Risk & Portfolio

Stocks that are each 30% volatile, with an average correlation of 0.3, held in equal weights. One stock: 30%. Five: 19.9%. Twenty: 17.4%. A hundred: 16.6%. Ten thousand: 16.4%. The 1/√N rule only holds for uncorrelated assets; with correlation there is a floor — σ√ρ, here 16.4% — that no number of similar stocks removes.

• “15–20 stocks is enough” comes from the 1960s and 80s (Evans & Archer 1968; Statman 1987). As individual stocks became more volatile, more were needed for the same effect (Campbell et al. 2001).
• Volatility is not the only risk: a portfolio of 30 stocks can still miss the few big winners that drive index returns.
• Correlations rise in crises (see Correlation Risk), so the floor moves up when it matters.

sigma, rho = 0.30, 0.30                                   # each stock 30% volatile, average correlation 0.3
n = np.array([1, 5, 20, 100, 10_000])
port = sigma * np.sqrt(rho + (1 - rho) / n)                # equal weights
floor = sigma * np.sqrt(rho)                               # what no number of such stocks removes

Assumes import numpy as np and import pandas as pd. Each snippet carries its own example numbers; the comments say which are made up or simulated.