r"""
Hicks: duration as an elasticity (modified duration)
=====================================================

John Hicks (1939) found Macaulay's average period again as the
elasticity of a capital value with respect to the discount factor. Measured
per unit of yield, that elasticity is *modified duration*,

.. math:: D_{mod} = -\frac{1}{P}\frac{dP}{dy} = \frac{D_{mac}}{1 + y/m},

for a yield compounded :math:`m` times a year. The script checks the
relation across compounding frequencies and shows that :math:`-D_{mod}`
is the slope of the log-price curve.
"""

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

from financialkit.fixed_income import bond_cash_flows, macaulay_duration, modified_duration, price_from_yield

cf = bond_cash_flows(100.0, 0.05, 15.0, freq=2)
y0 = 0.05

# %%
# Macaulay and modified duration by compounding frequency
# -------------------------------------------------------

for m in (1, 2, 4, 12, None):
    d_mac = macaulay_duration(cf, y0, m)
    d_mod = modified_duration(cf, y0, m)
    ratio = 1.0 if m is None else 1 + y0 / m
    print(f"m={str(m):>4}: D_mac={d_mac:.4f}  D_mod={d_mod:.4f}  D_mac/(1+y/m)={d_mac / ratio:.4f}")

# %%
# Modified duration is the slope of log-price against yield
# ---------------------------------------------------------

y = np.linspace(0.0, 0.10, 200)
log_p = np.log(price_from_yield(cf, y, 2))
d_mod = modified_duration(cf, y0, 2)
p0 = np.log(price_from_yield(cf, y0, 2))
fig, ax = plt.subplots()
ax.plot(100 * y, log_p, label="log price")
ax.plot(100 * y, p0 - d_mod * (y - y0), ls="--", label=f"slope $-D_{{mod}}$ = {-d_mod:.2f}")
ax.set_xlabel("yield, semiannual (%)")
ax.set_ylabel("log price")
ax.legend()
plt.show()