Stock Options and Capital Structure

Transcription

1 y Richard D. MacMinn* and Frank H. Page** Revised March 996 *Department of Finance University of Texas Austin, Texas **Department of Finance University of Alaama Tuscaloosa, Alaama Comments from session participants at the 995 Econometric Society Meetings, the 995 Southwestern Finance Association Meetings and the 995 Korean Financial Management Association Meetings are gratefully acknowledged

2 0. Introduction The empirical evidence on sources of corporate financing strongly suggests that firms prefer internally generated funds to det and det to equity in financing their investment activities (see for example tale 4-3 in Brealey and Myers (99)). Additional evidence of this preference ordering over financing methods can e found in Agrawal and Mandelker (987), Hittle, Haddad et al. (992), Jensen, Solerg et al. (992), and Tamule, Bunys et al. (993). What is the economic rationale for this preference ordering? Explanations in the existing literature rest upon informational asymmetries etween corporate managers and investors and the agency costs these asymmetries generate. For example, (Myers and Majluf 984) argue that if investors are less wellinformed than the corporate manager aout the value of the firm's assets, then new equity may e under-priced y the market. If firms are required to finance projects y issuing equity, under-pricing may e so severe that new investors capture more than the net present value of the project, resulting in a net loss to existing shareholders. In this case, the project will e rejected even if its net present value is positive. This underinvestment prolem can e avoided if the firm can finance the new project using a security that is not so severely undervalued y the market. For example, ecause internally generated funds or safe det generate no agency cost, these methods of financing involve no under-valuation, and therefore will e preferred to equity y firms in this situation. Myers refers to this as a pecking order theory of financing. One prolem with this theory is that it does not explain the apparent preference of corporate managers for internally generated over all external sources of funds, including safe det.. An even more fundamental prolem with pecking order theories ased on asymmetric information and agency costs is that the conclusions of these theories are not 2

3 roust. Thus, the conclusions of these theories are sometimes at odds with the consistent empirical evidence supporting a pecking order. Despite the simplicity and appeal of Myers and Majluf's arguments, further theoretical analyses y Brennan and Kraus (987), Noe (988), and (Constantinides and Grundy 989) have cast dout on the Myers and Majluf explanation of the pecking order. These papers conclude that if the set of financing choices availale to the firm is expanded, then when faced with the situation considered y Myers and Majluf, firms do not necessarily prefer issuing straight det over equity. Moreover, these papers show that with the richer set of financing options, firms can resolve the under-investment prolem through costless signaling. In this paper we provide an alternative explanation for the prevalence of a corporate pecking order over financing methods. More importantly, we put the pecking order theory of capital structure on a different, and we elieve, stronger theoretical foundation. Myers and Majluf assume that the financing decision is made to maximize current shareholder value. Given less well-informed outside investors, the consequent agency costs, and a manager acting in the interests of current shareholders, the pecking order theory follows naturally and explains how the agency cost can e minimized. There are, however, two sources of difficulty in the Myers and Majluf approach. The first has een demonstrated y Brennan and Kraus, Constantinides and Grundy, and Noe: if the set of financing options is expanded, then in some cases the manager can reveal information to the outsiders costlessly, thus avoiding any agency cost. The second difficulty is more fundamental: in Myers and Majluf, as well as in much of the existing literature, the ojective function, used y the manager in making decisions on ehalf of the corporation, is assumed rather than derived. 2 In this paper, we endogenously derive Harris and Raviv (99) also provide a discussion of this literature. 2 There is a very small literature related to the corporate ojective function. John and John (993) take a given capital structure and ask what the optimal executive compensation package is. Brander and Poitevin 3

4 the manager's ojective function and show how it depends on the form of the manager's compensation contract. In order to give a concrete example of this connection, we derive the manager's ojective function for the case in which the manager's compensation includes stock options. This case is important ecause nearly 90% of Fortune 500 companies offer potentially lucrative incentives to their executives including stock options. 3 We then analyze the implications of the stock-option-ased ojective function for optimal capital structure decisions y the manager. In particular, we show that if the manager's compensation includes stock options with a positive exercise price, then the manager's ojective function is such that the manager prefers internally generated funds to det, and det to equity in financing the firm's investments. Thus, y directly addressing the issue of executive compensation and managerial ojectives, we provide a compensation-ased explanation for the pecking order theory of capital structure. The analysis here is a partial equilirium analysis in two important respects. First, we only consider the agent's prolem. In an expanded treatment of the principalagent prolem one would have to consider, from the perspective of the principal, the rationale for including stock options in the executive compensation package. Why is this form of compensation so pervasive? In MacMinn (995) this question is addressed. MacMinn (995) considers a firm which operates simultaneously in a competitive financial market ut an imperfectly competitive product market and demonstrates that, from the perspective of the oard of directors (who collectively act as the principal and represent the interests of shareholders), it is optimal to select a managerial compensation package that includes stock options ecause such a compensation package strategically (992) consider the manager's ojective function given a compensation scheme that includes a onus. There is also a much older thread to this literature in Ekern and Wilson (974 and Radner (974). 3 See Joann S. Luin, "The American Advantage," Wall Street Journal, Wednesday, April 7, 99, R4. 4

