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Latent Variable Estimation in Bayesian Black-Litterman Models
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Thomas Y.L. Lin$^{\dagger}$\footnote{[email removed]}}
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Jerry Yao-Chieh Hu$^{\ddag}$\footnote{[email removed]}}
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Paul W. Chiou$^{\flat}$\footnote{[email removed]}}
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Peter Lin$^{\natural\S}$\footnote{[email removed]}}
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$^\dagger\;$Department of Physics, National Taiwan University, Taipei 106319, Taiwan\\
$^\ddag\;$Department of Computer Science, Northwestern University, Evanston, IL 60208, USA\\
\hphantom{$^\ddag\;$}Center for Foundation Models and Generative AI, Northwestern University, Evanston, IL 60208, USA\\
$^\flat\;$D'Amore-McKim School of Business, Northeastern University, Boston, MA 02115, USA\\
$^\natural\;$Whiting School of Engineering, Johns Hopkins University, Baltimore, MD 21218, USA\\
$^\S\;$Gamma Paradigm Capital, New York 10019, NY, USA
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\footnotetext{Preliminary version accepted at International Conference on Machine Learning (ICML) 2025.}
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We revisit the Bayesian Black–Litterman (BL) portfolio model and remove its reliance on subjective investor views.
Classical BL requires an investor “view”: a forecast vector $q$ and its uncertainty matrix $\Omega$ that describe how much a chosen portfolio should outperform the market.
Our key idea is to treat $(q,\Omega)$ as latent variables and learn them from market data within a single Bayesian network.
Consequently,
the resulting posterior estimation admits closed-form expression, enabling fast inference and stable portfolio weights.
Building on these, we propose two mechanisms to capture how features interact with returns: shared-latent parametrization and feature-influenced views; both recover classical BL and Markowitz portfolios as special cases.
Empirically, on 30-year Dow-Jones and 20-year sector-ETF data, we improve Sharpe ratios by 50% and cut turnover by 55% relative to Markowitz and the index baselines.
This work turns BL into a fully data-driven, view-free, and coherent Bayesian framework for portfolio optimization.
Keywords: Black-Litterman Models, Portfolio Optimization, Bayesian Networks, Graphical Models, Hierarchical Models, Latent Variable Models, Uncertainty Quantification
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Introduction
We propose a Bayesian reformulation of the Black-Litterman model for portfolio optimization.
Our motivation comes from the early works of the model black1992global,lee2000theory,salomons2007black,idzorek2007step where human experts are required to specify the value of investor views and corresponding uncertainties $(q, \Omega)$.
For example, the $i$-th views “the 2nd asset will outperform the 1st asset by $9\pm3$%” is encoded as $\{P_i=[-1,1,...,0]; q_i=0.09; \Omega_{ii}=0.03^2\}$.
Across decades, this heuristic framework attracts many research beach2007application,palomba2008multivariate,duqi2014black,silva2017more,deng2018generalized,kara2019hybrid,kolm2021factor working on this estimation.
Among them, one common approach is to take asset features and generate $(q,\Omega)$ by external models.
However, relying on external estimators in these methods leads to incoherent parameter learning and error propagation across the separate models.
In this work, we offer a new approach.
Our reformulation recasts the Black-Litterman model as a Bayesian network to integrate the features.
Under different scenarios, we identify two potential effects caused by the features and accordingly present this network as specific models.
Under the scenarios without subjective investor views, this network treats $q$ and $\Omega$ as latent --- rather than externally estimated --- parameters, and thereby estimates posterior distribution over both asset returns $r$ and their parameters $\theta$ directly from data, i.e., features.
In summary, our approach provides a unification of feature integration and parameter inference within a single framework, ensuring coherent estimations and mitigating error propagation.
Contributions.
Our contributions include:
itemize• Eliminating Subjective Human Input.
We introduce a Bayesian network formulation of the Black-Litterman model, treating $(q, \Omega)$ as latent variables. This enables direct estimation from feature data, bypassing heuristic human inputs and potential bias while subsuming the classical Black-Litterman model as a special case.
• Unified Feature Integration and Parameter Inference.
Unlike prior works, our approach avoids error propagation from external estimators by unifying feature integration and parameter inference into a single framework.
• Empirical Outperformance.
{
Our model achieves a 49.8% mean improvement in Sharpe ratios over the Markowitz model (0.66–0.87 vs. 0.35–0.62) and market indices (S&P 500, DJIA) on 20-year and 30-year datasets, respectively.
It achieves a 55.1% reduction in turnover rates while showing robustness to hyperparameters.
}
Organization.
(ref) includes preliminaries.
(ref) introduces our models and their theoretical analysis.
(ref) presents empirical studies to backup our work.
(ref) offers a practical guide for our models.
We defer conclusions and related works to (ref).
Preliminaries
Consider $m$ assets, and let $r\in\mathbb{R}^m$ be the returns of the $m$ assets.
Consider $k$ sets of specified portfolio weights on the $m$ assets, and encode each weight into each row of a portfolio weight matrix $P \in \mathbb{R}^{k\times m}$.
Let $q \in \mathbb{R}^k$ be the investor views on the $k$ specified portfolio returns and encode the variance of each view into diagonal elements of a diagonal uncertainty matrix $\Omega \in \mathbb{R}^{k \times k}$.
Larger $\Omega_{ii}$ implies greater uncertainty in $(P \mathbb{E}[r])_i$ and $\Omega_{ii} = 0$ implies absolute certainty.
markowitz1952portfolio introduces the theory of portfolio optimization, suggesting a suitable portfolio weight is the optimal trade-off between the mean and variance of the portfolio.
To be concrete, we provide a formal definition below.
definition[Unconstrained Risk-Adjusted Mean-Variance Optimization]
Let $r \in \mathbb{R}^m$ be the returns of the $m$ assets and $\tilde{r}$ denote the unobserved (or future) asset returns.
The goal of the optimization is to find $w \in \mathbb{R}^m$ maximizing the objective function:
\begin{align*}
\max_{w} \left\{ w^T \mathbb{E}[\tilde{r}] - \frac{\delta}{2}w^T \mathrm{Cov}[\tilde{r}] w \right\},
\end{align*}
where $\delta \in [0,\infty]$ is a risk-adjusted coefficient.
One major challenge of this framework is the reliance on estimating $\tilde{r}$:
problem[Predictive Estimation of Unobserved Asset Returns]
Let $r \in \mathbb{R}^m$ represent the returns of the $m$ assets and $\tilde{r}$ denote the unobserved (or future) asset returns.
Precisely, given observed data $D$, the goal is to estimate unobserved asset returns $\tilde{r} \sim p(r|D)$.
In this work, we refer to methods addressing such estimation challenge ((ref)) as {portfolio models} or simply {portfolios}. Following the predictive estimation of $\tilde{r}$, we apply the mean-variance optimization framework to obtain a decision vector $w$, referred to as {portfolio weights}.
