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NBER WORKING PAPER SERIES

ALPHAGLASS: INTERPRETABLE CHARACTERISTIC-BASED PORTFOLIO CHOICE
Sebastian Bell
Ali Kakhbod
Martin Lettau
Abdolreza Nazemi
Working Paper 35186
http://www.nber.org/papers/w35186

NATIONAL BUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
May 2026

The authors have nothing to disclose. The views expressed herein are those of the authors and do
not necessarily reflect the views of the National Bureau of Economic Research.
NBER working papers are circulated for discussion and comment purposes. They have not been
peer-reviewed or been subject to the review by the NBER Board of Directors that accompanies
official NBER publications.
© 2026 by Sebastian Bell, Ali Kakhbod, Martin Lettau, and Abdolreza Nazemi. All rights reserved.
Short sections of text, not to exceed two paragraphs, may be quoted without explicit permission
provided that full credit, including © notice, is given to the source.

AlphaGlass: Interpretable Characteristic-Based Portfolio Choice
Sebastian Bell, Ali Kakhbod, Martin Lettau, and Abdolreza Nazemi
NBER Working Paper No. 35186
May 2026
JEL No. C14, C45, G10, G11, G12
ABSTRACT
We propose AlphaGlass, an inherently interpretable machine-learning framework for constructing
portfolios that directly optimize investment objectives. AlphaGlass maps stock characteristics into
additive signals with sparse interactions and converts these signals into long-short portfolios
through a differentiable rank-and-mask layer. This end-to-end design allows the model to optimize
objectives such as the Sharpe ratio or mean-variance utility while keeping portfolio weights
interpretable and traceable to specific characteristics and interactions. We show theoretically that
in-sample objective maximization consistently estimates the population objective and that the
differentiable rank-and-mask layer is a faithful smooth proxy for the corresponding conventional
long-short quantile portfolio. In U.S. equities, AlphaGlass delivers strong out-of-sample
performance and reveals economically interpretable drivers of long and short positions.

Sebastian Bell
School of Economics and Management,
Karlsruhe Institute of Technology
sebastian.bell@kit.edu
Ali Kakhbod
University of California, Berkeley
Haas School of Business
akakhbod@berkeley.edu

Martin Lettau
University of California, Berkeley
Haas School of Business
and CEPR
and also NBER
lettau@haas.berkeley.edu
Abdolreza Nazemi
Karlsruhe Institute of Technology
nazemi@kit.edu

1

Introduction

A fundamental question in empirical asset pricing is how an investor should map a large set of
firm characteristics into portfolios that maximize a chosen investment objective. The traditional
characteristic-sorting approach yields transparent portfolio rules and remains the workhorse of the
field. However, it limits interactions among characteristics and does not optimize the investment
objective. Modern machine-learning methods can incorporate many predictors, nonlinearities, and
high-dimensional interactions, yet much of the literature focuses on return prediction with complex
“black-box” models, such as neural networks, tree-based ensembles, and generative architectures.1
This creates two challenges. First, return prediction is only an intermediate target: forecasting returns need not align with risk-adjusted portfolio objectives once signals are converted into portfolios. Second, the opacity of black-box models limits economic interpretability, making it difficult to
understand which model features drive the results.
We address this central problem by developing AlphaGlass, an interpretable framework for
characteristic-based portfolio construction for general investment objective functions. AlphaGlass
maps firm characteristics and their pairwise interactions into additive signals and then converts
these signals into portfolio weights. This design links portfolio weights to stock characteristics
through a separable parameterization, enabling an exact decomposition of portfolio scores and allocations into characteristic- and interaction-level contributions. The key innovation is that AlphaGlass
jointly estimates the signals and maximizes the investment objective in a single step, while maintaining economic transparency, thereby integrating elements that existing asset-pricing approaches
typically treat separately: end-to-end investor-objective optimization, rank-based portfolio formation, and maintaining inherent interpretability.
Our contribution is threefold. First, we introduce a unified methodology for learning interpretable characteristic-based portfolio rules that directly optimize investor objectives such as the
Sharpe ratio or mean-variance preferences. Unlike the standard prediction-then-sort pipeline, AlphaGlass embeds a differentiable rank-and-mask approximation of conventional quantile sorting
within an additive, sparse-interaction architecture. This makes the characteristic-to-portfolio mapping trainable end-to-end while preserving the ranking logic of empirical asset pricing. At the same
time, the additive structure provides exact economic attribution: each univariate shape function and
selected interaction surface contributes directly to the portfolio score, allowing the model’s long and
short decisions to be decomposed into characteristic- and interaction-level components. An important aspect of the model is that the mapping from signal to portfolio weights is differentiable, unlike
traditional “hard” cutoffs based on characteristic quantiles. We construct a smooth rank-and-mask
layer that approximates traditional sorted portfolio weights.
Second, we show that the approach delivers strong risk-adjusted performance while producing
transparent economic diagnostics. In U.S. equities from 2000 to 2021, AlphaGlass generates a pronounced spread in returns and Sharpe ratios across signal-sorted portfolios, achieving a monthly
1

Other approaches include regularized linear models (e.g. Feng et al., 2020; Freyberger et al., 2020; Kozak et al., 2020)

and PCA-based methods (e.g., Kelly et al., 2019; Lettau and Pelger, 2020; Kelly et al., 2023; Lettau, 2023).

1

out-of-sample Sharpe ratio of 0.47, outperforming benchmark characteristic-sorted portfolios and
decile portfolios formed on Random Forest and neural network-based return predictions. Moreover, the performance comes with ex ante transparency: the glass-box structure yields interpretable
shape functions and interaction surfaces that map directly into portfolio composition. In our main
specification, the most important univariate shapes identify economically plausible drivers such as
industry concentration (herf), operating leverage dynamics (pchsale_pchxsga), momentum (mom12m),
and industry-adjusted labor adjustment (chempia), while the sparse interaction block highlights conditional effects such as the acc × hire interaction. Because AlphaGlass is separable by design, we
can further decompose long and short allocations into signed, term-level contributions and identify
which characteristics most strongly push assets into the long and short tails. This decomposition provides a direct link between a high-performing portfolio rule and economically interpretable drivers.
We also consider several variations and extensions of the method, including alternative data splits,
subsample analyses (e.g., removing small firms), constrained portfolio optimization (e.g., drawdown
penalization), and other investor objectives (e.g., mean-variance utility).
Finally, we establish theoretical properties that relate AlphaGlass’s learning behavior to the underlying economic objective. We show that maximizing the sample Sharpe ratio is argmax-consistent
for maximizing the population Sharpe ratio within the AlphaGlass function class. Asymptotically, the
fitted model therefore attains the best-in-class out-of-sample Sharpe ratio available within the chosen interpretable architecture. We also formalize that the differentiable rank-and-mask procedure
is a faithful surrogate for the conventional hard sort-and-cut portfolio construction. In particular,
when signals are well separated, the “soft” weights converge to the hard quantile weights, and the
corresponding Sharpe ratios coincide asymptotically under mild conditions.
We complement these theoretical results with simulation evidence from a controlled data generating process. Because the conditional mean and volatility structures are known in that environment,
we can evaluate each fitted strategy on an arbitrarily large independent sample, yielding an arbitrarily
accurate approximation to its population Sharpe ratio and enabling comparison with oracle portfolio rules. We find that AlphaGlass consistently outperforms benchmark learners, capturing a larger
share of the highest attainable population Sharpe ratio.
Taken together, AlphaGlass provides an end-to-end approach to portfolio construction that combines the flexibility of state-of-the-art ML with the transparency required for empirical asset pricing.
The framework is designed to be modular and extensible: the rank-and-mask procedure can accommodate alternative quantile definitions and constraints, and the additive structure can incorporate
different regularization schemes and pruning rules to control complexity. Most importantly, the
combination of end-to-end maximization of flexible objective functions with an exact decomposability of portfolio signals yields a portfolio construction method that is simultaneously high-performing,
economically grounded, and transparent.
In asset pricing, the most popular ML application is predicting returns in high-dimensional panels of firm characteristics. Gu et al. (2020) and related work document that nonlinear learners, including bagging or boosting ensembles and neural networks, often outperform unregularized linear

2

benchmarks and can generate economically meaningful spreads when forecasts are translated into
trading strategies. Classical evidence documents a large set of anomalies and the empirical difficulty of summarizing cross-sectional expected returns with small linear specifications (Fama and
French, 1993; Harvey et al., 2016; Jensen et al., 2023). Modern ML work shows that regularization
and nonlinear function approximation can improve predictive performance relative to traditional
linear benchmarks, and that economically meaningful portfolio spreads can be generated when forecasts are translated into trading strategies (Gu et al., 2020; Bryzgalova et al., 2025). This includes
penalized linear methods (Feng et al., 2020; Freyberger et al., 2020; Kozak et al., 2020), dimensionreduction, factor-based, and embedding approaches that summarize high-dimensional firm information (Kelly et al., 2019; Giglio and Xiu, 2021; Kakhbod et al., 2024; Gabaix et al., 2025), and nonlinear
models such as tree ensembles and neural networks that capture interactions and state dependence
(Gu et al., 2020, 2021).2 Bell et al. (2025) use the Explainable Boosting Machine (EBM), a sequential
boosting technique, to predict bond returns. In contrast to this return-prediction focus, we target
the investor’s end objective by directly optimizing the portfolio Sharpe ratio within an inherently
interpretable model class.
In doing so, we contribute to a related strand of literature that employs ML techniques to construct optimal portfolios, emphasizing that predictive models are ultimately judged by the riskadjusted performance they enable. Common simple approaches, such as plug-in portfolios, can perform poorly out-of-sample because estimation error in expected returns and covariances is first order
and investor utility is not taken into account when estimating portfolio weights (Jobson and Korkie,
1980; Michaud, 1989; Kan and Zhou, 2007; Kelly and Xiu, 2023). This motivates shrinkage and robustification, including covariance regularization (Ledoit and Wolf, 2004, 2012) and Bayesian/BlackLitterman style approaches (Jorion, 1986; Black and Litterman, 1992). More directly, an influential
literature integrates estimation and optimization by parameterizing portfolio weights as functions
of observable characteristics and estimating the weight function by maximizing in-sample utility
(Brandt, 1999; Ait-Sahalia and Brandt, 2001; Brandt and Santa-Clara, 2006; Brandt et al., 2009). In
mean-variance utility settings, this approach is also known as maximum Sharpe ratio regression
(MSRR), which connects tangency-portfolio estimation to regression and facilitates regularization
and model selection (see, e.g., Britten-Jones, 1999; DeMiguel et al., 2020; Ao et al., 2019). Recent
work explores high-capacity MSRR architectures (e.g., neural networks) and clarifies conditions under
which greater model complexity can improve out-of-sample portfolio efficiency (Simon et al., 2025;
Didisheim et al., 2023). Relatedly, Cong et al. (2026) use deep reinforcement learning to learn goaloriented portfolio policies with flexible path-dependent objectives, whereas our approach focuses
on interpretable characteristic-based portfolio rules with transparent main effects, sparse interactions, and rank-based long-short portfolio formation. Finally, portfolio choice is closely linked to
SDF estimation: maximizing the Sharpe ratio corresponds to identifying the most stringent HansenJagannathan pricing restriction within a candidate payoff/SDF class, and regularized SDF estimation
2

Recent works expand the information set beyond standard characteristics to text and other unstructured data, includ-

ing applications to news, disclosures, and language-model-based signals (Ke et al., 2019; Chen et al., 2022; Fedyk et al.,
2024; Kakhbod et al., 2025).

3

brings ML tools directly into the fundamentals of asset pricing (Hansen and Jagannathan, 1997;
Kozak et al., 2020; Chen et al., 2024).
This literature motivates our end-to-end design: rather than forecasting returns and then forming portfolios, we directly optimize the investor’s objective within an inherently interpretable characteristic based framework. Relative to existing portfolio-optimization approaches, AlphaGlass combines neural-network flexibility with a separable rank-based architecture, so that portfolio signals
and long-short allocations can be decomposed into explicit characteristic- and interaction-level contributions. This design yields transparent decision rules and reveals the economically interpretable
drivers of long and short positions.
The rest of the paper is organized as follows. Section 2 details the estimation methodology and
training algorithm. Section 3 describes the data. Section 4 presents empirical results, while Section
5 provides insights into portfolio composition. Section 6 extends the model to a mean-variance objective. Section 7 outlines the model’s theoretical properties. Section 8 presents simulation evidence
comparing AlphaGlass to benchmark learners and oracle portfolio rules. Section 9 concludes.

2

Methodology

We develop a customized machine learning estimator that enables flexible portfolio construction
while retaining model interpretability. AlphaGlass is a neural network-based additive model, designed to combine the interpretability of Generalized Additive Models (GAMs) with the flexibility of
neural networks. Like the Explainable Boosting Machine (EBM) (Lou et al., 2013; Nori et al., 2019),
the model assumes an additive decomposition of the prediction function into main effects and a
sparse set of pairwise interactions, but distinctly differs in the way these effects are represented
and estimated.
Let 𝑟𝑙,𝑡 be the excess return of firm 𝑙=1,…,𝐿 at time 𝑡=1,…,𝑇. 𝑥𝑙,𝑡 is a vector of 𝑁 characteristics
of firm 𝑙 at 𝑡. The standard approach (e.g., Gu et al. (2020)) is as follows.
1. Predict returns as a function of lagged characteristics using nonlinear ML methods:
𝑟𝑙,𝑡+1 = 𝑔(𝑥𝑖,𝑙,𝑡 ) + 𝑒𝑙,𝑡+1 .

(1)
1

2
𝑔(𝑥𝑖,𝑙,𝑡 ) is estimated by minimizing the mean-square-error loss function: ℒMSE = 𝐿𝑇 ∑𝑙,𝑡 𝑒𝑙,𝑡
.
HL
̂𝑙𝑡+1 = ̂
2. Each 𝑡, form 𝑀 portfolios based on the predicted returns 𝑟
be the return
𝑔(𝑥𝑖,𝑙,𝑡 ). Let 𝑓𝑡+1

of the long-short (𝑀-minus-1) portfolio in 𝑡 + 1 that is formed at 𝑡.
HL
.
3. Compute the Sharpe ratio (SR) of 𝑓𝑡+1

(1) is estimated in a training sample 𝒯1 and validated in a validation sample 𝒯2 . The Sharpe ratio
HL
is computed in the test sample 𝒯3 . Gu et al. (2020) compare various methods, including linear
of 𝑓𝑡+1

models, boosted regression trees, random forests, and neural networks, using this methodology.
Note that the three steps are performed sequentially. Moreover, while the final objective is to
create a factor 𝑓𝑡 with a high Sharpe ratio, predicted returns in (1) are estimated using a different
4

objective function. Instead, the AlphaGlass model estimates all three steps jointly, using a single
objective function, as follows.
1. Asset-specific signals 𝑠𝑙,𝑡 are a function of lagged characteristics:
𝑠𝑙,𝑡 = 𝑔(𝑥𝑖,𝑙,𝑡 ) + 𝑒𝑙,𝑡+1 .

