Optimal Impact Portfolios with General Dependence and Marginals

Andrew Lo, Lan Wu, Ruixun Zhang, Chaoyi Zhao

Abstract: Impact investing typically involves ranking and selecting assets based on a non-financial impact factor, such as the environmental, social, and governance (ESG) score and the prospect of developing a disease-curing drug. We develop a framework for constructing optimal impact portfolios and quantifying their financial performances. Under general bivariate distributions of the impact factor and residual returns from a multi-factor asset-pricing model, the construction and performance of optimal impact portfolios depend critically on the dependence structure (copula) between the two. We derive a representation theorem to characterize the distribution of induced order statistics (returns of impact-ranked assets), which allows us to explicitly and efficiently compute the optimal portfolio weights under any copula. The optimal weights depend on the tail characteristics of the copula, as well as whether the marginal distribution of residual returns is skewed or heavy-tailed. Our framework requires the estimation of only a constant number of parameters as the number of assets grows, providing a more regularized and robust approach compared to traditional Markowitz portfolios.