We define debt ratio as the market value of a firm’s debt divided by the market value of the firm. In a perfect market with corporate taxes, given that the cost of debt is increasing and concave up and that the firm rebalances its debt, the cost of equity is an increasing and concave up function of the debt ratio if and only if the third derivative of the cost of debt is non-negative; otherwise, the cost of equity is increasing but its exact shape cannot be ascertained. In all cases, however, the cost of equity must be concave up initially. Also in this world, the weighted average cost of capital of the firm, WACC, is decreasing and concave down. In an imperfect market, the WACC may not have an absolute minimum between zero and 100 percent debt. Even if it does, the minimum may not occur at the debt ratio that maximizes firm value. The “pure-play” method to determine a new project’s discount rate is correct only if the opportunity cost of capital of the comparable firm remains constant with respect to the debt ratio or if the debt ratio of the comparable firm is equal to the target debt ratio of the firm evaluating the project. Strictly speaking, even if the two debt ratios are the same, the opportunity cost of capital of the comparable firm is not necessarily equal to that of the project unless the two costs of capital are identical functions of the debt ratio. Therefore, this method may not be valid.

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