Various theories have been proposed to explain momentum in stock returns. We provide evidence in favor of risk-based explanations. Specifically, we construct self-financing market neutral portfolios that take long positions in past winners and short positions in past losers. We show that the return spreads between past winners and losers in the first year are driven primarily by high volatility stocks. Momentum investment strategies, which buy past winning stocks and sell past losing stocks, are significantly less profitable once we control for volatility. We also show that momentum profits appear to be associated with economic cycles as proxied by the prime rate. The momentum strategies are substantially stronger during expansions than during recessions. This is mainly due to the relatively poor performance of past losers (rather than superior performance of the past winners) during expansions.

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.

In previous work, we have found statistically significant structures in tick-by-tick interbank foreign exchange (FX) price series on various time scales. These structures include negative autocorrelations in successive tick-by-tick returns and positive autocorrelations (trends) on longer time scales. To account for the observed structures, we propose state space models for financial time series in which the observed price is a noisy version of an unobserved, less-noisy "true price" process. The "true prices" in our models are stochastic processes with short-term, long-term, or multi-scale memory structures. The processes we consider include random walks, random trends, and fractional Brownian motions. For fractional Brownian motion processes, we represent the multi-scale correlational structures using self-similar wavelet decompositions. We estimate the state space model parameters using the Kalman filter and EM algorithms. Since both the observational noise and the changes in the true price series have non-gaussian distributions, the Kalman filter and EM algorithms are not able to completely separate the observational noise from the true price components. To improve this separation, we perform a neural-network-based independent component analysis (ICA) using algorithms developed for the blind separation of signals. Statistical analysis of our true price models supports our assertion that the estimated true prices are significantly different from the observed prices, and that significant non-random structures may exist in the FX markets.

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 algorithms. The objective functions that we consider as evaluation functions 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. 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 derive a smoothing regularizer for dynamic network models by requiring robustness in prediction performance to perturbations of the training data. The regularizer can be viewed as a generalization of the first-order Tikhonov stabilizer to dynamic models. For two layer networks with recurrent connections described by the training criterion with the regularizer is where Φ = {U, V, W} is the network parameter set, Z(t) are the targets, I(t) = {X(s), s = 1,2, …, t} represents the current and all historical input information, N is the size of the training data set, [Formula: see text] is the regularizer, and λ is a regularization parameter. The closed-form expression for the regularizer for time-lagged recurrent networks is where ‖ ‖ is the Euclidean matrix norm and γ is a factor that depends upon the maximal value of the first derivatives of the internal unit activations f(). Simplifications of the regularizer are obtained for simultaneous recurrent nets (τ ↦ 0), two-layer feedforward nets, and one layer linear nets. We have successfully tested this regularizer in a number of case studies and found that it performs better than standard quadratic weight decay.

Our previous analysis of tick-by-tick interbank foreign exchange (FX) rates has suggested that the market is not efficient on short time scales. We find that the price changes show mean-reverting rather than random-walk behavior (Moody and Wu, 1994). The results of rescaled range and Hurst exponent analysis presented in the first part of this paper further confirms the mean-reverting attribute in the FX data. The second part of this paper reports on the highly significant correlations between Bid/Ask spreads, volatility and forecastability found in the FX data. These interactions show that higher volatility results in higher forecast error and increased risk for market makers, and that, to compensate for this increase in risk, market makers increase their Bid/Ask spreads.

Selecting a "best subset" of input variables is a critical issue in forecasting. This is especially true when the number of available input series is large, and an exhaustive search through all combinations of variables is computationally infeasible. Inclusion of irrelevant variables not only doesn't help prediction, but can reduce forecasting accuracy through added noise or systematic bias. We demonstrate a technique called "sensitivity-based pruning" (SBP) that removes irrelevant input variables from a nonlinear forecasting or regression model. The technique makes use of a saliency measure computed for each input variable and uses estimates of prediction risk for determining the number of input variables to prune. We present preliminary results of the SBP technique applied to neural network predictors of a key business cycle measure, the US Index of Industrial Production.

The authors present learning rate schedules for fast adaptivek-means clustering which surpass the standard MacQueen learning rate schedule (J. MacQeen, 1967) in speed and quality of solution by several orders of magnitude for largek. The methods accomplish this by largely overcoming the problems of metastable local minima and nonstationarity of cluster region boundaries which plague the MacQueen approach. The authors use simulation results to compare the clustering performances of four learning rate schedules applied to independently sampled data from a uniform distribution in one and two dimensions

We propose a network architecture which uses a single internal layer of locally-tuned processing units to learn both classification tasks and real-valued function approximations (Moody and Darken 1988). We consider training such networks in a completely supervised manner, but abandon this approach in favor of a more computationally efficient hybrid learning method which combines self-organized and supervised learning. Our networks learn faster than backpropagation for two reasons: the local representations ensure that only a few units respond to any given input, thus reducing computational overhead, and the hybrid learning rules are linear rather than nonlinear, thus leading to faster convergence. Unlike many existing methods for data analysis, our network architecture and learning rules are truly adaptive and are thus appropriate for real-time use.

This paper mainly investigates the influences of compressible parameters on the vertical vibration of a pile embedded in layered poroelastic soil media. The pile is treated as a 1D elastic bar by the finite element method, and fundamental solutions for the layered poroelastic soils due to a vertical dynamic load are obtained by the analytical layer element method. Based on the compatibility conditions, the pile-soil dynamic interaction problem is solved. The numerical scheme has been compiled into a Fortran program for numerical calculation. Influences of the pile-soil stiffness ratio, compressible parameters, vibration frequency and the soil stratification are discussed.