7.18 EXAMPLE: RISK-FREE FIRM DEBT

Suppose that an all-equity firm A has the state contingent cash flows shown in Table 7.4. We consider two capital structures. In the first, the firm has issued 5 risk-free bonds, while in the second, it has issued 10 such bonds.

Table 7.4

Cash Flows for Firm A

 State End-of-Period Cash Flows 1 2 3 Firm A 10 20 30 Capital Structure I Financial Claims on the Real Assets: Stock A 5 15 25 Risk-free Debt (5 units) 5 5 5 Capital Structure II Financial Claims on the Real Assets: Stock A 0 10 20 Risk-free Debt (10 units) 10 10 10

Under capital structure I (II) let the current market price of equity be EI (EII). Assume further that investors can buy or sell (i.e., short) risk-free bonds, such as Treasury bills, with a face value of \$1, and suppose the current market price of these bills is T.

Suppose that investors are willing to pay a premium for capital structure II, i.e. they feel that that value of the firm is greater under II. This means that

10T + EII > 5T + EI, or

EII > EI- 5T

At these prices, the management of firm A can increase A's market value by altering its capital structure.

If you were an investor in this market, you could achieve the same result at less cost. Consider purchasing some share a of the firm's existing stock and selling short 5a units of T-bills.

The cost of this portfolio to you is:

a(EI - 5T)

It is clear that this cost is strictly less than aEII even though the end-of-period cash flows are identical (in terms of timing, magnitude, and default risk). Therefore, neither you nor any other investor would ever pay a premium for capital structure II, which contradicts the initial assumption that capital structure matters.

An important by-product of the CAPM is that the model tells us how to discount risky cash flows. This capital budgeting implication is described in the topic Risk-Adjusted Discount Rates.

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