A basic approach, termed the {traditional Markowitz model} markowitz1952portfolio, involves predicting the expected returns and the covariance matrix of asset returns directly from historical data using the sample mean and sample covariance. This method relies on the assumption that historical estimates are accurate representations of future parameters. However, in practice, estimation errors in the expected returns and covariance matrix lead to extreme and highly sensitive portfolio weights michaud1989markowitz, demiguel2009optimal. To mitigate this issue, the {Black-Litterman model} integrates the market equilibrium with investor views by {Black-Litterman formula}, thereby producing more stable and diversified portfolio weights black1992global. The following context elaborates on this model in detail.
Black-Litterman.
Black-Litterman (BL) model outputs a posterior of the asset returns mean $\mathbb{E}[r]$, termed Black-Litterman formula, by Bayes' theorems, taking investor views and market equilibrium price as input. Upon this, the model offers a predictive estimate $\tilde{r}$ on asset returns:
theorem[Black-Litterman (BL) Formula and Predictive Estimation, Theorem 1 of satchell2007demystification]
Let $r \in \mathbb{R}^m$ be the vector of asset returns with covariance $\Sigma \coloneqq \mathrm{Cov}[r]$.
Let $P \in \mathbb{R}^{k \times m}$ be the portfolio weight matrix for $k$ specified portfolios, and $(q, \Omega) \in \mathbb{R}^k \times \mathbb{R}^{k \times k}$ represent investor views and their uncertainty.
Let $\Pi \in \mathbb{R}^m$ represent the market equilibrium price and $\tau > 0$ be a scaling factor.
Assume a prior $P \,\mathbb{E}[r] \sim {N}(q, \Omega)$ and a likelihood $\Pi \mid \mathbb{E}[r] \sim {N}\bigl(\mathbb{E}[r], \tau\,\Sigma\bigr)$, then the posterior mean of $r$ given $\Pi$ is
\begin{align*}
\mathbb{E}[r \mid \Pi]
&\sim
{N}\Bigl(
G_{\tau}^{-1} \bigl[(\tau\Sigma)^{-1}\Pi + P^\top\Omega^{-1} \,q\bigr],
G_{\tau}^{-1}
\Bigr),
\end{align*}
where $G_{\tau} \coloneqq (\tau\,\Sigma)^{-1} + P^\top\,\Omega^{-1}\,P$.
Moreover, the predictive distribution $\tilde{r} \coloneqq r|\Pi$ is
\begin{align*}
\tilde{r} &\sim
{N}\Bigl(
G_{\tau}^{-1} \bigl[(\tau\Sigma)^{-1}\Pi + P^\top\Omega^{-1} q\bigr],
\Sigma + G_{\tau}^{-1}
\Bigr).
\end{align*}
The Black-Litterman formula ((ref)) is a well-known result of the Black-Litterman model.
However, the derivation lacks explanations of the assumptions used.
Most early works, including the original paper black1992global, provide heuristic derivation, while many lee2000theory,salomons2007black,idzorek2007step share different underlying assumptions.
This inconsistency leads to confusion for both the analysis of the model and a rigorous interpretation with Bayesian statistics.
To solve the issues, the Black-Litterman-Bayes (BLB) model kolm2017bayesian provides a reformulation of the Black-Litterman model.
Black-Litterman-Bayes (BLB).
kolm2017bayesian introduce the Black-Litterman-Bayes model to perform Bayesian inference on $\theta$, treating the market equilibrium as prior and the investor views as likelihood:
definition[BLB Model $(\theta,r,q,\Omega)$, Modified from Definition 1 of kolm2017bayesian]
Let $r\in\mathbb{R}^m$ be the returns of the $m$ assets, parametrized by $\theta$, with $r \sim p(r | \theta)$.
Let $q \in \mathbb{R}^k$ represent the views on the returns of the $k$ specified portfolio and $\Omega \in \mathbb{R}^{k \times k}$ be the uncertainty matrix.
The Black-Litterman-Bayes (BLB) model is a portfolio model composed of three fundamental density functions:
\begin{enumerate}
• {Parametrized Asset Returns}: $p(r | \theta)$, the distribution of asset returns given the parameter.
• {Prior}: $\pi(\theta)$, representing market equilibrium.
• {Likelihood}: $L(\theta | q, \Omega)
\coloneqq p(q, \Omega | \theta)$, capturing the relationship between the parameter and the investor views.
\end{enumerate}
(ref) details modelings of the prior and likelihood.
To recap, the {Black-Litterman-Bayes model} is a {portfolio model} aiming to address the estimation challenge of asset returns ((ref)).
It solves the problem by using Bayesian inference to obtain the posterior of the model parameter $p(\theta | q,\Omega)$, and subsequently produce the predictive estimation of unobserved asset returns $\tilde{r}$.
Following the estimation, we apply the mean-variance optimization framework ((ref)) to obtain the {portfolio weights} $w_{\rm BLB}$.
The intuition of the model is to simulate the dynamics of how the market adapts to new information after observing it. Here, the market equilibrium represents the initial state, and the investor views approximates the unobserved information.
When there is no investor views, the model remains in the initial state of market equilibrium.
Here we show the Black-Litterman formula ((ref)) is the posterior estimation on $\theta$ of the Black-Litterman-Bayes model under (ref):
lemma[Estimations by BLB Model]
Let the market capitalization weight on $m$ assets be $w_{\rm cap} \in \mathbb{R}^m$ and $\delta \in [0,\infty]$ be a risk-adjusted coefficient.
Let $P \in \mathbb{R}^{k\times m}$ be the portfolio weight matrix for $k$ specified portfolios.
Given a BLB model $(\theta,r,q,\Omega)$ ((ref)), assume
\begin{align}
&r\sim N(\theta, \Sigma), \\
&\theta \sim N(\theta_0, \Sigma_0), \\
&P \theta = q + \epsilon, \quad \epsilon \sim N(0, \Omega),
\end{align}
where $\Sigma, \Sigma_0 \in \mathbb{R}^{m \times m}$ are given intrinsic and prior covariance.
The {posterior mean} is
\begin{align}
p(\theta | q, \Omega) = N\!\left( \theta; G^{-1} \!\left( \Sigma_0^{-1} \theta_0 + P^ \mathsf{T} \Omega^{-1} q \right), G^{-1} \right),
\end{align}
where $G \coloneqq \Sigma_0^{-1} + P^ \mathsf{T} \Omega^{-1} P$.
The {predictive estimation of unobserved asset returns} $\tilde{r} \coloneqq r | q, \Omega$ is
\begin{align}
\tilde{r}
\!\sim\! N \!\left( \tilde{r}; G^{-1} \! \left[ \delta (\Sigma_0^{-1}\Sigma \!+\! I) w_{\rm cap} + P^ \mathsf{T} \Omega^{-1} q \right]\!, \Sigma +G^{-1}\right). \!
