A state preference model of optimal financial leverage

1. A STATE-PREFERENCE MODEL OF
OPTIMAL FINANCIAL LEVERAGE
ALAN KRAUS AND ROBERT H. LITZENBERGER*
1. 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

2. 912 The Journal oj 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. 264J,!
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(O ~ P, ~ 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 X, denote the earnings before interest and taxes the
firm will achieve in state V 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:
(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 X, 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 hoth direct and indirect costs of
bankruptcy.
2. The assumption is made that earnings in each state, Xi' 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.

3. State-Preference and Optimul Financial Leverage 913
debt holders. Let C, denote the cost of being insolvent in state j. Given the
limited liability feature of corporate securities, 0 ~ C, ~ X, holds for every
state j. Letting Y, be the amount received by the debt holders if state j occurs,
it follows that
{
D bD~~
v,= (2)
~-~ bD>~
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 Y, 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 as"
n
B(D) = L:YjPj =
j=1
k-1 n
L: (Xj - Cj)Pj +D L:P j
j=1 j=k
for Xk - 1 < D ~ Xk (k = 2, ... , n)
(3 )
n
L: rx, - Cj)Pj for D > x,
j=1
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 T j, with
T, > 0.4 The earnings remaining after taxes and payment to the debt holders
is assumed to be paid out in full to equity holders. Let Z, be the amount paid to
equity holders if state j occurs. Then
{
Xj(1 - Tj) +TjD - D for D ~ x,
Zj = (4)
o for D > x,
The term Xj(l-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, SeD), 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.

4. 914
n
S(D) = L:: ZjPj =
J=1
The Journal of Finance
n
L [Xj(l - Tj) +TjD - D]Pj
j=1
n
L:: [Xj(l - Tj) +TjD - D]Pj (5)
j=k
for Xk - 1 < D ~ Xk (k = 2, ... , n)
o for D > x,
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) = L rv, +Zj)Pj
J=1
n
L [(l - Tj)Xj +TjD]Pj
J=1
k-l n
L:: rx, - Cj)Pj + L:: [(1- Tj)Xj +TjD]Pj (6)
j=1 j=k
for Xk - 1 < D ~ Xk (k = 2, ... , n)
for D > x,
The market value of an unlevered firm is
n
V(O) = L (l - Tj)XjPj.
j=1
(7)
Substituting (7) into (6) yields the relationship between the market value of
the firm unlevered and its market value levered:
V(D) = V(O) +
k-l n
L:: rr,x, - Cj)PJ + D L::TJPJ
j=1 j=k
(8)
n
L (TjXj - Cj)Pj
J=1
for Xk_l<D~Xk(k=2, ... ,n)
for D > x,
Relation (8) states that the market value of a levered firm is equal to its

5. 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):
o for O~D ~Xl
V(D) = V(O) + T B(O) - (1- T)
(9)
for Xk - l <D ~ Xk (k = 2, ... ,n)
for D > 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." 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 Xl, X 2, ••• , X n •
Since this proposition is an obvious implication of relation (6), a formal
proof is omitted. If Xj _ l < D ~ Xj , the present value of tax savings increases
as D approaches X, 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: Xl' ... , Xn •
Letting V(Xk ) denote the market value of the firm when D = Xk,
S. 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 1"
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.

6. 916 The Journal of Finance
k-l n
V(Xk) = L (x, - Cj)Pj + L [(1 - Tj)Xj +TjXk]Pj.
j=l j=k
Proposition II: Suppose that for some value of k,"
n
Ck-lPk- l > rx, - Xk-d L TjPj
j~k
(10)
and
n
(Xk+l - Xk) L TjPj > CkPk,
j=k+l
then V(Xk- l) > V(Xk) < V(Xk+l)'
Proof: Under these conditions it follows directly from (10) that
n
V(Xk-d - V(Xk) = Ck-lPk- l - ex, - Xk- l) L TjPj > 0
j=k
n
V(Xk+d - V(Xk) = (Xk+l - x.: L TjPj - CkPk > O.
j=k+l
Proposition II states that it is possible for V(X,') 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
Xk-l < D ~ Xk , V(D) cannot be made arbitrarily close to V(Xk - l) (unless
Ck_l =0 or Pk-l =0). Lim [V(Xk _. 1 +E) - V(Xk_1) ] = - Ck-lPk - 1 is the
£->0
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'eD) denote the derivative of V(D) with respect to D,
for 0 ~ D ~ Xl
V'eD) =
6. See footnote 3.
o
for Xk - 1 < D ~ X, (k = 2, ... , n)
for D > x,
(11 )

