Online portfolio selection: A survey
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paper_title: Capital Growth Theory

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The following sections are included:IntroductionOrigins of the modelThe model and its basic propertiesConditions for capital growthRelationship to other long-run investment modelsRelationship to intertemporal consumption-investment modelsGrowth vs. securityApplicationsSummaryReferences
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paper_title: Portfolio Selection Efficient Diversification Of Investments

paper_content:

Thank you for reading portfolio selection efficient diversification of investments. As you may know, people have look numerous times for their favorite novels like this portfolio selection efficient diversification of investments, but end up in infectious downloads. Rather than reading a good book with a cup of tea in the afternoon, instead they cope with some malicious bugs inside their desktop computer.
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paper_title: Mean-Variance Analysis in Portfolio Choice and Capital Markets

paper_content:

The general portfolio selection model preliminary results solution to the general portfolio selection model special cases a portfolio selection programme.
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paper_title: OPTIMAL INVESTMENT AND CONSUMPTION STRATEGIES UNDER RISK FOR A CLASS OF UTILITY FUNCTIONS

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This paper develops a sequential model of the individual's economic decision problem under risk. On the basis of this model, optimal consumption, investment, and borrowing-lending strategies are obtained in closed form for a class of utility functions. For a subset of this class the optimal consumption strategy satisfies the permanent income hypothesis precisely. The optimal investment strategies have the property that the optimal mix of risky investments is independent of wealth, noncapital income, age, and impatience to consume. Necessary and sufficient conditions for long-run capital growth are also given.
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paper_title: Optimal Strategies for Repeated Games

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We extend the optimal strategy results of Kelly and Breiman and extend the class of random variables to which they apply from discrete to arbitrary random variables with expectations. Let Fn be the fortune obtained at the nth time period by using any given strategy and let $F_{n}^{*}$ be the fortune obtained by using the Kelly–Breiman strategy. We show (Theorem 1(i)) that $F_{n} / F_{n}^{*}$ is a supermartingale with $E(F_{n} / F_{n}^{*}) \leqq 1$ and, consequently, $E(\lim {F_{n} / F_{n}^{*}) \leqq 1$. This establishes one sense in which the Kelly–Breiman strategy is optimal. However, this criterion for ‘optimality’ is blunted by our result (Theorem 1(ii)) that $E(F_{n} / F_{n}^{*}) = 1$ for many strategies differing from the Kelly–Breiman strategy. This ambiguity is resolved, to some extent, by our result (Theorem 2) that $F_{n}^{*} / F_{n}$ is a submartingale with $E(F_{n}^{*} / F_{n}) \geqq 1$ and $E(\lim {F_{n}^{*}} / F_{n}) \geqq 1$; and $E(F_{n}^{*} / F_{n}) = 1$ if and only if at each time period j, 1≦j≦n, the strategies leading to Fn and $F_{n}^{*}$ are ‘the same’.
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paper_title: Portfolio Choice And The Kelly Criterion

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The following sections are included:IntroductionSamuelson's objections to logarithmic utilityAn outline of the theory of logarithmic utility as applied to portfolio selectionRelation to the Markowitz theory; solution to problems thereinThe theory in action: results for a real institutional portfolioConcluding remarksFOOTNOTESREFERENCES
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paper_title: On the Competitive Theory and Practice of Portfolio Selection (Extended Abstract)

paper_content:

Given a set of say m stocks (one of which may be “cash”), the online portfolio selection problem is to determine a portfolio for the i th trading period based on the sequence of prices for the preceding i – 1 trading periods. Competitive analysis is based on a worst case perspective and such a perspective is inconsistent with the more widely accepted analyses and theories based on distributional assumptions. The competitive framework does (perhaps surprisingly) permit non trivial upper bounds on relative performance against CBAL-OPT, an optimal offline constant rebalancing portfolio. Perhaps more impressive are some preliminary experimental results showing that certain algorithms that enjoy “respectable” competitive (i.e. worst case) performance also seem to perform quite well on historical sequences of data. These algorithms and the emerging competitive theory are directly related to studies in information theory and computational learning theory and indeed some of these algorithms have been pioneered within the information theory and computational learning communities. We present a mixture of both theoretical and experimental results, including a more detalied study of the performance of existing and new algorithms with respect to a standard sequence of historical data cited in many studies. We also present experiments from two other historical data sequences.
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paper_title: Prediction in financial markets: The case for small disjuncts

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Predictive models in regression and classification problems typically have a single model that covers most, if not all, cases in the data. At the opposite end of the spectrum is a collection of models, each of which covers a very small subset of the decision space. These are referred to as “small disjuncts.” The trade-offs between the two types of models have been well documented. Single models, especially linear ones, are easy to interpret and explain. In contrast, small disjuncts do not provides as clean or as simple an interpretation of the data, and have been shown by several researchers to be responsible for a disproportionately large number of errors when applied to out-of-sample data. This research provides a counterpoint, demonstrating that a portfolio of “simple” small disjuncts provides a credible model for financial market prediction, a problem with a high degree of noise. A related novel contribution of this article is a simple method for measuring the “yield” of a learning system, which is the percentage of in-sample performance that the learned model can be expected to realize on out-of-sample data. Curiously, such a measure is missing from the literature on regression learning algorithms. Pragmatically, the results suggest that for problems characterized by a high degree of noise and lack of a stable knowledge base it makes sense to reconstruct the portfolio of small rules periodically.
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paper_title: The Encyclopedia of Trading Strategies

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Preface. Part One: Tools of the Trade. Chapter 1: Data. Chapter 2: Simulators. Chapter 3: Optimizers and Optimazation. Chapter 4: Statistics. Part Two: The Study of Entries. Chapter 5: Breakout Models. Chapter 6: Moving-Average Models. Chapter 7: Oscillator-Based Entries. Chapter 8: Seasonality. Chapter 9: Lunar and Solar Rhythms. Chapter 10: Cycle-Based Entries. Chapter 11: Neural Networks. Chapter 12: Genetic Algorithms. Part Three: The Study of Exits. Chapter 13: The Standard Exit Strategy. Chapter 14: Improvements on the Standard Exit. Chapter 15: Adding Artificial Intelligence to Exits. Conclusion. Notice. Appendix. Index.
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paper_title: INVESTMENT POLICIES FOR EXPANDING BUSINESSES OPTIMAL IN A LONG-RUN SENSE

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Publisher Summary ::: Entrepreneurs have a given initial fortune and faces the following situation: during any time period, they may invest various amounts of their available fortune in various alternatives differing from interval to interval and keep the remainder of their money as a reserve fund. The alternatives materialize and pay off according to a probability distribution in which occurrences in different time intervals are independent. A repeated gambling of all available funds on this alternative will, with probability one, lead to ultimate ruin. In this situation, the right thing to do during every investment period is to hold part of one's capital in reserve and invest the remainder in the alternative with the maximum expected return. The obvious objection to this is that the investor is no longer following his optimum policy; besides, question arise regarding the criterion being used to decide what funds to hold in reserve. The chapter explores the problem of finding the investment policies, that is, the division of funds during every time interval between the investment alternatives and the reserve that will make the long-run growth of the entrepreneur's fortune as rapid as possible.
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paper_title: Financial time series forecasting using independent component analysis and support vector regression

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As financial time series are inherently noisy and non-stationary, it is regarded as one of the most challenging applications of time series forecasting. Due to the advantages of generalization capability in obtaining a unique solution, support vector regression (SVR) has also been successfully applied in financial time series forecasting. In the modeling of financial time series using SVR, one of the key problems is the inherent high noise. Thus, detecting and removing the noise are important but difficult tasks when building an SVR forecasting model. To alleviate the influence of noise, a two-stage modeling approach using independent component analysis (ICA) and support vector regression is proposed in financial time series forecasting. ICA is a novel statistical signal processing technique that was originally proposed to find the latent source signals from observed mixture signals without having any prior knowledge of the mixing mechanism. The proposed approach first uses ICA to the forecasting variables for generating the independent components (ICs). After identifying and removing the ICs containing the noise, the rest of the ICs are then used to reconstruct the forecasting variables which contain less noise and served as the input variables of the SVR forecasting model. In order to evaluate the performance of the proposed approach, the Nikkei 225 opening index and TAIEX closing index are used as illustrative examples. Experimental results show that the proposed model outperforms the SVR model with non-filtered forecasting variables and a random walk model.
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paper_title: The Symmetric Downside-Risk Sharpe Ratio and the Evaluation of Great Investors and Speculators

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The Sharpe ratio is a very useful measure of investment performance. However, it is based on mean-variance theory and thus is basically valid only for quadratic preferences or normal distributions. Hence skewed investment returns can lead to misleading conclusions. This is especially true for superior investors such as Warren Buffett and others with a large number of high returns. Many of these superior investors use capital growth wagering ideas to implement their strategies which leads to higher growth rates but also higher variability of wealth. A simple modification of the Sharpe ratio to assume that the upside deviation is identical to the downside risk provides a useful modification that gives more realistic results.
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paper_title: Portfolio Selection Efficient Diversification Of Investments

paper_content:

Thank you for reading portfolio selection efficient diversification of investments. As you may know, people have look numerous times for their favorite novels like this portfolio selection efficient diversification of investments, but end up in infectious downloads. Rather than reading a good book with a cup of tea in the afternoon, instead they cope with some malicious bugs inside their desktop computer.
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paper_title: Support vector machine with adaptive parameters in financial time series forecasting

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A novel type of learning machine called support vector machine (SVM) has been receiving increasing interest in areas ranging from its original application in pattern recognition to other applications such as regression estimation due to its remarkable generalization performance. This paper deals with the application of SVM in financial time series forecasting. The feasibility of applying SVM in financial forecasting is first examined by comparing it with the multilayer back-propagation (BP) neural network and the regularized radial basis function (RBF) neural network. The variability in performance of SVM with respect to the free parameters is investigated experimentally. Adaptive parameters are then proposed by incorporating the nonstationarity of financial time series into SVM. Five real futures contracts collated from the Chicago Mercantile Market are used as the data sets. The simulation shows that among the three methods, SVM outperforms the BP neural network in financial forecasting, and there are comparable generalization performance between SVM and the regularized RBF neural network. Furthermore, the free parameters of SVM have a great effect on the generalization performance. SVM with adaptive parameters can both achieve higher generalization performance and use fewer support vectors than the standard SVM in financial forecasting.
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paper_title: A learning-based contrarian trading strategy via a dual-classifier model

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Behavioral finance is a relatively new and developing research field which adopts cognitive psychology and emotional bias to explain the inefficient market phenomenon and some irrational trading decisions. Unlike the experts in this field who tried to reason the price anomaly and applied empirical evidence in many different financial markets, we employ the advanced binary classification algorithms, such as AdaBoost and support vector machines, to precisely model the overreaction and strengthen the portfolio compositions of the contrarian trading strategies. The novelty of this article is to discover the financial time-series patterns through a high-dimensional and nonlinear model which is constructed by integrated knowledge of finance and machine learning techniques. We propose a dual-classifier learning framework to select candidate stocks from the past results of original contrarian trading strategies based on the defined learning targets. Three different feature extraction methods, including wavelet transformation, historical return distribution, and various technical indicators, are employed to represent these learning samples in a 381-dimensional financial time-series feature space. Finally, we construct the classifier models with four different learning kernels and prove that the proposed methods could improve the returns dramatically, such as the 3-year return that improved from 26.79p to 53.75p. The experiments also demonstrate significantly higher portfolio selection accuracy, improved from 57.47p to 66.41p, than the original contrarian trading strategy. To sum up, all these experiments show that the proposed method could be extended to an effective trading system in the historical stock prices of the leading U.S. companies of S&P 100 index.
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paper_title: Neuro-genetic system for stock index prediction

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This goal of the paper is introduction and experimental evaluation of neuro-genetic system for short-term stock index prediction. The system works according to the following scheme: first, a pool of input variables are defined through technical data analysis. Then GA is applied to find an optimal set of input variables for a one day prediction. Due to the high volatility of mutual relations between input variables, a particular set of inputs found by the GA is valid only for a short period of time and a new set of inputs is calculated every 5 trading days. The data is gathered from the German Stock Exchange (being the target market) and two other markets (Tokyo Stock Exchange and New York Stock Exchange) together with EUR/USD and USD/JPY exchange rates. The method of selecting input variables works efficiently. Variables which are no longer useful are exchanged with the new ones. On the other hand some, particularly useful, variables are consequently utilized by the GA in subsequent independent prediction periods. The proposed system works well in cases of both upward or downward trends. The effectiveness of the system is compared with the results of four other models of stock market trading.
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paper_title: Growth Versus Security In Dynamic Investment Analysis

