Avoiding the Downside: A Practical Review of the Critical Line Algorithm for Mean–Semivariance Portfolio Optimization

Abstract

Optimizing a portfolio to reduce exposure to downside risk can be difficult, and usually involves third or higher order statistical moments of the portfolio’s return distribution. Mean–semivariance optimization simplifies this problem by using only the first two moments of the distribution and by penalizing returns below a predetermined reference. Although this penalty introduces a nonlinearity, mean–semivariance optimization can be performed easily and efficiently using the critical line algorithm (CLA) provided that the covariance matrix is estimated from a historical record of asset returns. In practice, this proviso is not restrictive. This chapter reviews the theory of the CLA and presents sample computer code for applying the algorithm to mean–variance and mean–semivariance portfolio optimization. It also reviews a method for finding the efficient mean–semivariance portfolio for any given feasible desired expected portfolio return.