(2)

𝑚
2. Portfolio weights 𝑤𝑙,𝑡
of 𝑚 = 1,…,𝑀 portfolios are functions of the signals:
𝑚
𝑤𝑙,𝑡
= ℎ𝑚 (𝑠𝑙,𝑡 ).

(3)

HL
HL
3. The weights 𝑤𝑙,𝑡
and returns 𝑓𝑡+1
of the long-short portfolio are
HL
𝑀
1
𝑤𝑙,𝑡
= 𝑤𝑙,𝑡
− 𝑤𝑙,𝑡
,

(4)

HL
HL
𝑓𝑡+1
= ∑ 𝑤𝑙,𝑡
𝑟𝑙,𝑡+1 .

(5)

𝑙
HL
4. Portfolio weights 𝑤𝑙,𝑡
,𝑡 = 1,…,𝑇 − 1 are obtained by minimizing the loss function
HL
̂ 𝑙,𝑡
{𝑤
} = argmin ℒ(𝑓2HL ,...,𝑓𝑇HL ).

(6)

HL
{𝑤𝑙,𝑡
}

Several points are worth noting. First, (2)–(6) are solved jointly rather than sequentially yielding
HL
̂𝑖,𝑙,𝑡 , 𝑠̂𝑙,𝑡 and 𝑤
̂ 𝑙,𝑡
𝑔
. Hence, the signals 𝑠𝑙,𝑡 in (2) are estimated with respect to the loss function (6),

rather than by minimizing the MSE of the errors 𝑒𝑙,𝑡+1 . Second, the specification of the loss function
is flexible. In principle, the loss function can be any well-defined (dis)utility function. In the main
part of the paper, we focus on the negative of the Sharpe ratio of 𝑓𝑡HL : ℒ = −SR(𝑓𝑡HL ), which has
been the standard specification in the literature. We also consider mean-variance preferences: ℒ =
−E(𝑓𝑡HL ) + (𝛾/2)Var(𝑓𝑡HL ). Our approach also allows for portfolio constraints and penalties. Third,
gradient-based ML methods require that the minimization problem (2)–(6) is differentiable. However,
𝑚
are non-differentiable step
standard portfolio construction relies on simple rankings, so that 𝑤𝑙,𝑡

functions with “hard” cutoffs. We replace such “hard” cutoffs with smooth functions, as explained
𝑚
below. Next, we describe the estimation of the signals 𝑠𝑙,𝑡 in (2) and portfolio weights 𝑤𝑙,𝑡
in (3).

2.1 Estimation of signals
Following Hastie and Tibshirani (1986), Lou et al. (2013), and Bell et al. (2025), we estimate signals
in (2) using a General Additive Model (GAM) with interactions3 :
𝑁

𝑠𝑙,𝑡 = 𝛽0 + ∑ 𝑓𝑖 (𝑥𝑖,𝑙,𝑡 ) + ∑ 𝑓𝑖𝑗 (𝑥𝑖,𝑙,𝑡 ,𝑥𝑗,𝑙,𝑡 ),

(7)

𝑖>𝑗

𝑖=1

where
• 𝑥𝑖,𝑙,𝑡 is the characteristic 𝑖 for firm 𝑙 at time 𝑡,
3

Note that we use the GAM-like structure deliberately to ensure inherent interpretability, but in principle the AlphaGlass

mechanism can be used with other architectures.

5

• 𝑓𝑖 represents the main effect of feature 𝑥𝑖 ,
• 𝑓𝑖𝑗 represents the interaction effect between features 𝑥𝑖 and 𝑥𝑗 .
The advantage of this specification is that it allows nonlinear effects while remaining inherently
interpretable. For example, the influence of each variable 𝑥𝑖,𝑙,𝑡 is given by the function 𝑓𝑖 (𝑥𝑖,𝑙,𝑡 )
and possible interactions with other variables via 𝑓𝑖𝑗 (𝑥𝑖,𝑙,𝑡 ,𝑥𝑗,𝑙,𝑡 ). Within this additive structure, the
AlphaGlass model represents each 𝑓𝑖 and 𝑓𝑖𝑗 as a small feed-forward neural network, trained under smoothness and regularization constraints. This neural representation enables the model to
produce smooth, differentiable effect functions and to incorporate additional constraints such as
monotonicity and sparsity.
The training procedure implements three main stages:
1. Main effect learning: For each feature 𝑖, a neural subnetwork 𝑓𝑖 is trained while minimizing
the chosen loss function. Each subnet consists of one input neuron for 𝑥𝑖 , a hidden layer
of 20 units, and an output neuron. All subnets use Rectified Linear Unit (ReLU) activations4
and are trained in parallel via gradient descent. Training stops after a pre-specified number of
epochs or when performance no longer improves meaningfully (as defined by an early stopping
threshold).
2. Interaction learning: After selecting the interaction pairs, neural subnetworks 𝑓𝑖𝑗 are trained
while keeping the main effect subnetworks fixed. The interaction subnetworks have two input
neurons (for 𝑥𝑖 and 𝑥𝑗 ), two hidden layers of 20 neurons each, and one output neuron.
3. Joint fine-tuning: Once all main effect and interaction subnetworks are trained, a global optimization step is performed in which all parameters are updated simultaneously (with a lower
learning rate in order to avoid losing information from the previous training steps). Due to the
structural separability of the neural subnetworks, this stage does not compromise the interpretability of individual shape functions while enabling small, coordinated adjustments that
improve performance.
We provide additional information on these training stages and other details related to the estimation technique in Appendix A.

2.2 Estimation of portfolio weights
In each period 𝑡, we construct 𝑀 portfolios as functions of signals 𝑠𝑙,𝑡 and compute their returns
in 𝑡 + 1. Since we are interested in the long-short portfolio, defined as the difference between the
portfolios 𝑚 = 𝑀 and 𝑚 = 1, we focus on the weights of these two portfolios. As mentioned above,
a key challenge in our setting is that portfolio weights must be differentiable with respect to the
signals. Rank-sorting with “hard” cutoffs, which is commonly used in portfolio construction, is
non-differentiable, rendering gradient-based estimators in ML methods, such as neural networks,
infeasible. To address this, we replace “hard” portfolio weights based on discrete rankings and
4

ReLU activation is based on the max operator: ℎ(𝑧) = max(0,𝑧).

6

portfolio inclusions with smooth “soft” weights.5 These “soft” portfolio weights can be interpreted
as the probability that a given stock is included in a given portfolio. The estimation proceeds in two
steps: first, transforming discrete ranks, and then computing portfolio weights.
First, in each month 𝑡, we replace discrete rankings with smooth pairwise comparisons, yielding
differentiable approximations of the cross-sectional order of signals. Let 𝑛𝑡 be the number of stocks
in month 𝑡. The relative ordering of signals for stocks 𝑙 and 𝑙′ in each month 𝑡 is approximated via
pairwise logistic comparisons
𝑆𝑙,𝑙′ ,𝑡 = 𝜎(

𝑠𝑙,𝑡 − 𝑠𝑙′ ,𝑡
),
𝜏s

1

where 𝜎(𝑥) = 1+𝑒−𝑥 ∈ (0,1) is the sigmoid function. The “temperature” hyperparameter 𝜏s > 0 (and
𝜏m > 0 below) controls the sharpness of the soft approximation. Smaller values make the sigmoids
steeper, yielding rankings that behave more like hard, discrete assignments, whereas larger values
produce smoother, more gradual transitions. The sigmoid output 𝑆𝑙,𝑙′ ,𝑡 can be interpreted as the
probability that asset 𝑙 outranks asset 𝑙′ based on the difference between their signals. When the
signals are far apart, the sigmoid is close to 0 or 1, whereas for signals that are close together, the
output lies near 0.5, reflecting uncertainty about their relative ordering. Note that 𝑆𝑙,𝑙′ ,𝑡 converges
to the (non-differentiable) indicator function as 𝜏s goes to zero, which is equivalent to the standard
“hard” comparison:
𝑆𝑙,𝑙′ ,𝑡 = 𝜎(

𝑠𝑙,𝑡 − 𝑠𝑙′ ,𝑡
) −−→ 𝟣{𝑠𝑙,𝑡 ≥ 𝑠𝑙′ ,𝑡 }.
𝜏s
𝜏s ↘0

The “soft” rank of asset 𝑙 in month 𝑡 is then defined as the sum of all pairwise comparisons of 𝑙 and
the other assets:
rank𝑙,𝑡 = 1 + ∑ 𝑆𝑙,𝑙′ ,𝑡 ∈ [1,𝑛𝑡 ].
𝑙′ ≠𝑙

The rank can be interpreted as the expected position of asset 𝑙 in the cross-sectional ordering. High
signals correspond to high probabilities 𝑆𝑙,𝑙′ ,𝑡 and therefore a high rank close to 𝑛𝑡 , while low signals
correspond to low ranks closer to 1. Hence, the soft rank behaves like a smooth analog of the usual
ordinal rank while remaining differentiable. The smooth formulation ensures that the mapping of
signals 𝑠𝑙,𝑡 to weights 𝑤𝑙,𝑡 is differentiable, allowing gradients of the Sharpe-based loss to propagate
back to the signal-generating model.
In the second step, we select the highest- and lowest-ranked assets for each month to form high
(𝑚 = 𝑀) and low (𝑚 = 1) portfolios. Once again, we define a “soft” approximation. Let 𝑘𝑡 = ⌊𝑛𝑡 /𝑀⌋
denote the cutoff size in month 𝑡. Consider the high portfolio first. We shift the rank𝑙,𝑡 so that stocks
with ranks around the cutoff have values close to 0. For example, if 𝑛𝑡 = 200 and 𝑀 = 10, the cutoff
is 𝑘𝑡 = 20, hence we subtract rank181 from rank𝑙,𝑡 . For the low portfolio, we first take the negative
rank𝑙,𝑡 and add rank𝑘,𝑡 . Then, we apply the sigmoid transformation:
5

In Section 7, we prove that the soft portfolio weights represent a faithful surrogate for the “hard” weights obtained

by exact ranking and truncation (Theorem 1).

7

H
𝑚𝑙,𝑡
= 𝜎(

rank𝑙,𝑡 − rank𝑛𝑡 −𝑘𝑡 +1,𝑡
),
𝜏m

L
𝑚𝑙,𝑡
= 𝜎(

rank𝑘𝑡 ,𝑡 − rank𝑙,𝑡
).
𝜏m

H
L
𝑚𝑙,𝑡
and 𝑚𝑙,𝑡
can be interpreted as probabilities that a stock 𝑙 is in the high and low portfolios,
H
L
respectively. For example, for stocks with high ranks 𝑚𝑙,𝑡
≈ 1 and 𝑚𝑙,𝑡
≈ 0, and vice versa for stocks

with low ranks. As before, the standard “hard” case is a special case when 𝜏s goes to zero:
𝐻
𝑚𝑙,𝑡
−−−→ 𝟣{rank𝑙,𝑡 ≥ rank𝑛𝑡 −𝑘𝑡 +1,𝑡 },
𝜏m ↘0

𝐿
𝑚𝑙,𝑡
−−−→ 𝟣{rank𝑘𝑡 ,𝑡 ≤ rank𝑙,𝑡 }.
𝜏m ↘0

H
L
To obtain portfolio weights, we normalize 𝑚𝑙,𝑡
and 𝑚𝑙,𝑡
, so that each leg sums to one within each

month:
H
𝑤𝑙,𝑡
=

H
𝑚𝑙,𝑡

L
𝑤𝑙,𝑡
=

H ,
∑𝑗 𝑚𝑗,𝑡

L
𝑚𝑙,𝑡
L
∑𝑗 𝑚𝑗,𝑡

.

Finally, the portfolio weights of the “high-minus-low” portfolio are given by
HL
H
L
𝑤𝑙,𝑡
= 𝑤𝑙,𝑡
− 𝑤𝑙,𝑡
.

Figure 1 compares the “soft” portfolio weights to “hard” weights for an example of 𝐿=200 stocks
and 𝑀 = 10 decile portfolios, so that the cutoff value 𝑘 = 20. Signals 𝑠𝑙 are uniformly distributed
between 0 and 1 and then normalized. The parameters of the sigmoid functions are 𝜏rank = 0.1
and 𝜏mask = 0.5. The orange line represents the “hard” equal-weighted weights of the long-short
portfolios based on a decile sort. The blue line shows the “soft” portfolio weights obtained via
sigmoid transformations. The 𝑥-axis shows stocks around the cutoffs 𝑙 = 20 and 𝑙 = 181.6

2.3 The loss function
In addition to the baseline loss function ℒ(𝑓𝑡HL ), we incorporate regularizations to control model
complexity and induce sparsity in the estimation of the signals in (7). Specifically, we augment the
loss function with ℓ1 -penalties on effect-specific weights as follows. Instead of estimating the 𝑓𝑖 and
𝑓𝑖𝑗 functions in (7) directly, we introduce regularization parameters 𝜃𝑖 and 𝛾𝑖𝑗 and define the GAM
model as
𝑁

̃ 𝑖,𝑙,𝑡 ) + ∑ 𝛾𝑖𝑗 𝑓𝑖𝑗̃ (𝑥𝑖,𝑙,𝑡 ,𝑥𝑗,𝑙,𝑡 ),
𝑠𝑙,𝑡 = 𝛽0 + ∑ 𝜃𝑖 𝑓𝑖(𝑥
𝑖=1
6

𝜃𝑖 ,𝛾𝑖𝑗 ≥ 0.

(8)

𝑖>𝑗

One could, in principle, map the AlphaGlass signal directly into weights over the full universe. We use a rank-and-

mask construction because the signal is best viewed as an ordinal portfolio score rather than a calibrated expected-return
forecast. The ordering of firms is economically meaningful, but the cardinal scale and spacing of scores are not uniquely
pinned down. Directly using score levels as weights would therefore require additional assumptions about the signal
scale and could make exposures sensitive to arbitrary transformations. The rank-and-mask layer instead converts the
ordering into normalized long and short legs with a fixed gross exposure, preserving dollar neutrality and remaining
well-defined as the stock universe changes. It also focuses the portfolio on the tails of the score distribution, where the
model produces the clearest separation, mimicking conventional portfolio construction. For completeness, we consider
an alternative weighting scheme over the full cross-section in Appendix E, resulting in lower Sharpe ratios.

