This paper revisits the cross-sectional approach to the performance analysis of multifactor investment strategies. Its main contributions are threefold: first, the use of a cross-sectional projection of asset returns onto the factor portfolio weights to form approximate portfolio returns; second, the introduction of nonlinear interaction terms between factors that faithfully reproduce the investment portfolio construction; and third, a natural and intuitive decomposition of the portfolio performance as the sum of factor contributions. The method we propose has several advantages over other time-series-based or general cross-sectional regression models: it faithfully reflects the current state of the investment portfolio, it is parsimonious in the number of explanatory variables, it leads to an approximation of the portfolio returns that has a small residual error and it provides a straightforward interpretation of the portfolio performances in terms of the factors it is designed from. The method we advocate is first presented and explained in detail, and then concrete applications to multifactor equity strategies are presented.