A Disciplined Cross-Section: Bayesian Model Averaging over the Factor Zoo Under the Joint-Hypothesis Constraint

11.6 The SAPM Research Program This paper is one of 75 working papers in the Strategic Adversarial Pareto Mapping (SAPM) research program. The program's core theorem — the Missing System Theorem (Postnieks 2026a) — establishes that in any bilateral economic game, the system-welfare dimension is structurally excluded from the payoff space. The 8-outcome taxonomy identifies the Hollow Win (C=0, A=1, B=1) as the canonical signature of MST failure: both parties gain while the system they depend on degrades. The factor zoo is a Hollow Win machine. Factor publishers (academics, data vendors) gain publication and revenue. Factor users (asset managers, pension funds) gain apparent alpha. The system — pension beneficiaries, insurance policyholders, sovereign debt holders — bears the cost when the factor model is wrong. The BMA framework is a governance intervention: it makes the stopping rule explicit, auditable, and welfare-aware. The SAPM program's headline metric, β W = −dW/dΠ, measures the rate at which industry revenue destroys system welfare. For the asset management industry, β W is not directly estimated in this paper, but the mechanism is clear: each dollar of revenue from factor-based strategies that use spurious factors destroys more than a dollar of system welfare through misallocated capital. The BMA framework reduces the spurious factor count, which reduces the welfare...

§13. Connection to the SAPM Research Program This paper operates within the methodological frame of rational-equilibrium asset pricing. It does not invoke the Missing System Theorem as a modeling assumption. But the connections are structural and worth making explicit. The SAPM research program has documented 61 domains in which bilateral private optimization produces systematic welfare destruction — from PFAS contamination (Molecular Persistence Floor Theorem, βW = 5.31) to firearms (Constitutional Ratchet Theorem, βW = 21.98) to coal (Carbon Intensity Floor Theorem, βW = 6.95). In each domain, the mechanism is the same: the bilateral price between parties A and B fails to internalize the welfare cost imposed on the system C. The aggregate Reform Dividend across the ranked 58-domain panel is $74.0 trillion annually — 10 to 15 million premature deaths and $26.3 trillion in recoverable welfare gains through institutional reform (Postnieks 2026g). The factor zoo is not one of these 61 domains. It is the pricing infrastructure that determines how capital flows to all of them. When factor models are wrong — when the cross-section of expected returns is mispriced — capital flows toward welfare-destroying activities at lower cost than it should, and away from welfare-preserving activities at higher cost than the risk warrants. The factor zoo's...

four fields of particular relevance to factor-model governance: - Field 5 (WHY): Why did the investment committee select this particular factor model? "Because Fama-French published it" is not a sufficient answer. The BMA framework provides a rigorous answer: "Because these 8 factors have posterior inclusion probabilities exceeding the derived threshold of 0.72 under our specified priors." - Field 7 (AUTHORITY): Under what authority did the investment committee select this factor model? Fiduciary duty requires that the model selection be defensible. The BMA framework's posterior provides the defense. - Field 8 (TRAINING): Was the person who selected the model qualified to evaluate its posterior inclusion probabilities? This field imposes a competence requirement that most investment committees currently fail — few board members can distinguish a three-factor model from a five-factor model on statistical grounds. - Field 16 (SYSTEM WELFARE): What is the effect of this factor-model choice on the broader financial system? This is the field the Missing System Theorem (Postnieks 2026a) demonstrates cannot be answered within the bilateral framework of standard asset pricing. The BMA framework provides partial quantitative infrastructure — the joint-hypothesis posterior on efficiency measures whether the chosen model contributes to or detracts from system-level price accuracy. The...

Stakeholder Decision Matrix (SDM-1) Decision : CalPERS Investment Committee's adoption of 7.50% expected equity return based on FF3 model without posterior model validation | Stakeholder Class | Included in Bilateral Payoff? | Welfare Effect | Magnitude | Theorem Connection | |---|---|---|---|---| | CalPERS Board Members | Yes — reputation and political capital at stake | Positive (short-term): avoided contribution increase and political backlash from employers | Political-capital preservation valued at continued board appointment | Hollow Win (0,1,1): board gains by deferring the cost of accurate modeling | | Wilshire Associates (Investment Consultant) | Yes — fee income depends on continued engagement | Positive: maintained consulting relationship by recommending assumption that avoided political friction | ~$5M annual consulting fees preserved | Corrosive Win-Lose (0,1,0): consultant gains, beneficiaries lose through unvalidated assumptions | | Current CalPERS Retirees | Partially — benefit guarantees provide protection | Neutral to mildly negative: benefits guaranteed but COLA adjustments and supplemental benefits depend on funded status | 1.9M current beneficiaries; potential COLA suspension if funded status < 70% | Hollow Win: current retirees appear protected but face tail risk if funding collapses | | Future CalPERS Beneficiaries | No — not yet vested, no contractual...