8

Figure 1: Soft vs. hard portfolio weights 𝑤𝑙HL

wlHL hard
wlHL soft

0.04

wl

0.02
0.00
0.02
0.04
10

15

20

25

30 170

Stocks

175

180

185

190

Note: This figure shows the weights of the high-low portfolio 𝑤𝑙HL for an example with 𝐿 = 200 stocks and 𝑀 = 10.
The blue and orange lines show the soft and hard weights, respectively. Signals 𝑠𝑙 are evenly distributed between
0 and 1. The parameters of the sigmoid functions are 𝜏rank = 0.1 and 𝜏mask = 0.5.

The model estimates (𝜃𝑖 , 𝑓𝑖̃ ), as well as (𝛾𝑖𝑗 , 𝑓𝑖𝑗̃ ), separately. When we interpret the model output,
however, we focus on the total effects 𝑓𝑖 (𝑥𝑖,𝑙,𝑡 ) = 𝜃𝑖 𝑓𝑖̃ (𝑥𝑖,𝑙,𝑡 ) and 𝑓𝑖𝑗 (𝑥𝑖,𝑙,𝑡 ,𝑥𝑗,𝑙,𝑡 ) = 𝛾𝑖𝑗 𝑓𝑖𝑗̃ (𝑥𝑖,𝑙,𝑡 ,𝑥𝑗,𝑙,𝑡 ).
The loss function incorporates ℓ1 -regularization of the main and interaction effects:
(𝑓𝑡HL |𝜆main ,𝜆inter ) =

ℒ

𝑁
HL
(𝑓𝑡 ) + 𝜆main ∑ |𝜃𝑖 | + 𝜆inter ∑ |𝛾𝑖𝑗 |.
𝑖>𝑗
𝑖=1

ℒ

(9)

The ℓ1 -penalties induce shrinkage towards zero during optimization. Exact zeros correspond to
effects that are effectively removed from the forward pass, and the hyperparameters 𝜆main and 𝜆inter
allow us to control the relative size of the regularization penalty during main effect and interaction
training, respectively. Further details on the selection and pruning mechanisms are provided in
Appendix A.1.

3

Data

We obtain monthly stock return data from CRSP and limit our stock universe to the three major U.S.
stock exchanges, NYSE, NASDAQ, and AMEX. These data are merged with the 91 firm characteristics
from Green et al. (2017) as provided by Gu et al. (2020). The characteristics are lagged relative to
returns so that returns in month 𝑡 are matched with monthly characteristics data at the end of month
𝑡 − 1, quarterly data by the end of 𝑡 − 5, and the most recent annual data by the end of 𝑡 − 7.
We remove observations for which return or market-cap data are unavailable, yielding a total of
1,656,664 observations over the sample period from January 2000 to December 2021. Full variable
definitions are available in Appendix B. Similar to Kelly et al. (2019) and Freyberger et al. (2020), all
characteristics are cross-sectionally ranked and mapped to the interval [0, 1]. In the following main

9

analysis, we split our data into training, validation, and test sets 𝒯1 , 𝒯2 , and 𝒯3 of lengths 10 years,
2 years, and 10 years, respectively. We consider other splits as robustness tests.
Table 1 reports means, standard deviations, and Sharpe ratios of long-short portfolios based on
univariate characteristic-sorts. Each month, we sort stocks according to a given characteristic and
form value-weighted decile portfolios. The long-short portfolio is the difference between portfolios
10 and 1. Panel A shows results for the combined train and validation samples, and Panel B shows
results for the test samples. We include the 10 characteristics with the highest in-sample Sharpe ratios: pchcurrat (%change in current ratio (current assets/liabilities)), pchquick (%change in quick ratio
(current near-cash (quick) assets/liabilities)), chcsho (change in shares outstanding), mvel1 (market
value (size)), pchsale_pchrect (%change in sales minus %change in accounts receivable), egr (growth
in common shareholder equity), tb (tax income to book income ratio), cashpr (cash productivity (net
operating CF minus capital expenditures, divided by adjusted net income)), salecash (sales to cash
ratio), sp (sales to price ratio). Note that many of the characteristics with high Sharpe ratios in the
training/validation samples have substantially lower Sharpe ratios in the test sample, for example,
pchcurrat, pchquick, mvel1 (size). The only two exceptions are chcsho and pchsale_pchrect, which

have out-of-sample Sharpe ratios of 0.235 and 0.196, respectively.

Table 1: Returns of characteristic-sorted portfolios
pchcurrat

pchquick

chcsho

mvel1

pchsale_pchrect

egr

tb

cashpr

salecash

sp

0.83
3.88
0.21

0.63
3.07
0.21

0.80
3.99
0.20

1.12
5.85
0.19

1.15
6.03
0.19

0.20
2.46
0.08

0.18
2.10
0.08

0.09
2.75
0.03

0.31
4.03
0.08

0.30
4.14
0.07

Panel A: In-sample
Mean
S.D.
SR

0.58
1.96
0.30

0.50
1.93
0.26

0.94
3.66
0.26

1.53
7.04
0.22

0.54
2.51
0.22

Panel B: Out-of-sample
Mean
S.D.
SR

0.14
1.79
0.08

0.18
1.83
0.10

0.44
1.89
0.24

0.02
5.43
0.00

0.41
2.11
0.20

Note: This table reports monthly means, standard deviations, and Sharpe ratios of high-minus-low portfolios based
on decile characteristic sorts. We sort characteristic-sorted portfolios by their Sharpe ratios in the combined training and validation samples.

4

Empirical results

Following the literature (e.g., Gu et al. (2020)), our benchmark estimation of the AlphaGlass model
assumes that the loss function is the negative of the Sharpe ratio of the long/short portfolio 𝑓𝑡HL :

ℒ(𝑓𝑡HL ) = −SR(𝑓𝑡HL ),

(10)

with ℓ1 -regularizations as in (9). As an alternative, we consider mean-variance preferences in Section
6. We benchmark the AlphaGlass model against several methods studied in the literature: random
forest (RF), neural network (NN), and explainable boosting machine (EBM). All three models follow the
10

standard approach outlined in Section 2, i.e., using the mean-square-error loss function to estimate
predicted returns in (1). The explainable boosting machine (EBM) is an inherently interpretable ML
model based on the GAM model in (7) to predict returns, but uses the MSE loss function rather than
the Sharpe ratio (see Lou et al. (2013) and Bell et al. (2025) for details). In all cases, we construct
decile portfolios and compute the 10-minus-1 factor 𝑓𝑡HL . We evaluate all approaches based on the
out-of-sample Sharpe ratios of the estimated long/short factors.

4.1 AlphaGlass vs. benchmark models
Table 2 reports the means, standard deviations, and Sharpe ratios of the 10 portfolios as well as
the 10-1 portfolio based on the AlphaGlass estimation. In-sample mean returns and Sharpe ratios
of the decile portfolios are nearly monotonic, with a monthly in-sample Sharpe ratio for the 10-1
portfolio of 0.54. The out-of-sample results largely mirror the in-sample results, indicating that the
model yields stable performance. The monthly Sharpe ratio of the high/low portfolio is 0.47 (1.62
p.a.), slightly lower than the in-sample counterpart. We discuss the model details and interpretation
in the following sections.
Results of the benchmark models are reported in Table 3. In-sample mean returns and Sharpe
ratios are (mostly) monotonic across the decile portfolios; however, the out-of-sample results are
mixed. The random forest model performs the worst. The mean return of the 10-1 portfolio 𝑓𝑡HL is
positive, but its Sharpe ratio is only 0.14. The corresponding Sharpe ratios for the neural network
and EBM are 0.34 and 0.30, respectively.7 All three models are outperformed by AlphaGlass.8
Figure 2 plots the cumulative log returns of the estimated 𝑓𝑡HL factors, as well as the CRSP-VW
index, in the test sample. Returns based on AlphaGlass and the neural net are comparable until
2017, but the AlphaGlass factor dominates afterward. The returns of the EBM portfolio are similar
to those of the CRSP-VW returns, whereas the cumulative return of the RF portfolio is nearly flat
until 2020.
A potential concern in portfolio construction is that results are driven by small, illiquid stocks.
To address this issue, we form long/short factors by size deciles. Each month, we sort stocks into size
quintiles and create factors using only stocks in a given quintile. Table 4 shows the out-of-sample
means, standard deviations, and Sharpe ratios for the four models by size quintile. Consider first the
results of the AlphaGlass model in Panel A. The AlphaGlass factors for smaller stocks have higher
mean returns than those for larger stocks; however, they are also more volatile. The Sharpe ratios are
high even for larger stocks, ranging from 0.25 to 0.4, indicating that the AlphaGlass model’s results
are not driven by small stocks. However, this is not the case for the other three models, as only the
7

Gu et al. (2020) also find that neural networks yield a higher Sharpe ratio than other ML methods, such as elastic nets,

RF, and gradient boosted regression trees. Note that they refit models every 12 months, so the results are not directly
comparable to those in this paper.
8
As an additional benchmark, Appendix D considers an SR-maximizing combination of characteristic-sorted long-short
portfolios. The poor out-of-sample results underline importance of optimization techniques that learn generalizable
portfolio rules rather than merely fit in-sample mean and covariance estimates.

11

Table 2: Returns of AlphaGlass portfolios
1

2

3

4

5

6

7

8

9

10

10-1

1.01
6.49
0.16

0.95
6.55
0.15

1.04
6.85
0.15

1.57
7.71
0.20

1.87
3.44
0.54

1.31
5.40
0.24

1.23
5.50
0.22

1.38
5.90
0.23

1.55
6.43
0.24

1.10
2.35
0.47

Panel A: In-sample
Mean
S.D.
SR

−0.30
7.63
−0.04

0.37
5.26
0.07

0.33
5.66
0.06

0.54
6.43
0.08

0.76
6.44
0.12

0.79
6.35
0.13

Panel B: Out-of-sample
Mean
S.D.
SR

0.46
5.79
0.08

0.67
4.40
0.15

0.92
3.64
0.25

0.95
4.40
0.22

1.04
4.94
0.21

1.19
5.18
0.23

Note: This table compares monthly means, standard deviations, and Sharpe ratios for ten equal-weighted portfolios
constructed based on predicted AlphaGlass signal deciles in the training/validation (Panel A) and testing sample
(Panel B). The last columns show the results of the 10-1 portfolio, which is long in portfolio 10 and short in portfolio 1.

Table 3: Returns of benchmark models
1

2

3

4

Mean
S.D.
SR

−3.20
9.47
−0.34

−1.15
6.66
−0.17

−1.46
7.14
−0.21

−0.65
6.22
−0.11

Mean
S.D.
SR

0.94
5.32
0.18

0.89
4.89
0.18

0.81
4.64
0.18

0.94
4.06
0.23

Mean
S.D.
SR

−1.54
10.89
−0.14

−0.18
8.75
−0.02

0.90
4.57
0.20

1.00
3.97
0.25

Mean
S.D.
SR

0.60
7.31
0.08

0.72
6.40
0.11

1.02
3.19
0.32

1.03
3.57
0.29

5

6

7

8

9

10

10-1

0.98
5.27
0.19

1.38
5.65
0.24

2.25
6.84
0.33

4.04
10.13
0.40

7.24
9.71
0.75

1.33
5.68
0.24

1.00
6.11
0.16

1.56
7.96
0.20

0.62
4.33
0.14

0.88
6.09
0.15

0.76
6.71
0.11

2.11
11.12
0.19

3.66
5.45
0.67

1.24
5.30
0.23

1.09
5.49
0.20

1.50
7.04
0.21

0.90
2.67
0.34

2.07
6.24
0.33

2.89
7.33
0.40

5.07
11.81
0.43

8.48
10.91
0.78

1.00
5.37
0.19

1.17
6.15
0.19

1.94
7.86
0.25

1.35
4.52
0.30

Panel A: RF In-sample
−0.13
5.51
−0.02

0.53
5.15
0.10

Panel B: RF Out-of-sample
0.96
4.11
0.23

1.04
4.45
0.23

1.22
5.33
0.23

Panel C: NN In-sample
1.05
4.43
0.24

1.03
5.03
0.21

1.05
5.50
0.19

Panel D: NN Out-of-sample
1.05
4.19
0.25

1.24
4.66
0.27

1.24
4.97
0.25

Panel E: EBM In-sample
Mean
S.D.
SR

−3.41
8.53
−0.40

−1.56
6.88
−0.23

−0.58
6.15
−0.10

−0.15
6.05
−0.03

0.37
5.65
0.07

0.94
5.66
0.17

1.44
5.65
0.25

Panel F: EBM Out-of-sample
Mean
S.D.
SR

0.59
5.79
0.10

0.85
4.51
0.19

0.87
4.11
0.21

0.91
4.11
0.22

0.94
4.45
0.21

1.11
4.72
0.24

1.32
5.44
0.24

Note: This table compares monthly means, standard deviations, and Sharpe ratios for ten equal-weighted portfolios
constructed based on Random Forest (RF), neural network (NN), and explainable boosting machine (EBM) prediction
deciles in the training/validation and testing sample. The last columns show the results of the 10-1 portfolio, which
is long in portfolio 10 and short in portfolio 1.

12

Figure 2: Cumulative returns of 𝑓𝑡HL of benchmark models

0.7

AlphaGlass
NN
RF

cum. log returns

0.6
0.5

EBM
CRSP-VW

0.4
0.3
0.2
0.1
0.0

2010

2012

2014

2016

2018

2020

2022

Note: This figure shows cumulative log returns of the 𝑓𝑡HL factors based on the AlphaGlass (blue), neural net (NN,
orange), random forest (RF, black), and the explainable boosting machine (EBM, light blue) model, as well as the
CRSP-VW index (red dashed) in the test sample.

factors with the smallest stocks in the first size quintile have Sharpe ratios above 0.2. Factors with
larger stocks have means and Sharpe ratios that are close to zero. Hence, AlphaGlass is the only
model that generates factors with high means and Sharpe ratios for stocks across the size spectrum.
The results in Table 4 are based on the AlphaGlass estimation for the entire sample. As an alternative exercise, we exclude small stocks in the bottom size quintile in each month and estimate
AlphaGlass using the remaining sample. The full results are in Table F.1. The Sharpe ratio of AlphaGlass is 0.43, slightly lower than the SR of 0.54 for the full sample that includes small stocks.
The performance of the other models is more sensitive to the exclusion of small stocks. The out-ofsample Sharpe ratios of the neural net and EBM models drop from 0.34 to 0.12, and from 0.30 to
0.13, respectively.

Our baseline design uses a 10-year training window, a 2-year validation window, and a 10-year
testing window. For robustness, we use a longer estimation period, with 15 years of training data,
while keeping the validation window fixed at two years and, correspondingly, shortening the test
period; see Table G.1 in Appendix G. The out-of-sample Sharpe ratio of the long/short portfolio is
0.49, slightly higher than in the benchmark case, suggesting that AlphaGlass is reasonably stable
across alternative train/test splits.