7. State-Preference and Optimal Financial Leverage 917
In the range 0 ~ D ~ Xl, 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 x, < D ~ Xk+l cannot exceed V'(D)
for x.,.,< D ~ x; since all 'r, and r, 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)
yeO)
L- ..L.- ......_ ..... ..... D
the intercept is V(0), 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 [YeO), V(XI ) , V(X2 ) , V(X3 ) , V(X4 ) ] that is not concave between
X2 and ~. 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.

8. 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-e-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 value. However, Hirshleifer and Robichek
and Myers have noted that both taxes and bankruptcy penalties should be
considered in the determination of optimal leverage.
The present paper formally introduces the tax advantage of debt and bank-
ruptcy penalties into a state preference framework. The market value of a
levered firm is shown to equal the unlevered market value, plus the corporate
tax rate times the market value of the firm's debt, less the complement of the
corporate tax rate times the present value of bankruptcy costs. Contrary to the
traditional net income approach to valuation, if the firm's debt obligation ex-
ceeds its earnings in some states the firm's market value is not necessarily a
concave function of its debt obligation.
APPENDIX A-SOLUTION BY DYNAMIC PROGRAMMING
To apply dynamic programming efficiently to the problem, it is necessary to define some
additional functions and establish a third proposition. First, since subtracting a constant from
a function does not change the abscissa value at which the function reaches a maximum, the
value of D that maximizes V(D) is equal to the value of D that minimizes the following
function.
where, using (2) and (4),
n
WeD) = - [V(D) - L:XjPj ]
j=l
n
= L: rx, - v, - Zj)Pj
j=l
(12)
for D > XJ
(13)

9. State-Preference and Optimal Financial Leverage 919
(14)
Although it is defined primarily for analytical convenience, the function W(D) has a
direct intuitive meaning. In a world without taxes or bankruptcy penalties the total market
value of the firm would equal ~XjPj, independent of its capital structure. W(D) equals
the amount by which the value of the firm is less than ~XjPj because of taxes and bank-
ruptcy penalties. The goal of minimizing W (D) with respect to D, therefore, is simply a
restatement of the optimal capital structure problem as that of finding the amount of debt
that entails the smallest reduction for taxes and bankruptcy penalties.
Let Wk(D) be defined as the value of W(D) when the only states of the world considered
are k, k + 1, ... , n.
D
Wk(D) =L: (Xj - Yj - Zj)Pj,
j=k
The optimal capital structure can be defined at the value of D that minimizes W1(D),
since this value of D also maximizes V(D).
For each value of k there is some value of D that minimizes Wk(D). By Proposition I
this value of D must equal one of the values Xk, Xk+1, . . . , Xn • Let mk be defined
as that integer among k, k + 1, ... , n, such that Wk(D) is minimized when D =Xmk. In
order to make the value of mk unique, it may be formally defined for k = 1, 2, ... , n - 1
as the smallest integer that satisfies
for all values of D. (15)
Obviously, m, =n.
Proposition Ill: Either mk =mk+1 or mk =k.
Proof: From (13) and (14),
Wk(D) = Wk+1(D) + CkPk for D > x;
Therefore, using (15),
Wk(Xmk+1) ~ Wk(D) for D > x;
Since Proposition I implies Xmk ~ Xk, it follows that
(16)
from which Proposition III follows.
Proposition III is by no means a surprising result. If the amount of debt that minimizes
Wk+1(D) exceeds Xk, this amount of debt would cause insolvency in state k. However,
any amount of debt exceeding Xk (i.e., Xk+1, ... , Xn ) would cause insolvency in state k
so that no such amount could result in a value of Wk(D) lower than Wk(Xmk+1). Over
this range of debt, in other words, the function Wk(D) differs from Wk+1(D) only by a
constant, CkPk. Since the amount of debt that minimizes Wk(D) cannot be less than Xk,
this amount must be either Xmk+1 or Xk.
The dynamic programming formulation is based directly on Proposition III, particularly
as expressed in (16). The latter may be rewritten as
(17)

