A State-Preference Model of Optimal Financial Leverage

American Finance Association A State-Preference Model of Optimal Financial Leverage Author(s): Alan Kraus and Robert H. Litzenberger Source: The Journal of Finance, Vol. 28, No. 4 (Sep., 1973), pp. 911-922 Published by: Wiley for the American Finance Association Stable URL: http://www.jstor.org/stable/2978343 . Accessed: 01/04/2013 02:57 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp . JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. . Wiley and American Finance Association are collaborating with JSTOR to digitize, preserve and extend access to The Journal of Finance. http://www.jstor.org This content downloaded from 121.194.98.241 on Mon, 1 Apr 2013 02:57:16 AM All use subject to JSTOR Terms and Conditions A STATE-PREFERENCE MODEL OF OPTIMAL FINANCIAL LEVERAGE ALAN KRAUS AND ROBERT H. LITZENBERGER* I. INTRODUCTION IN COMPLETE and perfect capital markets, Hirshleifer [6, 7], Robichek and Myers [13], and Stiglitz [15] have shown that the firm's market value is inde- pendent of its capital structure. Although firms may issue conventional types of complex securities, such as common stocks and bonds, if the number of distinct complex securities equals the number of states of nature, individuals are able to create primitive securities. A primitive security represents a dollar claim contingent on the occurrence of a specific state of nature and can be created by purchasing and selling short given amounts of complex securities. Since in a perfect market the firm is a price taker, the market prices of these primitive securities are unaffected by the firm's financing mix. Therefore, given the firm's capital budgeting decisions which determine the firm's returns in each state, the firm's market value is independent of its capital structure. The market value of the firm equals the summation over states of the product of the dollar return contingent on a state and the market price of the primitive security representing a dollar claim contingent on the occurrence of that state. The proof of the Modigliani-Miller [8] independence thesis in a state- preference framework does not depend upon the assumption that the firm will earn its debt obligation with certainty. The firm may not earn the "promised" return on its bonds in some states of the world and would be bankrupt. In these states the firm's bonds are claims on the residual value of the firm. Although the firm's financing mix determines the states in which the firm is insolvent, the value of the firm is not affected since bankruptcy penalties would not exist in a perfect market. Therefore, sufficient conditions for the Modigliani-Miller inde- pendence thesis are complete and perfect capital markets. The taxation of corporate profits and the existence of bankruptcy penalties are market imperfections that are central to a positive theory of the effect of capital structure on valuation. A tax advantage to debt financing arises since interest charges are tax deductible. Assuming that the firm earns its debt obligation, financial leverage decreases the firm's corporate income tax liability and increases its after-tax operating earnings. However, a corporate bond is not merely a bundle of contingent claims but is a legal obligation to pay a fixed * The authors are, respectively, Associate Professor of Finance, Faculty of Commerce and Business Administration, University of British Columbia, and Associate Professor of Finance, Graduate School of Business, Stanford University. The research for this paper was supported in part by a grant from the Dean Witter Foundation. The co-authors benefitted from the comments of Joel Demski, Nestor Gonzalez, James W. Hoag, William F. Sharpe, Richard E. Stehle, and James C. Van Horne on earlier drafts of this paper. The comments of the reviewer are also gratefully acknowledged. The co-authors are, of course, jointly and equally responsible for the content of the paper. 