Nonlinear methods trees and neural networks place great emphasis on exactly those predictors ignored by linear methods, such as term spreads and issuance activity. Many accounting characteristics are not available at the monthly frequency, which might explain their low importance in Figure 5. To investigate this, Figure A.
Price trend variables become less important compared to the liquidity and risk measures, although they are still quite influential. The characteristics that were ranked in the bottom half of predictors at the monthly horizon remain largely unimportant at the annual horizon. The exception is industry sic2 , which shows substantial predictive power at the annual frequency.
Figure 6 traces out the model-implied marginal impact of individual characteristics on expected excess returns. Our data transformation normalizes characteristics to the -1,1 interval, and holds all other variables fixed at their median value of zero.
We choose four illustrative characteristics for the figure, including size mvel1 , momentum mom12m , stock volatility retvol , and accruals acc. First, Figure 6 illustrates that machine learning methods identify patterns similar to some well-known empirical phenomena.
For example, expected stock returns are decreasing in size, increasing in past 1-year return, and decreasing in stock volatility. And, interestingly, all methods agree on a nearly exact zero relationship between accruals and future returns. Second, the penalized linear model finds no predictive association between returns and either size or volatility, while trees and neural networks find large sensitivity of expected returns to both of these variables. For example, a firm that drops from median size to the 20th percentile of the size distribution experiences an increase in its annualized expected return of roughly 2.
The inability of linear models to capture nonlinearities can lead them to prefer a zero association, and this can in part explain the divergence in the performance of linear and nonlinear methods. The panels should the sensitivity of expected monthly percentage returns vertical axis to the individual characteristics holding all other covariates fixed at their median values.
The favorable performance of trees and neural networks indicates a benefit to allowing for potentially complex interactions among predictors. They are, however, complex, and this is the source of both their power and their opacity. Any exploration of interaction effect is vexed by vast possibilities for identity and functional forms for interacting predictors.
In this section, we present a handful of interaction results to help illustrate the inner workings of one black box method, the NN3 model. As a first example, we examine a set of pairwise interaction effects in NN3.
Figure 7 reports how expected returns vary as we simultaneously vary values of a pair of characteristics over their support [-1,1], while holding all other variables fixed at their median value of zero. We show interactions of stock size mvel1 with four other predictors: short-term reversal mom1m , momentum mom12m , and total and idiosyncratic volatility retvol and idiovol, respectively. The upper-left figure shows that the short-term reversal effect is strongest and is essentially linear among small stocks blue line.
Among large stocks green line , reversal is concave, occurring primarily when the prior month return is positive. The upper-right figure shows the momentum effect, which is most pronounced among large stocks for the NN3 model. For small stocks, the volatility effect is hump shaped. Finally, the lower-right side shows that NN3 estimates no interaction effect between size and accruals—the size lines are simply vertical shifts of the univariate accruals curve.
The panels should the sensitivity of the expected monthly percentage returns vertical axis to the interactions effects for mvel1 with mom1m, mom12m, retvol, and acc in model NN3 holding all other covariates fixed at their median values. The panels show the sensitivity of expected monthly percentage returns vertical axis to interactions effects for mvel1 and retvol with bm and ntis in model NN3 holding all other covariates fixed at their median values.
Figure 8 illustrates interactions between stock-level characteristics and macroeconomic indicator variables. It shows, for example, that the size effect is more pronounced when aggregate valuations are low bm is high and when equity issuance ntis is low, while the low volatility anomaly is especially strong in high valuation and high issuance environments.
Furthermore, the dominant macroeconomic interactions are stable over time, as illustrated in Figure A. So far, we have analyzed predictability of individual stock returns. Next, we compare forecasting performance of machine learning methods for aggregate portfolio returns. Analyzing forecasts at the portfolio-level comes with a number of benefits. First, because all of our models are optimized for stock-level forecasts, portfolio forecasts provide an additional indirect evaluation of the model and its robustness.
Second, aggregate portfolios tend to be of broader economic interest because they represent the risky-asset savings vehicles most commonly held by investors via mutual funds, ETFs, and hedge funds. Third, the distribution of portfolio returns is sensitive to dependence among stock returns, with the implication that a good stock-level prediction model is not guaranteed to produce accurate portfolio-level forecasts.
The final advantage of analyzing predictability at the portfolio level is that we can assess the economic contribution of each method via its contribution to risk-adjusted portfolio return performance. This bottom-up approach works for any target portfolio whose weights are known a priori. In all cases, we create the portfolios ourselves using CRSP market equity value weights.
In contrast, all nonlinear models have substantial positive predictive performance. Nonlinear methods excel. In short, machine learning methods and nonlinear methods, in particular, produce unusually powerful out-of-sample portfolio predictions.
For characteristic-based portfolios, nonlinear machine learning methods help improve Sharpe ratios by anywhere from a few percentage points to over 24 percentage points. Campbell and Thompson also propose evaluating the economic magnitude of portfolio predictability with a market timing trading strategy. Table 6 reports the annualized Sharpe ratio gains relative to a buy-and-hold strategy for timing strategies based on machine learning forecasts.