10. 920 The Journal of Finance
(18)
where the summation is from k + 1 because Xk - Yk - Zk =0 in (13) for D =Xk·
Letting W*k denote Wk(Xmk) to simplify notation, (17) can be rewritten as
{
W*k+i + CkPk
W*k = min
Fk+i - Gk+iXk
where
n
Fk =L TjXjPj =Fk+i + TkXkPk
j=k
n
Gk =L TjPj =Gk+ i + TkPk
j=k
for k = 1, 2, ... , n - 1, and
W*n = 0,
(19)
(20)
(21)
In (18), the upper term implies mk =mk+i and the lower term implies mk =k. Thus,
given the value of mk+i, solution of (18) determines the value of mk'
In the dynamic programming formulation the choice of the optimal amount of debt can
be stated as the problem of determining m.. The solution is obtained by using (21) in (18)
to determine W*n-i' and hence mn_ i. Then the values calculated at this step are used to
evaluate W*n-2' and so on until mi is determined by the calculation of W*i' These steps
are shown in Appendix B for a simple numerical example.
APPENDIX B
NUMERICAL EXAMPLE
The model of Section II is illustrated in the following example. For simplicity, we assume
there are only four possible states of the world and that the tax rate is constant across
states. The data assumed for the example are summarized below.
k Xk Pk Ck Tk
1 200 0.30 200 0.50
2 400 0.40 400 0.50
3 1600 0.05 700 0.50
4 2000 0.20 800 0.50
The value of V(D) at D =0 (all equity) and at each of the values Xk is calculated from
relation (6).
V(O)
V(Xi )
V(X2)
V(Xa)
V(X4 )
=350
= 445
= 450
=440
= 445
In this simple example the optimal capital structure is highly leveraged. The maximum
value of the firm (450) occurs at D = X2, where the promised amount of debt payment is
400. Using (3), the market value of this amount of debt is 260. The market value of equity
in this capital structure is, of course, 450 - 260 = 190. This value can also be calculated
directly from (5).
Not by coincidence, it happens that this example is one in which the conditions of Prop-
osition II are met. This is reflected in the result that V(X2) exceeds V(Xa) but V(Xa)

11. State-Preference and Optimal Financial Leverage 921
is less than V(X4) . In other words, although V*(D) has a local maximum at D =X4 , the
global maximum occurs at D =X2• The following steps show the application to this ex-
ample of the dynamic programming formulation of Appendix A.
k=4
W*4 =0
F4 =(.5)(2000)(.20) =200
G4 = (.5)(.20) = .10
k=3
{
0 + (700) (.05) =35
W*a=min
200 - (.10) (1600) = 40
Fa =200 + (.5)(1600)(.05) =240
Ga =.10 + (.5) (.05) =.125
k=2
{
35 + (400)(.40) = 195
W*2 = min
240 - (.125)(400) =190
F2 =240 + (.5)(400)(.40) =320
G2 = .125 + (.5) (.40) =.325
k=l
{
190 + (200)(.30) =250
W*l = min
320 - (.325) (200) = 255
Since m1 =2, the optimal capital structure of the firm in the example is at D =X2, at
which point the total market value of the firm is maximized. From (14) the maximum total
value is found to be
D
V(X2) = L::x.r, - W(X2)
j=l
=600 - 250 =450.
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