\end{align}
proofSee (ref) for a detailed proof.
(ref) provides a solution to (ref).
With the predictive estimation of asset returns $\tilde{r}$, the mean-variance optimization framework ((ref)) determines the portfolio weights $w_{\rm BLB}$.
However, it relies on the subjective investor views and corresponding uncertainty $(q,\Omega)$\footnote{Besides providing the Bayesian inference formulation of the Black-Litterman model ((ref)), the original work kolm2017bayesian and its follow-up work kolm2021factor consider external data, specifically factors in Arbitrage Pricing Theory (APT) model.
Yet, these works focus on how their Black-Litterman-Bayes (BLB) approach applies to the APT model and do not address the issues of subjective investor views.}.
To address this issue, in this work, we propose a Bayesian reformulation of the Black-Litterman model without the need for subjective $(q, \Omega)$ from humans.
Methodology
In this work, we recast the Black-Litterman model as Bayesian networks for principled estimation of both investor views and asset returns, eliminating the need for subjective inputs.
This Bayesian formulation serves as a conceptual baseline for subsequent portfolio model specifications.
In (ref), we introduce the Bayesian Black-Litterman network, which underpins the Black-Litterman-Bayes model kolm2017bayesian.
Building on this, (ref) extends the network to incorporate external features, yielding the feature-integrated Black-Litterman network.
We then examine two scenarios:
itemize• In (ref), where investor views are observed, we illustrate the corresponding network ((ref)) and define the Mixed-effect Black-Litterman (M-BL) model ((ref)).
• In (ref), where no subjective views are given, we present two alternative probabilistic graphical models ((ref)) and define the Shared-Latent-Parametrization \underline{B}lack-\underline{L}itterman (SLP-BL) and \underline{F}eature-\underline{I}nfluenced-\underline{V}iews \underline{B}lack-\underline{L}itterman (FIV-BL) models ((ref)).
Bayesian Black-Litterman Network
We introduce a Black-Litterman network $(\theta, r, q, \Omega)$ with the two causal relationships.
First, the asset returns $r$ are realizations of the process governed by its parameter $\theta$. Second, the investor views $q$ are formed based on $\theta$ with an associated error term $\epsilon \sim N(0,\Omega)$.
We visualize this conceptual network in (ref).
figure[figure omitted — 424 chars of source]
Bayesian Feature-Integrated Black-Litterman Network
Building upon the Black-Litterman Network, we introduce a feature-integrated Black-Litterman network as a Black-Litterman network that integrates features $F$ and their effects.
Specifically, features $F$ exert two causal effects:
itemize• Effect 1: Features $F$ are extracted from the parameter $\theta$.
• Effect 2: Features $F$ influence the formation of views $q$.
We quantitatively specify Effect 1 and 2 in (ref).
General Scenario: Features with Observed Views
In this section, we discuss the general scenario where views are observed ((ref)) by specifying the two causal effects introduced in (ref).
Incorporating these effects into the network,
we showcase it in (ref) and define the Mixed-effect Black-Litterman (M-BL) model ((ref)) based on it.
Then, we estimate posterior distribution over both asset returns $r$ and their parameter $\theta$ ((ref)).
Consider the following problem:
problem[Feature-and-Views Hybrid Predictive Estimation]
Let $r \in \mathbb{R}^m$ represent the returns of the $m$ assets and $\tilde{r}$ denote the unobserved (or future) asset returns.
Let $q \in \mathbb{R}^k$ represent the views on returns of the $k$ portfolios.
Let $f_i \in \mathbb{R}^{d}$ represent the features on the $i$-th asset and $F \in \mathbb{R}^{m\times dm}$ be a block-diagonal matrix defined as
\begin{align*}
F \coloneqq \mathop{\rm{diag}}(f_1^ \mathsf{T} , f_2^ \mathsf{T} , \dots, f_m^ \mathsf{T} ).
\end{align*}
Given $D\coloneqq (q, \Omega, F,\Omega^F)$ where $(q,\Omega,F)$ are estimated from observations of asset returns, views, and features $\{(r_{l}, q_{l}, F_{l})\}_{l=1}^{n}$ and $\Omega^F$ is the homoscedastic error matrix corresponding to the observations $\{F_{l}\}_{l=1}^{n}$,
the goal is to estimate unobserved asset returns $\tilde{r} \sim p(r|D)$.
We aim to solve (ref) by feature-integrated Black-Litterman network, incorporating a mix of two causal effects of features in (ref).
Specifically,
itemize• Effect 1: Features $F$ are extracted from the parameter $\theta$. Consequently, the features $F$, along with their error term $\epsilon^F \sim N(0,\Omega^F)$, share the common parameter $\theta$ with asset returns $r$ and investor views $q$.
• Effect 2: Features $F$ influence the formation of views $q$. Consequently, the features $F$ and the parameter $\theta$ jointly determine the views $q$ with an uncertain $\epsilon \sim N(0,\Omega)$.
We visualize the network in (ref) under this general scenario.
figure[figure omitted — 692 chars of source]
To capture \hyperref[item:eff1]{Effect 1}, we define a $\theta\leftrightarrow F$ relationship:
definition[$\theta\leftrightarrow F$ Linear Model]
Given features $F \in \mathbb{R}^{m \times dm}$, let $r\in\mathbb{R}^m$ be the returns of the $m$ assets, parametrized by $\theta$, with $r \sim p(r | \theta)$.
Define regression intercept vector $\alpha^F \in \mathbb{R}^m$, regression coefficient vector $\beta^F \in \mathbb{R}^{dm}$, random error $\epsilon^F \in \mathbb{R}^m$, and error matrix $\Omega^F \in \mathbb{R}^{m \times m}$ such that:
\begin{align*}
\theta = \alpha^F + F\beta^F + \epsilon^F, \quad \epsilon^F \sim N(0, \Omega^F).
\end{align*}
To capture \hyperref[item:eff2]{Effect 2}, we define a $q\leftrightarrow F\leftrightarrow \theta$ relationship:
definition[$q\leftrightarrow F\leftrightarrow \theta$ Linear Model]
Given features $F \in \mathbb{R}^{m \times dm}$ and portfolio weight matrix for $k$ specified portfolios $P \in \mathbb{R}^{k \times m}$, let $r\in\mathbb{R}^m$ be the returns of the $m$ assets, parametrized by $\theta$, with $r \sim p(r | \theta)$ and $q \in \mathbb{R}^k$ be the views on returns of the $k$ portfolios.