4.2 Predictor importance
The additive structure of the AlphaGlass signal estimation in (7) allows us to investigate the role
of individual characteristics in the estimation. First, we construct a measure of the overall “importance” of predictors. The next section focuses on how predictors translate into signals through the
estimated “shape” functions 𝑓𝑖 (𝑥𝑖 ) and 𝑓𝑖𝑗 (𝑥𝑖 ,𝑥𝑗 ).
13

Table 4: Out-of-sample factors by size quintiles
Size quintile
Small

2

3

4

Big

0.81
3.21
0.25

0.59
2.20
0.27

0.33
3.39
0.10

0.06
4.02
0.01

−0.03
3.83
−0.01

0.06
3.16
0.02

−0.10
4.19
−0.02

0.18
4.63
0.04

Panel A: AlphaGlass
Mean
S.D.
SR

1.34
4.26
0.31

1.32
3.27
0.40

1.15
3.54
0.33

Panel B: Random forest
Mean
S.D.
SR

1.54
5.57
0.28

Mean
S.D.
SR

1.48
4.71
0.32

−0.06
4.43
−0.01

0.20
4.18
0.05

Panel C: Neural Net
0.69
3.75
0.19

0.44
3.83
0.11

Panel D: EBM
Mean
S.D.
SR

2.28
4.93
0.46

0.48
4.47
0.11

0.86
4.93
0.18

Note: This table compares monthly out-of-sample means, standard deviations, and Sharpe ratios for long/short
factors based on AlphaGlass, Random Forest (RF), neural network (NN), and explainable boosting machine (EBM)
predictions. The factors include only stocks in a size quintile, with quintiles formed each month.

We define the overall importance of a predictor by the mean absolute score 𝑆(𝑖) as absolute
values of 𝑓𝑖 (𝑥𝑖,𝑙,𝑡 ) averaged over the training sample:
𝑆(𝑖) =

1
∑ |𝑓𝑖 (𝑥𝑖,𝑙,𝑡 )|.
|𝒯1 | (𝑙,𝑡)∈𝒯

(11)

1

𝑆(𝑖) captures the average effect of variable 𝑖, taking the distribution over the sample into account,
and is therefore similar to the absolute value of a (standardized) coefficient in a linear regression.
Therefore, the mean absolute scores capture the average “size” of the effect of an individual variable
on the model’s prediction. The mean absolute scores of the interaction terms are defined analogously:
𝑆(𝑖,𝑗) =

1
∑ |𝑓𝑖𝑗 (𝑥𝑖,𝑙,𝑡 ,𝑥𝑗,𝑙,𝑡 )|.
|𝒯1 | (𝑙,𝑡)∈𝒯

(12)

1

We consider both individual effects and effects by characteristic categories. Figure 3 presents
the 15 most important effects according to the mean absolute score measure. We find that, overall,
univariate effects are the most important contributors, with the exception of the interaction between
acc and hire. The ranking is led by herf (index of industry sales concentration) and pchsale_pchxsga

(% change in sales - % change in SG&A), with mean absolute scores of 1.78 and 1.54, respectively.
This is followed by a noticeable drop to the next two variables, mom12m (12-month momentum) and
chempia (industry-adjusted change in employees), both with scores around 0.86. Beyond these top

14

four effects, the importance values become relatively compressed: cash (cash holdings), securedind
(secured debt indicator), egr (growth in common shareholder equity), pchcurrat (% change in current
ratio), operprof (operating profitability), and realestate (real estate holdings) all lie in a fairly narrow
range between roughly 0.65 and 0.75, followed by sp (sales to price ratio), the acc × hire (accrualemployee growth rate) interaction, chinv, cashdebt, and gma. This pattern suggests that the model
places particularly strong weight on a small set of leading univariate effects, while still combining
them with a broader range of medium-sized signals rather than relying on a single characteristic
alone. The fact that the acc × hire term ranks near the middle of the top 15 indicates that interaction
effects are economically relevant, but remain selective relative to the dominant main effects.
Figure 3: Importance of predictors

herf
pchsale_pchxsga
mom12m
chempia
cash
securedind
egr
pchcurrat
operprof
realestate
sp
acc x hire
chinv
cashdebt
gma
0.00

0.25

0.50

0.75

1.00

Mean Absolute Scores

1.25

1.50

1.75

Note: This figure shows the most important characteristics for the AlphaGlass model based on the mean absolute
scores 𝑆(𝑖) and 𝑆(𝑖,𝑗). Importance values are calculated as mean absolute scores on the training data.

In nonlinear models, it is usually not possible to determine with certainty which predictors are
relevant and which are not. Conceptually, variable selection should be related to mean returns and
Sharpe ratios associated with characteristics, as well as the dependence structure of the data. Table 5
reports properties of high/low characteristic portfolios that are informative about these dimensions.
The first column shows the 12 characteristics with the highest Sharpe ratios. Four of the 15 most
important predictors have among the highest Sharpe ratios across all 91 characteristics: pchcurrat,
egr, sp, and herf. Next, we estimate the first four principal components of the matrix consisting

of the long/short characteristic portfolios and compute the sum of the (absolute) loadings by characteristic. Smaller loadings indicate that a characteristic is less relevant to the common movements
captured by PCA factors. The second column of Table 5 shows the predictors with the smallest
loadings. Five of the most important characteristics have small PCA loadings: pchcurrat, chtoia,
chempia, pchsale_pchxsga, and herf. Finally, we compute the tangency mean-variance portfolio gen-

erated from the 91 high/low characteristic portfolios. The third column shows the characteristics

15

with the highest weights. Again, five of the most important predictors have high tangency weights:
herf, egr, cash, operprof, and sp. Although there is no clear interpretation of the predictors’

importance, these results suggest that many of the top predictors are associated with properties
relevant to portfolio optimization.
Table 5: Properties of characteristic portfolios
Sharpe ratios
1
2
3
4
5
6
7
8
9
10
11
12

pchcurrat
pchquick
chcsho
mvel1
pchsale_pchrect
egr
tb
cashpr
salecash
sp
agr
herf

PCA weights
pchcurrat
pricedelay
pchquick
chatoia
cinvest
pchsale_pchrect
aeavol
chempia
pchsale_pchinvt
pchsale_pchxsga
herf
pchcapx_ia

0.30
0.26
0.26
0.22
0.22
0.21
0.21
0.20
0.19
0.19
0.18
0.18

MV weights
0.04
0.05
0.05
0.05
0.08
0.08
0.10
0.10
0.11
0.11
0.11
0.12

pchsale_pchrect
tb
herf
egr
nincr
ear
roaq
cash
indmom
operprof
sp
mvel1

3.80%
3.70%
3.35%
3.06%
2.94%
2.62%
2.62%
2.49%
2.31%
1.97%
1.96%
1.93%

Note: This table reports the 12 characteristics with the highest Sharpe ratios of long/short characteristic portfolios,
the lowest sum of absolute loadings of the first four principal components of long/short characteristic portfolios,
and the highest mean-variance weights of long/short characteristic portfolios. Characteristics in the top-10 by
mean score are bolded.

So far, we have described the importance of individual characteristics. Next, we group the characteristics into categories and assess the total importance of all characteristics within each category.
Table C.2 lists the categories and included characteristics. Figure 4 shows the 10 categories with the
highest mean scores. The most important characteristics are those related to profitability and its
changes, followed by those related to investment, external financing, and the “classic” value/growth
and momentum/reversal characteristics.

4.3 Alternative importance measure
The mean absolute score 𝑆(𝑖) captures the average effect of predictor 𝑖 on signal estimation using
the training sample. An alternative measure of a variable’s importance is its direct effect on portfolio
Sharpe ratios in the test sample. Given the estimated model, we compute the change in the out-ofsample Sharpe ratio ΔSR when an individual term 𝑓𝑖 or 𝑓𝑖𝑗 is removed from the model. We construct
−𝑖
decile portfolios on signals 𝑠𝑙,𝑡
when the univariate effect of variable 𝑖 is eliminated from (7), and

compute the corresponding Sharpe ratio of the 10-1 long-short portfolio SR−𝑖 . The joint effect of
variables 𝑖 and 𝑗 is computed analogously. The effect of variable 𝑖 and interaction (𝑖,𝑗) on the
explanatory power is the reduction relative to the SR of the full model:
ΔSR𝑖 = SR − SR−𝑖 ,

(13)

ΔSR𝑖𝑗 = SR − SR−𝑖𝑗 .

(14)

16

Figure 4: Importance by categories

profitability
profitability
investment
external financing
mom-reversal
value-growth
asset composition
leverage
earnings
liquidity
0.0

0.5

1.0

1.5

2.0

Mean Absolute Scores

2.5

3.0

Note: This figure shows the most important categories for the AlphaGlass model based on the mean absolute scores
𝑆(𝑖) and 𝑆(𝑖,𝑗). Importance values are calculated as mean absolute scores on the training data and aggregated by
category.

Figure 5 shows the 15 most important predictors according to the Sharpe ratio criterion. When
maxret or mom12m are removed, the out-of-sample Sharpe ratio drops from 0.47 to 0.41. The effects

of removing herf, retvol, or egr are only slightly smaller. 10 characteristics are among the top 15
according to both criteria, indicating substantial overlap, even though the mean absolute score is
based on the training sample, whereas the Sharpe ratio criterion is computed on the test sample.
Figure 5: Importance of predictors: ΔSR

maxret
mom12m
herf
retvol
egr
pchsale_pchxsga
cash
rd
pchcurrat
acc x hire
chinv
agr
gma
baspread
mom6m
0.00

0.01

0.02

0.03

SR

0.04

0.05

0.06

Note: This figure shows the most important model terms for the AlphaGlass model according to the Sharpe ratio
criterion (13) and (14).

17

4.4 Shape functions
Due to the separable structure of the model for estimating signals in (7), we can analyze the (potentially) nonlinear effects of individual input characteristics or interaction terms directly from the
estimated shape functions 𝑓𝑖 and 𝑓𝑖𝑗 . In contrast to black-box models, these functions capture the
influence of each model term without the approximation error inherent in post-hoc methods such
as SHAP.9
The four most important univariate shape functions of our main model (in terms of mean absolute scores) are presented in Figure 6, with additional shape functions provided in Appendix H. Note
that the output of each function represents the contribution of an input variable to an asset-specific
signal 𝑠𝑙,𝑡 . As outlined in Section 2, this signal is transformed into portfolio weights through a soft
ranking and decile cut procedure, so that high (low) signals are associated with high (low) weights,
i.e., membership in the long (short) leg of the final portfolio. The mean and standard deviation of
the estimated signals are 0.74 and 6.06, respectively, with an interquartile range of (-4.38, 4.98).

We first examine the effect associated with the most important univariate effect (Panel A), the industry sales concentration measure herf, as defined in Hou and Robinson (2006). Intuitively, higher
values of herf indicate that industry sales are concentrated in the hands of a few firms (i.e., a more
concentrated and less competitive industry structure), whereas lower values indicate a more competitive industry with sales distributed more evenly across firms. The estimated shape function is
monotonically decreasing over the full range of herf values. It starts at a strongly positive value of
about 4 when herf = 0 and then declines steeply, approximately linearly, as herf increases. Around
herf ≈ 0.6, the function exhibits a kink: beyond this point, the function continues to decrease, but
the slope becomes noticeably less steep. At the upper end of the support, the function falls to below −2 at herf = 1, implying that very high concentration is associated with a substantially negative
contribution to the model’s output score. Economically, this shape indicates that industry structure
is an important driver of the model’s score, which is subsequently mapped into portfolio weights.
Low concentration (low herf) contributes positively to the score and therefore increases the portfolio
weight assigned to such stocks, whereas higher concentration (high herf) contributes negatively and
therefore reduces the portfolio weight. In this sense, the Sharpe ratio-optimal portfolio implied by
the model weights toward stocks operating in more competitive industries and away from stocks
in highly concentrated industries, with the strongest marginal shift occurring as concentration rises
from very low to moderate levels and a smaller marginal adjustment once concentration is already
high.
9

Post-hoc explainability methods approximate the decision process of a fitted black-box model and can therefore

introduce approximation error or instability (Rudin, 2019). In high-dimensional stock-level applications, methods such
as SHAP can also become computationally costly and often rely on sampling or approximations, which may introduce
estimation noise and make explanations sensitive to implementation choices (Slack et al., 2020). These distinctions are
especially important in our setting because AlphaGlass provides exact decomposition by construction rather than an
ex-post explanation of an opaque model.

18

Figure 6: Univariate functions 𝑓𝑖 (𝑥𝑖 )
A: herf

B: pchsale_pchxsga

2

4

1
0

fi(xi)

fi(xi)

2
0

1
2
3

2
0.0

0.2

0.4

herf

0.6

0.8

0.0

1.0

0.2

2.0
1.5
1.0
0.5
0.0
0.5
1.0

0.0

0.2

0.4

0.6

0.8

1.0

0.6

0.8

1.0

D: chempia

fi(xi)

fi(xi)

C: mom12m

0.4

pchsale_pchxsga

0.6

mom12m

0.8

1.0

1.5
1.0
0.5
0.0
0.5
1.0
1.5

0.0

0.2

0.4

chempia

Note: This figure shows the estimated 𝑓𝑖 (𝑥𝑖 ) functions of the most important characteristics. The 𝑥-axis plots the
input variable 𝑥𝑖 , cross-sectionally ranked and normalized to the [0,1] interval. The 𝑦-axis represents the additive
contribution to the prediction output through the shape function 𝑓𝑖 (𝑥𝑖 ).