911 This content downloaded from 121.194.98.241 on Mon, 1 Apr 2013 02:57:16 AM All use subject to JSTOR Terms and Conditions 912 The Journal of Finance amount. If the firm cannot meet its debt obligation, it is forced into bankruptcy and incurs the associated penalties. Robichek and Myers [12, p. 20] have noted that the optimization of capital structure involves a tradeoff between "the present value of the tax rebate associated with a marginal increase in leverage . . . [and] the present value of the marginal cost of the disadvantages of leverage." Similarly, Hirshleifer has suggested that "even within complete capital markets, allowing for considera- tions such as taxes and bankruptcy penalties would presumably permit the determination of an optimal debt-equity mix for the firm." [7, p. 264].' The present paper formally introduces corporate taxes and bankruptcy penalties into a single-period valuation model in a complete capital market. The firm's financing mix determines the states in which the firm will earn its debt obligation and receive the tax savings attributable to debt financing. The firm's financing mix also determines the states in which the firm is insolvent and incurs bankruptcy penalties. The problem of optimal capital structure is, there- fore, formulated as the determination of that level of debt such that the result- ing division of states (into those in which the firm is solvent and those in which it is insolvent) yields the maximum market value of the firm. It is shown that the total market value of the firm is not in general a concave function of financial leverage. II. FORMULATION OF THE PROBLEM Assume there are n possible states of the world and that capital markets are complete. Let Pj(0 < Pj < 1) denote the market price of the primitive security that consists of a claim on one dollar in state j. For the individual firm being considered, let Xj denote the earnings before interest and taxes the firm will achieve in state j.2 It is convenient, since the numbering of states is arbitrary, to assume that states are ordered by the values of Xj. Therefore, let the numbering of states be such that: X1 -<, X2 < * * * <, Xn-1 <, Xne(1) Although primitive securities are assumed to exist, it is not necessary to assume that firms issue such securities. It is sufficient to assume that firms issue only two claims, which may be designated as "debt" and "equity." Debt is a promise to pay a fixed amount, D, irrespective of the state that occurs. The ability of the firm to honor its promised debt payment in a given state, and, hence, the market price of this promised payment depends on the size of D relative to Xj. If a state occurs in which the firm cannot fully honor its debt claim, the firm is, by definition, insolvent. If such a state occurs, the firm enters bankruptcy and incurs the associated penalties (cost of insolvency). After paying the cost of insolvency, any remaining earnings are distributed to 1. See also Hirshleifer [6, p. 268] and Robichek and Myers [12, pp. 13-22]. Baxter [2] has presented empirical evidence consistent with the existence of both direct and indirect costs of bankruptcy. 2. The assumption is made that earnings in each state, Xj, are independent of the market value of the firm, V. This is consistent with the Modigliani-Miller [8, 9] world in which the effect of leverage on the firm's market value is examined for a given investment policy. This content downloaded from 121.194.98.241 on Mon, 1 Apr 2013 02:57:16 AM All use subject to JSTOR Terms and Conditions State-Preference and Optimul Financial Leverage 913 debt holders. Let Cj denote the cost of being insolvent in state j. Given the limited liability feature of corporate securities, 0 < Cj < Xj holds for every state j. Letting Yj be the amount received by the debt holders if state j occurs, it follows that rD for D Xj yj - (2) xi X Ci for D > Xj Relation (2) indicates that the amount actually paid to debt holders will fall short of the promised amount in states in which the firm is insolvent. Naturally, the market value of the debt will depend on the amounts that will actually be paid in the various states. The firm's debt may be viewed as a complex security consisting of a bundle of contingent claims of the form Yj dollars in state j. Since the values Yj depend on the promised amount, D, the market value of the firm's debt, B (D), may be expressed as3 n D Pi for O Xn j=l The description of equity in this framework requires some additional assump- tions. Assume that all payments to debt claims are tax deductible and that the tax rate applicable to earnings net of debt payments in state j is Tj, with Tj > 0.' The earnings remaining after taxes and payment to the debt holders is assumed to be paid out in full to equity holders. Let Zj be the amount paid to equity holders if state j occurs. Then r Xj(- Tj) + TjD - D for D ?Xj Z { = T (4) 0 O for D > Xj The term Xj(1-Tj) represents the firm's after tax earnings under an all equity capital structure, TjD is the tax saving attributable to debt financing and D is the payment to bondholders. The market value of the firm's equity, S(D), may be expressed as 3. The notation in relation (3) and subsequent relations assume, for convenience, that the inequalities in (1) are strong inequalities. These expressions must be modified slightly in the special cases in which earnings are identical in some states. 