Define regression intercept vector $\alpha \in \mathbb{R}^m$, regression coefficient vector $\beta \in \mathbb{R}^{dm}$, scale constant $\gamma \in \mathbb{R}$, random error $\epsilon \in \mathbb{R}^m$, and uncertainty matrix $\Omega \in \mathbb{R}^{m \times m}$ such that:
\begin{align}
q + \epsilon = P (\alpha + F\beta + \gamma \theta), \quad \epsilon \sim N(0, \Omega).
\end{align}
remark[Rationale]
(ref) extends the classical noisy views model $q+\epsilon= P\theta$ black1992global to incorporate the features $F$, where the LHS remains the $k$-dimensional noisy views and the RHS generalizes the classical $P\theta$ term by introducing a feature-driven term $\alpha+F\beta$, and a scaled parameter $\gamma \theta$.
It recovers the classical model when $\alpha=0$, $\beta=0$, and $\gamma=1$.
By mixing two effects of the features characterized by the two linear models ((ref)), we showcase the feature-integrated Black-Litterman network as the following Mixed-effect Black-Litterman (M-BL) model:
definition[Mixed-effect Black-Litterman (M-BL) Model $(\theta,r,q,\Omega,F,\Omega^F)$]
Let $r\in\mathbb{R}^m$ be the returns of the $m$ assets, parametrized by $\theta$, with $r \sim p(r | \theta)$.
Let $q \in \mathbb{R}^k$ represent the views on the returns of the $k$ specified portfolio and $\Omega \in \mathbb{R}^{k \times k}$ be the uncertainty matrix.
Let $F \in \mathbb{R}^{m \times dm}$ be the features of the $m$ assets and error matrix $\Omega^F \in \mathbb{R}^{m \times m}$
The M-BL model is a portfolio model composed of four fundamental density functions:
\begin{enumerate}
• {Parametrized Asset Returns}: $p(r | \theta)$, the distribution of asset returns given the parameter.
• {Prior}: $\pi(\theta)$, representing market equilibrium.
• {Likelihood of Features}: $L(\theta|F,\Omega^F) \coloneqq p(F,\Omega^F|\theta)$, the $\theta\leftrightarrow F$ relationship ((ref)).
• {Observation Likelihood}: $L(\theta, F|q, \Omega)$ $\coloneqq p(q, \Omega|\theta, F)$, the $q\leftrightarrow F\leftrightarrow \theta$ relationship ((ref)).
\end{enumerate}
We show the posterior estimation on $\theta$ of the M-BL model:
theorem[Parameter Estimation of the M-BL Model]
Given a M-BL model $(\theta,r,q,\Omega,F,\Omega^F)$ ((ref)) and regression parameters $(\alpha^F, \beta^F \alpha,\beta,\gamma) \in \mathbb{R}^m \times \mathbb{R}^{dm} \times \mathbb{R}^m \times \mathbb{R}^{dm} \times \mathbb{R}$, assume
\begin{align}
&\theta \sim N(\theta_0, \Sigma_0), \\
&\theta = \alpha^F + F \beta^F+ \epsilon^F, \quad \epsilon^F \sim N(0, \Omega^F), \\
&q + \epsilon = P (\alpha + F\beta + \gamma \theta), \quad \epsilon \sim N(0, \Omega).
\end{align}
Define $G^M \coloneqq \Sigma_0^{-1} + (\Omega^F)^{-1} + \gamma^2 P^ \mathsf{T} \Omega^{-1} P$. The {posterior mean} is
\begin{align*}
p(\theta | q, \Omega, F, \Omega^F)
=N(\theta; \mu_{\theta \mid q,\Omega,F,\Omega^F}, (G^M)^{-1}),
\end{align*}
where $\mu_{\theta \mid q,\Omega,F,\Omega^F} = (G^M)^{-1} \left[ \Sigma_0^{-1} \theta_0 + (\Omega^F)^{-1} (\alpha^F + F\beta^F) + \gamma P^ \mathsf{T} \Omega^{-1} (q - P\alpha - PF\beta) \right]$.
proofSee (ref) for a detailed proof.
The posterior estimation of parameter $\theta$ enables a predictive estimation on $\tilde{r}$:
corollary[Predictive Estimation by the M-BL Model]
Define $G^M \coloneqq \Sigma_0^{-1} + (\Omega^F)^{-1} + P^ \mathsf{T} \Omega^{-1} P$.
Assume $r\sim N(\theta, \Sigma)$.
Then, under (ref), M-BL model gives the predictive estimation of unobserved asset returns $\tilde{r} \coloneqq r \mid q, \Omega, F,\Omega^F$ as
\begin{align*}
&\tilde{r}
\sim N(\theta; \mu_{\theta \mid q,\Omega,F,\Omega^F}, \Sigma + (G^M)^{-1}),
\end{align*}
where $\mu_{\theta \mid q,\Omega,F,\Omega^F} = (G^M)^{-1} \left[ \Sigma_0^{-1} \theta_0 + (\Omega^F)^{-1} (\alpha^F + F\beta^F) + \gamma P^ \mathsf{T} \Omega^{-1} (q - P\alpha - PF\beta) \right]$.
remark[Classical Black-Litterman Recovery]
M-BL model recovers classical Black-Litterman model when:
\begin{enumerate}
• Features becomes uninformative, i.e., uncertainty approaches infinity: $\Omega^F \to \infty$, equivalently $(\Omega^F)^{-1} \to 0$, so the error matrix $(\Omega^F)^{-1}$ disappears from $G^M$.
• The $q\leftrightarrow F\leftrightarrow\theta$ linear model ((ref)) reduces to the classical noisy views model black1992global: $(\alpha,\beta,\gamma) = (0,0,1)$, so the residual $(q - P\alpha - PF\beta) \to q$.
\end{enumerate}
Under these conditions, we have
\begin{align}
G^M &\to {\Sigma_0^{-1}} + {P^ \mathsf{T} \Omega^{-1}P} = G, \nonumber
\end{align}
and thus, we recover (ref):
\begin{align*}
r | q,\Omega \sim N\Bigl(G^{-1}\left[\Sigma_0^{-1}\theta_0 + P^ \mathsf{T} \Omega^{-1}q\right], \Sigma+G^{-1}\Bigr).
\end{align*}
remark[Ground-Truth Limit]
(ref) accurately and precisely predict ground-truth asset returns with:
\begin{enumerate}
• Perfect information: $\Omega \to 0$, equivalently $\Omega^{-1} \to \infty$.
• Accurate views: $q \to Pr^\star$ where $r^\star$ is true asset returns.
• The $q\leftrightarrow F\leftrightarrow\theta$ linear model ((ref)) reduces to the classical noisy views model black1992global: $(\alpha,\beta,\gamma) = (0,0,1)$, so the residual $(q - P\alpha - PF\beta) \to q$.
