THEORY OF CAPITAL STRUCTURE
Determination of an optimal capital structure has frustrated theoreticians for
decades. The early work made numerous assumptions in order to simplify the problem
and assumed that both the cost of debt and the cost of equity were independent of
capital structure and that the relevant figure for consideration was the net income of the
firm. Under these assumptions, the average cost of capital decreased with the use of
leverage and the value of the firm (the value of the debt and equity combined) increased
while the value of the equity remained constant.
%
Ks
Ka
Kd
Debt/Equity
Modigliani and Miller showed that this could not be the case. Their contention
was that two identical firms, differing only in their capital structure, must have identical
total values. If they did not, individuals would engage in arbitrage and create the market
forces that would drive the two values to be equal.
Their proof of this proposition was based upon several assumptions (many of
which have subsequently been relaxed without changing the results):







All investors have complete knowledge of what future returns will be
All firms within an industry have the same risk regardless of capital structure
No taxes (we will relax this assumption subsequently)
No transactions costs
Individuals can borrow as easily and at the same rate of interest as the
corporation
All earnings are paid out as dividends (thus, earnings are constant and there
is no growth)
The average cost of capital is constant
Since no taxes has been assumed, the operating income (EBIT) is equivalent to
the net income which is all paid out as dividends. Thus, the value of the firm is equal to
V
EBIT
ka
Since the value of the firm is equal to the sum of the value of the debt and equity,
V  D E
then
k aV  k a ( D  E )
and
E
D
ka  k s (
)  kd (
)
D E
D E
Substituting the last equation into the preceding equation and solving for Ks
k s  ka  ( ka  kd )
D
E
Thus, ks must go up as debt is added to the capital structure.
%
Ks
Ka
Kd
Debt/Equity
To prove their point, they assumed two identical firms, an unlevered firm (all equity) and
a levered firm with $4 million of debt carrying an interest rate of 7.5%, both firms
generating an operating income (EBIT) of $900,000 annually. They adopted the
assumption that stockholders of both firms would have the same required rate of return
of 10% which, as previously mentioned, was the standard assumption at the time (that
the cost of equity was constant regardless of capital structure).
EBIT
-Interest
Income
Unlevered Firm
$ 900,000
0
$ 900,000
Levered Firm
$ 900,000
300,000
$ 600,000
Since the required rate of return of shareholders is 10% in both cases
Unlevered Firm
Value of Equity =
Value of Debt =
Total Value of Firm =
Levered Firm
$900, 000
 $9, 000, 000
.10
$600,000
 $ 6,000,000
.10
$
0
$9,000,000
$ 4,000,000
$10,000,000
If this were true, then someone who owns 10% of the levered firm would have
income of $60,000 ($600,000 * 10%) and could sell it for $600,000 ($6 million * 10%).
With this $600,000 the individual could borrow another $300,000 at 7.5% and buy 10%
of the unlevered firm for $900,000 ($9 million * 10%). What would this individual’s
income be now?
EBIT
-Interest
Income
$ 90,000
22,500
$ 67,500
($900,000 * 10%)
($300,000 * 7.5%)
Thus, the income would be greater by buying the unlevered firm’s stock and borrowing
money to finance the purchase. As other individuals see this opportunity, they also will
sell the stock of the levered firm (driving its price down) and buy the stock of the
unlevered firm (using some borrowed money) and thereby driving the value of the
unlevered firm’s stock up. As the price of the unlevered firm is bid up, the value of the
unlevered firm increases above $9 million dollars, while the selling of the levered firm’s
stock drives the equity value below $6 million (which decreases the total firm value,
including the $4 million of debt, below $10 million) until the two firms’ values, in
equilibrium, are equal and no opportunity to arbitrage the difference exists.
Consequently, if the total value of the two firms is equal, then the average cost of capital
must be equal. And if the average cost of capital is equal, then it must be true that the
cost of equity rises in such a manner as to exactly offset the increased use of cheaper
debt financing (and we end up with the previous graph showing this).
As we previously uncovered when we looked at financial leverage, this is not a
surprising result. As a firm increases its use of debt, the risk to the stockholder
increases and, as a consequence, the stockholder’s required rate of return will increase.
Modigliani and Miller simply defined how the stockholder’s required rate of return should
increase with increased financial leverage.
The lesson that is intended by this is that value cannot be created by simply
substituting one form of financing for another.
Subsequent to this analysis, it was pointed out that corporate taxes have an
impact on the valuation. Without going through the mathematics (which is in your
textbook), suffice it to say that the result was that the value of the firm increased with
increased leverage. Specifically,
VL  VU  t * D
The fact that the government is a “partner” in the business results in a subsidy when
debt financing is used and a deductible expense (unlike equity payments). When
corporate taxes were taken into account, the average cost of capital was found to
decrease with increased leverage:
%
Ks
Ka before-tax
Ka after tax
Kd before-tax
Kd after-tax
Debt/Equity
This implies that a firm should use as much debt as possible. Yet, we do not see
companies using 100% debt. It might be pointed out that during the late 1980s there
was a considerable amount of substitution of debt for equity among firms, particularly in
the case of leveraged buyouts. However, many of those firms subsequently failed (for
example, Unocal) and the typical debt/equity ratio today is similar to earlier levels.
So why do we not see more debt employed by companies? The answer to this
question has been sought by many and two primary proposals have been put forth.
First, bankruptcy costs were invoked as a factor. That is, the more debt a firm uses, the
higher the probability that the firm would default and go into bankruptcy. Therefore, the
present value of bankruptcy costs had to be deducted from the value of the firm. A