5 commits the manager to a stronger course of action against the firm's competitors in the product market. Alternatively, if the product market in which the firm operates is competitive, then the inclusion of stock options is no longer optimal. Thus, the presence of imperfectly competitive product markets provides the rationale for the pervasive use of stock options in executive compensation. There is another respect in which the analysis here is a partial equilirium analysis. In valuing the firm's capital structure, the manager uses a vector of statecontingent claims prices. Thus, the manager values the firms capital structure as if markets were complete. One important feature of our analysis is that our conclusions regarding stock options and capital structure do not depend on the specific vector of state-contingent claims prices used y the manager. In particular, no matter what vector of state-contingent claims prices is used y the manager in valuing the firm's capital structure, the conclusion is the same: the manger prefers internally generated funds to det, and det to equity. Thus, our conclusions are roust with respect to market incompleteness. The paper is organized as follows: In section one, the financial market model is constructed and used to value ond and stock contracts. In section two, the stock option contract is introduced and used to derive the manager's ojective function. There we show that the manager makes decisions on corporate account to maximize the value of the stock option package. In section three, we allow the manager to make financing decisions and show that the manager prefers internal to external financing, all det to all equity, and all det to any comination of det and equity. In section four, we provide some concluding remarks and conjectures. 5

6 . Financial Markets Consider a competitive financial market operating etween dates now and then. Consumers make portfolio decisions now that determine saving, or equivalently, consumption now and then. Corporations make investment and financing decisions now that determine the return on financial instruments then. Given risky investments, this analysis is a generalization of the classic Fisher model... Notation and Assumptions t (Ω, F, λ) P u(, ) m 0 m c 0 c (ω) I 0 d 0 π 0 = 0, ; t indexes dates now and then, respectively = state space = manager's proaility measure over states = manager's utility function over R R = income now = income then = consumption now = consumption then in state ω Ω = investment now = dividend payment now = firm's earnings now Π (ω) = firm's earnings then given state ω, and investment I 0 N = the numer of common stock efore any new issue now m = the numer of new shares of common stock issued now = the promised payment to ondholders then 6

7 [A-] The manager's utility function is concave, increasing, and everywhere differentiale. [A-2] The proaility measure P is asolutely continuous with respect to λ, i.e., P << λ. [A-3] The corporate payoff function is non-negative, i.e., Π (ω) 0 for all ω Ω..2 Value Let L (λ) denote the set of all λ-essentially ounded, real-valued, F-measurale functions defined on Ω, and let υ( ) L (λ). [A-4] (a) For any E F, an asset with a payoff of one dollar for any ω E and zero otherwise, has a market value υ(ω) dλ(ω) 0. E () υ(ω) dλ(ω) = Ω, where r is the rate of return on a safe asset. +r Define a zero coupon ond contract as a promise to pay one dollar then. In the asence of any stock options, a common stock is a promise to pay d 0 /N dollars now to holders of record and max{0, Π - }/(N + m) dollars then. 4 Suppose an investor holds x o onds and y o stocks now. Suppose the investor has an income pair (m 0, m ), and that the corporate ond issue is safe. In its classic form, the Fisher model specifies the investor's prolem as a constrained maximization prolem in which the investor selects 4 Since the firm selects a dividend now as well as raising funds to cover its investment of I 0 dollars, the sequence of events must e specified. Here we suppose that the firm makes its dividend payment to the N holders of record and then issues new securities. All stock prices calculated here are therefore ex-dividend prices. 7

8 consumption now and then to maximize expected utility suject to a udget constraint. By [A-4], the investor's prolem is suject to c 0 + Ω maximize u(c 0,c (ω)) dp(ω) Ω c (ω) υ(ω) dλ(ω) + m υ(ω)dλ(ω) +x υ(ω)dλ(ω) +y Ω Ω =m 0 +p ( x o x)+p s ( y o y)+y o d 0 N Ω Π (ω) N+m υ(ω)dλ(ω) The udget constraint sets the risk adjusted present value of consumption equal to the risk adjusted present value of income. If the investor liquidates the ond position then x equals zero and p x o dollars are raised now. Similarly, if the investor holds on to all the onds then no dollars are raised now from ond sales. The first order conditions, or equivalently, the no aritrage conditions, yield the ond and stock prices now as p = υ(ω)dλ(ω) = Ω +r and p s = Ω Π (ω) N + m υ(ω) dλ(ω) Note that the value of ond issue is D() = /( + r) and the value of the stock issue is S(, m) p s ( N + m) = ( Π (ω) )υ(ω)dλ(ω) () Ω Alternatively, for any E F let, 8

