r"""
Williams's dividend discount model
==================================

John Burr Williams (1938) defined the *investment value* of a stock as
the present value of all the dividends it will ever pay,

.. math::

   V_0 = \sum_{t=1}^{\infty} \frac{D_t}{(1 + r)^t}.

Earnings matter only through the dividends they eventually fund. In
practice an analyst forecasts a few years of dividends explicitly and
then assumes steady growth, so most of the value often sits in the
terminal value beyond the forecast horizon.
"""

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

from financialkit.asset_pricing import dividend_discount_value

r = 0.09
# Five years of fast growth from a dividend of 1, then 3% forever.
divs = 1.0 * 1.12 ** np.arange(1, 6)
value = dividend_discount_value(divs, r, terminal_growth=0.03)
explicit = dividend_discount_value(divs, r)
print(f"value = {value:.2f}, of which explicit dividends {explicit:.2f} ({explicit / value:.0%})")

# %%
# Where the value comes from
# --------------------------
#
# Truncating the sum after :math:`n` years shows how slowly the present
# value of the dividends accumulates: decades of dividends are needed to
# reach the full value.

years = np.arange(1, 151)
path = np.concatenate([divs, divs[-1] * 1.03 ** np.arange(1, years.size - divs.size + 1)])
partial = np.array([dividend_discount_value(path[:n], r) for n in years])
fig, ax = plt.subplots(figsize=(7, 4))
ax.plot(years, partial, label="PV of the first n dividends")
ax.axhline(value, color="C3", ls="--", label=f"Williams's value = {value:.2f}")
ax.set_xlabel("years of dividends counted")
ax.set_ylabel("present value")
ax.legend()
fig.tight_layout()

# %%
# Sensitivity to the discount rate
# --------------------------------

rates = np.linspace(0.06, 0.14, 50)
fig, ax = plt.subplots(figsize=(7, 3.5))
ax.plot(rates, [dividend_discount_value(divs, k, terminal_growth=0.03) for k in rates])
ax.set_xlabel("required return r")
ax.set_ylabel("value")
fig.tight_layout()

plt.show()