Abstract:  We develop a mathematical framework for constructing optimal impact portfolios and quantifying their financial performance by characterizing the returns of impact-ranked assets using induced order statistics and copulas. The distribution of induced order statistics can be represented by a mixture of order statistics and uniformly distributed random variables, where the mixture function is determined by the dependence structure between residual returns and impact factors—characterized by copulas—and the marginal distribution of residual returns. This representation theorem allows us to explicitly and efficiently compute optimal portfolio weights under any copula. This framework provides a systematic approach for constructing and quantifying the performance of optimal impact portfolios with arbitrary dependence structures and return distributions.