PanelGP combines stock-return and risk forecasts, but understates risk differences

A stock portfolio needs more than a list of promising investments. It also needs estimates of how uncertain those investments are and how their returns move together. In a November 2026 preprint, EPFL researchers Andrea Ruglioni and Damir Filipovic introduce PanelGP, a model designed to estimate expected returns and their covariance matrix within the same framework.

Their central idea is to connect company characteristics with exposure to shared return factors, while connecting economic conditions with the expected values of those factors. The historical tests suggest that flexible company-level relationships improve stock rankings. But the model remains less convincing at estimating the full spread of individual risks.

One model, two forecasting jobs

Factor models describe returns through a small number of common influences and each stock’s exposure, or loading, to them. PanelGP lets those loadings depend on firm characteristics. A separate relationship maps macro-financial conditions, which the authors call the context, to expected factors.

Both relationships use Gaussian-process priors: distributions over possible functions whose structure is controlled by covariance kernels. An affine kernel produces loadings resembling those of instrumented principal component analysis, or IPCA. Radial basis function kernels allow smooth nonlinear relationships instead.

The model also distinguishes stock-specific loading shocks from factor shocks shared across stocks. This separates sources of individual risk from sources of common movement. Uncertainty about the estimated relationships enters its forecasts too.

Fitting the model requires an approximation. The authors use sparse variational inference with a joint Gaussian approximation that retains dependence between loadings and factors. Conditional on this approximation, predictive covariances and portfolio cumulants—quantities describing features such as variance and asymmetry—have closed-form expressions. The resulting return distribution need not be Gaussian, although it cannot produce power-law tails.

Testing on historical index constituents

The empirical evaluation uses historical S&P 500 constituents, rather than restricting the sample to current members. Its evaluation panel covers input months from January 1995 through December 2024, with each month’s characteristics and context predicting the following month’s excess return.

Inputs include 137 firm characteristics and six macro-financial variables. The researchers compare four combinations of loading and factor kernels, alongside IPCA benchmarks with and without an intercept. Their main comparisons use five factors, selected as a compromise based on validation-period performance and complexity. PanelGP results average five optimization seeds.

In the fixed-training experiment, models are fitted on 1960–1989 data, validated on 1990–1994, and then left unchanged during evaluation. Replacing affine loadings with nonlinear loadings significantly improves the average monthly stock-ranking correlation and Gaussian log score under either factor kernel, after adjustment for multiple comparisons. That score evaluates expected returns and covariances together, not the distribution’s higher moments.

Adding context-dependent factor forecasts also improves several point estimates, but those incremental differences are not statistically distinguishable from zero. The clearest evidence therefore concerns nonlinear loadings, not a proven advantage from conditioning factors on economic context.

Portfolio gains need qualification

The authors also refit their preferred nonlinear model annually using the latest 360 input months. Its tangency portfolio, which uses both expected returns and covariances, records an annualized Sharpe ratio of 1.90 before costs, versus 1.74 for rolling IPCA with an intercept. The statistical tests cannot distinguish those Sharpe ratios.

Two other differences against that benchmark are significant: PanelGP has a higher Gaussian log score and lower minimum-variance portfolio volatility. Annualized volatility is 12.3%, compared with 13.2% for the benchmark.

With an assumed proportional trading cost of 10 basis points per dollar traded, the tangency Sharpe ratios become 1.52 and 1.35, respectively. These are historical results under a simplified cost model that excludes stock-specific spreads, shorting fees and price impact. Short positions are allowed in the portfolio construction.

The return-accuracy result is less favorable. Under annual re-estimation, PanelGP’s pooled predictive R-squared falls below both rolling IPCA benchmarks, although those differences are not significant. All models also show lower Sharpe ratios and ranking correlations in 2010–2024 than in 1995–2009.

Risk ordering is not risk calibration

PanelGP generally ranks riskier stocks and months above safer ones, but its predicted variances are too compressed. Risk rises more sharply in realized forecast errors than in its predictions; across stocks, the safest group also receives too much predicted variance.

The authors identify a structural constraint: several variance terms create a floor shared by every stock in a month, limiting adaptation to stock-specific idiosyncratic risk. Their inference approximation may contribute additional compression. The preprint thus offers evidence for joint forecasting and covariance-based portfolio construction, while leaving individual risk calibration as a substantial unresolved weakness.