r"""
Grinold's fundamental law of active management
==============================================

Richard Grinold (1989) decomposed an active manager's information ratio
into skill and breadth:

.. math::

   IR \approx IC \sqrt{BR},

where the information coefficient :math:`IC` is the correlation between
forecasts and outcomes and the breadth :math:`BR` is the number of
independent forecasts a year. A tiny edge applied to many independent
bets beats a large edge applied to a few: an IC of 0.05 on 400 stocks
gives the same IR as an IC of 0.5 on four markets.
"""

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

from financialkit.portfolio import fundamental_law, simulate_active_management

# %%
# Simulated managers against the law
# ----------------------------------
#
# Each manager makes ``breadth`` independent bets a year for 400 years.
# The exact IR of this model, :math:`IC\sqrt{BR}/\sqrt{1 + IC^2}`, is
# barely below Grinold's first-order law.

breadths = np.array([4, 16, 64, 256, 1024])
fig, ax = plt.subplots(figsize=(7, 4))
for k, ic in enumerate([0.02, 0.05, 0.1]):
    sims = [simulate_active_management(ic, int(b), 400, rng=k * 10 + i) for i, b in enumerate(breadths)]
    ax.errorbar(breadths, [s.ir for s in sims], yerr=[2 * s.ir_se for s in sims], fmt="o", color=f"C{k}", capsize=3)
    bb = np.logspace(0.5, 3.1, 100)
    ax.plot(bb, fundamental_law(ic, bb), color=f"C{k}", label=f"IC = {ic}")
ax.set_xscale("log")
ax.set_xlabel("breadth (independent bets per year)")
ax.set_ylabel("information ratio")
ax.legend()
fig.tight_layout()

# %%
# Same IR, different businesses
# -----------------------------

for ic, br in [(0.5, 4), (0.1, 100), (0.05, 400), (0.02, 2500)]:
    res = simulate_active_management(ic, br, 2000, rng=1)
    print(f"IC {ic:<4} x sqrt({br:>4}) = {fundamental_law(ic, br):.2f}; simulated IR {res.ir:.2f} +/- {res.ir_se:.2f}")
plt.show()