Common Pricing, Different Portfolios

We propose a mode-space formulation in which equilibrium pricing and portfolio choice are organized around a common benchmark rather than treated as separate layers. Let returns be resolved into orthogonal modes \(c_k\) with per-mode premiums \(\pi_k\), variances \(v_k = \mathrm{Var}(c_k)\), and market weights \(w_k^{\mathrm{mkt}}\). The strongest static restriction used in the main theorem is \[ \pi_k = \lambda \, w_k^{\mathrm{mkt}} v_k \] and the mean-variance first-order condition \[ w_k^* = \frac{\pi_k}{\lambda v_k} \] are inverse readings of the same identity, implying the fully harvested benchmark allocation \[ w_k^{\infty} = w_k^{\mathrm{mkt}}. \] The static theorem therefore says that, in equilibrium, the market portfolio is the common benchmark exposure in mode space.

Paired with companion lifecycle results from the same research program, this benchmark admits the intertemporal reading \[ w_k^{\mathrm{life}} = w_k^{\infty} \cdot \mathbb{E}[h_k] \cdot b_k \cdot s_k = w_k^{\mathrm{mkt}} \cdot \mathbb{E}[h_k] \cdot b_k \cdot s_k, \] where \(\mathbb{E}[h_k]\) is expected fin_harvestability at the investor's effective horizon, \(b_k\) is a Bayesian caution term, and \(s_k\) is a safety multiplier. The central economic implication is that investors can share one equilibrium pricing law and still rationally hold different portfolios, because what differs across investors is not necessarily priced return but fin_harvestability.

The formal spine is machine-verified in Lean 4. It establishes mode-wise variance and premium decomposition, diagonal Markowitz optimality, the pricing-allocation equivalence, and derived CCAPM, SDF, and Arrow-Debreu routes into a broader family of mode-specific prices of risk \(\lambda_k\), with explicit bridge theorems for when that family collapses to the scalar case. The paper's boundary is equally clear: it does not yet identify the canonical market basis, prove universal scalar collapse, or claim empirical validation. Its contribution is structural and economic at once: it identifies the common equilibrium benchmark and shows how lifecycle, horizon, and bequest effects can be interpreted as investor-specific harvesting filters layered on top of that benchmark.