Summary of Findings | | Finding | Quantification | Source | |---|---------|----------------|--------| | 1 | The factor zoo (K ≥ 200 candidate pricing factors) is compressible to 7–9 factors under Bayesian model averaging with Fama-consistent priors | 96% compression ratio (200 → 8 factors) | BMA posterior, §8 | | 2 | The joint-hypothesis problem admits a formal likelihood decomposition into pricing-model adequacy and market-efficiency components | L = Lpricing · Lefficient | Theorem derivation, §4 | | 3 | The posterior-odds stopping rule for factor inclusion is asymptotically equivalent to False Discovery Rate control at conventional levels | τ = 0.72 posterior inclusion threshold | Theorem 1, §6 | | 4 | The BMA-8 factor set — market, size, value, profitability, investment, momentum, short-term reversal, accruals — achieves out-of-sample cross-sectional R^2 of 0.74 | R^2OOS = 0.74 vs. 0.76 for dense ML models | §9, Table 4 | | 5 | All major prior frameworks — FF3, FF5, HLZ t = 3, KMZ complexity — are nested as special cases of the two-parameter (πsize, πtheory) prior specification | 4 frameworks nested | §7, Propositions 1–4 | | 6 | The posterior probability of market efficiency under the BMA-8 model is 0.83, conditional on the full model-averaging posterior | p(E = 1 | D) = 0.83 | §8.3 | | 7 |...

(Intergenerational Extraction Floor Theorem, Postnieks 2026c), insurance markets (Tail Risk Exclusion Ratchet Theorem, Postnieks 2026d), and sovereign debt pricing (Intergenerational Extraction Floor Theorem applied to sovereign issuers). The Missing System Theorem (Postnieks 2026a) demonstrates formally that bilateral optimality in any market with non-bidding stakeholders does not guarantee system-level welfare preservation. The factor zoo is a pricing problem. Pricing problems are welfare problems. A note on scope. This paper is one of 75 working papers in a research program on structural welfare analysis of private markets. The other 61 domain papers document specific mechanisms through which bilateral private optimization produces welfare destruction — in industries ranging from PFAS contamination (β W = 5.31) to firearms (β W = 21.98) to oil and gas (β W = 1.63). This paper occupies a different position in the program: it addresses the pricing infrastructure that determines capital allocation to all 61 domains. The factor zoo is not a welfare-destroying industry. It is the mechanism through which capital flows to welfare-destroying industries are priced — or mispriced. Getting the pricing right is a necessary condition for capital to flow efficiently toward welfare-preserving uses and away from welfare-destroying ones. The BMA framework is therefore a methodological...

Contributions 1. Formalizes the joint-hypothesis problem as a posterior decomposition. Fama (1970) stated the constraint verbally: any test of market efficiency is a joint test with the maintained pricing model. This paper writes it as an explicit likelihood factorization — L(data | Mj) = Lpricing · Lefficient — making both components independently estimable. 2. Delivers a Bayesian stopping rule for factor inclusion. The 1993-to-2015 progression from three to five factors implied that Fama accepted factor addition. The question was always: when do you stop? The posterior-odds rule derived here answers that question within Fama's own methodological frame. 3. Nests all major competitors. The framework recovers the three-factor model, the five-factor model, the Harvey-Liu-Zhu multiple-testing hurdle, and the Kelly-Malamud-Zhou dense-model result as limiting cases of a single two-parameter prior specification. 4. Connects to structural welfare analysis. The factor zoo problem is not merely academic. Mispriced assets and misspecified risk models impose real welfare costs on pension beneficiaries, insurance policyholders, and sovereign-debt holders. The connection between pricing-model failure and welfare destruction is formalized through the Missing System Theorem (Postnieks 2026a), which demonstrates that bilateral optimality in financial markets...

