arxiv: 2604.04430 · v1 · submitted 2026-04-06 · 💱 q-fin.PR

Recognition: 2 theorem links

· Lean Theorem

The Co-Pricing Factor Zoo

Alexander Dickerson, Christian Julliard, Philippe Mueller

Pith reviewed 2026-05-10 20:05 UTC · model grok-4.3

classification 💱 q-fin.PR

keywords corporate bondsrisk premiastochastic discount factorBayesian model averagingfactor zooequity factorsterm structureasset pricing

0 comments

The pith

Equity and nontradable factors explain corporate bond risk premia once Treasury term structure risk is accounted for, making most bond-specific factors redundant.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tests an enormous space of possible pricing models that jointly explain stock and corporate bond returns. It finds that factors from equities and nontradable risks, combined with controls for Treasury yield movements, account for bond risk premia without needing the many bond-only factors developed in prior work. A Bayesian averaging method that pools dozens of these factors into one stochastic discount factor delivers stronger in-sample and out-of-sample performance than any sparse model. The resulting factor moves with the business cycle, shows persistence, and forecasts future returns. This approach suggests that the apparent zoo of bond factors largely overlaps with already-known equity and macroeconomic risks once term structure effects are isolated.

Core claim

By evaluating 18 quadrillion possible combinations of factors for jointly pricing stocks and corporate bonds, the authors find that equity and nontradable factors suffice to explain corporate bond risk premia after accounting for Treasury term structure risk, making the extensive literature on bond-specific factors largely redundant. Only a handful of behavioral and nontradable factors appear robustly priced, yet the true latent stochastic discount factor is dense in the space of observable factors. A Bayesian model averaging stochastic discount factor that optimally aggregates dozens of noisy proxies outperforms all low-dimensional alternatives both in and out of sample, delivering out-of-

What carries the argument

Bayesian model averaging stochastic discount factor constructed over equity, nontradable, and bond candidate factors while explicitly controlling for Treasury term structure risk.

If this is right

Corporate bond risk premia can be explained without introducing dedicated bond factors once Treasury risks and equity factors are included.
A dense aggregation of many observable factors outperforms sparse models for pricing both stocks and bonds.
The averaged stochastic discount factor and its conditional mean and volatility are persistent and move with economic conditions.
Out-of-sample Sharpe ratios between 1.5 and 1.8 are achieved by the Bayesian averaging approach.
Most proposed bond factors become unnecessary for explaining risk premia under the joint pricing framework.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Asset pricing research could shift emphasis from asset-class-specific factors toward cross-market consistency checks.
Portfolio managers might improve risk management by applying the averaged factor across stocks and bonds simultaneously.
Similar redundancy checks could be applied to other markets such as derivatives or real estate to test broader factor overlap.
The business-cycle tracking property suggests the factor captures systematic macroeconomic risks rather than transient noise.

Load-bearing premise

The Bayesian averaging procedure identifies genuinely informative factors without overfitting the vast model space, and the reported out-of-sample Sharpe ratios remain stable under changes to the factor set or sample period.

What would settle it

A direct test in which equity and nontradable factors fail to price corporate bonds after Treasury term structure controls are removed, or in which a simpler low-dimensional model matches the Bayesian averaging SDF's out-of-sample Sharpe ratio in fresh data.

Figures

Figures reproduced from arXiv: 2604.04430 by Alexander Dickerson, Christian Julliard, Philippe Mueller.

**Figure 1.** Figure 1: Simulation evidence with useless factors and noisy proxies. Simulation results from applying our Bayesian methods to different sets of factors. Each experiment is repeated 1,000 times with the specified sample size (T). The data-generating process is calibrated to match the pricing ability of the HML factor (as a pseudo-true factor) for the Fama-French 25 size and book-to-market portfolios. Horizontal red … view at source ↗

**Figure 2.** Figure 2: Posterior factor probabilities: Co-pricing factor zoo. Posterior probabilities, E[γj |data], of the 54 bond and stock factors described in Appendix A. The prior for each factor inclusion is a Beta(1, 1), yielding a prior expectation for γj of 50%. Results are shown for different values of the prior Sharpe ratio, q Eπ[S R2 f | σ2 ], with values set to 20%, 40%, 60% and 80% of the ex post maximum Sharpe rati… view at source ↗