second factor was that of “agency” costs, such as the necessity of reporting regularly to
lenders (audited financial statements, bank “monitoring” and other bank fees, trustees
for debt payments, etc.) that accompany the use of debt. Both of these costs increase in
present value of expected costs terms as the proportion of debt increases. Another way
of viewing these costs is that the risk of receiving full interest and principal payments
increases and thus the required rate of return of lenders increases. (For example, “junk”
bonds often yield higher rates of interest than the required rate of return on equity for
companies with very little debt.) Consequently, the cost of debt increases and the
average cost of capital will ultimately increase.
%
Ks
Ka after-tax
Kd after-tax
Debt/Equity
As can be observed from the graph, a minimum average cost of capital exists,
but exactly where it should be has yet to be determined within a theoretical framework.
So what are the insights that we can gain from this theoretical view of capital
structure? First, we should note that, while debt financing is “cheap” in the sense that
required rates of return on equity will always be higher than the interest rate on debt,
there is a “hidden” cost in that the cost of equity rises as we utilize more debt financing.
This is one reason that using the average cost of capital in valuing a project or company
is more appropriate, even if we intend to borrow all of the money to finance it. While we
may use cheap debt to finance a project, the increased risk to shareholders from
increasing our financial leverage results in an increase in the cost of equity. The
average cost of capital reflects both the cost of debt as well as the cost of equity and
thus will reflect the increased cost of equity associated with the use of more debt
financing.
The second important concept is that tax-deductible debt financing results in a
tax subsidy by the government. This subsidy adds value to the firm. For example, what
is the “advantage” of being a home owner with a mortgage rather than leasing a home?
It is the taxes that you will save. The reason that Congress eliminated the deductibility
of credit card interest is that it did not want to encourage, through a tax subsidy, the
financing of purchases purely for consumption. On the other hand, the purchase of a
home (which is still tax-deductible) is an “investment”, not to mention the political
consequences of voting to end the subsidy of the American Dream of home ownership.
Current Thoughts on the Effect of Leverage on the Cost of Capital
The evidence quoted in the textbook suggests that the costs of an increased
probability of bankruptcy and the increasing agency costs associated with debt increase
at such a rate as to (approximately) just offset the tax advantage of using increasing
amounts of cheap, tax-deductible debt. Thus, on an after-tax basis, the weighted
average cost of capital remains relatively stable and is roughly the same as the required
return on equity for a firm that does not utilize any leverage.
%
Ks
Ka after-tax
Kd after-tax
Debt/Equity
Note that there is no effect on the cost of equity. The incremental costs
associated with increased bankruptcy risk and the agency costs imposed on the firm can
be reflected as a cost of using additional debt financing.
As an example, consider a California firm that was in violation of its existing bank
loan covenants of maintaining a specified quick ratio and debt service coverage ratio.
Naturally, the bank would prefer not to pursue bankruptcy proceedings since its recovery
would be minimal. Instead, the bank was willing to accept roughly 50% of the funds
owed to it plus preferred stock payable over a five-year period at a 10% (low) dividend
yield. Overall, the bank "foregave" approximately 25% of what was owed to it (which
resulted in extraordinary income for the firm). In order to generate the funds to pay the
bank, a second bank was found that was willing to make both a term loan and a
revolving line of credit secured by inventories and receivables. While the new loans had
no covenants regarding ratios, the bank charged a minimum $30,000 per month of bank
fees which included the interest on the debt as well as a fee based upon the amount of
inventories and receivables available (but not necessarily borrowed against) for the
revolver. The conclusion of the firm was that the optimal debt policy was to pay down
the new loan only to the extent that the interest and fees amounted to $30,000 per
month since any additional pay down would yield no decrease below the minimum
$30,000 per month of charges. It was estimated that the effective rate of interest on the
loan was in the neighborhood of 24% even under the optimal policy. Thus, some
lenders are willing to make high-risk loans, but the cost is incorporated under various
guises of interest, bank and commitment fees, resulting in an effective rate of interest
that borders on usurious (i.e, above legal limits).
As the risk rises for lenders as more debt is employed, so does the risk to the
stockholders. The textbook derives a relationship based upon the assumption of a riskfree corporate debt, although there is no such thing. Alternatively, one can consider the
fact that the cost of debt reflects the market's assessment of the risk to lenders and that
the cost of equity should include an additional risk-premium based upon the fact that an
equity position is more risky than that of debt even for a highly levered firm. With this
perspective, and referring back to the review on risk and the Capital Asset Pricing
Model, Hamada has shown that the beta of the leveraged firm is a function of the extent
to which debt is used relative to equity.
Debt
(1  t)]
Equity
 L   U [1 
Of course, the factor (1-t) reflects the fact that increased profitability from the use of
leverage is reduced by the "partnership" that the firm has with government in that
increased profits are reduced by their taxation. Similarly, increased losses are offset by
tax-loss carrybacks and carryforwards. Moreover, substituting for the definition of beta
shows that this relationship is true with respect to total risk as well as systematic risk
since the correlation between the market and an unlevered firm is the same as the
correlation between the market and a levered firm.
 L   U [1 
Debt
(1  t)]
Equity
or
 i,M  iL M
 M2
 i,M  iU  M 