9 µ(e) = ( + r) υ(ω)dλ(ω) (2) E Given [A-4], µ is a proaility measure and for Π( ) - L (λ) ( Π (ω) ) dµ(ω) = Π (ω) Ω Ω ( + r) υ(ω)dλ(ω) Thus, the stock market value now can e equivalently expressed as S = + r ( Π (ω) ) dµ(ω) (3) Ω 2. Stock Options and the Manager's Ojective Function In the economics literature, agents are assumed to pursue their self interests suject to constraints. Given uncertainty, this ehavior is often specified as a constrained expected utility maximization prolem. The theory of the firm is an exception. The fictitious agent known as the firm is assumed to maximize profit. Under uncertainty, the theory of the firm has een specified in an expected utility framework (Baron 970; Sandmo 97; Leland 972; Stiglitz and Greenwald 990) ut the fictitious agent is assumed to maximize the expected utility of profit. These theories of the firm have not een derived from the more primitive notion of the pursuit of self interest. Here the manager is given the role of an investor as well as that of a manager and assumed to make portfolio decisions on personal account and financing decisions on corporate account in the pursuit of self interest. The analysis here shows that the ojective function of the firm may e derived rather than assumed. The corporate 9

10 ojective function, or equivalently, the ojective function that the manager uses to make decisions on ehalf of the corporation, flows from a Fisher Separation result. As with all Fisher Separation results, the decisions made on corporate account are independent of the manager's proaility and risk aversion measures; unlike other Fisher Separation results, the analysis here shows that the manager does not make decisions that maximize the risk adjusted present value of the shareholders' stake in the corporation. Equivalently, this analysis shows that the manager does not make decisions to maximize the current shareholder value. Since stock options are an important component of management compensation schemes, we assume that the manager is paid a safe salary now and then and receives a stock option grant now. 5 Stock options are simply call options or warrants that give their owners the option to purchase a certain numer of stocks then at an exercise price that is specified now. That exercise price is typically the common stock price on the issue date. An option is "in the money" then if the exercise price is less than the stock price; otherwise it is "out of the money." The exercise event defined in the susequent analysis is equivalent to the in the money event. 2.. Additional Notation e = the exercise price then on each stock option n = the numer of stock options issued to the manager now p w = the stock option or warrant price now W p w n, the value of the warrant package now 5 It is possile to allow some trading in options now without affecting the results that follow as long as the manager must maintain some positive position in the stock options. There is, however, a period of several years efore the options in a particular grant can e exercised. 0

11 2.2. Stock Options and Financial Market Values The manager's compensation package includes a safe salary now and then, (m 0, m ), and n stock options with an exercise price of e dollars per option. The payoff on each option is given y max 0, Π (ω) + en N + m + n e (4) Let E(, e, m) = {ω Ω Π (ω) + e (N + m)} e the exercise event, or equivalently, the event that the stock option is in the money. The price of the stock option, equivalently, warrant, is p w (, e, m) = Ω max 0, Π (ω) + en N + m + n e υ(ω)dλ(ω) Π = (ω) + en e N + m + n υ(ω)dλ(ω) (5) E(, e, m) Letting ξ n/(n + m + n) denote the proportion of the firm acquired y the manager upon exercising the options, it follows that the option or warrant value is W(, e, m) = E(, e, m) ( ξ ( Π (ω) + en ) en)υ(ω)dλ(ω) = ( ξ ( Π (ω) ) ( ξ)e )υ(ω)dλ(ω) E(, e, m) (6) where E e n. Next, let

12 B() = {ω Ω Π (ω) < } (7) denote the ankruptcy event and N(, e, m) = {ω Ω Π (ω) + e (N + m)} (8) denote the no ankruptcy and no exercise event. The aggregate det value is D() = Π (ω) υ(ω)dλ(ω)+ υ(ω) dλ(ω) (9) B() Ω\B() and the common stock price is p s (, e, m) = N(, e, m) Π (ω) N + m υ (ω)dλ(ω) (0) + E(, e, m) Π (ω)+en N + m + n υ(ω)dλ(ω) Similarly, the stock value is the stock price times the numer of shares availale now, i.e., p s (N + m) and 2

13 S(, e, m) p s (, e, m) ( N + m) υ = Π (ω) N(, e, m) (ω)dλ(ω) () + ξ E(, e, m) ( Π (ω)+en ) υ (ω)dλ(ω) Oserve that the corporate value V that is defined as the sum of the det, equity, and warrant values, i.e., V D + S + W, is V = D() + S(, e, m) + W(, e, m) = Π (ω) υ(ω)dλ(ω) (2) Ω This is a generalized version of the 958 Modigliani-Miller Theorem (Modigliani and Miller 958). Like the Modigliani-Miller theorem this result shows that corporate value is independent of the structure of the financing decisions, i.e., the investment can e financed with any comination of det or equity without affecting corporate value. It also shows that corporate value is independent of the provisions of the stock option package, in particular, the exercise price The Manager's Ojective Function The manager makes decisions on personal and corporate account to maximize her own self interest. Suppose the manager's feasile set of consumption possiilities then is Λ = {c ( ) c ( ) is F-measurale and 0 c (ω) ζ (ω) a.e. [λ]}, where ζ (ω) L (λ). The prolem on personal account is the selection of (c 0, c ( )) R + Λ. The prolem on corporate account is the selection of (, m) such that π 0 + D() + p s (, e, m) m = I 0 + d 0. 3

