Deep Learning, Predictability, and Optimal Portfolio Returns: Babiak & Barunik (2026)

What this is. The paper’s core results, the investor model, and the neural network forecasting method with its defining equations: enough to know what was found and how, without reading the full 22 pages. To replicate or extend, read the source at the original.

TL;DR

The paper asks whether deep neural networks can generate economically meaningful portfolio gains for a long-horizon investor allocating between the US stock market (S&P 500) and a risk-free Treasury bill. Using 12 monthly macroeconomic and financial predictor variables from Welch and Goyal (2008), the paper trains feedforward networks (NN1, NN2, NN3) and an LSTM recurrent network and compares their out-of-sample portfolio performance to linear OLS benchmarks and the no-predictability expectations hypothesis (EH), rebalancing quarterly or annually over February 1955 to December 2018. All deep learning architectures achieve positive out-of-sample R-squared statistics where OLS models uniformly fail, and deliver certainty-equivalent returns and Sharpe ratios above the EH benchmark. For quarterly rebalancing, the LSTM has particularly high CERs and Sharpe ratios during recessions and an incremental advantage with more frequent rebalancing, but feedforward networks perform similarly overall and the LSTM does not dominate uniformly. The paper interprets predictor time-series structure as a source of gains, while also discussing forecast averaging and momentum; its dependence evidence is consistent with, but does not prove, a time-series mechanism. Earlier empirical papers by Feng et al. (2018) and Gu et al. (2020) established that deep learning improves statistical predictions for stock returns; this paper demonstrates the translation into economically significant portfolio performance gains.

Core results

Magnitudes are as reported. Locators point into the source PDF.

#	Result	Locator	Magnitude
R1	Deep NNs achieve positive OOS R-squared; all OLS models fail to beat the historical mean	Table 1, Panel A, p.8	LSTM OOS-R2-q = 1.6% (p=0.002); NN1 = 7.1%, NN2 = 5.1%, NN3 = 5.6% (all p<0.01); OLS1 = -2.5%, OLS2 = -3.6%, OLS3 = -8.0%, OLS4 = -2.5%
R2	LSTM roughly doubles the investor’s CER and triples the Sharpe ratio vs no-predictability benchmark (quarterly rebalancing, whole sample)	Table 2, Panel A, p.10	LSTM CER = 10.007% (p=0.000, SR = 0.175); EH CER = 4.737% (SR = 0.049); NN1 CER = 7.295%, NN2 CER = 6.984%
R3	Deep NN gains are especially large during NBER recessions: LSTM CER about 8.1x higher than EH in downturns	Table 2, Panel A, p.10	LSTM recession CER = 26.770% (p=0.000, SR = 0.358); EH recession CER = 3.311% (SR = -0.193); LSTM expansion CER = 7.998%
R4	Annual rebalancing: NNs still outperform but gains are smaller with less frequent portfolio revision	Table 2, Panel B, p.10	LSTM CER = 5.622% (p=0.012, SR = 0.118); EH CER = 4.542% (SR = 0.048); NN2 CER = 6.879%
R5	Results robust to transaction costs: LSTM CER exceeds 7.9% even at high TC	Table 7, p.17	Low TC (tau = 0.1%): LSTM CER = 9.592% (p=0.000, SR = 0.169); High TC (tau = 0.5%): LSTM CER = 7.910% (p=0.000, SR = 0.145)
R6	Results robust to borrowing and short-selling constraints: NNs maintain higher CERs than EH	Table 8, Panel C, p.18	LSTM with both constraints: CER = 7.775% (p=0.000, SR = 0.150) vs EH CER = 4.737% (SR = 0.049)

Overall (paper’s conclusion). Deep neural networks deliver economically meaningful portfolio gains over the full 1955-2018 sample, with benefits in both business-cycle states and particularly large gains during recessions. In Table 2 Panel A, CER improvements over EH are statistically significant for NN1, NN2, and LSTM, but not NN3; the recession comparison for LSTM is significant. Gains survive transaction costs, borrowing and short-selling constraints, alternative rebalancing horizons, and a post-1969 20-year rolling window robustness check. Feedforward and recurrent networks perform similarly overall; the LSTM often does especially well during recessions and with quarterly rebalancing, but does not uniformly dominate. The authors argue that deep learning exploits nonlinearities and time-series structure in standard predictors without explicitly modeling conditional volatility, while treating the specific mechanism as suggestive rather than conclusively identified.

Theory / model

The paper formulates a standard power utility portfolio choice problem; it proposes no formal asset-pricing model and takes the 12 predictor variables as given. A representative investor with a T-period horizon maximizes expected utility by choosing an allocation $\omega$ between the S&P 500 index and a risk-free Treasury bill.

