ABSTRACT

We derive sharp bounds on the sensitivity of Markowitz-optimal portfolio weights to changes in expected returns. Unlike the seminal Best–Grauer (1991) bounds, our results remain valid under arbitrary convex constraints, and the analysis does not require an invertible covariance matrix. The bounds remain governed by extremal eigenvalues and unify stability analysis across constrained and unconstrained settings. We further show that the standard budget and nonnegativity constraints induce an implicit ℓ1-regularization, yielding sparse and stable portfolio solutions. These findings clarify the structural drivers of portfolio instability and reinforce the theoretical rationale for constrained optimization and shrinkage in noisy, data-limited environments. – with Elfred John C. Abacan and Maria Margarita Debuque-Gonzales