Operating leverage, financial leverage, and equity risk

Journal of Banking and Finance 7 (1983) 197-212. North-Holland Publishing Company

OPERATING LEVERAGE, FINANCIAL LEVERAGE, AND EQUITY RISK Lucy H U F F M A N The City University of New York, New York, N Y 10021, USA Received June 1981, final version received June 1982 The analysis investigates the combined leverage effect of a fixed capacity decision (fixed cost) plus debt on the risk of equity returns. It is argued that the traditional DOL-DFL calculation is incorrect. A correct calculation is given, using the fact that the capacity decision is endogenous to the firm's decision process. The analysis reveals that the capacity decision partially offsets the effect on equity risk of increasing business risk or debt. However, this ability is lost at high levels of debt.

1. Introduction

A firm which makes a fixed commitment out of uncertain revenues has thereby increased the risk of the cash flow to equity. The effect, referred to as 'leverage', is measured as an elasticity defined by writing the risk of the cash flow to equity as a multiple of the revenue risk, the 'business risk'. That is, trresid~ =r/a,

(1)

where tr is the variance of percentage changes in underlying revenues, O'residual is the variance of percentage changes in cash flow to equity, r/ is the risk elasticity, the percentage change in the cash flow to equity per unit percentage change in revenues. In particular, the firm with cost per unit of (variable) production factors v, fixed production cost F, selling its product at price p, and producing x units, is traditionally said to have degree of operating leverage, DOL, 1

x(p-o)

DOL=x(p-v)-F"

(2)

Thus if the firm fixed a production cost by fixing the production capacity, the elasticity > 1. 1See Weston and Brigham (1981, pp. 571-574) for a discussion of the traditional form of the leverage effects. 0378-4266/83/$3.00 © Elsevier Science Publishers B.V. (North-Holland)

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L. Huffman, Operating leverage,financial leverage, and equity risk

Similarly, a firm having promised a payment D to its outstanding bondholders out of an uncertain profit EBIT= x ( p - v ) - F has increased the risk of the cashflow to its equity holders which is usually measured by the degree of financial leverage, DFL, DFL=

EBIT (EBIT-D)"

(3)

If the firm commits itself to both types of fixed payment so that the financial leverage is superimposed on a profit which is itself riskier because of the commitment to a fixed cost F, the elasticity for the 'combined leverage effect' is traditionally given by the product of eqs. (2) and (3) for DOL and DFL, eq. (4): z

(combined leverage effect)=

x(p-v) x(p- v)- F- D"

(4)

The formula for r/in eq. (4) suffers from three inaccuracies. The first is that the DOL calculation assumes the fixity has no effect on output or output margin. In fact, the essence of fixity is that it places an upper bound on the firm's ability to produce, i.e., a capacity production level. Consequently, prices are forced to adjust to restore equilibrium when realized demand at price p is larger than the capacity production level. The true formulation for t/cannot be a simple timeless function of revenue and fixed costs. The second flaw in eq. (4) is that the fixity is considered exogenous whereas, in fact, it is endogenous to the decision environment of the rationally managed firm. Since F is endogenously determined, it is likely that F differs for each level of business risk. Consequently, r/ is likely to be a function of a. The third arises from the fact that, in reality, an interaction exists between the outstanding debt and subsequent investment decisions. Conditions sufficient to produce this interaction have been described by Myers (1977). In particular, given a debt payment due in one period, the equity will refuse to undertake current investment projects whose PV is insufficient to cover the investment cost plus the forthcoming debt charge. The bondholders are assumed unable to intervene) Again, since F is endogenously determined, it 2The 'combined leverage effect' [Weston and Brigham (1981, p. 573)] shows the total leveraging effect on cash flow to equity of a given percentage change in revenues. The operating leverage causes such a change to have a magnified effect on EBIT. If financial leverage is superimposed on operating leverage, changes in EBIT have a magnified effect on cash flow to equity. air has been pointed out that ff there are only two claimants to the firm's assets, an agreement satisfactory to both can be concluded such that the equity is induced to take on all positive NPV projects. However, recent analysis has demonstrated that this is not guaranteed if there are more than two sources of capital. See Aivazian and Callen (1980) for a discussion of these issues.

