r"""
Tobin separation and the tangency portfolio
===========================================

James Tobin (1958) added a riskless asset to Markowitz's problem.
Combinations of cash and any risky portfolio lie on a straight line, and
the steepest such line touches the frontier at the *tangency portfolio*.
Every mean-variance investor holds that same mix of risky assets and
differs only in how much cash goes with it: the risky choice separates
from the attitude to risk. The line is the capital market line, and its
slope is the highest attainable Sharpe ratio.
"""

# %%
import matplotlib.pyplot as plt
import numpy as np

from financialkit.portfolio import efficient_frontier, tangency_portfolio, two_fund_allocation
from financialkit.portfolio.visualizers import plot_frontier, plot_weights

names = ["bonds", "credit", "US equity", "intl equity", "emerging"]
mu = np.array([0.03, 0.045, 0.07, 0.075, 0.09])
vol = np.array([0.05, 0.08, 0.16, 0.18, 0.25])
corr = np.array(
    [
        [1.0, 0.6, -0.1, -0.1, 0.0],
        [0.6, 1.0, 0.3, 0.3, 0.4],
        [-0.1, 0.3, 1.0, 0.8, 0.7],
        [-0.1, 0.3, 0.8, 1.0, 0.75],
        [0.0, 0.4, 0.7, 0.75, 1.0],
    ]
)
cov = corr * np.outer(vol, vol)
rf = 0.02

# %%
# The capital market line
# -----------------------

t = tangency_portfolio(mu, cov, rf)
ax = plot_frontier(efficient_frontier(mu, cov, 60, max_return=0.12), labels=names, rf=rf)
print(f"tangency portfolio: mean {t.expected_return:.2%}, volatility {t.volatility:.2%}, Sharpe {t.sharpe_ratio(rf):.3f}")

# %%
# Three investors, one risky portfolio
# ------------------------------------
#
# The optimal allocation :math:`\Sigma^{-1}(\mu - r_f)/\gamma` scales with
# risk tolerance; normalized, it is the tangency portfolio every time.

fig, ax = plt.subplots(figsize=(7, 3.5))
for i, gamma in enumerate([4.0, 8.0, 16.0]):
    res = two_fund_allocation(mu, cov, rf, gamma)
    ax.bar(np.arange(6) + 0.27 * (i - 1), np.append(res.weights, 1 - res.weights.sum()), width=0.27, label=f"gamma = {gamma:g}")
    print(f"gamma = {gamma:g}: risky mix {np.round(res.weights / res.weights.sum(), 3)}, cash {1 - res.weights.sum():+.2f}")
ax.set_xticks(range(6), names + ["cash"])
ax.axhline(0, color="0.3", lw=0.8)
ax.set_ylabel("weight")
ax.legend()
fig.tight_layout()

# %%

plot_weights(t.weights, labels=names).set_title("tangency portfolio")
plt.tight_layout()
plt.show()