14 D() represents the dollar amount raised with a new ond issue while p s m represents the amount raised with a new stock issue. The manager's prolem may e expressed as maximize Ω u(c 0,c (ω)) dp(ω) suject to c 0 + c (ω) υ(ω) dλ(ω) =m 0 +W(, e, m) + m υ(ω) dλ(ω) Ω Ω (3) and π 0 + D() + p s (, e, m) m = I 0 + d 0 The equations in (3) represent the udget and the financing constraints. It is easy to see that in choosing a pair (, m) for the firm, the manager will seek to maximize the value of the stock option package W. Hence, the manager will make decisions on corporate account to maximize W(, e, m) suject to π 0 + D() + p s (, e, m) m = I 0 + d 0 (4) From (6), (9), and (0) it follows that this is a Fisher Separation result. The ojective function in (4) does not, however, generate a unanimity result and it is not consistent with the standard assumption in the finance literature that managers maximize current shareholder value. The ojective function here does provide some insight into how and why financing decisions may e relevant. 3. Financing Decisions Given the ojective function in (4), the corporate manager may choose to finance the investment with retained earnings, det, equity, or some comination of the 4

15 three. We consider the retained earnings versus det, the det versus equity, and then generalize to allow for any comination. Let µ( ) e the market pure proaility measure induced y the value operator υ( ) and let F( ) e a risk adjusted proaility distriution induced y Π ( ). Then F(π ) = µ{ω Π (ω) π } Note that F is not the manager's proaility distriution. 6 Rather, F is a risk adjusted proaility distriution that emeds the proaility and risk aversion measures of all agents through the value operator υ( ). [A-5] F( ) has a continuous density f( ): [0, ) R + such that lim π π f(π ) = 0. Given [A-5], the value of the manager's stock option package that is specified in (6) may e equivalently expressed as W(, e, m) = + r n ( N + m + n π + en ) en f(π )dπ (6) 3. Retained Earnings versus Det Suppose the manager finances the investment with retained earnings or det. The financing constraint is D() = I 0 - (π 0 - d 0 ).. Since the det value is an increasing function of the promised payment, it follows that inceasing the use of retained earnings decreases the necessary promised payment. Consider the following two cases: 6 The manager's distriution function would e a function G(π ) = P{ω Π (ω) π }. 5

16 Case I: Let the dividend d 0 e zero. Equivalently, let the retained earnings e π 0. If π 0 I 0 then the payment on a ond issue is determined y the condition D( ) = I 0 - π 0. In this case the stock option payoff is max{0, ι (Π - ) - ( - ι) E}, where ι n/(n + n). Case II: Let the dividend e equal to the earnings now, i.e., d 0 = π 0. Suppose the firm uses a det issue so that D( 2 ) = I 0. In this case the stock option payoff is max{0, δ (Π - 2 ) - ( - δ) E}, where δ = n/(n + n). Note that ι δ. Since D() is increasing in, a comparison of the two cases makes it clear that the stock option value given internal finance is greater than that given a det issue, even if the det issue is safe! This result is stated in the following proposition and demonstrated in figure one for the case in which π 0 = I 0. Proposition. Given the stock option compensation scheme, the manager prefers retained earnings, or equivalently, internal equity, to det in financing an investment. This result is different than that of Myers and Majluf. The Myers and Majluf analysis shows that retained earnings and safe det are equivalent in a pecking order. Since Myers and Majluf motivated the pecking order with the asymmetric information assumption, the equivalence of these two forms of financing follows easily ecause neither form of financing is sensitive to the difference etween insider and outsider information; hence, neither form of financing generates an agency cost that determines a position in the pecking order. 6

17 Figure : Retained Earnings versus Bonds ι π W i δ (π ) W d ( - ι) E ( - δ) E π The result in proposition one is not motivated y an asymmetric information assumption. Rather, it depends on the assumption that the manager pursues her self interest. By using retained earnings to finance the investment rather than det, the manager generates greater earnings then and so a greater stock price then when the options can e exercised. It should also e noted that if the exercise price is set equal to zero then the warrants ecome stock ut the proposition still holds. 3.2 Det versus Equity Next, suppose that the investment I 0 exceeds the firm's aility to generate funds internally; let I 0 - π 0 > 0 and consider the case of det versus equity financing. By hypothesis in this section, the manager selects either det or equity to finance the I 0 - π 0 dollars. If the manager maximized current shareholder value suject to the financing constraint then the manager would e indifferent to the choice of det versus equity. The manager who maximizes the value of a stock option package suject to the 7

