Portfolio Optimization with Factors, Scenarios, and Realistic Short Positions

Abstract

This paper presents fast algorithms for calculating mean-variance efficient frontiers when the investor can sell securities short as well as buy long, and when a factor and/or scenario model of covariance is assumed. Currently, fast algorithms for factor, scenario, or mixed (factor and scenario) models exist, but (except for a special case of the results reported here) apply only to portfolios of long positions. Factor and scenario models are used widely in applied portfolio analysis, and short sales have been used increasingly as part of large institutional portfolios. Generally, the critical line algorithm (CLA) traces out mean-variance efficient sets when the investor's choice is subject to any system of linear equality or inequality constraints. Versions of CLA that take advantage of factor and/or scenario models of covariance gain speed by greatly simplifying the equations for segments of the efficient set. These same algorithms can be used, unchanged, for the long-short portfolio selection problem provided a certain condition on the constraint set holds. This condition usually holds in practice.

Trimability and Fast Optimization of Long–Short Portfolios

“…. ., y K in terms of the real securities as follows: 10 (12) With this definition, the portfolio variance can be written (see Jacobs, Levy, and Markowitz 2005) in the form (13) where W k is the variance of f k . Equation 13 expresses V P as a positively weighted sum of squares in the n Scenario Models.…”

“…. ., y S as follows: (15) With this definition, the variance of the portfolio's return can be written (see Jacobs, Levy, and Markowitz 2005) as (16) where Thus, portfolio variance can be expressed as a positively weighted sum of squares in the n original securities and S new fictitious securities, which are linearly related to the original securities by Equation 15. Again, therefore, portfolio variance can be written in terms of a diagonal covariance matrix.…”

“…From this definition, it follows (see Jacobs, Levy, and Markowitz 2005) that the variance of the portfolio's return is (22) Equation 22 is the expression for the variance of the return of a long-short portfolio when a multifactor covariance model is assumed. Note that, with the exception of the cross-product terms, Equation 22 has exactly the same form as Equation 13, which applied exclusively to long-only portfolios.…”

“…Equation 18 for the historical model is analogous to Equations 13 and 16 for the factor and scenario models. However, the equation for the historical model, unlike those for the other models, contains only a term involving the fictitious securities, no term involving V i (see Jacobs, Levy, and Markowitz 2005). Therefore, Here, we are making no assumption about the constraint set or expected returns other than the background assumptions that the model is feasible (i.e., meets the specified constraints) and has efficient portfolios.…”

“…If the trimability condition holds in the original model, then each efficient (E P ,V P ) combination has one and only one trim portfolio with the same (E P ,V P ), although there may be efficient portfolios with this (E P ,V P ) that are not trim (see Jacobs, Levy, and Markowitz 2005). Also, if the trimability condition holds in the original model, then the modified model has the same set of efficient (E P ,V P ) combinations as does the original model, and it has a unique set of efficient portfolios [one for each efficient (E P ,V P ) combination] that is the same as the unique set of trim efficient portfolios in the original model.…”