Efficient Universal Portfolios for Past-Dependent Target Classes

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.

Annals of Operations Research, 2000

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 ...

Proceedings of the 23rd international conference on Machine learning - ICML '06, 2006

We experimentally study on-line investment algorithms first proposed by Agarwal and Hazan and extended by Hazan et al. which achieve almost the same wealth as the best constant-rebalanced portfolio determined in hindsight. These algorithms are the first to combine optimal logarithmic regret bounds with efficient deterministic computability. They are based on the Newton method for offline optimization which, unlike previous approaches, exploits second order information. After analyzing the algorithm using the potential function introduced by Agarwal and Hazan, we present extensive experiments on actual financial data. These experiments confirm the theoretical advantage of our algorithms, which yield higher returns and run considerably faster than previous algorithms with optimal regret. Additionally, we perform financial analysis using mean-variance calculations and the Sharpe ratio.

Journal of Economic Dynamics and Control, 2003

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 speciÿed 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 di ers 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 in uence 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 speciÿc 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.

2016

In this paper, we consider the problem of portfolio optimization. The risk will be measured by conditional variance or semivariance. It is known that the historical returns used to estimate expected ones provide poor guides to future returns. Consequently, the optimal portfolio asset weights are extremely sensitive to the return assumptions used. Getting informations about the future evolution of different asset returns, could help the investors to obtain more efficient portfolio. The solution will be reached under conditional mean estimation and prediction. This strategy allows us to take advantage from returns prediction which will be obtained by nonparametric univariate methods. Prediction step uses kernel estimation of conditional mean. Application on Chinese and American markets are presented and discussed.

2009

We present a novel efficient algorithm for portfolio selection which theoretically attains two desirable properties: 1. Worst-case guarantee: the algorithm is universal in the sense that it asymptotically performs almost as well as the best constant rebalanced portfolio determined in hindsight from the realized market prices. Furthermore, it attains the tightest known bounds on the regret, or the log-wealth difference relative to the best constant rebalanced portfolio. We prove that the regret of algorithm is bounded by O(log Q), where Q is the quadratic variation of the stock prices. This is the first improvement upon Cover's [Cov91] seminal work that attains a regret bound of O(log T ), where T is the number of trading iterations. 2. Average-case guarantee: in the Geometric Brownian Motion (GBM) model of stock prices, our algorithm attains tighter regret bounds, which are provably impossible in the worst-case. Hence, when the GBM model is a good approximation of the behavior of market, the new algorithm has an advantage over previous ones, albeit retaining worst-case guarantees. We derive this algorithm as a special case of a novel and more general method for online convex optimization with exp-concave loss functions. 1

Mutual Funds, 2019

Since decades, the data science community tries to propose prediction models of financial time series. Yet, driven by the rapid development of information technology and machine intelligence, the velocity of today's information leads to high market efficiency. Sound financial theories demonstrate that in an efficient marketplace all information available today, including expectations on future events, are represented in today prices whereas future price trend is driven by the uncertainty. This jeopardizes the efforts put in designing prediction models. To deal with the unpredictability of financial systems, today's portfolio management is largely based on the Markowitz framework which puts more emphasis in the analysis of the market uncertainty and less in the price prediction. The limitation of the Markowitz framework stands in taking very strong ideal assumptions about future returns probability distribution. To address this situation we propose PAGAN, a pioneering methodo...

2011 IEEE 23rd International Conference on Tools with Artificial Intelligence, 2011

In this paper, we present a novel portfolio optimization method that aims to generalize the delta changes of future returns, based on historical delta changes of returns learned in a past window of time. Our method addresses two issues in portfolio optimization. First, we observe that daily returns of stock prices are very noisy and often non-stationary and dependent. In addition, they do not follow certain welldefined distribution functions, such as the Gaussian distribution. To address this issue, we first aggregate the return values over a multi-day period into an average return in order to reduce the noise of daily returns. We further propose a pre-selection scheme based on stationarity, normality and independence tests in order to select a subset of stocks that have promising statistical properties. Second, we have found that optimizing the average risk in a past window does not typically generalize to future returns with minimal risks. To this end, we develop a portfolio optimization method that uses the delta changes of aggregated returns in a past window to optimize the delta changes of future expected returns. Our experimental studies show that data conditioning and aggregation in our proposed method is an effective means of improving the generalizability while simultaneously minimizing the risk of the portfolio.