Debt

1

(1
t)


 M2
 Equity

or

 iL   iU 1 


Debt
(1 - t) 
Equity

Thus, CAPM gives us insight into the manner in which leverage affects risk, and hence
required rates of return, regardless of whether total risk or systematic risk is of more
concern to the decision-maker. As will be seen in the textbook, the unleveraging of
corporate betas and averaging in order to derive an industry unlevered beta and using
the CAPM to yield an industry required return on unlevered equity as a proxy for the
average cost of capital is central to the text’s approach of valuing the firm as a whole.
To see the more applied implications, consider the effect on a firm’s cost of
equity resulting from a change in capital structure. As an example, reconsider the firm
from the Cost of Capital handout in which the weighted average cost of capital was
calculated. Suppose the company intended to issue an additional $200,000 of debt and
use the proceeds to repurchase $200,000 of common stock. (For simplicity, assume
that there are no transactions costs involved.) In addition, assume that the cost of debt
rises from the market rate of 9% to 10% as a result of the higher amount of leverage.
Assume also that the required return for preferred stockholders increases from 8% to
8.4% as a result of the higher leverage as well. Finally, assume that the repurchase of
shares is estimated to result in driving the price per share up to $11.41 each.
Since new debt is generally priced and issued such that the coupon rate of
interest equals the market rate of interest, the new debt will need to carry a 10% coupon
rate of interest and will sell at par. The existing debt, however, must be repriced to
reflect the higher required return of debtholders. Discounting the annual interest
payments of $80 per bond and the par value of $1,000 at the end of ten years at the new
market rate of interest of 10% yields a new value of $877 per bond. The entire old bond
issue therefore will drop in value from $187,200 to $175,400.
Similarly, the preferred stock must be revalued. If the required rate of return on
the preferred stock has increased to 8.4%, the price will have decreased to $83.33 per
share or a total of $83,330.
To estimate the how the stockholders’ required rate of return will change, we will
need to do two things: first, we need to estimate the beta of the company prior to the
financial restructuring. Then we need to unlever the beta and relever it to reflect the new
financial structure. The stockholders’ required rate of return of 12% was previously
calculated utilizing the Gordon Growth Model which should be, conceptually, equivalent
to the CAPM. Then, if the risk-free rate is 4% and the expected return to the market is
10%, we can estimate the pre-refinancing beta as
K S  R F   LOld ( R M  R F )
or
12%  4%   (10% - 4%)
 LOld  1.3333
The next step is to unlever the beta to estimate the what the beta would be if the firm
had no debt. In this instance, the preferred stock also needs to be accounted for as if it
were debt, although without the tax-deductibility that interest payments enjoy, in order to
reflect the increased variability that common stockholders experience as a result of the
fact that preferred stockholders must be paid first; i.e., the leverage effect.
 LOld   U [1 
Debt
P/S
(1  t) 
]
Equity
Equity
1.3333   U [1 
187,200
87,500
(1  .4) 
]
1,125,000
1,125,000
or
 U  1.1322
The all-equity, or unlevered, beta is often referred to as the asset beta. In order to
relever the beta, we need to insert the new market values of the claims on the firm. The
new market value of the stock must first be calculated by taking into account the number
of shares repurchased as well as the new stock price.
No. of Shares Repurchase d 
200,000
 17,513 shares
11.42
Remaining shares  100,000 - 17,513  82,487 shares
New Market Value  82,487 shares * $11.42/sha re  $942,000
 LNew   U [1 
Debt
P/S
(1  t) 
]
Equity
Equity
 LNew  1.1322 * [1 
375,400
83,330
(1  .4) 
]
942,000
942,000
or
 LNew  1.5031
Using the CAPM, the new cost of common equity can be estimated as 13.02%
K S  R F   LNew ( R M  R F )
 4%  1.5031 * (10% - 4%)
 13.02%
The new estimate of the weighted average cost of capital is then calculated as was
previously done.
K Avg  K d (1  t )
Debt
P/S
C/S
 KP
 KS
Debt + P/S + C/S
Debt + P/S + C/S
Debt + P/S + C/S
 10% * (1 - .4) *
375,400
83,330
 8.4% *
375,400  83,330  942,000
375,400  83,330  942,000
 13.02% *
942,000
375,400  83,330  942,000
 6.0% * 0.268  8.4% * 0.0595  13.02% * 0.6725
 10.86%
Notice that the total value of the firm has remained constant at approximately $1.4
million and the average cost of capital remains the same as well.