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AbstractThis paper concerns the problem of optimal dynamic choice in discrete time for an investor. In each period the investor is faced with one or more risky investments. The maximization of the expected logarithm of the period by period wealth, referred to as the Kelly criterion, is a very desirable investment procedure. It has many attractive properties, such as maximizing the asymptotic rate of growth of the investor's fortune. On the other hand, instead of focusing on maximal growth, one can develop strategies based on maximum security. For example, one can minimize the ruin probability subject to making a positive return or compute a confidence level of increasing the investor's initial fortune to a given final wealth goal. This paper is concerned with methods to combine these two approaches. We derive computational formulas for a variety of growth and security measures. Utilizing fractional Kelly strategies, we can develop a complete tradeoff of growth versus security. The theory is applicable to favorable investment situations such as blackjack, horseracing, lotto games, index and commodity futures and options trading. The results provide insight into how one should properly invest in these situations.
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paper_title: Buy low, sell high

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We consider online trading in a single security with the objective of getting rich when its price ever exhibits a large upcrossing without risking bankruptcy. We investigate payoff guarantees that are expressed in terms of the extremity of the upcrossings. We obtain an exact and elegant characterisation of the guarantees that can be achieved. Moreover, we derive a simple canonical strategy for each attainable guarantee.
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paper_title: Computational learning techniques for intraday FX trading using popular technical indicators

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We consider strategies which use a collection of popular technical indicators as input and seek a profitable trading rule defined in terms of them. We consider two popular computational learning approaches, reinforcement learning and genetic programming, and compare them to a pair of simpler methods: the exact solution of an appropriate Markov decision problem, and a simple heuristic. We find that although all methods are able to generate significant in-sample and out-of-sample profits when transaction costs are zero, the genetic algorithm approach is superior for non-zero transaction costs, although none of the methods produce significant profits at realistic transaction costs. We also find that there is a substantial danger of overfitting if in-sample learning is not constrained.
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paper_title: Efficiency of the Market for Racetrack Betting

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AbstractMany racetrack bettors have systems. Since the track is a market similar in many ways to the stock market one would expect that the basic strategies would be either fundamental or technical in nature. Fundamental strategies utilize past data available from racing forms, special sources, etc. to “handicap” races. The investor then wagers on one or more horses whose probability of winning exceeds that determined by the odds by an amount sufficient to overcome the track take. Technical systems require less information and only utilize current betting data. They attempt to find inefficiencies in the “market” and bet on such “overlays” when they have positive expected value. Previous studies and our data confirm that for win bets these inefficiencies, which exist for underbet favorites and overbet longshots, are not sufficiently great to result in positive profits. This paper describes a technical system for place and show betting for which it appears to be possible to make substantial positive profits and thus to demonstrate market inefficiency in a weak form sense. Estimated theoretical probabilities of all possible finishes are compared with the actual amounts bet to determine profitable betting situations. Since the amount bet influences the odds and theory suggests that to maximize long run growth a logarithmic utility function is appropriate the resulting model is a nonlinear program. Side calculations generally reduce the number of possible bets in any one race to three or less hence the actual optimization is quite simple. The system was tested on data from Santa Anita and Exhibition Park using exact and approximate solutions (that make the system operational at the track given the limited time available for placing bets) and found to produce substantial positive profits. A model is developed to demonstrate that the profits are not due to chance but rather to proper identification of market inefficiencies.
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paper_title: Model Calibration and Automated Trading Agent for Euro Futures

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We explored the application of a machine learning method, Logitboost, to automatically calibrate a trading model using different versions of the same technical analysis indicators. This approach takes advantage of boosting's feature selection capability to select an optimal combination of technical indicators and design a new set of trading rules. We tested this approach with high-frequency data of the Dow Jones EURO STOXX 50 Index Futures (FESX) and the DAX Futures (FDAX) for March 2009. Our method was implemented with different learning algorithms and outperformed a combination of the same group of technical analysis indicators using the parameters typically recommended by practitioners. We incorporated this method of model calibration in a trading agent that relies on a layered structure consisting of the machine learning algorithm described above, an online learning utility, a trading strategy, and a risk management overlay. The online learning layer combines the output of several experts and suggests a short or long position. If the expected position is positive (negative), the trading agent sends a buy (sell) limit order at prices slightly lower (higher) than the bid price at the top of the buy (sell) order book less (plus) transaction costs. If the order is not 100% filled within a fixed period (i.e. 1 minute) of being issued, the existent limit orders are cancelled, and limit orders are reissued according to the new experts' forecast. As part of its risk management capability, the trading agent eliminates any weak trading signal. The trading agent algorithm generated positive returns for the two major European index futures (FESX and FDAX) and outperformed a buy-and-hold strategy.
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paper_title: Optimal Gambling Systems For Favorable Games

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The following sections are included:IntroductionThe nature of Λ*The asymptotic time minimization problemAsymptotic magnitude problemProblems with finite goals in coin tossingREFERENCES
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paper_title: Capital Growth Theory and Practice

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Publisher Summary This chapter is a survey of the theoretical results and practical uses of the capital growth approach. It presents the alternative formulations for capital growth models in discrete and continuous time. Various criteria for performance and requirements for feasibility are related in an expected utility framework. Typically, there is a trade-off between growth and security with a fraction invested in an optimal growth portfolio determined by the risk aversion criteria. Models for calculating the optimal fractional Kelly investment with alternative performance criteria are formulated. In capital accumulation under uncertainty, a decision maker must determine how much capital to invest in riskless and risky investment opportunities over time. The investment strategy yields a steam of capital, with investment decisions made so that the dynamic distribution of wealth has desirable properties. The distribution of accumulated capital to a fixed point in time and the distribution of the first passage time to a fixed level of accumulated capital are variables controlled by the investment decisions. An investment strategy which has many attractive and some not attractive properties is the optimal strategy, where the expected logarithm of wealth is maximized. This strategy is also referred to as the Kelly strategy. It maximizes the rate of growth of accumulated capital asymptotically. With the Kelly strategy, the first passage time to arbitrary large wealth targets is minimized, and the probability of reaching those targets is maximized. However, the strategy is very aggressive since the Arrow-Prat risk aversion index is essentially zero. Hence, the changes of losing a substantial portion of wealth are very high, particularly if the estimates of the returns distribution are in error. In the time domain, the chances are high that the first passage to subsistence wealth occurs before achieving the established wealth goals
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paper_title: Competitive solutions for online financial problems

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This article surveys results concerning online algorihtms for solving problems related to the management of money and other assets. In particular, the survey focucus us search, replacement, and portfolio selection problems
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paper_title: Learning to Trade via Direct Reinforcement

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We present methods for optimizing portfolios, asset allocations, and trading systems based on direct reinforcement (DR). In this approach, investment decision-making is viewed as a stochastic control problem, and strategies are discovered directly. We present an adaptive algorithm called recurrent reinforcement learning (RRL) for discovering investment policies. The need to build forecasting models is eliminated, and better trading performance is obtained. The direct reinforcement approach differs from dynamic programming and reinforcement algorithms such as TD-learning and Q-learning, which attempt to estimate a value function for the control problem. We find that the RRL direct reinforcement framework enables a simpler problem representation, avoids Bellman's curse of dimensionality and offers compelling advantages in efficiency. We demonstrate how direct reinforcement can be used to optimize risk-adjusted investment returns (including the differential Sharpe ratio), while accounting for the effects of transaction costs. In extensive simulation work using real financial data, we find that our approach based on RRL produces better trading strategies than systems utilizing Q-learning (a value function method). Real-world applications include an intra-daily currency trader and a monthly asset allocation system for the S&P 500 Stock Index and T-Bills.
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paper_title: Capital Growth and the Mean-Variance Approach to Portfolio Selection

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Three main approaches to the problem of portfolio selection may be discerned in the literature. The first of these is the mean-variance approach, pioneered by Markowitz [21], [22], and Tobin [30]. The second approach is that of chance-constrained programming, apparently initiated by Naslund and Whinston [26]. The third approach, LatanA© [19] and Breiman [6], [7], has its origin in capital growth considerations. The purpose of this paper is to contrast the mean-variance model, by far the most well-known and most developed model of portfolio selection, with the capital growth model, undoubtedly the least known. In so doing, we shall find the mean-variance model to be severely compromised by the capital growth model in several significant respects.
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paper_title: The Kelly Criterion in Blackjack Sports Betting, and the Stock Market

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The central problem for gamblers is to find positive expectation bets. But the gambler also needs to know how to manage his money, i.e., how much to bet. In the stock market (more inclusively, the securities markets) the problem is similar but more complex. The gambler, who is now an “investor”, looks for “excess risk adjusted return”. In both these settings, we explore the use of the Kelly criterion, which is to maximize the expected value of the logarithm of wealth (“maximize expected logarithmic utility”). The criterion is known to economists and financial theorists by names such as the “geometric mean maximizing portfolio strategy”, maximizing logarithmic utility, the growth-optimal strategy, the capital growth criterion, etc. The author initiated the practical application of the Kelly criterion by using it for card counting in blackjack. We will present some useful formulas and methods to answer various natural questions about it that arise in blackjack and other gambling games. Then we illustrate its recent use in a successful casino sports betting system. Finally, we discuss its application to the securities markets where it has helped the author to make a thirty year total of 80 billion dollars worth of “bets”.
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paper_title: Universal prediction

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This paper consists of an overview on universal prediction from an information-theoretic perspective. Special attention is given to the notion of probability assignment under the self-information loss function, which is directly related to the theory of universal data compression. Both the probabilistic setting and the deterministic setting of the universal prediction problem are described with emphasis on the analogy and the differences between results in the two settings.
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paper_title: Modified support vector machines in financial time series forecasting

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This paper proposes a modified version of support vector machines, called C-ascending support vector machine, to model non-stationary financial time series. The C-ascending support vector machines are obtained by a simple modification of the regularized risk function in support vector machines, whereby the recent e-insensitive errors are penalized more heavily than the distant e-insensitive errors. This procedure is based on the prior knowledge that in the non-stationary financial time series the dependency between input variables and output variable gradually changes over the time, specifically, the recent past data could provide more important information than the distant past data. In the experiment, C-ascending support vector machines are tested using three real futures collected from the Chicago Mercantile Market. It is shown that the C-ascending support vector machines with the actually ordered sample data consistently forecast better than the standard support vector machines, with the worst performance when the reversely ordered sample data are used. Furthermore, the C-ascending support vector machines use fewer support vectors than those of the standard support vector machines, resulting in a sparser representation of solution.
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paper_title: PERFORMANCE FUNCTIONS AND REINFORCEMENT LEARNING FOR TRADING SYSTEMS AND PORTFOLIOS

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We propose to train trading systems and portfolios by optimizing objective functions that directly measure trading and investment performance. Rather than basing a trading system on forecasts or training via a supervised learning algorithm using labelled trading data, we train our systems using recurrent reinforcement learning (RRL) algorithms. The performance functions that we consider for reinforcement learning are profit or wealth, economic utility, the Sharpe ratio and our proposed differential Sharpe ratio. The trading and portfolio management systems require prior decisions as input in order to properly take into account the effects of transactions costs, market impact, and taxes. This temporal dependence on system state requires the use of reinforcement versions of standard recurrent learning algorithms. We present empirical results in controlled experiments that demonstrate the efficacy of some of our methods for optimizing trading systems and portfolios. For a long/short trader, we find that maximizing the differential Sharpe ratio yields more consistent results than maximizing profits, and that both methods outperform a trading system based on forecasts that minimize MSE. We find that portfolio traders trained to maximize the differential Sharpe ratio achieve better risk-adjusted returns than those trained to maximize profit. Finally, we provide simulation results for an S&P 500/TBill asset allocation system that demonstrate the presence of out-of-sample predictability in the monthly S&P 500 stock index for the 25 year period 1970 through 1994. Copyright © 1998 John Wiley & Sons, Ltd.
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paper_title: Optimal Dynamic Portfolio Selection: Multiperiod Mean‐Variance Formulation

paper_content:

The mean-variance formulation by Markowitz in the 1950s paved a foundation for modern portfolio selection analysis in a single period. This paper considers an analytical optimal solution to the mean-variance formulation in multiperiod portfolio selection. Specifically, analytical optimal portfolio policy and analytical expression of the mean-variance efficient frontier are derived in this paper for the multiperiod mean-variance formulation. An efficient algorithm is also proposed for finding an optimal portfolio policy to maximize a utility function of the expected value and the variance of the terminal wealth. Copyright Blackwell Publishers Inc. 2000.
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paper_title: Stock market prediction system with modular neural networks

paper_content:

