The geometry of higher order modern portfolio theory

Battle Count: 102

The paper provides important theoretical foundations for higher-order portfolio optimization, which is relevant to quantitative trading strategies that go beyond mean-variance optimization. Understanding the geometry of critical points and feasible sets is crucial for portfolio construction. However, the paper is purely mathematical with no direct trading signals, backtesting, or implementation guidance. Its relevance is foundational rather than immediately actionable for practitioners.

The paper is highly theoretical, requiring advanced knowledge of algebraic geometry (Bézout's theorem, discriminant loci, Jacobian matrices, homogenization, Vandermonde matrices). Implementing the theoretical results would require symbolic computation tools (Macaulay2) and significant mathematical expertise. The degree of the feasible portfolio variety grows rapidly (d·(d-1)·...·(d-n+2)), making computation intractable for large portfolios. No practical implementation code is provided.

The paper is purely theoretical with complete proofs. All computations are verifiable using standard algebraic geometry tools (Macaulay2 is referenced). Theorems are stated with precise conditions. However, there is no code repository provided, and the paper does not include numerical experiments on real data. The mathematical derivations are self-contained and reproducible by a reader with algebraic geometry background.