Consistent with our other results, the strongest and most consistent trading strategies are those based on nonlinear models, with neural networks the best overall. As a robustness test, we also analyze bottom-up predictions for annual rather than monthly returns.
The comparative patterns in predictive performance across methods is the same in annual and monthly data. In this table, we report the performance of prediction-sorted portfolios over the year out-of-sample testing period.
All stocks are sorted into deciles based on their predicted returns for the next month. All portfolios are value weighted. Next, rather than assessing forecast performance among prespecified portfolios, we design a new set of portfolios to directly exploit machine learning forecasts. At the end of each month, we calculate 1-month-ahead out-of-sample stock return predictions for each method. We reconstitute portfolios each month using value weights.
Finally, we construct a zero-net-investment portfolio that buys the highest expected return stocks decile 10 and sells the lowest decile 1. Table 7 reports results. Out-of-sample portfolio performance aligns very closely with results on machine learning forecast accuracy reported earlier. Realized returns generally increase monotonically with machine learning forecasts from every method with occasional exceptions, such as decile 8 of NN1.
Neural network models again dominate linear models and tree-based approaches. In particular, for all but the most extreme deciles, the quantitative match between predicted returns and average realized returns using neural networks is extraordinarily close. The best 10—1 strategy comes from NN4, which returns on average 2.
Its monthly volatility is 5. While value-weight portfolios are less sensitive to trading cost considerations, it is perhaps more natural to study equal weights in our analysis because our statistical objective functions minimize equally weighted forecast errors. The qualitative conclusions of this table are identical to those of Table 7 , but the Sharpe ratios are substantially higher.
For example, the long-short decile spread portfolio based on the NN4 model earns an annualized Sharpe ratio of 2. To identify the extent to which equal-weight results are driven by micro-cap stocks, Table A. In this case, the NN4 long-short decile spread earns a Sharpe ratio of 1. As recommended by Lewellen , the OLS-3 model is an especially parsimonious and robust benchmark model.
He also recommends somewhat larger OLS benchmark models with either 7 or 15 predictors, which we report in Table A. The larger OLS models improve over OLS-3, but are nonetheless handily outperformed by tree-based models and neural networks. Drawdowns, turnover, and risk-adjusted performance of machine learning portfolios. Not only do neural network portfolios have higher Sharpe ratios than alternatives, they also have comparatively small drawdowns, particularly for equal-weight portfolios.
The maximum drawdown experienced for the NN4 strategy is In contrast, the maximum drawdown for OLS-3 are Equal-weight neural network strategies also experience the most mild 1-month losses. The bottom panel of Table 8 reports risk-adjusted performance of machine learning portfolios based on factor pricing models. In a linear factor model, the tangency portfolio of the factors themselves represents the maximum Sharpe ratio portfolio in the economy.
Any portfolio with a higher Sharpe ratio than the factor tangency portfolio possesses alpha with respect to the model. From prior work, the out-of-sample factor tangency portfolios of the Fama-French three and five-factor models have Sharpe ratios of roughly 0. It is unsurprising then that portfolios formed on the basis of machine learning forecasts earn large and significant alphas versus these models. As a result, neural networks have information ratios ranging from 0. Test statistics for the associated alphas are highly significant for both tree models and all neural network models.
The figure shows the cumulative log returns of portfolios sorted on out-of-sample machine learning return forecasts. The solid and dashed lines represent long top decile and short bottom decile positions, respectively. The shaded periods show NBER recession dates.
The results of Table 8 are visually summarized in Figure 9 , which reports cumulative performance for the long and short sides for select strategies, along with the cumulative market excess return as a benchmark. NN4 dominates the other models by a large margin in both directions. Interestingly, the short side of all portfolios is essentially flat in the post sample.
Finally, we consider two metastrategies that combine forecasts of all machine learning portfolios. The first is a simple equally weighted average of decile long-short portfolios from our eleven machine learning methods.
The value weighted decile spread Sharpe ratio is 1. Using the empirical context of return prediction as a proving ground, we perform a comparative analysis of methods in the machine learning repertoire.
At the highest level, our findings demonstrate that machine learning methods can help improve our empirical understanding of asset prices. Neural networks and, to a lesser extent, regression trees, are the best performing methods.
We track down the source of their predictive advantage to accommodation of nonlinear interactions that are missed by other methods. Machine learning methods are most valuable for forecasting larger and more liquid stock returns and portfolios. Lastly, we find that all methods agree on a fairly small set of dominant predictive signals, the most powerful predictors being associated with price trends including return reversal and momentum.
The next most powerful predictors are measures of stock liquidity, stock volatility, and valuation ratios. The overall success of machine learning algorithms for return prediction brings promise for both economic modeling and for practical aspects of portfolio choice.
With better measurement through machine learning, risk premiums are less shrouded in approximation and estimation error, thus the challenge of identifying reliable economic mechanisms behind asset pricing phenomena becomes less steep.
Finally, our findings help justify the growing role of machine learning throughout the architecture of the burgeoning fintech industry.