\end{enumerate}
The M-BL posterior mean satisfies:
\begin{align*}
\lim_{\substack{\Omega \to 0 \\ q \to r^\star}} \mu_{\theta \mid q,\Omega,F,\Omega^F}
&= (G^M)^{-1} \bigg[ \underbrace{P^ \mathsf{T} \Omega^{-1}(Pr^\star - P\alpha - PF\beta)}_{dominant term} + \underbrace{\Sigma_0^{-1}\theta_0 + (\Omega^F)^{-1}(\alpha^F + F\beta^F)}_{bounded} \bigg] \\
&= (G^M)^{-1} [P^ \mathsf{T} \Omega^{-1}Pr^\star + o(\Omega^{-1})] \\
&= r^\star \quad (a.s.)
\end{align*}
where the last step follows $G^M = \Sigma_0^{-1} + (\Omega^F)^{-1} + P^ \mathsf{T} \Omega^{-1}P \to P^ \mathsf{T} \Omega^{-1}P,
\quad (G^M)^{-1} o(\Omega^{-1}) \to 0$.
As a result, (ref) becomes
\begin{align*}
\tilde{r} \xrightarrow[]{d} \delta_{r^\star} \quad \text{as} \quad \Omega \to 0,\ q \to Pr^\star
\end{align*}
where $\delta_{r^\star}$ denotes the Dirac measure at $r^\star$.
(ref) estimates asset returns under the general scenario where views are observed, i.e., solves (ref).
It generalizes the classical Black-Litterman model, recovering its form when features are uninformative, and approaches the true returns with accurate and precise views ((ref)).
With the predictive estimation of asset returns $\tilde{r}$, the mean-variance optimization framework ((ref)) determines the portfolio weights $w_{\rm M-BL}$.
Scenario: Features with Latent Views
In the previous section, we solve (ref) with features and observed views.
However, an investor using the Black-Litterman model may not be an expert at quantifying the views.
In this section, we discuss the scenario without the views ((ref)) by specifying two effects of the features introduced in (ref) and treating $q$ and $\Omega$ as latent variables.
Incorporating the effects into the network,
we showcase two graphical forms in (ref) and define the Shared-Latent-Parametrization Black-Litterman (SLP-BL) and Feature-\underline{I}nfluenced-\underline{V}iews \underline{B}lack-\underline{L}itterman (FIV-BL) models ((ref)) based on them.
Then, we estimate posterior distribution over both asset returns $r$ and their parameter $\theta$ ((ref)).
Consider the following problem:
problem[Feature-Integrated Predictive Estimation]
Let $r \in \mathbb{R}^m$ represent the returns of the $m$ assets and $\tilde{r}$ denote the unobserved (or future) asset returns.
Let $f_i \in \mathbb{R}^{d}$ represent the features of the $i$-th asset and $F \in \mathbb{R}^{m\times dm}$ be a block-diagonal matrix defined as
\begin{align*}
F \coloneqq \mathop{\rm{diag}}(f_1^ \mathsf{T} , f_2^ \mathsf{T} , \dots, f_m^ \mathsf{T} ).
\end{align*}
Given $D\coloneqq (F,\Omega^F)$ where $F$ is estimated from observations of asset returns and features $\{(r_{l}, F_{l})\}_{l=1}^{n}$ and $\Omega^F$ is the homoscedastic error matrix corresponding to the observations $\{F_{l}\}_{l=1}^{n}$,
the goal is to estimate unobserved asset returns $\tilde{r} \sim p(r|D)$.
We aim to solve (ref) by feature-integrated Black-Litterman network.
We approach this by considering the two causal effects in (ref) and treating the views and uncertainty matrix $(q,\Omega)$ as latent parameters.
Specifically,
itemize• Effect 1:
Features $F$ are extracted from the parameter $\theta$. Consequently, the features $F$, along with their error term $\epsilon^F \sim N(0,\Omega^F)$, share the common parameter $\theta$ with asset returns $r$ and investor views $q$.
• Effect 2:
Features $F$ influence the formation of views $q$. In this scenario, the features $F$, along with their error term $\epsilon^F \sim N(0,\Omega^F)$, are related to the latent views $q$ through a separate equation from the parameter $\theta$.
However, in this section, we do not mix the two effects.
remark[Rationale of Differentiating Effect 1 and 2]
We differentiate the two effects because, in the scenario without investor views, the previous M-BL model ((ref)) estimates the parameter $\theta$ directly by $(F,\Omega)$, meaning Effect 1 dominates over Effect 2 when both are present.
If, in the general scenario where views are observed, Effect 2 is more significant than Effect 1,
then, when views are latent, the ignorance of Effect 2 leads to biased estimation.
This matches the intuition:
if we select the features not directly related to the asset (Effect 1) but highly influence the investor views (Effect 2),
such as macroeconomic indicators like interest rates or CPI,
then using these features to estimate asset returns directly is biased.
To avoid this bias, we differentiate the two effects with two modeling strategies.
One handles the case where Effect 1 dominates, and the other handles the case where Effect 2 is more significant.
We showcase the feature-integrated Black-Litterman network as two configurations: one incorporating Effect 1 and another incorporating Effect 2.
Intuitively, the first better captures generic features while the second more effectively handles the non-asset-related features.
{This implies that, in practice, if an investor takes generic features of assets (e.g. indicators derived from the time series of each asset, as shown in our experiment), configuration 1 should be used. If an investor takes features not specific to individual assets (e.g. interest rates), configuration 2 should be used. The two configurations are not contradicting, so one can take both types of features and incorporate them correspondingly.}
We visualize two configurations of the network in (ref) and define one model for each configuration accordingly.
figure[figure omitted — 1,442 chars of source]
Configuration 1: Shared Latent Parametrization
To capture \hyperref[item:eff1]{Effect 1}, we follow the $\theta\leftrightarrow F$ relationship ((ref)).
By incorporating Effect 1 of the features, we showcase the feature-integrated Black-Litterman network as Shared-Latent-Parametrization Black-Litterman (SLP-BL) model:
definition[SLP-BL model $(\theta,r,q,\Omega,F,\Omega^F)$]
Let $r\in\mathbb{R}^m$ be the returns of the $m$ assets, parametrized by $\theta$, with $r \sim p(r | \theta)$.
Let $q \in \mathbb{R}^k$ represent the views on the returns of the $k$ specified portfolio and $\Omega \in \mathbb{R}^{k \times k}$ be the uncertainty matrix.
Let $F \in \mathbb{R}^{m \times dm}$ be the features of the $m$ assets and error matrix $\Omega^F \in \mathbb{R}^{m \times m}$
The SLP-BL model is a portfolio model composed of four fundamental density functions:
\begin{enumerate}
• {Parametrized Asset Returns}: $p(r | \theta)$, the distribution of asset returns given the parameter.
• {Prior}: $\pi(\theta)$, representing market equilibrium.
• {Likelihood of Views}: $L(\theta | q, \Omega)
\coloneqq p(q, \Omega | \theta)$, the relationship between the parameter and the views.
