The Cost of Capital, Corporation Finance and the Theory of Investment

Questions & answers

Q1. What is wrong, in the authors’ telling, with how the “cost of capital” had previously been treated by economic theorists, and what alternative approach do they adopt?

The authors argue that the standard theorist’s approach – treating physical assets as yielding “known, sure streams,” which makes the cost of capital simply the bond rate of interest regardless of financing mix – “has admittedly been useful in dealing with some of the grosser aspects” of investment and capital accumulation but has “little descriptive value” once uncertainty is taken seriously, since under uncertainty “the profit outcome…has become a random variable” and profit maximization “no longer has an operational meaning.” Rather than extend the certainty framework into a subjective-utility approach (which the authors say has “serious drawbacks” because the cost of capital becomes dependent on the unobservable risk preferences of whichever owners happen to hold the stock), they adopt the market-value-maximization criterion: “any investment project and its concomitant financing plan must pass only the following test: will the project, as financed, raise the market value of the firm’s shares?” – a test that is independent of current owners’ particular tastes, since market prices reflect the preferences of all potential owners.

Q2. What are the paper’s key simplifying assumptions, and what is the “homemade leverage” mechanism underlying Proposition I?

The analysis groups firms into equivalent-risk classes (firms whose income streams are perfectly correlated, differing only by a scale factor), assumes perfect capital markets (investors can borrow and lend at the same rate as corporations, with no transaction costs), and assumes bonds within a class are perfect substitutes trading at a common interest rate r. Proposition I’s proof is a pure arbitrage argument, not an appeal to risk preferences: if a levered firm (company 2) is worth more than an otherwise identical unlevered firm (company 1) in the same class, an investor holding company 2’s stock can sell it, borrow on personal account in the same proportion as company 2’s own leverage, and buy a larger stake in company 1 for the same net outlay, earning a strictly higher return with no change in risk – and conversely, if the levered firm is worth less, an investor can sell unlevered shares and buy a mix of the levered firm’s stock and bonds to “undo” the leverage, again earning more. This homemade-leverage mechanism is what makes personal and corporate borrowing perfect substitutes, and the authors stress the mechanism is fundamentally different from the conventional appeal to investors “switching” between stocks and bonds as relative risk changes – “regarding Proposition I as resting on investors’ attitudes toward risk leads inevitably to a misunderstanding.”

Q3. What does Proposition II say about the relationship between a levered firm’s stock yield and its leverage, and why do the authors consider this result new?

Proposition II states that the expected yield on a firm’s common stock, i_j, equals the capitalization rate ρ_k appropriate to an all-equity firm in its risk class, plus a premium proportional to its debt-to-equity ratio: i_j = ρ_k + (ρ_k − r)·D_j/S_j (eq. 8) – a linear relationship, derived as a direct algebraic consequence of Proposition I. The authors note “a number of writers have stated close equivalents of our Proposition I…by appealing to intuition rather than by attempting a proof,” but say “Proposition II, however, so far as we have been able to discover, is new,” and illustrate it numerically: a firm with expected operating income of 1,000, debt of 4,000, interest rate 5 percent, and ρ_k of 10 percent has (by Proposition I) a market value of 10,000 and equity value of 6,000, so its stock yield is (1,000 − 200)/6,000 = 13.3 percent.

Q4. What empirical evidence do the authors offer for Propositions I and II, and what do the results show?

Using Allen’s data on 43 electric utilities (1947-48) and Smith’s data on 42 oil companies (1953), the authors test Proposition I by correlating the overall cost of capital (X^T/V) with the leverage ratio D/V: the traditional view predicts a significantly negative correlation, while their model predicts a correlation “not significantly different from zero” – and the actual correlations are r = .12 (utilities) and r = −.04 (oil companies), both statistically insignificant, which the authors call “clearly favorable to our hypothesis.” Testing Proposition II by correlating stock yield with the debt-to-equity ratio D/S, their model predicts a significantly positive relationship, and the data show r = .53 for both samples, “positive and highly significant” – and the fitted constant terms (5.3 percent for utilities, 8.5-8.7 percent for oil companies) are read as plausible estimates of ρ_k for each class, “accord[ing] well with a priori expectations.”