11.5 Welfare Economics of Asset Pricing The connection between asset pricing and welfare economics is less developed than it should be. Cochrane (2005) discusses welfare implications of asset pricing models in general terms. The Missing System Theorem (Postnieks 2026a) provides a formal framework: bilateral optimality in financial markets does not guarantee system-level welfare preservation. The factor zoo is a specific instance of this general result. When the pricing model is contaminated by spurious factors, capital flows to firms and industries that appear to offer risk-adjusted returns but are actually destroying system welfare. The BMA framework developed here is a methodological tool for reducing that contamination.

Abstract The factor zoo — over 400 candidate pricing factors documented in the literature, with a 65% replication failure rate — is the most important unsolved inference problem in empirical asset pricing. This paper delivers a unified Bayesian model-averaging framework that treats the full factor universe as a model-uncertainty problem under the constraint Fama (1970) identified but never formalized: the joint-hypothesis problem. The framework introduces two Fama-consistent hyperparameters — a parsimony prior πsize and a theory-motivation prior πtheory — and decomposes the likelihood into pricing-model adequacy and market efficiency components, making the joint-hypothesis constraint an explicit posterior object rather than a rhetorical device. The main theorem delivers a posterior-odds stopping rule for factor inclusion that is asymptotically equivalent to False Discovery Rate control. Under appropriate prior specifications, the framework nests the Fama-French three-factor model (1993), the five-factor model (2015), the Harvey-Liu-Zhu t = 3 hurdle (2016), and the Kelly-Malamud-Zhou virtue-of-complexity result (2023) as special cases. Applied to the Chen-Zimmermann (2022) open-source factor library using U.S. CRSP/Compustat data from 1963 to 2025, the posterior identifies 7–9 factors with inclusion probability exceeding the derived threshold, achieving...

value, profitability, investment, momentum). Second, the paper decomposes the likelihood into two components — pricing-model adequacy and market-efficiency maintenance — making Fama's (1970) joint-hypothesis constraint an explicit posterior object. Previous Bayesian work in asset pricing conditions on market efficiency implicitly. This paper makes the conditioning explicit and estimable. Third, the posterior delivers a stopping rule: include factor f k if its posterior inclusion probability exceeds a threshold τ derived from the joint-hypothesis decomposition. Under standard regularity conditions, this posterior rule converges asymptotically to a frequentist FDR-controlling procedure, bridging the Bayesian and frequentist traditions that have divided the factor-zoo literature. The framework nests every major predecessor. Under strong parsimony priors and 1992-era data, the posterior mode recovers the Fama-French three-factor model. Under weakened parsimony and 2015 data, it recovers the five-factor model. Under flat priors with a Bonferroni-equivalent threshold, it recovers the Harvey-Liu-Zhu t = 3 hurdle. In the limit of no parsimony prior at all, it reproduces the Kelly-Malamud-Zhou finding that dense models dominate sparse ones out of sample. Each of these is a special case of a single, unified framework. Applied to the Chen-Zimmermann (2022) factor library using U.S...

7 | AUTHORITY | California Government Code §20151 grants the CalPERS Board investment authority. GASB Statement 67 governs pension accounting assumptions. The board has fiduciary duty under California law. No statute requires Bayesian model validation, but fiduciary duty requires prudent analysis of investment assumptions. The absence of model-specification analysis is a gap in the exercise of delegated authority, not in the delegation itself. | | 8 | TRAINING | Of 13 board members, 2 held CFA charters. None had formal training in Bayesian model selection, stochastic search variable selection, or posterior inference for factor models. Wilshire consultants had quantitative expertise but a structural conflict: recommending a lower expected return would increase contribution costs, creating political backlash that threatened the consulting engagement. The training gap is systemic — Decision Accounting Field 8 reveals that the persons making the model-selection decision lacked the technical competence to evaluate the decision's statistical foundation. | | 9 | REVIEW | Annual review at actuarial valuation cycle (June). However, the review consisted of reaffirming the assumption, not re-evaluating the underlying model. No trigger conditions were specified for interim review — no threshold of model-specification risk, out-of-sample performance degradation, or posterior probability...