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199

cannot be the same for different levels of outstanding debt. Thus the dependence of r/ on D is more complex than as specified by eq. (4). Because of these three flaws the DOL-DFL treatment cannot be the correct analysis of the effect of debt and production fixity on the risk of equity returns given a level of business risk. The intent of this paper is to correctly specify the functional form of r/in eq. (1) and to examine the effect of the exogenous parameters, business risk a and outstanding debt D, on r/. To accomplish this, the analysis proceeds in two stages. In the first stage, the level of capacity and its comparative static properties are derived. In the second stage, the correct functional form of r/is derived and the behavior of r/ with respect to the exogenous parameters D and a is examined.

2. The model

Suppose the firm, a monopolist, faces a random (after-tax) revenue realization process from sales given by

(5) in which q is output quantity, p(q) the (after tax) price as a function of quantity, equals ~f(q) and ~ is a random demand parameter independent of the firm, changing through time. Changes in ~ are presumed to follow a stationary random walk. The firm faces a linear cost function cq. The firm is additionally assumed to have debt outstanding now at time t = 0 such that the principal D1 plus interest 11 i s due at t = 1.4 The firm thus has a tax shield from the interest equal to its tax rate T multiplied by 11. Assume the shield to be riskless. 5 Then the effect debt repayment is D = ( 1 - T)I1 +D1. The fixity in the decision structure arises from the need to decide on the level of production capacity in advance of the demand realization. To model this, assume the equity must choose at t--0 the level of capacity x with which to produce at t = l . Then, one period from now, the demand parameter gl at t---1 is realized and at that time the equity makes an optimal output decision q*(1) with q*
200

L. Huffman, Operating leverage, financial leverage, and equity risk

2.1. The production decision q* at t = 1

At t = 1, cash flow to the firm as a function of production q is stated by eq. (6), given demand parameter realization al: (6)

R(q),= 1 - - c q = a l q f ( q ) - - c q = q P l ( q ) - - c q . The decision q* is made so as to maximize equity cashflow at t = 1, max R(q), =1 - c q - D,

(7)

q

subject to q < x. The first-order conditions are alf(q*)+alq*f'(q*)--c--21=O,

(8)

21(q*-x) =0,

where 21 is the Kuhn-Tucker multiplier, the shadow price of extra capacity. It is 0 for q * < x and is positive and e q u a l to ~ l f ( x ) + a l x f ' ( x ) - c if production is constrained to be at capacity, q * = x . Thus the first-order conditions (8) are equivalent to the familiar marginal revenue = marginal cost condition, in which marginal cost properly equals operating cost c plus capacity boundedness cost 21. On substitution of eq. (8) into eq. (6), cashflow to the firm is found to be ~1 = R l ( q * ) - c q * = 21q* + alq*2f'(q*),

=o+~lq,2lf,(q,)l

for

q*
21 = 0 ,

=21x+~lxZlf'(x)

for

q*=x,

21>0.

(9)

From eq. (9) it is seen that the fh-m's cashflow consists of two components, (1) a 'marginal cost' component (generated if the firm priced at marginal cost), plus (2) a monopoly rent component dependent on if(q). These two components are graphically depicted as functions of al contingencies in fig. 1. The second component exhibits a pattern of behavior across al contingencies similar to that of the 'marginal cost' component (although more analytically complicated). Thus the results of the analysis of both components can be achieved by restricting the analysis to either component. Since the intent of this paper is to develop the effect of the fixity in as simple a way as possible, the analysis will focus on the first component only. 7 Then 7For the remainder of the paper, the firm is assumed to choose quantity (subject to capacity limit) such that price is set equal to marginal cost. See Huffman and Thomadakis (1982) for the analysis of the true monopolist finn (ignoring the capital structure problem).