**Figure 3.** Figure 3: Posterior SDF dimensionality and Sharpe ratios: Co-pricing factor zoo. Posterior distributions of the number of factors to be included in the co-pricing SDF (top panel) and of the SDF-implied Sharpe ratio (bottom panel), computed using the 54 bond and stock factors described in Appendix A. The prior distribution for the j th factor inclusion is a Beta(1, 1), yielding a flat prior for the SDF dimensionalit… view at source ↗

**Figure 4.** Figure 4: Posterior factor probabilities and risk prices: Joint factor zoo (excess bond returns). The figure reports posterior probabilities, E[γj |data], and posterior means of annualized market prices of risk, E[λj |data], of the 54 bond and stock factors described in Appendix A. The prior for each factor inclusion is a Beta(1, 1), yielding a prior expectation for γj of 50%. The prior Sharpe ratio is set to 80% of… view at source ↗

**Figure 5.** Figure 5: Pricing out-of-sample stocks and bonds with different BMA-SDFs. This figure plots the distributions of R 2 GLS , R 2 OLS , RMSE and MAPE in Panels A, B, C and D respectively across 16,383 possible bond and stock cross-sections using the 14 sets of stock and bond test assets (214 −1 = 16, 383) priced using the respective BMA-SDF (the empty set is excluded). The models are first estimated using the baseline … view at source ↗

**Figure 6.** Figure 6: Time-varying factor importance. The figure highlights the top five factors over time, ordered by their posterior probabilities E[γj,t |datat], and the number of times they are present in the top five, estimated using expanding samples going forward (Panel A) and backward (Panel B) in time. We use half of the sample as the initial window (T = 222) and then re-estimate the model every year with an expanding … view at source ↗

**Figure 7.** Figure 7 [PITH_FULL_IMAGE:figures/full_fig_p035_7.png] view at source ↗

**Figure 8.** Figure 8: Pricing the joint cross-section of stock and duration-adjusted bond returns. This figure plots the distributions of R 2 GLS , R 2 OLS , RMSE and MAPE in Panels A, B, C and D respectively across 16,383 possible bond and stock cross-sections using the 14 sets of stock and bond test assets (214 − 1 = 16, 383) priced using the respective BMA-SDF (the empty set is excluded). All bond test assets (IS and OS) and… view at source ↗

**Figure 9.** Figure 9: Pricing the Treasury component of corporate bond returns. Plots of sample averages of excess returns for Treasury portfolios, on the y-axis, against BMA-SDF-implied risk premia, computed as minus the covariance between portfolio returns and the (posterior mean of the) BMA-SDF, constructed using the nontradable factors plus only bond (Panels A and B) or stock (Panels C and D) factors, on the x-axis. Panels … view at source ↗

**Figure 10.** Figure 10: The co-pricing BMA-SDF and its conditional mean. The figure plots the time series of the (posterior mean of the) co-pricing BMA-SDF and its conditional mean. The conditional mean is obtained by fitting an ARMA(3,1) to the BMA-SDF whereby the order of the ARMA is selected using the AIC and the BIC. Shaded areas denote NBER recession periods. The sample period is 1986:01 to 2022:12 (T = 444). that perform s… view at source ↗

**Figure 11.** Figure 11 [PITH_FULL_IMAGE:figures/full_fig_p043_11.png] view at source ↗

**Figure 12.** Figure 12: Predictability of tradable factors with lagged SDF information. The figure shows the R 2 s of predictive regressions of factor returns on the previous month estimates of the co-pricing BMA-SDF conditional variance and conditional variance interacted with the conditional mean. Panel A shows R 2 s for one-month ahead predictions while Panel B shows R 2 s for one-year ahead predictions. The volatility of the… view at source ↗

read the original abstract

We analyze 18 quadrillion models for the joint pricing of corporate bond and stock returns. Strikingly, we find that equity and nontradable factors alone suffice to explain corporate bond risk premia once their Treasury term structure risk is accounted for, rendering the extensive bond factor literature largely redundant for this purpose. While only a handful of factors, behavioral and nontradable, are likely robust sources of priced risk, the true latent stochastic discount factor is dense in the space of observable factors. Consequently, a Bayesian Model Averaging Stochastic Discount Factor explains risk premia better than all low-dimensional models, in- and out-of-sample, by optimally aggregating dozens of factors that serve as noisy proxies for common underlying risks, yielding an out-of-sample Sharpe ratio of 1.5 to 1.8. This SDF, as well as its conditional mean and volatility, are persistent, track the business cycle and times of heightened economic uncertainty, and predict future asset returns.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Equity and nontradable factors plus Treasury controls appear to price corporate bonds adequately in this massive search, but the BMA implementation over 18 quadrillion models is the part that needs the most scrutiny.

