Geometric portfolio equilibria: constraint curvature and asset pricing

Institutional investors optimise over feasible sets whose shape is dictated by regulation and mandate. This paper asks whether the intrinsic geometry of those sets, as distinct from the shadow cost of the constraints that define them, carries independent information for equilibrium prices. We answer the question in two steps. First, we show that the position limits and linear matching requirements most commonly invoked in this literature generate polyhedral feasible sets whose smooth strata are affine and therefore intrinsically flat, so that no curvature mechanism can be attributed to them; the correct analytical tools there are tangent and normal cones and Karush-Kuhn-Tucker multipliers. Second, we isolate a class of genuinely nonlinear constraints, of which the active boundary of a quadratic rebalancing budget is the leading case, whose transversal intersection with the wealth constraint is a smooth submanifold of dimension N-2 with non-vanishing sectional curvature. On that class we formulate a competitive equilibrium with price impact, prove existence and uniqueness under an explicit curvature domination condition, and derive a three-term decomposition of expected excess returns into a zero-beta rate, a classical covariance premium, and a curvature term that is written explicitly as a second fundamental form contracted with the multiplier vector. A fully specified four-asset example computes every object in the decomposition in closed form and shows that the curvature term vanishes when the constraint is replaced by its tangent hyperplane, which separates it from the shadow cost of a binding constraint. An empirical exercise on S&P 500 index funds documents a priced constraint-proximity factor that is not spanned by the Fama-French-Carhart factors; we interpret this factor as a measure of crowding near regulatory limits rather than as an estimate of Riemannian curvature, and we state the conditions under which the two coincide.