4. A fixed capital repayment which is non-taxable either as a liquidation dividend and/or repay- ment of principal on the firm's debt could be introduced into the single-period model. This would complicate the notation without producing additional insights. This content downloaded from 121.194.98.241 on Mon, 1 Apr 2013 02:57:16 AM All use subject to JSTOR Terms and Conditions 914 The Journal of Finance n Z [Xj(l - T) +TjD - D]Pj Jj= for D Xn The total market value of the firm V(D) is the sum of the market value of its debt, B (D), and the market value of its equity, S (D). n V(D) - (Yi + Zi)Pi J=1 n [(1- Tj)Xj+TjD]Pj for 0 Xn J=1 The market value of an unlevered firm is n V(O) (1 - Tj)XjPj. (7) i=1 Substituting (7) into (6) yields the relationship between the market value of the firm unlevered and its market value levered: n D TjPj for O < D Xn JR=f Relation (8) states that the market value of a levered firm is equal to its This content downloaded from 121.194.98.241 on Mon, 1 Apr 2013 02:57:16 AM All use subject to JSTOR Terms and Conditions State-Preference and Optimal Financial Leverage 915 unlevered market value plus the present value over all states of the difference between the tax advantage of leverage and bankruptcy costs. Assuming a constant tax rate, T, across states and substituting (3) into (8): 0 for 0?X. In the absence of bankruptcy penalties, relation (9) is consistent with the M&M tax correction model [9]. However, in contrast to their analysis, the derivation of relation (9) does not assume existence of homogeneous risk classes, identical probability beliefs, absence of personal income taxes, or that all corporate bonds are free of default risk. Several authors have argued reductio ad absurdum that the M&M tax cor- rection model is unreasonable since it implies that the firm should utilize the maximum amount of debt in its capital structure [2, 12, 14, 16], Robichek and Myers [12, pp. 38-42] and Baxter, [2, p. 395] have noted that such a con- clusion has little intuitive appeal since it ignores the existence of bankruptcy penalties. From (9) it is apparent that taking explicit cognizance of bankruptcy penalties the maximization of the firm's market value is equivalent to neither the maximization of leverage nor the maximization of the market value of the firm's debt.5 The subsequent analysis examines the functional rela- tionship between the market value of the firm and the size of its "promised" debt payment. The optimization of the firm's financial structure involves a trade-off between the tax advantage of debt and bankruptcy penalties. III. PROPERTIES OF THE PROBLEM Two characteristics of the structure of the problem are of particular interest. Proposition I: The optimal value of D is one of the values X1, X2,... ,Xn. Since this proposition is an obvious implication of relation (6), a formal proof is omitted. If Xjj < D < Xj, the present value of tax savings increases as D approaches Xj while the present value of bankruptcy penalties is un- changed. One implication of Proposition I is that the problem of optimal capital structure can be solved by enumeration; it is necessary only to calculate the value of the firm for a finite number of possible values of debt: X1, . . ., Xn. Letting V (Xk) denote the market value of the firm when D = Xk, 5. Solomon [14, p. 103] argues that the Modigliani-Miller tax correction model implies that "the recipe for optimal leverage . . . is that companies ought to be financed 99.9 per cent with pure debt!" It should be noted that the market value of the firm's debt in a world of bankruptcy penalties does reflect the present value of these penalties. See relation (3) above. This content downloaded from 121.194.98.241 on Mon, 1 Apr 2013 02:57:16 AM All use subject to JSTOR Terms and Conditions 916 The Journal of Finance k-1 n V(Xk) = Z (Xi - Ci)Pi + Z [(1 - Tj)Xj + TjXk]Pj. (10) Proposition II: Suppose that for some value of k,6 n Ck-lPk-1 > (Xk -Xk.1) ZTjPj j=k n and (Xk + - Xk) TjPj > CkPk, J=k+l then V(Xkl1) > V(Xk) < V(Xk+l). Proof: Under these conditions it follows directly from (10) that n V(Xk-1) - V(Xk) = Ck-lPk-1 (Xk - Xk-1) Z TP > 0 J=k n V(Xk+l) - V(Xk) = (Xk+l - Xk) Z TP - CkPk > 0. j=k+l Proposition II states that it is possible for V (Xk) to be strictly less than both V(Xk-l) and V(Xk+l). In order to contrast Proposition II with divergent views of the effect of leverage on valuation, it is necessary to examine in more detail the form of the function that emerges from the present model. The traditional or modified net income appoach