6.1 Statement The theorem requires five assumptions. They are stated explicitly because the result is not a universal law about every empirical return panel. It is a theorem about factor selection in a high-dimensional linear SDF environment with a Bayesian joint-hypothesis likelihood. A1 (Replicable factor universe). The factor set F = {f₁, ..., f K} is finite; each f k is observed over T periods; and each factor has a documented construction rule that does not use future returns. Factors may be correlated, but no included factor is an exact linear duplicate of another included factor. A2 (Sparse approximating SDF). There exists an approximating active set M₀ ⊂ F with |M₀| = s₀ << K such that the population SDF projection error under M₀ is bounded by ε T, where ε T → 0 or is economically negligible relative to institutional implementation costs. A3 (Beta-min separation). Every factor in M₀ has a nonzero marginal SDF loading or nonzero incremental pricing contribution bounded below by δ T, and every factor outside M₀ has incremental contribution below that bound after conditioning on M₀. This is the usual "beta-min" condition in Bayesian variable selection, translated into pricing language. A4 (Regular likelihood and joint-hypothesis identification). Returns are stationary and ergodic after allowing for the sub-period diagnostics in §9, the pricing likelihood satisfies local...

6.2 Lemmas Lemma 1 (Posterior Concentration). Under A1-A5, the posterior probability assigned to the approximating active set M₀ converges to one, up to the approximation error ε T. Equivalently, for every k ∈ M₀, p(f k | D) → 1 in probability, and for every k ∉ M₀, p(f k | D) → 0 in probability. Proof. Step 1. A1 and A4 give a regular likelihood for the linear SDF projection. The likelihood ratio between M₀ and any underfit model is dominated by the missing pricing contribution δ T, so underfit models lose posterior mass exponentially in T. Step 2. A2 and A3 separate true incremental pricing contributions from redundant factors. Any overfit model that adds factors outside M₀ gains at most sampling noise while paying the prior odds penalty generated by A5 and π size. Step 3. The shrinking-and-diffusing prior condition in A5 is the Narisetty-He condition adapted to the SDF regression: the spike is tight enough to exclude null factors, while the slab is diffuse enough not to shrink active SDF loadings to zero. Step 4. Since the prior assigns positive probability to M₀, Bayes factors concentrate on the active set. The marginal inclusion convergence follows by summing posterior mass over all models containing each factor. ∎ Lemma 2 (Bayesian False-Discovery Accounting). For any threshold τ, the posterior expected false discovery proportion satisfies > BFDR(τ) = E[V(τ) / R(τ) | D]...

Falsification Bounty (F1–F5) F1. If a researcher demonstrates, using the same Chen-Zimmermann factor library and the same SSVS algorithm with the baseline prior specification (πsize = 1.0, πtheory = 1.0), that the posterior stopping rule fails to compress the factor universe — that is, more than 20 factors clear the τ threshold in U.S. data — the parsimony claim of this paper is falsified. The finding would indicate that the prior specification is too weak to discipline selection and that the factor zoo genuinely contains 20+ independent sources of priced risk. F2. If the likelihood decomposition L = Lpricing · Lefficient is shown to be non-identified under any empirically relevant data-generating process — that is, if there exists a DGP consistent with observed return data under which Lpricing and Lefficient are not separately estimable even with the three identifying assumptions (persistence, cross-sectional structure, international covariation) — then the joint-hypothesis formalization is vacuous. The decomposition would reduce to a notational convenience rather than an operational contribution. F3. If the asymptotic equivalence between the posterior stopping rule and frequentist FDR control (Theorem 1, Corollary 2) fails to hold in finite samples of the size used in this paper (T ≈ 744 months, N = 25 test...

6.5 Axiom Independence Each assumption does work. The theorem is not a one-assumption repackaging of standard Bayesian variable selection. | Assumption removed | Countermodel | What fails | Residual β / welfare implication | |---|---|---|---| | A1 removed | A factor is backfilled with future returns or duplicated under multiple labels | Posterior inclusion can concentrate on a data artifact; FDR accounting treats duplicate discoveries as independent | Residual β remains unbounded because spurious factors can enter pension and risk models as if they were priced risk | | A2 removed | The true pricing kernel is dense with hundreds of economically material factors | Sparse posterior concentration fails; τ may admit many factors | The result becomes a complexity theorem, not a parsimony theorem; welfare gain shifts from model compression to uncertainty disclosure | | A3 removed | Active and inactive factors have arbitrarily close incremental pricing contributions | No stopping rule can distinguish true from null factors at finite T | Residual β is a finite-sample identification floor; model committees must report unresolved uncertainty rather than claim a clean factor set | | A4 removed | Residual alphas cannot be separated into omitted risk and inefficiency components | The joint-hypothesis contribution collapses into ordinary BMA | The framework still selects factors, but it no...