We gratefully acknowledge the computing support from the Research Computing Center at the University of Chicago. The views and opinions expressed are those of the authors and do not necessarily reflect the views of AQR Capital Management, its affiliates, or its employees; do not constitute an offer, solicitation of an offer, or any advice or recommendation, to purchase any securities or other financial instruments, and may not be construed as such.
Supplementary data can be found on The Review of Financial Studies web site. One may be interested in potentially distinguishing between different components of expected returns, such as those due to systematic risk compensation, idiosyncratic risk compensation, or even due to mispricing.
They note that this is only a subset of those studied in the literature. Welch and Goyal analyze nearly twenty predictors for the aggregate market return. In both stock and aggregate return predictions, there presumably exists a much larger set of predictors that were tested but failed to predict returns and were thus never reported. For an application of the kernel trick to the cross-section of returns, see Kozak This usage is related to, but different from, its meaning in Bayesian statistics as a parameter of a prior distribution.
Although most theoretical analysis from high-dimensional statistics assume that data have sub-Gaussian or subexponential tails, Fan et al.
Freyberger, Neuhierl, and Weber offers a similar model in the return prediction context. Breiman et al. Ensemble methods demonstrate more reliable performance and are scalable for very large data sets, leading to their increased popularity in recent literature. Friedman et al. Ever since the seminal work of Hinton, Osindero, and Teh , the machine learning community has experimented and adopted deeper and wider networks, with as many as layers for image recognition see, e. See Wilson and Martinez In certain circumstances, early stopping and weight decay are shown to be equivalent.
See, for example, Bishop and Goodfellow, Bengio, and Courville Note that, because of nondifferentiabilities in tree-based models, the Dimopoulos, Bourret, and Lek method is not applicable. Therefore, when we conduct this second variable importance analysis, we measure variable importance for random forests and boosted trees using mean decrease in impurity see, e. We select the largest possible pool of assets for at least three important reasons. Moreover, because we aggregate individual stock return predictions to predict the index, we cannot omit such stocks.
Second, our results are less prone to sample selection or data-snooping biases that the literature see, e. Third, using a larger sample helps avoid overfitting by increasing the ratio of observation count to parameter count. That said, our results are qualitatively identical and quantitively unchanged if we filter out these firms. Our data construction differs by more closely adhering to variable definitions in original papers. For example, we construct book-equity and operating profitability following Fama and French Most of these characteristics are released to the public with a delay.
To avoid the forward-looking bias, we assume that monthly characteristics are delayed by at most 1 month, quarterly with at least 4 months lag, and annual with at least 6 months lag. Another issue is missing characteristics, which we replace with the cross-sectional median at each month for each stock, respectively. Evidently, the historical mean is such a noisy forecaster that it is easily beaten by a fixed excess return forecasts of zero.
Doing so distorts the size of tests through a selection bias. We also note that false discoveries in multiple comparisons should be randomly distributed. The rank of important characteristics top-third of the covariates is remarkably stable over time. This is true for all models, though we show results for one representative model NN3 in the interest of space. Bishop, C. Neural networks for pattern recognition.
Google Scholar. Google Preview. Box, G. Non-normality and tests on variances. Biometrika 40 : — Breiman, L. Random forests. Machine Learning 45 : 5 — Classification and regression trees. Butaru, F. Risk and risk management in the credit card industry.
Campbell, J. Predicting excess stock returns out of sample: Can anything beat the historical average? Review of Financial Studies 21 : — Cochrane, J. The dog that did not bark: A defense of return predictability. Cybenko, G. Approximation by superpositions of a sigmoidal function.
Mathematics of Control, Signals and Systems 2 : — Chemometrics and Intelligent Laboratory Systems 18 : — Diebold, F. Comparing Predictive Accuracy. Dietterich, T. Ensemble methods in machine learning. In International workshop on multiple classier systems , eds. Schwenker F. New York : Springer. Dimopoulos, Y. Use of some sensitivity criteria for choosing networks with good generalization ability. Neural Processing Letters 2 : 1 — 4. Eldan, R. The power of depth for feedforward neural networks.
In 29th Annual Conference on Learning Theory , eds. Feldman V. Brookline, MA : Microtome Publishing. Fama, E. Common risk factors in the returns on stocks and bonds. Journal of Financial Economics 33 : 3 — Dissecting anomalies. Journal of Finance 63 : — A five-factor asset pricing model. Journal of Financial Economics : 1 — When you copy a file by using BITS in background mode, the file is copied in multiple small parts.
If you're behind a proxy server or behind a firewall that removes this header, the file copy operation is unsuccessful. Modify the proxy server settings to support HTTP 1.
If you can't modify the proxy server in this manner, configure BITS to work in foreground mode. To do this, follow these steps:. Click Start , click Run , type one of the following commands, and then click OK. I guess its just The warning from easy anticheat that ur running to many ". You can see all this. For me it worked to close a few gameluncher i didnt use and close a few tabs at google, because every tab from your Internet explorer is running a own ".
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