• {Likelihood of Features}: $L(\theta|F,\Omega^F) \coloneqq p(F,\Omega^F|\theta)$, the $\theta\leftrightarrow F$ relationship ((ref)).
\end{enumerate}
We show the posterior estimation on $\theta$ of the SLP-BL model:
theorem[Parameter Estimation of the SLP-BL Model]
Given a SLP-BL model $(\theta,r,q,\Omega,F,\Omega^F)$ ((ref)) and regression parameters $(\alpha^F,\beta^F) \in \mathbb{R}^m \times \mathbb{R}^{dm}$, assume
\begin{align}
&\theta \sim N(\theta_0, \Sigma_0), \\
&\theta = \alpha^F + F \beta^F + \epsilon^F, \quad \epsilon^F \sim N(0, \Omega^F).
\end{align}
Define $G^F \coloneqq \Sigma_0^{-1} + (\Omega^F)^{-1}$. The {posterior mean} is
\begin{align*}
& p(\theta | F, \Omega^F)
= N\left( \theta; (G^F)^{-1}\left[ \Sigma_0^{-1} \theta_0 + (\Omega^F)^{-1} (\alpha^F + F\beta^F) \right], (G^F)^{-1} \right). \nonumber
\end{align*}
proofSee (ref) for a detailed proof.
The posterior estimation of parameter $\theta$ enables a predictive estimation on $\tilde{r}$:
corollary[Predictive Estimation by the SLP-BL Model]
Define $G^F \coloneqq \Sigma_0^{-1} + (\Omega^F)^{-1}$.
Assume $r\sim N(\theta, \Sigma)$.
Then, under (ref), SLP-BL model gives the predictive estimation of unobserved asset returns $\tilde{r} \coloneqq r | F,\Omega^F$ as
\begin{align*}
&\tilde{r}
\sim N\left(G^F\left[ \Sigma_0^{-1} \theta_0 + (\Omega^F)^{-1} (\alpha^F + F\beta^F) \right], \Sigma + (G^F)^{-1} \right).
\end{align*}
remark[Features Replace Views]
(ref) is equialvent to classical Black-Litterman model if:
\begin{enumerate}
• Features recover investor views:
$\alpha^F+F\beta^F \to P^{-1}q$
• Error matrix recovers uncertainty: $(\Omega^F)^{-1} \to P^ \mathsf{T} \Omega^{-1} P$
\end{enumerate}
Under these conditions, we have $G^F \to {\Sigma_0^{-1}} + {P^ \mathsf{T} \Omega^{-1}P} = G$,
and thus recover (ref):
\begin{align*}
r | q,\Omega \sim N\Bigl(G^{-1}\left[\Sigma_0^{-1}\theta_0 + P^ \mathsf{T} \Omega^{-1}q\right], \Sigma+G^{-1}\Bigr).
\end{align*}
(ref) estimates asset returns without the views, i.e., solves (ref).
With the predictive estimation of asset returns $\tilde{r}$, the mean-variance optimization framework ((ref)) outputs the portfolio weights $w_{\rm SLP-BL}$.
See (ref) for the selection of $\{\Sigma, \Sigma_0, \theta_0, \alpha^F, \beta^F, \Omega^F\}$.
Configuration 2: Feature-Influenced Views
To capture \hyperref[item:eff2]{Effect 2}, we define a $q\leftrightarrow F$ relationship by a multivariate linear model with local dependency:
definition[$q\leftrightarrow F$ Linear Model]
Given features $F \in \mathbb{R}^{m \times dm}$ and portfolio weight matrix for $k$ specified portfolios $P \in \mathbb{R}^{k \times m}$, let $q \in \mathbb{R}^k$ be the views on returns of the $k$ portfolios. Define regression intercept vector $\alpha \in \mathbb{R}^m$, regression coefficient vector ${\beta} \in \mathbb{R}^{dm}$, random error $\epsilon^F \in \mathbb{R}^m$, and error matrix $\Omega^F \in \mathbb{R}^{m \times m}$ such that:
\begin{align*}
q = P (\alpha + F{\beta} + \epsilon^F), \quad \epsilon^F \sim N(0, \Omega^F),
\end{align*}
Furthermore, define $\beta_1, \beta_2, \dots, \beta_m \in \mathbb{R}^d$ as $m$ partitions of the vector ${\beta}$ such that:
\begin{align*}
{\beta} = \left[\beta_1^ \mathsf{T} , \beta_2^ \mathsf{T} , \dots, \beta_m^ \mathsf{T} \right]^ \mathsf{T} .
\end{align*}
This captures the relationship without the loss of generality:
remark[Noisy Implied Asset Returns]
Based on the intuition that the views $q$ are formed on the returns of the $k$ specified portfolios, define the noisy implied asset returns
\begin{align*}
r^F \coloneqq \alpha + F {\beta} + \epsilon^F
\end{align*}
such that: $q = P r^F$.
Then, the dependency between each element of this implied asset returns and $d$ features becomes local:
\begin{align*}
r^F_i
= \alpha_i + F_{i,:} {\beta} + \epsilon^F_i
= \alpha_i + \beta_i^ \mathsf{T} f_i + \epsilon^F_i, \quad i \in [m].
\end{align*}
(ref) allows estimations of regression parameters and the error matrix $(\alpha,{\beta},\Omega^F)$ based on the observations $\{(r_{l}, F_{l})\}_{l=1}^{n}$. See (ref) for details.
\phantomsection
We now introduce the final piece in this configuration:
Regarding the uncertainty matrix $\Omega$, a simplified assumption is that when an investor forms views based on features, the error matrix $\Omega^F$ captures all the information about this uncertainty.
This would suggest omitting $\Omega$ from our model due to the replacement with $\Omega^F$.
Yet, in the general case, $\Omega$ remains necessary as it represents the intrinsic uncertainty of the views, regardless of the $(F,\Omega^F)$\footnote{This concept is similar to the existence of the intrinsic covariance $\Sigma$ regardless of the prior parameter $(\theta_0, \Sigma_0)$ of $\theta$ in (ref).}.
Additionally, retaining $\Omega$ allows our model to remain applicable when both $(q,\Omega)$ and $F$ are observed.
To treat $\Omega$ as a latent parameter, a prior $\pi(\Omega)$ must be specified to enable Bayesian inference.
By incorporating \hyperref[item:eff2]{Effect 2} of the features characterized by the $q\leftrightarrow F$ linear models ((ref)) and a given prior $\pi(\Omega)$, we showcase the feature-integrated Black-Litterman network as the following Feature-Influenced-Views Black-Litterman (FIV-BL) model:
definition[FIV-BL model $(\theta,r,q,\Omega,F,\Omega^F)$]
Let $r\in\mathbb{R}^m$ be the returns of the $m$ assets, parametrized by $\theta$, with $r \sim p(r | \theta)$.