L. Huffman, Operating leverage,financial leverage, and equity risk

I

i

201

(2)

ot1 Fig. 1. The behavior of the 'marginal cost' component (1) and the monopoly rent component (2) as a function of el.

cashflow to the firm as a function of ~1 contingencies is given by eq. (10), derived from eq. (9) by setting f'(q)= O, nl = R l ( q * ) - c q * = 0

for

q*
=21x

for

q* = x,

where 21 = e t f ( x ) - c = p t ( x ) - c. The cashflow to the equity at t = 1 follows from the solution to eq. (10). For those states of revenue such that q* O. Optimal production q*, firm cashflow rr and equity cashflow ne are depicted as a function of demand parameter at in figs. 2, 3 and 4 respectively. The level ~*(x) is that value at which capacity is fully utilized and revenue just covers cost cx; Rl'*(x) =

p'f*(x)x-cx=O.

(11)

The level ct~(x) is that value at which capacity is fully utilized and revenue is SNote that the income to the firm in this case is zero whether the production is actually undertaken or not. 9The equity owners, of course, will not make any production decision q * ~ 0 as long as q* is such that their return is zero; i.e., as long as

pl(x)- cx- D< O. Since the debt is due at t = 1, if the equity owners elect to go into default on the debt, the debt owners are able to gain control and choose q* optimally. Thus the decision q* is made so as to maximize the value of the firm at t = 1.

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L. Huffman, Operating leverage, financial leverage, and equity risk

q~

I ×

..................

[

Or*

Fig. 2. The behavior of optimal production q* as a function of ~q.

7r

/

tx1

£l(e -N-

Fig. 3. The behavior of firm cashflow n as a function of ~1.

!

11"e

Or* *

Fig. 4. The behavior of equity caslfflow n, as a function of 0tz.

sufficient to cover both cost cx and the debt obligation D: R'~(x)= p ' ~ ( x ) x - c x - D =O.

(12)

2.2. The capacity decision x* at t = 0

As seen b'y fig. 4, the equity cashflow follows the pattern of returns to a simple option with exercise date t = 1. At that time cashflow to equity is zero unless demand is such that revenue at capacity production exceeds operating cost plus debt (or ~1 >~*)- For demand levels such that revenue exceeds operating cost plus debt, equity cashflow is linearly proportional to the realization of the ~1 parameter. For demand such that ct1 <~t]', revenues at capacity production do not exceed operating cost plus debt and the equity receives nothing. At t = 0 then, the equity possesses an option on the underlying revenue at capacity production from a demand realization ~. The exercise time is t = 1 and the exercise price is operating cost at capacity production plus debt, c x + D . Assume that the variance of revenue is known, that revenue and aggregate wealth are jointly lognormally distributed and that utility functions

L. Huffman, Operating leverage,financial leverage, and equity risk

203

exhibit constant proportional risk aversion. Then the value of this option at t may be expressed as 1°

S(x, t) = r- 1[R(x)N(dl) - (cx + D)N(d2)],

where

(13)

R(x) =p(x)x=the current value of revenue from capacity production given the current demand realization, = the standardized normal cumulative probability density function, N(d)

dl = 1/a[lnR~x)/(cx + O)+a2/2], d2 =dl-~x/t, t o.2 r

= number of periods to expiration = 1 in this model, = the variance of In [1 + the rate of change in R(x)], = 1 + the risk free rate of return over the period.

The optimal capacity commitment is determined at a level x* such that the NPV of the decision is maximized. 11 If production capacity is assumed to cost k per unit at t = 0, the N P V of the decision to the equity is S ( x ) - k x . Thus the optimal x* is the solution to the first-order eq. (14)

OS(x*)/Ox=k

(14)

together with the second-order condition (15)

t?2S/cgx2 < 0. In appendix A it is shown that may be written as

(15)

~2S/~x2 is

less than zero and that eq. (14)

r- 1[p(x*)N(d*)- cN(d*)] = k.

(16)

Eq. (16) yields the optimum capacity level x*.

