast by smoothing historical data. These models can be very simple, sometimes trivially so, and are usually used for long term forecasts. The simplest method of all is the mean value calculation. ȳ = 1 T y i (2.1) t The next-step forecast can be taken to be the mean value. ŷ t+1 = ȳ (2.2) This simple model assumes that the time series is stationary 1 and without seasonal or cyclical components. It is also unable to predict short term fluctuations, but if the assumptions hold provides a simple long term forecast. A slight extension to this is the simple Moving Average, where only the last k observations are used to construct the mean value, MA(k). ŷ t+1 = 1 t y i (2.3) k i=t k+1 The value k determines how responsive the forecasts are. The larger the value of k, the less responsive the forecast to recent changes in the series. Correct selection of the value k can also smooth out seasonal effects. In the case where daily data is being used MA(7) could 1 For the purpose of this thesis a series is considered stationary if the generating distribution (including its location) doesn t change over time.