The Theory Underlying the Cost of Capital

Chapter 3 Chapter 3

II. The Theory Underlying the Cost of Capital

This section highlights the stochastic nature of the cost o f capital. The emphasis is on the relationship between cost of capital and the corporations’ internal characteristics, rather than on the general financing environment. The frst ^{part of} this section is devoted to demonstrating that the weighted average cost o f capital can theoretically be an unknown function o f capital structure (as measured by debt to total capitaliza- tion). This would imply that managers would not have the knowledge as to whether new financing would be more or less expensive. The second part provides empirical evidence in support of this theme.

Consider a possibility in which a f m ’ s current cost of debt and cost of equity are given by the linear forms:

kb = a1

+

p,L (1)

where kb, the cost of debt, and k, the cost of equity, are positive percentage rates, and L is the debt to total capitalization ratio of the

f a .

The positive constants, a, and a2, represent the cost of debt and cost o f equity if the fm carries no leverage, and

pb

^and

p,

represent the bondholder and stockholder sensitivity to leverage.’

Since the after-tax weighted average cost of capital, W?, is given by

W‘ = kJl-L)

+

kbL(I-rc), ( 3

where 2, is the applicable corporate tax rate, substituting from (l), (2) to (3)2

Now consider the following po~sibilities:~

Possibility I:

pb

=

p,

⁼ 0 or kp k, are notfictions of L;

Possibility 2:

pb

0,

p,=

0 ^or only kb is afunction of L;

Possibility 3:

p,

^> 0,

&=

0 or onZy ks is afunction of L;

Possibility 4:

P, , p,

0 or kb _and k, are functions of L.

If Possibility 1 describes the capital market environmenf (4) simplifies to

W’ = at

+

(a,

-

^at

-

^{a,+ (5)}

Leverage has an impact on the average cost of capital as long as ^al/at >< (I-rc). In the event that, (a, /al >

(I-rc),

the L coefficient is negative, and the well ^known Modigliani and Miller (1963) implication will hold wherein the average cost of capital is a linear declining function of le~erage.~ Any equilibrium with positive IeveIs of equity and [(%/al > (I-rc)] and

[pyps=O]

will violate the managerial principle of value maximization.

If Possibility 2 describes the capital market environment, from (4) we obtain W = a2

+

(a,

-

^a2

-

alrc)L

+

P,(I

-

^{T~)L’. (6)}

The sufficient conditions for a U-shaped ^{W g} function (SW’AL < 0, 62mX/6Lt ^> 0) is L < (az

-

^{a, (I- zc )) /2& (I}

-

^rc). The second order condition

is

^{satisfied as long}

as 0 C T, C I, since by definition,

pb

^> 0. A static tradeoff-consistent (convex) cost of capital function is thus feasible without the cost of equity being a function of L.

If Possibility ³ describes the capital market environment,

W‘ = at

+

(a,

+ p, -

^a2

-

^{a, s c ) ~}

- p&’.

(7) The sufficient condition for a negatively sloped W’ function is

L

C Ipx-az-al (I-t,)J/2ps.5 The second order condition for a convex W’ function

is

not met since SzmX/8L2=-2p, (and since

p,

^> 0 by definition). Thus _{a convex}

F

function is not feasible if ^{kb is} not a func6ion o f L.

The above illustrations of the cost of capital function under alternate sets o f financing environments helps to highlight the importance of (i) understanding that a relationship between W‘ and

L

does not necessarily imply a relationship between

L

and the individual parts (k,

,

kb) in W

‘;

and, (ii) understanding that the conditions for the convexity of the W’ function may be far more complex than the simple argument that cost o f equity and debt rise with the issuance of debt. For instance, it has been demonstrated that if possibility 4 describes the capital market environment, the necessary condition for convexity of the ^W regression is that cost of debt is more sensitive to leverage than cost of equity.

We now present some evidence relating to the temporal instability of the cost of capital function for a sample o f 15 1 U.S. companies over the 1973 through 1990 period. The 15 1 companies represent NYSE f m s for which monthly stock price information is continuously available on the tapes provided by the Center of Research in Security Prices (CRSP) over the 1971 through 1990 period.

Tests are conducted in the framework of the regression,

n

W'= CLo

+ p, ^L, + p* Lf +c ^{pi c, +}

^E,

^,

S

where

FF

represents the cost of capital, L represents financial leverage, and C represents the set of firm-specific variables thought to impactthe f m s ' cost of capital.

The squared leverage tern allows for the possibility o f ^a nonlinear relationship between leverage and cost of capital, as implied by distress cost theories.

Cost o f capital is defmed as Wv'= k,(S/(S+B))+k,(l-r)

@/(S+B)),

where

k,

_{is the}

estimated cost of equity, kd is the cost of debt (interest expense/debt),z ) is the marginal tax rate, S is the market value of equity, and B is total debt. Financial leverage is measured by @B/(S+B)). The variables B, kd, and C, over the interval are obtained from the 1991 COMPUSTAT tapes. The time series on the cost of equity for each fmis estimated from monthly CRSP data by employing the Capital Asset Pricing Model, k~=RF+pi(Rm-RF), where

RF

represents the yield on the one year t-bill, and

%,

the return on the S&P 500 index (e-g., Lintner (1965)).'

Table 1 presents the results from the regression model for data that is aggregated within four intervals. The computed F-values from the Chow (1960) and Fisher (1 970) tests for the equality of coefficients from contiguous regressions are provided along- side the results from each regression.

The coefficients for

L

are negative for all the interval regressions, consistent with the hypothesis of a cost savings from leverage. However, the notion of a convex relationship is not suggested. The

L

coefficient is insjgnificant for 3 of the 4 regressions, and is negative for the 1979-1981 regression. Several of the control variables ^are found to contain explanatory power. F& instance, the size coefficient is significantly negative for 3 of the 4 regressions,,and the growth coefficient is significantly positive in 2 regressions. However, it is notable that the coefficients for some of these variables are mixed. For instance, the uniqueness coefficient is alternately negative and positive.

Therefore, casual analysis o f the t-statistics would indicate that the relationship between W and

L,

and between W and C are unstable over time. There is also considerable fluctuation in the adjusted-R2 statistics, indicating that the system has temporally inconsistent explanatory power. The results from the Chow-Fisher tests further verify the instability of the systems estimated. The statistics that test the null hypothesis that all the coefficients fiom contiguous regressions are identical (Chowl), and the statistics that test the null that the

L

_{and L2} coefficients from contiguous

The Chow1 and Chow2 F-statistics are computed to be significant at the 1 percent level of significance. Thus, there is strong evidence to indicate that both the determi- nants of W1 and the relationship between WL and

L

are unstable over time.

The instability of the cost of capital function established in the present study does mean that even neutral leverage decisions that seem to serve no material purpose in one environment, may suddenly acquire or lose value in another environment. Given the generally episodic and costly nature of recapitalizations, the observation of an unstable

FP

function suggests that f m s may not realistically be able to sustain value-neutral capital structures, ^as envisioned by the Static Tradeoff framework.