r"""
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 :math:`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%}")

# %%
# 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))

# %%
# 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()