19

The second most important effect is associated with pchsale_pchxsga, which denotes the accounting signal from Abarbanell and Bushee (1998), defined as the percentage change in sales minus
the percentage change in selling, general, and administrative (SG&A) expenses. The variable captures
whether a firm’s revenue growth is accompanied by relatively contained overhead costs. Higher values indicate that sales are growing faster than SG&A, consistent with improving operating leverage
and cost discipline, whereas lower values indicate that SG&A is rising at least as quickly as sales,
suggesting weaker cost control or deterioration in operating efficiency.
The corresponding AlphaGlass shape function for pchsale_pchxsga, displayed in Panel B, is
monotonically increasing, starting at a strongly negative value slightly above −4 at an input value
of 0 and rises steeply as the characteristic increases, reaching close to +2 at an input value of 1.
The increase is particularly pronounced at low to moderate values, after which the function exhibits
a kink around the midrange (between approximately 0.5 and 0.6). Beyond this point, the function
continues to increase, but at a noticeably lower slope, implying diminishing marginal contributions
at higher values of the characteristic. This shape implies that the model assigns higher scores to
firms whose sales growth outpaces SG&A growth, and lower scores to firms whose sales growth is not
supported by comparatively restrained overhead expenses. Since the model score is subsequently
transformed into portfolio weights, the function indicates that the Sharpe ratio-optimal portfolio
places greater weight on stocks with high pchsale_pchxsga and reduces exposure to stocks with low
values. The steep rise at the lower end suggests that moving from weak to moderate relative cost
efficiency produces the largest increase in the model-implied score, while improvements at already
high levels of pchsale_pchxsga still raise the score but with smaller incremental effects.
The shape function in Panel C is associated with 12-month stock return momentum as in Jegadeesh and Titman (1993), measured as a stock’s past return over the previous year (excluding
the most recent month) and commonly interpreted as capturing the tendency of recent winners to
continue outperforming recent losers in the near term. Higher values of the momentum input, therefore, indicate stronger positive past performance over the lookback window, whereas lower values
indicate weaker or negative past performance. In line with this observation, the estimated shape
function is monotonically increasing across the input range. At an input value of 0, the function
output is moderately negative at around −1. It then rises only gradually, reaching roughly −0.5 at a
momentum of about 0.5, implying that variations in momentum within the lower half of the distribution have only a limited influence on the model score. Beyond this midrange level, the slope increases
markedly, and the function rises more rapidly, reaching a strongly positive output of approximately
+2 at the maximum input value. This pattern indicates a convex response, with substantially larger
score contributions for very high-momentum realizations. Economically, the shape implies that the
model rewards exposure to stocks with strong 12-month momentum, whereas assigning lower scores
to stocks with weak momentum. This indicates that the Sharpe ratio-optimal portfolio tilts toward
high-momentum stocks and away from low-momentum stocks. The relatively flat profile over the
lower half of the input range suggests that moderately low momentum is penalized, but not strongly
differentiated, whereas exceptionally strong momentum is treated as particularly beneficial and receives a disproportionately large increase in score and hence portfolio weight.
20

Finally, we examine the effect of chempia, the industry-adjusted change in employees following
Asness et al. (2000) (Panel D). It captures a firm’s growth in headcount relative to the typical change
in headcount among its industry peers over the same period. Higher values, therefore, indicate
comparatively stronger employment growth than the industry benchmark, whereas lower values
indicate comparatively weaker employment growth (or employment contraction) relative to peers.
The associated shape function is monotonically decreasing and approximately symmetric, with
no pronounced kinks or sharp changes in slope. It declines in an almost linear fashion across the
input range, taking a positive value of about +1.5 at chempia =0 and falling steadily to about −1.5 at
chempia = 1. Therefore, the contribution to the model score moves smoothly from positive to nega-

tive as the industry-adjusted change in employees increases. Economically, this pattern implies that
lower industry-adjusted employee growth is associated with higher model scores, while relatively
high employee growth is associated with lower scores. Since the model score is subsequently converted into portfolio weights, the shape indicates that, all else equal, the Sharpe ratio–optimal portfolio increases weight on firms with low (or negative) industry-adjusted changes in employees and
reduces weight on firms exhibiting unusually strong headcount expansion relative to their industries.
The near-linear, smooth decline suggests that this preference is broadly proportional: incremental
increases in relative employee growth are associated with a correspondingly lower contribution to
the score and therefore a lower portfolio tilt.
We conclude that the AlphaGlass model identifies several economically plausible effects that are
exploited to maximize the portfolio’s Sharpe ratio.

4.5 Interactions
In addition to the univariate effects presented so far, the AlphaGlass optimizer models interactions
between pairs of input characteristics. We show the effect associated with the most important such
term in Figure 7. Specifically, the shape function 𝑓𝑖𝑗 (𝑥𝑖 ,𝑥𝑗 ) for the interaction between working
capital accruals (acc) and the employee growth rate (hire), which we call “acc&hire”, is presented in
Panel A of Figure 7 in the form of a heatmap. Working capital accruals summarize changes in noncash working capital components and can be interpreted as a measure of the extent to which current
earnings are supported by accruals rather than cash flows. The employee growth rate captures the
firm’s hiring intensity and thus its expansion in labor inputs.
The estimated interaction surface indicates that the contribution of hiring to the model score
depends strongly on accruals levels. When acc is low, the interaction output is negative for low
hire, reaching values below −1, but it increases considerably as hire rises, ultimately exceeding

+2.5 at high hiring rates. Hence, conditional on low accruals, higher employee growth is associated
with a substantially more positive contribution to the score, whereas low hiring is penalized. As
acc increases, however, this relationship diminishes: the gradient with respect to hire becomes

progressively weaker, and for large acc the interaction output is relatively small (between 0 and 1)
across the hiring spectrum. At high acc, there is also a mild reversal in the pattern, with low hire
yielding slightly higher outputs than high hire.
21

Another way to illustrate interaction effects is to plot 𝑥𝑖 conditional on different values of 𝑥𝑗 , i.e.,
𝑓𝑖𝑗 (𝑥𝑖 |𝑥𝑗 ). Panel B shows the effect of acc on the 𝑥-axis for five different values of hire, equivalent
to taking horizontal cuts of the heatmap in Panel A. Panel C presents the opposite case of holding acc
constant at different levels and observing the resulting contributions as a function of hire (equivalent
to taking vertical cuts in the heatmap).
In economic terms, this interaction suggests that the portfolio-relevant information in hiring is
concentrated among firms with low working capital accruals. In that region, high hiring is associated
with a large positive contribution to the score and therefore a higher model-implied portfolio weight,
while low hiring reduces the score and thus lowers the weight. When accruals are high, by contrast,
hiring conveys little incremental benefit for the score: the interaction contribution becomes small
and fairly uniform, and the slight reversal implies that high hiring is no longer rewarded and may
even be modestly disfavored relative to low hiring. Overall, the model therefore puts weight on
high-hiring firms only when such hiring occurs alongside low accruals, while it largely neutralizes
(or slightly reverses) the hiring tilt when accruals are elevated.

5

Economic insights from interpretability and portfolio properties

So far, we have demonstrated AlphaGlass’ capabilities in terms of performance and structural interpretability. Next, we detail how the model provides insights into portfolio composition through
transparent trading rules derived from its shape functions. In particular, we focus on the composition of the long and short legs of the 10-1 portfolio, which is constructed from the model’s output
signals.
In contrast to black-box approaches, which allow only ex-post analysis of selected assets, our
interpretable model provides an ex-ante understanding of how portfolios are constructed. In other
words, the AlphaGlass model produces transparent rules describing which values of characteristics
lead to higher output scores and, therefore, feature in the portfolios.
We illustrate this using the most important effects presented in Section 4.4. For the most important characteristic, herf, the shape function decreases monotonically, indicating that output scores
are increased for low-herf stocks and decreased for high-herf stocks. In terms of portfolio composition, this suggests that low-herf stocks are expected to appear in the long leg (decile portfolio
10) constructed from the AlphaGlass scores, whereas high-herf stocks are expected to populate the
short leg (decile portfolio 1). Panel A of Figure 8 shows that this prediction is indeed reflected in
the composition, as monthly mean herf is considerably higher in portfolio 1 compared to portfolio
10.10 This pattern holds consistently throughout the sample period.
Similar conclusions can be drawn from the pchsale_pchxsga and mom12m shape functions: Both
increase monotonically and are, therefore, consistent with stocks with high (low) characteristic values appearing in the long (short) leg. Panels B and C of Figure 8 confirm this expectation, showing
10

Means within portfolios are scaled by the monthly mean across all stocks to eliminate changes in the characteristic

levels.

22

Figure 7: Interaction – acc&hire
A: 𝑓𝑖𝑗 (𝑥𝑖 ,𝑥𝑗 )

2.5
0.8

2.0
1.5

0.6

hire

1.0
0.5

0.4

0.0
0.5

0.2

1.0
0.2

0.4

acc

0.6

0.8

B: acc by hire interaction

3

hire

0
0.25
0.5
0.75
1

2
1
0
1
0.0

0.2

0.4

acc

0.6

0.8

1.0

0.8

1.0

C: hire by acc interaction

3

acc

0
0.25
0.5
0.75
1

2
1
0
1
0.0

0.2

0.4

hire

0.6

Note: Panel A displays the shape function of the interaction between acc and hire. The scaled values of acc and
hire are shown on the 𝑥- and the 𝑦-axis, respectively. The color-coded axis represents the additive contribution
to the signal output associated with combinations of 𝑥 and 𝑦 values in basis points. In Panels B and C, conditional
contributions are derived from the shape function by separately holding each variable involved in the interaction
constant at different levels.

23

clear separation between the two legs over the sample period.
In contrast, Panel D presents the same analysis for the least important univariate effect, namely
the change in 6-month momentum (chmom). We observe little separation between the two legs, confirming the expectation that chmom does not meaningfully influence portfolio composition.

Figure 8: Characteristics of AlphaGlass portfolios
A: herf in portfolios 1 and 10

B: pchsale_pchxsga in portfolios 1 and 10

0.8

0.8

1
10

0.7
0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2 2000

2004

2008

2012

2016

0.2 2000

2020

C: mom12m in portfolios 1 and 10

0.8

0.7

0.7

0.5

2008

2012

2016

2020

1
10

0.6

1
10

0.5

0.4

0.4

0.3

0.3

0.2 2000

2004

D: chmom in portfolios 1 and 10

0.8
0.6

1
10

0.7

2004

2008

2012

2016

0.2 2000

2020

2004

2008

2012

2016

2020

Note: This figure shows the characteristics of the top and bottom deciles of the AlphaGlass portfolios. Panels A to
C show the monthly means of the three characteristics with the highest 𝑆(𝑖). Panel D shows the characteristic with
the lowest 𝑆(𝑖).

5.1 Signal decomposition
Next, we examine which characteristics explain asset selection within the AlphaGlass portfolio. To
this end, we measure which model terms increase the signals, pushing assets toward the long
leg, and which decrease the signals, most likely leading to inclusion in the short leg. Let 𝒫𝑞 ⊂

𝒯1 be the set of training-set indices (𝑙,𝑡) that fall into decile 𝑞 ∈ {1,…,10} at their respective formation month. To isolate signed contributions, we decompose each shape function output 𝑎 =
𝑓𝑖 (𝑥𝑖,𝑙,𝑡 ) (for some feature 𝑖), or 𝑎=𝑓𝑖𝑗 (𝑥𝑖,𝑙,𝑡 ,𝑥𝑗,𝑙,𝑡 ) (for some pair (𝑖,𝑗)) into its positive and negative
parts:
[𝑎]+ = max{𝑎,0},

[𝑎]− = max{−𝑎,0}.

24

We define directional mean scores on any decile 𝑞 as the portfolio-size–normalized averages of the
positive and negative parts of the shape functions:
𝑆𝑖+ (𝑞) =

1

∑ [𝑓𝑖 (𝑥𝑖,𝑙,𝑡 )]+
|𝒫𝑞 | (𝑙,𝑡)∈𝒫

(15)

𝑞

𝑆𝑖− (𝑞) = −

1
∑ [𝑓𝑖 (𝑥𝑖,𝑙,𝑡 )]−
|𝒫𝑞 | (𝑙,𝑡)∈𝒫

(16)

𝑞

+
𝑆𝑖𝑗
(𝑞) =

1

∑ [𝑓𝑖𝑗 (𝑥𝑖,𝑙,𝑡 ,𝑥𝑗,𝑙,𝑡 )]+
|𝒫𝑞 | (𝑙,𝑡)∈𝒫

(17)

𝑞

−
𝑆𝑖𝑗
(𝑞) = −

1

∑ [𝑓𝑖𝑗 (𝑥𝑖,𝑙,𝑡 ,𝑥𝑗,𝑙,𝑡 )]− .
|𝒫𝑞 | (𝑙,𝑡)∈𝒫

(18)

𝑞

By construction, 𝑆+ summarizes how strongly (on average) a feature or interaction contributes positively within the chosen decile, whereas 𝑆− summarizes the average negative contribution and is
reported with a minus sign so its magnitude is directly comparable to |𝑆+ |. For our long-short analysis, we set
𝑆𝑖+ = 𝑆𝑖+ (10),

+
+
𝑆𝑖𝑗
= 𝑆𝑖𝑗
(10)

and

𝑆𝑖− = 𝑆𝑖− (1),

−
−
𝑆𝑖𝑗
= 𝑆𝑖𝑗
(1).

Note that this step, once again, relies on the separable model structure, which allows for exact
decomposition of the model output (the portfolio signals) into term-level contributions (the shape
function outputs).
The resulting importance rankings are presented in Figure 9. Panel A reports the top 15 effects
+ 11
in terms of 𝑆𝑖+ and 𝑆𝑖𝑗
. The most prominent driver of long signals is industry sales concentration

herf, with 𝑆+ = 2.42, far exceeding the remaining effects. This aligns with the previous finding that
herf is the most important contributor to model signals overall. The decomposition clarifies that its

importance is mainly due to positive contributions toward the long leg, i.e., the low-herf region (see
Figure 6). Beyond this leading term, the next tier of effects have more comparable magnitudes (𝑆+ ≈
0.79–1.37), suggesting that the model’s long selection is supported by a diverse set of reinforcing
signals rather than a single-factor rule.
The leading contributors fall into several intuitive themes. First, the inclusion of mom12m among
the top effects indicates that recent price trends materially strengthen long signals, consistent with
momentum serving as an important selection criterion. Second, the presence of rd (R&D increase)
and egr (growth in common shareholder equity) indicates that the long leg is tilted toward firms with
characteristics associated with investment and growth. Third, fundamentals and operating dynamics
play a central role: nincr (number of earnings increases) contributes positively, pointing to a preference for firms exhibiting improving earnings patterns, and pchsale_pchxsga (percentage change
in sales minus percentage change in SG&A) suggests that sales growth outpacing overhead growth
is rewarded in the signal, consistent with operating leverage or efficiency considerations. Relatedly,
chinv (change in inventory) appears among the top long contributors, indicating that inventory dy11

Figure I.1 in the Appendix also reports leg-wise mean score decompositions when positive and negative contributions

are taken into account, confirming the main drivers.

25

Figure 9: Importance of predictors for long/short allocation
A: Mean positive scores 𝑆+ (𝑖) and 𝑆+ (𝑖,𝑗) in portfolio 10

herf
pchsale_pchxsga
rd
divo
mom12m
chempia
acc x hire
cash
egr
pchcurrat
nincr
secured
securedind
chinv
sp
0.0

0.5

1.0

1.5

2.0

Mean Positive Scores

2.5

B: Mean negative scores 𝑆− (𝑖) and 𝑆− (𝑖,𝑗) in portfolio 1

sin
pchsale_pchxsga
divi
realestate
herf
operprof
mom12m
securedind
salerec
maxret
tang
cash
gma
cashdebt
chinv
0

1

2

3

Mean Negative Scores

4

5

Note: This figure shows the most important effects of the AlphaGlass model for allocation in the long (Panel A) and
short leg (Panel B) of the 10-1 portfolio. Importance values are calculated as mean positive (negative) scores on the
training data, respectively.

