Modigliani-Miller

Franco Modigliani and Merton Miller (1958) showed that, without taxes or frictions, a firm’s value does not depend on how it is financed (Proposition I). Debt is cheaper than equity, but borrowing makes the remaining equity riskier, and its required return rises exactly enough to keep the weighted average cost of capital constant (Proposition II):

\[r_E = r_A + (r_A - r_D)\,\frac{D}{E}.\]

In 1963 they added corporate tax: interest is deductible, so each unit of permanent debt adds a tax shield worth \(t D\).

import matplotlib.pyplot as plt
import numpy as np

from financialkit.asset_pricing import levered_cost_of_equity, levered_firm_value, wacc

rA, rD = 0.10, 0.05
L = np.linspace(0.0, 0.8, 100)  # debt / value
de = L / (1 - L)

fig, axes = plt.subplots(1, 2, figsize=(10, 4))
for t, ls in [(0.0, "-"), (0.3, "--")]:
    rE = levered_cost_of_equity(rA, rD, de, t)
    axes[0].plot(L, rE, "C3", ls=ls, label=f"r_E, t = {t:.0%}")
    axes[0].plot(L, wacc(rE, rD, L, t), "C0", ls=ls, label=f"WACC, t = {t:.0%}")
axes[0].axhline(rD, color="0.5", lw=1, label="r_D")
axes[0].set_xlabel("D / V")
axes[0].set_ylabel("required return")
axes[0].set_title("Proposition II")
axes[0].legend(fontsize=8)

<matplotlib.legend.Legend object at 0x116bed7f0>

Proposition I

VU = 100.0
D = np.linspace(0, 80, 50)
for t in (0.0, 0.2, 0.35):
    axes[1].plot(D, levered_firm_value(VU, D, t), label=f"t = {t:.0%}")
axes[1].set_xlabel("debt D")
axes[1].set_ylabel("firm value V_L")
axes[1].set_title("Proposition I: V_L = V_U + t D")
axes[1].legend()
fig.tight_layout()

plt.show()