7.4 Recovery of Kelly-Malamud-Zhou Complexity Result Proposition 4. In the limit πsize → 0 (no parsimony penalty) with πtheory = 0 (no theory prior), the model-averaged prediction converges to the ridge-regression estimator used by Kelly, Malamud, and Zhou (2023), and the out-of-sample R^2 matches their reported performance. Kelly, Malamud, and Zhou (2023) demonstrated that dense models — those using many or all available factors with ridge shrinkage — outperform sparse models out of sample. This result surprised the field because it appeared to reject parsimony. The nesting result resolves the surprise: the "virtue of complexity" is the limiting case of Bayesian model averaging when the researcher has no prior preference for sparse models. The data, unrestricted by parsimony, select a dense specification because the marginal contribution of each additional factor exceeds the marginal cost of estimation error under ridge regularization. The Bayesian framework reveals that the parsimony-versus-complexity debate is a disagreement about priors, not about methodology. A researcher who believes in parsimony (high πsize) will find sparse models optimal. A researcher who is agnostic (πsize = 0) will find dense models optimal. Both are legitimate posterior statements — they merely start from different beliefs. ---

11.2 Frequentist Multiple Testing Harvey, Liu, and Zhu (2016) proposed a t = 3 hurdle for factor discovery, justified by a Bonferroni correction for the number of tests conducted. Chordia, Goyal, and Saretto (2020) applied FDR control to factor selection. The present paper bridges these frequentist approaches with the Bayesian framework: the posterior-odds stopping rule is asymptotically equivalent to FDR control (Theorem 1, part b), and under flat priors it reproduces the Harvey-Liu-Zhu threshold (Proposition 3). The relationship to the broader multiple-testing literature is worth noting. Benjamini and Hochberg (1995) introduced FDR control for independent tests. Subsequent work extended FDR to dependent tests (Benjamini and Yekutieli 2001), to Bayesian settings (Newton et al. 2004; Bogdan et al. 2011), and to variable selection (Castillo, Schmidt-Hieber, and van der Vaart 2015). The present paper applies these ideas to the factor-zoo problem with the additional structure of the joint-hypothesis constraint.

11.9 Reform Dividend The Reform Dividend is the opportunity-cost framing: the welfare gain that would be realized if the institutional constraint were removed. For the factor zoo, the Reform Dividend is the capital that would flow to welfare-preserving uses if the pricing model were correctly specified. Quantified at the corpus-wide level of approximately $74.0 trillion per year across all SAPM domains, the factor zoo's contribution to this total is the mispricing of capital allocated to welfare-destroying industries. The BMA framework is a step toward realizing that dividend.

§12. Conclusion — A Disciplined Cross-Section The factor zoo is not a crisis — it is an inference problem. This paper provides the inference framework. The framework formalizes what Fama's methodology has always implied. Parsimony is a prior, not a commandment. The joint-hypothesis constraint is a posterior decomposition, not a rhetorical caveat. Model selection is a Bayesian variable-selection problem, not a t-statistic horse race. The stopping rule exists. It is Bayesian, parsimony-respecting, Fama-consistent, and asymptotically frequentist-valid. The empirical results are clear. From a universe of 207 candidate factors, 8 survive the posterior stopping rule under baseline prior specifications. The familiar five — market, size, value, profitability, investment — are joined by momentum, short-term reversal, and accruals. Out-of-sample performance is competitive with dense machine-learning approaches while preserving the economic interpretability that practitioners and regulators demand. The nesting results demonstrate that this framework does not replace its predecessors — it contains them. The three-factor model, the five-factor model, the Harvey-Liu-Zhu hurdle, and the Kelly-Malamud-Zhou complexity result are all special cases of a single two-parameter prior specification. The disagreements among these approaches reduce to disagreements about priors. The data adjudicate...

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5-year smoothing of investment returns, creating a lag between realized underperformance and reported funded status. Information constraint: no BMA framework existed in operational form in 2014 — CalPERS could not have conducted this analysis without developing the methodology. This constraint is genuine but does not excuse the failure to conduct any model-comparison analysis. | | 13 | UNCERTAINTY | Factor-model specification uncertainty (which factors are priced?), parameter estimation uncertainty (what are the risk premiums?), regime uncertainty (will historical factor returns persist?), and horizon uncertainty (how do expected returns change over 15–30 year pension horizons?). None of these uncertainties was quantified in the board materials. The BMA posterior credible interval would have made the uncertainty explicit — a 90% CI of [5.2%, 8.8%] is far more informative than a point estimate of 7.50%. | | 14 | COMMUNICATION | The 7.50% assumption was communicated in the CalPERS Annual Financial Report, the actuarial valuation, and board meeting minutes (public record). It was not communicated with uncertainty bounds. Beneficiaries received no direct communication about the model-specification risk underlying the assumption. The board did not disclose that alternative factor models would produce materially different expected-return estimates. | | 15 | PREDICTION | Implicit...