26

namics contain incremental information for long selection in the learned portfolio mapping.
The long-leg ranking also highlights balance-sheet and financing characteristics. cash (cash holdings) and pchcurrat (percentage change in current ratio) both contribute positively, consistent with
the model assigning higher long signals to firms with stronger liquidity positions or improving shortterm solvency. At the same time, secured (secured debt) and securedind (secured debt indicator)
feature among the top positive contributors, implying that secured borrowing is, in this learned
allocation, associated with higher signals for long-decile firms, potentially capturing differences in
collateralization, financing constraints, or capital structure that the model exploits together with
other characteristics.
Moreover, we find further evidence that AlphaGlass leverages nonlinear interaction information.
The strong ranking of acc & hire indicates that the model conditions the contribution of workingcapital accruals (acc) on the firm’s employee growth (hire). In other words, the long signal is not
driven by accruals or hiring alone, but by particular combinations of these variables in regions where
the learned bivariate shape function contributes positively. Likewise, chempia (industry-adjusted
change in employees) underscores that labor-force adjustments relative to industry peers are informative for selection in the long leg.
Panel B presents the analogous ranking for the short leg. The dominant driver of short signals
is sin, the sin stock indicator, with a magnitude of |𝑆− | = 5.60, substantially larger than all other
terms. This implies that, among bottom-decile assets, the learned sin shape function frequently
takes values that strongly reduce the signal, making sin the single most influential downward contributor to short-leg inclusion. A second large effect is pchsale_pchxsga (|𝑆− |=2.57), indicating that
the same operating-dynamics variable that supports long signals in the top decile also meaningfully
contributes to low signals in the bottom decile, consistent with the shape function described above
in which unfavorable regions of sales-minus-overhead dynamics push assets toward the short leg.
Several additional terms with sizable negative contributions reflect payout and balance-sheet
characteristics. divi (dividend initiation) appears prominently (|𝑆− | = 2.01), suggesting that, within
the short decile, dividend initiations are associated with signal-reducing regions of the learned shape
function. realestate (real estate holdings) also ranks highly (|𝑆− | = 1.55), and herf (industry sales
concentration) again appears (|𝑆− | = 1.04), indicating that industry structure and asset composition
features contribute materially to the model’s negative scoring within the short-decile set.
Profitability-related characteristics also feature among the main downward contributors: operprof (operating profitability) and gma (gross profitability) both appear in the top 15. Their presence

in the short-leg ranking should be interpreted through the lens of the learned nonlinear shape functions: among bottom-decile stocks, the realized profitability values tend to fall in regions that further
reduce the signal. Relatedly, salerec (sales-to-receivables) contributes negatively, consistent with the
model using working-capital/receivables information when forming low signals.
The short-leg ranking also highlights risk and financing variables. maxret (maximum daily return)
enters among the top effects, suggesting that extreme daily moves (a proxy for lottery-like behavior or
crash/optionality exposure) are associated with lower signals in the short decile. securedind (secured

27

debt indicator) again appears, indicating that the secured-debt dimension is informative for both legs,
though in different regions of the covariate space. Finally, several liquidity and debt-service measures
(cash and cashdebt, cash flow to debt) contribute to signal reductions in the bottom decile, consistent
with the model penalizing weaker financing capacity or adverse liquidity/debt-service configurations
among short candidates. Inventory dynamics (chinv) also appear again, suggesting that inventory
changes contain incremental information for both extreme tails of the signal distribution.
Two patterns stand out when comparing Panels A and B. First, some characteristics are important
on both sides, most notably pchsale_pchxsga, herf, mom12m, securedind, cash, and chinv. This is consistent with a nonlinear score function that uses the same covariates to separate winners from losers
by assigning opposite-signed contributions in different regions of the feature space (and/or by interacting with other terms), rather than employing disjoint sets of predictors for longs and shorts. Second, each leg also has clear distinctive drivers: the long leg features growth/investment and improving fundamentals (e.g., rd, egr, nincr) as well as a salient interaction (acc x hire), whereas the short
leg is dominated by sin and places greater emphasis on payout/initiation and asset-composition variables (divi, realestate), together with measures related to tail risk (maxret) and profitability/quality
(operprof, gma).
Overall, the signal decomposition indicates that AlphaGlass’s asset selection is driven by a combination of broad economic themes, such as industry structure, operating dynamics, profitability/quality, liquidity/financing, and risk, while allowing for strong nonlinearities and interaction
effects. Importantly, the overlap of key terms across the long and short tails suggests that the
model primarily differentiates positions through directional contributions of the same underlying
characteristics, rather than by relying on entirely separate sets of variables for the two legs.

5.2 Interactions
In the previous sections, we found that, in addition to the univariate effects discussed above, the
AlphaGlass model relies on interactions in its decision process, particularly the acc&hire interaction.
In what follows, we study the relationship between this interaction and AlphaGlass portfolios. Recall
the heatmap and the conditional effects displayed in Figure 7. Analogously to the univariate effects,
we can derive implications from this learned shape function. We observe the highest interaction
scores when low acc values coincide with high hire values. When acc is high, on the other hand,
we find the lowest interaction outputs for low-hire stocks. This indicates that stocks with higher
interaction scores, which are therefore more likely to be included in the long leg, should exhibit a
negative correlation between acc and hire. The interaction shape suggests the opposite pattern for
low score (likely shorted) stocks. Low interaction outputs are observed for stocks in which acc and
hire are both low or both high, indicating a positive correlation between the two characteristics in

these stocks.
The described effects are reflected in AlphaGlass portfolios. Panel A of Figure 10 shows the
scatter plot of acc on the 𝑥-axis and hire of the first portfolio (the short leg of the AlphaGlass
strategy) on the 𝑦-axis. The plot also shows the orange regression line with 95% confidence intervals
28

and the regression coefficient in the top corner. Consistent with the expectation, there is a positive
relationship between acc and hire with a highly significant regression coefficient of 0.54 (t = 11.4).
Panel B shows the same plot for portfolio 10 (the long leg of the AlphaGlass strategy), with the
coefficient of −0.38 (t = −3.3) confirming the expected negative relationship.
Figure 10: acc&hire within AlphaGlass portfolios
A: acc&hire in portfolio 1

0.58
0.57

B: acc&hire in portfolio 10

0.50

= 0.54

0.55

hire

hire

0.56
0.54

0.46
0.44

0.53

0.42

0.52

0.40

0.51
0.52

0.54

= -0.38

0.48

0.56

0.58

acc

0.60

0.62

0.27 0.28 0.29 0.30 0.31 0.32 0.33 0.34 0.35

acc

Note: This figure shows scatter plots of acc and hire in AlphaGlass portfolios 1 and 10. The orange line is the
regression line with 95% confidence intervals.

6

Mean-variance preferences

AlphaGlass allows flexible specification of the loss function. In principle, any differentiable function of the factor 𝑓𝑡HL is permissible. For example, the estimation allows for portfolio constraints
or penalties, see Appendix J, which considers drawdown penalties. While the SR criterion used in
the baseline analysis is natural in asset pricing and portfolio construction, it is useful to consider
alternative objective functions that more flexibly reflect investor preferences. In particular, meanvariance utility provides a standard formulation in which expected returns are traded off against
return variance through a risk-aversion parameter. This allows us to examine whether the AlphaGlass framework continues to perform well when the estimation target is no longer scale-free but
instead depends explicitly on the investor’s risk tolerance.12

𝒰(𝑓𝑡 ) = E(𝑓𝑡 ) −
12

𝛾
Var(𝑓𝑡 ),
2

(19)

In Appendix K, we show how the Sharpe ratio and mean-variance objectives complement each other. Although they

select the same portfolio rule under certain assumptions, the equivalence is not general, so it is informative to study
the mean-variance case separately. In this section, we therefore estimate the model for mean-variance preferences over
returns 𝑓𝑡 :

29

where 𝛾 is the coefficient of absolute risk aversion. We estimate the AlphaGlass model using the
(dis)utility of the long/short portfolio, 𝑓HL , as the loss function:

ℒ = −𝒰(𝑓𝑡HL ).

(20)

Table 6 shows the maximized in-sample and out-of-sample utilities, 𝒰, for 𝛾 = 1 and 𝛾 = 3 for
AlphaGlass and the three benchmark models. AlphaGlass delivers the strongest out-of-sample performance for both levels of risk aversion. For 𝛾 = 1, its out-of-sample utility is 1.12, compared with
0.99 for EBM, 0.86 for NN, and 0.53 for RF. For 𝛾 = 3, the ranking is unchanged: AlphaGlass again
performs best, with an out-of-sample utility of 1.23, followed by EBM (0.87), NN (0.79), and RF (0.34).
Therefore, AlphaGlass remains the most effective method even when the objective is changed from
Sharpe ratio maximization to a utility criterion. Moreover, the results reveal a pronounced contrast
between in-sample and out-of-sample performance. The benchmark models achieve substantially
larger in-sample utility values than AlphaGlass. For example, when 𝛾 = 1, RF and EBM reach insample utilities of 8.56 and 7.88, respectively, far above the AlphaGlass value of 1.81. A similar
pattern holds for 𝛾=3, where RF and EBM again produce much larger in-sample utilities than AlphaGlass. However, these in-sample gains do not survive out-of-sample. Instead, all three benchmark
models experience large declines, whereas AlphaGlass retains a much larger share of its in-sample
performance.
Overall, these results underscore the broader point that AlphaGlass is not confined to maximizing the Sharpe ratio. Rather, the framework can accommodate alternative portfolio objectives
while preserving its central features: end-to-end estimation, economic interpretability, and strong
out-of-sample performance.
Table 6: Maximized utility 𝒰 of mean-variance preferences
In-sample

Out-of-sample

Panel A: 𝛾 = 1
AlphaGlass
RF
NN
EBM

1.81
8.56
3.51
7.88

1.12
0.53
0.86
0.99

Panel B: 𝛾 = 3
AlphaGlass
RF
NN
EBM

1.43
7.48
3.21
6.70

1.23
0.34
0.79
0.87

Note: This table compares maximized mean-variance utility for 𝛾 = 1,3. The AlphaGlass model is estimated using
a mean-variance loss function (20). The RF, NN, and EBM models use the MSE loss function. The results in the table
are based on the long/short portfolios implied by the models and multiplied by 100.

30

7

AlphaGlass theoretical properties

In this section, we derive important properties of the AlphaGlass approach with respect to its learning behavior. Complete proofs are in Appendix K.1, while Appendix K.2 provides additional theoretical results. In particular, it clarifies the relationship between the Sharpe ratio and mean-variance
objectives, and justifies the advantages of our implementation compared to a softmax-based formulation.13
As detailed in Section 2, AlphaGlass is trained through a differentiable approximation to the
standard hard rank-and-cut construction. This raises a practical issue: the economically meaningful
object is the hard-sorted quantile portfolio, while optimization uses a smooth surrogate. First, we
justify our training approach by showing that the differentiable soft sorts we employ serve as a
faithful proxy for the hard portfolio procedures applied to the test data, and that their gradients
used during backpropagation are appropriate.
We fix the temperature parameters 𝜏 = (𝜏rank ,𝜏mask . For each 𝜃 and 𝑡, let 𝑤𝑡hard (𝜃|𝜏) be the
portfolio weights obtained by exact ranking and truncation (e.g., invest only in the top-ranked assets
that pass the long/short mask). Let 𝑤𝑡soft (𝜃|𝜏) be the corresponding differentiable approximation
with temperature vector 𝜏 entering the soft ranking and masking. We denote the associated empirical
hard
soft
̂ 𝑇 (𝜃) and SR
̂ 𝑇 (𝜃|𝜏).
Sharpe ratios by SR
To quantify how clearly the model separates selected and non-selected assets, we introduce a
(𝑡)

margin: for each (𝜃,𝑡), let Δ𝜃 > 0 measure the smallest pairwise gap between signals in month 𝑡.
Theorem 1 formalizes that our differentiable portfolio construction is a faithful surrogate for the
usual hard top-bottom sort:
Theorem 1 (Soft-hard equivalence)
(𝑡)

For fixed 𝜃, the random variables Δ𝜃 have a continuous density at 0 (no point mass at ties). Then:
(𝑡)

1. Pointwise convergence given a margin: Fix 𝑡 and suppose Δ𝜃 ≥𝛿>0. Then there exist constants
𝐶1 ,𝐶2 ,𝑐 > 0 depending only on 𝑞 and 𝑛max such that for all sufficiently small 𝜏,
‖𝑤𝑡soft (𝜃|𝜏) − 𝑤𝑡hard (𝜃)‖1 ≤ 𝐶1 𝑒−𝑐𝛿/𝜏rank + 𝐶2 𝑒−𝑐/𝜏mask .
2. Averaged convergence and Sharpe ratio: Let 𝜏𝑇 ↓ 0. Then
𝑇

𝑝
1
soft
hard
⟶ 0,
∑ ‖𝑤𝑡 (𝜃|𝜏𝑇 ) − 𝑤𝑡 (𝜃)‖1 −
𝑇 𝑡=1

soft

hard

̂
̂𝑇
SR𝑇 (𝜃|𝜏𝑇 ) − SR

𝑝

(𝜃) −
⟶ 0.

(𝑡)

3. Gradient consistency at non-tie points): Fix 𝑇 and 𝜃 such that Δ𝜃 > 0 for all 𝑡 = 1,…,𝑇 (no ties
13

Note that, although the formal statements below are written for the SR criterion, the arguments underlying The-

orems 1–3 are more general. Theorem 1 is primarily a statement about the soft rank-and-mask construction and its
approximation of the corresponding hard-sorted portfolio and for Theorems 2 and 3 the proofs establish convergence
of the empirical portfolio moments 𝜇̂𝑇 (𝜃) and 𝜎̂ 𝑇 (𝜃) and then pass these moment bounds to the objective of interest.
The same logic extends to other criteria of the form 𝑄(𝜃) = 𝜙(𝜇(𝜃),𝜎(𝜃)), provided the objective is well behaved (e.g.,
𝛾

preserves continuity and bounded gradients). Mean-variance utility, 𝑄(𝜃) = 𝜇(𝜃) − 2 𝜎2 (𝜃), is a simple example covered
by this extension.

31

hard

in the sample at 𝜃) and 𝜎̂ 𝑇hard (𝜃) > 0. Then ̂
SR𝑇
hard
Clarke ̂
hence 𝜕
SR𝑇 (𝜃) = {0}, and

(𝜃) is locally constant in a neighborhood of 𝜃,

soft

SR𝑇 (𝜃|𝜏)‖ ⟶ 0
‖∇ ̂

as 𝜏 ↓ 0.