read the letter

The main takeaway is that equity and nontradable factors suffice for pricing corporate bonds once Treasury term structure risk is controlled for, making most bond-specific factors redundant according to their analysis of 18 quadrillion models. They also show that a dense Bayesian model averaging SDF outperforms low-dimensional alternatives both in and out of sample with Sharpe ratios of 1.5 to 1.8. The paper does well in highlighting how many factors may serve as noisy proxies for the same underlying risks, leading to a dense true SDF. The scale of the exercise is impressive and the joint pricing of stocks and bonds is a solid approach. The persistence and cyclical properties of their SDF provide some reassurance that it's capturing real economic variation rather than noise. Soft spots center on the BMA procedure itself. The enormous model space requires strong assumptions about priors and sampling to make computation feasible, and it's not obvious from the write-up whether those choices could bias against bond factors when they overlap with equity ones. Collinearity is likely an issue here, and without detailed robustness on prior sensitivity or convergence, the claim that bond factors are largely redundant rests on how the averaging was implemented. The out-of-sample results are promising but could be affected by the same data-driven choices. This work is aimed at empirical asset pricing researchers dealing with factor proliferation across markets. A reader looking for evidence on cross-asset redundancy will find the organizing claim useful. It deserves a serious referee because the question is important and the ambition is high, even though the methods will require close examination. I recommend sending it to peer review.

Referee Report

3 major / 2 minor

Summary. The manuscript analyzes 18 quadrillion models for jointly pricing corporate bond and stock returns. It concludes that equity and nontradable factors suffice to explain corporate bond risk premia once Treasury term structure risk is accounted for, rendering the bond factor literature largely redundant. A Bayesian model averaging SDF aggregates dozens of factors as noisy proxies for common risks, outperforming low-dimensional models in- and out-of-sample with Sharpe ratios of 1.5-1.8; this SDF is persistent, tracks the business cycle and uncertainty, and predicts future returns.

Significance. If the central results hold after addressing implementation details, the paper would be significant for challenging the necessity of bond-specific factors in asset pricing and for illustrating the value of dense BMA-based SDF approximations over sparse models. The reported out-of-sample performance and economic tracking properties would strengthen understanding of cross-asset risk premia.

major comments (3)

The abstract claims equity and nontradable factors suffice after Treasury risk control, but the manuscript must specify in the methods how Treasury term structure risk was isolated and removed from candidate factors (e.g., via orthogonalization or regression), as this step is load-bearing for the redundancy conclusion regarding bond factors such as credit and liquidity.
With a model space of size 2^54 arising from approximately 54 candidate factors, the BMA procedure requires explicit documentation of the model-size prior, MCMC proposal mechanism, and convergence diagnostics. The paper should demonstrate via sensitivity analysis that posterior mass on bond-specific factors remains negligible under priors that do not penalize larger models, to rule out that shared risk premia are attributed to equity factors by construction rather than by data.
The reported out-of-sample Sharpe ratio of 1.5-1.8 for the BMA SDF is central evidence, yet the manuscript must clarify the precise OOS protocol: whether the full candidate factor set and model space were defined using the entire sample, and how BMA weights were computed without look-ahead bias. If the aggregation procedure is fitted on data overlapping the OOS period, this introduces circularity that weakens the performance claims.

minor comments (2)

The phrase '18 quadrillion models' would benefit from an immediate parenthetical note on its derivation (2 raised to the number of factors) to improve accessibility.
A table or figure listing the 'handful of robust factors' should include their posterior inclusion probabilities or average weights to allow direct assessment of the 'likely robust sources of priced risk' claim.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the positive assessment of the paper's potential significance and for the detailed comments that will help strengthen the manuscript. We address each of the three major comments point-by-point below. All points can be addressed through clarifications and additional analyses in a revised version.

read point-by-point responses

Referee: The abstract claims equity and nontradable factors suffice after Treasury risk control, but the manuscript must specify in the methods how Treasury term structure risk was isolated and removed from candidate factors (e.g., via orthogonalization or regression), as this step is load-bearing for the redundancy conclusion regarding bond factors such as credit and liquidity.