In single-period portfolio optimization several facets of the problem may influence the goodness of the portfolios selected. Despite that, some of these facets are frequently ignored when the optimization problem is solved. In this thesis, we aim at investigating the impact of these facets on the optimization problem and on the performances of the portfolios selected. Firstly, we consider the problem of generating scenarios. In the domain of single-period portfolio optimization, scenarios are used to compute the expected value of the portfolio return and the value of a risk (or safety) measure. Historical data observations, taken as equally probable scenarios, are frequently used to this aim. However, several other parametric and non-parametric methods can be alternatively applied. Specifically, when dealing with scenario generation techniques practitioners are mainly concerned on how reliable and effective such methods are when embedded into portfolio selection models. To this aim, we survey different techniques to generate scenarios for the rates of return. We also compare these techniques by providing in-sample and out-of-sample analysis of the portfolios selected using these techniques to generate the rates of return. Evidence on the computational burden required to generate scenarios by the different techniques is also provided. As reference model we use the Conditional Value-at-Risk (CVaR) model with transaction costs. Extensive computational results based on different historical data sets from the London Stock Exchange Market (FTSE) are presented and some interesting financial conclusions are drawn. Secondly, we analyze portfolio optimization when data uncertainty is taken into consideration. Data uncertainty is a common feature in most of real-life optimization problems. In deterministic mathematical optimization, it is assumed that all the input data are known with certainty and equal to some nominal values. Nevertheless, the optimal solution of the nominal problem can reveal itself sub-optimal or even infeasible when some of the data take values different from the nominal ones. An area where data uncertainty is a natural concern is portfolio optimization. As a matter of fact, in portfolio selection every optimization model deals with the estimate of the portfolio rate of return. In the literature several techniques that are immune to data uncertainty have been proposed. These techniques are called robust. We investigate the effectiveness of two well-known robust optimization techniques when applied to a specific portfolio selection problem, and compare the portfolios selected by the corresponding robust counterparts. As reference model we consider the portfolio optimization problem with the CVaR as performance measure. We carried out extensive computational experiments based on real-life data from the London Stock Exchange Market (FTSE) under different market behaviors. Thirdly, we study the optimal portfolio selection problem in a rebalancing framework, considering fixed and proportional transaction costs and evaluating how much they affect a re-investment strategy. Specifically, we modify the single-period portfolio optimization model with transaction costs, based on the CVaR as performance measure, to introduce portfolio rebalancing. The aim is to provide investors and financial institutions with an effective tool to better exploit new information made available by the market. We then suggest a procedure to use the proposed optimization model in a rebalancing framework. Extensive computational results based on different historical data sets from the German Stock Exchange Market (XETRA) are presented. The last part of the thesis is devoted to the problem of replicating the performances of a stock market index, but considering transaction costs and without purchasing all the securities that constitute the index, i.e. the index tracking problem. Furthermore, we also consider the problem of out-performing a market index, i.e. the so-called enhanced index tracking problem. We present mixed-integer linear programming formulations of these two problems. Our formulations include both fixed and proportional transaction costs, a cardinality constraint limiting the number of securities that constitute the portfolio, upper and lower limits on the investment in the securities, and a threshold on the total transaction costs paid. Both models can be used to rebalance a current portfolio composition. We also introduce a heuristic framework, called Enhanced Kernel Search, to solve the problem of tracking an index. We test and analyze the behavior of several implementations of the Enhanced Kernel Search framework. We show the effectiveness and efficiency of the framework comparing the performances of the tested heuristics with those of a general-purpose solver. The computational experiments are carried out using benchmark instances for the index tracking problem.

Proceedings of the …, 2011

Portfolio allocation theory has been heavily influenced by a major contribution of Harry Markowitz in the early fifties: the mean-variance approach. While there has been a continuous line of works in on-line learning portfolios over the past decades, very few works have really tried to cope with Markowitz model. A major drawback of the mean-variance approach is that it is approximation-free only when stock returns obey a Gaussian distribution, an assumption known not to hold in real data. In this paper, we first alleviate this assumption, and rigorously lift the mean-variance model to a more general mean-divergence model in which stock returns are allowed to obey any exponential family of distributions. We then devise a general on-line learning algorithm in this setting. We prove for this algorithm the first lower bounds on the most relevant quantity to be optimized in the framework of Markowitz model: the certainty equivalents. Experiments on four real-world stock markets display its ability to track portfolios whose cumulated returns exceed those of the best stock by orders of magnitude.