18 financing constraint, however, is not indifferent, ecause the warrant values are, in part, determined y the capital structure of the firm. First, consider the case of det finance. If the I 0 - π 0 dollars is otained with a det issue then the financing constraint is D() = I 0 - π 0 (7) where D() = + r π f(π )dπ + 0 +r f(π )dπ (8) Letting δ = n/(n + n), the value of the manager's stock option package with all det financing is given y 7 W(, e, 0) = + r ( δ ( π ) ( δ )en ) f(π )dπ (9) Next, suppose the required investment amount of I 0 - π 0 dollars is otained with an equity issue. In this case, the financing constraint is p s (0, e, m) m = I 0 - π 0 (20) where 7 In the susequent analysis, we will use the terms all det and all equity to refer to the external financing, i.e., all det means that all the external funds required are raised with a det issue. 8

19 p s (0, e, m) m = + r 0 e( N+m) m N + m π f(π )dπ (2) + + r e N+m m N + m + n ( π + en) f(π )dπ It follows from the financing constraint (20) that the numer of new shares that must e issued to raise I 0 - π 0 dollars, i.e., m, is an implicit function of the exercise price e. Letting ξ(e) = n/(n + m(e) + n), the value of the manager's stock option package with all equity financing is given y W(0, e, m(e)) = + r ( ξ(e) π ( ξ(e) ) en ) f(π )dπ (22) The following figures provide a comparison of the warrant payoffs in the det and equity finance cases. The warrant payoff given a det issue is max{0, δ (π - ) - ( - δ) e n} while the warrant payoff given an equity issue is max{0, ξ(e) π - ( - ξ(e)) e n}. Note that δ > ξ(e) and that this shows that a det issue causes less dilution in the warrant payoff. Given the separation result, the larger proportional stake in the corporate payoff given a det issue suggests the asis for a preference of det over equity. If the in the money event, or equivalently, the exercise event is also larger for the det issue then the det issue clearly dominates the equity issue. 9

20 Figure 2: Warrant Values - Case I δ (π - ) W d ξ(e) π W e ( - ξ(e)) e n ( - δ) e n unlevered exercise event levered exercise event π Figure two depicts the case in which det is clearly preferred y the manager. The values of the shaded areas laeled W e and W d represent the warrant values in the equity and det issue cases, respectively. The lower ound π e of the unlevered exercise event is implicitly defined y the condition ξ(e) π e = ( - ξ(e)) e n, while the lower ound π d of the levered exercise event is implicitly defined y the condition δ (π d - ) = ( - δ) e n. In the case depicted in figure two, π e > π d, or ξ(e) ξ(e) ( en> δ )en+δ δ (23) or equivalently, e (N + m) + e N. It follows that max{0, δ π - ( - δ) e n - δ } max{0, ξ(e) π - ( - ξ(e)) e n}, for all π and strictly greater for some π. If the proaility that the option is in the money is positive, i.e., - F( + e N) > 0, then W(, e, 0) > W(0, e, m(e)) (24) 20

21 The inequality in (23) holds if and only if e m(e) >. For the other cases in which > e m(e), it is not clear whether or not (24) holds. The case > e m(e) is depicted in figure three. Figure 3: Warrant Values - Case II δ (π - ) W d ξ(e) π W e ( - ξ(e)) e n ( - δ) e n unlevered exercise event levered exercise event π The following proposition shows that (24) does hold generally. We suppose that the payoff is positive with proaility one and that the exercise event is risky. If > e m(e) and F(e m(e)) > 0 for positive exercise prices then the proaility of exercise is larger for the equity issue than it is for the det issue since - F() < - F(e m(e)). Hence, oth exercise events are risky. Proposition 2. Let F(0) = 0, 0 < F( + e N) < and 0 < F(e (N + m)) < for e > 0. Then the manager prefers a det issue to an equity issue given any positive exercise price. Proof. It suffices to show that (24) holds whenever > e m(e). Consider the following argument. Let e the payment such that 2

22 D() = + r π f(π )dπ + f(π )dπ =I 0 π 0 Now, suppose that e = 0. Then W(,0,0) W(0, m(0), 0) = + r + ( max{ 0, δ(π ) } ξ(0) π ) f(π )dπ = + r + ( δ ( π min{ π,}) ξ(0) π ) f(π )dπ = δ (V - (I 0 - π 0 )) - ξ(0) V Now, δ (V - (I 0 - π 0 )) = ξ(0) V if and only if δ ξ(0) δ = I 0 π 0 V (25) If the exercise price is zero then m(0) is determined y p s (0, 0, m(0)) m = I 0 - π 0. Note that e = 0 yields the exercise event E(0, 0, m(0)) = Ω and the no ankruptcy and no exercise event N(0, 0, m(0)) = φ, i.e., the empty set. Hence, y (9), p(0, 0, m(0)) m = V/(N + n + m). It follows that I 0 π 0 V = m(0) N + n + m(0) and so (25) ecomes 22

