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Capital structure is a firm's mix of debt and equity financing. Franco Modigliani and Merton Miller showed in 1958 that in perfect capital markets, with no taxes, transaction costs, or bankruptcy costs, a firm's total value does not depend on that mix (Proposition I). Debt is cheaper than equity, but adding debt makes equity riskier, so the cost of equity rises with leverage, r_E = r_U + (D/E)(r_U − r_D), and the weighted average cost of capital stays constant (Proposition II). Real-world capital structure choices therefore matter only because of the market imperfections the theorems assume away, chiefly the tax deductibility of interest and the costs of financial distress.
Work one firm through two financing plans, all equity and half debt, and compute the cash flows to investors in good and bad years. Show that an investor can recreate either plan's payoffs by borrowing or lending on personal account, which is the "homemade leverage" argument. Then add a corporate tax and compute the interest tax shield, so the departure from irrelevance is concrete.
From the weighted average cost of capital you know that a firm's financing mix appears in its discount rate. A natural question follows: can a firm raise its value just by changing that mix? Debt usually has a lower required return than equity, so it seems swapping equity for debt should lower the WACC and raise value.
Modigliani and Miller showed in 1958 that in perfect capital markets this reasoning fails. Proposition I says firm value depends only on the cash flows its assets generate, not on how those cash flows are divided between creditors and shareholders. The argument uses arbitrage. If a levered firm were worth more than an identical unlevered one, an investor could buy shares of the unlevered firm, borrow personally in the same proportion, and receive the same payoffs as a shareholder of the levered firm at lower cost. This homemade leverage keeps the two values equal.
Proposition II explains why cheap debt does not help. Debt holders have a prior claim, so adding debt shifts risk onto shareholders, who then demand a higher return. With no taxes, r_E = r_U + (D/E)(r_U − r_D). Take a firm with r_U = 10% that borrows half its value at 5%. The cost of equity rises to 15%, and the WACC stays at 0.5 × 15% + 0.5 × 5% = 10%.
The theorems matter because their assumptions point to the real reasons financing matters. The first is taxes. Interest is deductible from corporate income in many tax systems, so debt reduces taxes. In their 1963 paper, Modigliani and Miller showed that with permanent debt D and corporate tax rate τ_c, firm value becomes V_U + τ_c D. The second is financial distress. Heavy debt raises the risk of bankruptcy, with legal costs, lost customers, and poor investment decisions under pressure. The tradeoff theory says firms choose debt where the marginal tax benefit equals the marginal expected distress cost. The pecking order theory of Stewart Myers and Nicolas Majluf (1984) adds that managers know more than outside investors, so firms prefer internal funds, then debt, and issue equity last.
Modigliani received the Nobel Memorial Prize in Economic Sciences in 1985 and Miller in 1990, both partly for this work. The same no-arbitrage logic underlies the dividend irrelevance result.