MM Proposition II: Leverage and the Cost of Equity

Modigliani-Miller's first proposition says that in a frictionless world, how a company splits its financing between debt and equity does not change the total value of the firm. That is a statement about the whole pie. Proposition II asks the next question: if the pie's total value is fixed, what happens to the slice that goes to shareholders as the company borrows more?

The answer is that the cost of equity rises, mechanically, with leverage. Debt does not make the firm riskier — MM I already ruled that out — but it does make the equity riskier, because debt holders get paid first. Equity absorbs all of the firm's operating risk on top of a fixed claim ahead of it, so as debt grows, that risk gets concentrated onto a shrinking sliver of equity.

The formula, one piece at a time

$$r_E = r_U + (r_U - r_D) \times \frac{D}{E}$$

Reading it left to right: $r_E$ is the cost of equity — what shareholders require. $r_U$ is the unlevered cost of capital, the return the business would need to offer if it had no debt at all; think of it as the pure operating risk of the assets. $r_D$ is the cost of debt. $D/E$ is the debt-to-equity ratio.

In plain English: the cost of equity equals the unlevered cost of capital, plus a risk premium that grows in direct proportion to how much debt sits ahead of equity. The premium term, $(r_U - r_D) \times D/E$, is the extra compensation shareholders demand for taking on the financial risk that leverage adds to the underlying business risk.

Worked example

A firm's assets would require a 9% return if financed entirely with equity ($r_U = 9\%$). It can borrow at 5% ($r_D = 5\%$). Currently it has $40 million of debt and $60 million of equity, so $D/E = 40/60 = 0.667$.

$$r_E = 9\% + (9\% - 5\%) \times 0.667 = 9\% + 2.67\% = 11.67\%$$

Shareholders now require 11.67%, not 9%, purely because of the leverage — nothing about the underlying business changed. If the firm doubles its borrowing to $80 million against $20 million of equity, $D/E$ jumps to 4.0, and $r_E = 9\% + 4\% \times 4.0 = 25\%$. The equity has become far riskier even though the firm's assets are identical.

What this means in practice

Two things worth checking against intuition. First, WACC itself does not move as leverage changes in this frictionless setting — the rising cost of equity exactly offsets the extra weight of cheap debt, so the blended cost of capital stays at $r_U$. Second, once you add real-world frictions like the interest tax shield, the line gets a downward kink: WACC actually falls with leverage up to a point, because debt is subsidized by the tax deduction, even as $r_E$ keeps rising for the same reason as before.