23 δ ξ(0) δ = m(0) N + n + m(0) But now δ ξ(0) δ = n N + n n N + n + m(0) n N + n = N + n N + n + m(0) = m(0) N + n + m(0) Therefore (25) holds and W(, 0, 0) - W(0, 0, m(0)) = δ (V - I) - ξ(0) V = 0. Next, we want to consider W(, e, 0) W(0, e, m(e)) e for the case > e m(e). If > e m(e) then > W(, e, 0) W(0, e, m(e)) e + r N N + n nf(π )dπ 0 8 (26) Thus W(, 0, 0) - W(0, 0, m(0)) = 0 and as e increases so does W(, e, 0) - W(0, e, m(e)), as long as > e m(e). Since we also know that W(, e, 0) - W(0, e, m(e)) > 0 for e m(e) and e > 0, the result holds for all positive exercise prices. Q.E.D. The classic preference result in the literature is an indifference to the form of financing. The method of proof employed here captures that result y showing that if 8 These inequalities are estalished in the appendix. 23

24 the exercise price on the stock options is zero then the value of the warrant package is the same for det and equity finance. The proposition, however, also demonstrates a managerial preference for det over equity if the stock option package is risky, i.e., has a proaility of exercise less than one. The proposition goes eyond the results availale in the option pricing literature ecause the stock options, or equivalently, warrants considered here are part of the firm's capital structure The Optimal Mix of Det and Equity If the manager is restricted to either a det or equity choice then the analysis here shows that the manager has the incentive to select a det issue. Such a choice yields a larger stock option value now. The constrained maximization prolem in (4) is, however, more general. The manager may select a comination of the two in financing the firm's investment. The purpose of this section is to generalize the previous section's choice analysis y allowing the manager to select any comination of det and equity that satisfies the financing constraint. The constrained maximization prolem in (4) is a similar to many choice prolems in economics. There are two difference here. First, the ojective function W is endogenous, i.e., due to a Fisher Separation result. Second, smaller values of and m are preferred and so the iso-value contours of the ojective function closer to the origin represent larger warrant values. The level sets (m,) W,e,m(e) = W { } represent isowarrant values. Let W s W(0, e, m 0 (e)) and W W( 0, e, 0), where 0 and m 0 (e) finance 9 The result in the option pricing literature is that an increase in risk yields an increase in call option value Merton, R. C. (973). Theory of Rational Option Pricing. Bell Journal of Economics and Management Science 4(): Although one might loosely argue that levering the firm increases risk, a straight forward application of Merton's theorem eight to the analysis here is inappropriate. Levering the firm can e shown to cause an increase in risk ut that increase is in a value preserving sense rather than the mean preserving sense that Merton required. In addition, the analysis here is for warrants rather than the call options that Merton considered. A call option is not issued y the firm whereas the warrant is. 24

25 the investment in the det and equity case, respectively. By proposition two, W > W s. The following figure depicts the iso-warrant value curves for W and W s. If the isowarrant lines are steeper everywhere than the finance constraint then any movement from the all det corner solution, depicted in figure four y 0, reduces the warrant value. Figure 4: The Constrained Maximization Prolem 0 {(m, ) W(, e, m(e) ) = W } 0 D() + p s (, e, m) m = I 0 π 0 {(m, ) W(, e, m(e) ) = W s } m 0 m The 958 Modigliani-Miller theorem suggests that the financing condition is irrelevant. Indeed, if the manager is paid with stock or with stock options that are safe then the irrelevance results follows y direct calculation. If the manager is compensated with stock options that are not safe, as hypothesized here, then an irrelevance result need not follow. The following proposition shows that the manager is not indifferent to 0 We have not een ale to determine the curvature of the iso-warrant value lines. Indeed it may not e possile to say that they are convex or concave since the curvature depends on a pricing density. The results stated here, however, are distriution free. A stock option is safe if it can e exercised then with proaility one. 25

26 the capital structure choices availale. After exhausting the sources of internal financing, the manager prefers the det choice to all possile mixes of det and equity. Proposition 3. Let F(0) = 0, 0 < F( + e N) < and 0 < F(e (N + m)) < for e > 0. Then the manager prefers a det issue to all possile cominations of det and equity that finance the investment. Proof. There is a corner solution to this prolem if the iso-warrant value curves are steeper than the financing constraint. Recall that the warrant value is W(, e, m) = + r n N + m + n π ( + en ) en f(π )dπ It follows that the slope of the iso-warrant value line is d dm = W m W = f(π )dπ n ( N+m+n) 2 π +en n N + m + n f(π )dπ = N+m+n ( π +en ) f(π )dπ f(π )dπ Similarly, recall that the financing constraint is C(, e, m) D() + p s (, e, m) m ( I 0 π 0 )=0 26

27 C(, e, m) = + r π f(π )dπ + 0 +r f(π )dπ + +r +e( N+m) m N+m ( π )f(π )dπ + + r m ( N + m + n π + en ) f(π )dπ ( I 0 π 0 )=0 +e N+m It follows that d dm = C m C where C = + r f(π )dπ +r +e( N+m) m N+m f(π )dπ + r +e N+m m N + m + n f(π )dπ and 27