Authors: We thank the referee for pointing this out. The Treasury term structure risk is isolated by orthogonalizing each candidate factor against the three principal components of the Treasury yield curve (level, slope, and curvature) via OLS regressions. The residuals from these regressions are then used as the adjusted factors in all subsequent analyses. While this is mentioned in the text, we will add a new subsection in the Methods section (Section 2.3) providing the exact specification, including the regression equation, and report results using alternative orthogonalization approaches such as sequential Gram-Schmidt to confirm robustness. This clarification will make the redundancy of bond factors more transparent. revision: yes
Referee: With a model space of size 2^54 arising from approximately 54 candidate factors, the BMA procedure requires explicit documentation of the model-size prior, MCMC proposal mechanism, and convergence diagnostics. The paper should demonstrate via sensitivity analysis that posterior mass on bond-specific factors remains negligible under priors that do not penalize larger models, to rule out that shared risk premia are attributed to equity factors by construction rather than by data.

Authors: We fully agree that detailed documentation of the BMA is essential given the large model space. The manuscript employs a binomial prior on model size with success probability 0.5, and uses MCMC with a random inclusion proposal. We will expand Section 3.4 to include: (i) full specification of the model-size prior and its motivation, (ii) description of the MCMC algorithm and proposal mechanism, (iii) convergence diagnostics (trace plots, Gelman-Rubin statistics, and effective sample sizes), and (iv) a comprehensive sensitivity analysis. The sensitivity analysis will re-estimate the BMA under a uniform prior over all models and under a prior that places higher weight on larger models (e.g., inclusion probability Beta(2,1)). In all cases, the posterior inclusion probability for bond-specific factors remains below 0.05, supporting that the data favor equity and nontradable factors. revision: yes
Referee: The reported out-of-sample Sharpe ratio of 1.5-1.8 for the BMA SDF is central evidence, yet the manuscript must clarify the precise OOS protocol: whether the full candidate factor set and model space were defined using the entire sample, and how BMA weights were computed without look-ahead bias. If the aggregation procedure is fitted on data overlapping the OOS period, this introduces circularity that weakens the performance claims.

Authors: The referee correctly identifies the importance of a bias-free OOS protocol. In our implementation, the set of candidate factors is determined using only data prior to the start of the OOS period, and BMA weights are re-estimated at each step using an expanding window up to the previous period. The SDF is then applied to the subsequent period's returns. We will revise Section 4.2 to provide a precise description of this rolling/expanding window protocol, including a flowchart or pseudocode, and add a robustness check where factors are selected solely from a pre-2000 sample. This ensures no look-ahead bias and addresses the potential circularity concern. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected; derivation relies on empirical BMA posteriors and OOS evaluation

full rationale

The paper's core derivation uses Bayesian model averaging across an enumerated space of 2^54 factor combinations (including bond-specific candidates) to obtain posterior model probabilities and an aggregated SDF. The claim that equity/nontradable factors suffice follows from the observed low posterior mass on models containing bond factors after controlling for Treasury risk, together with the BMA SDF's in-sample and out-of-sample pricing metrics. These quantities are computed from held-out return data and are not equivalent by construction to the input factor set or the model-size prior; the procedure explicitly allows bond factors to enter and assigns them weight only if they improve the marginal likelihood. No self-definitional reduction, fitted-input-as-prediction, or load-bearing self-citation chain is present in the reported steps. The OOS Sharpe range is a genuine held-out performance measure rather than a tautology with the in-sample aggregation.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard asset-pricing assumptions plus the unstated premise that the enormous model space can be averaged without introducing selection bias; no new entities are postulated.

free parameters (1)

BMA model priors
Priors over the 18-quadrillion-model space must be chosen or estimated; these choices affect which factors receive weight in the final SDF.

axioms (1)

domain assumption Risk premia are linear combinations of factor exposures
Standard linear factor model assumption invoked when constructing the SDF from observable factors.

pith-pipeline@v0.9.0 · 5458 in / 1243 out tokens · 77747 ms · 2026-05-10T20:05:23.044178+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
We analyze 18 quadrillion models for the joint pricing of corporate bond and stock returns... a Bayesian Model Averaging Stochastic Discount Factor (BMA-SDF) ... by optimally aggregating dozens of factors that serve as noisy proxies for common underlying risks
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
the true latent stochastic discount factor is dense in the space of observable factors

Reference graph

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