The Sherman-Morrison-Markowitz Portfolio
Steven E. Pav
TL;DR
The paper develops the Sherman-Morrison-Markowitz portfolio, showing that replacing the covariance with the second moment in multi-period settings yields an optimal direction proportional to $\mathsf{A}_2^{-1}(x)\boldsymbol{\mu}(x)$, with a scalar downscale given by constraints and risk budgets. It unifies unconditional (multi-period) Sharpe optimization with classical mean-variance goals through Sherman-Morrison identities, introduces Hansen ratio as a natural objective, and extends the theory to discrete features, basis portfolios, and linear conditioning models. Key contributions include a closed-form form for the unconstrained and constrained optimal allocations, a Pythagorean decomposition of squared Hansen ratios under hedging constraints, and several practical guidance items such as neural-net estimation strategies and leverage analysis. Overall, the Sherman-Morrison-Markowitz portfolio provides a principled, down-levered alternative to Markowitz with modest practical gains, particularly when conditioning on features and when using uncentered moments in multi-period optimization. The work highlights avenues for estimation, bounds on performance gaps, and scalable implementations in linear and neural-network based models.
Abstract
We show that the Markowitz portfolio is a scalar multiple of another portfolio which replaces the covariance with the second moment matrix, via simple application of the Sherman-Morrison identity. Moreover it is shown that when using conditional estimates of the first two moments, this "Sherman-Morrison-Markowitz" portfolio solves the standard unconditional portfolio optimization problems. We argue that in multi-period portfolio optimization problems it is more natural to replace variance and covariance with their uncentered counterparts. We extend the theory to deal with constraints in expectation, where we find a decomposition of squared effects into spanned and orthogonal components. Compared to the Markowitz portfolio, the Sherman-Morrison-Markowitz portfolio downlevers by a small amount that depends on the conditional squared maximal Sharpe ratio; the practical impact will be fairly small, however. We present some example use cases for the theory.