2.3. Properties of the optimal fixed capacity decision Given the level x* implicit in eq. (16), attention is directed to the comparative static properties of x* with respect to the exogenous parameters, outstanding debt D and business risk a.

2.3.1. The effect of debt The effect of the presence of D on the capacity decision x* made by the 1°See Brennan (1979) for the necessity of these assumptions. 11This decision is made so as to maximize the net value to equity, not to the firm, since it arises prior to the time at which the debt can legally assume control.

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L. Huffman, Operating leverage, financial leverage, and equity risk

equity owners at t = 0 can be analyzed by examination of the derivative ax*/aD. Since x* is not explicitly obtained, the effect is examined by taking the implicit derivative of eq. (16) with respect to D and substituting eq. (B.2) from appendix B, yielding eq. (17),

ax* -(~/ao)[aS(x*)/ax*] r- lS(d';) a--b- = (a/ax*)EaS(x*)/ax*] = a~S/ax .2"

(17)

Since g2S/ax 2 was shown to be negative [see eq. (15)!, ax*/aD is negative. An increase in outstanding debt D causes a decrease in the optimal fixed capacity commitment made by the equity.

2.3.2. The effect of a The effect of a change in the levered firm's business risk is given by the derivative ax*/~tr. The partial ax*/atr is again developed by taking the implicit derivative of eq. (16) with respect to a, and substituting from appendix B eq. (B.4), yielding eq. (18),

ax* aa

-(~/a~)[aS(x*)/ax*] (O/Ox*)[aS(x*)/ax*]

r- lZ(d'~)(1)/x*~)(cl';-~cx/O) a~S(x*)/ax .2

(18)

Since a2S(x)/dx2O so long as d~ < trcx*/D, or so long as

R(x*) < (cx* + D)exp (a2(~ + cX */D) }.

(19)

If D = 0, ax*/&r is always greater than 0. However, an increase in the level of debt relative to the value of d* leads to the reverse behavior if d~ > cx*D.

3. The development of The first stage of the analysis having been completed, it is now possible to derive the correct formulation of t / a n d to examine its behavior with respect to the firm's exogenous parameters D and tr.

3.1. The formulation of rl The correct formulation of t/is given by the relation

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205

r/= percentage change in value of equity divided by percentage change in revenue,

=(dS/S)/d(xp(x)/p(x)), =1/(1

-

(20)

(cx+D)N(d2)~

This formulation is different from the usual DOL-DFL analysis given in eq. (4) in three essential ways. The first is that t/is not timeless but varies with the time-related arguments dx and d2 of the cumulative normal probability density functions. The arguments arise from the effect of the fixed capacity production decision on output. The second is that r/ is not independent of business risk a as in eq. (4) but is rather a function of a. This functional relationship arises from the influence of a on both x* and dx and d2. The third is that, since x* is also a function of debt D, r/ is a more complex function of D than specified by eq. (4).

3.2. The behavior of rl To see more clearly the behavior of r/ with respect to the exogenous parameters D and a, the comparative static properties of r/are developed.

3.2.1. The behavior of r~ with respect to debt The behavior of r/ with respect to debt is given by the derivative dr//dD, presented in eq. (21) dr/ dr/ dr/ dx* -- -d D - 0D ax* dD"

(21)

The total derivative is the sum of two effects. The direct effect is 8rl/dD. The indirect effect on r/, ar//dx*.ax*/dD, arises from an optimal change in x* as D is increased. The direct effect component Or//dD is derived and is shown to be greater than zero in appendix C, eq. (C.6). The indirect component dr//dx*, dx*/OD is a product of eq. (17) and eq. (C.2) from appendix C. The sign of eq. (17) is negative. The sign of eq. (C.2) is a function of the magnitude of D. For D less than D + = ]p'(x)]cx2/p(x)xlp'(x) I the sign is positive. Conversely, for D>D +, the sign is negative. Therefore the indirect effect is negative for D D ÷, the optimal capacity decision exacerbates, rather than attenuates, the effect of the increase in outstanding debt on equity risk.

