Asset Pricing Anomalies as Latent SDF Decomposition: A Formal Derivation of Multi-Factor Models

The Capital Asset Pricing Model (CAPM) predicts that a single market factor prices all assets. Empirically, this fails: value stocks (high book-to-market) earn ~5% annual premium, small-cap stocks earn ~3%, and momentum strategies earn ~8%. The Fama-French three-factor model captures these anomalies empirically but lacks theoretical foundation. We provide a formal derivation from the Latent framework. The stochastic discount factor (SDF) admits a spectral decomposition M = Σᵢ aᵢφᵢ where φᵢ are Latent basis functions with weights decaying as ρ⁻ᵏ. When the spectral gap parameter ρ > 1 but finite, higher-order components contribute non-negligibly: φ₁ corresponds to the market factor, φ₂ to value (HML), and φ₃ to size (SMB). We prove that three factors capture >98% of variance when ρ ≈ 2, matching the empirical range of 3-5 significant factors. The CAPM emerges as the ρ → ∞ limit. We formalize 17 theorems establishing multi-factor pricing, spectral weight decay, value and size premium bounds, factor number determination, and momentum as weight-shift dynamics.