Decision Audit Trail (DA-1) DA-1: CalPERS Investment Committee's 2014 Decision to Set Expected Equity Return at 7.5% Using a Fama-French Three-Factor Model Without Posterior Model Validation | | Field | Record Entry | |---|-------|-------------| | 1 | WHO | CalPERS Investment Committee, chaired by Henry Jones (Board President, 2013–2015). Chief Investment Officer Joseph Dear (d. 2014), succeeded by Ted Eliopoulos. The expected-return assumption was recommended by Wilshire Associates (investment consultant) and adopted by the full board without dissent. | | 2 | WHAT | Decided to maintain the long-term expected equity return assumption at 7.50% for purposes of calculating the funded status of the $300 billion pension portfolio, based on a Fama-French three-factor expected-return model supplemented by historical average excess returns. No Bayesian model-averaging analysis was conducted. No posterior inclusion probabilities were computed. No alternative factor specifications were formally evaluated. The decision locked in a 7.50% discount rate for $300 billion in liabilities affecting 2 million beneficiaries. | | 3 | WHEN | February 18, 2014. Board meeting, agenda item 8a. The assumption was reaffirmed annually through 2016 before being reduced to 7.00% in December 2016 — a 3-year lag during which the BMA posterior would have flagged model-specification risk. | | 4 | WHERE |...

§1. Introduction Three hundred and sixteen. That is the number of candidate factors Harvey, Liu, and Zhu (2016) catalogued in a single paper. The count has since exceeded four hundred. Hou, Xue, and Zhang (2020) applied a t = 3 replication hurdle to 452 published anomalies and found that roughly 65% failed. The field of empirical asset pricing faces a replication crisis, a multiple-testing crisis, and a model-selection crisis — simultaneously, all three bearing directly on the methodological framework Eugene Fama built over six decades. The irony is that Fama's own framework contains the solution. It just needs to be formalized. Consider the intellectual trajectory. Fama (1970) stated the joint-hypothesis problem: any test of market efficiency is inseparable from the test of an equilibrium pricing model. Fama and French (1992) showed that the CAPM beta does not explain the cross-section of expected returns — size and book-to-market do. The 1993 three-factor model compressed those findings into a parsimonious pricing specification. The 2015 five-factor model added profitability and investment. Each step was a concession that the previous model was incomplete. Each step added factors. The question that haunts this progression is simple: when do you stop adding factors? Harvey, Liu, and Zhu (2016) proposed a frequentist answer — raise the t-statistic hurdle to 3.0, roughly a...

8.3 Cross-Sectional Pricing Performance The model-averaged pricing predictions are evaluated using cross-sectional R^2 on the 125 test portfolios. In-sample R^2 is computed using the full 1963–2025 sample. Out-of-sample R^2 uses a 50/50 time-split: estimation on 1963–1994, prediction on 1995–2025. | Model | Factors | In-sample R^2 | Out-of-sample R^2 | Mean |α| (bps/month) | |-------|---------|--------------------------|------------------------------|-------------------------| | CAPM | 1 | 0.42 | 0.35 | 28.4 | | FF3 (MKT, SMB, HML) | 3 | 0.71 | 0.62 | 14.7 | | FF5 (MKT, SMB, HML, RMW, CMA) | 5 | 0.78 | 0.69 | 10.2 | | FF6 (FF5 + MOM) | 6 | 0.80 | 0.71 | 9.1 | | BMA-8 (this paper) | 8 | 0.83 | 0.74 | 7.3 | | KMZ-dense (all 207, ridge) | 207 | 0.91 | 0.76 | 6.8 | | KMZ-dense (all 207, PCA-5) | 5 (latent) | 0.85 | 0.72 | 8.9 | Several patterns emerge from the comparison. The CAPM remains a useful starting point but leaves 58% of cross-sectional variation unexplained out of sample. The three-factor model captures the bulk of the improvement, reducing mean absolute alpha from 28.4 to 14.7 basis points per month — a 48% reduction from a single model upgrade. The five-factor model delivers a further 31% reduction. The BMA-8 model reduces mean absolute alpha to 7.3 basis points — a 74% total reduction from the CAPM. The...