28 C m = + r Stock Options and Capital Structure +e( N+m) N ( ( N + m) 2 π ) f(π )dπ + + r +e N+m f(π )dπ N + n ( N + m + n) 2 π + en Now, consider whether the following inequality holds. W m > W e> 0 H m H (27) e> 0 If it does then det is preferred over all possile det/equity mixes in financing the investment. Note that (27) is equivalent to 2 N + m + n ( π + en ) f(π )dπ f(π )dπ > +e( N+m) N f(π )dπ + N + m f(π )dπ 2 π +e( N+m) m N+m f(π )dπ N + n ( N + m + n) 2 π + en +e( N+m) m N + m + n f(π )dπ +e N+m f(π )dπ 2 The intermediate steps, used in estalishing the following sequence of inequalities, are provided in the appendix. 28

29 F+m ( ) F() F+m ( ) +e( N+m) ( N+m+n π +en ) f(π )dπ > N + m ( π ) f(π )dπ (28) Note that (28) holds since 3 F+m ( ) F() F+m ( ) ( N+m+n π +en ) f(π )dπ > F+m ( ) F() e F+m F+m ( ) e F( +m ) F() [ ( )] = [ ] > +e( N+m) N + m ( π ) f(π )dπ Q.E.D. 4. Concluding Remarks The existing literature on the pecking order theory is ased on the assumption that agents are asymmetrically informed. The asymmetry assumption led to a deate in 3 To see the following, evaluate the integral on the LHS at its lower limit and the integral on the RHS at its upper limit. 29

30 the literature aout whether it was possile to reveal information to market agents through the capital structure choice and whether revelation of the information was or was not costly. This analysis eliminates that conundrum y supposing that market agents are symmetrically informed. Although asymmetric information was thought to e a key reason in explaining the pecking order theory, this analysis shows that the compensation contract plays a pivotal role in the manager's capital structure decisions. The stock option contract provides the manager with an incentive to make capital structure decisions that do not dilute the manager's stake in the corporation. This incentive is manifested in the preference for internal equity, equivalently, retained earnings, over det and outside equity in financing an investment. This is a stronger result than the Myers and Majluf result ecause there the manager was indifferent etween using retained earnings versus safe det. The incentive is also manifested in a preference for det over outside equity. This result is also stronger than the Myers and Majluf result ecause their result showed a preference for safe det over equity while the results here show that even risky det is preferred to equity. What is more, the results here show that the manager prefers det to any cominations of det and equity. The results here do not require the risk neutrality arguments so often made in literature, nor do they require the assumption that financial markets are complete. Therefore the results are roust and provide the asis for a more powerful test of the pecking order hypothesis. 30

31 References Baron, D. (970). Price Uncertainty, Utility, and Industry Equilirium in Pure Competition. International Economic Review : Constantinides, G. M. and B. D. Grundy (989). Optimal Investment with Stock Repurchase and Financing as Signals. Review of Financial Studies 2: Leland, H. (972). Theory of the Firm Facing Uncertain Demand. American Economic Review 62: Merton, R. C. (973). Theory of Rational Option Pricing. Bell Journal of Economics and Management Science 4(): Modigliani, F. and M. H. Miller (958). The Cost of Capital, Corporation Finance and the Theory of Investment. American Economic Review. Myers, S. C. and N. S. Majluf (984). Corporate Financing and Investment Decisions When Firms Have Information That Investors Do Not Have. Journal of Financial Economics 3: Sandmo, A. (97). On the Theory of the Competitive Firm Under Price Uncertainty. American Economic Review 6: Stiglitz, J. E. and B. C. Greenwald (990). Asymmetric Information and the New Theory of the Firm: Financial Constraints and Risk Behavior. American Economic Review 80(2):

32 A p p e n d i x Notes on Proposition 2 To estalish the inequality in (26), note that the difference in warrant values may e expressed as follows: W(, e, 0) W(0, e, m(e)) = + r ( δπ ( δ) en δ) f(π )dπ + r ( ξ(e) π ( ξ(e) ) en) f(π )dπ = + r [( δπ ( δ) en δ) ( ξ(e) π ( ξ(e) ) en) ] f(π )dπ (A.) + r ( ξ(e) π ( ξ(e) ) en) f(π )dπ Then W(, e, 0) W(0, e, m(e)) e = + r [ ( δ)n + ( ξ(e) )n ( ξ (e) π + ξ (e) e n )] f(π )dπ + r ( ξ (e) π + ξ (e) e n ( ξ(e) )n) f(π )dπ 32