to valuation is that the value of the firm is a concave function of debt (resembling an inverted U). Under this approach, the slope of the function would be positive for very low levels of debt, decrease monotonically with leverage, and eventually become negative as leverage becomes extreme. An implication of the concavity of the function is that if the value is not at its maximum, the firm's market value can be increased by a small change in leverage. Contrary to the traditional view, V(D) is not, in general, a continuous func- tion of D. This can be seen from (6) and (10) by noting that, for Xkl < D < Xk, V(D) cannot be made arbitrarily close to V(Xk-l) (unless Ck1 = 0 or Pk-1 = 0). Lim [V(Xk.l + e) - V(Xk-1)] Ck-lPk-1 is the 6-40 reduction in market value of the firm as debt is increased just enough to change state (k - 1) from a state in which the firm would be solvent to one in which it would be insolvent. Letting V'(D) denote the derivative of V (D) with respect to D, n TiP for O < D < X J=1 V'(D)= , n 2.TjPj for Xkl < D Xn 6. See footnote 3. This content downloaded from 121.194.98.241 on Mon, 1 Apr 2013 02:57:16 AM All use subject to JSTOR Terms and Conditions State-Preference and Optimal Financial Leverage 917 In the range 0 < D < X1, and under the assumption of a constant tax rate, the slope V'(D) is equal to the tax rate divided by unity plus the default-free rate of interest. This result is consistent with the M&M tax correction model under the assumption that the firm earns its debt obligation with certainty. Relation (11) shows that the slopes of successive linear segments of V(D) are non-increasing. That is, V'(D) for Xk < D < Xk+1 cannot exceed V'(D) for Xk-1 < D < Xk, since all Tj and Pj are assumed non-negative. In general, successive slopes decrease as in the traditional view. The general form of V (D) is depicted in Figure 1 by the solid line segments. As seen in relation (8) V(D) V(X2) - V(X) V(X 4) V(X1) --t V(O) I r I 1 1 3 D X1 X2 X3 X4 FIGURE 1 the intercept is V(O), the market value of the firm under an all equity capital structure. A direct implication of Proposition II is that even connecting the values V(Xk) to produce a continuous function does not necessarily produce a concave function of D. This is shown in Figure 1 by the dotted lines that produce a function [V(W), V(X1), V(X2), V(X3), V(X4)] that is not concave between X2 and X4. Since the maximum of V(D) must occur at one of the values Xk, the function connecting adjacent V(Xk) points provides a better comparison between the present model and the traditional view than does V(D). While the traditional view is not associated with a specific algebraic relationship between debt and the value of the firm, proponents of the traditional view of the effect of leverage on valuation have agreed that the basic shape of the relation is concave. A major implication of the model proposed here is that the relation between debt and the value of the firm may be of a fundamentally different shape, over at least part of its range, from that envisioned by the traditional view. A numerical illustration of the model presented above is given in Appendix B. This content downloaded from 121.194.98.241 on Mon, 1 Apr 2013 02:57:16 AM All use subject to JSTOR Terms and Conditions 918 The Journal of Finance Another implication of the properties of the present model is that any solution technique that assures finding the optimal solution (i.e., the value of D for which V(D) is a maximum) must involve consideration of all possible solutions. As noted earlier, Proposition I implies that the optimal solution can always be located by complete enumeration-i.e., comparison of values of V(Xk) for all values of k. However, it is shown in Appendix A that dynamic programming techniques of partial enumeration can also locate the optimal solution in all cases, including situations in which the function is not concave and simpler iterative techniques may fail. IV. CONCLUSION In complete and perfect capital markets the firm's market value is indepen- dent of its capital structure. The taxation of corporate profits and the existence of bankruptcy penalties are market imperfections that are central to a positive theory of the effect of leverage on the firm's market value. In their tax correc- tion article, Modigliani-Miller [9] have shown that, assuming the firm earns its debt obligation with certainty, the firm's market value would be a linear func- tion of the amount of debt used in its capital structure. Hirshleifer [6] has shown that in absence of bankruptcy penalties any tax minimizing procedure would increase the firm's market val