L. Huffman, Operating leverage,financial leverage, and equity risk

206

The total derivative, the sum of these two effects, is r/2N(dz) [- . Z(dl) Z(d2.] r -1N(d2) d~l dD xp(x)N(dl)a Ltr +N(dl) N(d2)_] a2S/Sx 2 X

~12(cx+ D)N(d2) F

D

[p'(x){7{- .

Z(dO Z(d2)J N(d~) N(d2) "

t22)

By multiplying out and rearranging, eq. (22) simplifies to

an

n:N(d~) [

Z(dO z(a~)-]

d--D=xp(x)N(dO "La4 N(dl) N(d2)J x {p'(x)N(dt)4 ×I1

Z(d2)D~ax

x(cx-+DD) IP'(X)}']jp-~ r -t

(23)

N(d2)x(cx + D) l / ~2S/~x2

The sign of drl/dD is unambiguously positive for D>D ÷. To establish the sign of drl/dD for D
,tZn(d~) [

Z(d~) Zfd~)]

xp(x)N(dO~rLa 4 N(,lO N--~2)J is examined. Since [a+Z(dO/N(dl)-Z(d2)/N(d2)] is greater than zero [see Galai and Masulis (1976, app. I)], this coefficient is positive. Next, the sign of the term in curly brackets is examined. The denominator is less than zero [see eq. (15)]. The numerator is less than a2S/Ox2 (see appendix A for the expression 82S/0x2). Since both numerator and denominator are negative, the term in curly brackets is positive. Therefore, drl/dD is positive for all levels of debt. For D < D ÷ the direct effect is only partially attenuated by the indirect component, causing an overall increase in equity risk as debt increases.

Property 1. The direct effect of an increase in debt on equity risk is partially offset by the change in capacity in response to the increase in debt. The ability of the capacity decision to attenuate the increase in equity risk due to the increased outstanding debt is lost above a critical level of debt D ÷

3.2.2. The behavior of r1 with respect to tr The behavior of r/of a levered firm with respect to a is given by the total

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207

derivative dr//da, presented in eq. (24) dr/

Or/

Or/ ~3x*

d--a=a--a + ax* aa"

(24)

The total derivative is again the sum of two effects. The direct effect is Orl/Oa. The indirect effect on q, Otl/3X*.Ox*/Oa, arises from an optimal change in x* as business risk is increased. The direct component is derived and is shown to be less than zero in appendix C, eq. (C.4). The indirect component is a product of eq. (18) and eq. (C.2) from appendix C. The sign of eq. (18) is positive for current values of revenue R(x) and debt combinations such that eq. (19) holds. The sign of (C.2) is positive for D
dtl

tl2(CX+ D)N(d2) [Z(dz)da Z(d,)d2l

d--a=

xp(x)N(dOa L N---~2) N(dl) 3

,l~(cx+O)N(d2)[

D

xp(x)N(dOa Lx(cx+D)

Ip'(x)llF z(a,) z(d~)]

-p--~-j~a-~ N(d,)

N(d2)A

(25)

x [r. - l(Z(d2)D/ax)(d2-acx/D)] 02S/Ox2

On multiplying out terms and rearranging, using eq. (A.3), eq. (25) can be written as

drl d-a =

tl2(CX+ D)N(d2)[ Ad 2 + B ] xp(x)N(dOa L~~I3' where

LN(d2) N--~)A Ip'(x)N(dO LN(dz) N(d,)_][P'(x)lN(dO

(26)

L p(x) x(cx+O) '

N--~) ]

+[-Z(d2) . f Z(dl) )~ , ]p'(x)l'~qZ(d2) D" L xU(d2) +k-N-~l) +a cx+ D - V - - ~ ) J x The derivative dtl/da is negative as long as d2 > - B / A , or

R(x) >-(cx + D)r - x exp {-(0"2/2 + B/A)},

(27)

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208

that is, the revenue is larger than the discounted value of operating cost plus debt. Property 2. The direct effect of an increase in business risk on equity risk of the levered firm is partially offset by the change in capacity in response to the increase in a. This ability of the capacity decision to attenuate the effect on equity risk is lost if eq. (27) does not hold. 12 As long as eq. (27) holds, therefore, equity risk of the levered firm does not increase proportionately to business risk.