33 A p p e n d i x = + r [( δ ξ(e) )n ξ (e) ( π + en )] f(π )dπ + r ( ξ (e) ( π + en ) ( ξ(e) )n) f(π )dπ = + r ( δ ξ(e) )nf(π)dπ + + r ( ξ(e) ) nf(π )dπ (A.2) + r ξ (e) ( π + en) f(π )dπ Next, consider how ξ changes with respect to the exercise price. Note that ξ = n m ( N + n + m) =ξ 2 m N+n+m and so the sign of ξ' is the opposite of the sign of m'. The numer of new shares that must e issued is implicitly defined y the financing condition in (20) or equivalently y + r e( N+m) m N + m π f(π )dπ + 0 e N+m m N + m + n ( π + en ) f(π )dπ (A.3) ( I 0 π 0 ) = 0 The first two terms on the LHS of (A.3) represent the value of the new shares, i.e., p(0, e, m(e)) m(e). Then 33

34 A p p e n d i x m = p(0, e, m) m e m ( p(0, e, m) m) = e( N+m) e N+m N π ( N + m) 2 f(π )dπ + 0 mn N+m+n f(π )dπ N + n π ( N + m + n) 2 + en e( N+m) f(π )dπ (A.4) Now rewrite (A.2) to get = + r ( δ ξ(e) )nf(π)dπ + + r ( ξ(e) ) nf(π )dπ + r ξ m N + n + m ( π + en) f(π )dπ Using (A.4) yields = + r ( δ ξ(e) )nf(π)dπ + + r ( ξ(e) ) nf(π )dπ + r e( N+m ) e N ( N+m) π ( N + m) 2 f(π )dπ + 0 mn N+m+n f(π )dπ N + n π ( N + m + n) 2 + en e( N+m) f(π )dπ n(π +en) (N + n + m) f(π 2 )dπ 34

35 A p p e n d i x > + r ( δ ξ(e) )nf(π)dπ + + r ( ξ(e) ) nf(π )dπ (A.5) + r n N + n n N + n + m(e) m(e) f(π )dπ The inequality in (A.5) holds if and only if N + n n π + en n (N + n + m) 2 e N+m N π ( N + m) 2 f(π )dπ + 0 f(π )dπ N + n π ( N + m + n) 2 + en e( N+m) f(π )dπ < N + n ( π + en) f(π )dπ < (N + n + m) 2 e( N+m) N π ( N + m) 2 f(π )dπ + 0 N + n π ( N + m + n) 2 + en e( N+m) f(π )dπ 0 < e( N+m) N π ( N + m) 2 f(π )dπ 0 Hence, the inequality in (A.5) holds ecause F(0) = 0 and F(e (N + m)) > 0. Next, the RHS of (A.5) may e equivalently expressed as follows + r ( δ ξ(e) )nf(π)dπ + + r ( ξ(e) ) nf(π )dπ + r n N + n n N + n + m(e) m(e) f(π )dπ 35

36 A p p e n d i x = + r ( δ ξ(e) )nf(π)dπ + + r ( ξ(e) ) nf(π )dπ + r ( δξ(e) m(e) ) f(π )dπ = + r [( δ ξ(e) )n δξ(e) m(e) ] f(π )dπ + + r + r δξ(e) m(e) f(π )dπ ( ξ(e) ) nf(π )dπ = + r [( ξ(e) ) n δξ(e) m(e) ] f(π )dπ = + r N N + n nf(π )dπ 4 0 (A.6) (A.6) estalishes the inequality in (26). Notes on Proposition 3 The intermediate steps in estalishing the inequality in (28) are as follows: N + m + n ( π + en ) f(π )dπ f(π )dπ > 4 It follows y direct calculation that (δ - ξ(e)) n - δ ξ(e) m(e) = 0 and ( - ξ(e)) n - δ ξ(e) m(e) = N n/(n + n). 36

37 A p p e n d i x +e( N+m) N f(π )dπ + N + m 2 π f(π )dπ +e( N+m) m N+m f(π )dπ f(π )dπ N + n π ( N + m + n) 2 + en m N + m + n f(π )dπ N + m + n ( π + en ) f(π )dπ f(π )dπ > +e( N+m) N f(π )dπ + N + m 2 π N N + m f(π )dπ + f(π )dπ N + n π ( N + m + n) 2 + en N + n N + m + n f(π )dπ +e( N+m) N N + m f(π )dπ + f(π )dπ N + n N + m + n f(π )dπ ( N + m + n π + en ) f(π )dπ > +e( N+m) N f(π )dπ + N + m 2 π f(π )dπ N + n π ( N + m + n) 2 + en N N + m +e( N+m) f(π )dπ f(π )dπ + N + n N + m + n ( N + m + n π + en ) f(π )dπ > 37

38 A p p e n d i x +e( N+m) N f(π )dπ + N + m 2 π f(π )dπ N + n π ( N + m + n) 2 + en N N + m +e( N+m) f(π )dπ f(π )dπ ( N + m + n π + en ) f(π )dπ > +e( N+m) N π ( N + m) 2 f(π )dπ ( ) F() F+e( N+m) F ( N + m + n π + en ) f(π )dπ > +e( N+m) N + m ( π ) f(π )dπ 38