Risk & Portfolio

Two assets, each 20% volatile, held 50/50. At a correlation of 0.2 the portfolio’s volatility is 15.5%. If the correlation rises to 0.9 in a crisis it is 19.5% — almost the volatility of either asset alone. The diversification you counted on in calm markets is mostly gone exactly when losses come.

• Correlations measured in high-volatility periods are biased upwards even if the underlying link has not changed (Forbes & Rigobon 2002); some of the “breakdown” is a measurement effect.
• The rise is real in the tails, though: equity markets are more correlated in big down-moves than in big up-moves (Longin & Solnik 2001; Ang & Chen 2002).
• One number cannot describe dependence in the tails; that is what copulas and stress scenarios are for.

vol = np.array([0.20, 0.20])                               # two assets, 20% volatility each
w = np.array([0.5, 0.5])

def port_vol(rho):
    cov = np.outer(vol, vol) * np.array([[1, rho], [rho, 1]])
    return np.sqrt(w @ cov @ w)

calm, stress = port_vol(0.2), port_vol(0.9)

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.