4. Conclusion The traditional specification of r/ is oversimplified due to the fact that it does not consider the endogenous nature of the capacity decision and its relation to production. By correcting for these weaknesses, r/is shown to be an explicit function of the business risk of the firm. It is also both an implicit, as well as explicit, function of outstanding debt. The implicit relationship arises from the dependence of the endogenous fixed capacity level on the debt. The analysis reveals two properties of the capacity decision made by the levered firm. Firstly, the capacity decision partially offsets the increase in equity risk caused by an increase in outstanding debt. This effect diminishes as the size of the outstanding debt increases. Secondly, the capacity decision partially offsets the effect of an increase in business risk. This ability diminishes if either revenue declines or the level of outstanding debt is increased.

Appendix A A.1. The derivation of eq. (16) From eq. (13) S(x) = r - 1 [ R ( x ) N ( d l ) - ( c x + O)N(d2)]; then eq. (14) becomes 0S k=~xx = r- 1[R'(x)N(dl)-cN(d2)] (A.1) + r- 1 R(x)Z(dl)-~x - (cx + D)Z(d2) Ox _J 12It can be shown that eq. (27) holds if D is not large compared to fixed cost cx.

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209

where Z(d)=the standardized normal density function= 1/x//~-exp {-d2/2}. Using the facts

OdI

Od2

I FR'(x)

-0x=aLR--~

c ] cxT+D ' and

R(x)Z(dl)=(cx + D)Z(d2)

(A.2)

(A.3)

[see Galai and Masulis (1976), app. I)] eq. (A.1) may be written as (A.4)

k=

OS =r-X[R'(x)N(dO-cN(d2)]. 0x

(A.4)

Since this analysis is not examining the monopoly rent component, R'(x) is set equal to p(x). Then (A.4) becomes eq. (16),

OS k = Ox = r- X[p(x)N(dl)-cN(d2)].

(16)

A.2. The sign of O2S/gx 2 The second derivative 02S/Ox 2, obtained from (16), is

e:s ~[p'(x)N(dO+p(x)Z(aO~f_~_cZ(d:)~d: F~x~=~Ox]]"

(A.5)

Note that Op(x)/Ox is not zero since 02S/Ox 2 is the description of the rate of change of the equilibrium condition, eq. (16). On using (A.2) and (A.3), eq. (A.5) can be written as (A.6) after combining terms

Ox--~=~ p'(x)N(aO+ Z(d~)[-p-~+x(cx-+D) " The second derivative is negative for

D Z(d=) [

D

]p'(x)r]

Since N(d 0 > N(d2) and d 2 > - Z(d2)/N(d2) [see Galai and Masulis (1976,

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L. Huffman, Operating leverage, financial leverage, and equity risk

app. I)], the above inequality holds if (A.7) holds,

d2>

IP'(X)lCr

D

(A.7)

D

p(x) J On substituting for d2, eq. (A.7) can be written as

R(x) > (cx + D) e-¢~r,

where

(A.8)

K =½-~ x

(cx--+D)

p-~d

That is, the sign of the second derivative is negative as long as revenue is greater than the discounted future operating cost + debt repayment.

