The effect of diversification on firm performance has been debated. We reexamine the effect using a sample of 44,248 observations of non-financial US firms for the 1997–2009 period employing the quantile regression approach. Our empirical results show that the effect of diversification on firm performance is not homogeneous across various quantile levels: the diversification discount (premium) shows up in firms with high (low) RoE quantiles. Further, we find that diversification affects firm risk as well. Therefore, we consider a risk-adjusted performance measure and find that both diversification discount and premium disappear, which is consistent with the risk-return trade-off principle.

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.