Lesson from Naïve Diversification About the Risk-Reward Trade-off

Studies of naïve diversification show that average total portfolio risk declines asymptotically as number of stocks increases. Recent work by Bennett and Sias (2011) and Haensly (2020) shows that a significant amount of idiosyncratic risk remains, even for portfolios with large numbers of stocks. The corresponding shocks are non-trivial. For example, more than half of all equal-weighted portfolios with 100 stocks have better than a 16 percent chance of an annual shock at least as large as about half of the annualized mean excess return on the U.S. total stock market index over July 1963–June 2018. I perform a simulation analysis of portfolio reward-to-risk as well as the components of total portfolio risk. The results confirm recent work that a significant amount of idiosyncratic risk remains even for portfolios as large as 500 stocks. I also show that, on average, investors do not appear to be rewarded for exposure to this non-systematic risk. The cross-sectional distribution of the true Sharpe ratio rises and its dispersion shrinks significantly as the number of stocks in the portfolio increases, whereas the cross-sectional distribution of the true non-systematic risk falls and its dispersion shrinks significantly as the number of stocks in the portfolio increases. This pattern (as number of stocks increases) is the opposite of what we would expect if exposure to non-systematic risk were compensated. In particular, relatively undiversified portfolios (those with small numbers of stocks) have higher levels of non-systematic risk that better diversified portfolios but exhibit lower reward for risk as measured by the Sharpe ratio. The pattern appears regardless of the true asset pricing model for generating security returns, the portfolio weighting method, or specification of security alphas