Let $q \in \mathbb{R}^k$ represent the views on the returns of the $k$ specified portfolio and $\Omega \in \mathbb{R}^{k \times k}$ be the uncertainty matrix.
Let $F \in \mathbb{R}^{m \times dm}$ be the features of the $m$ assets and error matrix $\Omega^F \in \mathbb{R}^{m \times m}$
The FIV-BL model is a portfolio model composed of five fundamental density functions:
\begin{enumerate}
• {Parametrized Asset Returns}: $p(r | \theta)$, the distribution of asset returns given the parameter.
• {Prior}: $\pi(\theta)$, representing market equilibrium.
• {Likelihood of Views}: $L(\theta | q, \Omega)
\coloneqq p(q, \Omega | \theta)$, the relationship between the parameter and the views.
• {Views given Features}: $p(q|F,\Omega^F)$, the $q\leftrightarrow F$ relationship ((ref)).
• {Prior on Uncertainty Matrix}: $\pi(\Omega)$, representing intrinsic uncertainty of the views.
\end{enumerate}
The FIV-BL model marginalizing out latent parameters $(q,\Omega)$ to estimate the posterior of $\theta$:
theorem[Parameter Estimation of the FIV-BL Model]
Let $P \in \mathbb{R}^{k\times m}$ be the portfolio weight matrix for $k$ specified portfolios.
Given a FIV-BL model $(\theta,r,q,\Omega,F,\Omega^F)$ ((ref)), regression parameters $(\alpha,{\beta}) \in \mathbb{R}^m \times \mathbb{R}^{dm}$, and a prior $\pi(\Omega)$, assume
\begin{align}
&\theta \sim N(\theta_0, \Sigma_0) \\
&P \theta = q + \epsilon, \quad \epsilon \sim N(0,\Omega), \\
&q = P(\alpha + F {\beta} + \epsilon^F), \quad \epsilon^F \sim N(0, \Omega^F),
\end{align}
where $\theta_0 \in \mathbb{R}^m$ and $\Sigma_0 \in \mathbb{R}^{m \times m}$ are given prior mean and covariance, and $(\epsilon, \epsilon^F)$ are mutually independent.
Define $G \coloneqq \Sigma_{0}^{-1} + P^{ \mathsf{T} }\Omega^{-1}P$.
The {posterior mean distribution} is:
\begin{align}
p(\theta | F, \Omega^F)
&= \int N \left(\theta; \mu_{\theta | \Omega, F, \Omega^F},\Sigma_{\theta | \Omega, F, \Omega^F}\right) \pi(\Omega) d\Omega, \\
where
&\begin{cases}
\mu_{\theta | \Omega, F, \Omega^F} = G^{-1}\left(\Sigma_{0}^{-1}\theta_{0} + P^{ \mathsf{T} }\Omega^{-1} P(\alpha + F{\beta}) \right), \nonumber \\
\Sigma_{\theta | \Omega, F, \Omega^F}
= G^{-1} + G^{-1}P^{ \mathsf{T} }\Omega^{-1}\bigl(P\Omega^FP^{ \mathsf{T} }\bigr)\Omega^{-1}PG^{-1}. \nonumber
\end{cases}
\end{align}
proofSee (ref) for a detailed proof.
remark[Posterior Collapse Under Perfect Views]
If we omit the intrinsic uncertainty matrix by $\Omega \to 0$, or equivalently $\Omega^{-1} \to 0$, we have
\begin{align*}
G &\to {P^ \mathsf{T} \Omega^{-1}P}, \\
\mu_{\theta | \Omega, F, \Omega^F} &\to G^{-1}\left(P^{ \mathsf{T} }\Omega^{-1} P(\alpha + F{\beta}) \right), \\
\Sigma_{\theta | \Omega, F, \Omega^F} &\to G^{-1} + G^{-1}P^{ \mathsf{T} }\Omega^{-1}\bigl(P\Omega^FP^{ \mathsf{T} }\bigr)\Omega^{-1}PG^{-1}.
\end{align*}
Thus, the posterior collapses to
\begin{align*}
\theta | F, \Omega^F = \theta | \Omega, F, \Omega^F \sim N(\alpha + F{\beta}, \Omega^F),
\end{align*}
effectively recovering the $\theta\leftrightarrow F$ relationship $\theta = \alpha + F\beta + \epsilon^F$ ((ref)) except losing the prior information on $\theta$.
Furthermore, reintroducing this prior leads to (ref).
The integral (ref) is a form of Infinite Gaussian Mixture model (IGMM) rasmussen1999infinite.
In general, there is no further closed‐form solution for it unless $\Omega$ is restricted to a special conjugate family or effectively collapses to a point mass (i.e., $\Omega$ is known and fixed). In non‐conjugate settings, the expression remains a continuous mixture of Gaussian distributions, and must be evaluated or approximated numerically (e.g. via Monte Carlo or approximation methods newman1999monte,kruschke2010bayesian,wainwright2008graphical,blei2017variational).
Since there is no trivial conjugate prior $\pi(\Omega)$ for the likelihood $\theta | \Omega, F, \Omega^F$, here we offer an approximation method.
We first substitute $\Omega$ with $\Sigma_{\theta | \Omega, F, \Omega^F}$.
Then, we approximate the mean of the likelihood $\mu_{\theta | \Omega, F, \Omega^F}$ as a constant.
Finally, we assign a conjugate prior to $\Sigma_{\theta | \Omega, F, \Omega^F}$ as an Inverse-Wishart (IW) distribution.
This allows us to obtain a tractable joint distribution $p(\theta, F, \Omega^F)$ --- specifically a Normal-Inverse-Wishart (NIW) distribution.
As a result, the posterior mean distribution $p(\theta | F, \Omega^F)$ follows a student-t distribution after marginalizing out $\Sigma_{\theta | \Omega, F, \Omega^F}$:
corollary[Conjugate Prior]
Consider a FIV-BL model $(\theta, r, q, \Omega, F, \Omega^F)$ ((ref)) with constants $(P,\Sigma,\theta_0,\Sigma_0,\alpha,\beta, \Omega_0)
\in
\mathbb{R}^{k\times m} \times \mathbb{R}^{m\times m} \times \mathbb{R}^m \times \mathbb{R}^{m\times m} \times \mathbb{R}^m \times \mathbb{R}^{dm} \times \mathbb{R}^{k\times k}$.
Define $G \coloneqq \Sigma_{0}^{-1} + P^ \mathsf{T} \Omega^{-1}P$.
