Harry Markowitz transformed finance by framing the portfo- lio 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 nu- merical 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 an- alytical 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 computa- tional 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 rele- vant to investment practice illustrating our formulas and the asso- ciated algorithms. Theoretical aspects of our algorithms (a work in progress) are briefly discussed.
Clarke, R., De Silva, R. & Thorley, S. (2011), ‘Minimum-variance portfolio composition’, Journal of Portfolio Management 2(37), 31–45.