The power utility function is (p.6):

$$U(r_{p,t+T}) = \frac{r_{p,t+T}^{1-\gamma}}{1-\gamma}, \qquad \gamma = 4 \tag{U}$$

where $r_{p,t+T}$ is the cumulative portfolio return and $\gamma$ is the coefficient of relative risk aversion, calibrated to 4 following Johannes et al. (2014). The optimization problem is (Eq. 7, p.5):

$$\max_\omega \mathbb{E}_t\!\left[U(r_{p,t+T})\right] \tag{7}$$

subject to the cumulative portfolio return (Eq. 8, p.5):

$$r_{p,t+T} = \prod_{\tau=1}^{T}\!\left[(1-\omega_{t+\tau-1})\exp\!\left(r^f_{t+\tau}\right) + \omega_{t+\tau-1}\exp\!\left(r^f_{t+\tau} + r_{t+\tau}\right)\right] \tag{8}$$

where $r^f_{t+\tau}$ is the risk-free rate and $r_{t+\tau}$ is the excess equity log return. Two horizon settings: T = 6 months with quarterly rebalancing, and T = 24 months with annual rebalancing. Stock weights are bounded to $-1 \leq \omega_{t+\tau} \leq 2$.

Each predictor variable $x^i_t$ follows an AR(1) process (Eq. 12, p.6):

$$x^i_{t+\tau} = \alpha^{x^i} + \beta^{x^i} x^i_{t+\tau-1} + \varepsilon^{x^i}_{t+\tau} \tag{12}$$

The joint forecast errors $\varepsilon_t = (\varepsilon^r_t, \varepsilon^x_t)$ are drawn from a multivariate normal with estimated covariance $\hat{\Sigma}_t$, enabling simulation of future return paths for utility evaluation.

Method

Linear benchmark. The standard predictive regression (Eq. 1, p.3):

$$r_{t+1} = \alpha + \beta x_t + \varepsilon^r_{t+1} \tag{1}$$

where $x_t = (x^1_t, \ldots, x^n_t)$ is the vector of 12 Welch and Goyal (2008) predictors. OLS1-OLS4 vary by predictor set (single dividend yield vs. kitchen-sink) and estimation window (expanding vs. rolling 10-year).

Deep feedforward neural network. The NN approximation replaces the linear link function (Eq. 2-3, p.4):

$$r_{t+1} = \hat{f}_{W,b}(x_t) + \varepsilon^r_{t+1} \tag{2}$$ $$\hat{r}_{t+1} = f^{(L)}_{W^{(L)},b^{(L)}} \circ \cdots \circ f^{(1)}_{W^{(1)},b^{(1)}}(x_t) \tag{3}$$

where each layer $f^{(\ell)}$ applies a sigmoid or ReLU activation to an affine transformation. With L = 2 and linear activations, Eq. (3) reduces to OLS. Three feedforward architectures: NN1 (16 neurons, 1 hidden layer), NN2 (32-16, 2 layers), NN3 (32-16-8, 3 layers).

LSTM recurrent network. The LSTM builds on lstm-recurrent-network by adding gated temporal state. A recurrent network introduces serial dependence into Eq. (3) via a lagged hidden state (Eq. 4, p.4):

$$h_t = f(W_h h_{t-1} + W_x x_t + b_0) \tag{4}$$

The LSTM variant extends this with gated memory cells. The hidden state (output gate) is (Eq. 5, p.5):

$$h_t = \sigma\!\left(W_h^{(o)} h_{t-1} + W_x^{(o)} x_t + b_0^{(o)}\right) \circ \tanh(c_t) \tag{5}$$

The memory cell combines a forget gate and an input gate (Eq. 6, p.5):

$$c_t = \sigma\!\left(W_h^{(g)} h_{t-1} + W_x^{(g)} x_t + b_0^{(g)}\right) \circ c_{t-1} + \sigma\!\left(W_h^{(i)} h_{t-1} + W_x^{(i)} x_t + b_0^{(i)}\right) \circ \tanh(k_t) \tag{6}$$

where $\sigma(\cdot)$ is the sigmoid function and $k_t$ is the new information flow. The forget gate term $\sigma(\cdot) c_{t-1}$ enables long-range dependence across many lags, while the input gate $\sigma(\cdot)\tanh(k_t)$ incorporates new predictor information. The LSTM uses three recurrent layers with 32-16-8 neurons and LSTM cells in the last layer, building on time-series-forecasting by learning nonlinear time-series patterns in the Welch and Goyal (2008) predictors.

Estimation uses the Adam optimizer with weight decay regularization (Kingma and Ba 2014), dropout (0-60%), and Huber loss. Hyperparameters (learning rate, weight decay, dropout, activation) are selected from 100 random combinations on a rolling validation sample each quarter, or each year for annual rebalancing, using statistical and economic criteria.

