Risk & Portfolio

A 60/40 portfolio of stocks (16% volatile) and bonds (5%) looks balanced by money, but stocks supply 94% of its risk. Weighting by inverse volatility gives 24% stocks, 76% bonds, with each contributing 50% of the risk. That portfolio is much less volatile, so to match 60/40’s risk it needs about 1.77× leverage — which is where the bond leverage in risk parity comes from.

• Inverse volatility only equalizes risk for two assets or when correlations are equal; with more assets you need to solve for equal contributions.
• It relies on leverage being cheap and available; the bet is that low-volatility assets pay more per unit of risk (Asness, Frazzini & Pedersen 2012).
• When bonds and stocks fall together, as in 2022, the “balance” offers little protection.

vol = np.array([0.16, 0.05])                               # stocks, bonds
cov = np.outer(vol, vol) * np.array([[1, 0.1], [0.1, 1]])

def contributions(w):                                      # each asset's share of portfolio variance
    return w * (cov @ w) / (w @ cov @ w)

sixty_forty = contributions(np.array([0.6, 0.4]))
w_rp = (1 / vol) / (1 / vol).sum()                         # inverse volatility: equal risk for two assets
parity = contributions(w_rp)
rp_vol = np.sqrt(w_rp @ cov @ w_rp)
leverage = np.sqrt(np.array([0.6, 0.4]) @ cov @ np.array([0.6, 0.4])) / rp_vol   # to match 60/40's risk

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.