Sensitivity Analysis of Portfolio Properties with Budget Constraints

In this paper, we develop a framework that allows one to understand how variations in a portfolio property are apportioned to changes in portfolio composition as trading is performed. This investigation requires a new mathematical setting. On the one hand, partial derivatives, traditionally employed as portfolio sensitivity measures, do not allow to embed the sensitivity in the context of the trading strategy. On the other hand, the budget constraint turns the analysis into a constrained sensitivity problem. We propose a general result that allows to solve the pivotal asset choice problem for any budget constraint. In addition, we formulate our approach so that the simultaneous sensitivity of portfolio properties w.r.t. assets groups becomes straightforward. Analytical expressions for the sensitivity of portfolio returns, variance, GARCH volatility and VaR are derived. A numerical exemplification is proposed with reference to the 30 assets of the Dow Jones Index.