Estimating the covariance of asset returns, i.e., the risk model, is a key component of financial portfolio construction and evaluation. Most risk modeling approaches produce a factor model that decomposes the asset variability into two components: the first attributed to a small number of factors that are common among the assets and the second attributed to the idiosyncratic behavior of each asset. Third-party providers typically provide risk models to investors, and while these models are typically of high quality, they may fail to capture important information, e.g., changing market regimes and transient factors. To overcome these limitations, we propose a systematic method based on maximum likelihood estimation to enhance an existing factor model by both refining the given model and adding new statistical factors. Our approach relies only on the observed sequence of realized returns and on the choice of two hyperparameters: the number of additional factors and the half-life parameter that determines the weights assigned to returns in the log-likelihood objective. Importantly, our methodology applies to the situation where asset returns may be missing, making it suitable for typical equity datasets. We demonstrate our approach on the Barra short-term US risk model, a high-quality risk model used in practice, for a universe of US high-capitalization equities. We show that the proposed extension captures structure in the returns that is missed by the original model.

We propose a parsimonious quantile regression framework to learn the dynamic tail behaviors of financial asset returns.

We propose a parsimonious quantile regression framework to learn the dynamic tail behaviors of financial asset returns.

Neural Information Processing Systems http://nips.cc/

We propose a robust portfolio optimization approach based on quantile statistics. The proposed method is robust to extreme events in asset returns, and accommodates large portfolios under limited historical data. Specifically, we show that the risk of the estimated portfolio converges to the oracle optimal risk with parametric rate under weakly dependent asset returns. The theory does not rely on higher order moment assumptions, thus allowing for heavy-tailed asset returns. Moreover, the rate of convergence quantifies that the size of the portfolio under management is allowed to scale exponentially with the sample size of the historical data. The empirical effectiveness of the proposed method is demonstrated under both synthetic and real stock data. Our work extends existing ones by achieving robustness in high dimensions, and by allowing serial dependence.

We propose a parsimonious quantile regression framework to learn the dynamic tail behaviors of financial asset returns.

We propose a parsimonious quantile regression framework to learn the dynamic tail behaviors of financial asset returns.

This work introduces a new framework for modeling financial markets through an interpretable probabilistic state machine. By clustering historical returns based on momentum and risk features across multiple time horizons, we identify distinct market states that capture underlying regimes, such as expansion phase, contraction, crisis, or recovery. From a transition matrix representing the dynamics between these states, we construct a probabilistic state machine that models the temporal evolution of the market. This state machine enables the generation of a custom distribution of returns based on a mixture of Gaussian components weighted by state frequencies. We show that the proposed benchmark significantly outperforms the traditional approach in capturing key statistical properties of asset returns, including skewness and kurtosis, and our experiments across random assets and time periods confirm its robustness.

We show its flexible tail dependence structure through simulation.

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 $Ω$ that describe how much a chosen portfolio should outperform the market. Our key idea is to treat $(q,Ω)$ 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.