Optimal Impact Portfolios with General Dependence and Marginals

80 Pages Posted: 9 Aug 2022 Last revised: 29 Oct 2022

See all articles by Andrew W. Lo

Andrew W. Lo

Massachusetts Institute of Technology (MIT) - Laboratory for Financial Engineering; Santa Fe Institute

Lan Wu

Peking University

Ruixun Zhang

Peking University; MIT Laboratory for Financial Engineering

Chaoyi Zhao

Peking University

Date Written: July 31, 2022

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, which reduces to a correlation under normality assumptions. More generally, we explicitly derive the optimal portfolio weights under two widely-used copulas---the Gaussian copula and the Archimedean copula family, and find that the optimal weights depend on the tail characteristics of the copula. In addition, when the marginal distribution of residual returns is skewed or heavy-tailed, assets with the most extreme impact factors have lower weights than non-extreme assets due to their high risk. Our framework requires the estimation of only a constant number of parameters as the number of assets grow, an advantage over traditional Markowitz portfolios. Overall, these results provide a recipe for constructing and quantifying the performance of optimal impact portfolios with arbitrary dependence structures and return distributions.

Keywords: Impact Investing; Environmental, Social, and Governance (ESG); Portfolio Theory; Copula; Heavy Tail

JEL Classification: C10, C20, G11, G12

Suggested Citation

Lo, Andrew W. and Wu, Lan and Zhang, Ruixun and Zhao, Chaoyi, Optimal Impact Portfolios with General Dependence and Marginals (July 31, 2022). Available at SSRN: https://ssrn.com/abstract=4177277 or http://dx.doi.org/10.2139/ssrn.4177277

Andrew W. Lo

Massachusetts Institute of Technology (MIT) - Laboratory for Financial Engineering ( email )

100 Main Street
E62-618
Cambridge, MA 02142
United States
617-253-0920 (Phone)
781 891-9783 (Fax)

HOME PAGE: http://web.mit.edu/alo/www

Santa Fe Institute

1399 Hyde Park Road
Santa Fe, NM 87501
United States

Lan Wu

Peking University ( email )

No. 38 Xueyuan Road
Haidian District
Beijing, Beijing 100871
China

Ruixun Zhang (Contact Author)

Peking University ( email )

5 Yiheyuan Road
Beijing, Beijing 100871
China

HOME PAGE: http://www.math.pku.edu.cn/teachers/ZhangRuixun%20/index.html

MIT Laboratory for Financial Engineering

77 Massachusetts Ave. E62-663
Cambridge, MA 02142
United States

Chaoyi Zhao

Peking University ( email )

No. 5 Yiheyuan Road
Beijing
China

HOME PAGE: http://zhaochaoyi.github.io/

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