This theorem formalizes multiple desirable properties:
• Soft approximation: If, at time 𝑡, the signal scores exhibit a nontrivial margin, then the soft
weights 𝑤𝑡soft (𝜃|𝜏) are exponentially close (in ℓ1 ) to the hard equal-weight portfolio 𝑤𝑡hard (𝜃)
as 𝜏rank goes to zero. Intuitively, once the scores are well separated everywhere, the sigmoids in
the soft ranking/masking behave like indicators, so the differentiable construction reproduces
the hard rank-and-cut portfolio.
• Time-average and Sharpe ratio consistency: Under stationarity/weak dependence and no mass
at exact ties, an annealing schedule 𝜏𝑇 ↓ 0 yields
𝑇

𝑝
1
soft
hard
⟶ 0,
∑ ‖𝑤𝑡 (𝜃|𝜏) − 𝑤𝑡 (𝜃)‖1 −
𝑇 𝑡=1

soft

hard

̂ 𝑇 (𝜃) − SR
̂𝑇
SR

𝑝

(𝜃) −
⟶ 0.

Therefore, in large samples, the differentiable construction targets the same portfolio and the
same sample Sharpe ratio as the discrete benchmark.14
In other words, Theorem 1 provides a direct bridge between the portfolio procedure used for optimization and the hard quantile portfolios used for evaluation: whenever the model produces clear
cross-sectional separation, the soft weights are exponentially close to the hard weights, and the
corresponding Sharpe ratios match in large samples. This ensures that the population and sample
Sharpe ratios appearing in our theoretical results can be interpreted as Sharpe ratios of the standard hard-sorted long-short portfolios, up to a vanishing approximation error. At the same time, at
parameter values where the ranking is stable (no ties), the hard Sharpe ratio is locally flat, and the
gradient of the soft Sharpe vanishes as the temperature goes to zero. This formalizes the idea that
the soft relaxation correctly captures the non-smooth structure of the true optimization problem,
while still allowing gradient-based training.
We formalize that, within the AlphaGlass class, maximizing the sample Sharpe ratio is asymptotically aligned with maximizing the population Sharpe ratio.15 Given a sample of length 𝑇, we define
the population (or out-of-sample) mean and volatility of the portfolio return as
𝜇(𝜃) = 𝔼[𝜋𝑡 (𝜃)],

𝜎2 (𝜃) = Var(𝜋𝑡 (𝜃)),

and the population Sharpe ratio as
SR(𝜃) =
14

𝜇(𝜃)
.
𝜎(𝜃)

We also show the complementary property of gradient validity: The gradient of the soft objective converges (as

temperatures go to zero) to a valid Clarke (1975) subgradient of the non-differentiable hard-sort objective. Away from
ties, the hard objective is locally flat, and the soft gradient vanishes, indicating that using the smooth surrogate provides
principled gradients for training against a hard-sorting target (see Appendix K.1 for more details).
15
For clarity, we work with a simplified notation in this section and refer to the technical assumptions and detailed
proofs in the Appendix.

32

Given a sample of length 𝑇, the empirical analogs are
𝑇

𝜇
̂𝑇 (𝜃) =

𝑇

1
∑ 𝜋𝑡 (𝜃),
𝑇 𝑡=1

̂
𝜎𝑇2 (𝜃) =

1
2
̂𝑇 (𝜃)) ,
∑ (𝜋𝑡 (𝜃) − 𝜇
𝑇 𝑡=1

and the empirical Sharpe ratio
̂𝑇 (𝜃)
̂ 𝑇 (𝜃) = 𝜇
.
SR
̂
𝜎𝑇 (𝜃)
Theorem 2 formally justifies optimizing the Sharpe ratio directly:16
Theorem 2 (Consistency of Sharpe maximization within the AlphaGlass class)
Fix 𝜏rank ,𝜏mask > 0 and define the population and empirical maximizers of the Sharpe ratio as
𝜃̂𝑇 ∈ argmax ̂
SR𝑇 (𝜃).

𝜃⋆ ∈ argmax SR(𝜃),
𝜃∈Θ

𝜃∈Θ

Then:
𝑝

1. sup𝜃∈Θ | ̂
SR𝑇 (𝜃) − SR(𝜃)| −
⟶ 0.
𝑝

2. Any sequence of empirical maximizers satisfies dist(𝜃̂𝑇 ,argmax SR) −
⟶ 0. If 𝜃⋆ is unique,
𝑝
𝜃̂𝑇 −
⟶ 𝜃⋆ .
𝑝

̂ 𝑇 (𝜃̂𝑇 ) −
3. Consequently, SR
⟶ SR(𝜃⋆ ).
In other words, asymptotically, AlphaGlass attains the best Sharpe available within its interpretable
class. The theorem shows that our training objective (the sample Sharpe ratio computed from the
soft long-short portfolio) is statistically aligned with the economic target (the population Sharpe
ratio). In particular, it provides:
̂ 𝑇 (𝜃) converges uniformly
• Uniform learnability: As 𝑇 → ∞, the entire empirical estimate 𝜃 ↦ SR
(in probability) to the true 𝜃↦SR(𝜃). In other words, wherever we look in the model space, the
sample Sharpe ratio is a reliable proxy for the population Sharpe ratio.
̂ 𝑇 (𝜃) converges to
• Consistency: For any sequence of empirical maximizers 𝜃̂𝑇 ∈ argmax𝜃∈Θ SR
the set of population maximizers argmax𝜃∈Θ SR(𝜃) in probability. If the maximizer is unique,
then 𝜃̂𝑇 →𝜃⋆ . Therefore, maximizing the sample Sharpe ratio asymptotically recovers the bestin-class model (at the population level).
• Value optimality: The achieved Sharpe ratio at the fitted model converges to the best attainable
Sharpe ratio within the class:
̂ 𝑇 (𝜃̂𝑇 ) ⟶ SR(𝜃⋆ ).
SR
In practice, this means that training AlphaGlass by maximizing the in-sample Sharpe ratio is
asymptotically “correct”: With enough data, the resulting parameter vector yields a portfolio whose
population Sharpe ratio approaches the highest Sharpe ratio achievable within the chosen architecture.
16

For clarity, we work with a simplified notation of the mathematical properties in this section and refer to the technical

assumptions and detailed explanations in Appendix K.1.

33

This raises the question of how reliably the AlphaGlass portfolio that maximizes the in-sample
Sharpe ratio performs out-of-sample. Even if maximizing sample Sharpe ratio targets the population
optimum (Theorem 2), a finite training sample can still produce over-optimistic in-sample Sharpe
ratios. The next result gives a finite-sample reliability statement: with high probability, the population Sharpe ratio of the fitted AlphaGlass strategy is close to the best attainable population Sharpe
ratio within the class, with a gap controlled by an explicit measure of effective strategy complexity,
sample length, and temporal dependence.
For this, we consider the class of portfolio returns
Π = {𝜋𝑡 (𝜃) = 𝑤𝑡 (𝜃)⊤ 𝑟𝑡+1 ∶ 𝜃 ∈ Θ}.
For a block length 𝑏 ∈ {1,…,⌊𝑇/2⌋}, let
𝑚=⌊

𝑇
⌋,
2𝑏

and denote by ℜblk
𝑇,𝑏 (Π) the blocked (dependent-data) Rademacher complexity of Π, as defined in Appendix K.1. Intuitively, ℜblk
𝑇,𝑏 (Π) measures the effective richness of the model class after accounting
for temporal dependence through blocking.17
Theorem 3 (Oracle inequality for out-of-sample Sharpe ratio)
Assume that returns are bounded and weakly dependent, that the AlphaGlass map 𝜃 ↦ 𝑤𝑡 (𝜃) is Lipschitz, and that the portfolio volatility is uniformly bounded away from zero. Then for every 𝛿 ∈ (0,1),
𝑏
log(8/𝛿)
⎛ +
⎞⎞ ≥ 1 − 𝛿 − 2𝑚𝛽(𝑏),
Pr⎛SR(𝜃⋆ ) − SR(𝜃̂𝑇 ) ≤ 𝐶1 ℜblk
𝑇,𝑏 (Π) + 𝐶2
𝑇
𝑚
√
⎝
⎝
⎠⎠
for some constants 𝐶1 ,𝐶2 >0 depending only on basic properties of the data and the model. Here 𝛽(𝑏)
denotes the 𝛽-mixing coefficient at lag 𝑏.
In particular, if 𝑏 = 𝑏𝑇,𝛿 is chosen so that
2𝑚𝛽(𝑏) ≤ 𝛿/2,
then
𝑏
log(8/𝛿)
⎛ +
⎞⎞ ≥ 1 − 𝛿.
Pr⎛SR(𝜃⋆ ) − SR(𝜃̂𝑇 ) ≤ 𝐶1 ℜblk
𝑇,𝑏 (Π) + 𝐶2
𝑚
⎝
⎝𝑇 √
⎠⎠
This property is important because it guarantees that the empirically Sharpe-optimal AlphaGlass
portfolio generalizes: its out-of-sample Sharpe ratio is close to that of the best possible portfolio
within the model class. The gap is controlled by three components: the effective complexity of the
strategy class, captured by ℜblk
𝑇,𝑏 (Π), a finite-sample remainder 𝑏/𝑇, and a concentration term based
on the effective number of approximately independent blocks, 𝑚 = ⌊𝑇/(2𝑏)⌋.
This result matters economically because it converts a statistical training objective into a reliability statement about investment performance. Optimizing the sample Sharpe ratio does not
17

Intuitively, ℜblk
𝑇,𝑏 (Π) measures how well the class Π can fit random noise in the data after accounting for weak de-

pendence through block aggregation: larger values mean higher effective model capacity and thus a greater potential for
overfitting. See, for example, Shalev-Shwartz and Ben-David (2014) for a general overview.

34

merely fit noise: with high probability, the resulting portfolio’s population Sharpe ratio is near-bestin-class, and the gap vanishes as the sample length grows provided the effective complexity remains
controlled and dependence is not too strong. Therefore, the bound can be read as a transparent datasufficiency condition. To make the SR shortfall small, the sample length 𝑇 must be large relative to
the model’s effective complexity and relative to the dependence-adjusted block structure. This formalizes a fundamental trade-off in portfolio construction: increasing model flexibility can raise the
attainable in-class Sharpe ratio, but it also requires more data for the same level of out-of-sample
reliability.
In particular, the number of effective model parameters influences reliability through its effect on
the complexity term ℜblk
𝑇,𝑏 (Π). Activating many terms, especially interactions, increases the number
of ways the strategy can adapt to the sample, so more data are required for the same level of outof-sample reliability. At the same time, this underscores that sparsity and pruning are not merely
interpretability devices. They directly reduce effective model complexity and thereby tighten the
bound, making it more likely that in-sample Sharpe improvements persist out-of-sample.
Taken together, these results provide a unified theoretical justification highlighting the approach’s
practical benefits. The differentiable rank-and-mask procedure faithfully approximates hard quantile portfolios (Theorem 1), maximizing sample Sharpe targets the population Sharpe ratio within
the AlphaGlass class (Theorem 2), and finite-sample generalization guarantees explain when a high
in-sample Sharpe ratio is informative about out-of-sample performance (Theorem 3).

8

Monte Carlo simulation

While AlphaGlass is guaranteed to target the maximum attainable Sharpe ratio within its model class,
this guarantee does not, by itself, provide a numerical comparison against richer population benchmarks. In simulation, we can go one step further. Because the data-generating process is known, we
can evaluate each fitted model on an arbitrarily large independent test sample, so that the resulting
out-of-sample Sharpe ratio is an arbitrarily accurate approximation to the model’s population Sharpe
ratio. This allows us to study not only whether AlphaGlass outperforms benchmark learners in finite
samples, but also how close it comes to the simulated population optimum and to oracle portfolio
rules that are infeasible in practice.

8.1 Simulation design
Concretely, our simulations are designed as follows. In each month 𝑡, we draw a cross section of
𝑛 = 500 assets with 𝑝 = 6 characteristics. Let
𝑥𝑙,𝑡 = (𝑥𝑙,𝑡,1 ,…,𝑥𝑙,𝑡,𝑝 )⊤ ,

𝑙 = 1,…,𝑛,

and assume that the raw characteristics are jointly Gaussian with unit variances and common pairwise correlation 0.25. As in the empirical analysis, each characteristic is then transformed into a

35

cross-sectional rank on [0,1]:
𝑢𝑙,𝑡,𝑗 =

rank(𝑥𝑙,𝑡,𝑗 )
∈ [0,1],
𝑛

𝑈𝑙,𝑡 = (𝑢𝑙,𝑡,1 ,…,𝑢𝑙,𝑡,𝑝 )⊤ .

This normalization keeps the simulated design close to the empirical implementation and removes
any role for the marginal scale of the characteristics. Monthly returns are generated according to
𝑖.𝑖.𝑑.

𝑟𝑙,𝑡+1 = 𝜇sig 𝑚(𝑈𝑙,𝑡 ) + 𝜎(𝑈𝑙,𝑡 )𝜀𝑙,𝑡+1 ,

𝜀𝑙,𝑡+1 ∼ 𝑁(0,1),

(21)

𝑎1 = 1.

(22)

with 𝜇sig = 0.018, where the conditional mean component is
3

𝑚(𝑈𝑙,𝑡 ) = 𝑎1 (2𝑢𝑙,𝑡,1 − 1) ,

The conditional idiosyncratic volatility is characteristic-dependent and given by
1 2

log𝜎(𝑈𝑙,𝑡 )2 = 𝑔0 + 𝑔1 𝑢𝑙,𝑡,2 + 𝑔2 (𝑢𝑙,𝑡,3 − 2 ) ,

(23)

with (𝑔0 ,𝑔1 ,𝑔2 ) = (−4.6, 2.5, 1.2).
For each Monte Carlo replication, we simulate a training panel of length 𝑇 ∈ {120,240} months
and estimate four models: AlphaGlass, Explainable Boosting Machines (EBM), Random Forests and a
neural network. Consistent with our empirical implementation, EBM, Random Forest, and the neural
net are estimated as pooled return-prediction models, whereas AlphaGlass is trained directly on
the time-series Sharpe-ratio objective. We then evaluate each fitted model on an independent large
population sample of 1000 months. This separation ensures that the reported performance reflects
out-of-sample portfolio quality rather than in-sample fit.
To evaluate all methods on a common footing, we translate fitted scores into the same portfolio
rule used in the empirical analysis: in each month, assets are sorted into deciles by model scores,
and we form the equal-weight long-short 10-1 portfolio. Let 𝐷1,𝑡 (𝑠) and 𝐷10,𝑡 (𝑠) denote the bottom
and top deciles under score 𝑠. The corresponding portfolio return is
10−1
𝑅𝑡+1
(𝑠) =

1

𝑟𝑙,𝑡+1 −

∑

|𝐷10,𝑡 (𝑠)| 𝑙∈𝐷

10,𝑡 (𝑠)

1
∑ 𝑟𝑙,𝑡+1 ,
|𝐷1,𝑡 (𝑠)| 𝑙∈𝐷 (𝑠)
1,𝑡

10−1
and performance is measured by the annualized Sharpe ratio of {𝑅𝑡+1
(𝑠)}.

