Analytical Solutions to the Constrained Markowitz Problem via Fixed Point Theory




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 Alex Shkolnik, Department of Statistics and Applied Probability, University of California Santa Barbara


Harry Markowitz transformed finance by framing the portfolio construction as a trade-off between the mean and variance of return. The classic Markowitz problem, as solved by every investor in the Capital Asset Pricing Model for instance, may be expressed in closed form. But when the portfolio weights face inequality constraints, as typically required for practical investments, a numerical optimizer must be used.

A standard example is one that prohibits "short sales" as in the original portfolio selection problem considered in Markowitz’s seminal paper from 1952. No general analytical results or closed form formulas for this classic optimization problem appear in the literature.


Building on a special case solved (or rather, guessed) in Clarke, De Silva & Thorley (2011), we develop an analytical formula for the solution to the Markowitz problem with no short sales. Our results make use of fixed point theory to characterize the solution in a way that reveals its geometric properties. When the covariance has an underlying low-dimensional factor structure, significant computational gains in run-time and accuracy are achieved. Moreover, our closed-form formulae allow us to study the composition and the sensitivities of the portfolio weights with respect to various model parameters including factor variances, idiosyncratic variances, and security-level factor exposures.

We present several examples relevant to investment practice illustrating our formulas and the associated algorithms.


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