Portfolio Mathematics with General Linear and Quadratic Constraints

Abstract

Optimizing investment portfolios is one of the oldest research areas in finance. It has been studied most prolifically in the context of mean/variance optimization problems. Modern investment management is facilitated primarily by delegated managers in principal/agent relationships. Agents optimize not in mean and variance but in excess return above a benchmark and tracking error to a benchmark. Although the mathematics behind such relationships are similar to the mean/variance problem, the differences are subtle and significant. In this paper, I derive the general portfolio optimization problems with a constraint to tracking error (a quadratic constraint), the most pervasive constraint placed on delegated portfolio managers. I also analyze the general linear constraint when applied to Sharpe Ratio maximization. General Linear and Quadratic Constraints Abstract Optimizing investment portfolios is one of the oldest research areas in finance. It