Empirical specifications

Data and rolling estimation. Predictors and returns span January 1945 to December 2018. Portfolio evaluation starts February 1955 (after an initial 10-year training window). Neural networks and rolling-window OLS specifications use a 10-year rolling window, re-estimated quarterly (annually for annual rebalancing); OLS1 and OLS3 instead use expanding-window estimates. The investor’s models and portfolio weights are updated at the corresponding rebalancing frequency.

Statistical accuracy: OOS R-squared. Following Campbell and Thompson (2008) (Eq. 14-15, p.7), the OOS R-squared for quarterly rebalancing is:

$$R^2_{q,oos} = 1 - \frac{\displaystyle\sum_{t \in \mathcal{T}_q}(r_t - \hat{r}_t^{\mathcal{M}_s})^2}{\displaystyle\sum_{t \in \mathcal{T}_q}(r_t - \bar{r}_t)^2} \tag{15}$$

where $\mathcal{T}_q$ is the set of end-of-quarter months when the investor reoptimizes and $\bar{r}_t$ is the historical mean over the same sample. The Clark-West (2007) test evaluates $H_0: R^2_{oos} \leq 0$ (Table 1, p.8). All NNs achieve positive OOS R-squared; all OLS models are negative (R1).

Portfolio performance: CER and Sharpe ratio. Expected utility is computed by simulating 1,000,000 draws from $(\varepsilon_{r,t+\tau}, \varepsilon^x_{t+\tau}) \sim N(0, \hat{\Sigma}_t)$, iterating Eqs. (10)-(13) forward. The annualized certainty-equivalent return CER solves $U(1 + \text{CER}/100) = \mathbb{E}_t[U(r_{p,t+T})]$. Statistical significance for CER differences from EH is assessed by the one-sided Diebold and Mariano (2002) test (Table 2, p.10). Sharpe ratios are computed monthly.

Robustness checks. Transaction costs: $\hat{r}^{\tau,\mathcal{M}_s}_t = \hat{r}^{\mathcal{M}_s}_t - \tau|\omega_t - \omega_{t-1} \hat{r}^{\mathcal{M}_s}_{t-1}|$ with $\tau \in \{0.1\%, 0.5\%\}$ (Table 7, p.17). Borrowing constraint (non-negative allocation to risk-free) and short-selling constraint ($0 \leq \omega_t \leq 1$) imposed separately and jointly (Table 8, p.18). A 20-year rolling window provides out-of-sample evidence from February 1969 onward (Table 9, p.19). A timing strategy (Table 10, p.20) benchmarks monthly Sharpe ratios across specifications and compares to Rossi (2018).

Datasets used

Dataset	Role in paper	Wiki page
Welch-Goyal monthly predictors	12 monthly predictor variables (dividend yield, log earnings-price ratio, dividend payout ratio, book-to-market, net equity expansion, treasury bill rate, term spread, default yield spread, default return spread, cross-sectional premium, inflation growth, stock variance) for S&P 500 excess log returns, Jan 1945-Dec 2018; from Amit Goyal’s website	No page yet
NBER business cycle reference dates	Define expansion and recession subsamples in Tables 2, 4, and Figure 2	NBER cycles

Sample: monthly, January 1945 to December 2018 (888 monthly observations). Portfolio evaluation: February 1955 to December 2018. Estimation run on GPU cluster (2 servers with 2 NVIDIA GeForce RTX 2080 Ti GPUs; rolling window calibration takes approximately 2 days).

When to read the full paper

Read the original if you are: building a two-asset dynamic portfolio strategy with aggregate return predictors and want the exact hyperparameter grid and GPU estimation setup; comparing LSTM vs. feedforward NN architectures (NN1-NN3) with a consistent utility-based criterion; comparing the reported results with Rossi’s (2018) boosted-tree portfolio evidence or Johannes et al.’s (2014) Bayesian predictive framework on the Welch and Goyal (2008) predictor set; or studying the subperiod and decade performance in Tables 4 and 9 (where the Gu et al. (2020) cross-section evidence is separately in the literature). Table 2 gives the headline CER and SR by NBER phase; Tables 7-9 contain the robustness evidence.

Attribution and rights

Source: peer-reviewed, Journal of Empirical Finance 87 (2026) 101705. This distillation was extracted by an LLM on 2026-06-25 and is not human-verified or independently reproduced. The paper is paywalled (Elsevier B.V., all rights reserved, including text and data mining and AI training); only short extracts are reproduced here.

Babiak, Mykola, and Jozef Baruník. “Deep learning, predictability, and optimal portfolio returns.” Journal of Empirical Finance 87 (2026) 101705. DOI: 10.1016/j.jempfin.2026.101705. © 2026 Elsevier B.V. All rights reserved.