r"""
Leland's endogenous default and optimal leverage
================================================

Leland (1994) priced perpetual debt on a firm whose coupons are tax
deductible and whose bankruptcy destroys half its assets. Shareholders
keep paying coupons as long as equity is worth something and default at
the asset level :math:`V_B` where equity hits zero with zero slope.
Firm value, unlevered assets plus tax shields minus expected bankruptcy
costs, is maximised at a coupon that sets the optimal leverage; in
Leland's base case this is about 75% with a 75 bp spread.
"""

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

from financialkit.credit import leland_model, leland_optimal_coupon

r, sigma, tau, alpha = 0.06, 0.20, 0.35, 0.50
C_star, best = leland_optimal_coupon(100.0, r, sigma, tau, alpha)
print(f"optimal coupon {C_star:.2f}: leverage {best.leverage:.1%}, spread {1e4 * best.credit_spread:.0f} bp, V_B = {best.default_barrier:.1f}")

# %%
# Equity is an option that shareholders exercise by defaulting
# ------------------------------------------------------------

V = np.linspace(30, 160, 300)
fig, ax = plt.subplots(figsize=(7, 4))
for VB, ls in ((best.default_barrier, "-"), (0.8 * best.default_barrier, "--"), (1.2 * best.default_barrier, ":")):
    E = [leland_model(v, C_star, r, sigma, tau, alpha, VB).equity if v > VB else 0.0 for v in V]
    ax.plot(V, E, ls=ls, label=f"default at V_B = {VB:.1f}")
ax.axhline(0, color="0.5", lw=0.8)
ax.set_xlabel("asset value V")
ax.set_ylabel("equity")
ax.set_ylim(-10, 80)
ax.set_title("only the endogenous barrier meets zero smoothly")
ax.legend()

# %%
# The trade-off of capital structure
# ----------------------------------

coupons = np.linspace(0.2, 14, 120)
res = [leland_model(100.0, c, r, sigma, tau, alpha) for c in coupons]
fig, ax = plt.subplots(1, 2, figsize=(11, 4))
ax[0].plot(coupons, [x.firm_value for x in res], label="firm value")
ax[0].plot(coupons, [x.debt for x in res], label="debt")
ax[0].plot(coupons, [x.equity for x in res], label="equity")
ax[0].axvline(C_star, color="k", ls=":")
ax[0].set_xlabel("coupon")
ax[0].legend()
ax[1].plot([x.leverage for x in res], [1e4 * x.credit_spread for x in res])
ax[1].axvline(best.leverage, color="k", ls=":")
ax[1].set_xlabel("leverage")
ax[1].set_ylabel("credit spread (bp)")

plt.show()