We show theoretically that the usual estimated investment strategies will not achieve the optimal Sharpe ratio when the dimensionality is high relative to sample size, and the $ 1/N $ rule is optimal in a 1-factor model with diversifiable risks as dimensionality increases, which explains why it is difficult to beat the $ 1/N $ rule in practice. We also explore conditions under which it can be beaten, and find that we can outperform it by combining it with the estimated rules when $ N $ is small, and by combining it with anomalies or machine learning portfolios, conditional on the profitability of the latter, when $ N $ is large.

This paper examines the impact of the local opioid epidemic on corporate innovation. Utilizing a large sample of U.S. public firms from 2003 to 2017, we find that firms located in counties with higher death rates caused by opioid overdoses are significantly less innovative as measured by their patenting activities. To establish causality, we exploit the state implementations of the Prescription Drug Monitoring Programs (PDMPs) as quasi-experiments and an instrumented variable approach. We find suggestive evidence that the opioid epidemic hinders local firms' innovation by increasing healthcare costs, decreasing productivity, and through the exodus of local inventors.

We explore the evolution of financial innovation using US finance patents. Patented financial innovations are substantial and increasingly economically important. Their subject matter has changed, consistent with the industry’s shift toward household investors and borrowers. Information technology (IT) and other nonfinancial firms drove the surge in financial patenting. The location of innovation shifted, with banks moving activity away from states with tight financial regulation and high-tech regions attracting innovation by payments, IT, and nonfinancial firms. Analyses of returns suggest that the social value of these innovations is higher than their private value. We present a simple model to explain these trends.