Assume $\theta | \Omega,F,\Omega^F \sim N(\mu^\prime,\Sigma^\prime)$,
\begin{align*}
where
\begin{cases}
\mu^\prime
\coloneqq
G^{-1}\Bigl(\Sigma_{0}^{-1}\theta_{0} + P^ \mathsf{T} \Omega_0^{-1}P[\alpha + F\beta]\Bigr), \\
\Sigma^\prime
\coloneqq G^{-1} + G^{-1}P^ \mathsf{T} \Omega^{-1}\bigl(P\Omega^FP^ \mathsf{T} \bigr)\Omega^{-1}PG^{-1}.
\end{cases}
\end{align*}
Assume $\Sigma^\prime$ have an Inverse-Wishart prior:
\begin{align}
\pi(\Sigma^\prime)
=
IW(\Sigma^\prime;\Psi^\prime,\nu^\prime), \quad (\Psi^\prime,\nu^\prime)\in \mathbb{R}^{m}\times \mathbb{R}
\end{align}
Then the marginal posterior of $\theta$ given $(F,\Omega^F)$ follows a multivariate-$t$ distribution:
\begin{align*}
p(\theta \mid F,\Omega^F)
&\sim
t_{\nu^\prime}\Bigl(\theta;\mu^\prime,\tfrac{\Psi^\prime}{\nu^\prime - m + 1}\Bigr).
\end{align*}
proofSee (ref) for a detailed proof.
Since $t$-distribution lacks a conjugate prior, we omit intrinsic covariance $\Sigma$ in estimating unobserved asset returns $\tilde{r}$:
corollary[Approximated Predictive Estimation by the FIV-BL Model]
Assume $r\sim N(\theta, \Sigma)$.
Under (ref),
considering $\theta|\Omega,F,\Omega^F \sim N\left(\mu^\prime,\Sigma^\prime\right)$, assume:
\begin{align*}
&\Sigma^\prime \sim IW(\Psi^\prime, \nu^\prime), \\
&\mu^\prime is a constant G^{-1}\left(\Sigma_{0}^{-1}\theta_{0} + P^{ \mathsf{T} }\Omega^{-1}_0 P(\alpha + F{\beta}) \right).
\end{align*}
Then, as $\Sigma \to 0$, FIV-BL model gives the predictive estimation $\tilde{r} \coloneqq r | F,\Omega^F$ as
\begin{align*}
\tilde{r}
&\sim t_{\nu^\prime}\left(\tilde{r}; \mu^\prime, \frac{\Psi^\prime}{\nu^\prime - m + 1}\right).
\end{align*}
(ref) also estimates asset returns without the views, i.e., solves (ref).
With the predictive estimation of asset returns $\tilde{r}$, the mean-variance optimization framework ((ref)) determines the portfolio weights $w_{\rm FIV-BL}$.
See (ref) for the selection of $\{\Sigma, \Sigma_0, \theta_0, P, \alpha, \beta, \Omega^F, \Psi^\prime, \nu^\prime, \Omega_0\}$.
Proof-of-Concept Experiments
Depart from the classic Black-Litterman model that relies on subjective investor views, our model estimates the posterior distribution over asset returns directly from the feature data.
To demonstrate this concept, we consider the setting without subjective investor views.
Specifically, we focus on integrating asset-specific features, as discussed in (ref), and choose the SLP-BL model ((ref)) accordingly.
We show the model works under this setting and consistently outperforms the benchmarks.
Dataset I: SPDR Sector ETFs.
We collect adjusted daily closing prices and volume for 11 Sector ETFs ((ref)) from April 13, 2004 to February 22, 2024 (20 years).
To avoid selection bias, the portfolio selection list is updated in sync with the introduction of new sectors.
Dataset II: Dow Jones Index.
{We collect adjusted daily closing prices and volume for 41 stocks ((ref)) that have been part of the Dow Jones index from January 5, 1994 to February 22, 2024 (30 years).}
To avoid selection bias, the portfolio selection list is updated in sync with the index.
Backtest Task.
{We backtest our SLP-BL model for each dataset period.}
On each monthly rebalance day, the model outputs a portfolio weight $w_{\rm SLP-BL}$ that maximizes the Sharpe ratio ((ref)), a standardized mean-variance optimization framework ((ref)).
In the model, the prior is set as traditional Markowitz model and the features are selected based on nine generic indicators ((ref)) derived from asset-specific data.
We follow (ref) for the choice of $\{\Sigma, \Sigma_0, \theta_0, \alpha^F, \beta^F, \Omega^F\}$ except that, to avoid the issues of mismatch scale, we use price and indicators data to derive the regression parameters.
Benchmarks and Evaluation.
The benchmarks of our portfolio model are set as (i) market index (e.g. S&P500 and DJIA) (ii) equal-weighted portfolio model (iii) traditional Markowitz model.
We evaluate the models with the following metrics: Cumulative Return, Compound Annual Growth Rate (CAGR), Sharpe Ratio, Maximum Drawdown, and Volatility.
We present the results for five pairs of traditional Markowitz model and our Black-Litterman model with varying rolling window lengths of historical returns:
50 days, 80 days, 100 days, 120 days, and 150 days.
Results.
The SLP-BL model consistently outperforms both traditional Markowitz model and market indices across two datasets
{((ref)).
This is attributed to the more stable portfolio weights based on Bayesian framework, as shown by (ref).
}
table[table omitted — 1,572 chars of source]
table[table omitted — 1,633 chars of source]
Discussion and Conclusion
We propose a Bayesian reformulation of the Black-Litterman model for portfolio optimization without the need for subjective investor views.
Our key contribution is a unified Bayesian network that integrates features and infers parameters.
In the case of observed views ((ref)), the network estimates asset returns based on a mix of two feature effects ((ref), (ref)), generalizing the classical Black-Litterman model and recovering ground-truth estimation with perfect views ((ref)).
In the case of latent views ((ref)), we differentiate the feature effects to handle distinct features ((ref)).
Accordingly, we present two models: the first provides closed-form asset return estimation ((ref), (ref)), while the second results in a mixture model that requires numerical methods ((ref), (ref), (ref)).
Numerically, our model works without investor views and demonstrates consistent, hyperparameter-robust improvements over the Markowitz model and market indices across long-term, real-world datasets ((ref)).
Impact Statement
This work improves portfolio optimization by reducing subjective human inputs. It enhances transparency and promotes data-driven decision-making. The framework benefits both institutional and individual investors with more reliable and fair strategies. However, data-driven models may amplify biases, so careful evaluation is needed for fair outcomes. Overall, this work advances financial modeling and emphasizes ethical implementation.
Acknowledgments
TL would like to thank Gamma Paradigm Research and NTU ABC Lab for support.
JH would like to thank Han Liu, Mimi Gallagher, Sara Sanchez, Dino Feng and Andrew Chen for enlightening discussions on related topics, and the Red Maple Family for support.
The authors would like to thank the anonymous reviewers and program chairs for constructive comments.
JH is supported by the Northwestern University.
The content is solely the responsibility of the authors and does not necessarily represent the official views of the funding agencies.