r"""
Jensen's alpha
==============

Michael Jensen (1968) measured a mutual fund's skill as the intercept
of its excess return regressed on the market's,

.. math::

   r_{pt} - r_{ft} = \alpha_p + \beta_p (r_{mt} - r_{ft}) + \varepsilon_{pt}.

A manager who takes more market risk earns more on average but no
alpha; only returns beyond the security market line count. Over 1945-1964
Jensen found the average alpha of 115 funds to be negative after fees,
and no more funds with significant positive alpha than chance allows.
"""

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

from financialkit.asset_pricing import jensen_alpha, simulate_factor_model

rng = np.random.default_rng(1968)
n_funds, T = 115, 240
betas = rng.uniform(0.7, 1.3, n_funds)
# Fees of 1% a year and no skill, except five genuinely skilled funds.
true_alpha = np.full(n_funds, -0.01 / 12)
true_alpha[:5] = 0.04 / 12
panel = simulate_factor_model(T, betas, [0.006], [[0.0020]], idio_vol=0.02, alphas=true_alpha, rng=rng)
res = jensen_alpha(panel.excess_returns, panel.factors[:, 0])
print(f"mean annual alpha: {12 * res.alphas.mean():+.2%}")
print(f"funds with alpha t > 2: {(res.alpha_t > 2).sum()} (true skilled: 5)")

# %%
fig, axes = plt.subplots(1, 2, figsize=(10, 4))
axes[0].hist(res.alpha_t, bins=25, color="0.7")
axes[0].axvline(2, color="C3", ls="--")
axes[0].set_xlabel("t-statistic of alpha")
axes[0].set_title("most funds: no skill after fees")
axes[1].scatter(res.betas[:, 0], 12 * res.alphas, s=10)
axes[1].scatter(res.betas[:5, 0], 12 * res.alphas[:5], s=25, color="C3", label="skilled funds")
axes[1].axhline(0, color="k", lw=0.8)
axes[1].set_xlabel("estimated beta")
axes[1].set_ylabel("annual alpha")
axes[1].set_title("alpha is unrelated to beta")
axes[1].legend()
fig.tight_layout()

plt.show()