Maillard-Roncalli-Teiletche risk parity

An equally weighted stock-bond portfolio takes nearly all its risk from the stocks. Sébastien Maillard, Thierry Roncalli and Jérôme Teiletche (2010) studied the equal risk contribution portfolio, in which each asset’s Euler contribution \(w_i(\Sigma w)_i/\sigma_p\) to the portfolio volatility is the same. It always exists and is unique, and its volatility lies between the minimum-variance and 1/N portfolios’. It is computed here from Spinu’s convex reformulation.

import matplotlib.pyplot as plt
import numpy as np

from financialkit.portfolio import equal_weight, min_variance_portfolio, risk_parity
from financialkit.risk import euler_allocation

names = ["govt bonds", "credit", "equities", "commodities"]
vol = np.array([0.05, 0.08, 0.17, 0.22])
corr = np.array([[1.0, 0.5, -0.2, 0.0], [0.5, 1.0, 0.4, 0.2], [-0.2, 0.4, 1.0, 0.4], [0.0, 0.2, 0.4, 1.0]])
cov = corr * np.outer(vol, vol)

Weights and risk contributions

ports = {
    "1/N": equal_weight(4),
    "minimum variance": min_variance_portfolio(cov, long_only=True).weights,
    "risk parity": risk_parity(cov).weights,
}
fig, axes = plt.subplots(1, 2, figsize=(10, 3.8))
bottom_w, bottom_r = np.zeros(3), np.zeros(3)
for i, name in enumerate(names):
    w = np.array([p[i] for p in ports.values()])
    rc = np.array([euler_allocation(p, cov, measure="volatility").shares[i] for p in ports.values()])
    axes[0].bar(list(ports), w, bottom=bottom_w, label=name)
    axes[1].bar(list(ports), rc, bottom=bottom_r, label=name)
    bottom_w += w
    bottom_r += rc
axes[0].set_title("capital")
axes[1].set_title("risk contribution")
axes[1].legend(fontsize=8, loc="upper left", bbox_to_anchor=(1, 1))
fig.tight_layout()
for label, w in ports.items():
    print(f"{label:>16}: vol {np.sqrt(w @ cov @ w):.2%}")

             1/N: vol 9.17%
minimum variance: vol 4.46%
     risk parity: vol 5.73%

Risk budgets

Unequal budgets give each asset a chosen share of the risk; the contributions match them exactly.

budgets = np.array([0.4, 0.2, 0.3, 0.1])
w = risk_parity(cov, budgets).weights
print("weights:", np.round(w, 3), " risk shares:", np.round(euler_allocation(w, cov, measure="volatility").shares, 3))

weights: [0.637 0.158 0.156 0.049]  risk shares: [0.4 0.2 0.3 0.1]

Risk parity and correlation

With two assets, ERC is inverse volatility whatever the correlation; with more, correlations matter. Here the equity-bond correlation is varied.

rhos = np.linspace(-0.6, 0.8, 30)
W = []
for r in rhos:
    c = corr.copy()
    c[0, 2] = c[2, 0] = r
    W.append(risk_parity(c * np.outer(vol, vol)).weights)
fig, ax = plt.subplots(figsize=(7, 3.5))
ax.stackplot(rhos, np.array(W).T, labels=names)
ax.set_xlabel("bond-equity correlation")
ax.set_ylabel("weight")
ax.set_ylim(0, 1)
ax.legend(fontsize=8, loc="upper left", bbox_to_anchor=(1, 1))
fig.tight_layout()
plt.show()