Q5. How does introducing a corporate income tax change Proposition I, and what does the paper conclude about the actual size of the tax-driven advantage to debt?

Because interest payments are tax-deductible, the arbitrage proof must be adjusted to show that firm values are proportional to income net of taxes (interest paid plus after-tax stockholder income) rather than to pre-tax operating income; working through the algebra, the cost of capital X/V now falls as the debt ratio D/V rises, at the constant rate τr/(1−τ), where τ is the corporate tax rate (eq. 31) – so, unlike in the no-tax case, debt financing does create a real, if modest, gain to stockholders even in otherwise perfect capital markets. In a numerical illustration with a 52 percent corporate rate, a 40 percent personal tax rate on dividends, and a 20 percent capital-gains rate, the required rates of return work out to about 20 percent for new-equity-financed investment, 16 percent for debt-financed investment, and 15 percent for investment from internal funds – leading the authors to flag “the remarkably small difference between the ‘cost’ of equity funds and debt funds. With the numerical values assumed, equity money turned out to be only 25 per cent more expensive than debt money, rather than something on the order of 5 times as expensive as is commonly supposed to be the case.” They further note a “paradoxical implication” that the stockholders’ incentive to use debt financing is smaller the lower the interest rate is, since in the limit of a near-zero borrowing rate the tax advantage of debt would also be practically zero.

Q6. What is Proposition III, and what is its illustrative numerical example?

Proposition III states that, for a firm acting in its owners’ interest, an investment is worth undertaking if and only if its expected yield, p*, exceeds the firm’s class capitalization rate ρ_k – the appropriate “cutoff” rate for investment decisions – regardless of whether the specific investment is financed by retained earnings, new debt, or new common stock. The authors illustrate the common-stock case with a numerical example: a firm with ρ_k = 10 percent, interest rate r = 4 percent, expected income of 1,000, and an initial 50/50 debt-equity structure (1,000 shares, each worth 5) considers a 100-dollar investment yielding 12 percent, financed by issuing 20 new shares at the current 5-dollar price. Although earnings per share on the existing 1,000 shares would appear to fall from .80 to .796 – making it look like the 12 percent project dilutes a stock currently yielding 16 percent – the firm’s total value rises to 10,120 and its equity value to 5,120 across 1,020 shares, so each share is worth 5.02: “the dilution in expected earnings per share…has been more than offset, in its effect upon the market price of the shares, by the decrease in leverage,” confirming that 10 percent (ρ_k), not the current 16 percent stock yield, is the correct hurdle rate.

Q7. The authors explicitly anticipate that Proposition III could be “easily misinterpreted” – what exactly does it NOT imply, and what still makes some financing plans preferable to others?

The authors warn that “read hastily, Proposition III seems to imply that the capital structure of a firm is a matter of indifference” and that the optimal-capital-structure problem “is no problem at all” – a misreading they move immediately to correct: the proposition says only that the financing instrument is irrelevant to whether an investment is worth undertaking, not that owners or managers have no grounds for preferring one financing plan to another. They give concrete reasons financing choice still matters: the market is likely to weight current and recent earnings heavily in forming expectations, so a firm with a major, not-yet-reflected investment opportunity might prefer debt financing (to be retired later from the resulting higher earnings or via a later equity issue at better prices) over issuing underpriced stock immediately; and managers have their own objectives, distinct from shareholders’, such as resisting loan covenants that “infringe on prerogatives” or restrict freedom to maneuver, which make some financing arrangements more attractive to management even when shareholders would be indifferent between them.

Q8. What do the authors say about the scope and limitations of their own analysis?

They describe their framework as “static, partial equilibrium analysis,” explicitly built on assumptions such as “a state of atomistic competition in the capital markets and an ease of access to those markets which only a relatively small (though important) group of firms even come close to possessing,” and defer a full general-equilibrium treatment – showing how the risk-class capitalization rates themselves are determined – to a subsequent paper. They close by framing Propositions I-III as “the foundations of a theory of the valuation of firms and shares in a world of uncertainty” and an “operational definition of the cost of capital,” while cautioning that “much remains to be done before the cost of capital can be put away on the shelf among the solved problems,” and inviting relaxation of the paper’s “drastic simplifications…in the direction of greater realism and relevance.”