A discussion is presented of a buying- and selling-time prediction system for stocks on the Tokyo Stock Exchange and the analysis of internal representation. The system is based on modular neural networks. The authors developed a number of learning algorithms and prediction methods for the TOPIX (Tokyo Stock Exchange Prices Indexes) prediction system. The prediction system achieved accurate predictions, and the simulation on stocks trading showed an excellent profit
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paper_title: On the Competitive Theory and Practice of Portfolio Selection (Extended Abstract)

paper_content:

Given a set of say m stocks (one of which may be “cash”), the online portfolio selection problem is to determine a portfolio for the i th trading period based on the sequence of prices for the preceding i – 1 trading periods. Competitive analysis is based on a worst case perspective and such a perspective is inconsistent with the more widely accepted analyses and theories based on distributional assumptions. The competitive framework does (perhaps surprisingly) permit non trivial upper bounds on relative performance against CBAL-OPT, an optimal offline constant rebalancing portfolio. Perhaps more impressive are some preliminary experimental results showing that certain algorithms that enjoy “respectable” competitive (i.e. worst case) performance also seem to perform quite well on historical sequences of data. These algorithms and the emerging competitive theory are directly related to studies in information theory and computational learning theory and indeed some of these algorithms have been pioneered within the information theory and computational learning communities. We present a mixture of both theoretical and experimental results, including a more detalied study of the performance of existing and new algorithms with respect to a standard sequence of historical data cited in many studies. We also present experiments from two other historical data sequences.
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paper_title: Empirical Bayes stock market portfolios

paper_content:

We exhibit a compound sequential Bayes portfolio selection algorithm based solely on the past which not only lives off market fluctuations but follows the drift as well. In fact, this sequential portfolio performs as well (up to first order terms in the exponent) as the optimal portfolio based on advance knowledge of the n-period empirical distribution of the market. Moreover, to first order in the exponent, the capital resulting from this portfolio will be no less than the best of the available stocks. This is a result that holds for every sample sequence. Thus bull markets and bear markets can not fool the investor into over-committing or under-committing his capital to the risky alternatives available to him. The goal is accomplished by a choice of portfolio which is robust with respect to futures that may differ drastically from the past.
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paper_title: Growth optimal investment in horse race markets with costs

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We formulate the problem of growth optimal investment in horse race markets with proportional costs and study growth optimal strategies both for stochastic horse races as well as races where one does not make any distributional assumptions. Our results extend all known results for frictionless horse race markets to their natural analog in markets with costs.
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paper_title: Portfolio Selection with Transaction Costs

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In this paper, optimal consumption and investment decisions are studied for an investor who has available a bank account paying a fixed rate of interest and a stock whose price is a log-normal diffusion. This problem was solved by Merton and others when transactions between bank and stock are costless. Here we suppose that there are charges on all transactions equal to a fixed percentage of the amount transacted. It is shown that the optimal buying and selling policies are the local times of the two-dimensional process of bank and stock holdings at the boundaries of a wedge-shaped region which is determined by the solution of a nonlinear free boundary problem. An algorithm for solving the free boundary problem is given.
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paper_title: Growth Optimal Investment with Transaction Costs ⋆

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Discrete time infinite horizon growth optimal investment in stock markets with transactions costs is considered. The stock processes are modelled by homogeneous Markov processes. Assuming that the distribution of the market process is known, we show two recursive investment strategies such that, in the long run, the growth rate on trajectories (in "liminf" sense) is greater than or equal to the growth rate of any other investment strategy with probability 1.
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paper_title: Universal Portfolios With and Without Transaction Costs

paper_content:

A constant rebalanced portfolio is an investment strategy which keeps the same distribution of wealth among a set of stocks from period to period. Recently there has been work on on-line investment strategies that are competitive with the best constant rebalanced portfolio determined in hindsight (Cover, 1991, 1996; Helmbold et al., 1996; Cover & Ordentlich, 1996a, 1996b; Ordentlich & Cover, 1996). For the universal algorithm of Cover (Cover, 1991),we provide a simple analysis which naturallyextends to the case of a fixed percentage transaction cost (commission ), answering a question raised in (Cover, 1991; Helmbold et al., 1996; Cover & Ordentlich, 1996a, 1996b; Ordentlich & Cover, 1996; Cover, 1996). In addition, we present a simple randomized implementation that is significantly faster in practice. We conclude by explaining how these algorithms can be applied to other problems, such as combining the predictions of statistical language models, where the resulting guarantees are more striking.
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paper_title: Dynamic Optimization of Long-Term Growth Rate for a Portfolio with Transaction Costs and Logarithmic Utility

paper_content:

We study the optimal investment policy for an investor who has available one bank account and n risky assets modeled by log‐normal diffusions. The objective is to maximize the long‐run average growth of wealth for a logarithmic utility function in the presence of proportional transaction costs. This problem is formulated as an ergodic singular stochastic control problem and interpreted as the limit of a discounted control problem for vanishing discount factor. The variational inequalities for the discounted control problem and the limiting ergodic problem are established in the viscosity sense. The ergodic variational inequality is solved by using a numerical algorithm based on policy iterations and multigrid methods. A numerical example is displayed for two risky assets.
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paper_title: Performance Analysis of Log-Optimal Portfolio Strategies with Transaction Costs

paper_content:

In this paper we introduce an empirical approximation of the log-optimal investment strategy that guarantees an almost optimal growth rate of investments. The proposed strategy also considers the effects of portfolio rearrangement costs on growth optimality and recommends a suboptimal portfolio for discrete investment periods. We do not assume any parametric structure for the market process, only a first-order Markov property. The model introduced is based on kernel-based agents' (experts') approximation of the maximum theoretical growth rate with transaction costs. Although the optimal solution is theoretically a complex Bellman programming problem, our suboptimal empirical result appears to be attractive for Dow Jones 30 shares. The paper presents a performance analysis where the return of the empirical log-optimal portfolio is compared with passive portfolio counterparts compiled from similar components using the CAPM, the three-factor model and the four-factor model. The proposed methods, in the presence of transaction costs, provide a significant positive abnormal return compared with the preceding equilibrium models, and is even a survivorship bias-free setup.
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paper_title: Empirical Bayes stock market portfolios

paper_content:

We exhibit a compound sequential Bayes portfolio selection algorithm based solely on the past which not only lives off market fluctuations but follows the drift as well. In fact, this sequential portfolio performs as well (up to first order terms in the exponent) as the optimal portfolio based on advance knowledge of the n-period empirical distribution of the market. Moreover, to first order in the exponent, the capital resulting from this portfolio will be no less than the best of the available stocks. This is a result that holds for every sample sequence. Thus bull markets and bear markets can not fool the investor into over-committing or under-committing his capital to the risky alternatives available to him. The goal is accomplished by a choice of portfolio which is robust with respect to futures that may differ drastically from the past.
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paper_title: On-Line Portfolio Selection Using Multiplicative Updates

paper_content:

We present an on-line investment algorithm that achieves almost the same wealth as the best constant-rebalanced portfolio determined in hindsight from the actual market outcomes. The algorithm employs a multiplicative update rule derived using a framework introduced by Kivinen and Warmuth. Our algorithm is very simple to implement and requires only constant storage and computing time per stock in each trading period. We tested the performance of our algorithm on real stock data from the New York Stock Exchange accumulated during a 22-year period. On these data, our algorithm clearly outperforms the best single stock as well as Cover's universal portfolio selection algorithm. We also present results for the situation in which the investor has access to additional "side information." Copyright Blackwell Publishers Inc 1998.
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paper_title: Internal Regret in On-Line Portfolio Selection

paper_content:

This paper extends the game-theoretic notion of internal regret to the case of on-line potfolio selection problems. New sequential investment strategies are designed to minimize the cumulative internal regret for all possible market behaviors. Some of the introduced strategies, apart from achieving a small internal regret, achieve an accumulated wealth almost as large as that of the best constantly rebalanced portfolio. It is argued that the low-internal-regret property is related to stability and experiments on real stock exchange data demonstrate that the new strategies achieve better returns compared to some known algorithms.
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paper_title: From external to internal regret

paper_content:

External regret compares the performance of an online algorithm, selecting among N actions, to the performance of the best of those actions in hindsight. Internal regret compares the loss of an online algorithm to the loss of a modified online algorithm, which consistently replaces one action by another. ::: ::: In this paper we give a simple generic reduction that, given an algorithm for the external regret problem, converts it to an efficient online algorithm for the internal regret problem. We provide methods that work both in the full information model, in which the loss of every action is observed at each time step, and the partial information (bandit) model, where at each time step only the loss of the selected action is observed. The importance of internal regret in game theory is due to the fact that in a general game, if each player has sublinear internal regret, then the empirical frequencies converge to a correlated equilibrium. ::: ::: For external regret we also derive a quantitative regret bound for a very general setting of regret, which includes an arbitrary set of modification rules (that possibly modify the online algorithm) and an arbitrary set of time selection functions (each giving different weight to each time step). The regret for a given time selection and modification rule is the difference between the cost of the online algorithm and the cost of the modified online algorithm, where the costs are weighted by the time selection function. This can be viewed as a generalization of the previously-studied sleeping experts setting.
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paper_title: ASYMPTOTICALLY OPTIMAL PORTFOLIOS

paper_content:

This paper extends to continuous time the concept of universal portfolio introduced by Cover (1991). Being a performance weighted average of constant rebalanced portfolios, the universal portfolio outperforms constant rebalanced and buy-and-hold portfolios exponentially over the long run. an asymptotic formula summarizing its long-term performance is reported that supplements the one given by Cover. A criterion in terms of long-term averages of instantaneous stock drifts and covariances is found which determines the particular form of the asymptotic growth. A formula for the expected universal wealth is given. Copyright 1992 Blackwell Publishers.
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paper_title: Prediction, Learning, and Games

paper_content:

1. Introduction 2. Prediction with expert advice 3. Tight bounds for specific losses 4. Randomized prediction 5. Efficient forecasters for large classes of experts 6. Prediction with limited feedback 7. Prediction and playing games 8. Absolute loss 9. Logarithmic loss 10. Sequential investment 11. Linear pattern recognition 12. Linear classification 13. Appendix.
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paper_title: Universal portfolio selection

paper_content:

A typical problem in portfolio selection in stock markets is that it is not clear which of the many available strategies should be used. We apply a general algorithm of prediction with expert advice (the Aggregating Algorithm) to two different idealizations of the stock market. One is the well-known game introduced by Cover in connection with his “universal portfolio” algorithm; the other is a more realistic modification of Cover’s game introduced in this paper, where market’s participants are allowed to take “short positions”, so that the algorithm may be applied to currency and futures markets. Besides applying the Aggregating Algorithm to a countable (or finite) family of arbitrary investment strategies, we also apply it, in the case of Cover’s game, to the uncountable family of “constant rebalanced portfolios” considered by Cover. We generalize Cover’s worst-case bounds for his “universal portfolio” algorithm (which can be regarded as a special case of the Aggregating Algorithm corresponding to learning rate 1) to the case of learning rates not exceeding 1. Finally, we discuss a general approach to designing investment strategies in which, instead of making statistical or other assumptions about the market, natural assumptions of computability are made about possible investment strategies; this approach leads to natural extensions of the notion of Kolmogorov complexity.
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paper_title: Efficient algorithms for universal portfolios

paper_content:

A constant rebalanced portfolio is an investment strategy which keeps the same distribution of wealth among a set of stocks from day to day. There has been much work on Cover's Universal algorithm, which is competitive with the best constant rebalanced portfolio determined in hindsight (D. Helmbold et al., 1995; A. Blum and A. Kalai, 1999; T.M. Cover and E. Ordentlich, 1996). While this algorithm has good performance guarantees, all known implementations are exponential in the number of stocks, restricting the number of stocks used in experiments. We present an efficient implementation of the Universal algorithm that is based on non-uniform random walks that are rapidly mixing (D. Applegate and R. Kannanm, 1991). This same implementation also works for non-financial applications of the Universal algorithm, such as data compression (T.M. Cover, 1886) and language modeling (A. Kalai et al., 1999).
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paper_title: Universal portfolios with side information

paper_content:

We present a sequential investment algorithm, the /spl mu/-weighted universal portfolio with side information, which achieves, to first order in the exponent, the same wealth as the best side-information dependent investment strategy (the best state-constant rebalanced portfolio) determined in hindsight from observed market and side-information outcomes. This is an individual sequence result which shows the difference between the exponential growth wealth of the best state-constant rebalanced portfolio and the universal portfolio with side information is uniformly less than (d/(2n))log (n+1)+(k/n)log 2 for every stock market and side-information sequence and for all time n. Here d=k(m-1) is the number of degrees of freedom in the state-constant rebalanced portfolio with k states of side information and m stocks. The proof of this result establishes a close connection between universal investment and universal data compression.
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paper_title: Portfolio Selection Efficient Diversification Of Investments

paper_content:

Thank you for reading portfolio selection efficient diversification of investments. As you may know, people have look numerous times for their favorite novels like this portfolio selection efficient diversification of investments, but end up in infectious downloads. Rather than reading a good book with a cup of tea in the afternoon, instead they cope with some malicious bugs inside their desktop computer.
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paper_title: Universal data compression and portfolio selection

paper_content:

The authors consider universal data compression, universal portfolio selection (online portfolio algorithms) and the relationship of both to information theory. Apparently the fundamental minimax redundancy game in data compression and the minimax regret game for the growth rate of wealth in investment have the same answer. There is also a duality between entropy rate and the growth rate of wealth.
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paper_title: A universal data compression system

paper_content:

A universal data compression algorithm is described which is capable of compressing long strings generated by a "finitely generated" source, with a near optimum per symbol length without prior knowledge of the source. This class of sources may be viewed as a generalization of Markov sources to random fields. Moreover, the algorithm does not require a working storage much larger than that needed to describe the source generating parameters.
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paper_title: On-Line Portfolio Selection Using Multiplicative Updates

paper_content:

We present an on-line investment algorithm that achieves almost the same wealth as the best constant-rebalanced portfolio determined in hindsight from the actual market outcomes. The algorithm employs a multiplicative update rule derived using a framework introduced by Kivinen and Warmuth. Our algorithm is very simple to implement and requires only constant storage and computing time per stock in each trading period. We tested the performance of our algorithm on real stock data from the New York Stock Exchange accumulated during a 22-year period. On these data, our algorithm clearly outperforms the best single stock as well as Cover's universal portfolio selection algorithm. We also present results for the situation in which the investor has access to additional "side information." Copyright Blackwell Publishers Inc 1998.
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paper_title: Fast Universalization of Investment Strategies with Provably Good Relative Returns

paper_content:

A universalization of a parameterized investment strategy is an online algorithm whose average daily performance approaches that of the strategy operating with the optimal parameters determined offline in hindsight. We present a general framework for universalizing investment strategies and discuss conditions under which investment strategies are universalizable. We present examples of common investment strategies that fit into our framework. The examples include both trading strategies that decide positions in individual stocks, and portfolio strategies that allocate wealth among multiple stocks. This work extends Cover's universal portfolio work. We also discuss the runtime efficiency of universalization algorithms. While a straightforward implementation of our algorithms runs in time exponential in the number of parameters, we show that the efficient universal portfolio computation technique of Kalai and Vempala involving the sampling of log-concave functions can be generalized to other classes of investment strategies.
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paper_title: FAST UNIVERSALIZATION OF INVESTMENT STRATEGIES ∗

paper_content:

A universalization of a parameterized investment strategy is an online algorithm whose average daily performance approaches that of the strategy operating with the optimal parameters determined offline in hindsight. We present a general framework for universalizing investment strategies and discuss conditions under which investment strategies are universalizable. We present examples of common investment strategies that fit into our framework. The examples include both trading strategies that decide positions in individual stocks, and portfolio strategies that allocate wealth among multiple stocks. This work extends in a natural way Cover's universal portfolio work. We also discuss the runtime efficiency of universalization algorithms. While a straightforward implementation of our algorithms runs in time exponential in the number of parameters, we show that the efficient universal portfolio computation technique of Kalai and Vempala [Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science, Redondo Beach, CA, 2000, pp. 486--491] involving the sampling of log-concave functions can be generalized to other classes of investment strategies, thus yielding provably good approximation algorithms in our framework.
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paper_title: Universal Portfolios With and Without Transaction Costs

paper_content:

A constant rebalanced portfolio is an investment strategy which keeps the same distribution of wealth among a set of stocks from period to period. Recently there has been work on on-line investment strategies that are competitive with the best constant rebalanced portfolio determined in hindsight (Cover, 1991, 1996; Helmbold et al., 1996; Cover & Ordentlich, 1996a, 1996b; Ordentlich & Cover, 1996). For the universal algorithm of Cover (Cover, 1991),we provide a simple analysis which naturallyextends to the case of a fixed percentage transaction cost (commission ), answering a question raised in (Cover, 1991; Helmbold et al., 1996; Cover & Ordentlich, 1996a, 1996b; Ordentlich & Cover, 1996; Cover, 1996). In addition, we present a simple randomized implementation that is significantly faster in practice. We conclude by explaining how these algorithms can be applied to other problems, such as combining the predictions of statistical language models, where the resulting guarantees are more striking.
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paper_title: Universal prediction of individual sequences

paper_content:

The problem of sequentially determining the next, future, outcome of a specific binary individual sequence, based on its observed past, using a finite state predictor is considered. The authors define the finite state predictability of the (infinite) sequence x/sub 1/ . . . z/sub n/ . . . as the minimum fraction of prediction errors that can be made by any such predictor, and prove that this can be achieved, upto an arbitrary small prescribed distance, for each individual sequence, by fully sequential guessing schemes. The rate at which the sequential guessing schemes approach the predictability is calculated. An efficient guessing procedure is based on the incremental parsing algorithm used in Lempel-Ziv data compression, and its fraction of errors also approaches the predictability of the sequence. Some relations between compressibility and predictability are discussed and use of the predictability as an additional measure for the complexity, or randomness, of the sequence is suggested. >
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paper_title: Efficient Universal Portfolios for Past-Dependent Target Classes

paper_content:

We present a new universal portfolio algorithm that achieves almost the same level of wealth as could be achieved by knowing stock prices ahead of time. Specifically the algorithm tracks the best in hindsight wealth achievable within target classes of linearly parameterized portfolio sequences. The target classes considered are more general than the standard constant rebalanced portfolio class and permit portfolio sequences to exhibit a continuous form of dependence on past prices or other side information. A primary advantage of the algorithm is that it is easily computable in a polynomial number of steps by way of simple closed-form expressions. This provides an edge over other universal algorithms that require both an exponential number of computations and numerical approximation.
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paper_title: Meta optimization and its application to portfolio selection

paper_content:

Several data mining algorithms use iterative optimization methods for learning predictive models. It is not easy to determine upfront which optimization method will perform best or converge fast for such tasks. In this paper, we analyze Meta Algorithms (MAs) which work by adaptively combining iterates from a pool of base optimization algorithms. We show that the performance of MAs are competitive with the best convex combination of the iterates from the base algorithms for online as well as batch convex optimization problems. We illustrate the effectiveness of MAs on the problem of portfolio selection in the stock market and use several existing ideas for portfolio selection as base algorithms. Using daily S\&P500 data for the past 21 years and a benchmark NYSE dataset, we show that MAs outperform existing portfolio selection algorithms with provable guarantees by several orders of magnitude, and match the performance of the best heuristics in the pool.
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paper_title: On-line portfolio selection using stochastic programming

paper_content:

Abstract This paper is dedicated to the problem of dynamic portfolio optimization for the case when the number of decision periods is large and new information about market arrives during each such period. We propose the family of adaptive portfolio selection policies which rebalance the current portfolio during each decision period by adopting portfolio from a specified family with the best performance on the past data. In the absence of transaction costs the general conditions are found under which this policy yields asymptotically the same performance as the best portfolio from the same family constructed with the full knowledge of the future. These results are extended for the case of nonzero transaction costs by introducing a class of threshold portfolio optimization policies which rebalance current portfolio only when its performance differs from performance of the best portfolio by a given threshold. The value of this threshold is adapted to the changing market conditions. We show that it is possible to select a sequence of threshold values in such a way that the asymptotic influence of transaction costs on portfolio performance is negligible and overall portfolio performance is asymptotically the same as the performance of portfolio with the perfect knowledge of the future. We do not assume neither specific probabilistic structure of the market data nor their stationarity. Our theory is illustrated by numerical experiments with real data. Finally, we discuss the relevance of our results in the context of high performance computing.
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paper_title: Stochastic Nonstationary Optimization for Finding Universal Portfolios

paper_content:

We apply ideas from stochastic optimization for defining universal portfolios. Universal portfolios are that class of portfolios which are constructed directly from the available observations of the stocks behavior without any assumptions about their statistical properties. Cover [7] has shown that one can construct such portfolio using only observations of the past stock prices which generates the same asymptotic wealth growth as the best constant rebalanced portfolio which is constructed with the full knowledge of the future stock market behavior.In this paper we construct universal portfolios using a different set of ideas drawn from nonstationary stochastic optimization. Our portfolios yield the same asymptotic growth of wealth as the best constant rebalanced portfolio constructed with the perfect knowledge of the future and they are less demanding computationally compared to previously known universal portfolios. We also present computational evidence using New York Stock Exchange data which shows, among other things, superior performance of portfolios which explicitly take into account possible nonstationary market behavior.
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paper_title: A Comparison of New and Old Algorithms for a Mixture Estimation Problem

paper_content:

We investigate the problem of estimating the proportion vector which maximizes the likelihood of a given sample for a mixture of given densities. We adapt a framework developed for supervised learning and give simple derivations for many of the standard iterative algorithms like gradient projection and EM. In this framework, the distance between the new and old proportion vectors is used as a penalty term. The square distance leads to the gradient projection update, and the relative entropy to a new update which we call the exponentiated gradient update (EGή). Curiously, when a second order Taylor expansion of the relative entropy is used, we arrive at an update EMή which, for ή=1, gives the usual EM update. Experimentally, both the EMή-update and the EGή-update for ή > 1 outperform the EM algorithm and its variants. We also prove a polynomial bound on the rate of convergence of the EGή algorithm.
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paper_title: Introduction to Stochastic Programming

paper_content:

The aim of stochastic programming is to find optimal decisions in problems which involve uncertain data. This field is currently developing rapidly with contributions from many disciplines including operations research, mathematics, and probability. At the same time, it is now being applied in a wide variety of subjects ranging from agriculture to financial planning and from industrial engineering to computer networks. This textbook provides a first course in stochastic programming suitable for students with a basic knowledge of linear programming, elementary analysis, and probability. The authors aim to present a broad overview of the main themes and methods of the subject. Its prime goal is to help students develop an intuition on how to model uncertainty into mathematical problems, what uncertainty changes bring to the decision process, and what techniques help to manage uncertainty in solving the problems.In this extensively updated new edition there is more material on methods and examples including several new approaches for discrete variables, new results on risk measures in modeling and Monte Carlo sampling methods, a new chapter on relationships to other methods including approximate dynamic programming, robust optimization and online methods.The book is highly illustrated with chapter summaries and many examples and exercises. Students, researchers and practitioners in operations research and the optimization area will find it particularly of interest. Review of First Edition:"The discussion on modeling issues, the large number of examples used to illustrate the material, and the breadth of the coverage make'Introduction to Stochastic Programming' an ideal textbook for the area." (Interfaces, 1998)
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paper_title: On-Line Portfolio Selection Using Multiplicative Updates

paper_content:

We present an on-line investment algorithm that achieves almost the same wealth as the best constant-rebalanced portfolio determined in hindsight from the actual market outcomes. The algorithm employs a multiplicative update rule derived using a framework introduced by Kivinen and Warmuth. Our algorithm is very simple to implement and requires only constant storage and computing time per stock in each trading period. We tested the performance of our algorithm on real stock data from the New York Stock Exchange accumulated during a 22-year period. On these data, our algorithm clearly outperforms the best single stock as well as Cover's universal portfolio selection algorithm. We also present results for the situation in which the investor has access to additional "side information." Copyright Blackwell Publishers Inc 1998.
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paper_title: Online Lazy Updates for Portfolio Selection with Transaction Costs

paper_content:

A major challenge for stochastic optimization is the cost of updating model parameters especially when the number of parameters is large. Updating parameters frequently can prove to be computationally or monetarily expensive. In this paper, we introduce an efficient primal-dual based online algorithm that performs lazy updates to the parameter vector and show that its performance is competitive with reasonable strategies which have the benefit of hindsight. We demonstrate the effectiveness of our algorithm in the online portfolio selection domain where a trader has to pay proportional transaction costs every time his portfolio is updated. Our Online Lazy Updates (OLU) algorithm takes into account the transaction costs while computing an optimal portfolio which results in sparse updates to the portfolio vector. We successfully establish the robustness and scalability of our lazy portfolio selection algorithm with extensive theoretical and experimental results on two real-world datasets.
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paper_title: Empirical properties of asset returns: stylized facts and statistical issues

paper_content:

We present a set of stylized empirical facts emerging from the statistical analysis of price variations in various types of financial markets. We first discuss some general issues common to all statistical studies of financial time series. Various statistical properties of asset returns are then described: distributional properties, tail properties and extreme fluctuations, pathwise regularity, linear and nonlinear dependence of returns in time and across stocks. Our description emphasizes properties common to a wide variety of markets and instruments. We then show how these statistical properties invalidate many of the common statistical approaches used to study financial data sets and examine some of the statistical problems encountered in each case.
---
paper_title: Meta optimization and its application to portfolio selection

paper_content:

Several data mining algorithms use iterative optimization methods for learning predictive models. It is not easy to determine upfront which optimization method will perform best or converge fast for such tasks. In this paper, we analyze Meta Algorithms (MAs) which work by adaptively combining iterates from a pool of base optimization algorithms. We show that the performance of MAs are competitive with the best convex combination of the iterates from the base algorithms for online as well as batch convex optimization problems. We illustrate the effectiveness of MAs on the problem of portfolio selection in the stock market and use several existing ideas for portfolio selection as base algorithms. Using daily S\&P500 data for the past 21 years and a benchmark NYSE dataset, we show that MAs outperform existing portfolio selection algorithms with provable guarantees by several orders of magnitude, and match the performance of the best heuristics in the pool.
---
paper_title: PORTFOLIO SELECTION AND ONLINE LEARNING

paper_content:

This paper studies a new strategy for selecting portfolios in the stock market. The strategy is inspired by two streams of previous work: (1) work on universalization of strategies for portfolio selection, which began with Thomas Cover's work on constant rebalanced portfolios, published in 1991,4 and (2) more general work on universalization of online algorithms,17,21,23,30 especially Vladimir Vovk's work on the aggregating algorithm and Markov switching strategies.32 The proposed investment strategy achieves asymptotically the same exponential rate of growth as the portfolio that turns out to be best expost in the long run and does not require any underlying statistical assumptions on the nature of the stock market.
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paper_title: The Variation of Certain Speculative Prices

paper_content:

The purpose of this chapter is to present and test a new model of price behavior in speculative markets. The principal feature of this model is that starting from the Bachelier process as applied to InZ(t) instead of Z(t), the Gaussian distribution is replaced throughout by another family of probability laws to be referred to as stable Paretian. In a somewhat complex way, the Gaussian is a limiting case of this new family, so the new model proposed in this chapter is actually a generalization of the continuous random walk of Bachelier.
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paper_title: Brownian Motion in the Stock Market

paper_content:

It is shown that common-stock prices, and the value of money can be regarded as an ensemble of decisions in statistical equilibrium, with properties quite analogous to an ensemble of particles in statistical mechanics. If Y = loge[P(t + r)/P0(t)], where P(t + r) and P0(t) are the price of the same random choice stock at random times t + r and t, then the steady state distribution function of Y is \documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $\varphi(Y)=\exp(-Y^{2}/2\sigma^{2}\tau)/\sqrt{2\pi \sigma^{2}\tau}$ \end{document}, which is precisely the probability distribution for a particle in Browman motion, if σ is the dispersion developed at the end of unit time. A similar distribution holds for the value of money, measured approximately by stoc...
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paper_title: Derandomizing Stochastic Prediction Strategies

paper_content:

AbstractIn this paper we continue study of the games of prediction with expert advice with uncountably many experts. A convenient interpretation of such games is to construe the pool of experts as one “stochastic predictor”, who chooses one of the experts in the pool at random according to the prior distribution on the experts and then replicates the (deterministic ) predictions of the chosen expert. We notice that if the stochastic predictor's total loss is at most L with probability at least p then the learner's loss can be bounded by cL + aln ::: $$\frac{1}{{\text{P}}}$$ ::: for the usual constants c and a. This interpretation is used to revamp known results and obtain new results on tracking the best expert. It is also applied to merging overconfident experts and to fitting polynomials to data.
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paper_title: Universal portfolio selection

paper_content:

A typical problem in portfolio selection in stock markets is that it is not clear which of the many available strategies should be used. We apply a general algorithm of prediction with expert advice (the Aggregating Algorithm) to two different idealizations of the stock market. One is the well-known game introduced by Cover in connection with his “universal portfolio” algorithm; the other is a more realistic modification of Cover’s game introduced in this paper, where market’s participants are allowed to take “short positions”, so that the algorithm may be applied to currency and futures markets. Besides applying the Aggregating Algorithm to a countable (or finite) family of arbitrary investment strategies, we also apply it, in the case of Cover’s game, to the uncountable family of “constant rebalanced portfolios” considered by Cover. We generalize Cover’s worst-case bounds for his “universal portfolio” algorithm (which can be regarded as a special case of the Aggregating Algorithm corresponding to learning rate 1) to the case of learning rates not exceeding 1. Finally, we discuss a general approach to designing investment strategies in which, instead of making statistical or other assumptions about the market, natural assumptions of computability are made about possible investment strategies; this approach leads to natural extensions of the notion of Kolmogorov complexity.
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paper_title: Online Convex Programming and Generalized Infinitesimal Gradient Ascent

paper_content:

Convex programming involves a convex set F ⊆ Rn and a convex cost function c : F → R. The goal of convex programming is to find a point in F which minimizes c. In online convex programming, the convex set is known in advance, but in each step of some repeated optimization problem, one must select a point in F before seeing the cost function for that step. This can be used to model factory production, farm production, and many other industrial optimization problems where one is unaware of the value of the items produced until they have already been constructed. We introduce an algorithm for this domain. We also apply this algorithm to repeated games, and show that it is really a generalization of infinitesimal gradient ascent, and the results here imply that generalized infinitesimal gradient ascent (GIGA) is universally consistent.
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paper_title: Universal Constant Rebalanced Portfolios with Switching

paper_content:

In this paper, we consider online (sequential) portfolio selection in a competitive algorithm framework. We construct a sequential algorithm for portfolio investment that asymptotically achieves the wealth of the best piecewise constant rebalanced portfolio tuned to the underlying individual sequence of price relative vectors. Without knowledge of the investment duration, the algorithm can perform as well as the best investment algorithm that can choose both the partitioning of the sequence of the price relative vectors as well as the best constant rebalanced portfolio within each segment based on knowledge of the sequence of price relative vectors in advance. We use a transition diagram similar to that in F.M.J. Willems, (1996) to compete with an exponential number of switching investment strategies, using only linear complexity in the data length for combination. The regret with respect to the best piecewise constant strategy is at most O(ln(n)) in the exponent, where n is the investment duration. This method is also extended in S.S. Kozat and A.C. Singer, (2006) to switching among a finite collection of candidate algorithms, including the case where such transitions are represented by an arbitrary side-information sequence.
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paper_title: Logarithmic regret algorithms for online convex optimization

paper_content:

In an online convex optimization problem a decision-maker makes a sequence of decisions, i.e., chooses a sequence of points in Euclidean space, from a fixed feasible set. After each point is chosen, it encounters a sequence of (possibly unrelated) convex cost functions. Zinkevich (ICML 2003) introduced this framework, which models many natural repeated decision-making problems and generalizes many existing problems such as Prediction from Expert Advice and Cover's Universal Portfolios. Zinkevich showed that a simple online gradient descent algorithm achieves additive regret $O(\sqrt{T})$ , for an arbitrary sequence of T convex cost functions (of bounded gradients), with respect to the best single decision in hindsight. ::: ::: In this paper, we give algorithms that achieve regret O(log?(T)) for an arbitrary sequence of strictly convex functions (with bounded first and second derivatives). This mirrors what has been done for the special cases of prediction from expert advice by Kivinen and Warmuth (EuroCOLT 1999), and Universal Portfolios by Cover (Math. Finance 1:1---19, 1991). We propose several algorithms achieving logarithmic regret, which besides being more general are also much more efficient to implement. ::: ::: The main new ideas give rise to an efficient algorithm based on the Newton method for optimization, a new tool in the field. Our analysis shows a surprising connection between the natural follow-the-leader approach and the Newton method. We also analyze other algorithms, which tie together several different previous approaches including follow-the-leader, exponential weighting, Cover's algorithm and gradient descent.
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paper_title: The Random Character Of Stock Market Prices

paper_content:

Thank you very much for downloading the random character of stock market prices. As you may know, people have search hundreds times for their chosen novels like this the random character of stock market prices, but end up in harmful downloads. Rather than enjoying a good book with a cup of coffee in the afternoon, instead they juggled with some harmful virus inside their desktop computer. the random character of stock market prices is available in our book collection an online access to it is set as public so you can get it instantly. Our book servers spans in multiple locations, allowing you to get the most less latency time to download any of our books like this one. Kindly say, the the random character of stock market prices is universally compatible with any devices to read.
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paper_title: Empirical Bayes stock market portfolios

paper_content:

We exhibit a compound sequential Bayes portfolio selection algorithm based solely on the past which not only lives off market fluctuations but follows the drift as well. In fact, this sequential portfolio performs as well (up to first order terms in the exponent) as the optimal portfolio based on advance knowledge of the n-period empirical distribution of the market. Moreover, to first order in the exponent, the capital resulting from this portfolio will be no less than the best of the available stocks. This is a result that holds for every sample sequence. Thus bull markets and bear markets can not fool the investor into over-committing or under-committing his capital to the risky alternatives available to him. The goal is accomplished by a choice of portfolio which is robust with respect to futures that may differ drastically from the past.
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paper_title: Multi-domain learning by confidence-weighted parameter combination

paper_content:

State-of-the-art statistical NLP systems for a variety of tasks learn from labeled training data that is often domain specific. However, there may be multiple domains or sources of interest on which the system must perform. For example, a spam filtering system must give high quality predictions for many users, each of whom receives emails from different sources and may make slightly different decisions about what is or is not spam. Rather than learning separate models for each domain, we explore systems that learn across multiple domains. We develop a new multi-domain online learning framework based on parameter combination from multiple classifiers. Our algorithms draw from multi-task learning and domain adaptation to adapt multiple source domain classifiers to a new target domain, learn across multiple similar domains, and learn across a large number of disparate domains. We evaluate our algorithms on two popular NLP domain adaptation tasks: sentiment classification and spam filtering.
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paper_title: Efficient projections onto the l1-ball for learning in high dimensions

paper_content:

We describe efficient algorithms for projecting a vector onto the l1-ball. We present two methods for projection. The first performs exact projection in O(n) expected time, where n is the dimension of the space. The second works on vectors k of whose elements are perturbed outside the l1-ball, projecting in O(k log(n)) time. This setting is especially useful for online learning in sparse feature spaces such as text categorization applications. We demonstrate the merits and effectiveness of our algorithms in numerous batch and online learning tasks. We show that variants of stochastic gradient projection methods augmented with our efficient projection procedures outperform interior point methods, which are considered state-of-the-art optimization techniques. We also show that in online settings gradient updates with l1 projections outperform the exponentiated gradient algorithm while obtaining models with high degrees of sparsity.
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paper_title: Confidence Weighted Mean Reversion Strategy for Online Portfolio Selection

paper_content:

Online portfolio selection has been attracting increasing attention from the data mining and machine learning communities. All existing online portfolio selection strategies focus on the first order information of a portfolio vector, though the second order information may also be beneficial to a strategy. Moreover, empirical evidence shows that relative stock prices may follow the mean reversion property, which has not been fully exploited by existing strategies. This article proposes a novel online portfolio selection strategy named Confidence Weighted Mean Reversion (CWMR). Inspired by the mean reversion principle in finance and confidence weighted online learning technique in machine learning, CWMR models the portfolio vector as a Gaussian distribution, and sequentially updates the distribution by following the mean reversion trading principle. CWMR’s closed-form updates clearly reflect the mean reversion trading idea. We also present several variants of CWMR algorithms, including a CWMR mixture algorithm that is theoretical universal. Empirically, CWMR strategy is able to effectively exploit the power of mean reversion for online portfolio selection. Extensive experiments on various real markets show that the proposed strategy is superior to the state-of-the-art techniques. The experimental testbed including source codes and data sets is available online.
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paper_title: Can We Learn to Beat the Best Stock

paper_content:

A novel algorithm for actively trading stocks is presented. While traditional universal algorithms (and technical trading heuristics) attempt to predict winners or trends, our approach relies on predictable statistical relations between all pairs of stocks in the market. Our empirical results on historical markets provide strong evidence that this type of technical trading can "beat the market" and moreover, can beat the best stock in the market. In doing so we utilize a new idea for smoothing critical parameters in the context of expert learning.
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paper_title: Multi-Class Confidence Weighted Algorithms

paper_content:

The recently introduced online confidence-weighted (CW) learning algorithm for binary classification performs well on many binary NLP tasks. However, for multi-class problems CW learning updates and inference cannot be computed analytically or solved as convex optimization problems as they are in the binary case. We derive learning algorithms for the multi-class CW setting and provide extensive evaluation using nine NLP datasets, including three derived from the recently released New York Times corpus. Our best algorithm out-performs state-of-the-art online and batch methods on eight of the nine tasks. We also show that the confidence information maintained during learning yields useful probabilistic information at test time.
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paper_title: When are Contrarian Profits Due to Stock Market Overreaction

paper_content:

The profitability of contrarian investment strategies need not be the result of stock market overreaction. Even if returns on individual securities are temporally independent, portfolio strategies that attempt to exploit return reversals may still earn positive expected profits. This is due to the effects of cross-autocovariances from which contrarian strategies inadvertently benefit. We provide an informal taxonomy of return-generating processes that yield positive [and negative] expected profits under a particular contrarian portfolio strategy, and use this taxonomy to reconcile the empirical findings of weak negative autocorrelation for returns on individual stocks with the strong positive autocorrelation of portfolio returns. We present empirical evidence against overreaction as the primary source of contrarian profits, and show the presence of important lead-lag relations across securities.
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paper_title: An Anatomy of Trading Strategies

paper_content:

In this paper, we use a single unifying framework to analyze the sources of profits to a wide spectrum of return-based trading strategies implemented in the literature. We show that less than 50 percent of the 120 strategies implemented in the paper yield statistically significant profits and, unconditionally, momentum and contrarian strategies are equally likely to be successful. However, when we condition on the return-horizon (short, medium, or long) of the strategy, or the time-period during which it is implemented, two patterns emerge. A momentum strategy is usually profitable at the medium (three- to 12-month) horizon, while a contrarian strategy nets statistically significant profits at long-horizons but only during the 1926-1947 subperiod. More importantly, our results show that the cross-sectional variation in the mean returns of individual securities included in these strategies plays an important role in the profitability of the strategies. The cross-sectional variation can potentially account for the profitability of momentum strategies and it is also responsible for attenuating the profits from price reversals to long-horizon contrarian strategies.
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paper_title: Universal Investment in Markets with Transaction Costs

paper_content:

In this paper we investigate growth optimal investment in two-asset discrete-time markets with proportional transaction costs and no distributional assumptions on the market return sequences. We construct a policy with growth rate at least as large as any interval policy. Since interval policies are mu-optimal for independent and identically distributed (i.i.d.) markets (Iyengar 2002), it follows that our policy when employed in an i.i.d. market is able to "learn" the optimal interval policy and achieve growth optimality; in other words, it is a universal growth optimal policy for i.i.d. markets.
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paper_title: Does the Stock Market Overreact?

paper_content:

Research in experimental psychology suggests that, in violation of Bayes' rule, most people tend to "overreact" to unexpected and dramatic news events. This study of market efficiency investigates whether such behavior affects stock prices. The empirical evidence, based on CRSP monthly return data, is consistent with the overreaction hypothesis. Substantial weak form market inefficiencies are discovered. The results also shed new light on the January returns earned by prior "winners" and "losers." Portfolios of losers experience exceptionally large January returns as late as five years after portfolio formation. As ECONOMISTS INTERESTED IN both market behavior and the psychology of individual decision making, we have been struck by the similarity of two sets of empirical findings. Both classes of behavior can be characterized as displaying overreaction. This study was undertaken to investigate the possibility that these phenomena are related by more than just appearance. We begin by describing briefly the individual and market behavior that piqued our interest. The term overreaction carries with it an implicit comparison to some degree of reaction that is considered to be appropriate. What is an appropriate reaction? One class,,of tasks which have a well-established norm are probability revision problems for which Bayes' rule prescribes the correct reaction to new information. It has now been well-established that Bayes' rule is not an apt characterization of how individuals actually respond to new data (Kahneman et al. [14]). In revising their beliefs, individuals tend to overweight recent information and underweight prior (or base rate) data. People seem to make predictions according to a simple matching rule: "The predicted value is selected so that the standing of the case in the distribution of outcomes matches its standing in the distribution of impressions" (Kahneman and Tversky [14, p. 416]). This rule-of-thumb, an instance of what Kahneman and Tversky call the representativeness heuristic, violates the basic statistical principal that the extremeness of predictions must be moderated by considerations of predictability. Grether [12] has replicated this finding under incentive compatible conditions. There is also considerable evidence that the actual expectations of professional security analysts and economic forecasters display the same overreaction bias (for a review, see De Bondt [7]). One of the earliest observations about overreaction in markets was made by J. M. Keynes:"... day-to-day fluctuations in the profits of existing investments,
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paper_title: Robust Median Reversion Strategy for Online Portfolio Selection

paper_content:

Online portfolio selection has attracted increasing attention from data mining and machine learning communities in recent years. An important theory in financial markets is mean reversion, which plays a critical role in some state-of-the-art portfolio selection strategies. Although existing mean reversion strategies have been shown to achieve good empirical performance on certain datasets, they seldom carefully deal with noise and outliers in the data, leading to suboptimal portfolios, and consequently yielding poor performance in practice. In this paper, we propose to exploit the reversion phenomenon by using robust $L_1$ -median estimators, and design a novel online portfolio selection strategy named “Robust Median Reversion” (RMR), which constructs optimal portfolios based on the improved reversion estimator. We examine the performance of the proposed algorithms on various real markets with extensive experiments. Empirical results show that RMR can overcome the drawbacks of existing mean reversion algorithms and achieve significantly better results. Finally, RMR runs in linear time, and thus is suitable for large-scale real-time algorithmic trading applications.
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paper_title: Online Passive-Aggressive Algorithms

paper_content:

We present a unified view for online classification, regression, and uni-class problems. This view leads to a single algorithmic framework for the three problems. We prove worst case loss bounds for various algorithms for both the realizable case and the non-realizable case. A conversion of our main online algorithm to the setting of batch learning is also discussed. The end result is new algorithms and accompanying loss bounds for the hinge-loss.
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paper_title: Kernel-based semi-log-optimal empirical portfolio selection strategies

paper_content:

The purpose of this paper is to introduce an approximation of the kernel-based log-optimal investment strategy that guarantees an almost optimal rate of growth of the capital under minimal assumptions on the behavior of the market. The new strategy uses much less knowledge on the distribution of the market process. It is analyzed both theoretically and empirically. The theoretical results show that the asymptotic rate of growth well approximates the optimal one that one could achieve with a full knowledge of the statistical properties of the underlying process generating the market, under the only assumption that the market is stationary and ergodic. The empirical results show that the proposed semi-log-optimal and the log-optimal strategies have essentially the same performance measured on past NYSE data.
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paper_title: Growth Optimal Investment with Transaction Costs ⋆

paper_content:

Discrete time infinite horizon growth optimal investment in stock markets with transactions costs is considered. The stock processes are modelled by homogeneous Markov processes. Assuming that the distribution of the market process is known, we show two recursive investment strategies such that, in the long run, the growth rate on trajectories (in "liminf" sense) is greater than or equal to the growth rate of any other investment strategy with probability 1.
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paper_title: An Asymptotic Analysis of the Mean-Variance Portfolio Selection

paper_content:

This paper gives an asymptotic analysis of the mean-variance (Markowitz-type) portfolio selection under mild assumptions on the market behavior. Theoretical results show the rate of underperformance of the risk aware Markowitz-type portfolio strategy in growth rate compared to the log-optimal portfolio strategy, which does not have explicit risk control. Statements are given with and without full knowledge of the statistical properties of the underlying process generating the market, under the only assumption that the market is stationary and ergodic. The experiments show how the achieved wealth depends on the coefficient of risk aversion measured on past NYSE data.
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paper_title: Nonparametric kernel-based sequential investment strategies

paper_content:

The purpose of this paper is to introduce sequential investment strategies that guarantee an optimal rate of growth of the capital, under minimal assumptions on the behavior of the market. The new strategies are analyzed both theoretically and empirically. The theoretical results show that the asymptotic rate of growth matches the optimal one that one could achieve with a full knowledge of the statistical properties of the underlying process generating the market, under the only assumption that the market is stationary and ergodic. The empirical results show that the performance of the proposed investment strategies measured on past nyse and currency exchange data is solid, and sometimes even spectacular.
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paper_title: Nonparametric sequential prediction of time series

paper_content:

Time series prediction covers a vast field of every-day statistical applications in medical, environmental and economic domains. In this paper we develop nonparametric prediction strategies based on the combination of a set of 'experts' and show the universal consistency of these strategies under a minimum of conditions. We perform an in-depth analysis of real-world data sets and show that these nonparametric strategies are more flexible, faster and generally outperform ARMA methods in terms of normalized cumulative prediction error.
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paper_title: CORN: Correlation-driven nonparametric learning approach for portfolio selection

paper_content:

Machine learning techniques have been adopted to select portfolios from financial markets in some emerging intelligent business applications. In this article, we propose a novel learning-to-trade algorithm termed CORrelation-driven Nonparametric learning strategy (CORN) for actively trading stocks. CORN effectively exploits statistical relations between stock market windows via a nonparametric learning approach. We evaluate the empirical performance of our algorithm extensively on several large historical and latest real stock markets, and show that it can easily beat both the market index and the best stock in the market substantially (without or with small transaction costs), and also surpass a variety of state-of-the-art techniques significantly.
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paper_title: Meta optimization and its application to portfolio selection

paper_content:

Several data mining algorithms use iterative optimization methods for learning predictive models. It is not easy to determine upfront which optimization method will perform best or converge fast for such tasks. In this paper, we analyze Meta Algorithms (MAs) which work by adaptively combining iterates from a pool of base optimization algorithms. We show that the performance of MAs are competitive with the best convex combination of the iterates from the base algorithms for online as well as batch convex optimization problems. We illustrate the effectiveness of MAs on the problem of portfolio selection in the stock market and use several existing ideas for portfolio selection as base algorithms. Using daily S\&P500 data for the past 21 years and a benchmark NYSE dataset, we show that MAs outperform existing portfolio selection algorithms with provable guarantees by several orders of magnitude, and match the performance of the best heuristics in the pool.
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paper_title: Prediction, Learning, and Games

paper_content:

1. Introduction 2. Prediction with expert advice 3. Tight bounds for specific losses 4. Randomized prediction 5. Efficient forecasters for large classes of experts 6. Prediction with limited feedback 7. Prediction and playing games 8. Absolute loss 9. Logarithmic loss 10. Sequential investment 11. Linear pattern recognition 12. Linear classification 13. Appendix.
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paper_title: Universal portfolio selection

paper_content:

A typical problem in portfolio selection in stock markets is that it is not clear which of the many available strategies should be used. We apply a general algorithm of prediction with expert advice (the Aggregating Algorithm) to two different idealizations of the stock market. One is the well-known game introduced by Cover in connection with his “universal portfolio” algorithm; the other is a more realistic modification of Cover’s game introduced in this paper, where market’s participants are allowed to take “short positions”, so that the algorithm may be applied to currency and futures markets. Besides applying the Aggregating Algorithm to a countable (or finite) family of arbitrary investment strategies, we also apply it, in the case of Cover’s game, to the uncountable family of “constant rebalanced portfolios” considered by Cover. We generalize Cover’s worst-case bounds for his “universal portfolio” algorithm (which can be regarded as a special case of the Aggregating Algorithm corresponding to learning rate 1) to the case of learning rates not exceeding 1. Finally, we discuss a general approach to designing investment strategies in which, instead of making statistical or other assumptions about the market, natural assumptions of computability are made about possible investment strategies; this approach leads to natural extensions of the notion of Kolmogorov complexity.
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paper_title: Efficient learning algorithms for changing environments

paper_content:

We study online learning in an oblivious changing environment. The standard measure of regret bounds the difference between the cost of the online learner and the best decision in hindsight. Hence, regret minimizing algorithms tend to converge to the static best optimum, clearly a suboptimal behavior in changing environments. On the other hand, various metrics proposed to strengthen regret and allow for more dynamic algorithms produce inefficient algorithms. We propose a different performance metric which strengthens the standard metric of regret and measures performance with respect to a changing comparator. We then describe a series of data-streaming-based reductions which transform algorithms for minimizing (standard) regret into adaptive algorithms albeit incurring only poly-logarithmic computational overhead. Using this reduction, we obtain efficient low adaptive-regret algorithms for the problem of online convex optimization. This can be applied to various learning scenarios, i.e. online portfolio selection, for which we describe experimental results showing the advantage of adaptivity.
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paper_title: Can We Learn to Beat the Best Stock

paper_content:

A novel algorithm for actively trading stocks is presented. While traditional universal algorithms (and technical trading heuristics) attempt to predict winners or trends, our approach relies on predictable statistical relations between all pairs of stocks in the market. Our empirical results on historical markets provide strong evidence that this type of technical trading can "beat the market" and moreover, can beat the best stock in the market. In doing so we utilize a new idea for smoothing critical parameters in the context of expert learning.
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paper_title: Fast Universalization of Investment Strategies with Provably Good Relative Returns

paper_content:

A universalization of a parameterized investment strategy is an online algorithm whose average daily performance approaches that of the strategy operating with the optimal parameters determined offline in hindsight. We present a general framework for universalizing investment strategies and discuss conditions under which investment strategies are universalizable. We present examples of common investment strategies that fit into our framework. The examples include both trading strategies that decide positions in individual stocks, and portfolio strategies that allocate wealth among multiple stocks. This work extends Cover's universal portfolio work. We also discuss the runtime efficiency of universalization algorithms. While a straightforward implementation of our algorithms runs in time exponential in the number of parameters, we show that the efficient universal portfolio computation technique of Kalai and Vempala involving the sampling of log-concave functions can be generalized to other classes of investment strategies.
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paper_title: FAST UNIVERSALIZATION OF INVESTMENT STRATEGIES ∗

paper_content:

A universalization of a parameterized investment strategy is an online algorithm whose average daily performance approaches that of the strategy operating with the optimal parameters determined offline in hindsight. We present a general framework for universalizing investment strategies and discuss conditions under which investment strategies are universalizable. We present examples of common investment strategies that fit into our framework. The examples include both trading strategies that decide positions in individual stocks, and portfolio strategies that allocate wealth among multiple stocks. This work extends in a natural way Cover's universal portfolio work. We also discuss the runtime efficiency of universalization algorithms. While a straightforward implementation of our algorithms runs in time exponential in the number of parameters, we show that the efficient universal portfolio computation technique of Kalai and Vempala [Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science, Redondo Beach, CA, 2000, pp. 486--491] involving the sampling of log-concave functions can be generalized to other classes of investment strategies, thus yielding provably good approximation algorithms in our framework.
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paper_title: Tracking the best expert

paper_content:

We generalize the recent worst-case loss bounds for on-line algorithms where the additional loss of the algorithm on the whole sequence of examples over the loss of the best expert is bounded. The generalization allows the sequence to be partitioned into segments and the goal is to bound the additional loss of the algorithm over the sum of the losses of the best experts of each segment. This is to model situations in which the examples change and different experts are best for certain segments of the sequence of examples. In the single expert case the additional loss is proportional to log n, where n is the number of experts and the constant of proportionality depends on the loss function. ::: ::: When the number of segments is at most k+ 1 and the sequence of length l then we can bound the additional loss of our algorithm over the best partitioning by O(klogn + klog(l/k)). Note that it takes the same order of bits to denote the sequence of experts and the boundaries of the segments. When the loss per trial is bounded by one then we obtain additional loss bounds that are independent of the length of the sequence. The bound becomes O(k logn + klog(L/k)), where L is the loss of the best partition into k+1 segments. ::: ::: Our algorithms for tracking the best expert are simple adaptations of Vovk's original algorithm for the single best expert case. These algorithms keep one weight per expert and spend (1) time per weight in each trial. ::: ::: Acknowledgments We would like to thank Peter Auer and Phillip Long for valuable discussions.
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paper_title: Portfolio Choice And The Kelly Criterion

paper_content:

The following sections are included:IntroductionSamuelson's objections to logarithmic utilityAn outline of the theory of logarithmic utility as applied to portfolio selectionRelation to the Markowitz theory; solution to problems thereinThe theory in action: results for a real institutional portfolioConcluding remarksFOOTNOTESREFERENCES
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paper_title: Capital Growth Theory

paper_content:

The following sections are included:IntroductionOrigins of the modelThe model and its basic propertiesConditions for capital growthRelationship to other long-run investment modelsRelationship to intertemporal consumption-investment modelsGrowth vs. securityApplicationsSummaryReferences
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paper_title: Optimal Gambling Systems For Favorable Games

paper_content:

The following sections are included:IntroductionThe nature of Λ*The asymptotic time minimization problemAsymptotic magnitude problemProblems with finite goals in coin tossingREFERENCES
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paper_title: Capital Growth and the Mean-Variance Approach to Portfolio Selection

paper_content:

Three main approaches to the problem of portfolio selection may be discerned in the literature. The first of these is the mean-variance approach, pioneered by Markowitz [21], [22], and Tobin [30]. The second approach is that of chance-constrained programming, apparently initiated by Naslund and Whinston [26]. The third approach, LatanA© [19] and Breiman [6], [7], has its origin in capital growth considerations. The purpose of this paper is to contrast the mean-variance model, by far the most well-known and most developed model of portfolio selection, with the capital growth model, undoubtedly the least known. In so doing, we shall find the mean-variance model to be severely compromised by the capital growth model in several significant respects.
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paper_title: On the Competitive Theory and Practice of Portfolio Selection (Extended Abstract)

paper_content:

Given a set of say m stocks (one of which may be “cash”), the online portfolio selection problem is to determine a portfolio for the i th trading period based on the sequence of prices for the preceding i – 1 trading periods. Competitive analysis is based on a worst case perspective and such a perspective is inconsistent with the more widely accepted analyses and theories based on distributional assumptions. The competitive framework does (perhaps surprisingly) permit non trivial upper bounds on relative performance against CBAL-OPT, an optimal offline constant rebalancing portfolio. Perhaps more impressive are some preliminary experimental results showing that certain algorithms that enjoy “respectable” competitive (i.e. worst case) performance also seem to perform quite well on historical sequences of data. These algorithms and the emerging competitive theory are directly related to studies in information theory and computational learning theory and indeed some of these algorithms have been pioneered within the information theory and computational learning communities. We present a mixture of both theoretical and experimental results, including a more detalied study of the performance of existing and new algorithms with respect to a standard sequence of historical data cited in many studies. We also present experiments from two other historical data sequences.
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paper_title: Mean reversion in stock prices: evidence from emerging markets

paper_content:

This paper investigates whether stock‐price indexes of emerging markets can be characterized as random walk (unit root) or mean reversion processes. We implement a panelbased test that exploits cross‐sectional information from seventeen emerging equity markets during the period January 1985 to April 2002. The gain in power allows us to reject the null hypothesis of random walk in favor of mean reversion at the 5 percent significance level. We find a positive speed of reversion with a half‐life of about 30 months. These results are similar to those documented for developed markets. Our findings provide an interesting comparison to existing studies on more matured markets and reduce the likelihood of earlier mean reversion findings as attributable to data mining.
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paper_title: Seasonality in Stock Price Mean Reversion: Evidence from the U.S. and the U.K.

paper_content:

The evidence of slowly mean-reverting components in stock prices has been controversial. The hypothesis of stock price mean-reversion is tested using a regression model that yields the highest asymptotic power among a class of regression tests. Although the evidence that the equally weighted index of stocks exhibits mean-reversion is significant in the period 1926-88, this phenomenon is entirely concentrated in January. In the postwar period, both the equally weighted and the value-weighted indices exhibit seasonal mean-reversion in January. A similar phenomenon is also observed for the equally weighted index of stocks traded on the London Stock Exchange. Copyright 1991 by American Finance Association.
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paper_title: The 52–week high and momentum investing

paper_content:

When coupled with a stock's current price, a readily available piece of information-the 52-week high price-explains a large portion of the profits from momentum investing. Nearness to the 52-week high dominates and improves upon the forecasting power of past returns (both individual and industry returns) for future returns. Future returns forecast using the 52-week high do not reverse in the long run. These results indicate that short-term momentum and long-term reversals are largely separate phenomena, which presents a challenge to current theory that models these aspects of security returns as integrated components of the market's response to news. Copyright 2004 by The American Finance Association.
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paper_title: Price Momentum and Trading Volume

paper_content:

This study shows that past trading volume provides an important link between “momentum” and “value” strategies. Specifically, we find that firms with high ~low! past turnover ratios exhibit many glamour ~value! characteristics, earn lower ~higher! future returns, and have consistently more negative ~positive! earnings surprises over the next eight quarters. Past trading volume also predicts both the magnitude and persistence of price momentum. Specifically, price momentum effects reverse over the next five years, and high ~low! volume winners ~losers! experience faster reversals. Collectively, our findings show that past volume helps to reconcile intermediate-horizon “underreaction” and long-horizon “overreaction” effects. FINANCIAL ACADEMICS AND PRACTITIONERS have long recognized that past trading volume may provide valuable information about a security. However, there is little agreement on how volume information should be handled and interpreted. Even less is known about how past trading volume interacts with past returns in the prediction of future stock returns. Stock returns and trading volume are jointly determined by the same market dynamics, and are inextricably linked in theory ~e.g., Blume, Easley, and O’Hara ~1994!!. Yet prior empirical studies have generally accorded them separate treatment. In this study, we investigate the usefulness of trading volume in predicting cross-sectional returns for various price momentum portfolios. The study is organized into two parts. In the first part, we document the interaction between past returns and past trading volume in predicting future returns
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paper_title: Market States and Momentum

paper_content:

We test overreaction theories of short-run momentum and long-run reversal in the cross section of stock returns. Momentum profits depend on the state of the market, as predicted. From 1929 to 1995, the mean monthly momentum profit following positive market returns is 0.93%, whereas the mean profit following negative market returns is −0.37%. The up-market momentum reverses in the long-run. Our results are robust to the conditioning information in macroeconomic factors. Moreover, we find that macroeconomic factors are unable to explain momentum profits after simple methodological adjustments to take account of microstructure concerns. SEVERAL BEHAVIORAL THEORIES have been developed to jointly explain the shortrun cross-sectional momentum in stock returns documented by Jegadeesh and Titman (1993) and the long-run cross-sectional reversal in stock returns documented by DeBondt and Thaler (1985). 1 Daniel, Hirshleifer, and Subrahmanyam (1998; hereafter DHS) and Hong and Stein (1999; hereafter HS) each employ different behavioral or cognitive biases to explain these anomalies. 2, 3
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paper_title: International Momentum Strategies

paper_content:

International equity markets exhibit short-term return continuation. Between 1980 and 1995 an internationally diversified portfolio of past short-term winners outperformed a portfolio of short-term losers by more than one percent per month, after correcting for risk. Return continuation is present in all twelve sample countries and lasts for about one year. Return continuation is negatively related to firm size but is not limited to small firms. The international evidence is remarkably similar to findings for the U.S. by Jegadeesh and Titman (1993) and makes it unlikely that the U.S. experience was simply due to chance. Because momentum strategies are relatively easy to implement, the results pose a
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paper_title: Isolated and Connected Word Recognition-Theory and Selected Applications

paper_content:

The art and science of speech recognition have been advanced to the state where it is now possible to communicate reliably with a computer by speaking to it in a disciplined manner using a vocabulary of moderate size. It is the purpose of this paper to outline two aspects of speech-recognition research. First, we discuss word recognition as a classical pattern-recognition problem and show how some fundamental concepts of signal processing, information theory, and computer science can he combined to give us the capability of robust recognition of isolated words and simple connected word sequences. We then describe methods whereby these principles, augmented by modern theories of formal language and semantic analysis, can be used to study some of the more general problems in speech recognition. It is anticipated that these methods will ultimately lead to accurate mechanical recognition of fluent speech under certain controlled conditions. I
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paper_title: Dynamic Volume-Return Relation of Individual Stocks

paper_content:

We examine the dynamic relation between return and volume of individual stocks. Using a simple model in which investors trade to share risk or speculate on private information, we show that returns generated by risk-sharing trades tend to reverse themselves while returns generated by speculative trades tend to continue themselves. We test this theoretical prediction by analyzing the relation between daily volume and first-order return autocorrelation for individual stocks listed on the NYSE and AMEX. We find that the cross-sectional variation in the relation between volume and return autocorrelation is related to the extent of informed trading in a manner consistent with the theoretical prediction.
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paper_title: EFFICIENT CAPITAL MARKETS: A REVIEW OF THEORY AND EMPIRICAL WORK*

paper_content:

Efficient Capital Markets: A Review of Theory and Empirical Work Author(s): Eugene F. Fama Source: The Journal of Finance, Vol. 25, No. 2, Papers and Proceedings of the Twenty-Eighth Annual Meeting of the American Finance Association New York, N.Y. December, 28-30, 1969 (May, 1970), pp. 383-417 Published by: Blackwell Publishing for the American Finance Association Stable URL: http://www.jstor.org/stable/2325486 Accessed: 30/03/2010 21:28
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paper_title: Do Seasonal Anomalies Still Work

paper_content:

Dzhabarov and Ziemba investigate whether traditional seasonal anomalies, such as the January effect,monthly effect, January barometer, sell-in-May-and-go-away phenomenon, holiday effect, and turn-of-the-month effect, still exist in the turbulent markets of the early part of the 21st century.The evidence indicates that there is still value in these anomalies. In their study, the authors use futures data from 1993 to 2009 and from 2004 to 2009 for small-cap stocks measured by the Russell 2000 Index and for large-cap stocks measured by the S&P 500 Index.As was true in the 1990s, the effects tend to be stronger in small-cap stocks.The results are useful for investors who wish to tilt portfolios and for speculators who wish to trade the effects.
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paper_title: Dynamic programming algorithm optimization for spoken word recognition

paper_content:

This paper reports on an optimum dynamic progxamming (DP) based time-normalization algorithm for spoken word recognition. First, a general principle of time-normalization is given using time-warping function. Then, two time-normalized distance definitions, called symmetric and asymmetric forms, are derived from the principle. These two forms are compared with each other through theoretical discussions and experimental studies. The symmetric form algorithm superiority is established. A new technique, called slope constraint, is successfully introduced, in which the warping function slope is restricted so as to improve discrimination between words in different categories. The effective slope constraint characteristic is qualitatively analyzed, and the optimum slope constraint condition is determined through experiments. The optimized algorithm is then extensively subjected to experimental comparison with various DP-algorithms, previously applied to spoken word recognition by different research groups. The experiment shows that the present algorithm gives no more than about two-thirds errors, even compared to the best conventional algorithm.
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paper_title: Evidence of Predictable Behavior of Security Returns

paper_content:

This paper presents new empirical evidence of predictability of individual stock returns. The negative first-order serial correlation in monthly stock returns is highly significant. Furthermore, significant positive serial correlation is found at longer lags, and the twelve-month serial correlation is particularly strong. Using the observed systematic behavior of stock return, one-step-ahead return forecasts are made and ten portfolios are formed from the forecasts. The difference between the abnormal returns on the extreme decile portfolios over the period 1934-87 is 2.49 percent per month. Copyright 1990 by American Finance Association.
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paper_title: Universal portfolios with side information

paper_content:

We present a sequential investment algorithm, the /spl mu/-weighted universal portfolio with side information, which achieves, to first order in the exponent, the same wealth as the best side-information dependent investment strategy (the best state-constant rebalanced portfolio) determined in hindsight from observed market and side-information outcomes. This is an individual sequence result which shows the difference between the exponential growth wealth of the best state-constant rebalanced portfolio and the universal portfolio with side information is uniformly less than (d/(2n))log (n+1)+(k/n)log 2 for every stock market and side-information sequence and for all time n. Here d=k(m-1) is the number of degrees of freedom in the state-constant rebalanced portfolio with k states of side information and m stocks. The proof of this result establishes a close connection between universal investment and universal data compression.
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paper_title: The evolution of the January effect

paper_content:

This paper extends recent studies of the January effect by investigating the evolution of the daily pattern of the effect across size deciles. Our evidence documents a sizable mean reverting component beginning in the latter part of January and a shorter duration of the seasonal effect. Despite lower abnormal returns in the second part of January, higher abnormal returns in the first part of January keep the magnitude of the January effect unchanged. Further analysis of daily trading volumes suggests a stable trading volume intensity in the first part of January and a substantial decline in trading volume intensity in the second part of January.
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paper_title: The persistent holiday effect: additional evidence

paper_content:

The holiday effect is one of the most perplexing of all seasonal anomalies. Based on evidence using pre-1987 equity returns, this anomaly has been shown to be responsible for somewhere between 30 to 50% of the total return on the market while exhibiting below average variances (Lakonishok and Smidt, 1988; Ariel, 1990). The purpose of this paper is to investigate and document the persistence of the holiday effect (or lack thereof) beyond the 1987 period for equities traded on the NYSE, AMEX, and NASDAQ exchanges. Has the holiday effect continued into the 1990s, or have efficient trading rules exploited the anomaly to the point where it no longer exists? The results provide additional evidence into the nature and potential causes of seasonal market anomalies.
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paper_title: The incredible January effect : the stock market's unsolved mystery

paper_content:

Based on historical research, the authors show readers that certain kinds of securities have been producing unaccountably high returns during the first month of the year for decades. The Incredible January Effect shows how to implement an investment strategy to capture the benefits of the January effect; and which investment opportunities are most lucrative in the following categories: financial futures, index options, mutual funds, or common stocks.
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paper_title: Exact indexing of dynamic time warping

paper_content:

The problem of indexing time series has attracted much interest. Most algorithms used to index time series utilize the Euclidean distance or some variation thereof. However, it has been forcefully shown that the Euclidean distance is a very brittle distance measure. Dynamic time warping (DTW) is a much more robust distance measure for time series, allowing similar shapes to match even if they are out of phase in the time axis. Because of this flexibility, DTW is widely used in science, medicine, industry and finance. Unfortunately, however, DTW does not obey the triangular inequality and thus has resisted attempts at exact indexing. Instead, many researchers have introduced approximate indexing techniques or abandoned the idea of indexing and concentrated on speeding up sequential searches. In this work, we introduce a novel technique for the exact indexing of DTW. We prove that our method guarantees no false dismissals and we demonstrate its vast superiority over all competing approaches in the largest and most comprehensive set of time series indexing experiments ever undertaken.
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paper_title: Capital market seasonality: The case of stock returns

paper_content:

Abstract In this paper we present evidence on the existence of seasonality in monthly rates of return on the New York Stock Exchange from 1904–1974. With the exception of the 1929–1940 period, there are statistically significant differences in mean returns among months due primarily to large January returns. Dispersion measures reveal no consistent seasonal patterns and the characteristic exponent seems invariant among months. We also explore possible implications of the observed seasonality for the capital asset pricing model and other research.
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paper_title: The Mathematics of Money Management: Risk Analysis Techniques for Traders

paper_content:

The Empirical Techniques. Characteristics of Fixed Fractional Trading and Salutary Techniques. Parametric Optimal f on the Normal Distribution. Parametric Techniques on Other Distributions. Introduction to Multiple Simultaneous Positions Under the Parametric Approach. Correlative Relationships and the Derivation of the Efficient Frontier. The Geometry of Portfolios. Risk Management. Appendices. Bibliography and Suggested Reading. Index.
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paper_title: Growth Optimal Investment with Transaction Costs ⋆

paper_content:

Discrete time infinite horizon growth optimal investment in stock markets with transactions costs is considered. The stock processes are modelled by homogeneous Markov processes. Assuming that the distribution of the market process is known, we show two recursive investment strategies such that, in the long run, the growth rate on trajectories (in "liminf" sense) is greater than or equal to the growth rate of any other investment strategy with probability 1.
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paper_title: Portfolio Management Formulas: Mathematical Trading Methods for the Futures, Options, and Stock Markets

paper_content:

The Random Process and Gambling Theory. Systems and Optimization. Reinvestment of Returns and Geometric Growth Concepts. Optimal Fixed Fractional Trading. Risk of Ruin. The Total Portfolio Approach. Covering the Periphery. Appendices. Bibliography and Suggested Reading. Index.
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paper_title: Universal Portfolios With and Without Transaction Costs

paper_content:

A constant rebalanced portfolio is an investment strategy which keeps the same distribution of wealth among a set of stocks from period to period. Recently there has been work on on-line investment strategies that are competitive with the best constant rebalanced portfolio determined in hindsight (Cover, 1991, 1996; Helmbold et al., 1996; Cover & Ordentlich, 1996a, 1996b; Ordentlich & Cover, 1996). For the universal algorithm of Cover (Cover, 1991),we provide a simple analysis which naturallyextends to the case of a fixed percentage transaction cost (commission ), answering a question raised in (Cover, 1991; Helmbold et al., 1996; Cover & Ordentlich, 1996a, 1996b; Ordentlich & Cover, 1996; Cover, 1996). In addition, we present a simple randomized implementation that is significantly faster in practice. We conclude by explaining how these algorithms can be applied to other problems, such as combining the predictions of statistical language models, where the resulting guarantees are more striking.
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paper_title: The Kelly Criterion in Blackjack Sports Betting, and the Stock Market

paper_content:

The central problem for gamblers is to find positive expectation bets. But the gambler also needs to know how to manage his money, i.e., how much to bet. In the stock market (more inclusively, the securities markets) the problem is similar but more complex. The gambler, who is now an “investor”, looks for “excess risk adjusted return”. In both these settings, we explore the use of the Kelly criterion, which is to maximize the expected value of the logarithm of wealth (“maximize expected logarithmic utility”). The criterion is known to economists and financial theorists by names such as the “geometric mean maximizing portfolio strategy”, maximizing logarithmic utility, the growth-optimal strategy, the capital growth criterion, etc. The author initiated the practical application of the Kelly criterion by using it for card counting in blackjack. We will present some useful formulas and methods to answer various natural questions about it that arise in blackjack and other gambling games. Then we illustrate its recent use in a successful casino sports betting system. Finally, we discuss its application to the securities markets where it has helped the author to make a thirty year total of 80 billion dollars worth of “bets”.
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paper_title: The Leverage Space Trading Model: Reconciling Portfolio Management Strategies and Economic Theory

paper_content:

Preface. Introduction. PART I The Single Component Case: Optimal f . CHAPTER 1 The General History of Geometric Mean Maximization. CHAPTER 2 The Ineluctable Coordinates. CHAPTER 3 The Nature of the Curve. PART II The Multiple Component Case: The Leverage Space Portfolio Model. CHAPTER 4 Multiple, Simultaneous f -"Leverage Space". CHAPTER 5 Risk Metrics in Leverage Space and Drawdown. PART III The Leverage Space Praxis. CHAPTER 6 A Framework to Satisfy Both Economic Theory and Portfolio Managers. CHAPTER 7 Maximizing the Probability of Profit. Bibliography. Index.
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paper_title: The New Money Management: A Framework for Asset Allocation

paper_content:

A New Framework. Laws of Growth, Utility, and Finite Streams. Conditional Probabilities Involving Correlation. A New Model. Money Management for Money Managers. Bibliography and Sources. Index.
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paper_title: Long/Short Equity Investing

paper_content:

Investors who have the flexibility to invest both long and short can benefit from both “winners” and “losers.” This will be especially advantageous if the latter — the short-sale candidates — are less efficiently priced than the winners — the purchase candidates. This is likely to be the case in markets in which investors hold diverse opinions and short selling is restricted. Short positions can be combined with long positions to create market-neutral, hedge, or equitized strategies. Practical issues include restrictions on shorting, trading requirements, custody issues, and tax treatment.
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Title: Online Portfolio Selection: A Survey

Section 1: INTRODUCTION
Description 1: Introduce the fundamental research problem of online portfolio selection, the major theories underpinning it, and an overview of different algorithm categories.

Section 2: Scope
Description 2: Clarify the specific focus of the survey, the aspects it will cover, and the related topics that are excluded.

Section 3: Related Surveys
Description 3: Summarize existing related surveys and explain how this survey is different and more comprehensive.

Section 4: Organization
Description 4: Provide an overview of how the survey is structured and what each section will cover.

Section 5: PROBLEM SETTING
Description 5: Formally define the online portfolio selection problem, introduce the financial market model, and discuss practical issues and assumptions.

Section 6: ONLINE PORTFOLIO SELECTION APPROACHES
Description 6: Survey the various online portfolio selection algorithms including benchmarks and approaches focusing on different strategies (Follow the Winner, Follow the Loser, Pattern-Matching, Meta-Learning).

Section 7: Benchmarks
Description 7: Introduce the common baseline strategies used for comparison in online portfolio selection.

Section 8: Follow-the-Winner Approaches
Description 8: Explain strategies that increase the weights of more successful stocks or experts, including Universal Portfolios and Exponential Gradient.

Section 9: Follow-the-Loser Approaches
Description 9: Discuss strategies that transfer wealth from winners to losers, following the principle of mean reversion, such as Anti-Correlation and PAMR.

Section 10: Pattern-Matching Based Approaches
Description 10: Describe nonparametric and pattern-matching strategies that use historical data to predict and optimize the portfolio.

Section 11: Meta-Learning Algorithms
Description 11: Cover approaches that combine multiple strategies or experts to form a final portfolio, focusing on learning from their performances.

Section 12: CONNECTION WITH CAPITAL GROWTH THEORY
Description 12: Connect the surveyed algorithms to Capital Growth Theory, illustrating the theoretical principles behind the trading strategies.

Section 13: CHALLENGES AND FUTURE DIRECTIONS
Description 13: Identify the current challenges in the field of online portfolio selection and propose future research directions for more effective solutions.

Section 14: CONCLUSIONS
Description 14: Summarize the main points of the survey, the current state of research, and the potential for future exploration in online portfolio selection.