Appendix B B.1. The derivative of OS/gx with respect to D Since from eq. (16), OS/Ox= r-l[xp(x)N(dO-cN(d2)], the derivative may be written as eq. (B.1),

8D Ox

xp(x)Z(dx)-~-cZ(d2)-~

,

(B.1)

where c~dl/OD=Od2/t~D = - 1/a(cx + D). Then on substituting (A.3), ~S

- r - t DZ(d2)

d---D0---~=

x(cx + D)"

(B.2)

B.2. The derivative of Six with respect to a The derivative may be written as (B.3)

00S

[

OtrOx =r-1 p(x)Z(dO

~

0d2],

-cZ(d2)--~a_]

(B.3)

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211

where Oda/&r=--d2/tr, Od2/Otr=-dl/a. On substituting eq. (A.3), eq. (B.3) may be expressed as 0 0S = r- 1 D Z(d2) Vacx ] X a [--D- - - d 2 "

(B.4)

O---aOX

Appendix C C.1. The derivative Orl/Ox Since

(cx + D)N(d2)~

xp(x)N(dl),l from eq. (20)

I/= 1/(1

0~]

?/2

{

(C.1)

Ox xp(x)N(dO eN(d2)+(cx + D)Z(d2) Od2

xp(x)N(d,)

(p(x)+ xp'(x))N(dO+ xp(x)Z(dO-~x j j.

On substituting eq. (A.2) and collecting terms, eq. (C.1) can be written as (C.2),

o ~=

Ip'( )l [

Z(d,)

xp(x)U(d0~ Lx(c~+O) ~ 3 L~4-g~ N(d~)3

(C.2)

Note that the term [ a + Z(dO/N(dl)-Z(d2)/N(d2)] >0 [see Galai and Masulis (1976, app. I)].

C.2. The derivative Orl/Otr The derivative &l/Oa is

Or/

r/2

[

ad2 (cx+O)U(dgZ(dl)(adl/a~)] ~ ]. ~-xp(x~(dO (cx+D)Z(d2) 0---~-

(C.3)

L. Huffman, Operating leverage,financial leverage, and equity risk

212

On substituting for Od2/~a and Odl/Oa from appendix B, eq. (C.3) may be written as 0q

rl2(CX+ D)N(d2) [g(dz)dl

Oa

xp(x)N(dOa 1_ N(d2)

Note that

Z(d2)dl

Z(dOd2 ] N(dO "

(C.4)

Z(dOd:q -J >0 [see Galai and Masulis (1976, app. I)].

C.3. The derivative Oq/OD The derivative Orl/ODis

07]

?/2 [

Od2

OD-xp(x)N(dO N(d2)+(cx +D)Z(d2)~ OD (cx + D)N(d2)Z(d~) Odx-] N(dl) ODJ"

(c.5)

On substituting for Od2/OD and Odl/OD from appendix B, eq. (C.5) may be written as Or/ r/2N(d2) [ Z(d,) Z(d2)] (C.6) on-xp(x)N(dOa a-~ N(da) N(d2)J" Note that, as above, the term in brackets is positive. References Aivazian, Varouj and Jeffrey L. Callen, 1980, Corporate leverage and growth: The gametheoretic issues, Journal of Financial Economics 8, Dec. Black, Fisher and Myron Scholes, 1973, The pricing of options and corporate liabilities, Journal of Political Economy 81, May-June. Brennan, M.J., 1979, The pricing of contingent claims in discrete time models, Journal of Finance 24, March. Galai, Dan and Ronald W. Masulis, 1976, The option pricing model and the risk factor of stock, Journal of Financial Economics 3, Jan.-March. Geske, Robert, 1977, The valuation of corporate liabilities as compound options, Journal of Financial and Quantitative Analysis 12, Nov. Geske, Robert, 1979, The valuation of compound options, Journal of Financial Economics 7, March. Huffman, L. and S.B. Thomadakis, 1982, On the theory of the firm under uncertainty: Fixed commitments and consequent asymmetries. Rubenstein, Mark, 1973, A mean-variance synthesis of corporate financial theory, Journal of Finance 28, March. Rubenstein, Mark, 1976, The valuation of uncertain income streams and the pricing of options, Bell Journal of Economics 7, Autumn. Weston, J. Fred and Eugene Brigham, 1981, Managerial finance, Seventh ed. (The Dryden Press, Hinsdale, N J). Myers, Stewart C., 1977, Determinants of corporate borrowing, Journal of Financial Economics 5, Nov.