Generalizing Markowitz Portfolio Optimization by a Quadratic Risk Measure

Ignas Gasparavi\v{c}ius and Andrius Grigutis

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Abstract: We show that the key optimization results of the classical Markowitz portfolio selection theory, originally formulated for variance as the risk measure, remain available in explicit closed form under a broader class of strictly convex quadratic risk measures. The proposed framework replaces the covariance matrix with an arbitrary symmetric positive definite matrix and allows additional linear and constant terms, thereby containing various models arising in transaction cost optimization, benchmark relative optimization, covariance regularization, and factor models. Closed-form formulas are obtained for the efficient frontier, the global minimum risk portfolio, the maximum Sharpe ratio portfolio, the Capital Market Curve, the tangency portfolio, and the maximum utility portfolio. In contrast to the classical Markowitz model, the tangency portfolio does not coincide with the maximum Sharpe ratio portfolio, revealing a new geometric phenomenon. A numerical example confirms the derived formulas.

Date: 2026-08
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