8.2 Oracle benchmarks
A central advantage of the simulation design is that it makes latent benchmarks observable. Because
the conditional mean and conditional volatility functions are known, we can construct oracle portfolio rules that are infeasible in practice and use them to assess how much of the Sharpe-relevant
signal each method recovers. Our simulation uses three such oracle benchmarks. The first is a
weighting oracle that uses the true conditional means and variances and may choose continuous
zero-investment weights. For each month 𝑡, let
𝜇𝑡 = (𝜇1,𝑡 ,…,𝜇𝑛,𝑡 )⊤ ,

36

𝜇𝑙,𝑡 = 𝜇sig 𝑚(𝑈𝑙,𝑡 ).

The weighting oracle solves
max
𝑤𝑡

𝑤𝑡⊤ 𝜇𝑡
⊤
√𝑤𝑡 Σ𝑡 𝑤𝑡

s.t.

𝟏⊤ 𝑤𝑡 = 0,

𝑤̃𝑙,𝑡 =

𝜇𝑙,𝑡 − 𝜆𝑡
,
2
𝜎𝑙,𝑡

‖𝑤𝑡 ‖1 = 2.

(24)

With diagonal Σ𝑡 , the solution is
W
𝑤𝑙,𝑡
=

2 𝑤̃𝑙,𝑡
,
𝑛
∑𝑗=1 |𝑤̃𝑗,𝑡 |

𝑛

𝜆𝑡 =

2
∑𝑗=1 𝜇𝑗,𝑡 /𝜎𝑗,𝑡
𝑛

2
∑𝑗=1 1/𝜎𝑗,𝑡

.

(25)

This is the natural upper bound in the simulated economy, but it is not directly comparable to the
estimated models because it uses continuous weights rather than equal-weight deciles.
The second oracle is a ranking oracle. It uses the same equal-weight 10-1 portfolio construction
as the estimated models, but sorts assets on the true risk-adjusted score
R
𝑞𝑙,𝑡
=

𝜇𝑙,𝑡 − 𝜆𝑡
.
2
𝜎𝑙,𝑡

(26)

This oracle is the most informative benchmark for our purposes because it holds equal-weight portfolio construction fixed, in line with model evaluation, and isolates pure ranking quality. Any difference
between AlphaGlass and this oracle, therefore, reflects imperfect recovery of the Sharpe-relevant
score, rather than gains from alternative weighting schemes.
The third benchmark is a mean oracle, which also uses equal-weight decile portfolios but sorts
on the true conditional mean only:
𝜇

𝑞𝑙,𝑡 = 𝜇𝑙,𝑡 .

(27)

This oracle is useful because it represents the best equal-weight decile strategy available to a method
that learns only expected returns and ignores characteristic-dependent volatility.

8.3 Results
Table 7 reports Monte Carlo means and standard deviations across 100 replications. We can draw
the following conclusions from the results. First, the oracle benchmarks are stable across the two
training lengths. The weighting oracle attains an annualized Sharpe ratio of about 3.0, the ranking
oracle about 2.38, and the mean oracle about 2.03. The gap between the ranking and mean oracles
is economically meaningful, around 0.34 Sharpe ratio units for both 𝑇 = 120 and 𝑇 = 240. This
is precisely the value of volatility adjustment in a decile-sort environment: even when the final
portfolio is equal-weighted within each tail, sorting on a risk-adjusted score dominates sorting on
expected return alone. The further gap between the weighting oracle and the ranking oracle, roughly
0.61, captures the additional gains from moving from equal-weight tail portfolios to fully optimal
continuous weights.
Second, AlphaGlass dominates all three benchmark learners at both sample sizes. For 𝑇 = 120,
AlphaGlass attains an average annualized Sharpe ratio of 1.774, compared with 1.568 for EBM, 1.449
for the neural net, and 1.411 for Random Forest. For 𝑇=240, these values rise to 1.970, 1.755, 1.557,
and 1.438, respectively. As expected, all models benefit from longer training samples, but Alpha37

Glass remains the top performer throughout, while EBM is the strongest of the return-prediction
benchmarks.
Third, the key comparison is the fraction of the ranking-oracle Sharpe ratio. AlphaGlass captures
74.3% of the ranking-oracle SR at 𝑇 = 120 and 83.4% at 𝑇 = 240. The corresponding values are 65.8%
and 74.2% for EBM, 60.7% and 65.9% for the neural net, and 59.1% and 60.8% for Random Forest.
Because all models and the ranking oracle are evaluated through the same equal-weight 10−1 decile
rule, these differences reflect ranking quality rather than differences in portfolio implementation.
The results, therefore, show that training directly on a Sharpe-ratio objective helps AlphaGlass learn
the part of the signal that matters most for tail-portfolio formation.
Fourth, the comparison with the mean oracle shows that AlphaGlass not only learns the nonlinear conditional mean well, but also recovers part of the incremental information contained in
the volatility-related characteristics. At 𝑇 = 120, AlphaGlass reaches 86.7% of the mean-oracle SR,
compared with 76.7% for EBM, 70.9% for the neural net, and 68.9% for Random Forest. At 𝑇 = 240,
AlphaGlass reaches 97.7%, while EBM, the neural net, and Random Forest reach 86.9%, 77.1%, and
71.1%, respectively. Thus, AlphaGlass nearly closes the gap to the best mean-only decile strategy in
the longer sample and simultaneously preserves a clear advantage relative to the alternative learners
in the Sharpe-relevant ranking benchmark.
Overall, the simulation supports the empirical analysis. In a setting where the Sharpe-optimal
portfolio depends on both expected return and characteristic-dependent risk, AlphaGlass learns a
more useful portfolio score than methods trained as generic return predictors. The gains are robust
across training lengths, become stronger with more data, and are particularly pronounced when
evaluated relative to the equal-weight ranking oracle, which most closely mirrors empirical portfolio
construction.

9

Conclusion

We demonstrate that portfolio objectives can be optimized directly using inherently interpretable
machine learning. The AlphaGlass framework bridges flexible nonlinear learning and the transparency requirements of empirical asset pricing by combining an additive, glass-box architecture
with a sparse set of pairwise interactions with an end-to-end differentiable portfolio formation layer
that maps scores into long-short quantile weights. This design preserves full interpretability by
construction: each portfolio decision admits an exact decomposition into main effects and interaction contributions, enabling economically meaningful attribution without relying on noisy post-hoc
explainability methods.
Empirically, AlphaGlass delivers strong out-of-sample performance in U.S. equities, producing a
pronounced return and Sharpe spread across signal-sorted portfolios and outperforming key benchmarks. Crucially, these gains come with ex ante transparency. The learned shape functions and
interaction surfaces reveal clear economic patterns in how characteristics map onto optimal allocations, including industry structure, operating leverage dynamics, momentum, labor adjustment, and

38

Table 7: Results of Monte Carlo simulations
𝑇 = 120

𝑇 = 240

Panel A: Oracle benchmarks
Weighting oracle
Ranking oracle
Mean oracle

2.995 (0.116)
2.388 (0.107)
2.047 (0.123)

2.982 (0.118)
2.368 (0.120)
2.021 (0.105)

Panel B: Model Sharpe ratios
AlphaGlass
EBM
Random Forest
Neural net

1.774
1.568
1.411
1.449

(0.340)
(0.219)
(0.236)
(0.204)

1.970
1.755
1.438
1.557

(0.226)
(0.173)
(0.205)
(0.191)

Panel C: Fraction of ranking-oracle SR
AlphaGlass
EBM
Random Forest
Neural net

0.743
0.658
0.591
0.607

(0.140)
(0.094)
(0.096)
(0.085)

0.834
0.742
0.608
0.659

(0.104)
(0.076)
(0.086)
(0.086)

Panel D: Fraction of mean-oracle SR
AlphaGlass
EBM
Random Forest
Neural net

0.867 (0.164)
0.767 (0.100)
0.689 (0.106)
0.709 (0.097)

0.977 (0.112)
0.869 (0.081)
0.711 (0.090)
0.771 (0.092)

This table reports Monte Carlo mean Sharpe ratios (annualized) with standard deviations in parentheses across 100
replications. Each replication simulates a panel with 𝑛 = 500 assets and 𝑝 = 6 characteristics per month. Models
are trained on 𝑇 ∈ {120,240} months and evaluated on an independent population sample of 1000 months. All
estimated models are evaluated through the same equal-weight long-short 10−1 decile portfolio.

39

economically intuitive conditional relationships captured by sparse interactions. The term-level decomposition further clarifies which characteristics systematically push assets toward the long and
short tails, providing a direct bridge from a high-performing portfolio rule to an interpretable set of
economic drivers.
Finally, we theoretically justify our approach. We show that maximizing the sample Sharpe ratio
is statistically aligned with maximizing the population Sharpe ratio within the AlphaGlass class and
that the differentiable rank-and-mask procedure is a faithful surrogate for conventional hard sorts.
Overall, AlphaGlass provides a practical template for learning tradable factors and portfolio rules
that are simultaneously high-performing, economically grounded, and transparent—and therefore
well suited for both portfolio construction and mechanism-based empirical asset-pricing inference.

40

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</reference>

<statements>
1. The core differences are therefore not merely “more or less sophisticated.” They concern variance versus tail and coherent risk, sample moments versus equilibrium-plus-views versus learned signals, and explicit weight vectors versus market-anchored tilts versus policy mappings.
2. A hybrid is feasible and already exists in several published forms: Black-Litterman or inverse-optimization backbones combined with machine-learning or LLM views, coherent-risk objectives replacing variance, and end-to-end interpretable architectures such as AlphaGlass.
3. AlphaGlass similarly optimizes investment objectives such as the Sharpe ratio or mean-variance utility end-to-end, while keeping portfolio weights traceable to characteristics and interactions.
4. The risk axis therefore moves from variance, to VaR/CVaR, to learned policies under a convex or investor-specified objective.
5. It maps firm characteristics into additive signals with sparse pairwise interactions and converts those signals into long-short portfolios through a differentiable rank-and-mask layer
6. The model is a neural-network-based additive design combining the interpretability of Generalized Additive Models with neural-network flexibility, so portfolio signals and allocations can be decomposed into explicit characteristic- and interaction-level contributions
7. It replaces hard rank cutoffs with smooth “soft” weights, interpreted as probabilities that a stock belongs to a given portfolio, and proves that the soft rank-and-mask procedure is a faithful surrogate for conventional hard sort-and-cut construction when signals are well separated
8. The key innovation is joint estimation: rather than forecasting returns and then forming portfolios, AlphaGlass directly optimizes the investor’s objective within an interpretable characteristic-based framework
9. Section 6 extends the same design to a mean-variance objective
10. The evidence supports hybrid frameworks, but not a symmetric merger of all three families: AlphaGlass fuses interpretable learning with portfolio objectives
11. The hybrids that appear strongest assign each component a specific job: a risk objective that is more faithful than variance supplied by CVaR, coherent risk, or Sharpe-ratio optimization
12. The hybrids that appear strongest assign each component a specific job: an interpretable architecture supplied by additive rank-based models
13. AlphaGlass | Interpretable additive neural architecture, sparse interactions, differentiable rank-and-mask, and direct Sharpe or mean-variance optimization | Predict-then-optimize mismatch and black-box opacity | May 2026 NBER working paper, not peer-reviewed; simulations and U.S. equity out-of-sample evidence
14. AlphaGlass is the deepest fusion because it collapses the prediction and optimization stages into one estimand
15. It is motivated by two defects in the standard machine-learning pipeline: return prediction is only an intermediate target, and forecasting returns need not align with risk-adjusted portfolio objectives once signals are converted into portfolios; separately, black-box opacity limits economic interpretability
16. AlphaGlass addresses both by jointly estimating signals and maximizing the investment objective in a single step while maintaining transparency through additive main effects and sparse interactions
17. In Monte Carlo simulations, AlphaGlass attains an average annualized Sharpe ratio of 1.774 at a 120-month training length, compared with 1.568 for EBM, 1.449 for a neural network, and 1.411 for Random Forest; at 240 months, the values rise to 1.970, 1.755, 1.557, and 1.438, respectively
18. It captures 74.3% of the ranking-oracle Sharpe ratio at 120 months and 83.4% at 240 months
19. In U.S. equities, it outperforms benchmark characteristic-sorted portfolios and decile portfolios formed on Random Forest and neural-network return predictions, while revealing interpretable drivers such as industry concentration, operating leverage dynamics, momentum, and industry-adjusted labor adjustment
20. AlphaGlass reports strong simulation and out-of-sample results, but it is a May 2026 NBER working paper and not yet a peer-reviewed journal article, and the paper itself notes that many characteristics with high in-sample Sharpe ratios have substantially lower out-of-sample Sharpe ratios
21. AlphaGlass treats opacity as a design defect to be engineered around, using an additive architecture with sparse interactions and a differentiable rank layer to make long and short positions decomposable
22. This supports the user’s premise that dense deep models suffer from poor interpretability, but refines it: the limitation is architectural, not inherent to machine learning as such
23. The same paper notes that when optimizing for drawdowns, a neural net can have a higher raw Sharpe ratio than AlphaGlass but a much larger drawdown penalty, and that EBM can match AlphaGlass’s Sharpe ratio while suffering a high drawdown penalty, so interpretability and performance are not automatically aligned
24. Deep learning and reinforcement learning measure risk by the objective chosen by the practitioner, predict returns through nonlinear characteristic interactions or skip return prediction entirely, and produce either opaque policy mappings or, when constrained, interpretable portfolio rules.
25. The most defensible framework combines a stable equilibrium or regularized mean-variance backbone, machine-learning or LLM-generated views with explicit uncertainty calibration, a tail-aware or coherent risk objective such as CVaR or an investor objective such as Sharpe ratio
26. The most defensible framework combines a stable equilibrium or regularized mean-variance backbone, machine-learning or LLM-generated views with explicit uncertainty calibration, a tail-aware or coherent risk objective such as CVaR or an investor objective such as Sharpe ratio, and an interpretable architecture that preserves attribution to characteristics and interactions.
27. AlphaGlass is the strongest current evidence that this can be done end-to-end: it directly optimizes the investor’s objective while keeping allocations decomposable, and it extends to a mean-variance objective.
28. What would change the judgment is not another backtest with a higher Sharpe ratio, but long-sample out-of-sample results after realistic costs, robustness across market regimes and holding horizons, transparent attribution that survives auditability, and deployment evidence showing that learned or hybrid allocators displace regularized mean-variance and Black-Litterman workflows in live robo-advisor or institutional asset-allocation settings.
</statements>

Begin the assessment now. Output only the JSON list, without any conversational text or explanations.